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Lecture notes: Structural Analysis II

2011 S. Parvanova, University of Architecture, Civil Engineering and Geodesy - Sofia 54

Influence lines for statically indeterminate structures I. Basic concepts about the application of method of forces. The plane frame structure given in Fig. 1 is statically indeterminate or redundant with degree of statical indeterminacy n=-2, this structure has two redundant members. The internal force at any section of the presented plane frame could be obtained by the following general equation, using the method of forces:

0,1 1 ,2 2" " " " " " "m m m mS S S X S X= + + "

",

, where Sm is bending moment, shear or normal force at section m (or support reaction if the problem is to find its influence line); is required internal force at section m for the corresponding simple statically determinate system; is the internal force at section m for the primary system caused by the force (or moment) X1, replacing the first eliminated constraint;

is the internal force at section m caused by the force (or moment) X2 introduced to replace the second eliminated constraint.

0mS

,1mS

,2mS

Figure 1 Statically indeterminate plane frame structure II. Determination of influence lines Let us construct the influence lines for the internal forces in section m of the plane frame structure given in Fig. 1. The bending moment, shear and normal forces at section m can be expressed as:

0,1 1 ,2 2

0,1 1 ,2 2

0,1 1 ,2 2

" " " " " " "

" " " " " " " ",

" " " " " " " ".

m m m m

m m m m

m m m m

M M M X M X

Q Q Q X Q X

N N N X N X

= + + = + + = + +

The following reasoning holds for the bending moment only, but they also could be done for shear and normal forces as well as for any support reaction. and are constants, which are dependant on the chosen simple system , and respectively on the introduced unknown forces X1 and X2, but they do not depend on the external loads, in that respect on the unit load moving across the road lane.

,1mM ,2mM

0mM depends on the position of moving load, and corresponding

X1

X2

m m

F=1 F=1

1 4

2 3

EI=const

GospodinovText BoxTable of contents



influence line for the primary system should be derived. The forces 1X and 2X , which are introduced to replace the eliminated constraints depend on the external loads, therefore they are dependant on the position of the unit load. In the method of forces the basic unknowns are obtained by the canonical equations having the following appearance:

11 1 12 2 1

21 1 22 2 2

0

0.f

f

X X

X X

+ + = + + =

The coefficients ij to the unknowns of the canonical equations represent the generalized displacements of the primary structure obtained by the elimination of the redundant members. These displacements being due to unit loads (forces or moments) acting along the direction of eliminated constraints. Numerically the values of these coefficients depend on the layout of the chosen primary structure and the cross sectional dimensions of its members. The coefficient ik represents the displacement along the direction i, induced by a unit action acting along the direction k. In line with the above statements the coefficients ij do not depend on the external loads, in that respect for influence line construction, they do not depend on the position of load unity. The coefficients indicate the displacements along the direction of eliminated constraints caused by the applied loads. As far as the applied loads are presented by the unit force moving along the road lane, for influence lines determination, these displacements, , depend on the

position of the load unity. In that respect in order to obtain the unknown forces

if

ifiX we should

first construct the influence lines for displacements if . The canonical equations written in matrix form take the following appearance: [ ] { } { }fX = , Wherefrom follows that: { } [ ] { }1 fX = . [ ] is so called compliance matrix of size 2x2 containing the coefficients ij , { }f is the vector of the free terms , {if }X is a vector which consists of unknown forces iX . Let us introduce a matrix [ ] in such a way that: [ ] [ ] 1 = . [ ] is the inverse matrix of [ ] , multiplied by -1. Then the unknowns of the method of forces can be expressed as: { } [ ] { }fX = , or in an expanded form:

1 11 1 12 2

2 21 1 22 2

" " " " " ",

" " " " " "f f

.f f

X

X

= + = +

The influence lines for the unknown forces iX could be derived from the above expressions.



The chosen primary statically determinate structure, obtained by eliminating all the redundant constraints, of the given one, is presented in Fig. 1. The forces 1X and 2X introduced to replace these constraints are depicted in the same figure. Next we should apply successively to the primary structure the unit actions and 1 1X = 2 1X = and trace the diagrams of the corresponding bending moments Mi. These diagrams are presented in Fig. 2.

Figure 2 Bending moment diagrams to the simple structure What follows is the calculation of coefficients ij , multiplying one by another the unit graphs M1 and M2, in order to compose of compliance matrix.

2 2

11

12

2

22

1 3 3 1 3 4 21 ,3 3

1 3 1 4 2 ,6

1 1 4 1.3333.3

EI EI EI

EI EI

EI EI

= + = = = = =

The compliance matrix [ ] multiplied by EI reads:

[ ] 21 22 1.3333

EI = . After inversing the matrix [ ] and multiplying the inverse matrix with -1 we get the matrix [ ] : [ ] 0.055556 0.083333

0.083333 0.875EI = .

Next we should construct the influence lines for the displacements if , as elastic curve of the road lane. For that purpose a unit load of the direction of required displacement is introduced. The unit load coincides with actions X1 and X2, the relevant bending moment diagrams are M1 and M2.

X1=1

X2=1 3

3 m m

1

Mm,1=-1.5

Qm,1=0.75 Nm,1=-1

1

0.75

0.75

Mm,1=0.5 Qm,1=0.25 Nm,1=0

0.25

0.25

M1 M2



The conjugate beams with the corresponding fictitious loads are given in Fig. 3. The bending moment diagrams in the conjugate beams are the required displacements influence lines " "1 f and 2" "f (Fig. 3).

Figure 3 Influence lines for the displacements if Now, the influence lines for unknown forces Xi should be derived as { } [ ] { }1 fX = . These lines are presented in Fig. 4.

Figure 4 Influence lines for the unknown forces iX Finally, the influence lines for internal forces at section m of simple system should be drawn. These graphics are depicted in Fig. 5.

X1 X2 0.03125

0.09375

0.16667

0.25

0.625

0.32813 0.60938

0.08333

-

++ -

X1=1

X2=1 3

3 m m

1

M1 M2

w1

1 2

Conjugate beam 3/(EI)

3/(EI) Staticaly determinate conjugate beam

1/(EI)

EI 1f EI 2f

3.0 1.875

2.625

2.0

0.66667

1.0 0.625 0.875 +

- -+

w1

1 2

1/(EI)



Figure 5 Influence lines for the internal forces in the primary statically determinate system The final influence lines, for required internal forces Mm, Qm and Nm at section m of the statically indeterminate structure, are obtained by the following equations:

01 2

01 2

01 2

" " " " 1.5 " " 0.5 " ",

" " " " 0.75 " " 0.25 " ",

" " " " 1 " " 0 " ".

m m

m m

m m

M M X X

Q Q X X

N N X X

= + = + + = +

These lines are given in Fig. 6.

Figure 6 Influence lines for the internal forces in the original plane frame

0.5625

0.14844 0.19531

0.125

0.406250.12109

0.26172

0.1875

0.59375

" "mQ

" "mN 0.031250.09375

0.16667

0.08333

" "mM

+

-

+

-

+

m

1.00.50.5

0.5

0.50.250.25

0.25 0.5

0" "mQ 0" "mN

0" "mM +

-+

-



III. Kinematical method of influence lines construction Kinematical way of influence line construction was examined in details for the trusses. Recall the Muller-Breslau principle here which states that: The influence line for a function (M, Q, N or support reaction) coincides with the graphics of vertical displacements of the road lane, obtained by application of unit virtual displacement at the point (or points) of application of the function. For statically determinate structure the deflected shape (respectively influence lines) are series of straight line segments. For statically indeterminate structures the influence lines are compound of several curves, one for each plate of the road lane. In order to obtain the influence line for bending moment at section m, using the kinematic method, the plane frame should be first cut at this section by placing a hinge (the constraint of the bending moment is eliminated). Next a unit mutual rotation is imposed in line of action of Mm in such a way that the moment performs negative work (Fig. 7). If the original frame has n redundant constraints, the new structure, with one eliminated constraint, has n-1 redundant members. Thus, we can obtain only the shape of influence line, or this is only qualitative representation of the influence line.

Figure 7 Kinematic method for bending moment influence line In order to obtain the value of influence line ordinates numerically two different approaches are applicable. According to the first approach for calculation of the influence line ordinate a unit load should be placed in the original frame structure and the bending moment in section m for this frame should be computed.

1

The second approach is based on the method of forces. We can consider the modified structure with eliminated constraint as a statically indeterminate simple system (similar to statically determinate simple system). The bending moment Mm is the basic unknown of the force method in this case, with other words: X1=Mm (Fig. 8), 11 is mutual rotation caused by Mm=1 (or X1=1). The applied load is load unity on the road line above the required ordinate, in that respect 1 f is a mutual rotation of the direction of Mm induced by the unit load and can be derived by the following expressions:

11

ff

M Mds

EI = ,

where 1M and fM are bending moment diagrams in (n-1) statically indeterminate system (the system with eliminated constraint). Alternatively the same mutual rotation can be obtained as:

2

1

2

Mm

1 2

1 2 11 = + = 1

Mm



01

1f

fM M

dsEI = ,

here 0fM is bending moment diagram in any statically determinate system, obtained from the corresponding (n-1) statically indeterminate system (Fig. 8). Therefore for convenience the unit force can be applied in any statically determinate simple system obtained by eliminating the redundant constraints of the modified system. The canonical equation reads:

11 1 1 11 1 1 110 respectively or finally /f m f mX M M + = = = f . The required ordinate becomes: 1

1 1 /f= 11 .

Figure 8 Determination of influence line ordinates

1 111 4.5714 /( )

M M ds EIEI= = ,

01 ,1

1 2.5715 /( )f

fM M

ds EIEI = = ,

01 ,2

2 0.57147 /( )f

fM M

ds EIEI = = ,

1 1 11 2 2 11/ 0.5625, / 0.125f f= = = = . The ordinate values are absolutely the same as those given in Fig. 4. In general if numerical values of influence line are to be determined, we can compute the displacements at successive points along the road lane, when the structure is subjected to the unit load placed at the eliminated constraint (mutual rotation caused by unit force is equal to the vertical displacement induced by the bending moment Mm according to the Maxwells theorem). Then each obtained value of vertical displacements, of the points belonging to the road lane, must be divided by the displacement at the point where the unit load acts, taken with negative sign (in our case this displacement is denoted 11 ). In conclusion it can be said that the kinematic method permits the easy determination of the shape of the influence line for any action, this shape being the same as that of the elastic curve of the corresponding structure with eliminated constraint. This analogy can be considerable value both in checking the accuracy of influence lines obtained by other methods and in seeking those parts

F=1 F=1 Mm=1

1.4286 1.0

.5714 0.5714

2.0 1.0

1.0

0,1

0

fM 0

,2fM 1M



of the structure which must be loaded in order to provide the maximum or minimum values of the required internal forces. Influence lines for X1 and X2 using the kinematic method.

1

X1

X1

1 1 =

X2

X2

+

+



References DARKOV, A. AND V. KUZNETSOV. Structural mechanics. MIR publishers, Moscow, 1969 WILLIAMS, . Structural analysis in theory and practice. Butterworth-Heinemann is an imprint of Elsevier , 2009 HIBBELER, R. C. Structural analysis. Prentice-Hall, Inc., Singapore, 2006 KARNOVSKY, I. A., OLGA LEBED. Advanced Methods of Structural Analysis. Springer Science+Business Media, LLC 2010

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