tripping of thin-walled stiffeners

Thin-Walled Structures 37 (2000) 1–26www.elsevier.com/locate/tws

Tripping of thin-walled stiffeners in the axiallycompressed stiffened panel with lateral pressure

Yuren Hu a,*, Bozhen Chena, Jiulong Sunb

a School of Naval Architecture and Ocean Engineering, Shanghai Jiaotong University, 1954 Hua ShanRoad, Shanghai 200030, China

b Shanghai Rules and Research Institute, China Classification Society, 1234 Pudong Ave., Shanghai200135, China

Received 13 August 1999; received in revised form 6 January 2000; accepted 31 January 2000

Abstract

Tripping of stiffeners in stiffened panels under combined loads of axial force and lateralpressure is studied. Firstly, on the basis of the Vlasov’s differential equation for torsionalbuckling of thin-walled bars, a generalized eigenvalue problem for tripping of stiffeners isderived by using the Galerkin’s Method. Then the effect of the lateral pressure (dead load)to the critical axial stress (live load) upon tripping is investigated by solving the eigenvalueproblem. The rotational restraint provided by the plate is taken into account. The effects ofthe compressive stress in the plate and the plate buckling mode are also discussed. Finally,an approximate equation to estimate the critical tripping stress with the effect of the lateralpressure is proposed. After some modifications, it can be applied in design rules for the purposeof checking the tripping strength of the stiffeners. 2000 Elsevier Science Ltd. All rightsreserved.

Keywords:Stiffened plate panel; Tripping; Torsional buckling

1. Introduction

The orthogonally stiffened panel is a fundamental structural component in shiphulls. It is also widely used in civil engineering, bridge, aerospace, offshore andother engineering fields. In modern ships with longitudinal framing system, stiffeners

* Corresponding author. Tel.:+86-21-6384-2238.E-mail address:[email protected] (Y. Hu).

0263-8231/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved.PII: S0263 -8231(00 )00010-0

2 Y. Hu et al. / Thin-Walled Structures 37 (2000) 1–26

are more closely spaced in the longitudinal direction than those in the transversedirection, On the other hand, the longitudinal stiffeners are weaker than the transverseones. When the panel is subjected to longitudinal compression, buckling of stiffenersbetween two adjacent transverse frames will be one of the most important failuremodes of the panel [1].

Buckling of stiffeners between two adjacent transverse frames can be further div-ided into two categories. One is the beam-column type flexural buckling of the stiff-ener-plate combination. The other is the tripping of a stiffener about its line of attach-ment to the plate as shown in Fig. 1. The stiffeners usually have thin-walled opencross-sections, such asT, angle cross-sections. Due to the low torsional rigidity ofthe thin-walled open cross-section, tripping will occur prior to beam-column typeflexural buckling when the stiffeners are subjected to axial compression. A stiffenerwill fail rapidly upon tripping, leading to the subsequent loss of the load carryingcapacity of the whole panel. Therefore, this kind of failure mode is thought to bevery dangerous and has drawn great attention of structural engineers. It has beenrecognized that tripping should be fully taken into consideration in estimating theultimate longitudinal strength of ship hulls [1–3]. Many classification societies havealso required in their design rules that tripping strength of stiffeners should bechecked [4–7].

Study on tripping of stiffeners in stiffened panels under pure axial compression(without lateral pressure) has a quite long history. Earlier work was summarized byBleich in his well-known book [8]. Further works were carried out by Faulkner,Adamchak and others in the 1970s and 1980s [9–12]. An elaboration of this problemcan also be found in Hughes’ book [13]. Some new works were reported by Daniel-son and others in the 1990s [14,15]. Most of the above mentioned works are basedon the classical theory of thin-walled bars. In addition, the folded plate analysis wasalso employed by Wittrick [16] and Smith [17]. In recent years, non-linear finiteelement method has become a powerful tool in research on this subject [18].

Another problem closely related to tripping of stiffeners under axial load is trip-ping of stiffeners under pure lateral pressure (without axial load). When the panelis subjected to lateral pressure only, stiffeners will trip if the lateral pressure reachesa critical value. This is the case when the stiffeners in the transverse bulkhead ofships are considered. This problem has been studied by many researchers. Among

Fig. 1. Tripping of stiffeners.

3Y. Hu et al. / Thin-Walled Structures 37 (2000) 1–26

others, one of the authors has published two papers on this subject [19,20], in whichtripping of stiffeners under lateral pressure was analyzed by using the Galerkin’smethod and design curves were proposed to estimate the critical value of the lat-eral pressure.

Generally, the two cases, i.e. under axial load and under lateral pressure, are stud-ied separately. However, the two kinds of load are applied simultaneously in manystructures. For instance, panels in ship bottom, side and longitudinal bulkhead aresubjected to not only the axial load caused by longitudinal bending of the hull, butalso the transverse load caused by the local water or cargo pressure. Therefore, it isnecessary to study the problem of tripping of stiffeners under the combined load ofaxial force and lateral pressure. An approximate equation as follows has been pro-posed by Adamchak to modify the critical axial stress by taking into account theeffect of the lateral pressure [2,10].

sT,EuqÞ05sT,Euq=0S12qqcrD (1)

where qcr is the critical value of the lateral pressure when no axial load exists.Recently, Hughes and Ma investigated the elastic and inelastic tripping of stiffenersunder axial force, end moment, lateral pressure and their combinations by usingenergy method [21,22].

The purpose of this paper is to study tripping of stiffeners in stiffened panelsunder combined loads of axial force and lateral pressure. Firstly, on the basis of theVlasov’s differential equation for torsional buckling of thin-walled bars, a gen-eralized eigenvalue problem for tripping of stiffeners is derived by using the Galerk-in’s Method. The method developed in the paper can be conveniently used to copewith the tripping problems of stiffeners under axial force, lateral pressure and theircombinations including two cases of live axial force with dead lateral pressure andlive lateral pressure with dead axial force. The method can also easily be extendedto the load case of end moment, which is not discussed in the paper. Then the effectof the lateral pressure (dead load) to the critical axial stress (live load) upon trippingis investigated by solving the eigenvalue problem. The rotational restraint providedby the plate is taken into account. The effects of the compressive stress in the plateand the plate buckling mode are also discussed. Finally, an approximate equation toestimate the critical tripping stress with the effect of the lateral pressure is proposed.After some modifications, it can be applied in design rules for the purpose of check-ing the tripping strength of the stiffeners.

2. Analytical model and differential equation

The stiffener considered is assumed to be a thin-walled bar under an axial forceof P=sA combined with a uniformly distributed lateral pressureq (force per unitlength). It can rotate about its line of attachment to the plate. The plate will bendwhen the stiffener rotates and provide a rotational restraint to the stiffener at the lineof attachment. The analytical model of the stiffener is shown in Fig. 2.


Fig. 2. Analytical model of a stiffener.

Further, a stiffener with general asymmetric cross-section as shown in Fig. 3 isconsidered.x axis andy axis are the principal axes of the cross-section. The anglebetween the plate and thex axis is denoted byq. The lateral pressure is assumed toapply at the line of attachment with the direction normal to the plate. This is areasonable assumption for panels subjected to uniformly distributed lateral pressure.The lateral pressure is defined positive when it points to the plate side as shown inFig. 3. The lateral pressureq can be resolved into two components in the directionof x axis andy axis, which are

qx5qsinq qy5qcosq (2)

The following differential equation is derived by Vlasov [23,24] for torsionalbuckling of the thin-walled bar with a fixed axis of rotation.

EIw,DjIV+(Ip,Ds−GJ)j0+[(Mxby,D+Mybx,D)j9]9+[qx(xd−ex)+qy(yd−ey)]j+kjj=0

(3)

wherej is the twisting angle of the stiffener;E is the elastic modulus of the material;

Fig. 3. General asymmetric cross-section of a stiffener.


Iw,D and Ip,D are the sectorial moment of inertia and the polar moment of inertia ofthe stiffener cross-section about the axis of rotation, respectively;J is the torsionalmoment of inertia of the cross-section;bx,D andby,D are another two sectional proper-ties whose definition can be found together with other’s in Appendix A of the paper;kf is the spring stiffness per unit length of the rotational restraint on the axis ofrotation; xd and yd are the coordinates of the axis of rotation;ex and ey are thecoordinates of the application point of the lateral pressure;Mx andMy are the bendingmoments caused by the lateral pressure. When the lateral pressure is uniformly dis-tributed,Mx, My and their derivativesM9

x, M9y (shear forces in the stiffener) are

Mx = −q cosqa2

2 Sza

−z2

a2D M9x = −

q cosqa2 S1−2

zaD

My = −q sinqa2

2 Sza

−z2

a2D M9y = −

q sinqa2 S1−2

zaD

(4)

According to the analytical model of the stiffener, the axis of rotation and theaxis of application of the lateral pressure coincide, so the differential equation canbe simplified to

EIw,DjIV1(Ip,Ds2GJ)j02qbDa2

2 Sza2

z2

a2Dj02qbDa

2 S122zaDj91kjj50 (5)

where

bD5 cosqby,D1 sinqbx,D (6)

The rotational restraint to the stiffener is provided by the plate. The rotationalspring stiffness per unit length,kj, can be derived by considering bending of theplate strip of an infinitesimal widthdz as an elastic beam [13,23], which is

kj5Et3

3b(1−n2)(7)

whereb is the stiffener spacing.t is the plate thickness.n is the Poisson’s ratio ofthe material. The effects of the compressive stress in the plate and plate bucklingmode are ignored in Eq. (7). We will discuss these effects in Section 5 of this paper.

3. Galerkin’s method and the eigenvalue problem

The Galerkin’s Method is used to solve the differential Eq. (5). The twisting angleof the stiffener is expressed in a series form as follows.

j(z)5On

rnyn(z) (8)


whereyn(z) is a function satisfying all the boundary conditions of the stiffener.According to the Galerkin’s method, integrating along the length of the stiffeneryields the following equation.

Ea

0

HEIw,DOn

rnyIVn 1(Ip,Ds2GJ)O

n

rny0n2

qbDa2

2 Sza2

z2

a2DOn

rny0n2

qbDa2 S1 (9)

22zaDO

n

rny9n1kjO

n

rnynJym dz50 m51,2,3,…

After integration and manipulation, a set of homogeneous linear equations in amatrix form can be obtained as follows:

([A]2q[B]2s[C]){ r} 5{0} (10)

The elements of the matrices [A], [B] and [C] are

Aij 5EIw,DEa

0

yIVj (z)·yi(z) dz2GJE

a

0

y0j (z)·yi(z) dz1kjE

a

0

yj (z)·yi(z) dz (11)

Bij 5bDa2 E

a

0

[aSza2

z2

a2Dy0j (z)1S122

zaDy9

j (z)]yi(z) dz (12)

Cij 52Ip,DEa

0

y0j (z)·yi(z) dz (13)

Generally speaking, whenq is a dead load ands is a live load, the critical stresscan be determined by setting the determinant of the coefficient matrix in Eq. (10),([A]2q[B]2s[C]), equal to zero and solving the equation fors. Then, among allthe solutions ofs the smallest one is the critical stresssT,E. However, the higherorder algebraic equation ofs is usually difficult to solve. Therefore there is a needto seek a more efficient method.

It can be found that Eq. (10) is a generalized eigenvalue problem. Since matrices[A], [B] and [C] are symmetric and matrix [C] is positively determined, Eq. (10)can be transformed to a standard form. Firstly, [C] is tridiagonalized by using theCholescki’s Method.

[C]5[LC][LC]T (14)

where [LC] is a tridiagonal matrix. Then the following standard eigenvalue problemcan be derived.

([K]2s[I]){ r} 5{0} or det([K]2s[I])50 (15)


where

[K]5[LC]T([A]2q[B])[LC] (16)

After suitably choosing the number of the terms of the twisting angle series,n, thestandard eigenvalue problem can be solved by using the well-established methodsuch as the Jacobi Method or the Method of Bisection. Then the smallest eigenvalueobtained is the critical stresssT,E.

Generally, the stiffener is assumed to be simply supported. Then the twisting angleseries can be written as

yn(z)5sinnpza

(17)

Substituting it into Eq. (13) yields

Aij 550 iÞj

a2[EIw,DSip

aD4

+GJSipaD2

+kj] i=j(18)

Bij 550 i−j=2k−1

bDa·ij (i2+j 2)(i2−j 2)2 i−j=2k

−bDa·(p2i2−3)

24i−j=0

(19)

Cij 550 iÞj

a2Ip,DSip

aD2

i=j(20)

Results of calculation show that the method developed above converges rapidly.In general cases, a rather accurate result can be obtained by choosing a twistingangle series with only a few terms.

An example is given below for illustration. Consider a stiffener with aT cross-section. The height and thickness of the web of the stiffener ared=35 cm andtw=0.6cm, respectively. The breadth and the thickness of the flange arebf=10 cm andtf=0.8cm, respectively. The length of the stiffener isa=600 cm. The spacing of stiffenersis b=60 cm. The thickness of the plate ist=0.6 cm. The critical stresses under lateral


pressure of different values and by choosing different numbers of series terms arelisted in Table 1. It can be seen from the table that generally a very accurate resultcan be obtained whenn=5. If the lateral pressure is not large, thenn=3 is enough.In fact, in many cases, only one term can yield a reasonably accurate result, but itshould be noted that the critical stress does not necessarily correspond to the firstterm of the series. That is to say, the tripping mode number is not necessarily equalto 1. The approximate result by taking only one term of the series will be discussedin Section 6 of the paper.

4. Critical value and effect of lateral pressure

If the stiffener is only subjected to lateral pressure, then it will trip when thelateral pressure reaches a certain value in the positive direction as shown in Fig. 3.Tripping of stiffeners under pure lateral pressure can also be analyzed by using thepresent method. For this purpose, lets=0 in Eq. (10) and the following equationcan be reduced.

([A]2q[B]({ r} 5{0} (21)

This is also a generalized eigenvalue problem and can be solved in a similar wayto that for solving Eq. (10). Now the matrix [B] is tridiagonalized, yielding

[B]5[LB][LB]T (22)

where [LB] is a tridiagonal matrix. Then the following standard eigenvalue problemcan be derived.

([K]2q[I]){ r} 5{0} or det([K]2q[I])50 (23)

where

[K]5[LB]T([A])[LB] (24)

After suitably choosingn, the standard eigenvalue problem can be solved by usingthe well-established method. Then the smallest eigenvalue obtained is the criticalvalue of the lateral pressureqcr.

Table 1Critical axial stresses upon trippingsT,E (MPa) for theT stiffener

q(N/mm) 228.52 221.39 214.26 27.13 0 7.13 14.26 21.39 28.52

n=3 418.41 375.14 331.87 288.60 245.33 202.06 158.78 115.51 72.24error 4.83% 3.05% 1.51% 0.45% 0.00% 0.64% 3.27% 10.61% 36.07%n=5 400.92 364.86 327.12 287.37 245.33 200.80 153.79 104.46 53.10error 0.44% 0.23% 0.09% 0.02% 0.00% 0.01% 0.03% 0.03% 0.03%n=15 399.15 364.03 326.83 287.32 245.33 200.78 153.75 104.43 53.09


Table 2Effect of lateral pressure to the critical axial stresssT,E (MPa)

q/qcr 21.0 20.8 20.6 20.4 20.2 0.0 0.2 0.4 0.6 0.8 1.0

sT,E 432.46 399.15 364.03 326.83 287.32 245.33 200.78 153.75 104.43 53.09 0.00error 243.27 238.54 232.61 224.94 214.61 0.0 22.19 59.56 134.92 362.92`

To investigate the effect of the lateral pressure to the critical axial stress, theexample in Section 3 of aT stiffener is considered again. The critical value of thelateral pressure for this stiffener can be obtained by solving the eigenvalue problemof Eq. (23), which isqcr=35.65 N/mm. The critical axial stresses under lateral press-ures with different values ranging fromq=2qcr, to q=qcr are listed in Table 2. In thetable, the term “error” denotes the relative error of the critical stress without lateralpressure to that with the effect of the lateral pressure, that is (sT,Euq=02sT,Euq=q)/sT,Euq=q. The results are also plotted in Fig. 4.

It can be seen from Table 2 and Fig. 4 that when the lateral pressureq is negativeor points to the stiffener flange side as shown in Fig. 2, the critical axial stress willincrease compared to that without the effect of the lateral pressure. On the otherhand it will decrease when the lateral pressure is positive or points to the plate side.The effect of the lateral pressure to the critical axial stress should not be neglectedif the absolute value ofq is not negligible compared to the critical valueqcr.

It can also be found that when the lateral pressure is positive, the critical stresssT,E varies almost linearly with the variation of the lateral pressureq. In this casethe approximate equation proposed by Adamchak can be applied. The result fromEq. (1) is slightly lower than that from the present method. However, when the

Fig. 4. Critical stress of theT stiffener under different lateral pressure.


lateral pressure is negative and is not very small, the result from Eq. (1) will behigher than that from the present method and the difference can not be neglected.

A series of calculations are also carried out for the sameT stiffener with differentlengths. The results are shown in Fig. 5. In the figure, the abscissa represents theslenderness,a/r, of the stiffener-plate combination, wherer is defined byr=√I/A. Iis the moment of inertia of the cross-section about the axis parallel to the plate andA is the cross-section area of the stiffener-plate combination. Curves ofsT,E versesa/r corresponding toq=220 N/mm, q=0 andq=20 N/mm as well asq=21

2qcr andq=1

2qcr are plotted in the figure. A curve of the critical value of the lateral pressureqcr versesa/r is also plotted in the figure.

It can be seen from Fig. 5 that for stiffeners with the same cross-section the criticalvalue of the lateral pressure,qcr, decreases with the increase of the stiffener length. Ifthe lateral pressure is constant, then its effect to the critical axial stress,sT,E, increaseswith the increase of the stiffener length. This is because the effect of the lateral pressureis basically determined by the ratioq/qcr, and with the increase of the lengthqcr decreases,leading to the increase of the ratioq/qcr. If q/qcr keeps constant, then the effect ofq tosT,E is almost the same for any stiffener length as shown in the figure.

Calculation has been carried out for a number of other stiffeners and similar pat-terns of the effect of the lateral pressure to the critical axial stress have been found.

5. Effects of compressive stress in plate and plate buckling mode

The expression for the spring stiffness per unit length of the rotational restraintprovided by the plate,kj, in Eq. (7) is derived from the elastic bending of the plate.If the effect of the compressive stress in the plate to the rotational restraint is takeninto account and the mode of plate buckling is considered, then determination ofkj

Fig. 5. Critical stress of theT stiffener with different length and under different lateral pressure.


becomes very complicated. To cope with the compressive stress in the plate, anapproximate method has been proposed by Faulkner [25]. The following effectivespring stiffnesskje is defined to replace the original spring stiffnesskj.

kje55kjS1 −sscr

D s,scr

0 s$scr

(25)

wherescr is the elastic buckling stress of the plate, which can be estimated by

scr52Ep2

12(1−n2)S tbD2Fm0b

a1

am0b

G2

(26)

wherem0 is the buckling mode number of the plate. This approximate method hasbeen adopted by many classification societies in their design rules [4–7]1.

By introducing the effective spring stiffnesskje into the present method, theelements of the matrix [A] and [C] in Eq. (10) become

Aij 55EIw,DEa

0

yIVj (z)·yi(z) dz−GJE

a

0

y0j (z)·yi(z) dz+kjE

a

0

yj (z)·yi(z) dz s,scr

EIw,DEa

0

yIVj (z)·yi(z) dz−GJE

a

0

y0j (z)·yi(z) dz s$scr

(27)

Cij 55−Ip,DEa

0

y0j (z)·yi(z) dz+

kjscr

Ea

0


−Ip,DEa

0


(28)

1 In some design rules effective spring stiffness is further defined by

kje55kjS1−sscr

Da s,scr

0 s$scr

wherea equals 1 or 2.


Taking the twisting angle as the series form of Eq. (17) for simply supported stiff-eners, the following equations for the matrix elements can be obtained.

Aij 550 i Þj

a2[EIw,DSip

aD4

+GJSipaD2

+kj] s,scr

a2[EIw,DSip

aD4

+GJSipaD2

] s$scr6 i=j(29)

Cij 550 iÞj

a2FIp,DSip

aD2

+kjscr

G s,scr

a2Ip,DSip

aD2

s$scr6 i=j(30)

The results for the sameT stiffener as in the example of Section 3 from thecalculation by adopting the effective rotational spring stiffness are shown in Fig. 6.The results for different stiffener lengths are shown in Fig. 7. From the figures itcan be seen that the critical axial stress will be much lower when the effective springstiffness of Eq. (25) is used than that when the constant spring stiffness of Eq. (7)is used. The reason is that by definitionkje is always smaller thankj when there

Fig. 6. The effect of compressive stress in plate and plate buckling mode (different lateral pressure).


Fig. 7. The effect of compressive stress in plate and plate buckling mode (different stiffener length).

exists compressive stress in the plate. Especially when the compressive stress reachesthe buckling stress,kje= 0, that is to say, no restraint is provided by the plate.

Buckling of the plate will occur prior to tripping of the stiffener in most cases.After buckling, one half-wave will form in the transverse direction of the plate anda number of half-waves in the longitudinal direction for the common long plate. Thehalf-waves in the longitudinal direction bend towards opposite side alternately (seeFig. 8). It is reasonable to assume that those half-waves of the plate that bend inthe same direction as the rotation of the stiffener, as shown in Fig. 8(A-A), willprovide no restraint to the stiffener, while other half-waves that bend in the oppositedirection to the rotation of the stiffener, as shown in Fig. 8(B-B), will providerestraint as the elastic plate. Obviously, buckling mode of the plate will affect thetripping behaviour of the stiffener.

Fig. 8. Buckling mode of plate and its effect to tripping of stiffeners.


An approximate method is used in this paper to take into account the effect ofthe plate buckling mode. Let the tripping mode number of the stiffener ism and thebuckling mode number in the longitudinal direction of the plate ism0. It is assumedin a conservative view of point that ifm0/m=1, then no restraint is provided by thewhole plate. If m0/m=2, then one-half of the plate provides the restraint. Ifm0/m=3, then only one-third of the plate provides the restraint. The proportion ofthe plate that provides the restraint can be deduced by analogy form0/m.3 and forthe cases in whichm0/m is not an integer. Accordingly, a deduction factorf can beintroduced into Eq. (25), yielding

kje55kj(1−sscr

f) s,scr

kj(1−f) s$scr

(31)

where f is defined by

f5

1−

Fm0

mG2m0

n

if Fm0

mG is even

1+Fm0

mG2m0

m

if Fm0

mG is odd

(32)

where [·] denotes eliminating decimals of the number.Adopting the effective spring stiffnesskje of Eq. (31), the elements of the matrix

[A] and [C] in Eq. (10) become

Aij 5 (33)

5EIw,DEa

0

yIVj (z)·y1(z) dz−GJE

a

0

y0j (z)·yi(z) dz+kjE

a

0


EIw,DEa

0

yIVj (z)·yi(z) dz·GJE

a

0

y0j (z)·yi(z) dz+kj(1−f)E

a

0

yj (z)·yi(z) dz s$scr


Cij 55−Ip,DEa

0

y0j (z)·yi(z) dz+

kjfscr

Ea

0


−Ip,DEa

0


(34)

For simply supported stiffeners, they are

Aij 550 iÞj

a2[EIw,DSip

aD4

+GJSipaD2

+kj] s,scr

a2[EIw,DSip

aD4

+GJSipaD2

+kj(1−f)] s$scr6 i=j(35)

Cij 550 iÞj

a2[Ip,DSip

aD2

+kjfscr

] s,scr

a2Ip,DSip

aD2

s$scr6 i=j(36)

The results for theT stiffener of the example by using the effective rotationalstiffness defined by Eq. (31) are also plotted in Figs. 6 and 7. It can be seen fromthe figures that the critical axial stress is lower than that by using the rotationalspring stiffness,kf, of Eq. (7), but it is higher than that using the effective springstiffness, kfe, of Eq. (25) because the plate will provide a portion of rotationalrestraint after buckling when the effect of the plate buckling mode is taken intoaccount. Apparently, Eq. (31) is more reasonable than the other two equations. Itcan also be seen from Fig. 7 that with the effect of the plate buckling mode, thepattern of variation of the critical stress for the same stiffener with different lengthwill become more complex.

It should be pointed out that when the effect of the plate buckling mode is takeninto account, the relationship between the critical stress and the lateral pressure isnot linear. The approximate Eq. (1) will lead to errors (see Fig. 6).

6. Approximate equation and its application in rules

If the absolute value of the lateral pressure is not very high, then the cross termsin Eq. (10) can be omitted, leading to an approximate equation of the following formfor calculation of the critical axial stress with the effect of the lateral pressure.


sT,E5 minm51,2,…

Am−qBmm

Cm

5

EIw,DSmpa D2

+GJ+[kj+2qbD

(p2m2−3)12

]S ampD2

Ip,D+kjfscr

S ampD2

(37)

The results for theT stiffener of the example by using this approximate equationand the comparison with those from the method of the paper are shown in Figs. 9and 10.

Further introduced into the approximate equation is an effective rotational springstiffness in which the effect of the compressive stress in the plate with a more com-plex pattern as expressed in the footnote of Section 4, and the effect of the lateralbending of the stiffener web (see Appendix B) are also taken into account. Theeffective rotational spring stiffness is defined as follows.

kje5F1−S sscr

DafGH1+

4t3d3t3wbF1−S sscr

DafGJkj (38)

From this the following equation can be derived.

sT51

Ip,D{ EIw,D

m2p2

a2 1GJ1F kjF1−SsT

scrDafG

1+4t3d3t3wbF1−SsT

scrDafG12qbD

(p2m2−3)24 G a2

m2p2} (39)

Fig. 9. Results and comparison of the approximate equation for theT stiffener.


Fig. 10. Results and comparison of the approximate equation for theT stiffener (different length).

This equation should be solved forsT by iteration. The smallest root ofsT is thecritical stress, that is

sT,E5 minm51,2,…

sT (40)

The results for theT stiffener by using Eqs. (39) and (40) with the effect of thelateral bending of the stiffener web are shown in Figs. 11 and 12. It can be seen

Fig. 11. Effect of stiffener web bending to the critical axial stress for theT stiffener.


Fig. 12. Effect of stiffener web bending to the critical axial stress for theT stiffener (different length).

from the figures that the critical axial stress will decrease with the effect of the lateralbending of the stiffener web. The amount of decrease is basically not affected bythe value of the lateral pressure. Calculation is also made for the same stiffener bytakinga=2 . Results show that the critical stress is different from that by takinga=1only when the critical stress is lower than the buckling stress of the plate, and thedifference is very small and can be neglected.

To further investigate the effect of the lateral bending of the stiffener web, a seriesof calculation is carried out for theT stiffener with different web thickness. Theresults are shown in Fig. 13. It can be found from the figure that the effect of thebending of the stiffener web is significant only when the ratiotw/d is small or theweb is relatively high.

It is usually required in design rules that tripping strength be checked for stiffenersunder given applied axial stresssapp. That is to say, the applied stresssapp shouldnot exceed the tripping stresssT,E given sapp. For this application, the approximateEqs. (39) and (40) can be modified by substitutingsT on the right side of Eq. (39)by sapp. Then the equation for estimating the tripping stress can be derived as follows.

sT,E51

Ip,DFEIw,D

p2

a2Sm21mm2D1GJ1

qbDa2

12 G (41)

where,

m5a4

EIw,Dp4HEt3

3b(1−n2)F1−Ssapp

scrDafG

1+4t3d3t3wbF1−Ssapp

scrDafG 2

qbD

4 J (42)

and the tripping mode numberm (number of half waves) is determined by


Fig. 13. Effect of stiffener web bending to the critical axial stress for theT stiffener (different webthickness).

m=1 0,m#4

m=2 4,m#36

m=3 36,m#144

…

m=k (k−1)2k2,m#k2(k+1)2

(43)

Compared to the equation in the present rules of the classification societies, theterms related to the lateral pressureq and the reduction factorf are added in Eqs.(41) and (42). The terms related toq represent the effect of the lateral pressure tothe tripping stress and the reduction factorf represents the effect of the plate bucklingmode. These effects should not be neglected from the discussion in previous sectionsof the paper.

7. Conclusions

1. Tripping of stiffeners in stiffened panels under combined loads of axial force andlateral pressure is studied in this paper. On the basis of the Vlasov’s differentialequation for torsional buckling of thin-walled bars, a generalized eigenvalue prob-lem for tripping of stiffeners is derived by using the Galerkin’s Method. Theeigenvalue problem can easily be solved by the existing well-established algor-ithms. The method developed in the paper can be conveniently applied to trippingof stiffeners under the following load cases:


O axial force only,O lateral pressure only,O live axial force with dead lateral pressure,O live lateral pressure with dead axial force.The method can also easily be extended to the load case of end moment, whichis not discussed in the paper.

2. Tripping of stiffeners under combined loads of live axial force and dead lateralpressure is investigated in detail by using the method developed in the paper. Theeffect of the lateral pressure to the critical axial stress upon tripping is examined.The following conclusions can be drawn from the study.O When the lateral pressure is negative or points to the stiffener flange side, the

critical axial stress will increase compared to that without the effect of thelateral pressure, while it will decrease when the lateral pressure is positive orpoints to the plate side. The effect of the lateral pressure to the critical axialstress should not be neglected if the absolute value of the lateral pressure isnot negligible compared to the critical valueqcr.

O When the lateral pressure is positive, the critical stresssT,E varies almost lin-early with the variation of the lateral pressureq. In this case the approximateEq. (1) proposed by Adamchak can be applied. The result from Eq. (1) isslightly lower than that from the present method. However, when the lateralpressure is negative and is not very small, the result from Eq. (1) will be higherthan that from the present method and the difference can not be neglected.

O For stiffeners with the same cross-section the critical value of the lateral press-ure,qcr, decreases with the increase of the stiffener length. If the lateral pressureis constant, then its effect to the critical axial stress,sT,E, increases with theincrease of the stiffener length. This is because the effect of the lateral pressureis basically determined by the ratioq/qcr, and with the increase of the lengthqcr decreases leading to the increase of the ratioq/qcr. If q/qcr keeps constant,then the effect ofq to sT,E is almost the same for any stiffener length.

3. The rotational restraint provided by the plate is considered in the present method.The effective rotational spring stiffness proposed by Faulkner is adopted toaccount for the effect of the compressive stress in the plate. On this basis, areduction factor is introduced to take into account the effect of the plate bucklingmode. The following conclusions can be drawn from the calculation for anexample of a stiffener withT cross-section. Similar conclusions can be obtainedfrom calculations for other stiffeners.O The critical axial stress will be much lower when the effective spring stiffness

of Eq. (25) is used than that when the constant spring stiffness of Eq. (7) isused. The reason is that by definitionkfe is always smaller thankf when thereexists compressive stress in the plate. Especially when the compressive stressreaches the buckling stress, no restraint is provided by the plate.

O The critical axial stress by using the effective rotational spring stiffness withthe effects of the compressive stress in the plate and the plate buckling modeis lower than that by using the rotational spring stiffness without any effect ofthe compressive stress, but it is higher than that using the effective spring


stiffness with only the effect of the compressive stress in the plate but withoutthe effect of the plate buckling mode. The reason is that the plate will providea portion of rotational restraint after buckling when the effect of the platebuckling mode is taken into account. With the effect of the plate bucklingmode, the pattern of variation of the critical stress for the same stiffener withdifferent length will become more complex.

O When the effect of the plate buckling mode is taken into account, the relation-ship between the critical stress and the lateral pressure is not linear. Theapproximate Eq. (1) will lead to errors.

4. An approximate equation to estimate the critical axial stress for tripping of stiff-eners in stiffened panels is proposed. The effects of the lateral pressure, the com-pressive stress in the plate and the plate buckling mode are taken into account inthe approximate equation. The effect of the lateral bending of the stiffener webis also taken into account. The following conclusions can be drawn from thecalculation by using the approximate equation.O When the absolute value of the lateral pressure is not very high, the approxi-

mate equation can predict the critical axial stress quite accurately.O The critical axial stress will decrease with the effect of the lateral bending of

the stiffener web. The amount of decrease is basically not affected by the valueof the lateral pressure. However, the effect of the lateral bending of the stiffenerweb is significant only when the web is relatively high.

O After some modifications, the approximate equation can be applied in designrules for the purpose of checking the tripping strength of the stiffeners. Com-pared to the equation in the present rules of the classification societies, theterms related to the lateral pressureq representing the effect of the lateralpressure to the tripping stress and the reduction factorf representing the effectof the plate buckling mode are added in the equation.

5. Finally, it should be noted that only elastic tripping of stiffeners is studied in thispaper. The method developed in the paper is applied only to estimate the elasticcritical stress. In many cases, the calculated elastic critical stress will exceed theelastic limit of the material. Therefore, inelastic tripping should be studied. In thisaspect, a widely used simplified method is to estimate the inelastic critical stressof tripping in a similar way to that used for inelastic beam-column buckling.According to the tangent modulus theory, the inelastic critical stress is

sT,cr=Et

EsT,E [2,9,10], whereEt is the tangent modulus, which can be obtained from

the Ostenfeld–Bleich parabola. If the method proposed by Faulkner for inelastic

buckling of plates is adopted, thensT,cr=!Et

EsT,E [26]. Inelastic tripping of stiff-

eners is beyond the scope of this paper and it will not be discussed in detail here.

Another point that should be mentioned is that the critical stress obtained by thepresent method is the stress in the stiffener. It is not the average stress of the stiffener-plate combination. To get the average stress of the stiffener-plate combination, as


usually required in estimation of the ultimate strength of ship hulls, the concept ofeffective width of the plate should be employed [9,26]. This will not be discussedin detail in this paper either.

Acknowledgements

The authors wish to express their appreciation to the China Classification Societyfor supporting this research. However, any views in this paper are those of the authorsand do not necessarily reflect the official views of the CCS.

Appendix A. Definition of sectional properties of thin-walled stiffeners

The sectional properties of thin-walled stiffeners in Eq. (3) will be defined in thisappendix.Iw,D and Ir,D are the sectorial moment of inertia and the polar moment ofinertia of the stiffener cross-section about the axis of rotation, respectively. They aredefined by

Iw,D5Iw1(xt2xd)2Ix1(yt2yd)2Iy (A1)

Ip,D5Ax2d1Ay2

d1Ix1Iy (A2)

where Iw is the sectorial moment of inertia of the cross-section about the twistingcenter;Ix and Iy are the moments of inertia of the cross-section about the principalaxes;A is the area of the cross-section;xt andyt are the coordinates of the twistingcenter in the principal coordinate system;xd and yd are the coordinates of the axisof rotation in the principal coordinate system. For open cross-sections such asT orangle, the twisting center is located at the intersection of the walls and the sectorialmoment of inertia about the twisting center is equal to zero.

J is the torsional moment of inertia of the cross-section, which is defined by

J513E

A

t2 dA513O

i

l it3i (A3)

where li and ti are the length and the thickness of theith wall of the cross-section.The other two sectional properties,bx,D andby,D, are defined by

by,D51Ix

[EA

y3 dA1EA

x2y dA]22yd (A4)

bx,D51Iy

[EA

x3 dA1EA

xy2 dA]22xd (A5)


Appendix B. Derivation of the effect of lateral bending of stiffener web

The rotational spring stiffness per unit length,kj, in Eq. (7) is derived under theassumption that the web of the stiffener is rigid. If the web is high enough, it willbend laterally upon tripping of the stiffener as shown in Fig. 14. In this case, therotational spring stiffness will be affected by the lateral bending of the stiffener web.

Sharp [27] has studied tripping of the lipped flanges of channel sections with theeffect of the lateral bending of the flange. The method Sharp developed can also beapplied to the tripping problem of stiffeners in the stiffened panels. It can be seenfrom Fig. 14 that the local angle of rotationj1 at the line of attachment of thestiffener to the plate when the web bends laterally is less than the global rotationanglej corresponding to a rigid web. Consider again a plate strip of a widthdzwith the thicknesst, the following relation between the local angle of rotation andthe moment can easily be derived.

Mj1

54E1i1

b5

Et3 dz3b(1−n2)

(A6)

where b is the stiffener spacing.E1=E

1−n2 is the equivalent elastic modulus.

i1=t3 dz12

is the moment of inertia of the cross-section of the plate strip.

Further consider the analytical model of Fig. 15. The displacement at the topmostpoint of the web can be determined by the beam theory, which is

D5j1d1Pd3

3E1iw5j1d1

4E1i1d2

3E1iwbj15jd (A7)

where iw=t3w dz12

is the moment of inertia of the cross-section of the web strip of the

width dz. Taking into account Eq. (A6),P can be obtained by equilibrium

P5Md

54E1i1bdj1 (A8)

Fig. 14. Lateral bending of stiffener web.


Fig. 15. Analytical model of stiffener web.

Then we get the following relationship between the local angle and global angleof rotation.

j5S114E1i1d3E1iwbDj15S11

4t3d3t3wbDj1 (A9)

From Eq. (A6) and Eq. (A9), the following rotational spring stiffness per unitlength with the effect of the lateral bending of the stiffener web can be derived.

kj5Mj

·1dz

5M

S1+4t3d3t3wbDj1

·1dz

5Et3

3b(1−n2)S1+4t3d3t3wbD

(A10)

It should be pointed out that Sharp’s conclusion for the lipped flange of channelsections was directly applied to the case of tripping of stiffeners in the stiffenedpanels by Hughes in his book [13]. Thus the constant in the parenthesis in thedenominator on the right side of Eq. (A10) is 2/3 according to Hughes. This is notcorrect and the correct value should be 4/3.

Eq. (A10) is derived under the assumption that no compressive stress exists inthe plate. To account for the effect of the compressive stress in the plate, it is assumedthat no restraint is provided by the plate after its buckling and the amount of therestraint decreases linearly or parabolically to zero with the increase of the compress-ive stress in the plate from zero to the value of the buckling stress of the plate.Therefore, Eq. (A6) becomes

Mj1

54E1i1

b F12S sscrDaG5

Et3 dz3b(1−n2)F12S sscr

DaG (A11)


Then, the following effective rotational spring stiffness can be derived without dif-ficulty.

kje5

Et3F1−S sscrDaG

3b(1−n2)H1+4t3d3t3wbF1−S sscr

DaGJ (A12)

This expression has been adopted by many classification societies in their designrules [4–7].

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tripping of thin-walled stiffeners

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