Research ArticleVibration Analysis of Steel-Concrete Composite BoxBeams considering Shear Lag and Slip

Zhou Wangbao1 Li Shu-jin1 Jiang Lizhong2 and Qin Shiqiang1

1School of Civil Engineering and Architecture Wuhan University of Technology Wuhan 430070 China2School of Civil Engineering Central South University Changsha 410075 China

Correspondence should be addressed to Zhou Wangbao zhwbwhuteducn

Received 21 January 2015 Revised 15 March 2015 Accepted 15 March 2015

Academic Editor Giovanni Garcea

Copyright copy 2015 Zhou Wangbao et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In order to investigate dynamic characteristics of steel-concrete composite box beams a longitudinal warping function of beamsection considering self-balancing of axial forces is established On the basis of Hamilton principle governing differential equationsof vibration and displacement boundary conditions are deduced by taking into account coupled influencing of shear lag interfaceslip and shear deformationTheproposedmethod shows an improvement over previous calculationsThe central differencemethodis applied to solve the differential equations to obtain dynamic responses of composite beams subjected to arbitrarily distributedloads The results from the proposed method are found to be in good agreement with those from ANSYS through numericalstudies Its validity is thus verified and meaningful conclusions for engineering design can be drawn as follows There are obviousshear lag effects in the top concrete slab and bottom plate of steel beams under dynamic excitation This shear lag increases withthe increasing degree of shear connections However it has little impact on the period and deflection amplitude of vibration ofcomposite box beamsThe amplitude of deflection and strains in concrete slab reduce as the degree of shear connections increasesNevertheless the influence of shear connections on the period of vibration is not distinct

1 Introduction

The advantages of fully making use of compressive strengthof concrete and tensile strength of steel make composite boxbeams popular in the bridge engineering For composite boxbeams with large flange width however there are shear lageffects in the top concrete slab and bottom plates of steel boxbeams due to the nonuniform transverse distribution of shearstress across section In addition due to the fact that the shearconnectors between the steel beam and concrete slab cannotbe absolutely rigid there exists relative slip between themeven for fully shear connected beamsTherefore the behaviorof steel-concrete composite box beams suffers from coupledeffects of both the shear lag and shear slip [1ndash5] Gara [6 7]presented a beam finite element in which the warping of theslab cross section was considered for the long-term analysisof steel-concrete composite decks taking into account theshear lag in the slab and the partial shear interaction atthe slab-beam interface By using energy variation method

Zhou et al [5] derived the governing differential equationsand boundary conditions of the steel-concrete composite boxbeams by considering the longitudinal warp caused by shearlag effects and slip between steel beams and concrete slabs

Morassi and Dilena [8ndash10] investigated an experimenton damage-induced changes in modal parameters of steel-concrete composite beams subject to small vibrations andthe experiments revealed that flexural frequencies showeda rather high sensitivity to damage and therefore can beconsidered as a valid indicator upon a diagnostic analysisTherefore it makes sense to study the vibration characteristicof composite beam Adam et al [11] analyzed the dynamicflexural behavior of elastic two-layer beams with interlayerslip by assuming the Bernoulli-Euler hypothesis to hold foreach layer separately and considering a linear constitutiveequation between the horizontal slip and the interlayershear force Biscontin et al [12] performed an experimentalanalytical investigation on the dynamic behavior of I-steel-concrete composite beams subject to small vibrations and a

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 601757 8 pageshttpdxdoiorg1011552015601757

2 Mathematical Problems in Engineering

one-dimensional model of a composite beam was presentedwhere the elements connecting the steel and reinforced con-crete slab were described by means of a strain energy densityfunction defined along the beam axis Berczynski [13 14] pre-sented a solution of the problem of free vibrations of I-steel-concrete composite beams and found that the results obtainedon the basis of the Timoshenko beam theory model achievedthe highest conformitywith the experimental results both forhigher and for lowermodes of flexural vibrations of the beamXu andWu [15] investigated the static dynamic and bucklingbehavior of partial-interaction T-composite members bytaking into account the influences of rotary inertia andshear deformations and obtained the analytical expressionsof the frequencies of the simply supported composite beamShen et al [16] studied the dynamic behavior of partial-interaction T-composite beams by using state-space methodwhich was properly established via selecting the appropriatestate variables and the characteristic equations of frequencyand the corresponding modal shapes of free vibration undergeneralized boundary conditions were then obtained Shenand Zhong [17] examined the deformation of partially I-steel-concrete composite beams under distributed loading andfree vibrations of partially I-steel-concrete composite beamsunder various boundary conditions where the weak-formquadrature element method was used

Considerable efforts have been put into the investigationof dynamic characteristics of composite beams includinginterface slip effects [18ndash22] Most of studies focus on the I-steel beam-concrete composite beams or T-type compositebeams The studies about the dynamic characteristics ofcomposite box beams are rare especially the studies involvingboth the interface slip and shear lag in the dynamic character-istics Based on a longitudinal warping function consideringself-balancing of axial forces and the Hamilton principlethis paper deduces the governing differential equations ofdynamic responses of composite box beams under arbitrarilydistributed loads It takes into account the influence of shearlag interface slip and shear deformation The equations aresolved by central difference methods Numerical studies arecarried out and a good agreement is achieved between resultsfrom the proposed method and finite element method usingANSYS The influences of the shear lag effect and the degreeof shear connections on the dynamic responses of compositebox beams are examined and meaningful conclusions forengineering design are drawn

2 Basic Assumptions

Figure 1 shows the section size and coordinate system ofcomposite box beams The parameters 119905

1 1199052 1199053 1199054 and

119905lowast

4are the thicknesses of the top concrete slab cantilever

plate bottom plate web and top flange of steel box beamrespectively the parameters 119887

1 1198872 1198873 1198874 and 119887

lowast

4are the

widths of half-top concrete slab cantilever plate half-bottomplate of steel beams web of steel beams and top flange ofsteel beams respectively the parameters ℎ

119888and ℎ

119904are the

distances between the centroids of the concrete slab andsteel beam to the slab-beam interface respectively and here

hc

hsy 0

z w

b1b2 b1 b2

b4

blowast

4

t4

t1

b3 b3

t3

tlowast

4h

t2

Figure 1 Schematic of section of steel-concrete composite boxbeams

ℎ = ℎ119888+ ℎ119904 Reasonable assumptions to simplify the analysis

model are made as follows(1) According to the displacement compatibility the

longitudinal warping function of concrete top slab cantileverplate bottom flange and web of steel beams is assumed as[5 23 24]

119892119894= 120595119894(119910)119880 (119909 119905) 119894 = 1 2 3 4 (1)

120595119894= 120572119894(1199102

1198872

119894

minus 1) + 119889 119894 = 1 2 3 4 (2)

Considering self-balancing of axial forces produced bylongitudinal displacement yields [5 23 24]

1205721= 1 120572

2=

1198872

2

1198872

1

1205723= 119911119887

1198872

3

(1198872

1119911119905) 120572

4= 0

(3)

Considering self-balancing of axial forces produced bylongitudinal warping function yields [5 23 24]

int

119860

120595119889119860 = 0 (4)

Substituting (2) into (4) obtains the constant term oflongitudinal warping displacements as follows

119889 =2 (12057211198601119899 + 120572

21198602119899 + 120572

31198603)

(31198600)

(5)

For 119894 = 2 replace 119910 with 119910 = 1198871+ 1198872minus 119910 119911

119905and

119911119887are the 119911-coordinates of centroids of the concrete slab

and bottom flange 1199110is the 119911-coordinate of the slab-beam

interface 119889 is the constant term of longitudinal warpingdisplacements 119880(119909 119905) is the function of the amplitude ofwarping displacements 120595

119894is the warping shape function of

the beam section as shown in Figure 2120572119894is the self-balancing

coefficient of section warping 119899 = 119864119904119864119888 and 119864

119904and 119864

119888are

the modulus of elasticity of steel and concrete respectively

1198601= 211988711199051 119860

2= 211988721199052

1198603= 211988731199053 119860

4= 2 (119887

41199054+ 119887lowast

4119905lowast

4)

1198600=

(1198601+ 1198602)

119899+ 1198603+ 1198604

(6)

Mathematical Problems in Engineering 3

A998400

1A998400

2

A998400

3

d

Figure 2 Schematic of warping shape function

(2) The longitudinal displacement of any point in thetransverse section of composite box beams is assumed as thesuperposition of the longitudinal displacement based on theplain section assumption the longitudinal warping displace-ment due to the shear lag and longitudinal displacement dueto the relative interface slip It can be expressed as [5 23 24]

119906119894= 119896119888120585 minus (119911 minus 119911

119905) 120579 + 119892

119894119894 = 1 2 (7)

119906119894= 119896119904120585 minus (119911 minus 119911

119904) 120579 + 119892

119894119894 = 3 4 (8)

119896119888= minus

119860119904

1198600

119896119904=

119860119888

(1198991198600) (9)

120577 (119909 119905) = 120585 + ℎ120579 (10)

where 120579(119909 119905) is the rotation of the beam section 119860119908= 211988741199054

119860119904= 1198603+ 1198604is the cross section area of steel beams 119860

119888=

1198601+ 1198602is the cross section area of concrete slabs 120585(119909 119905) is

the longitudinal displacement difference between centroidsof the concrete slab and steel beam 120577(119909 119905) is slab-beaminterface slip 119911

119904is the 119911-coordinate of the centroid of the

steel beam 119896119904is the ratio between steel beamrsquos longitudinal

displacement due to the relative interface slip and relativeinterface slip and 119896

119888is the ratio between concrete slabrsquos

longitudinal displacement due to the relative interface slipand relative interface slip

(3) The vertical compression and transverse strain ofconcrete slabs and steel beams are ignored [5 23]

3 Vibration Differential Equation andBoundary Conditions

31 The Strain of the Cross Section The sectional strain canbe obtained from the above longitudinal displacement ofcomposite beam sections as

120576119909119894

= 119896119888

120597120585

120597119909minus (119911 minus 119911

119905)120597120579

120597119909+ 120595119894

120597119880

120597119909119894 = 1 2

120576119909119894

= 119896119904

120597120585

120597119909minus (119911 minus 119911

119904)120597120579

120597119909+ 120595119894

120597119880

120597119909119894 = 3 4

120574119909119910119894

=120597119906119894

120597119910=

120597120595119894

120597119910119880 119894 = 1 2 3

120574119909119911

=120597119908

120597119909minus 120579

(11)

where 120576119894(119894 = 1 2 3 4) are the longitudinal strain of the top

concrete slab cantilever plate bottom flange and web of steelbeams respectively 120574

119894(119894 = 1 2 3) are the shear strain of

top concrete slab cantilever plate and bottom plate of steelbeams respectively 120574

119909119911is the shear strain of the web of steel

beams 119908(119909 119905) is the vertical deflection of composite boxbeams

32 Total Potential Energy of the Composite Box Beam Thestrain energy of composite box beams is defined as [5]

119881 = 05 int

119871

0

int

1198600

(1198641199041205762+ 1198661199041205742) 119889119860119889119909

+ 05 int

119871

0

119896sl1205772119889119909 +

051198661199041198600(1199081015840minus 120579)2

120572119904

(12)

Substituting (10)-(11) into (12) gives the strain energy ofcomposite box beams as

119881 = 05 int

119871

0

[

[

11986312058510158402+ 11986511988010158402+ 2119867119880

10158401205851015840+ 11986812057910158402+ 1198691198802

minus 211987811988010158401205791015840+ 119896sl120577

2+

1198661199041198600(1199081015840minus 120579)2

120572119904

]

]

119889119909

(13)

where119863 = 1198641198881198962

119888119860119888+1198641199041198962

119904119860119904 119865 = 119864

119888119861119888119891+119864119904119861119904119891119867 = 119864

119888119861119888ℎ+

119864119904119861119904ℎ 119868 = 119864

119888119861119888119894+119864119904119861119904119894 119869 = 119866

119888119861119888119895+119866119904119861119904119895 119878 = 119864

119888119861119888119904+119864119904119861119904119904

119861119888119891

= int119860119888

1205952119889119860 119861

119904119891= int119860119904

1205952119889119860 119861

119888ℎ= int119860119888

119896119888120595119889119860 119861

119904ℎ=

int119860119904

119896119904120595119889119860 119861

119888119894= int119860119888

(119911 minus 119911119905)2119889119860 119861

119904119894= int119860119904

(119911 minus 119911119904)2119889119860 119861

119888119895=

int119860119888

(120597120595120597119910)2119889119860 119861

119904119895= int1198603

(120597120595120597119910)2119889119860 119861

119888119904= int119860119888

(119911minus119911119905)120595 119889119860

119861119904119904

= int119860119904

(119911 minus 119911119904)120595 119889119860 and 120572

119904is the correction coefficient of

shear deformation Considering that the webs bear most ofthe vertical shear force in section here the value of 120572

119904is taken

as 1198600(211988741199054) and 2119887

41199054is the section area of webs 119896sl is the

slip stiffness between the concrete slab and the steel beam119871 isthe span of the composite box beam 119866

119904is the shear modulus

of the steel beamThe kinetic energy of the composite box beam is [5]

119879 =1

2int

119871

0

int

119860

(1205882+ 119898

2) 119889119860119889119909 (14)

Substituting of (7) and (8) yields

119879

=1

2int

119871

(1198982+ 1198631

1205852+ 11986512+ 21198671

120585 + 1198681

1205792minus 21198781 120579) 119889119909

(15)

where1198631= 1205881198881198962

119888119860119888+1205881199041198962

1199041198601199041198651= 120588119888119861119888119891+1205881199041198611199041198911198671= 120588119888119861119888ℎ+

120588119904119861119904ℎ 1198681= 120588119888119861119888119894+ 120588119904119861119904119894 1198781= 120588119888119861119888119904+ 120588119904119861119904119904119898 = 120588

119888119860119888+ 120588119904119860119904

119860 = 119860119904+ 119860119888 and 120588

119888and 120588119904are the density of concrete slabs

and steel beams respectively

4 Mathematical Problems in Engineering

The work done by the external loads can be expressed as

119882 = int

119871

119902 (119909 119905) 119908 119889119909 (16)

where 119902(119909 119905) is the distribution function of arbitrary load

33 VibrationDifferential Equation and Boundary ConditionsThegoverning equations of vibration of composite box beamsand corresponding boundary conditions can be deducedbased on Hamilton principle as

11986511988010158401015840+ 11986712058510158401015840minus 119869119880 minus 119878120579

10158401015840minus 1198651 minus 119867

1

120585 + 1198781

120579 = 0 (17)

11986711988010158401015840+ 11986312058510158401015840minus 119896sl120577 minus 119867

1 minus 119863

1

120585 = 0 (18)

1198661199041198600(11990810158401015840minus 1205791015840)

120572119904

minus 119898 + 119902 (119909 119905) = 0 (19)

1198661199041198600(1199081015840minus 120579)

120572119904

+ 11986812057910158401015840minus 11987811988010158401015840minus 119896sl120577ℎ + 119878

1 minus 1198681

120579 = 0 (20)

(1198651198801015840+ 1198671205851015840minus 1198781205791015840) 120575119880

10038161003816100381610038161003816

119871

0= 0 (21)

(1198631205851015840+ 119867119880

1015840) 120575120585

10038161003816100381610038161003816

119871

0= 0 (22)

1198661199041198600(1199081015840minus 120579)

120572119904

120575119908

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871

0

= 0 (23)

(1198681205791015840minus 1198781198801015840) 120575120579

10038161003816100381610038161003816

119871

0= 0 (24)

Taking a simply supported beam as example (21)ndash(24)give boundary conditions as

1198801015840(119871 119905) = 120585

1015840(119871 119905) = 120579

1015840(119871 119905) = 119908 (119871 119905) = 0

1198801015840(0 119905) = 120585

1015840(0 119905) = 120579

1015840(0 119905) = 119908 (0 119905) = 0

(25)

given the initial conditions as

119880 (119909 0) = 1198800(119909) (119909 0) = 119880

0(119909)

120585 (119909 0) = 1205850(119909)

120585 (119909 0) = 120585

0(119909)

119908 (119909 0) = 1199080(119909) (119909 0) = 119908

0(119909)

120579 (119909 0) = 1205790(119909) 120579 (119909 0) = 120579

0(119909)

(26)

4 Finite Difference Method ofVibration Differential Equation

41 Difference Scheme Let solution domain be 120590 = (119909 119905) |

0 le 119909 le 119871 0 le 119905 le 119879 119879 is the end time of solution arectangularmesh ismade in the solution area with a time stepof 120591 and space step of 120592 so that

119909119894= 119894120592 (119894 = 0 1 2 119868)

119905119895= 119895120591 (119895 = 0 1 2 119869)

(27)

where 119868 = 119871120592 119869 = 119879120591

Let

119880119895

119894= 119880 (119909

119894 119905119895) 120585

119895

119894= 120585 (119909

119894 119905119895)

119908119895

119894= 119908 (119909

119894 119905119895) 120579

119895

119894= 120579 (119909

119894 119905119895)

(28)

The central difference calculation of governing differen-tial equation yields

119865

119880119895

119894+1minus 2119880119895

119894+ 119880119895

119894minus1

1205922

+ 119867

120585119895

119894+1minus 2120585119895

119894+ 120585119895

119894minus1

1205922

minus 119878

120579119895

119894+1minus 2120579119895

119894+ 120579119895

119894minus1

1205922

minus 1198651

119880119895+1

119894minus 2119880119895

119894+ 119880119895minus1

119894

1205912

minus 119869119880119895

119894minus 1198671

120585119895+1

119894minus 2120585119895

119894+ 120585119895minus1

119894

1205912

+ 1198781

120579119895+1

119894minus 2120579119895

119894+ 120579119895minus1

119894

1205912

= 0

(119894 = 1 119868 minus 1 119895 = 1 2 119869)

(29)

119867

119880119895

119894+1minus 2119880119895

119894+ 119880119895

119894minus1

1205922

+ 119863

120585119895

119894+1minus 2120585119895

119894+ 120585119895

119894minus1

1205922

minus 119896sl (120585119895

119894+ ℎ120579119895

119894) minus 119867

1

119880119895+1

119894minus 2119880119895

119894+ 119880119895minus1

119894

1205912

minus 1198631

120585119895+1

119894minus 2120585119895

119894+ 120585119895minus1

119894

1205912

= 0

(119894 = 1 119868 minus 1 119895 = 1 2 119869)

(30)

1198661199041198600

120572119904

(

119908119895

119894+1minus 119908119895

119894minus1

2120592minus 120579119895

119894) + 119868

120579119895

119894+1minus 2120579119895

119894+ 120579119895

119894minus1

1205922

minus 119878

119880119895

119894+1minus 2119880119895

119894+ 119880119895

119894minus1

1205922

+ 1198781

119880119895+1

119894minus 2119880119895

119894+ 119880119895minus1

119894

1205912

minus 1198681

120579119895+1

119894minus 2120579119895

119894+ 120579119895minus1

119894

1205912

minus 119896slℎ (120585119895

119894+ ℎ120579119895

119894) = 0

(119894 = 1 119868 minus 1 119895 = 1 2 119869)

(31)

1198661199041198600

120572119904

(

119908119895

119894+1minus 2119908119895

119894+ 119908119895

119894minus1

1205922

minus

120579119895

119894+1minus 120579119895

119894minus1

2120592)

+ 119902119895

119894minus 119898

119908119895+1

119894minus 2119908119895

119894+ 119908119895minus1

119894

1205912

= 0

(119894 = 1 119868 minus 1 119895 = 1 2 119869)

(32)

Mathematical Problems in Engineering 5

The difference calculation of boundary conditions yields

119880119895

119868minus 119880119895

119868minus1

120592=

119880119895

1minus 119880119895

0

120592= 0 119895 = 0 1 119869

(33)

120585119895

119868minus 120585119895

119868minus1

120592=

120585119895

1minus 120585119895

0

120592= 0 119895 = 0 1 119869

(34)

119908119895

119868= 119908119895

0= 0 119895 = 0 1 119869 (35)

120579119895

119868minus 120579119895

119868minus1

120592=

120579119895

1minus 120579119895

0

120592= 0 119895 = 0 1 119869

(36)

The difference calculation of initial condition yields

1198800

119894= 1198800119894

1198801

119894minus 1198800

119894

120591= 1198800119894

119894 = 0 1 2 119868

1205850

119894= 1205850119894

1205851

119894minus 1205850

119894

120591= 1205850119894

119894 = 0 1 2 119868

1199080

119894= 1199080119894

1199081

119894minus 1199080

119894

120591= 1199080119894

119894 = 0 1 2 119868

1205790

119894= 1205790119894

1205791

119894minus 1205790

119894

120591= 1205790119894

119894 = 0 1 2 119868

(37)

42 Solution Step Let U119895 = 119880119895

0 119880119895

1 119880

119895

119868119879 120585119895 = 120585

119895

0 120585119895

1

120585119895

119868119879 w119895 = 119908

119895

0 119908119895

1 119908

119895

119868119879 and 120579119895 = 120579

119895

0 120579119895

1 120579

119895

119868119879

Given U119895minus1 120585119895minus1 w119895minus1 120579119895minus1 U119895 120585119895 w119895 and 120579119895 the solving ofU119895+1 120585119895+1 w119895+1 and 120579119895+1 follows the steps below

(a) Calculate U0 1205850 w0 1205790 U1 1205851 w1 and 1205791 from theinitial conditions given in (37)

(b) Given U119895minus1 120585119895minus1 120579119895minus1 U119895 120585119895 w119895 and 120579119895 calcu-late 119880

119895+1

1 119880119895+1

2 119880

119895+1

119868minus1 120585119895+1

1 120585119895+1

2 120585

119895+1

119868minus1 and

120579119895+1

1 120579119895+1

2 120579

119895+1

119868minus1 from (29)ndash(31)

(c) Calculate 119880119895+1

0 119880119895+1

119868 120585119895+1

0 120585119895+1

119868 and 120579

119895+1

0 120579119895+1

119868

from (33) (34) and (36) and the solution of U119895+1120585119895+1 and 120579119895+1 is obtained in combination withstep (b)

(d) Given w119895minus1 w119895 and 120579119895 determine 119908119895+1

1 119908119895+1

2

119908119895+1

119868minus1 from (32)

(e) Calculate 119908119895+1

0 119908119895+1

119868 from (35) and calculate w119895+1

combined with step (d)

43 Degeneration of theVibrationDifferential Equation Like-wise the governing equations of vibration of composite box

beams and boundary conditions without shear lag effects canbe deduced as

11986312058510158401015840minus 119896sl120577 minus 119863

1

120585 = 0 (38)

1198661199041198600(11990810158401015840minus 1205791015840)

120572119904

minus 119898 + 119902 (119909 119905) = 0 (39)

1198661199041198600(1199081015840minus 120579)

120572119904

+ 11986812057910158401015840minus 119896sl120577ℎ minus 119868

1120579 = 0 (40)

1205851015840120575120585

10038161003816100381610038161003816

119871

0= 0 120579

1015840120575120579

10038161003816100381610038161003816

119871

0= 0 (119908

1015840minus 120579) 120575119908

10038161003816100381610038161003816

119871

0= 0 (41)

For beamswith two ends simply supported the boundaryconditions can be expressed from (41) as

1205851015840(119871 119905) = 120579

1015840(119871 119905) = 119908 (119871 119905) = 0

1205851015840(0 119905) = 120579

1015840(0 119905) = 119908 (0 119905) = 0

(42)

The vibration differential equation of composite boxbeams without considering shear lag effects is a degenera-tion of the one considering the shear lag effects the solutionmethod of which can be referred to in Sections 41 and 42

5 Analysis of Examples

The validity of the proposed method is verified by compar-ing to numerical results from finite element method Thecomparisons are made on four simply supported compositebox beams with different degree of shear connections undersuddenly imposed distributed loads The dynamic responsesof beams with and without shear lag effects are analyzedThe distributed load is taken as 119902 = 300 kNm with a timestep of 120591 = 000002 s and a space step of 120592 = 03m Themechanical and geometrical parameters of composite boxbeams are taken as 119864

119904= 20 times 10

5MPa 119864119888= 45 times 10

4MPa119866119904= 771 times 10

4MPa 120583119904= 028 120583

119888= 018 120588

119904= 7900 kgsdotmminus3

120588119888= 2400 kgsdotmminus3 119887

1= 1198873= 25m 119887

2= 20m 119887

4= 30m

1198875= 02m 119905

1= 1199052= 03m 119905

3= 1199055= 006m 119905

4= 009m

119871 = 30m 119897 = 03m 119899119904= 2 and 119891

119904= 300MPa

The commercial finite element software ANSYS is used inthis study In the finite element model the concrete slab andsteel beam are modeled by SOLID65 and SHELL43 elementsrespectively Shear connector is modeled by COMBIN14elements being spring elements [25] The aspect ratio of themesh is kept close to one throughout the mean mesh sizevaries from 02m to 03m and about 9200 finite elementsin total are employed The transverse distributed loads areimposed on the top flanges The translational 119909 and 119910

degrees of freedom of all nodes at two ends of beams arerestrained The torsion at the beam ends is thus restrainedThe translational 119911 DOF of the left end is restrained to meetthe static determined requirement and allow rotation alongthe moment The results are shown in Figures 3 and 4

The 119896sl is calculated as [5 23 24]

119896sl =1198701

119897 (43)

6 Mathematical Problems in Engineering

000 005 010 015 020 0250

4

8

12

16

20

24

Time (s)

Disp

lace

men

t at m

idsp

an (m

m)

(a) 119903 = 025

000 005 010 015 020 0250

4

8

12

16

20

24

Disp

lace

men

t at m

idsp

an (m

m)

Time (s)

(b) 119903 = 05

000 005 010 015 020 0250

4

8

12

16

20

24

Time (s)

Disp

lace

men

t at m

idsp

an (m

m)

ANSYSThis paper 120582 = 1

This paper 120582 = 0

(c) 119903 = 10

000 005 010 015 020 0250

4

8

12

16

20

24

Time (s)

Disp

lace

men

t at m

idsp

an (m

m)

ANSYSThis paper 120582 = 1

This paper 120582 = 0

(d) 119903 = 15

Figure 3 Displacement time history at midspan of composite beams (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion ofshear lag)

The spring parameter of the COMBIN14 elements can becalculated as [25]

1198701= 066119899

119904119881119906 (44)

The degree of shear connections is calculated as [5]

119903 =119899119904119881119906119871

119860119904119891119904119897 (45)

where 119897 is the spacing of shear connectors 119903 is the degree ofshear connections 119891

119904is the tensile strength of steel beams 119899

119904

is the number of connectors across the beam section119881119906is the

ultimate shear strength of a single shear connectorFrom Figures 3 and 4 it can be seen that the results from

the proposed method agree well with those from ANSYSfor different degrees of shear connections This verifies theaccuracy of the proposed method and some meaningfulconclusions for engineering design can be drawn as follows

(1) For various degrees of shear connections the vibra-tion period of composite beams with and withoutshear lag effects is almost the same which indicatesthat the shear lag and shear connection degree havelittle impact on the vibration period

(2) The larger the degree of shear connections thesmaller the amplitude of deflection

(3) The influence of the shear lag on the amplitude ofdeflection is not significant indicating that it has littleeffect on the deflection of beams

(4) The longitudinal strain in the concrete slab and steelbeam bottom plate shows obvious shear lag effects

(5) The longitudinal strain in the concrete slab reducesobviously with an increment in the degree of shear

Mathematical Problems in Engineering 7

0 1 2 3 4 5100

120

140

160

180

200

minus5 minus4 minus3 minus2 minus1

ANSYS r = 025

This paper 120582 = 1 r = 025

This paper 120582 = 0 r = 025ANSYS r = 05

This paper 120582 = 1 r = 05

This paper 120582 = 0 r = 05ANSYS r = 1

This paper 120582 = 1 r = 1

This paper 120582 = 0 r = 1

Y-coordinate (m)

Stra

in d

istrib

utio

n (10minus6)

(a) Concrete plate

0 1 2 3minus3 minus2 minus1

ANSYS r = 025

This paper 120582 = 1 r = 025

This paper 120582 = 0 r = 025ANSYS r = 05

This paper 120582 = 1 r = 05

This paper 120582 = 0 r = 05ANSYS r = 1

This paper 120582 = 1 r = 1

This paper 120582 = 0 r = 1

Y-coordinate (m)

minus220

minus230

minus240

minus250

minus260

minus270

minus280

Stra

in d

istrib

utio

n (10minus6)

(b) Steel beam bottom plate

Figure 4 Strain distribution in the flanges of midspan section (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion of shear lag)

connections which indicates that the shear connec-tion degree has a great effect on the longitudinal strainof concrete slab Nevertheless the longitudinal strainin the steel beam bottom plate changes little with anincrement in the degree of shear connection

6 Conclusions

(1) Numerical analyses are carried out to verify theaccuracy of the proposed theory The comprehensiveconsiderations of shear lag shear deformation andinterface slip yield an accurate prediction of dynamicresponses of composite box beams

(2) The degree of shear connections has little impacton the vibration period of composite beams but hassignificant impact on the amplitude of deflection ofcomposite beams Nevertheless the shear lag effecthas limited contribution to the deflection or thevibration period of composite beams

(3) The longitudinal strain in the concrete slabs andbottom plate of steel beams shows an obvious shearlag effect

(4) The longitudinal strain in the concrete slabs reducesgreatly with increasing shear connection degree Nev-ertheless the longitudinal strain in the steel beambottom plate changes little with an increment in thedegree of shear connection

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Function of China(51408449 51378502) and the Fundamental Research Fundsfor the Central Universities of China (2014-IV-049)

References

[1] J Nie and C S Cai ldquoSteel-concrete composite beams consider-ing shear slip effectsrdquo Journal of Structural Engineering vol 129no 4 pp 495ndash506 2003

[2] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005

[3] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007

[4] Z Wangbao J Lizhong K Juntao and B Minxi ldquoDistortionalbuckling analysis of steel-concrete composite girders in negativemoment areardquoMathematical Problems in Engineering vol 2014Article ID 635617 10 pages 2014

[5] W-B Zhou L-Z Jiang Z-J Liu and X-J Liu ldquoClosed-form solution for shear lag effects of steel-concrete compositebox beams considering shear deformation and sliprdquo Journal ofCentral South University vol 19 no 10 pp 2976ndash2982 2012

[6] F Gara G Ranzi and G Leoni ldquoSimplified method of analysisaccounting for shear-lag effects in composite bridge decksrdquoJournal of Constructional Steel Research vol 67 no 10 pp 1684ndash1697 2011

[7] F Gara G Leoni and L Dezi ldquoA beam finite element includingshear lag effect for the time-dependent analysis of steel-concrete

8 Mathematical Problems in Engineering

composite decksrdquoEngineering Structures vol 31 no 8 pp 1888ndash1902 2009

[8] A Morassi and L Rocchetto ldquoA damage analysis of steel-concrete composite beams via dynamic methods part I Exper-imental resultsrdquo Journal of Vibration and Control vol 9 no 5pp 507ndash527 2003

[9] M Dilena and A Morassi ldquoA damage analysis of steel-concretecomposite beams via dynamic methods part II Analyticalmodels anddamage detectionrdquo Journal of Vibration andControlvol 9 no 5 pp 529ndash565 2003

[10] M Dilena and A Morassi ldquoVibrations of steelmdashconcretecomposite beamswith partially degraded connection and appli-cations to damage detectionrdquo Journal of Sound and Vibrationvol 320 no 1-2 pp 101ndash124 2009

[11] C Adam R Heuer and A Jeschko ldquoFlexural vibrations ofelastic composite beams with interlayer sliprdquo Acta Mechanicavol 125 no 1ndash4 pp 17ndash30 1997

[12] G Biscontin A Morassi and P Wendel ldquoVibrations of steel-concrete composite beamsrdquo Journal of Vibration and Controlvol 6 no 5 pp 691ndash714 2000

[13] S Berczynski and T Wroblewski ldquoVibration of steel-concretecomposite beams using the Timoshenko beam modelrdquo Journalof Vibration and Control vol 11 no 6 pp 829ndash848 2005

[14] S Berczynski and T Wroblewski ldquoExperimental verification ofnatural vibration models of steel-concrete composite beamsrdquoJournal of Vibration and Control vol 16 no 14 pp 2057ndash20812010

[15] R Xu and Y Wu ldquoStatic dynamic and buckling analysisof partial interaction composite members using Timoshenkorsquosbeam theoryrdquo International Journal of Mechanical Sciences vol49 no 10 pp 1139ndash1155 2007

[16] X Shen W Chen Y Wu and R Xu ldquoDynamic analysis ofpartial-interaction composite beamsrdquo Composites Science andTechnology vol 71 no 10 pp 1286ndash1294 2011

[17] Z Shen and H Zhong ldquoStatic and vibrational analysis ofpartially composite beams using the weak-form quadratureelement methodrdquo Mathematical Problems in Engineering vol2012 Article ID 974023 23 pages 2012

[18] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013

[19] J G S da Silva S A L de Andrade and E D C LopesldquoParametric modelling of the dynamic behaviour of a steel-concrete composite floorrdquo Engineering Structures vol 75 pp327ndash339 2014

[20] S Lenci and J Warminski ldquoFree and forced nonlinear oscil-lations of a two-layer composite beam with interface sliprdquoNonlinear Dynamics vol 70 no 3 pp 2071ndash2087 2012

[21] Q-H Nguyen M Hjiaj and P Le Grognec ldquoAnalyticalapproach for free vibration analysis of two-layer Timoshenkobeams with interlayer sliprdquo Journal of Sound and Vibration vol331 no 12 pp 2949ndash2961 2012

[22] W-A Wang Q Li C-H Zhao and W-L Zhuang ldquoDynamicproperties of long-span steel-concrete composite bridges withexternal tendonsrdquo Journal of Highway and TransportationResearch andDevelopment (English Edition) vol 7 no 4 pp 30ndash38 2013

[23] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013

[24] W-B Zhou L-Z Jiang Z-W Yu and Z Huang ldquoFree vibra-tion characteristics of steel-concrete composite continuous boxgirder considering shear lag and sliprdquo China Journal of Highwayand Transport vol 26 no 5 pp 88ndash94 2013

[25] J Nie J Fan and C S Cai ldquoStiffness and deflection of steel-concrete composite beams under negative bendingrdquo Journal ofStructural Engineering vol 130 no 11 pp 1842ndash1851 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

one-dimensional model of a composite beam was presentedwhere the elements connecting the steel and reinforced con-crete slab were described by means of a strain energy densityfunction defined along the beam axis Berczynski [13 14] pre-sented a solution of the problem of free vibrations of I-steel-concrete composite beams and found that the results obtainedon the basis of the Timoshenko beam theory model achievedthe highest conformitywith the experimental results both forhigher and for lowermodes of flexural vibrations of the beamXu andWu [15] investigated the static dynamic and bucklingbehavior of partial-interaction T-composite members bytaking into account the influences of rotary inertia andshear deformations and obtained the analytical expressionsof the frequencies of the simply supported composite beamShen et al [16] studied the dynamic behavior of partial-interaction T-composite beams by using state-space methodwhich was properly established via selecting the appropriatestate variables and the characteristic equations of frequencyand the corresponding modal shapes of free vibration undergeneralized boundary conditions were then obtained Shenand Zhong [17] examined the deformation of partially I-steel-concrete composite beams under distributed loading andfree vibrations of partially I-steel-concrete composite beamsunder various boundary conditions where the weak-formquadrature element method was used

Considerable efforts have been put into the investigationof dynamic characteristics of composite beams includinginterface slip effects [18ndash22] Most of studies focus on the I-steel beam-concrete composite beams or T-type compositebeams The studies about the dynamic characteristics ofcomposite box beams are rare especially the studies involvingboth the interface slip and shear lag in the dynamic character-istics Based on a longitudinal warping function consideringself-balancing of axial forces and the Hamilton principlethis paper deduces the governing differential equations ofdynamic responses of composite box beams under arbitrarilydistributed loads It takes into account the influence of shearlag interface slip and shear deformation The equations aresolved by central difference methods Numerical studies arecarried out and a good agreement is achieved between resultsfrom the proposed method and finite element method usingANSYS The influences of the shear lag effect and the degreeof shear connections on the dynamic responses of compositebox beams are examined and meaningful conclusions forengineering design are drawn

2 Basic Assumptions

Figure 1 shows the section size and coordinate system ofcomposite box beams The parameters 119905

1 1199052 1199053 1199054 and

119905lowast

4are the thicknesses of the top concrete slab cantilever

plate bottom plate web and top flange of steel box beamrespectively the parameters 119887

1 1198872 1198873 1198874 and 119887

lowast

4are the

widths of half-top concrete slab cantilever plate half-bottomplate of steel beams web of steel beams and top flange ofsteel beams respectively the parameters ℎ

119888and ℎ

119904are the

distances between the centroids of the concrete slab andsteel beam to the slab-beam interface respectively and here

hc

hsy 0

z w

b1b2 b1 b2

b4

blowast

4

t4

t1

b3 b3

t3

tlowast

4h

t2

Figure 1 Schematic of section of steel-concrete composite boxbeams

ℎ = ℎ119888+ ℎ119904 Reasonable assumptions to simplify the analysis

model are made as follows(1) According to the displacement compatibility the

longitudinal warping function of concrete top slab cantileverplate bottom flange and web of steel beams is assumed as[5 23 24]

119892119894= 120595119894(119910)119880 (119909 119905) 119894 = 1 2 3 4 (1)

120595119894= 120572119894(1199102

1198872

119894

minus 1) + 119889 119894 = 1 2 3 4 (2)

Considering self-balancing of axial forces produced bylongitudinal displacement yields [5 23 24]

1205721= 1 120572

2=

1198872

2

1198872

1

1205723= 119911119887

1198872

3

(1198872

1119911119905) 120572

4= 0

(3)

Considering self-balancing of axial forces produced bylongitudinal warping function yields [5 23 24]

int

119860

120595119889119860 = 0 (4)

Substituting (2) into (4) obtains the constant term oflongitudinal warping displacements as follows

119889 =2 (12057211198601119899 + 120572

21198602119899 + 120572

31198603)

(31198600)

(5)

For 119894 = 2 replace 119910 with 119910 = 1198871+ 1198872minus 119910 119911

119905and

119911119887are the 119911-coordinates of centroids of the concrete slab

and bottom flange 1199110is the 119911-coordinate of the slab-beam

interface 119889 is the constant term of longitudinal warpingdisplacements 119880(119909 119905) is the function of the amplitude ofwarping displacements 120595

119894is the warping shape function of

the beam section as shown in Figure 2120572119894is the self-balancing

coefficient of section warping 119899 = 119864119904119864119888 and 119864

119904and 119864

119888are

the modulus of elasticity of steel and concrete respectively

1198601= 211988711199051 119860

2= 211988721199052

1198603= 211988731199053 119860

4= 2 (119887

41199054+ 119887lowast

4119905lowast

4)

1198600=

(1198601+ 1198602)

119899+ 1198603+ 1198604

(6)

Mathematical Problems in Engineering 3

A998400

1A998400

2

A998400

3

d

Figure 2 Schematic of warping shape function

(2) The longitudinal displacement of any point in thetransverse section of composite box beams is assumed as thesuperposition of the longitudinal displacement based on theplain section assumption the longitudinal warping displace-ment due to the shear lag and longitudinal displacement dueto the relative interface slip It can be expressed as [5 23 24]

119906119894= 119896119888120585 minus (119911 minus 119911

119905) 120579 + 119892

119894119894 = 1 2 (7)

119906119894= 119896119904120585 minus (119911 minus 119911

119904) 120579 + 119892

119894119894 = 3 4 (8)

119896119888= minus

119860119904

1198600

119896119904=

119860119888

(1198991198600) (9)

120577 (119909 119905) = 120585 + ℎ120579 (10)

where 120579(119909 119905) is the rotation of the beam section 119860119908= 211988741199054

119860119904= 1198603+ 1198604is the cross section area of steel beams 119860

119888=

1198601+ 1198602is the cross section area of concrete slabs 120585(119909 119905) is

the longitudinal displacement difference between centroidsof the concrete slab and steel beam 120577(119909 119905) is slab-beaminterface slip 119911

119904is the 119911-coordinate of the centroid of the

steel beam 119896119904is the ratio between steel beamrsquos longitudinal

displacement due to the relative interface slip and relativeinterface slip and 119896

119888is the ratio between concrete slabrsquos

longitudinal displacement due to the relative interface slipand relative interface slip

(3) The vertical compression and transverse strain ofconcrete slabs and steel beams are ignored [5 23]

3 Vibration Differential Equation andBoundary Conditions

31 The Strain of the Cross Section The sectional strain canbe obtained from the above longitudinal displacement ofcomposite beam sections as

120576119909119894

= 119896119888

120597120585

120597119909minus (119911 minus 119911

119905)120597120579

120597119909+ 120595119894

120597119880

120597119909119894 = 1 2

120576119909119894

= 119896119904

120597120585

120597119909minus (119911 minus 119911

119904)120597120579

120597119909+ 120595119894

120597119880

120597119909119894 = 3 4

120574119909119910119894

=120597119906119894

120597119910=

120597120595119894

120597119910119880 119894 = 1 2 3

120574119909119911

=120597119908

120597119909minus 120579

(11)

where 120576119894(119894 = 1 2 3 4) are the longitudinal strain of the top

concrete slab cantilever plate bottom flange and web of steelbeams respectively 120574

119894(119894 = 1 2 3) are the shear strain of

top concrete slab cantilever plate and bottom plate of steelbeams respectively 120574

119909119911is the shear strain of the web of steel

beams 119908(119909 119905) is the vertical deflection of composite boxbeams

32 Total Potential Energy of the Composite Box Beam Thestrain energy of composite box beams is defined as [5]

119881 = 05 int

119871

0

int

1198600

(1198641199041205762+ 1198661199041205742) 119889119860119889119909

+ 05 int

119871

0

119896sl1205772119889119909 +

051198661199041198600(1199081015840minus 120579)2

120572119904

(12)

Substituting (10)-(11) into (12) gives the strain energy ofcomposite box beams as

119881 = 05 int

119871

0

[

[

11986312058510158402+ 11986511988010158402+ 2119867119880

10158401205851015840+ 11986812057910158402+ 1198691198802

minus 211987811988010158401205791015840+ 119896sl120577

2+

1198661199041198600(1199081015840minus 120579)2

120572119904

]

]

119889119909

(13)

where119863 = 1198641198881198962

119888119860119888+1198641199041198962

119904119860119904 119865 = 119864

119888119861119888119891+119864119904119861119904119891119867 = 119864

119888119861119888ℎ+

119864119904119861119904ℎ 119868 = 119864

119888119861119888119894+119864119904119861119904119894 119869 = 119866

119888119861119888119895+119866119904119861119904119895 119878 = 119864

119888119861119888119904+119864119904119861119904119904

119861119888119891

= int119860119888

1205952119889119860 119861

119904119891= int119860119904

1205952119889119860 119861

119888ℎ= int119860119888

119896119888120595119889119860 119861

119904ℎ=

int119860119904

119896119904120595119889119860 119861

119888119894= int119860119888

(119911 minus 119911119905)2119889119860 119861

119904119894= int119860119904

(119911 minus 119911119904)2119889119860 119861

119888119895=

int119860119888

(120597120595120597119910)2119889119860 119861

119904119895= int1198603

(120597120595120597119910)2119889119860 119861

119888119904= int119860119888

(119911minus119911119905)120595 119889119860

119861119904119904

= int119860119904

(119911 minus 119911119904)120595 119889119860 and 120572

119904is the correction coefficient of

shear deformation Considering that the webs bear most ofthe vertical shear force in section here the value of 120572

119904is taken

as 1198600(211988741199054) and 2119887

41199054is the section area of webs 119896sl is the

slip stiffness between the concrete slab and the steel beam119871 isthe span of the composite box beam 119866

119904is the shear modulus

of the steel beamThe kinetic energy of the composite box beam is [5]

119879 =1

2int

119871

0

int

119860

(1205882+ 119898

2) 119889119860119889119909 (14)

Substituting of (7) and (8) yields

119879

=1

2int

119871

(1198982+ 1198631

1205852+ 11986512+ 21198671

120585 + 1198681

1205792minus 21198781 120579) 119889119909

(15)

where1198631= 1205881198881198962

119888119860119888+1205881199041198962

1199041198601199041198651= 120588119888119861119888119891+1205881199041198611199041198911198671= 120588119888119861119888ℎ+

120588119904119861119904ℎ 1198681= 120588119888119861119888119894+ 120588119904119861119904119894 1198781= 120588119888119861119888119904+ 120588119904119861119904119904119898 = 120588

119888119860119888+ 120588119904119860119904

119860 = 119860119904+ 119860119888 and 120588

119888and 120588119904are the density of concrete slabs

and steel beams respectively

4 Mathematical Problems in Engineering

The work done by the external loads can be expressed as

119882 = int

119871

119902 (119909 119905) 119908 119889119909 (16)

where 119902(119909 119905) is the distribution function of arbitrary load

33 VibrationDifferential Equation and Boundary ConditionsThegoverning equations of vibration of composite box beamsand corresponding boundary conditions can be deducedbased on Hamilton principle as

11986511988010158401015840+ 11986712058510158401015840minus 119869119880 minus 119878120579

10158401015840minus 1198651 minus 119867

1

120585 + 1198781

120579 = 0 (17)

11986711988010158401015840+ 11986312058510158401015840minus 119896sl120577 minus 119867

1 minus 119863

1

120585 = 0 (18)

1198661199041198600(11990810158401015840minus 1205791015840)

120572119904

minus 119898 + 119902 (119909 119905) = 0 (19)

1198661199041198600(1199081015840minus 120579)

120572119904

+ 11986812057910158401015840minus 11987811988010158401015840minus 119896sl120577ℎ + 119878

1 minus 1198681

120579 = 0 (20)

(1198651198801015840+ 1198671205851015840minus 1198781205791015840) 120575119880

10038161003816100381610038161003816

119871

0= 0 (21)

(1198631205851015840+ 119867119880

1015840) 120575120585

10038161003816100381610038161003816

119871

0= 0 (22)

1198661199041198600(1199081015840minus 120579)

120572119904

120575119908

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871

0

= 0 (23)

(1198681205791015840minus 1198781198801015840) 120575120579

10038161003816100381610038161003816

119871

0= 0 (24)

Taking a simply supported beam as example (21)ndash(24)give boundary conditions as

1198801015840(119871 119905) = 120585

1015840(119871 119905) = 120579

1015840(119871 119905) = 119908 (119871 119905) = 0

1198801015840(0 119905) = 120585

1015840(0 119905) = 120579

1015840(0 119905) = 119908 (0 119905) = 0

(25)

given the initial conditions as

119880 (119909 0) = 1198800(119909) (119909 0) = 119880

0(119909)

120585 (119909 0) = 1205850(119909)

120585 (119909 0) = 120585

0(119909)

119908 (119909 0) = 1199080(119909) (119909 0) = 119908

0(119909)

120579 (119909 0) = 1205790(119909) 120579 (119909 0) = 120579

0(119909)

(26)

4 Finite Difference Method ofVibration Differential Equation

41 Difference Scheme Let solution domain be 120590 = (119909 119905) |

0 le 119909 le 119871 0 le 119905 le 119879 119879 is the end time of solution arectangularmesh ismade in the solution area with a time stepof 120591 and space step of 120592 so that

119909119894= 119894120592 (119894 = 0 1 2 119868)

119905119895= 119895120591 (119895 = 0 1 2 119869)

(27)

where 119868 = 119871120592 119869 = 119879120591

Let

119880119895

119894= 119880 (119909

119894 119905119895) 120585

119895

119894= 120585 (119909

119894 119905119895)

119908119895

119894= 119908 (119909

119894 119905119895) 120579

119895

119894= 120579 (119909

119894 119905119895)

(28)

The central difference calculation of governing differen-tial equation yields

119865

119880119895

119894+1minus 2119880119895

119894+ 119880119895

119894minus1

1205922

+ 119867

120585119895

119894+1minus 2120585119895

119894+ 120585119895

119894minus1

1205922

minus 119878

120579119895

119894+1minus 2120579119895

119894+ 120579119895

119894minus1

1205922

minus 1198651

119880119895+1

119894minus 2119880119895

119894+ 119880119895minus1

119894

1205912

minus 119869119880119895

119894minus 1198671

120585119895+1

119894minus 2120585119895

119894+ 120585119895minus1

119894

1205912

+ 1198781

120579119895+1

119894minus 2120579119895

119894+ 120579119895minus1

119894

1205912

= 0

(119894 = 1 119868 minus 1 119895 = 1 2 119869)

(29)

119867

119880119895

119894+1minus 2119880119895

119894+ 119880119895

119894minus1

1205922

+ 119863

120585119895

119894+1minus 2120585119895

119894+ 120585119895

119894minus1

1205922

minus 119896sl (120585119895

119894+ ℎ120579119895

119894) minus 119867

1

119880119895+1

119894minus 2119880119895

119894+ 119880119895minus1

119894

1205912

minus 1198631

120585119895+1

119894minus 2120585119895

119894+ 120585119895minus1

119894

1205912

= 0

(119894 = 1 119868 minus 1 119895 = 1 2 119869)

(30)

1198661199041198600

120572119904

(

119908119895

119894+1minus 119908119895

119894minus1

2120592minus 120579119895

119894) + 119868

120579119895

119894+1minus 2120579119895

119894+ 120579119895

119894minus1

1205922

minus 119878

119880119895

119894+1minus 2119880119895

119894+ 119880119895

119894minus1

1205922

+ 1198781

119880119895+1

119894minus 2119880119895

119894+ 119880119895minus1

119894

1205912

minus 1198681

120579119895+1

119894minus 2120579119895

119894+ 120579119895minus1

119894

1205912

minus 119896slℎ (120585119895

119894+ ℎ120579119895

119894) = 0

(119894 = 1 119868 minus 1 119895 = 1 2 119869)

(31)

1198661199041198600

120572119904

(

119908119895

119894+1minus 2119908119895

119894+ 119908119895

119894minus1

1205922

minus

120579119895

119894+1minus 120579119895

119894minus1

2120592)

+ 119902119895

119894minus 119898

119908119895+1

119894minus 2119908119895

119894+ 119908119895minus1

119894

1205912

= 0

(119894 = 1 119868 minus 1 119895 = 1 2 119869)

(32)

Mathematical Problems in Engineering 5

The difference calculation of boundary conditions yields

119880119895

119868minus 119880119895

119868minus1

120592=

119880119895

1minus 119880119895

0

120592= 0 119895 = 0 1 119869

(33)

120585119895

119868minus 120585119895

119868minus1

120592=

120585119895

1minus 120585119895

0

120592= 0 119895 = 0 1 119869

(34)

119908119895

119868= 119908119895

0= 0 119895 = 0 1 119869 (35)

120579119895

119868minus 120579119895

119868minus1

120592=

120579119895

1minus 120579119895

0

120592= 0 119895 = 0 1 119869

(36)

The difference calculation of initial condition yields

1198800

119894= 1198800119894

1198801

119894minus 1198800

119894

120591= 1198800119894

119894 = 0 1 2 119868

1205850

119894= 1205850119894

1205851

119894minus 1205850

119894

120591= 1205850119894

119894 = 0 1 2 119868

1199080

119894= 1199080119894

1199081

119894minus 1199080

119894

120591= 1199080119894

119894 = 0 1 2 119868

1205790

119894= 1205790119894

1205791

119894minus 1205790

119894

120591= 1205790119894

119894 = 0 1 2 119868

(37)

42 Solution Step Let U119895 = 119880119895

0 119880119895

1 119880

119895

119868119879 120585119895 = 120585

119895

0 120585119895

1

120585119895

119868119879 w119895 = 119908

119895

0 119908119895

1 119908

119895

119868119879 and 120579119895 = 120579

119895

0 120579119895

1 120579

119895

119868119879

Given U119895minus1 120585119895minus1 w119895minus1 120579119895minus1 U119895 120585119895 w119895 and 120579119895 the solving ofU119895+1 120585119895+1 w119895+1 and 120579119895+1 follows the steps below

(a) Calculate U0 1205850 w0 1205790 U1 1205851 w1 and 1205791 from theinitial conditions given in (37)

(b) Given U119895minus1 120585119895minus1 120579119895minus1 U119895 120585119895 w119895 and 120579119895 calcu-late 119880

119895+1

1 119880119895+1

2 119880

119895+1

119868minus1 120585119895+1

1 120585119895+1

2 120585

119895+1

119868minus1 and

120579119895+1

1 120579119895+1

2 120579

119895+1

119868minus1 from (29)ndash(31)

(c) Calculate 119880119895+1

0 119880119895+1

119868 120585119895+1

0 120585119895+1

119868 and 120579

119895+1

0 120579119895+1

119868

from (33) (34) and (36) and the solution of U119895+1120585119895+1 and 120579119895+1 is obtained in combination withstep (b)

(d) Given w119895minus1 w119895 and 120579119895 determine 119908119895+1

1 119908119895+1

2

119908119895+1

119868minus1 from (32)

(e) Calculate 119908119895+1

0 119908119895+1

119868 from (35) and calculate w119895+1

combined with step (d)

43 Degeneration of theVibrationDifferential Equation Like-wise the governing equations of vibration of composite box

beams and boundary conditions without shear lag effects canbe deduced as

11986312058510158401015840minus 119896sl120577 minus 119863

1

120585 = 0 (38)

1198661199041198600(11990810158401015840minus 1205791015840)

120572119904

minus 119898 + 119902 (119909 119905) = 0 (39)

1198661199041198600(1199081015840minus 120579)

120572119904

+ 11986812057910158401015840minus 119896sl120577ℎ minus 119868

1120579 = 0 (40)

1205851015840120575120585

10038161003816100381610038161003816

119871

0= 0 120579

1015840120575120579

10038161003816100381610038161003816

119871

0= 0 (119908

1015840minus 120579) 120575119908

10038161003816100381610038161003816

119871

0= 0 (41)

For beamswith two ends simply supported the boundaryconditions can be expressed from (41) as

1205851015840(119871 119905) = 120579

1015840(119871 119905) = 119908 (119871 119905) = 0

1205851015840(0 119905) = 120579

1015840(0 119905) = 119908 (0 119905) = 0

(42)

The vibration differential equation of composite boxbeams without considering shear lag effects is a degenera-tion of the one considering the shear lag effects the solutionmethod of which can be referred to in Sections 41 and 42

5 Analysis of Examples

The validity of the proposed method is verified by compar-ing to numerical results from finite element method Thecomparisons are made on four simply supported compositebox beams with different degree of shear connections undersuddenly imposed distributed loads The dynamic responsesof beams with and without shear lag effects are analyzedThe distributed load is taken as 119902 = 300 kNm with a timestep of 120591 = 000002 s and a space step of 120592 = 03m Themechanical and geometrical parameters of composite boxbeams are taken as 119864

119904= 20 times 10

5MPa 119864119888= 45 times 10

4MPa119866119904= 771 times 10

4MPa 120583119904= 028 120583

119888= 018 120588

119904= 7900 kgsdotmminus3

120588119888= 2400 kgsdotmminus3 119887

1= 1198873= 25m 119887

2= 20m 119887

4= 30m

1198875= 02m 119905

1= 1199052= 03m 119905

3= 1199055= 006m 119905

4= 009m

119871 = 30m 119897 = 03m 119899119904= 2 and 119891

119904= 300MPa

The commercial finite element software ANSYS is used inthis study In the finite element model the concrete slab andsteel beam are modeled by SOLID65 and SHELL43 elementsrespectively Shear connector is modeled by COMBIN14elements being spring elements [25] The aspect ratio of themesh is kept close to one throughout the mean mesh sizevaries from 02m to 03m and about 9200 finite elementsin total are employed The transverse distributed loads areimposed on the top flanges The translational 119909 and 119910

degrees of freedom of all nodes at two ends of beams arerestrained The torsion at the beam ends is thus restrainedThe translational 119911 DOF of the left end is restrained to meetthe static determined requirement and allow rotation alongthe moment The results are shown in Figures 3 and 4

The 119896sl is calculated as [5 23 24]

119896sl =1198701

119897 (43)

6 Mathematical Problems in Engineering

000 005 010 015 020 0250

4

8

12

16

20

24

Time (s)

Disp

lace

men

t at m

idsp

an (m

m)

(a) 119903 = 025

000 005 010 015 020 0250

4

8

12

16

20

24

Disp

lace

men

t at m

idsp

an (m

m)

Time (s)

(b) 119903 = 05

000 005 010 015 020 0250

4

8

12

16

20

24

Time (s)

Disp

lace

men

t at m

idsp

an (m

m)

ANSYSThis paper 120582 = 1

This paper 120582 = 0

(c) 119903 = 10

000 005 010 015 020 0250

4

8

12

16

20

24

Time (s)

Disp

lace

men

t at m

idsp

an (m

m)

ANSYSThis paper 120582 = 1

This paper 120582 = 0

(d) 119903 = 15

Figure 3 Displacement time history at midspan of composite beams (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion ofshear lag)

The spring parameter of the COMBIN14 elements can becalculated as [25]

1198701= 066119899

119904119881119906 (44)

The degree of shear connections is calculated as [5]

119903 =119899119904119881119906119871

119860119904119891119904119897 (45)

where 119897 is the spacing of shear connectors 119903 is the degree ofshear connections 119891

119904is the tensile strength of steel beams 119899

119904

is the number of connectors across the beam section119881119906is the

ultimate shear strength of a single shear connectorFrom Figures 3 and 4 it can be seen that the results from

the proposed method agree well with those from ANSYSfor different degrees of shear connections This verifies theaccuracy of the proposed method and some meaningfulconclusions for engineering design can be drawn as follows

(1) For various degrees of shear connections the vibra-tion period of composite beams with and withoutshear lag effects is almost the same which indicatesthat the shear lag and shear connection degree havelittle impact on the vibration period

(2) The larger the degree of shear connections thesmaller the amplitude of deflection

(3) The influence of the shear lag on the amplitude ofdeflection is not significant indicating that it has littleeffect on the deflection of beams

(4) The longitudinal strain in the concrete slab and steelbeam bottom plate shows obvious shear lag effects

(5) The longitudinal strain in the concrete slab reducesobviously with an increment in the degree of shear

Mathematical Problems in Engineering 7

0 1 2 3 4 5100

120

140

160

180

200

minus5 minus4 minus3 minus2 minus1

ANSYS r = 025

This paper 120582 = 1 r = 025

This paper 120582 = 0 r = 025ANSYS r = 05

This paper 120582 = 1 r = 05

This paper 120582 = 0 r = 05ANSYS r = 1

This paper 120582 = 1 r = 1

This paper 120582 = 0 r = 1

Y-coordinate (m)

Stra

in d

istrib

utio

n (10minus6)

(a) Concrete plate

0 1 2 3minus3 minus2 minus1

ANSYS r = 025

This paper 120582 = 1 r = 025

This paper 120582 = 0 r = 025ANSYS r = 05

This paper 120582 = 1 r = 05

This paper 120582 = 0 r = 05ANSYS r = 1

This paper 120582 = 1 r = 1

This paper 120582 = 0 r = 1

Y-coordinate (m)

minus220

minus230

minus240

minus250

minus260

minus270

minus280

Stra

in d

istrib

utio

n (10minus6)

(b) Steel beam bottom plate

Figure 4 Strain distribution in the flanges of midspan section (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion of shear lag)

connections which indicates that the shear connec-tion degree has a great effect on the longitudinal strainof concrete slab Nevertheless the longitudinal strainin the steel beam bottom plate changes little with anincrement in the degree of shear connection

6 Conclusions

(1) Numerical analyses are carried out to verify theaccuracy of the proposed theory The comprehensiveconsiderations of shear lag shear deformation andinterface slip yield an accurate prediction of dynamicresponses of composite box beams

(2) The degree of shear connections has little impacton the vibration period of composite beams but hassignificant impact on the amplitude of deflection ofcomposite beams Nevertheless the shear lag effecthas limited contribution to the deflection or thevibration period of composite beams

(3) The longitudinal strain in the concrete slabs andbottom plate of steel beams shows an obvious shearlag effect

(4) The longitudinal strain in the concrete slabs reducesgreatly with increasing shear connection degree Nev-ertheless the longitudinal strain in the steel beambottom plate changes little with an increment in thedegree of shear connection

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Function of China(51408449 51378502) and the Fundamental Research Fundsfor the Central Universities of China (2014-IV-049)

References

[1] J Nie and C S Cai ldquoSteel-concrete composite beams consider-ing shear slip effectsrdquo Journal of Structural Engineering vol 129no 4 pp 495ndash506 2003

[2] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005

[3] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007

[4] Z Wangbao J Lizhong K Juntao and B Minxi ldquoDistortionalbuckling analysis of steel-concrete composite girders in negativemoment areardquoMathematical Problems in Engineering vol 2014Article ID 635617 10 pages 2014

[5] W-B Zhou L-Z Jiang Z-J Liu and X-J Liu ldquoClosed-form solution for shear lag effects of steel-concrete compositebox beams considering shear deformation and sliprdquo Journal ofCentral South University vol 19 no 10 pp 2976ndash2982 2012

[6] F Gara G Ranzi and G Leoni ldquoSimplified method of analysisaccounting for shear-lag effects in composite bridge decksrdquoJournal of Constructional Steel Research vol 67 no 10 pp 1684ndash1697 2011

[7] F Gara G Leoni and L Dezi ldquoA beam finite element includingshear lag effect for the time-dependent analysis of steel-concrete

8 Mathematical Problems in Engineering

composite decksrdquoEngineering Structures vol 31 no 8 pp 1888ndash1902 2009

[8] A Morassi and L Rocchetto ldquoA damage analysis of steel-concrete composite beams via dynamic methods part I Exper-imental resultsrdquo Journal of Vibration and Control vol 9 no 5pp 507ndash527 2003

[9] M Dilena and A Morassi ldquoA damage analysis of steel-concretecomposite beams via dynamic methods part II Analyticalmodels anddamage detectionrdquo Journal of Vibration andControlvol 9 no 5 pp 529ndash565 2003

[10] M Dilena and A Morassi ldquoVibrations of steelmdashconcretecomposite beamswith partially degraded connection and appli-cations to damage detectionrdquo Journal of Sound and Vibrationvol 320 no 1-2 pp 101ndash124 2009

[11] C Adam R Heuer and A Jeschko ldquoFlexural vibrations ofelastic composite beams with interlayer sliprdquo Acta Mechanicavol 125 no 1ndash4 pp 17ndash30 1997

[12] G Biscontin A Morassi and P Wendel ldquoVibrations of steel-concrete composite beamsrdquo Journal of Vibration and Controlvol 6 no 5 pp 691ndash714 2000

[13] S Berczynski and T Wroblewski ldquoVibration of steel-concretecomposite beams using the Timoshenko beam modelrdquo Journalof Vibration and Control vol 11 no 6 pp 829ndash848 2005

[14] S Berczynski and T Wroblewski ldquoExperimental verification ofnatural vibration models of steel-concrete composite beamsrdquoJournal of Vibration and Control vol 16 no 14 pp 2057ndash20812010

[15] R Xu and Y Wu ldquoStatic dynamic and buckling analysisof partial interaction composite members using Timoshenkorsquosbeam theoryrdquo International Journal of Mechanical Sciences vol49 no 10 pp 1139ndash1155 2007

[16] X Shen W Chen Y Wu and R Xu ldquoDynamic analysis ofpartial-interaction composite beamsrdquo Composites Science andTechnology vol 71 no 10 pp 1286ndash1294 2011

[17] Z Shen and H Zhong ldquoStatic and vibrational analysis ofpartially composite beams using the weak-form quadratureelement methodrdquo Mathematical Problems in Engineering vol2012 Article ID 974023 23 pages 2012

[18] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013

[19] J G S da Silva S A L de Andrade and E D C LopesldquoParametric modelling of the dynamic behaviour of a steel-concrete composite floorrdquo Engineering Structures vol 75 pp327ndash339 2014

[20] S Lenci and J Warminski ldquoFree and forced nonlinear oscil-lations of a two-layer composite beam with interface sliprdquoNonlinear Dynamics vol 70 no 3 pp 2071ndash2087 2012

[21] Q-H Nguyen M Hjiaj and P Le Grognec ldquoAnalyticalapproach for free vibration analysis of two-layer Timoshenkobeams with interlayer sliprdquo Journal of Sound and Vibration vol331 no 12 pp 2949ndash2961 2012

[22] W-A Wang Q Li C-H Zhao and W-L Zhuang ldquoDynamicproperties of long-span steel-concrete composite bridges withexternal tendonsrdquo Journal of Highway and TransportationResearch andDevelopment (English Edition) vol 7 no 4 pp 30ndash38 2013

[23] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013

[24] W-B Zhou L-Z Jiang Z-W Yu and Z Huang ldquoFree vibra-tion characteristics of steel-concrete composite continuous boxgirder considering shear lag and sliprdquo China Journal of Highwayand Transport vol 26 no 5 pp 88ndash94 2013

[25] J Nie J Fan and C S Cai ldquoStiffness and deflection of steel-concrete composite beams under negative bendingrdquo Journal ofStructural Engineering vol 130 no 11 pp 1842ndash1851 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

A998400

1A998400

2

A998400

3

d

Figure 2 Schematic of warping shape function

(2) The longitudinal displacement of any point in thetransverse section of composite box beams is assumed as thesuperposition of the longitudinal displacement based on theplain section assumption the longitudinal warping displace-ment due to the shear lag and longitudinal displacement dueto the relative interface slip It can be expressed as [5 23 24]

119906119894= 119896119888120585 minus (119911 minus 119911

119905) 120579 + 119892

119894119894 = 1 2 (7)

119906119894= 119896119904120585 minus (119911 minus 119911

119904) 120579 + 119892

119894119894 = 3 4 (8)

119896119888= minus

119860119904

1198600

119896119904=

119860119888

(1198991198600) (9)

120577 (119909 119905) = 120585 + ℎ120579 (10)

where 120579(119909 119905) is the rotation of the beam section 119860119908= 211988741199054

119860119904= 1198603+ 1198604is the cross section area of steel beams 119860

119888=

1198601+ 1198602is the cross section area of concrete slabs 120585(119909 119905) is

the longitudinal displacement difference between centroidsof the concrete slab and steel beam 120577(119909 119905) is slab-beaminterface slip 119911

119904is the 119911-coordinate of the centroid of the

steel beam 119896119904is the ratio between steel beamrsquos longitudinal

displacement due to the relative interface slip and relativeinterface slip and 119896

119888is the ratio between concrete slabrsquos

longitudinal displacement due to the relative interface slipand relative interface slip

(3) The vertical compression and transverse strain ofconcrete slabs and steel beams are ignored [5 23]

3 Vibration Differential Equation andBoundary Conditions

31 The Strain of the Cross Section The sectional strain canbe obtained from the above longitudinal displacement ofcomposite beam sections as

120576119909119894

= 119896119888

120597120585

120597119909minus (119911 minus 119911

119905)120597120579

120597119909+ 120595119894

120597119880

120597119909119894 = 1 2

120576119909119894

= 119896119904

120597120585

120597119909minus (119911 minus 119911

119904)120597120579

120597119909+ 120595119894

120597119880

120597119909119894 = 3 4

120574119909119910119894

=120597119906119894

120597119910=

120597120595119894

120597119910119880 119894 = 1 2 3

120574119909119911

=120597119908

120597119909minus 120579

(11)

where 120576119894(119894 = 1 2 3 4) are the longitudinal strain of the top

concrete slab cantilever plate bottom flange and web of steelbeams respectively 120574

119894(119894 = 1 2 3) are the shear strain of

top concrete slab cantilever plate and bottom plate of steelbeams respectively 120574

119909119911is the shear strain of the web of steel

beams 119908(119909 119905) is the vertical deflection of composite boxbeams

32 Total Potential Energy of the Composite Box Beam Thestrain energy of composite box beams is defined as [5]

119881 = 05 int

119871

0

int

1198600

(1198641199041205762+ 1198661199041205742) 119889119860119889119909

+ 05 int

119871

0

119896sl1205772119889119909 +

051198661199041198600(1199081015840minus 120579)2

120572119904

(12)

Substituting (10)-(11) into (12) gives the strain energy ofcomposite box beams as

119881 = 05 int

119871

0

[

[

11986312058510158402+ 11986511988010158402+ 2119867119880

10158401205851015840+ 11986812057910158402+ 1198691198802

minus 211987811988010158401205791015840+ 119896sl120577

2+

1198661199041198600(1199081015840minus 120579)2

120572119904

]

]

119889119909

(13)

where119863 = 1198641198881198962

119888119860119888+1198641199041198962

119904119860119904 119865 = 119864

119888119861119888119891+119864119904119861119904119891119867 = 119864

119888119861119888ℎ+

119864119904119861119904ℎ 119868 = 119864

119888119861119888119894+119864119904119861119904119894 119869 = 119866

119888119861119888119895+119866119904119861119904119895 119878 = 119864

119888119861119888119904+119864119904119861119904119904

119861119888119891

= int119860119888

1205952119889119860 119861

119904119891= int119860119904

1205952119889119860 119861

119888ℎ= int119860119888

119896119888120595119889119860 119861

119904ℎ=

int119860119904

119896119904120595119889119860 119861

119888119894= int119860119888

(119911 minus 119911119905)2119889119860 119861

119904119894= int119860119904

(119911 minus 119911119904)2119889119860 119861

119888119895=

int119860119888

(120597120595120597119910)2119889119860 119861

119904119895= int1198603

(120597120595120597119910)2119889119860 119861

119888119904= int119860119888

(119911minus119911119905)120595 119889119860

119861119904119904

= int119860119904

(119911 minus 119911119904)120595 119889119860 and 120572

119904is the correction coefficient of

shear deformation Considering that the webs bear most ofthe vertical shear force in section here the value of 120572

119904is taken

as 1198600(211988741199054) and 2119887

41199054is the section area of webs 119896sl is the

slip stiffness between the concrete slab and the steel beam119871 isthe span of the composite box beam 119866

119904is the shear modulus

of the steel beamThe kinetic energy of the composite box beam is [5]

119879 =1

2int

119871

0

int

119860

(1205882+ 119898

2) 119889119860119889119909 (14)

Substituting of (7) and (8) yields

119879

=1

2int

119871

(1198982+ 1198631

1205852+ 11986512+ 21198671

120585 + 1198681

1205792minus 21198781 120579) 119889119909

(15)

where1198631= 1205881198881198962

119888119860119888+1205881199041198962

1199041198601199041198651= 120588119888119861119888119891+1205881199041198611199041198911198671= 120588119888119861119888ℎ+

120588119904119861119904ℎ 1198681= 120588119888119861119888119894+ 120588119904119861119904119894 1198781= 120588119888119861119888119904+ 120588119904119861119904119904119898 = 120588

119888119860119888+ 120588119904119860119904

119860 = 119860119904+ 119860119888 and 120588

119888and 120588119904are the density of concrete slabs

and steel beams respectively

4 Mathematical Problems in Engineering

The work done by the external loads can be expressed as

119882 = int

119871

119902 (119909 119905) 119908 119889119909 (16)

where 119902(119909 119905) is the distribution function of arbitrary load

33 VibrationDifferential Equation and Boundary ConditionsThegoverning equations of vibration of composite box beamsand corresponding boundary conditions can be deducedbased on Hamilton principle as

11986511988010158401015840+ 11986712058510158401015840minus 119869119880 minus 119878120579

10158401015840minus 1198651 minus 119867

1

120585 + 1198781

120579 = 0 (17)

11986711988010158401015840+ 11986312058510158401015840minus 119896sl120577 minus 119867

1 minus 119863

1

120585 = 0 (18)

1198661199041198600(11990810158401015840minus 1205791015840)

120572119904

minus 119898 + 119902 (119909 119905) = 0 (19)

1198661199041198600(1199081015840minus 120579)

120572119904

+ 11986812057910158401015840minus 11987811988010158401015840minus 119896sl120577ℎ + 119878

1 minus 1198681

120579 = 0 (20)

(1198651198801015840+ 1198671205851015840minus 1198781205791015840) 120575119880

10038161003816100381610038161003816

119871

0= 0 (21)

(1198631205851015840+ 119867119880

1015840) 120575120585

10038161003816100381610038161003816

119871

0= 0 (22)

1198661199041198600(1199081015840minus 120579)

120572119904

120575119908

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871

0

= 0 (23)

(1198681205791015840minus 1198781198801015840) 120575120579

10038161003816100381610038161003816

119871

0= 0 (24)

Taking a simply supported beam as example (21)ndash(24)give boundary conditions as

1198801015840(119871 119905) = 120585

1015840(119871 119905) = 120579

1015840(119871 119905) = 119908 (119871 119905) = 0

1198801015840(0 119905) = 120585

1015840(0 119905) = 120579

1015840(0 119905) = 119908 (0 119905) = 0

(25)

given the initial conditions as

119880 (119909 0) = 1198800(119909) (119909 0) = 119880

0(119909)

120585 (119909 0) = 1205850(119909)

120585 (119909 0) = 120585

0(119909)

119908 (119909 0) = 1199080(119909) (119909 0) = 119908

0(119909)

120579 (119909 0) = 1205790(119909) 120579 (119909 0) = 120579

0(119909)

(26)

4 Finite Difference Method ofVibration Differential Equation

41 Difference Scheme Let solution domain be 120590 = (119909 119905) |

0 le 119909 le 119871 0 le 119905 le 119879 119879 is the end time of solution arectangularmesh ismade in the solution area with a time stepof 120591 and space step of 120592 so that

119909119894= 119894120592 (119894 = 0 1 2 119868)

119905119895= 119895120591 (119895 = 0 1 2 119869)

(27)

where 119868 = 119871120592 119869 = 119879120591

Let

119880119895

119894= 119880 (119909

119894 119905119895) 120585

119895

119894= 120585 (119909

119894 119905119895)

119908119895

119894= 119908 (119909

119894 119905119895) 120579

119895

119894= 120579 (119909

119894 119905119895)

(28)

The central difference calculation of governing differen-tial equation yields

119865

119880119895

119894+1minus 2119880119895

119894+ 119880119895

119894minus1

1205922

+ 119867

120585119895

119894+1minus 2120585119895

119894+ 120585119895

119894minus1

1205922

minus 119878

120579119895

119894+1minus 2120579119895

119894+ 120579119895

119894minus1

1205922

minus 1198651

119880119895+1

119894minus 2119880119895

119894+ 119880119895minus1

119894

1205912

minus 119869119880119895

119894minus 1198671

120585119895+1

119894minus 2120585119895

119894+ 120585119895minus1

119894

1205912

+ 1198781

120579119895+1

119894minus 2120579119895

119894+ 120579119895minus1

119894

1205912

= 0

(119894 = 1 119868 minus 1 119895 = 1 2 119869)

(29)

119867

119880119895

119894+1minus 2119880119895

119894+ 119880119895

119894minus1

1205922

+ 119863

120585119895

119894+1minus 2120585119895

119894+ 120585119895

119894minus1

1205922

minus 119896sl (120585119895

119894+ ℎ120579119895

119894) minus 119867

1

119880119895+1

119894minus 2119880119895

119894+ 119880119895minus1

119894

1205912

minus 1198631

120585119895+1

119894minus 2120585119895

119894+ 120585119895minus1

119894

1205912

= 0

(119894 = 1 119868 minus 1 119895 = 1 2 119869)

(30)

1198661199041198600

120572119904

(

119908119895

119894+1minus 119908119895

119894minus1

2120592minus 120579119895

119894) + 119868

120579119895

119894+1minus 2120579119895

119894+ 120579119895

119894minus1

1205922

minus 119878

119880119895

119894+1minus 2119880119895

119894+ 119880119895

119894minus1

1205922

+ 1198781

119880119895+1

119894minus 2119880119895

119894+ 119880119895minus1

119894

1205912

minus 1198681

120579119895+1

119894minus 2120579119895

119894+ 120579119895minus1

119894

1205912

minus 119896slℎ (120585119895

119894+ ℎ120579119895

119894) = 0

(119894 = 1 119868 minus 1 119895 = 1 2 119869)

(31)

1198661199041198600

120572119904

(

119908119895

119894+1minus 2119908119895

119894+ 119908119895

119894minus1

1205922

minus

120579119895

119894+1minus 120579119895

119894minus1

2120592)

+ 119902119895

119894minus 119898

119908119895+1

119894minus 2119908119895

119894+ 119908119895minus1

119894

1205912

= 0

(119894 = 1 119868 minus 1 119895 = 1 2 119869)

(32)

Mathematical Problems in Engineering 5

The difference calculation of boundary conditions yields

119880119895

119868minus 119880119895

119868minus1

120592=

119880119895

1minus 119880119895

0

120592= 0 119895 = 0 1 119869

(33)

120585119895

119868minus 120585119895

119868minus1

120592=

120585119895

1minus 120585119895

0

120592= 0 119895 = 0 1 119869

(34)

119908119895

119868= 119908119895

0= 0 119895 = 0 1 119869 (35)

120579119895

119868minus 120579119895

119868minus1

120592=

120579119895

1minus 120579119895

0

120592= 0 119895 = 0 1 119869

(36)

The difference calculation of initial condition yields

1198800

119894= 1198800119894

1198801

119894minus 1198800

119894

120591= 1198800119894

119894 = 0 1 2 119868

1205850

119894= 1205850119894

1205851

119894minus 1205850

119894

120591= 1205850119894

119894 = 0 1 2 119868

1199080

119894= 1199080119894

1199081

119894minus 1199080

119894

120591= 1199080119894

119894 = 0 1 2 119868

1205790

119894= 1205790119894

1205791

119894minus 1205790

119894

120591= 1205790119894

119894 = 0 1 2 119868

(37)

42 Solution Step Let U119895 = 119880119895

0 119880119895

1 119880

119895

119868119879 120585119895 = 120585

119895

0 120585119895

1

120585119895

119868119879 w119895 = 119908

119895

0 119908119895

1 119908

119895

119868119879 and 120579119895 = 120579

119895

0 120579119895

1 120579

119895

119868119879

Given U119895minus1 120585119895minus1 w119895minus1 120579119895minus1 U119895 120585119895 w119895 and 120579119895 the solving ofU119895+1 120585119895+1 w119895+1 and 120579119895+1 follows the steps below

(a) Calculate U0 1205850 w0 1205790 U1 1205851 w1 and 1205791 from theinitial conditions given in (37)

(b) Given U119895minus1 120585119895minus1 120579119895minus1 U119895 120585119895 w119895 and 120579119895 calcu-late 119880

119895+1

1 119880119895+1

2 119880

119895+1

119868minus1 120585119895+1

1 120585119895+1

2 120585

119895+1

119868minus1 and

120579119895+1

1 120579119895+1

2 120579

119895+1

119868minus1 from (29)ndash(31)

(c) Calculate 119880119895+1

0 119880119895+1

119868 120585119895+1

0 120585119895+1

119868 and 120579

119895+1

0 120579119895+1

119868

from (33) (34) and (36) and the solution of U119895+1120585119895+1 and 120579119895+1 is obtained in combination withstep (b)

(d) Given w119895minus1 w119895 and 120579119895 determine 119908119895+1

1 119908119895+1

2

119908119895+1

119868minus1 from (32)

(e) Calculate 119908119895+1

0 119908119895+1

119868 from (35) and calculate w119895+1

combined with step (d)

43 Degeneration of theVibrationDifferential Equation Like-wise the governing equations of vibration of composite box

beams and boundary conditions without shear lag effects canbe deduced as

11986312058510158401015840minus 119896sl120577 minus 119863

1

120585 = 0 (38)

1198661199041198600(11990810158401015840minus 1205791015840)

120572119904

minus 119898 + 119902 (119909 119905) = 0 (39)

1198661199041198600(1199081015840minus 120579)

120572119904

+ 11986812057910158401015840minus 119896sl120577ℎ minus 119868

1120579 = 0 (40)

1205851015840120575120585

10038161003816100381610038161003816

119871

0= 0 120579

1015840120575120579

10038161003816100381610038161003816

119871

0= 0 (119908

1015840minus 120579) 120575119908

10038161003816100381610038161003816

119871

0= 0 (41)

For beamswith two ends simply supported the boundaryconditions can be expressed from (41) as

1205851015840(119871 119905) = 120579

1015840(119871 119905) = 119908 (119871 119905) = 0

1205851015840(0 119905) = 120579

1015840(0 119905) = 119908 (0 119905) = 0

(42)

The vibration differential equation of composite boxbeams without considering shear lag effects is a degenera-tion of the one considering the shear lag effects the solutionmethod of which can be referred to in Sections 41 and 42

5 Analysis of Examples

The validity of the proposed method is verified by compar-ing to numerical results from finite element method Thecomparisons are made on four simply supported compositebox beams with different degree of shear connections undersuddenly imposed distributed loads The dynamic responsesof beams with and without shear lag effects are analyzedThe distributed load is taken as 119902 = 300 kNm with a timestep of 120591 = 000002 s and a space step of 120592 = 03m Themechanical and geometrical parameters of composite boxbeams are taken as 119864

119904= 20 times 10

5MPa 119864119888= 45 times 10

4MPa119866119904= 771 times 10

4MPa 120583119904= 028 120583

119888= 018 120588

119904= 7900 kgsdotmminus3

120588119888= 2400 kgsdotmminus3 119887

1= 1198873= 25m 119887

2= 20m 119887

4= 30m

1198875= 02m 119905

1= 1199052= 03m 119905

3= 1199055= 006m 119905

4= 009m

119871 = 30m 119897 = 03m 119899119904= 2 and 119891

119904= 300MPa

The commercial finite element software ANSYS is used inthis study In the finite element model the concrete slab andsteel beam are modeled by SOLID65 and SHELL43 elementsrespectively Shear connector is modeled by COMBIN14elements being spring elements [25] The aspect ratio of themesh is kept close to one throughout the mean mesh sizevaries from 02m to 03m and about 9200 finite elementsin total are employed The transverse distributed loads areimposed on the top flanges The translational 119909 and 119910

degrees of freedom of all nodes at two ends of beams arerestrained The torsion at the beam ends is thus restrainedThe translational 119911 DOF of the left end is restrained to meetthe static determined requirement and allow rotation alongthe moment The results are shown in Figures 3 and 4

The 119896sl is calculated as [5 23 24]

119896sl =1198701

119897 (43)

6 Mathematical Problems in Engineering

000 005 010 015 020 0250

4

8

12

16

20

24

Time (s)

Disp

lace

men

t at m

idsp

an (m

m)

(a) 119903 = 025

000 005 010 015 020 0250

4

8

12

16

20

24

Disp

lace

men

t at m

idsp

an (m

m)

Time (s)

(b) 119903 = 05

000 005 010 015 020 0250

4

8

12

16

20

24

Time (s)

Disp

lace

men

t at m

idsp

an (m

m)

ANSYSThis paper 120582 = 1

This paper 120582 = 0

(c) 119903 = 10

000 005 010 015 020 0250

4

8

12

16

20

24

Time (s)

Disp

lace

men

t at m

idsp

an (m

m)

ANSYSThis paper 120582 = 1

This paper 120582 = 0

(d) 119903 = 15

Figure 3 Displacement time history at midspan of composite beams (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion ofshear lag)

The spring parameter of the COMBIN14 elements can becalculated as [25]

1198701= 066119899

119904119881119906 (44)

The degree of shear connections is calculated as [5]

119903 =119899119904119881119906119871

119860119904119891119904119897 (45)

where 119897 is the spacing of shear connectors 119903 is the degree ofshear connections 119891

119904is the tensile strength of steel beams 119899

119904

is the number of connectors across the beam section119881119906is the

ultimate shear strength of a single shear connectorFrom Figures 3 and 4 it can be seen that the results from

the proposed method agree well with those from ANSYSfor different degrees of shear connections This verifies theaccuracy of the proposed method and some meaningfulconclusions for engineering design can be drawn as follows

(1) For various degrees of shear connections the vibra-tion period of composite beams with and withoutshear lag effects is almost the same which indicatesthat the shear lag and shear connection degree havelittle impact on the vibration period

(2) The larger the degree of shear connections thesmaller the amplitude of deflection

(3) The influence of the shear lag on the amplitude ofdeflection is not significant indicating that it has littleeffect on the deflection of beams

(4) The longitudinal strain in the concrete slab and steelbeam bottom plate shows obvious shear lag effects

(5) The longitudinal strain in the concrete slab reducesobviously with an increment in the degree of shear

Mathematical Problems in Engineering 7

0 1 2 3 4 5100

120

140

160

180

200

minus5 minus4 minus3 minus2 minus1

ANSYS r = 025

This paper 120582 = 1 r = 025

This paper 120582 = 0 r = 025ANSYS r = 05

This paper 120582 = 1 r = 05

This paper 120582 = 0 r = 05ANSYS r = 1

This paper 120582 = 1 r = 1

This paper 120582 = 0 r = 1

Y-coordinate (m)

Stra

in d

istrib

utio

n (10minus6)

(a) Concrete plate

0 1 2 3minus3 minus2 minus1

ANSYS r = 025

This paper 120582 = 1 r = 025

This paper 120582 = 0 r = 025ANSYS r = 05

This paper 120582 = 1 r = 05

This paper 120582 = 0 r = 05ANSYS r = 1

This paper 120582 = 1 r = 1

This paper 120582 = 0 r = 1

Y-coordinate (m)

minus220

minus230

minus240

minus250

minus260

minus270

minus280

Stra

in d

istrib

utio

n (10minus6)

(b) Steel beam bottom plate

Figure 4 Strain distribution in the flanges of midspan section (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion of shear lag)

connections which indicates that the shear connec-tion degree has a great effect on the longitudinal strainof concrete slab Nevertheless the longitudinal strainin the steel beam bottom plate changes little with anincrement in the degree of shear connection

6 Conclusions

(1) Numerical analyses are carried out to verify theaccuracy of the proposed theory The comprehensiveconsiderations of shear lag shear deformation andinterface slip yield an accurate prediction of dynamicresponses of composite box beams

(2) The degree of shear connections has little impacton the vibration period of composite beams but hassignificant impact on the amplitude of deflection ofcomposite beams Nevertheless the shear lag effecthas limited contribution to the deflection or thevibration period of composite beams

(3) The longitudinal strain in the concrete slabs andbottom plate of steel beams shows an obvious shearlag effect

(4) The longitudinal strain in the concrete slabs reducesgreatly with increasing shear connection degree Nev-ertheless the longitudinal strain in the steel beambottom plate changes little with an increment in thedegree of shear connection

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Function of China(51408449 51378502) and the Fundamental Research Fundsfor the Central Universities of China (2014-IV-049)

References

[1] J Nie and C S Cai ldquoSteel-concrete composite beams consider-ing shear slip effectsrdquo Journal of Structural Engineering vol 129no 4 pp 495ndash506 2003

[2] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005

[3] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007

[4] Z Wangbao J Lizhong K Juntao and B Minxi ldquoDistortionalbuckling analysis of steel-concrete composite girders in negativemoment areardquoMathematical Problems in Engineering vol 2014Article ID 635617 10 pages 2014

[5] W-B Zhou L-Z Jiang Z-J Liu and X-J Liu ldquoClosed-form solution for shear lag effects of steel-concrete compositebox beams considering shear deformation and sliprdquo Journal ofCentral South University vol 19 no 10 pp 2976ndash2982 2012

[6] F Gara G Ranzi and G Leoni ldquoSimplified method of analysisaccounting for shear-lag effects in composite bridge decksrdquoJournal of Constructional Steel Research vol 67 no 10 pp 1684ndash1697 2011

[7] F Gara G Leoni and L Dezi ldquoA beam finite element includingshear lag effect for the time-dependent analysis of steel-concrete

8 Mathematical Problems in Engineering

composite decksrdquoEngineering Structures vol 31 no 8 pp 1888ndash1902 2009

[8] A Morassi and L Rocchetto ldquoA damage analysis of steel-concrete composite beams via dynamic methods part I Exper-imental resultsrdquo Journal of Vibration and Control vol 9 no 5pp 507ndash527 2003

[9] M Dilena and A Morassi ldquoA damage analysis of steel-concretecomposite beams via dynamic methods part II Analyticalmodels anddamage detectionrdquo Journal of Vibration andControlvol 9 no 5 pp 529ndash565 2003

[10] M Dilena and A Morassi ldquoVibrations of steelmdashconcretecomposite beamswith partially degraded connection and appli-cations to damage detectionrdquo Journal of Sound and Vibrationvol 320 no 1-2 pp 101ndash124 2009

[11] C Adam R Heuer and A Jeschko ldquoFlexural vibrations ofelastic composite beams with interlayer sliprdquo Acta Mechanicavol 125 no 1ndash4 pp 17ndash30 1997

[12] G Biscontin A Morassi and P Wendel ldquoVibrations of steel-concrete composite beamsrdquo Journal of Vibration and Controlvol 6 no 5 pp 691ndash714 2000

[13] S Berczynski and T Wroblewski ldquoVibration of steel-concretecomposite beams using the Timoshenko beam modelrdquo Journalof Vibration and Control vol 11 no 6 pp 829ndash848 2005

[14] S Berczynski and T Wroblewski ldquoExperimental verification ofnatural vibration models of steel-concrete composite beamsrdquoJournal of Vibration and Control vol 16 no 14 pp 2057ndash20812010

[15] R Xu and Y Wu ldquoStatic dynamic and buckling analysisof partial interaction composite members using Timoshenkorsquosbeam theoryrdquo International Journal of Mechanical Sciences vol49 no 10 pp 1139ndash1155 2007

[16] X Shen W Chen Y Wu and R Xu ldquoDynamic analysis ofpartial-interaction composite beamsrdquo Composites Science andTechnology vol 71 no 10 pp 1286ndash1294 2011

[17] Z Shen and H Zhong ldquoStatic and vibrational analysis ofpartially composite beams using the weak-form quadratureelement methodrdquo Mathematical Problems in Engineering vol2012 Article ID 974023 23 pages 2012

[18] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013

[19] J G S da Silva S A L de Andrade and E D C LopesldquoParametric modelling of the dynamic behaviour of a steel-concrete composite floorrdquo Engineering Structures vol 75 pp327ndash339 2014

[20] S Lenci and J Warminski ldquoFree and forced nonlinear oscil-lations of a two-layer composite beam with interface sliprdquoNonlinear Dynamics vol 70 no 3 pp 2071ndash2087 2012

[21] Q-H Nguyen M Hjiaj and P Le Grognec ldquoAnalyticalapproach for free vibration analysis of two-layer Timoshenkobeams with interlayer sliprdquo Journal of Sound and Vibration vol331 no 12 pp 2949ndash2961 2012

[22] W-A Wang Q Li C-H Zhao and W-L Zhuang ldquoDynamicproperties of long-span steel-concrete composite bridges withexternal tendonsrdquo Journal of Highway and TransportationResearch andDevelopment (English Edition) vol 7 no 4 pp 30ndash38 2013

[23] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013

[24] W-B Zhou L-Z Jiang Z-W Yu and Z Huang ldquoFree vibra-tion characteristics of steel-concrete composite continuous boxgirder considering shear lag and sliprdquo China Journal of Highwayand Transport vol 26 no 5 pp 88ndash94 2013

[25] J Nie J Fan and C S Cai ldquoStiffness and deflection of steel-concrete composite beams under negative bendingrdquo Journal ofStructural Engineering vol 130 no 11 pp 1842ndash1851 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

The work done by the external loads can be expressed as

119882 = int

119871

119902 (119909 119905) 119908 119889119909 (16)

where 119902(119909 119905) is the distribution function of arbitrary load

33 VibrationDifferential Equation and Boundary ConditionsThegoverning equations of vibration of composite box beamsand corresponding boundary conditions can be deducedbased on Hamilton principle as

11986511988010158401015840+ 11986712058510158401015840minus 119869119880 minus 119878120579

10158401015840minus 1198651 minus 119867

1

120585 + 1198781

120579 = 0 (17)

11986711988010158401015840+ 11986312058510158401015840minus 119896sl120577 minus 119867

1 minus 119863

1

120585 = 0 (18)

1198661199041198600(11990810158401015840minus 1205791015840)

120572119904

minus 119898 + 119902 (119909 119905) = 0 (19)

1198661199041198600(1199081015840minus 120579)

120572119904

+ 11986812057910158401015840minus 11987811988010158401015840minus 119896sl120577ℎ + 119878

1 minus 1198681

120579 = 0 (20)

(1198651198801015840+ 1198671205851015840minus 1198781205791015840) 120575119880

10038161003816100381610038161003816

119871

0= 0 (21)

(1198631205851015840+ 119867119880

1015840) 120575120585

10038161003816100381610038161003816

119871

0= 0 (22)

1198661199041198600(1199081015840minus 120579)

120572119904

120575119908

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871

0

= 0 (23)

(1198681205791015840minus 1198781198801015840) 120575120579

10038161003816100381610038161003816

119871

0= 0 (24)

Taking a simply supported beam as example (21)ndash(24)give boundary conditions as

1198801015840(119871 119905) = 120585

1015840(119871 119905) = 120579

1015840(119871 119905) = 119908 (119871 119905) = 0

1198801015840(0 119905) = 120585

1015840(0 119905) = 120579

1015840(0 119905) = 119908 (0 119905) = 0

(25)

given the initial conditions as

119880 (119909 0) = 1198800(119909) (119909 0) = 119880

0(119909)

120585 (119909 0) = 1205850(119909)

120585 (119909 0) = 120585

0(119909)

119908 (119909 0) = 1199080(119909) (119909 0) = 119908

0(119909)

120579 (119909 0) = 1205790(119909) 120579 (119909 0) = 120579

0(119909)

(26)

4 Finite Difference Method ofVibration Differential Equation

41 Difference Scheme Let solution domain be 120590 = (119909 119905) |

0 le 119909 le 119871 0 le 119905 le 119879 119879 is the end time of solution arectangularmesh ismade in the solution area with a time stepof 120591 and space step of 120592 so that

119909119894= 119894120592 (119894 = 0 1 2 119868)

119905119895= 119895120591 (119895 = 0 1 2 119869)

(27)

where 119868 = 119871120592 119869 = 119879120591

Let

119880119895

119894= 119880 (119909

119894 119905119895) 120585

119895

119894= 120585 (119909

119894 119905119895)

119908119895

119894= 119908 (119909

119894 119905119895) 120579

119895

119894= 120579 (119909

119894 119905119895)

(28)

The central difference calculation of governing differen-tial equation yields

119865

119880119895

119894+1minus 2119880119895

119894+ 119880119895

119894minus1

1205922

+ 119867

120585119895

119894+1minus 2120585119895

119894+ 120585119895

119894minus1

1205922

minus 119878

120579119895

119894+1minus 2120579119895

119894+ 120579119895

119894minus1

1205922

minus 1198651

119880119895+1

119894minus 2119880119895

119894+ 119880119895minus1

119894

1205912

minus 119869119880119895

119894minus 1198671

120585119895+1

119894minus 2120585119895

119894+ 120585119895minus1

119894

1205912

+ 1198781

120579119895+1

119894minus 2120579119895

119894+ 120579119895minus1

119894

1205912

= 0

(119894 = 1 119868 minus 1 119895 = 1 2 119869)

(29)

119867

119880119895

119894+1minus 2119880119895

119894+ 119880119895

119894minus1

1205922

+ 119863

120585119895

119894+1minus 2120585119895

119894+ 120585119895

119894minus1

1205922

minus 119896sl (120585119895

119894+ ℎ120579119895

119894) minus 119867

1

119880119895+1

119894minus 2119880119895

119894+ 119880119895minus1

119894

1205912

minus 1198631

120585119895+1

119894minus 2120585119895

119894+ 120585119895minus1

119894

1205912

= 0

(119894 = 1 119868 minus 1 119895 = 1 2 119869)

(30)

1198661199041198600

120572119904

(

119908119895

119894+1minus 119908119895

119894minus1

2120592minus 120579119895

119894) + 119868

120579119895

119894+1minus 2120579119895

119894+ 120579119895

119894minus1

1205922

minus 119878

119880119895

119894+1minus 2119880119895

119894+ 119880119895

119894minus1

1205922

+ 1198781

119880119895+1

119894minus 2119880119895

119894+ 119880119895minus1

119894

1205912

minus 1198681

120579119895+1

119894minus 2120579119895

119894+ 120579119895minus1

119894

1205912

minus 119896slℎ (120585119895

119894+ ℎ120579119895

119894) = 0

(119894 = 1 119868 minus 1 119895 = 1 2 119869)

(31)

1198661199041198600

120572119904

(

119908119895

119894+1minus 2119908119895

119894+ 119908119895

119894minus1

1205922

minus

120579119895

119894+1minus 120579119895

119894minus1

2120592)

+ 119902119895

119894minus 119898

119908119895+1

119894minus 2119908119895

119894+ 119908119895minus1

119894

1205912

= 0

(119894 = 1 119868 minus 1 119895 = 1 2 119869)

(32)

Mathematical Problems in Engineering 5

The difference calculation of boundary conditions yields

119880119895

119868minus 119880119895

119868minus1

120592=

119880119895

1minus 119880119895

0

120592= 0 119895 = 0 1 119869

(33)

120585119895

119868minus 120585119895

119868minus1

120592=

120585119895

1minus 120585119895

0

120592= 0 119895 = 0 1 119869

(34)

119908119895

119868= 119908119895

0= 0 119895 = 0 1 119869 (35)

120579119895

119868minus 120579119895

119868minus1

120592=

120579119895

1minus 120579119895

0

120592= 0 119895 = 0 1 119869

(36)

The difference calculation of initial condition yields

1198800

119894= 1198800119894

1198801

119894minus 1198800

119894

120591= 1198800119894

119894 = 0 1 2 119868

1205850

119894= 1205850119894

1205851

119894minus 1205850

119894

120591= 1205850119894

119894 = 0 1 2 119868

1199080

119894= 1199080119894

1199081

119894minus 1199080

119894

120591= 1199080119894

119894 = 0 1 2 119868

1205790

119894= 1205790119894

1205791

119894minus 1205790

119894

120591= 1205790119894

119894 = 0 1 2 119868

(37)

42 Solution Step Let U119895 = 119880119895

0 119880119895

1 119880

119895

119868119879 120585119895 = 120585

119895

0 120585119895

1

120585119895

119868119879 w119895 = 119908

119895

0 119908119895

1 119908

119895

119868119879 and 120579119895 = 120579

119895

0 120579119895

1 120579

119895

119868119879

Given U119895minus1 120585119895minus1 w119895minus1 120579119895minus1 U119895 120585119895 w119895 and 120579119895 the solving ofU119895+1 120585119895+1 w119895+1 and 120579119895+1 follows the steps below

(a) Calculate U0 1205850 w0 1205790 U1 1205851 w1 and 1205791 from theinitial conditions given in (37)

(b) Given U119895minus1 120585119895minus1 120579119895minus1 U119895 120585119895 w119895 and 120579119895 calcu-late 119880

119895+1

1 119880119895+1

2 119880

119895+1

119868minus1 120585119895+1

1 120585119895+1

2 120585

119895+1

119868minus1 and

120579119895+1

1 120579119895+1

2 120579

119895+1

119868minus1 from (29)ndash(31)

(c) Calculate 119880119895+1

0 119880119895+1

119868 120585119895+1

0 120585119895+1

119868 and 120579

119895+1

0 120579119895+1

119868

from (33) (34) and (36) and the solution of U119895+1120585119895+1 and 120579119895+1 is obtained in combination withstep (b)

(d) Given w119895minus1 w119895 and 120579119895 determine 119908119895+1

1 119908119895+1

2

119908119895+1

119868minus1 from (32)

(e) Calculate 119908119895+1

0 119908119895+1

119868 from (35) and calculate w119895+1

combined with step (d)

43 Degeneration of theVibrationDifferential Equation Like-wise the governing equations of vibration of composite box

beams and boundary conditions without shear lag effects canbe deduced as

11986312058510158401015840minus 119896sl120577 minus 119863

1

120585 = 0 (38)

1198661199041198600(11990810158401015840minus 1205791015840)

120572119904

minus 119898 + 119902 (119909 119905) = 0 (39)

1198661199041198600(1199081015840minus 120579)

120572119904

+ 11986812057910158401015840minus 119896sl120577ℎ minus 119868

1120579 = 0 (40)

1205851015840120575120585

10038161003816100381610038161003816

119871

0= 0 120579

1015840120575120579

10038161003816100381610038161003816

119871

0= 0 (119908

1015840minus 120579) 120575119908

10038161003816100381610038161003816

119871

0= 0 (41)

For beamswith two ends simply supported the boundaryconditions can be expressed from (41) as

1205851015840(119871 119905) = 120579

1015840(119871 119905) = 119908 (119871 119905) = 0

1205851015840(0 119905) = 120579

1015840(0 119905) = 119908 (0 119905) = 0

(42)

The vibration differential equation of composite boxbeams without considering shear lag effects is a degenera-tion of the one considering the shear lag effects the solutionmethod of which can be referred to in Sections 41 and 42

5 Analysis of Examples

The validity of the proposed method is verified by compar-ing to numerical results from finite element method Thecomparisons are made on four simply supported compositebox beams with different degree of shear connections undersuddenly imposed distributed loads The dynamic responsesof beams with and without shear lag effects are analyzedThe distributed load is taken as 119902 = 300 kNm with a timestep of 120591 = 000002 s and a space step of 120592 = 03m Themechanical and geometrical parameters of composite boxbeams are taken as 119864

119904= 20 times 10

5MPa 119864119888= 45 times 10

4MPa119866119904= 771 times 10

4MPa 120583119904= 028 120583

119888= 018 120588

119904= 7900 kgsdotmminus3

120588119888= 2400 kgsdotmminus3 119887

1= 1198873= 25m 119887

2= 20m 119887

4= 30m

1198875= 02m 119905

1= 1199052= 03m 119905

3= 1199055= 006m 119905

4= 009m

119871 = 30m 119897 = 03m 119899119904= 2 and 119891

119904= 300MPa

The commercial finite element software ANSYS is used inthis study In the finite element model the concrete slab andsteel beam are modeled by SOLID65 and SHELL43 elementsrespectively Shear connector is modeled by COMBIN14elements being spring elements [25] The aspect ratio of themesh is kept close to one throughout the mean mesh sizevaries from 02m to 03m and about 9200 finite elementsin total are employed The transverse distributed loads areimposed on the top flanges The translational 119909 and 119910

degrees of freedom of all nodes at two ends of beams arerestrained The torsion at the beam ends is thus restrainedThe translational 119911 DOF of the left end is restrained to meetthe static determined requirement and allow rotation alongthe moment The results are shown in Figures 3 and 4

The 119896sl is calculated as [5 23 24]

119896sl =1198701

119897 (43)

6 Mathematical Problems in Engineering

000 005 010 015 020 0250

4

8

12

16

20

24

Time (s)

Disp

lace

men

t at m

idsp

an (m

m)

(a) 119903 = 025

000 005 010 015 020 0250

4

8

12

16

20

24

Disp

lace

men

t at m

idsp

an (m

m)

Time (s)

(b) 119903 = 05

000 005 010 015 020 0250

4

8

12

16

20

24

Time (s)

Disp

lace

men

t at m

idsp

an (m

m)

ANSYSThis paper 120582 = 1

This paper 120582 = 0

(c) 119903 = 10

000 005 010 015 020 0250

4

8

12

16

20

24

Time (s)

Disp

lace

men

t at m

idsp

an (m

m)

ANSYSThis paper 120582 = 1

This paper 120582 = 0

(d) 119903 = 15

Figure 3 Displacement time history at midspan of composite beams (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion ofshear lag)

The spring parameter of the COMBIN14 elements can becalculated as [25]

1198701= 066119899

119904119881119906 (44)

The degree of shear connections is calculated as [5]

119903 =119899119904119881119906119871

119860119904119891119904119897 (45)

where 119897 is the spacing of shear connectors 119903 is the degree ofshear connections 119891

119904is the tensile strength of steel beams 119899

119904

is the number of connectors across the beam section119881119906is the

ultimate shear strength of a single shear connectorFrom Figures 3 and 4 it can be seen that the results from

the proposed method agree well with those from ANSYSfor different degrees of shear connections This verifies theaccuracy of the proposed method and some meaningfulconclusions for engineering design can be drawn as follows

(1) For various degrees of shear connections the vibra-tion period of composite beams with and withoutshear lag effects is almost the same which indicatesthat the shear lag and shear connection degree havelittle impact on the vibration period

(2) The larger the degree of shear connections thesmaller the amplitude of deflection

(3) The influence of the shear lag on the amplitude ofdeflection is not significant indicating that it has littleeffect on the deflection of beams

(4) The longitudinal strain in the concrete slab and steelbeam bottom plate shows obvious shear lag effects

(5) The longitudinal strain in the concrete slab reducesobviously with an increment in the degree of shear

Mathematical Problems in Engineering 7

0 1 2 3 4 5100

120

140

160

180

200

minus5 minus4 minus3 minus2 minus1

ANSYS r = 025

This paper 120582 = 1 r = 025

This paper 120582 = 0 r = 025ANSYS r = 05

This paper 120582 = 1 r = 05

This paper 120582 = 0 r = 05ANSYS r = 1

This paper 120582 = 1 r = 1

This paper 120582 = 0 r = 1

Y-coordinate (m)

Stra

in d

istrib

utio

n (10minus6)

(a) Concrete plate

0 1 2 3minus3 minus2 minus1

ANSYS r = 025

This paper 120582 = 1 r = 025

This paper 120582 = 0 r = 025ANSYS r = 05

This paper 120582 = 1 r = 05

This paper 120582 = 0 r = 05ANSYS r = 1

This paper 120582 = 1 r = 1

This paper 120582 = 0 r = 1

Y-coordinate (m)

minus220

minus230

minus240

minus250

minus260

minus270

minus280

Stra

in d

istrib

utio

n (10minus6)

(b) Steel beam bottom plate

Figure 4 Strain distribution in the flanges of midspan section (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion of shear lag)

connections which indicates that the shear connec-tion degree has a great effect on the longitudinal strainof concrete slab Nevertheless the longitudinal strainin the steel beam bottom plate changes little with anincrement in the degree of shear connection

6 Conclusions

(1) Numerical analyses are carried out to verify theaccuracy of the proposed theory The comprehensiveconsiderations of shear lag shear deformation andinterface slip yield an accurate prediction of dynamicresponses of composite box beams

(2) The degree of shear connections has little impacton the vibration period of composite beams but hassignificant impact on the amplitude of deflection ofcomposite beams Nevertheless the shear lag effecthas limited contribution to the deflection or thevibration period of composite beams

(3) The longitudinal strain in the concrete slabs andbottom plate of steel beams shows an obvious shearlag effect

(4) The longitudinal strain in the concrete slabs reducesgreatly with increasing shear connection degree Nev-ertheless the longitudinal strain in the steel beambottom plate changes little with an increment in thedegree of shear connection

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Function of China(51408449 51378502) and the Fundamental Research Fundsfor the Central Universities of China (2014-IV-049)

References

[1] J Nie and C S Cai ldquoSteel-concrete composite beams consider-ing shear slip effectsrdquo Journal of Structural Engineering vol 129no 4 pp 495ndash506 2003

[2] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005

[3] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007

[4] Z Wangbao J Lizhong K Juntao and B Minxi ldquoDistortionalbuckling analysis of steel-concrete composite girders in negativemoment areardquoMathematical Problems in Engineering vol 2014Article ID 635617 10 pages 2014

[5] W-B Zhou L-Z Jiang Z-J Liu and X-J Liu ldquoClosed-form solution for shear lag effects of steel-concrete compositebox beams considering shear deformation and sliprdquo Journal ofCentral South University vol 19 no 10 pp 2976ndash2982 2012

[6] F Gara G Ranzi and G Leoni ldquoSimplified method of analysisaccounting for shear-lag effects in composite bridge decksrdquoJournal of Constructional Steel Research vol 67 no 10 pp 1684ndash1697 2011

[7] F Gara G Leoni and L Dezi ldquoA beam finite element includingshear lag effect for the time-dependent analysis of steel-concrete

8 Mathematical Problems in Engineering

composite decksrdquoEngineering Structures vol 31 no 8 pp 1888ndash1902 2009

[8] A Morassi and L Rocchetto ldquoA damage analysis of steel-concrete composite beams via dynamic methods part I Exper-imental resultsrdquo Journal of Vibration and Control vol 9 no 5pp 507ndash527 2003

[9] M Dilena and A Morassi ldquoA damage analysis of steel-concretecomposite beams via dynamic methods part II Analyticalmodels anddamage detectionrdquo Journal of Vibration andControlvol 9 no 5 pp 529ndash565 2003

[10] M Dilena and A Morassi ldquoVibrations of steelmdashconcretecomposite beamswith partially degraded connection and appli-cations to damage detectionrdquo Journal of Sound and Vibrationvol 320 no 1-2 pp 101ndash124 2009

[11] C Adam R Heuer and A Jeschko ldquoFlexural vibrations ofelastic composite beams with interlayer sliprdquo Acta Mechanicavol 125 no 1ndash4 pp 17ndash30 1997

[12] G Biscontin A Morassi and P Wendel ldquoVibrations of steel-concrete composite beamsrdquo Journal of Vibration and Controlvol 6 no 5 pp 691ndash714 2000

[13] S Berczynski and T Wroblewski ldquoVibration of steel-concretecomposite beams using the Timoshenko beam modelrdquo Journalof Vibration and Control vol 11 no 6 pp 829ndash848 2005

[14] S Berczynski and T Wroblewski ldquoExperimental verification ofnatural vibration models of steel-concrete composite beamsrdquoJournal of Vibration and Control vol 16 no 14 pp 2057ndash20812010

[15] R Xu and Y Wu ldquoStatic dynamic and buckling analysisof partial interaction composite members using Timoshenkorsquosbeam theoryrdquo International Journal of Mechanical Sciences vol49 no 10 pp 1139ndash1155 2007

[16] X Shen W Chen Y Wu and R Xu ldquoDynamic analysis ofpartial-interaction composite beamsrdquo Composites Science andTechnology vol 71 no 10 pp 1286ndash1294 2011

[17] Z Shen and H Zhong ldquoStatic and vibrational analysis ofpartially composite beams using the weak-form quadratureelement methodrdquo Mathematical Problems in Engineering vol2012 Article ID 974023 23 pages 2012

[18] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013

[19] J G S da Silva S A L de Andrade and E D C LopesldquoParametric modelling of the dynamic behaviour of a steel-concrete composite floorrdquo Engineering Structures vol 75 pp327ndash339 2014

[20] S Lenci and J Warminski ldquoFree and forced nonlinear oscil-lations of a two-layer composite beam with interface sliprdquoNonlinear Dynamics vol 70 no 3 pp 2071ndash2087 2012

[21] Q-H Nguyen M Hjiaj and P Le Grognec ldquoAnalyticalapproach for free vibration analysis of two-layer Timoshenkobeams with interlayer sliprdquo Journal of Sound and Vibration vol331 no 12 pp 2949ndash2961 2012

[22] W-A Wang Q Li C-H Zhao and W-L Zhuang ldquoDynamicproperties of long-span steel-concrete composite bridges withexternal tendonsrdquo Journal of Highway and TransportationResearch andDevelopment (English Edition) vol 7 no 4 pp 30ndash38 2013

[23] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013

[24] W-B Zhou L-Z Jiang Z-W Yu and Z Huang ldquoFree vibra-tion characteristics of steel-concrete composite continuous boxgirder considering shear lag and sliprdquo China Journal of Highwayand Transport vol 26 no 5 pp 88ndash94 2013

[25] J Nie J Fan and C S Cai ldquoStiffness and deflection of steel-concrete composite beams under negative bendingrdquo Journal ofStructural Engineering vol 130 no 11 pp 1842ndash1851 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

The difference calculation of boundary conditions yields

119880119895

119868minus 119880119895

119868minus1

120592=

119880119895

1minus 119880119895

0

120592= 0 119895 = 0 1 119869

(33)

120585119895

119868minus 120585119895

119868minus1

120592=

120585119895

1minus 120585119895

0

120592= 0 119895 = 0 1 119869

(34)

119908119895

119868= 119908119895

0= 0 119895 = 0 1 119869 (35)

120579119895

119868minus 120579119895

119868minus1

120592=

120579119895

1minus 120579119895

0

120592= 0 119895 = 0 1 119869

(36)

The difference calculation of initial condition yields

1198800

119894= 1198800119894

1198801

119894minus 1198800

119894

120591= 1198800119894

119894 = 0 1 2 119868

1205850

119894= 1205850119894

1205851

119894minus 1205850

119894

120591= 1205850119894

119894 = 0 1 2 119868

1199080

119894= 1199080119894

1199081

119894minus 1199080

119894

120591= 1199080119894

119894 = 0 1 2 119868

1205790

119894= 1205790119894

1205791

119894minus 1205790

119894

120591= 1205790119894

119894 = 0 1 2 119868

(37)

42 Solution Step Let U119895 = 119880119895

0 119880119895

1 119880

119895

119868119879 120585119895 = 120585

119895

0 120585119895

1

120585119895

119868119879 w119895 = 119908

119895

0 119908119895

1 119908

119895

119868119879 and 120579119895 = 120579

119895

0 120579119895

1 120579

119895

119868119879

Given U119895minus1 120585119895minus1 w119895minus1 120579119895minus1 U119895 120585119895 w119895 and 120579119895 the solving ofU119895+1 120585119895+1 w119895+1 and 120579119895+1 follows the steps below

(a) Calculate U0 1205850 w0 1205790 U1 1205851 w1 and 1205791 from theinitial conditions given in (37)

(b) Given U119895minus1 120585119895minus1 120579119895minus1 U119895 120585119895 w119895 and 120579119895 calcu-late 119880

119895+1

1 119880119895+1

2 119880

119895+1

119868minus1 120585119895+1

1 120585119895+1

2 120585

119895+1

119868minus1 and

120579119895+1

1 120579119895+1

2 120579

119895+1

119868minus1 from (29)ndash(31)

(c) Calculate 119880119895+1

0 119880119895+1

119868 120585119895+1

0 120585119895+1

119868 and 120579

119895+1

0 120579119895+1

119868

from (33) (34) and (36) and the solution of U119895+1120585119895+1 and 120579119895+1 is obtained in combination withstep (b)

(d) Given w119895minus1 w119895 and 120579119895 determine 119908119895+1

1 119908119895+1

2

119908119895+1

119868minus1 from (32)

(e) Calculate 119908119895+1

0 119908119895+1

119868 from (35) and calculate w119895+1

combined with step (d)

43 Degeneration of theVibrationDifferential Equation Like-wise the governing equations of vibration of composite box

beams and boundary conditions without shear lag effects canbe deduced as

11986312058510158401015840minus 119896sl120577 minus 119863

1

120585 = 0 (38)

1198661199041198600(11990810158401015840minus 1205791015840)

120572119904

minus 119898 + 119902 (119909 119905) = 0 (39)

1198661199041198600(1199081015840minus 120579)

120572119904

+ 11986812057910158401015840minus 119896sl120577ℎ minus 119868

1120579 = 0 (40)

1205851015840120575120585

10038161003816100381610038161003816

119871

0= 0 120579

1015840120575120579

10038161003816100381610038161003816

119871

0= 0 (119908

1015840minus 120579) 120575119908

10038161003816100381610038161003816

119871

0= 0 (41)

For beamswith two ends simply supported the boundaryconditions can be expressed from (41) as

1205851015840(119871 119905) = 120579

1015840(119871 119905) = 119908 (119871 119905) = 0

1205851015840(0 119905) = 120579

1015840(0 119905) = 119908 (0 119905) = 0

(42)

The vibration differential equation of composite boxbeams without considering shear lag effects is a degenera-tion of the one considering the shear lag effects the solutionmethod of which can be referred to in Sections 41 and 42

5 Analysis of Examples

The validity of the proposed method is verified by compar-ing to numerical results from finite element method Thecomparisons are made on four simply supported compositebox beams with different degree of shear connections undersuddenly imposed distributed loads The dynamic responsesof beams with and without shear lag effects are analyzedThe distributed load is taken as 119902 = 300 kNm with a timestep of 120591 = 000002 s and a space step of 120592 = 03m Themechanical and geometrical parameters of composite boxbeams are taken as 119864

119904= 20 times 10

5MPa 119864119888= 45 times 10

4MPa119866119904= 771 times 10

4MPa 120583119904= 028 120583

119888= 018 120588

119904= 7900 kgsdotmminus3

120588119888= 2400 kgsdotmminus3 119887

1= 1198873= 25m 119887

2= 20m 119887

4= 30m

1198875= 02m 119905

1= 1199052= 03m 119905

3= 1199055= 006m 119905

4= 009m

119871 = 30m 119897 = 03m 119899119904= 2 and 119891

119904= 300MPa

The commercial finite element software ANSYS is used inthis study In the finite element model the concrete slab andsteel beam are modeled by SOLID65 and SHELL43 elementsrespectively Shear connector is modeled by COMBIN14elements being spring elements [25] The aspect ratio of themesh is kept close to one throughout the mean mesh sizevaries from 02m to 03m and about 9200 finite elementsin total are employed The transverse distributed loads areimposed on the top flanges The translational 119909 and 119910

degrees of freedom of all nodes at two ends of beams arerestrained The torsion at the beam ends is thus restrainedThe translational 119911 DOF of the left end is restrained to meetthe static determined requirement and allow rotation alongthe moment The results are shown in Figures 3 and 4

The 119896sl is calculated as [5 23 24]

119896sl =1198701

119897 (43)

6 Mathematical Problems in Engineering

000 005 010 015 020 0250

4

8

12

16

20

24

Time (s)

Disp

lace

men

t at m

idsp

an (m

m)

(a) 119903 = 025

000 005 010 015 020 0250

4

8

12

16

20

24

Disp

lace

men

t at m

idsp

an (m

m)

Time (s)

(b) 119903 = 05

000 005 010 015 020 0250

4

8

12

16

20

24

Time (s)

Disp

lace

men

t at m

idsp

an (m

m)

ANSYSThis paper 120582 = 1

This paper 120582 = 0

(c) 119903 = 10

000 005 010 015 020 0250

4

8

12

16

20

24

Time (s)

Disp

lace

men

t at m

idsp

an (m

m)

ANSYSThis paper 120582 = 1

This paper 120582 = 0

(d) 119903 = 15

Figure 3 Displacement time history at midspan of composite beams (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion ofshear lag)

The spring parameter of the COMBIN14 elements can becalculated as [25]

1198701= 066119899

119904119881119906 (44)

The degree of shear connections is calculated as [5]

119903 =119899119904119881119906119871

119860119904119891119904119897 (45)

where 119897 is the spacing of shear connectors 119903 is the degree ofshear connections 119891

119904is the tensile strength of steel beams 119899

119904

is the number of connectors across the beam section119881119906is the

ultimate shear strength of a single shear connectorFrom Figures 3 and 4 it can be seen that the results from

the proposed method agree well with those from ANSYSfor different degrees of shear connections This verifies theaccuracy of the proposed method and some meaningfulconclusions for engineering design can be drawn as follows

(1) For various degrees of shear connections the vibra-tion period of composite beams with and withoutshear lag effects is almost the same which indicatesthat the shear lag and shear connection degree havelittle impact on the vibration period

(2) The larger the degree of shear connections thesmaller the amplitude of deflection

(3) The influence of the shear lag on the amplitude ofdeflection is not significant indicating that it has littleeffect on the deflection of beams

(4) The longitudinal strain in the concrete slab and steelbeam bottom plate shows obvious shear lag effects

(5) The longitudinal strain in the concrete slab reducesobviously with an increment in the degree of shear

Mathematical Problems in Engineering 7

0 1 2 3 4 5100

120

140

160

180

200

minus5 minus4 minus3 minus2 minus1

ANSYS r = 025

This paper 120582 = 1 r = 025

This paper 120582 = 0 r = 025ANSYS r = 05

This paper 120582 = 1 r = 05

This paper 120582 = 0 r = 05ANSYS r = 1

This paper 120582 = 1 r = 1

This paper 120582 = 0 r = 1

Y-coordinate (m)

Stra

in d

istrib

utio

n (10minus6)

(a) Concrete plate

0 1 2 3minus3 minus2 minus1

ANSYS r = 025

This paper 120582 = 1 r = 025

This paper 120582 = 0 r = 025ANSYS r = 05

This paper 120582 = 1 r = 05

This paper 120582 = 0 r = 05ANSYS r = 1

This paper 120582 = 1 r = 1

This paper 120582 = 0 r = 1

Y-coordinate (m)

minus220

minus230

minus240

minus250

minus260

minus270

minus280

Stra

in d

istrib

utio

n (10minus6)

(b) Steel beam bottom plate

Figure 4 Strain distribution in the flanges of midspan section (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion of shear lag)

connections which indicates that the shear connec-tion degree has a great effect on the longitudinal strainof concrete slab Nevertheless the longitudinal strainin the steel beam bottom plate changes little with anincrement in the degree of shear connection

6 Conclusions

(1) Numerical analyses are carried out to verify theaccuracy of the proposed theory The comprehensiveconsiderations of shear lag shear deformation andinterface slip yield an accurate prediction of dynamicresponses of composite box beams

(2) The degree of shear connections has little impacton the vibration period of composite beams but hassignificant impact on the amplitude of deflection ofcomposite beams Nevertheless the shear lag effecthas limited contribution to the deflection or thevibration period of composite beams

(3) The longitudinal strain in the concrete slabs andbottom plate of steel beams shows an obvious shearlag effect

(4) The longitudinal strain in the concrete slabs reducesgreatly with increasing shear connection degree Nev-ertheless the longitudinal strain in the steel beambottom plate changes little with an increment in thedegree of shear connection

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Function of China(51408449 51378502) and the Fundamental Research Fundsfor the Central Universities of China (2014-IV-049)

References

[1] J Nie and C S Cai ldquoSteel-concrete composite beams consider-ing shear slip effectsrdquo Journal of Structural Engineering vol 129no 4 pp 495ndash506 2003

[2] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005

[3] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007

[4] Z Wangbao J Lizhong K Juntao and B Minxi ldquoDistortionalbuckling analysis of steel-concrete composite girders in negativemoment areardquoMathematical Problems in Engineering vol 2014Article ID 635617 10 pages 2014

[5] W-B Zhou L-Z Jiang Z-J Liu and X-J Liu ldquoClosed-form solution for shear lag effects of steel-concrete compositebox beams considering shear deformation and sliprdquo Journal ofCentral South University vol 19 no 10 pp 2976ndash2982 2012

[6] F Gara G Ranzi and G Leoni ldquoSimplified method of analysisaccounting for shear-lag effects in composite bridge decksrdquoJournal of Constructional Steel Research vol 67 no 10 pp 1684ndash1697 2011

[7] F Gara G Leoni and L Dezi ldquoA beam finite element includingshear lag effect for the time-dependent analysis of steel-concrete

8 Mathematical Problems in Engineering

composite decksrdquoEngineering Structures vol 31 no 8 pp 1888ndash1902 2009

[8] A Morassi and L Rocchetto ldquoA damage analysis of steel-concrete composite beams via dynamic methods part I Exper-imental resultsrdquo Journal of Vibration and Control vol 9 no 5pp 507ndash527 2003

[9] M Dilena and A Morassi ldquoA damage analysis of steel-concretecomposite beams via dynamic methods part II Analyticalmodels anddamage detectionrdquo Journal of Vibration andControlvol 9 no 5 pp 529ndash565 2003

[10] M Dilena and A Morassi ldquoVibrations of steelmdashconcretecomposite beamswith partially degraded connection and appli-cations to damage detectionrdquo Journal of Sound and Vibrationvol 320 no 1-2 pp 101ndash124 2009

[11] C Adam R Heuer and A Jeschko ldquoFlexural vibrations ofelastic composite beams with interlayer sliprdquo Acta Mechanicavol 125 no 1ndash4 pp 17ndash30 1997

[12] G Biscontin A Morassi and P Wendel ldquoVibrations of steel-concrete composite beamsrdquo Journal of Vibration and Controlvol 6 no 5 pp 691ndash714 2000

[13] S Berczynski and T Wroblewski ldquoVibration of steel-concretecomposite beams using the Timoshenko beam modelrdquo Journalof Vibration and Control vol 11 no 6 pp 829ndash848 2005

[14] S Berczynski and T Wroblewski ldquoExperimental verification ofnatural vibration models of steel-concrete composite beamsrdquoJournal of Vibration and Control vol 16 no 14 pp 2057ndash20812010

[15] R Xu and Y Wu ldquoStatic dynamic and buckling analysisof partial interaction composite members using Timoshenkorsquosbeam theoryrdquo International Journal of Mechanical Sciences vol49 no 10 pp 1139ndash1155 2007

[16] X Shen W Chen Y Wu and R Xu ldquoDynamic analysis ofpartial-interaction composite beamsrdquo Composites Science andTechnology vol 71 no 10 pp 1286ndash1294 2011

[17] Z Shen and H Zhong ldquoStatic and vibrational analysis ofpartially composite beams using the weak-form quadratureelement methodrdquo Mathematical Problems in Engineering vol2012 Article ID 974023 23 pages 2012

[18] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013

[19] J G S da Silva S A L de Andrade and E D C LopesldquoParametric modelling of the dynamic behaviour of a steel-concrete composite floorrdquo Engineering Structures vol 75 pp327ndash339 2014

[20] S Lenci and J Warminski ldquoFree and forced nonlinear oscil-lations of a two-layer composite beam with interface sliprdquoNonlinear Dynamics vol 70 no 3 pp 2071ndash2087 2012

[21] Q-H Nguyen M Hjiaj and P Le Grognec ldquoAnalyticalapproach for free vibration analysis of two-layer Timoshenkobeams with interlayer sliprdquo Journal of Sound and Vibration vol331 no 12 pp 2949ndash2961 2012

[22] W-A Wang Q Li C-H Zhao and W-L Zhuang ldquoDynamicproperties of long-span steel-concrete composite bridges withexternal tendonsrdquo Journal of Highway and TransportationResearch andDevelopment (English Edition) vol 7 no 4 pp 30ndash38 2013

[23] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013

[24] W-B Zhou L-Z Jiang Z-W Yu and Z Huang ldquoFree vibra-tion characteristics of steel-concrete composite continuous boxgirder considering shear lag and sliprdquo China Journal of Highwayand Transport vol 26 no 5 pp 88ndash94 2013

[25] J Nie J Fan and C S Cai ldquoStiffness and deflection of steel-concrete composite beams under negative bendingrdquo Journal ofStructural Engineering vol 130 no 11 pp 1842ndash1851 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

000 005 010 015 020 0250

4

8

12

16

20

24

Time (s)

Disp

lace

men

t at m

idsp

an (m

m)

(a) 119903 = 025

000 005 010 015 020 0250

4

8

12

16

20

24

Disp

lace

men

t at m

idsp

an (m

m)

Time (s)

(b) 119903 = 05

000 005 010 015 020 0250

4

8

12

16

20

24

Time (s)

Disp

lace

men

t at m

idsp

an (m

m)

ANSYSThis paper 120582 = 1

This paper 120582 = 0

(c) 119903 = 10

000 005 010 015 020 0250

4

8

12

16

20

24

Time (s)

Disp

lace

men

t at m

idsp

an (m

m)

ANSYSThis paper 120582 = 1

This paper 120582 = 0

(d) 119903 = 15

Figure 3 Displacement time history at midspan of composite beams (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion ofshear lag)

The spring parameter of the COMBIN14 elements can becalculated as [25]

1198701= 066119899

119904119881119906 (44)

The degree of shear connections is calculated as [5]

119903 =119899119904119881119906119871

119860119904119891119904119897 (45)

where 119897 is the spacing of shear connectors 119903 is the degree ofshear connections 119891

119904is the tensile strength of steel beams 119899

119904

is the number of connectors across the beam section119881119906is the

ultimate shear strength of a single shear connectorFrom Figures 3 and 4 it can be seen that the results from

the proposed method agree well with those from ANSYSfor different degrees of shear connections This verifies theaccuracy of the proposed method and some meaningfulconclusions for engineering design can be drawn as follows

(1) For various degrees of shear connections the vibra-tion period of composite beams with and withoutshear lag effects is almost the same which indicatesthat the shear lag and shear connection degree havelittle impact on the vibration period

(2) The larger the degree of shear connections thesmaller the amplitude of deflection

(3) The influence of the shear lag on the amplitude ofdeflection is not significant indicating that it has littleeffect on the deflection of beams

(4) The longitudinal strain in the concrete slab and steelbeam bottom plate shows obvious shear lag effects

(5) The longitudinal strain in the concrete slab reducesobviously with an increment in the degree of shear

Mathematical Problems in Engineering 7

0 1 2 3 4 5100

120

140

160

180

200

minus5 minus4 minus3 minus2 minus1

ANSYS r = 025

This paper 120582 = 1 r = 025

This paper 120582 = 0 r = 025ANSYS r = 05

This paper 120582 = 1 r = 05

This paper 120582 = 0 r = 05ANSYS r = 1

This paper 120582 = 1 r = 1

This paper 120582 = 0 r = 1

Y-coordinate (m)

Stra

in d

istrib

utio

n (10minus6)

(a) Concrete plate

0 1 2 3minus3 minus2 minus1

ANSYS r = 025

This paper 120582 = 1 r = 025

This paper 120582 = 0 r = 025ANSYS r = 05

This paper 120582 = 1 r = 05

This paper 120582 = 0 r = 05ANSYS r = 1

This paper 120582 = 1 r = 1

This paper 120582 = 0 r = 1

Y-coordinate (m)

minus220

minus230

minus240

minus250

minus260

minus270

minus280

Stra

in d

istrib

utio

n (10minus6)

(b) Steel beam bottom plate

Figure 4 Strain distribution in the flanges of midspan section (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion of shear lag)

connections which indicates that the shear connec-tion degree has a great effect on the longitudinal strainof concrete slab Nevertheless the longitudinal strainin the steel beam bottom plate changes little with anincrement in the degree of shear connection

6 Conclusions

(1) Numerical analyses are carried out to verify theaccuracy of the proposed theory The comprehensiveconsiderations of shear lag shear deformation andinterface slip yield an accurate prediction of dynamicresponses of composite box beams

(2) The degree of shear connections has little impacton the vibration period of composite beams but hassignificant impact on the amplitude of deflection ofcomposite beams Nevertheless the shear lag effecthas limited contribution to the deflection or thevibration period of composite beams

(3) The longitudinal strain in the concrete slabs andbottom plate of steel beams shows an obvious shearlag effect

(4) The longitudinal strain in the concrete slabs reducesgreatly with increasing shear connection degree Nev-ertheless the longitudinal strain in the steel beambottom plate changes little with an increment in thedegree of shear connection

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Function of China(51408449 51378502) and the Fundamental Research Fundsfor the Central Universities of China (2014-IV-049)

References

[1] J Nie and C S Cai ldquoSteel-concrete composite beams consider-ing shear slip effectsrdquo Journal of Structural Engineering vol 129no 4 pp 495ndash506 2003

[2] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005

[3] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007

[4] Z Wangbao J Lizhong K Juntao and B Minxi ldquoDistortionalbuckling analysis of steel-concrete composite girders in negativemoment areardquoMathematical Problems in Engineering vol 2014Article ID 635617 10 pages 2014

[5] W-B Zhou L-Z Jiang Z-J Liu and X-J Liu ldquoClosed-form solution for shear lag effects of steel-concrete compositebox beams considering shear deformation and sliprdquo Journal ofCentral South University vol 19 no 10 pp 2976ndash2982 2012

[6] F Gara G Ranzi and G Leoni ldquoSimplified method of analysisaccounting for shear-lag effects in composite bridge decksrdquoJournal of Constructional Steel Research vol 67 no 10 pp 1684ndash1697 2011

[7] F Gara G Leoni and L Dezi ldquoA beam finite element includingshear lag effect for the time-dependent analysis of steel-concrete

8 Mathematical Problems in Engineering

composite decksrdquoEngineering Structures vol 31 no 8 pp 1888ndash1902 2009

[8] A Morassi and L Rocchetto ldquoA damage analysis of steel-concrete composite beams via dynamic methods part I Exper-imental resultsrdquo Journal of Vibration and Control vol 9 no 5pp 507ndash527 2003

[9] M Dilena and A Morassi ldquoA damage analysis of steel-concretecomposite beams via dynamic methods part II Analyticalmodels anddamage detectionrdquo Journal of Vibration andControlvol 9 no 5 pp 529ndash565 2003

[10] M Dilena and A Morassi ldquoVibrations of steelmdashconcretecomposite beamswith partially degraded connection and appli-cations to damage detectionrdquo Journal of Sound and Vibrationvol 320 no 1-2 pp 101ndash124 2009

[11] C Adam R Heuer and A Jeschko ldquoFlexural vibrations ofelastic composite beams with interlayer sliprdquo Acta Mechanicavol 125 no 1ndash4 pp 17ndash30 1997

[12] G Biscontin A Morassi and P Wendel ldquoVibrations of steel-concrete composite beamsrdquo Journal of Vibration and Controlvol 6 no 5 pp 691ndash714 2000

[13] S Berczynski and T Wroblewski ldquoVibration of steel-concretecomposite beams using the Timoshenko beam modelrdquo Journalof Vibration and Control vol 11 no 6 pp 829ndash848 2005

[14] S Berczynski and T Wroblewski ldquoExperimental verification ofnatural vibration models of steel-concrete composite beamsrdquoJournal of Vibration and Control vol 16 no 14 pp 2057ndash20812010

[15] R Xu and Y Wu ldquoStatic dynamic and buckling analysisof partial interaction composite members using Timoshenkorsquosbeam theoryrdquo International Journal of Mechanical Sciences vol49 no 10 pp 1139ndash1155 2007

[16] X Shen W Chen Y Wu and R Xu ldquoDynamic analysis ofpartial-interaction composite beamsrdquo Composites Science andTechnology vol 71 no 10 pp 1286ndash1294 2011

[17] Z Shen and H Zhong ldquoStatic and vibrational analysis ofpartially composite beams using the weak-form quadratureelement methodrdquo Mathematical Problems in Engineering vol2012 Article ID 974023 23 pages 2012

[18] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013

[19] J G S da Silva S A L de Andrade and E D C LopesldquoParametric modelling of the dynamic behaviour of a steel-concrete composite floorrdquo Engineering Structures vol 75 pp327ndash339 2014

[20] S Lenci and J Warminski ldquoFree and forced nonlinear oscil-lations of a two-layer composite beam with interface sliprdquoNonlinear Dynamics vol 70 no 3 pp 2071ndash2087 2012

[21] Q-H Nguyen M Hjiaj and P Le Grognec ldquoAnalyticalapproach for free vibration analysis of two-layer Timoshenkobeams with interlayer sliprdquo Journal of Sound and Vibration vol331 no 12 pp 2949ndash2961 2012

[22] W-A Wang Q Li C-H Zhao and W-L Zhuang ldquoDynamicproperties of long-span steel-concrete composite bridges withexternal tendonsrdquo Journal of Highway and TransportationResearch andDevelopment (English Edition) vol 7 no 4 pp 30ndash38 2013

[23] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013

[24] W-B Zhou L-Z Jiang Z-W Yu and Z Huang ldquoFree vibra-tion characteristics of steel-concrete composite continuous boxgirder considering shear lag and sliprdquo China Journal of Highwayand Transport vol 26 no 5 pp 88ndash94 2013

[25] J Nie J Fan and C S Cai ldquoStiffness and deflection of steel-concrete composite beams under negative bendingrdquo Journal ofStructural Engineering vol 130 no 11 pp 1842ndash1851 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0 1 2 3 4 5100

120

140

160

180

200

minus5 minus4 minus3 minus2 minus1

ANSYS r = 025

This paper 120582 = 1 r = 025

This paper 120582 = 0 r = 025ANSYS r = 05

This paper 120582 = 1 r = 05

This paper 120582 = 0 r = 05ANSYS r = 1

This paper 120582 = 1 r = 1

This paper 120582 = 0 r = 1

Y-coordinate (m)

Stra

in d

istrib

utio

n (10minus6)

(a) Concrete plate

0 1 2 3minus3 minus2 minus1

ANSYS r = 025

This paper 120582 = 1 r = 025

This paper 120582 = 0 r = 025ANSYS r = 05

This paper 120582 = 1 r = 05

This paper 120582 = 0 r = 05ANSYS r = 1

This paper 120582 = 1 r = 1

This paper 120582 = 0 r = 1

Y-coordinate (m)

minus220

minus230

minus240

minus250

minus260

minus270

minus280

Stra

in d

istrib

utio

n (10minus6)

(b) Steel beam bottom plate

Figure 4 Strain distribution in the flanges of midspan section (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion of shear lag)

connections which indicates that the shear connec-tion degree has a great effect on the longitudinal strainof concrete slab Nevertheless the longitudinal strainin the steel beam bottom plate changes little with anincrement in the degree of shear connection

6 Conclusions

(1) Numerical analyses are carried out to verify theaccuracy of the proposed theory The comprehensiveconsiderations of shear lag shear deformation andinterface slip yield an accurate prediction of dynamicresponses of composite box beams

(2) The degree of shear connections has little impacton the vibration period of composite beams but hassignificant impact on the amplitude of deflection ofcomposite beams Nevertheless the shear lag effecthas limited contribution to the deflection or thevibration period of composite beams

(3) The longitudinal strain in the concrete slabs andbottom plate of steel beams shows an obvious shearlag effect

(4) The longitudinal strain in the concrete slabs reducesgreatly with increasing shear connection degree Nev-ertheless the longitudinal strain in the steel beambottom plate changes little with an increment in thedegree of shear connection

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Function of China(51408449 51378502) and the Fundamental Research Fundsfor the Central Universities of China (2014-IV-049)

References

[1] J Nie and C S Cai ldquoSteel-concrete composite beams consider-ing shear slip effectsrdquo Journal of Structural Engineering vol 129no 4 pp 495ndash506 2003

[2] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005

[3] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007

[4] Z Wangbao J Lizhong K Juntao and B Minxi ldquoDistortionalbuckling analysis of steel-concrete composite girders in negativemoment areardquoMathematical Problems in Engineering vol 2014Article ID 635617 10 pages 2014

[5] W-B Zhou L-Z Jiang Z-J Liu and X-J Liu ldquoClosed-form solution for shear lag effects of steel-concrete compositebox beams considering shear deformation and sliprdquo Journal ofCentral South University vol 19 no 10 pp 2976ndash2982 2012

[6] F Gara G Ranzi and G Leoni ldquoSimplified method of analysisaccounting for shear-lag effects in composite bridge decksrdquoJournal of Constructional Steel Research vol 67 no 10 pp 1684ndash1697 2011

[7] F Gara G Leoni and L Dezi ldquoA beam finite element includingshear lag effect for the time-dependent analysis of steel-concrete

8 Mathematical Problems in Engineering

composite decksrdquoEngineering Structures vol 31 no 8 pp 1888ndash1902 2009

[8] A Morassi and L Rocchetto ldquoA damage analysis of steel-concrete composite beams via dynamic methods part I Exper-imental resultsrdquo Journal of Vibration and Control vol 9 no 5pp 507ndash527 2003

[9] M Dilena and A Morassi ldquoA damage analysis of steel-concretecomposite beams via dynamic methods part II Analyticalmodels anddamage detectionrdquo Journal of Vibration andControlvol 9 no 5 pp 529ndash565 2003

[10] M Dilena and A Morassi ldquoVibrations of steelmdashconcretecomposite beamswith partially degraded connection and appli-cations to damage detectionrdquo Journal of Sound and Vibrationvol 320 no 1-2 pp 101ndash124 2009

[11] C Adam R Heuer and A Jeschko ldquoFlexural vibrations ofelastic composite beams with interlayer sliprdquo Acta Mechanicavol 125 no 1ndash4 pp 17ndash30 1997

[12] G Biscontin A Morassi and P Wendel ldquoVibrations of steel-concrete composite beamsrdquo Journal of Vibration and Controlvol 6 no 5 pp 691ndash714 2000

[13] S Berczynski and T Wroblewski ldquoVibration of steel-concretecomposite beams using the Timoshenko beam modelrdquo Journalof Vibration and Control vol 11 no 6 pp 829ndash848 2005

[14] S Berczynski and T Wroblewski ldquoExperimental verification ofnatural vibration models of steel-concrete composite beamsrdquoJournal of Vibration and Control vol 16 no 14 pp 2057ndash20812010

[15] R Xu and Y Wu ldquoStatic dynamic and buckling analysisof partial interaction composite members using Timoshenkorsquosbeam theoryrdquo International Journal of Mechanical Sciences vol49 no 10 pp 1139ndash1155 2007

[16] X Shen W Chen Y Wu and R Xu ldquoDynamic analysis ofpartial-interaction composite beamsrdquo Composites Science andTechnology vol 71 no 10 pp 1286ndash1294 2011

[17] Z Shen and H Zhong ldquoStatic and vibrational analysis ofpartially composite beams using the weak-form quadratureelement methodrdquo Mathematical Problems in Engineering vol2012 Article ID 974023 23 pages 2012

[18] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013

[19] J G S da Silva S A L de Andrade and E D C LopesldquoParametric modelling of the dynamic behaviour of a steel-concrete composite floorrdquo Engineering Structures vol 75 pp327ndash339 2014

[20] S Lenci and J Warminski ldquoFree and forced nonlinear oscil-lations of a two-layer composite beam with interface sliprdquoNonlinear Dynamics vol 70 no 3 pp 2071ndash2087 2012

[21] Q-H Nguyen M Hjiaj and P Le Grognec ldquoAnalyticalapproach for free vibration analysis of two-layer Timoshenkobeams with interlayer sliprdquo Journal of Sound and Vibration vol331 no 12 pp 2949ndash2961 2012

[22] W-A Wang Q Li C-H Zhao and W-L Zhuang ldquoDynamicproperties of long-span steel-concrete composite bridges withexternal tendonsrdquo Journal of Highway and TransportationResearch andDevelopment (English Edition) vol 7 no 4 pp 30ndash38 2013

[23] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013

[24] W-B Zhou L-Z Jiang Z-W Yu and Z Huang ldquoFree vibra-tion characteristics of steel-concrete composite continuous boxgirder considering shear lag and sliprdquo China Journal of Highwayand Transport vol 26 no 5 pp 88ndash94 2013

[25] J Nie J Fan and C S Cai ldquoStiffness and deflection of steel-concrete composite beams under negative bendingrdquo Journal ofStructural Engineering vol 130 no 11 pp 1842ndash1851 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

composite decksrdquoEngineering Structures vol 31 no 8 pp 1888ndash1902 2009

[8] A Morassi and L Rocchetto ldquoA damage analysis of steel-concrete composite beams via dynamic methods part I Exper-imental resultsrdquo Journal of Vibration and Control vol 9 no 5pp 507ndash527 2003

[9] M Dilena and A Morassi ldquoA damage analysis of steel-concretecomposite beams via dynamic methods part II Analyticalmodels anddamage detectionrdquo Journal of Vibration andControlvol 9 no 5 pp 529ndash565 2003

[10] M Dilena and A Morassi ldquoVibrations of steelmdashconcretecomposite beamswith partially degraded connection and appli-cations to damage detectionrdquo Journal of Sound and Vibrationvol 320 no 1-2 pp 101ndash124 2009

[11] C Adam R Heuer and A Jeschko ldquoFlexural vibrations ofelastic composite beams with interlayer sliprdquo Acta Mechanicavol 125 no 1ndash4 pp 17ndash30 1997

[12] G Biscontin A Morassi and P Wendel ldquoVibrations of steel-concrete composite beamsrdquo Journal of Vibration and Controlvol 6 no 5 pp 691ndash714 2000

[13] S Berczynski and T Wroblewski ldquoVibration of steel-concretecomposite beams using the Timoshenko beam modelrdquo Journalof Vibration and Control vol 11 no 6 pp 829ndash848 2005

[14] S Berczynski and T Wroblewski ldquoExperimental verification ofnatural vibration models of steel-concrete composite beamsrdquoJournal of Vibration and Control vol 16 no 14 pp 2057ndash20812010

[15] R Xu and Y Wu ldquoStatic dynamic and buckling analysisof partial interaction composite members using Timoshenkorsquosbeam theoryrdquo International Journal of Mechanical Sciences vol49 no 10 pp 1139ndash1155 2007

[16] X Shen W Chen Y Wu and R Xu ldquoDynamic analysis ofpartial-interaction composite beamsrdquo Composites Science andTechnology vol 71 no 10 pp 1286ndash1294 2011

[17] Z Shen and H Zhong ldquoStatic and vibrational analysis ofpartially composite beams using the weak-form quadratureelement methodrdquo Mathematical Problems in Engineering vol2012 Article ID 974023 23 pages 2012

[18] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013

[19] J G S da Silva S A L de Andrade and E D C LopesldquoParametric modelling of the dynamic behaviour of a steel-concrete composite floorrdquo Engineering Structures vol 75 pp327ndash339 2014

[20] S Lenci and J Warminski ldquoFree and forced nonlinear oscil-lations of a two-layer composite beam with interface sliprdquoNonlinear Dynamics vol 70 no 3 pp 2071ndash2087 2012

[21] Q-H Nguyen M Hjiaj and P Le Grognec ldquoAnalyticalapproach for free vibration analysis of two-layer Timoshenkobeams with interlayer sliprdquo Journal of Sound and Vibration vol331 no 12 pp 2949ndash2961 2012

[22] W-A Wang Q Li C-H Zhao and W-L Zhuang ldquoDynamicproperties of long-span steel-concrete composite bridges withexternal tendonsrdquo Journal of Highway and TransportationResearch andDevelopment (English Edition) vol 7 no 4 pp 30ndash38 2013

[23] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013

[24] W-B Zhou L-Z Jiang Z-W Yu and Z Huang ldquoFree vibra-tion characteristics of steel-concrete composite continuous boxgirder considering shear lag and sliprdquo China Journal of Highwayand Transport vol 26 no 5 pp 88ndash94 2013

[25] J Nie J Fan and C S Cai ldquoStiffness and deflection of steel-concrete composite beams under negative bendingrdquo Journal ofStructural Engineering vol 130 no 11 pp 1842ndash1851 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of