preprint series mechanics and applied mathematics university of oslo
TRANSCRIPT
No. 2ISSN 0809–4403 June 2006
Semi-analytical buckling strength analysis ofarbitrarily stiffened plates with varying thickness
by
Lars Brubak and Jostein Hellesland
PREPRINT SERIES
MECHANICS AND
APPLIED MATHEMATICS
UNIVERSITY OF OSLODEPARTMENT OF MATHEMATICS
UNIVERSITETET I OSLOMATEMATISK INSTITUTT
Dept. of Math., University of Oslo
Mechanics and Applied Mathematics
Preprint Series No. 2
ISSN 0809–4403 June 2006
Semi-analytical buckling strength analysis of
arbitrarily stiffened plates with varying thickness
Lars Brubak and Jostein Hellesland
Mechanics Division, Department of Mathematics,
University of Oslo, 0316 Oslo, Norway
Abstract
A computationally efficient method for elastic buckling and buckling strength analysis of biaxially loaded,stiffened plates with varying, stepwise constant thickness, are presented. The stiffeners may be sniped orend-loaded (continuous), and their orientations may be arbitrary. Both global and local plate bucklingmodes are captured. The method is semi-analytical and makes use of simplified displacement computationsthat involve the elastic buckling load (eigenvalue), determined using a Rayleigh-Ritz approach, and finallystress computations using large deflection theory in combination with strength assessment using vonMises’ yield criterion applied to membrane stresses. The displacements are represented by trigonometricfunctions, defined over the entire plate. The method is implemented into a Fortran computer code, andnumerical results, obtained for a variety of plate and stiffener geometries, are compared to fully nonlinearfinite element analysis results.
Key words: Stiffened plates; Stepped plate; Arbitrary stiffener orientations; Buckling strength; Elasticbuckling load; Semi-analytical method
1 Introduction
In many branches of engineering, stiffenedplates are used as main structural componentsin order to improve the strength/weight ratioand reduce costs. For analysis of large struc-tures, computationally efficient analysis toolsare useful for obtaining results within a rea-sonable time limit. Also, such tools may be anecessity in the design of structures with com-plex geometry and stiffener arrangements, for
Revision May 2007: An error in Eq. 10 is corrected. This errorwas not present in the computer program, and consequently,computed results in the paper are correct.
which explicit strength formulas [1, 2, 3] maynot be applicable. Nonlinear finite elementmethod analyses could be used in such cases.However, such analyses are often time con-suming to prepare, run and postprocess, andother approaches may be better suited in morepractical oriented design contexts.
Computationally efficient analysis tools us-ing semi-analytical methods for buckling andultimate strength predictions are becomingmore common. A rather advanced nonlinearbuckling model was developed by Byklum etal. [4, 5] with a basis in previous work by Steen
1
[6, 7]. These studies deal mainly with unstiff-ened and regularly stiffened plates. A relatedmodel presented in Brubak, Hellesland andSteen [8], deals with buckling strength analy-sis of constant thickness plates provided withsniped stiffeners with irregular orientations.These models have been adopted by the shipclassification and engineering services companyDet Norske Veritas (DNV), and implementedinto a computerised software code entitledPULS [9]. The semi-analytical models aboveare based on the Rayleigh-Ritz method. Amore detailed review of semi-analytical meth-ods that make use of that method as well as ofthe Galerkin method (Paik and Lee [10], etc.)is given in Brubak et al. [8].
The studies mentioned above deal withplates of constant thickness. Sometimes it isadvantageous to vary the plate thickness lo-cally to increase the buckling strength and ob-tain more cost efficient structures. Several ap-proaches have been formulated for such plates.For example, Azhari and Shahidi [11] pre-sented a semi-analytical method for analysingthe post-buckling behaviour of initially perfect,unstiffened stepped plates (i.e., with varying,stepwise constant thickness). Xiang and Wei[12] developed a method for linear elastic buck-ling and vibration analysis of such plates. Thesetwo papers summarise some earlier works onstepped plates. Most of these works deal mainlywith unstiffened plates and not with bucklingstrength of imperfect plates (onset of yielding,capacity etc.) as such.
The main objective of the present paper isto present and document the applicability ofa semi-analytical model for local and globalbuckling strength analysis of uni-directionallystepped plates. It represents an extension ofan earlier work [8] on constant thickness plateswith sniped stiffeners. The present study alsoincludes extensions to end-loaded (continuous)stiffeners and torsional stiffness considerations,but does not include local failure modes of thestiffeners. The plates may have regular or arbi-
Longitudinal stiffener(continuous)
Longitudinalgirder
Transversegirder
Snipedstiffener
Longitudinalstiffener
(b)
(a)
Figure 1. Examples of stiffened plates: (a) Stiff-ened plate enclosed by longitudinal and transversegirders, and (b) girder stiffened by sniped stiffen-ers.
trarily oriented stiffeners, and may have vari-ous restraints at plate edges and in the interiorof the plate.
2 Stiffened plate modelling
Two stiffened plate examples are shown inFig. 1. The stiffeners of the plate enclosed bythe strong longitudinal and transverse girdersin Fig. 1a, are typically continuous and sub-ject to axial loading at their ends (at the plateboundary). Unlike these, the sniped (“discon-tinuous”) stiffeners illustrated on the girderweb in Fig. 1b, will not be axially loaded attheir ends. Normally, this will neither be thecase for non-regular stiffeners, such as for in-stance in the stern and bow of a ship hull.
The plate considered can be defined with ref-erence to Fig. 2. The plate may consist of anarbitrary number of uni-directional plate stripsof different thickness (3 shown in the figure).It may be provided with one or several stiffen-ers with arbitrary orientations. The stiffeners
2
Sy
Syt1t3
Sx (mean)
Sx (mean)
SxSx
Sx
Stiffener
(x1, y1)
(x2, y2)L
b
b1 b2 b3
x
yt1 t2 t3
(a) (b)
ts
hw
tf
bf
tw
Figure 2. Simply supported plate with varying, stepwise constant plate thickness, subjected to externalin-plane compression or tension Sx and Sy in the x- and y-direction, respectively.
may be sniped at the ends or end-loaded (con-tinuous). They may have different cross-sectionprofiles, and may be eccentric, as in Fig. 2b, orsymmetric about the middle plane of the plate.
The stiffeners are modelled as simple beamswith flexural stiffness only against out-of-planebending. This implies that possible local buck-ling of stiffeners, including torsional instability,can not be predicted. This may not representa serious limitation in practical cases as de-sign rules generally impose constructional de-sign provisions that prevent local buckling ofstiffeners. Also, stiffeners are usually propor-tioned such as to provide sufficient strength toprevent a global plate buckling mode. Withoutthe additional axial stress from global bending,local buckling of stiffeners is even less likely.Thus, the simplified stiffener model seems likea reasonable one.
The torsional stiffness of the stiffeners maybe included, but their axial stiffness is ne-glected. The latter implies that the stiffenerseffect on the internal membrane stress distribu-tion in the plate is neglected. This is consideredan acceptable simplification.
The usual assumptions that the plate edgesremain straight (due to the neighbouringplates) while in-plane movements are allowed,
are adopted. Otherwise, all four plate edgesare supported in the out-of-plane direction. Anedge, or a part of an edge, may rotationallybe simply supported, partly or fully clamped,as illustrated in Fig. 3. Rotational and out-of-plane restraints, modelled by strong transla-tional and rotational springs, can also be addedalong specified lines with arbitrary orientationsin the interior of the plate.
The displacement field is defined over theentire plate. It was initially questioned whetherthis was adequate for stepped plates. Prelimi-nary eigenvalue and buckling strength resultsindicated that it is. The alternative of specify-ing one displacement field for each plate stripand impose continuity requirements along theboundaries between the plate strips, will beconsiderably more complicated and was notconsidered.
Hooke’s material law for plane stress for anelastic, isotropic material is adopted, and fur-ther, Kirchhoff’s deformation assumption (astraight line normal to the middle plane priorto loading, remains straight and normal to theplane after deformation). These are the usualthin plate assumptions [13]. The first impliesthat only in-plane stresses (σx, σy, τxy) andin-plane strains (εx, εy, γxy) are present. The
3
ss cc
sscc
-t1- -t2- -t3-
ss
Stiffeners
Figure 3. Stepped plate example with partly sim-ple (ss) and partly clamped supports (cc), and ir-regular stiffener arrangement.
second implies that strains, consisting of mem-brane strains (constant over the plate thick-ness) and bending strains, vary linearly acrossthe plate thickness.
The plate, Fig. 2, is subjected to the ex-ternal, in-plane biaxial compressive or tensileloading defined by the stresses shown in the fig-ure, and defined as positive when compressive.A consequence of the straight edge assump-tion is that the resultant of the mean externalstress Sx, in the x-direction, will initially be dis-tributed as illustrated by the varying, stepwiseconstant stress Sx. External shear stresses arenot included in the model at present. The ef-fect of out-of-plane displacements, that cause aredistribution of normal stresses and formationof shear stresses, is discussed later.
3 Major Computational steps
The computations are carried out in twomajor steps:1. In the first step, labelled the elastic bucklingstress limit (ESL) state, the elastic bucklingload (first eigenvalue) and corresponding buck-ling mode of the stiffened plate are calculated.2. In the second step, labelled the bucklingstrength limit (BSL) state, a buckling strength(capacity) assessment is made for the platewith a specified out-of-plane imperfection. Thisstep involves displacement computations usingan approximate displacement magnifier (basedon small deflection theory) that is a functionof the applied load and the elastic buckling
b1 b2 b3
-t1- -t2- -t3-L
x
y
d3d2d1
d4
Figure 4. Definition of in-plane degrees of freedomfor linear analysis of a stepped plate example withthree plate strips.
load (ESL), computation of stresses accordingto large deflection theory and, finally, strengthassessment using the von Mises’ yield criterionapplied to membrane stresses.
The essential parts of the model are de-scribed below. Additional details are given inBrubak [14].
4 Initial, reference membrane stresses
In order to determine the initial, referencebiaxial membrane stresses in the various platestrips, a linear static analysis is performed fora perfect plate with a chosen reference loading(Sx0 and Sy0, positive in compression). For il-lustration, the stepped plate shown in Fig. 4 isconsidered. It consists of three plate strips andhas four in-plane degrees of freedom d1 to d4.Stiffeners are not included (their axial stiffnessis neglected). The resulting axial plate stiffnessrelationship can be given in matrix form by
K0d = P0 (1)
where
K0 = E∗
∑2i=1
tiLbi
− t2Lb2
0 ν(t1 − t2)
− t2Lb2
∑3i=2
tiLbi
− t3Lb3
ν(t2 − t3)
0 − t3Lb3
t3Lb3
νt3
ν(t1 − t2) ν(t2 − t3) νt3∑3
i=1tibi
L
(2)
d = [d1, d2, d3, d4]T (3)
4
P0 = [0, 0,−Py0,−Px0]T (4)
Here, E∗ = E/(1 − ν2), E and ν are Young’smodulus and Poisson’s ratio, respectively,Px0 = Sx0(b1t1 + b2t2 + b3t3) and Py0 = Sy0Lt3are the resultant, reference forces acting on theplate edges. Once Eq. 1 is solved, the initialreference strains can be computed from
εx0 =d4
Land εy0 =
d1/b1
(d2 − d1)/b2
(d3 − d2)/b3
(5)
for plate strip 1,2 and 3, respectively Insertedinto Hooke’s law, these strains give initial, refer-ence membrane stresses σm
x0 and σmy0 in the plate
strips. These strains and stresses are defined aspositive when tensile (i.e., opposite of the exter-nally applied stress definition). The membraneshear stresses and strains (τm
xy0, γmx0) are zero
since external shear loading is not considered.
5 Elastic buckling stress limit (ESL)
In the first step of the analysis, the elasticbuckling load (first eigenvalue) of a perfect,stiffened plate is computed using the Rayleigh-Ritz method. For completeness and conve-nience of the reader, the major elements aresummarised below. The assumed out-of-planedisplacement field, which satisfies the bound-ary conditions of a simply supported plate, isgiven by
w(x, y) =M∑
i=1
N∑
j=1
aijsin(πix
L)sin(
πjy
b) (6)
where aij are amplitudes, L the plate length, bthe total plate width, 0 ≤ x ≤ L and 0 ≤ y ≤ b.
Although each component in a series of sinefunctions represents a simply supported condi-tion, added together they are, in combinationwith rotational springs along supports, nearlyable also to describe fully or partially restrained
conditions [4, 14]. Therefore, rather than tospecify different fields, such as for instance aseries of cosine functions for a clamped plate,the sine curve assumption is used for variousboundary conditions. To achieve the same ac-curacy, a higher number of degrees of freedom(number of terms) will normally be requiredwith a sine field than with a field that satisfiesthe kinematic boundary conditions more appro-priately.
Equilibrium of the loaded plate requires thatits total potential energy, Π = U + T , has astationary value, i.e., δΠ = δ(U +T ) = 0. Here,U is the strain energy and T is the potentialenergy of the external loads. This requirementleads to the eigenvalue problem
(KM + ΛeKG)ae = 0 (7)
where KM is the material stiffness matrix (dueto U), KG the geometrical stiffness matrix (dueto T for the initial, reference loading), Λe theeigenvalues (load factors at buckling) and ae
the eigenvectors containing all the displacementamplitudes aij that together define the buck-ling mode for a specific eigenvalue. Energy con-tributions used in the derivation of the eigen-value problem are given below. Strain energyfrom membrane stresses is not included as itdoes not affect computed eigenvalues.
The elastic strain energy from bending of theentire plate, U b
plate, is obtained by adding up thecontributions U b
s from each plate strip s. WithNs number of plate strips, the total bendingstrain energy becomes
U bplate =
Ns∑
s=1
U bs (8)
where
U bs =
∫ L
0
∫ ys2
ys1
Ds
2
(
(w,xx + w,yy)2
−2(1 − ν)(w,xxw,yy − w2,xy)
)
dx dy
(9)
5
is the contribution from plate strip number s,located between y = ys1 and y = ys2, withthickness ts and flexural plate stiffness Ds =Et3s/12(1 − ν2), and where the conventional“comma” notation w,xy for ∂2w/∂x∂y, etc., isadopted. By substituting the assumed displace-ment field, an analytical solution of this inte-gral may be derived and written in the form
U bs =
M∑
i=1
N∑
j=1
N∑
l=1
aijail
DsL
4
×
(
(πi
L)2 + (
πj
b)2
)(
(πi
L)2 + (
πl
b)2
)
Gj,l
−2(1 − ν)
(
(πi
L)2(
πl
b)2Gj,l − (
πi
L)2πj
b
πl
bHj,l
)
(10)
where Gjl and Hjl are given in Appendix A.Note that b and L are the total plate dimen-sions. For a constant thickness plate, i.e. a platewith only one plate strip, Gjl = Hjl = 0 if j 6= land Gjj = Hjj = b/2. Thus, Eq. 10 breaksdown into the more well known double sum forthat case.
Similarly, the potential energy of the exter-nal plate loads can be given by
T =Ns∑
s=1
Ts (11)
where
Ts = Λ∫ L
0
∫ ys2
ys1
ts2
(
σx0w2,x + σy0w
2,y
)
dy dx
(12)in the case of proportional loading. The analyt-ical solution of this integral can be written
Ts=ΛM∑
i=1
N∑
j=1
N∑
l=1
aijail
(
σmx0ts(iπ)2
4LGjl
+σm
y0tsLjlπ2
4b2Hjl
)
(13)
The effect of a constant uniaxial or biaxial pre-stress of the plate can readily be included above
by adding another term that is not increasedby the load factor Λ. In the same manner asfor the strain energy of the plate, Eq. 13 breaksdown into the more well known double sum forconstant thickness plates.
The curvature of a stiffener is equal to thecurvature of the plate along the stiffener. Then,for an arbitrarily oriented stiffener with lengthLs, end coordinates (x1, y1) and (x2, y2), andcross-section area As, the bending strain energydue to the stiffener can be given by
U bstiff =
EIe
2
∫
Ls
w2,ss dLs =
EIe
2L4s
∫
Ls
(
L2xw,xx + 2LxLyw,xy + L2
yw,yy
)2
dLs
(14)
where Ie is an effective moment of inertia aboutthe axis of bending, w,ss the curvature in thestiffener direction, Lx = (x2 − x1) and Ly =(y2 − y1). In the case of an eccentric stiffener,the stiffener will “lift” the axis of bending abovethe middle plane. Ie can be given by
Ie =∫
As
(z − zc)2 dAs + betz
2c (15)
where zc is the distance from the plate middleplane to the centroidal axis (through the centreof area) of a section consisting of the stiffenerand an effective plate area of width be. For astepped plate, t may conservatively be taken asthe thickness of the thinnest of the plate strips.Bending of the plate about its own axis (middleplane) is included in the plate strain energy.The strain energy integral above may be solvedanalytically or by numerical integration. Thelatter is chosen here.
For a symmetric stiffener, zc = 0 is the cor-rect solution as bending in this case will beabout the middle plane. For an eccentric stiff-ener, Eq. 15 is an approximation, whose accu-racy will depend on the assumed value of be. Itis found that zc = 0 is an acceptable value alsofor eccentric stiffeners in many practical cases.
6
In practical design work, a zc–value calculatedwith a be of about be = 20t has been suggested[8, 14].
The torsional stiffness of the stiffeners maybe accounted for by including the energy con-tribution (St. Venant torsion) given by
UTstiff =
GJ
2
∫
Ls
w2,nsdLs =
GJ
2L4s
∫
Ls
(
LxLy(w,yy − w,xx)
+ (L2x − L2
y)w,xy
)2
dLs
(16)
where J is the torsion constant, w,ns is the par-tial double derivative of w with respect to thedirections normal to and along the stiffener andG = E/2(1 + ν).
For end loaded (continuous) stiffeners, it isnecessary to include the potential energy due toexternal loads on the stiffener ends. For the ar-bitrarily oriented stiffener considered above, itcan be shown that its effect due to plate short-ening can be expressed by
Tstiff = −ΛPs0
2
∫
Ls
w2,s dLs =
− ΛPs0
2L2s
∫
Ls
(
Lxw,x + Lyw,y
)2
dLs
(17)
where Ps0 is the initial, resultant reference loadon the stiffener. It acts in the stiffener direc-tion and is defined positive in compression. Thetranslation of this load due to the rotation ofthe stiffener end is not included as it does notaffect computed eigenvalues.
If plate edges, or portions of edges, arepartly or fully clamped (modelled by rotationalsprings), additional strain energy contributionshave to be added. Similarly, contributions haveto be added for any rotational or translationalrestraints in the interior of the plate. For addi-tional details, see Brubak et al. [8, 14].
6 Buckling strength limit (BSL)
6.1 Load incrementation and strength crite-
rion
The present model aims at predicting an ap-proximate buckling strength limit (BSL) of astiffened plate with an initial displacement im-perfection (w0). Additional displacements (w)at a given load stage are estimated using theapproximate displacement magnifier
w =Λ
Λecr − Λ
w0 (18)
where Λecr is the load factor of the first eigen-
value and Λ the load factor at a given stage ofthe loading (ΛSx0, ΛSy0).
This, and similar magnifiers based on lin-earised elastic second order theory (small de-flection theory), are commonly used for ap-proximate analyses of both columns and plates[13]. The load factor, and the resulting dis-placements and stresses, redistributed due tothe out-of-plane displacements, are increaseduntil the adopted strength criterion (below) issatisfied.
Use of Eq. 18 implies that the bucklingstrength estimate will never exceed the elasticbuckling load limit. In order to capture theload carrying capability of slender (thin) platesbeyond this limit, often denoted the post-critical (or reserve) strength, displacementsmust be computed using large displacementtheory. However, in practical design, it may bedesirable not to utilise this reserve strength, inorder to limit the formation of significant plas-tic (permanent) deformations of slender plates.It is in such contexts that the present modelrepresents a sound alternative.
A buckling strength criterion, proposed anddiscussed previously [8], is adopted for the pre-sented model. According to this criterion, thebuckling strength is obtained at first yield ac-cording to the von Mises’ yield criterion [13]
7
applied to membrane stresses:
σme =
√
(σmx )2 + (σm
y )2 − σmx σm
y + 3(τmxy)
2 ≤ fY
(19)Here, fY is the yield strength. This membranestress criterion allows in an approximate man-ner for some additional strength due to redis-tribution of stresses after yielding in the outerplate fibres.
The critical stress points are typically lo-cated in the plate along the edges, along thestiffeners or at the boundary between twoneighbouring plate strips. The latter is typicalfor a plate strip subjected mainly to a stressin the x-direction and that is located betweentwo considerably thicker plate strips prevent-ing the out-of-plane displacements along itsboundaries.
6.2 Imperfection shape and amplitude
The imperfection shape can be taken accord-ing to any specified shape. Here, w0 is takenequal to the first buckling mode (eigenmode)from the ESL analysis. Then, w0 can be givenin the same form as Eq. 6, but with the coeffi-cients aij replaced by bij. The latter are scaledto give the chosen w0,max.
In design, maximum imperfection ampli-tudes w0,max will normally be taken accordingto values, w0,spec., specified in relevant designcodes [1, 2, etc.]. In order to compensate inpart for the conservativeness for slender platesimplied by the displacement magnifier (Eq.18), a slenderness dependent w0,max proposedby Brubak et al. [8] is adopted here. It is shownin Fig. 5 and defined by
w0,max
w0,spec.=
(1 − 112
λ4) if λ ≤√
1.56
3/λ4 if λ ≥√
1.56(20)
where
λ =
√
ΛY
Λecr
(21)
PSfrag replacements1
1
0.8
0.6
0.4
0.2
00 0.5 1.5 2
w0,max
w0,spec.
λ
Figure 5. Adopted maximum imperfection ampli-tude versus reduced slenderness.
is the reduced slenderness. It is defined in termsof the load factor ΛY at which the von Mises’yield stress is reached (σm
e = ΛY ) and the loadfactor Λe
cr of the first eigenvalue of the perfectplate.
6.3 Stress redistribution
Following the classical approach, large de-flection theory (large rotations, but small in-plane strains) is used to capture the redistribu-tion of stresses that takes place due to out-of-plane displacements. In this theory, the mem-brane strains in an imperfect plate can be writ-ten [15]
εmx =u,x +
1
2w2
,x + w0,xw,x (22)
εmy = v,y +
1
2w2
,y + w0,yw,y (23)
γmxy =u,y + v,x + w,xw,y + w0,xw,y + w0,yw,x(24)
where w0 is the initial imperfection, w is theadditional displacements, u and v are the x-and y- displacements at the middle plane of theplate, respectively.
The strain compatibility equation for im-perfect plates can now be obtained by differ-entiation and combination of Eqs. 22-24. Bysubstituting strains from Hooke’s law for planestress into this equation, and introducing Airy’sstress function F (x, y), defined by σm
x = F,yy,σm
y = F,xx, τmxy = −F,xy, the following nonlin-
8
ear plate compatibility equation results:
∇4F = E(w2,xy − w,xxw,yy
+ 2w0,xyw,xy − w0,xxw,yy − w0,yyw,xx)(25)
This equation was given by Marguerre [15], andrepresents an extension of von Karman’s platetheory.
The shape of the imperfection displacementfield (w0) is taken on the same form as Eq. 6,with amplitudes bij instead of aij. A solution ofEq. 25 on the form
F (x, y) = Λ(1
2σm
x0y2 +
1
2σm
y0x2)
+2M∑
i=0
2N∑
j=0
fijcos(iπ
Lx)cos(
jπ
by)
(26)
was given by Levy [16] for perfect plates (w0 =0). Byklum et al. [4] showed that the same formcan be used for imperfect plates and derivedthe stress amplitudes fij, given in Appendix A,for such plates. They are found by substitutingF (x, y), w and w0 into Eq. 25.
Airy’s stress function identically satisfies in-plane equilibrium within each plate strip. Todiscuss conditions at interfaces, consider Eq.26, where the first term is due to the externalstresses. The second (summation) term repre-sents the redistribution due to out-of-plane dis-placements (w0 and w), and is a function of theload stage factor Λ through the amplitudes aij
and bij. The second term varies smoothly, alsoacross an interface between two strips (of dif-ferent thickness), while the first term exhibitsstress jumps at interfaces. A consequence of thisis that equilibrium will be satisfied only on theaverage between two strips (of full length), andnot along limited strip lengths.
Local inaccuracies in stresses near interfacesincrease with increasing dominance of the sum-mation term in Eq. 26, i.e., with increasingplate slenderness since the displacements (w)increase with increasing slenderness. For veryslender plates, possible effects on strength pre-dictions are of minor concern as the present
PSfrag replacements
Ansys, sniped stiff.
model, sniped stiff.
model, sniped stiff.w/torsion
model, continuous stiff.
0.5
1.5
2
1
00 5 1510 252030
hw
tw
Sx
fY
Sx Sx
PSfrag replacements
Ansys, sniped stiff.model, sniped stiff.
model, sniped stiff.w/torsionmodel, continuous stiff.
0.51.5
2105
1510252030
Figure 6. Elastic buckling stress (ESL) in theglobal and local buckling range of a uniaxiallyloaded plate (L = b = 2000 mm, t = 20 mm)with an eccentric stiffener (tw = 12 mm, heighthw, be = 30t).
model is rather conservative for such platesdue to the neglect of the post-critical reservestrength.
7 Sniped vs. end-loaded stiffeners
Fig. 6 shows buckling stresses (ESL) fora uniaxially loaded (Sx), simply supported,quadratic plate having constant thickness andone regular, eccentric flat bar stiffener (tw, hw)in the middle (parallel to Sx). The stiffener iseither sniped or end-loaded (but with no rota-tional restraints at the ends). In the latter case,Sx is also applied to the stiffener area. The elas-tic material properties and modelling detailsare the same given below (Section 9). With in-creasing stiffener stiffness, the buckling stressincreases as the buckling mode changes froma fully global mode, through a mixed mode toan almost fully local mode at the “thresholdvalue” at which the rather flat plateau begins(at about hw/tw = 12-13).
The moment of inertia Ie (Eq. 15) is calcu-lated with an effective width be = 30t. In thelocal buckling range, the model results wouldhave increased slightly if the Ie-values instead
9
had been calculated about the middle plane ofthe plate.
The effect on results of end-loading on a con-tinuous stiffener is also seen to be rather smallin the considered case. This is particularly sofor local plate buckling cases, to the right of thethreshold value, where results for the snipedand the end-loaded stiffener (dashed line) co-incide almost exactly. This was to be expectedsince the stiffener-plate interface will remainnearly straight when local buckling governs.The external loads on a stiffener will then onlycontribute negligibly to the potential energy,T . As a consequence, it makes little differencefor the resulting buckling stresses acting onthe plate whether the stiffeners are sniped orend-loaded in local plate buckling cases.
In global buckling cases, the difference isseen to be somewhat greater, but not sig-nificantly so. In this case, the smaller globalbuckling stress in the end-loaded stiffenercase is almost completely compensated forby the greater load area. The resultant edgeloadings, Pcont. = Sx,cont.(bt + twhw) andPsniped = Sx,snipedbt are found to be almost thesame. For local plate buckling cases, which isthe target in most practical design situations,the difference in total loading is simply equalto the additional load carried by the stiffenerin the end loaded case.
8 Effect of torsional stiffness
The effect of including the torsional stiffness(J = hwt3w/3) of the stiffener can be seen inFig. 6 by comparing the results obtained bythe present model for the sniped stiffener casewith and without torsion included (full, thinline versus full, thick line). For global bucklingmodes, there is no effect since the stiffener doesnot rotate, due to symmetry. For local platebuckling, there is some, but very small benefi-cial effects. Local plate buckling results by thepresent model with torsional effects included,
are almost identical to comparable ANSYS re-sults, which are also shown in the figure (opendots). Such torsional effects are conservativelyneglected in the remainder of the paper.
9 Validation
The applicability of the presented model, in-corporated into a Fortran computer code, hasbeen assessed for many plate and stiffener di-mensions. Elastic buckling stress limits (ESL)are verified against ESL results obtained bythe finite element analysis computer programANSYS [17] using Shell93 elements. Bucklingstrength limits (BSL) are compared with ul-timate strength limits, here labelled USL, ob-tained from fully nonlinear ANSYS analyses(considering both geometric and material non-linearity).
Ultimate strength limits (USL) are the max-imum loads plates can carry without becom-ing unstable. For thin (slender) plates, they willgive information about the reserve strength, be-yond the BSL results. The BSL and the USL re-sults should ideally converge as the plate thick-ness increases
The specified imperfection is taken asw0,spec. = 5 mm for the purpose of makingcomparisons. In the fully nonlinear USL analy-ses, this value is used directly as the maximumimperfection amplitude (w0,max = w0,spec. =5 mm). In the BSL calculations, on the otherhand, the fictitious, slenderness dependentvalue w0,max according to Eq. 20 is used. Thecorresponding imperfection shapes are takenequal to the first eigenmode of the plate, ascomputed with ANSYS and the present model,respectively.
The elastic material properties used areE = 208000 MPa and ν = 0.3, and the yieldstrength is fY = 235 MPa. In the fully non-linear ANSYS analyses, a bilinear stress-strainrelationship is used that is defined by the prop-erties above, and additionally by a hardening
10
modulus ET = 1000 MPa. In the subsequentcomparisons, the moment of inertia of stiffen-ers is approximated by Ie about the middleplain (i.e., Eq. 15 with zc = 0) and the torsionalstiffness is neglected.
The displacement field in the present modelis defined with 15 terms in each direction(225 degrees of freedom). This generally pro-vides sufficient numerical accuracy. Resultingstrength predictions may be up to about 2%greater than those that would have been ob-tained with two to three times the number ofterms in each direction. In many cases, thenumber of terms could be significantly reducedwithout reducing the accuracy noticeable. Incomparison, in a typical USL analysis by AN-SYS, the number of degrees of freedom usedwas about 20000. Probably, sufficient accuracycould have been obtained with fewer degrees offreedom.
Results are presented in subsequent sections.They are limited to simply supported plateswith eccentric or symmetric sniped stiffeners(free and not loaded at the ends). To providesevere test cases for the stepped plates, the rela-tive difference in thickness between the thickestand thinnest strip in some of the plates is chosenrather large. Further, both regular and ratherstrongly irregular stiffener orientations are con-sidered. Additional comparisons with ANSYSresults are given by Brubak [14], where alsoclamped plates are considered.
10 ESL predictions
Typical elastic buckling stresses (eigenval-ues) are shown by the interaction curves in Fig.7, obtained for an unstiffened plate consisting oftwo very different plate strips. The present ESLmodel does not neglect any energy contributionin the case of unstiffened plates. Results willtherefore converge towards the exact solution asthe number of degrees of freedom increases. Theexcellent agreement with the ANSYS results
PSfrag replacements
ESL Ansys
ESL model
-1-1
1
1
-1.51.5
0
0
0.5
0.5
-0.5
-0.5
Sy
fY
Sx
fY
Figure 7. ESL interaction curves for a steppedplate consisting of two plate strips of unequalwidth and thickness (L/b1/t1 = 1000/1000/20mm, and L/b2/t2 = 1000/500/10 mm).
Table 1Dimensions [mm] of plates with stepwise constantplate thickness and two inclined, eccentric stiffen-ers.
L b t1 t2 t3 hw tw bf tf
Plate 1 1000 2400 25 20 15 200 15 150 25
Plate 2 1200 2400 25 20 15 200 15 150 25
Plate 3 1000 3000 20 18 16 200 16 100 16
shows that the assumed displacement field, de-fined over the entire plate, is clearly acceptablewith the present choice of degrees of freedom.
For stiffened plates, the agreement can notbe expected to be equally good due to the ne-glect of some energy contributions and approx-imations in initial, reference stress computa-tions (neglect of axial stiffener stiffness). How-ever, as will be seen in ESL results shown belowfor several stepped plates (Table 1 and 2), theagreement is still considered to be very goodfor biaxial stress combinations of interest (Fig.9 and 10).
11
b3
b3
b3
Stiffener Stiffener
L
x
y
−t1− −t2− −t3−
PSfrag replacements
ESL Ansys
ESL modelBSL modelUSL Ansys
von Mises-11
-1.51.5
00.5-0.5
2
(a) Plate geometry
PSfrag replacements
ESL Ansys
ESL model
BSL model
USL Ansys
von Mises
-1
-1
1
1-1.5
-1.5
1.5
1.5
0
0
0.5
0.5
-0.5
-0.5
2
(b) Plate 1
Sy
fY
Sx
fY
PSfrag replacements
ESL Ansys
ESL modelBSL modelUSL Ansys
von Mises-11
-1.51.5
00.5-0.5
2
ESL Ansys
ESL model
BSL model
USL Ansys
von Mises
-1
-1
1
1-1.5
-1.5
1.5
1.5
0
0
0.5
0.5
-0.5
-0.5
2
(c) Plate 2
PSfrag replacements
ESL Ansys
ESL modelBSL modelUSL Ansys
von Mises-11
-1.51.5
00.5-0.5
2
ESL Ansys
ESL model
BSL model
USL Ansys
von Mises
-1
-1
1
1-1.5
-1.5
1.5
1.5
0
0
0.5
0.5
-0.5
-0.5
2
(d) Plate 3
Sy
fY
Sy
fY
Sx
fY
Sx
fY
Figure 9. Interaction curves in the stress space Sx-Sy for plates no. 1, 2 and 3, with stepwise constantthickness and with two eccentric, inclined stiffeners.
11 BSL predictions – Inclined eccentricstiffeners
A number of plates with three strips and twoinclined, eccentric, sniped T-section stiffeners(Fig. 2b) have been analysed. Selected resultsfor plates defined in Table 1 and Fig. 9a will bepresented and discussed.
The first buckling mode calculated by thepresent model and by the finite element model
has been found to be quite similar in eachcase considered. A typical case is shown inFig. 8a and b. The displacements are largest inthe thinnest plate strip. The mode shown canbe considered as a local plate buckling mode,as the out-of-plane plate displacements alongthe stiffeners are small. This is a consequenceof the chosen stiffeners, which apparently aresufficiently stiff to prevent a global bucklingmode. The same is found to be the case for the
12
b1, t1 b2, t2 b3, t3
y
x
(a)
Stiffeners
y
x
(b)
Figure 8. First buckling mode of plate 1 subjectedto a uniaxial external stress Sy, calculated by (a)the present model and (b) Ansys.
other plates in Table 1. The main differencebetween the models is that the finite elementmodel accounts for the sideways deflections ofthe stiffeners and of their torsional and axialstiffness.
In Fig. 9, biaxial load interaction curvesare given for buckling strengths (BSL), elas-tic buckling stresses (ESL), which have beendiscussed above, and the “yield limit” accord-ing to the von Mises’ yield criterion for thematerial as such. The latter represents a maxi-mum strength limit. In the context of bucklingstrength, the most relevant loading situationsare those with uniaxial or biaxial compressivestresses (in the first quadrant).
Also shown in the figure are ultimatestrength (USL) results. These are seen to beclose to the BSL results for the consideredplates, which have relatively small to inter-mediate slenderness values. The slenderness λvaries along the interaction curves, but it issmaller than 1.05 for all load combinations.The corresponding relative imperfection am-plitude w0,max/w0,spec. varies between 0.90 and1. Thus, the reduction of the imperfection am-
Table 2Dimensions [mm] of plates with stepwise constantplate thickness and two symmetric stiffeners withregular orientation.
L b t1 t2 t3 hw tw bf tf
Plate 4 3000 3300 30 28 26 300 10 150 20
Plate 5 3000 2100 24 22 20 300 10 150 20
Plate 6 3000 2100 16 14 12 300 8 100 10
plitude in the BSL procedure is small in thesecases. More slender plates, for which BSL re-sults will become more conservative relative toUSL results, are discussed below.
12 BSL predictions – Regular, symmet-ric stiffeners
For uni-directionally stepped plates, it is be-lieved that regular stiffeners represent a morecommon stiffener arrangement than inclinedstiffeners. Similar results to those presentedabove have been obtained for a number of suchplates, of which preliminary results were givenin [18]. Selected results, for the plates withthree strips and two regular, symmetric stiffen-ers defined in Table 2 and Fig. 10a, are shownin Fig. 10b, c and d. Also, for the plates consid-ered here, closer examinations show that thefirst buckling modes are local plate bucklingmodes.
The BSL and USL results are close to eachother for plate 4 and 5, which have relativelysmall to intermediate reduced slenderness val-ues. For instance, for plate 5, the maximum re-duced slenderness is approximately λ = 1.17(for Sx = 1.67Sy). The corresponding imperfec-tion amplitude is w0,max = 0.84w0,spec. accord-ing to Eq. 20.
For plate 6, the maximum reduced slender-ness is approximately 2.36 (for Sx = 1.43Sy),which is representative of a very slender case.The corresponding imperfection amplitude isw0,max = 0.10w0,spec., which is physically speak-ing, unreasonably small. Even so, the BSL re-sults calculated by the present method are still
13
PSfrag replacements
ESL Ansys
ESL modelBSL modelUSL Ansys
von Mises-11
-1.51.5
00.5
-0.52
(a) Plate geometry
PSfrag replacements
ESL Ansys
ESL model
BSL model
USL Ansys
von Mises
-1
-1
1
1-1.5
-1.5
1.5
1.5
0
0
0.5
0.5
-0.5
-0.5
2
(b) Plate 4
b3
b3
b3
L3
L3
L3
x
y
Stiffeners
−t1− −t2− −t3−
Sy
fY
Sx
fY
PSfrag replacements
ESL Ansys
ESL modelBSL modelUSL Ansys
von Mises-11
-1.51.5
00.5-0.5
2
ESL Ansys
ESL model
BSL model
USL Ansys
von Mises
-1
-1
1
1-1.5
-1.5
1.5
1.5
0
0
0.5
0.5
-0.5
-0.5
2
(c) Plate 5
PSfrag replacements
ESL Ansys
ESL modelBSL modelUSL Ansys
von Mises-11
-1.51.5
00.5-0.5
2
ESL Ansys
ESL model
BSL model
USL Ansys
von Mises
-1
-1
1
1-1.5
-1.5
1.5
1.5
0
0
0.5
0.5
-0.5
-0.5
2
(d) Plate 6
Sy
fY
Sy
fY
Sx
fY
Sx
fY
Figure 10. Interaction curves in the stress space Sx-Sy for plates no. 4, 5 and 6, with stepwise constantthickness and with two symmetric, regular stiffeners.
conservative compared to the fully nonlinearUSL results. The latter is in this case twice aslarge as the BSL results.
The reason for the conservativeness in BSLresults for slender plates is, as mentioned be-fore, that the present BSL model is not ableto capture the post-critical (reserve) strength.Therefore, the present BSL model is most fea-sible in practical design cases in which loadingin the post-buckling range is not accepted.
13 Concluding remarks
An approximate method for global and localbuckling strength (BSL) analysis of steppedplates with arbitrary stiffener orientations hasbeen presented. Out-of plane displacementsare estimated using an approximate displace-ment magnifier and plate failure is estimatedusing von Mises’ yield criterion. The assumeddisplacement field implies that equilibrium is
14
only satisfied on the average between the platestrips.
The applicability and versatility of the pre-sented model are documented for a variousplate geometries and computed BSL resultsare validated against fully nonlinear USL anal-yses. The BSL predictions are generally con-servative, in particularly for slender plates.This is due to the use of the approximatedisplacement magnifier, which results in buck-ling strengths that never exceed the elasticbuckling loads. Consequently, the postbuck-ling (reserve) strengths for slender plates arenot accounted for. Although conservative, thismay be a sound theoretical treatment in designsituation in which structural elements are notaccepted to buckle elastically.
The method is computationally very effi-cient. Using a computer code that has not yetbeen optimised, the computer time for a BSLprediction for a given loading is typically 1-2seconds on a medium fast computer (1.5 GHzprocessor, 512 MB RAM). Compared to non-linear ultimate strength (USL) analyses by AN-SYS, the presented method is typically morethan 1000 times faster for the same problem.Speed is always an advantage, in particular inoptimisation, reliability and other situationsrequiring larger numbers of case studies. Thesize of the computer code of the method islimited and the number of input data requiredis minimal. Due to such factors, the methodis suitable for incorporation into computerisedanalysis and design codes. A constant thick-ness version is already included in a code la-belled PULS [9] and can be downloaded fromwww.dnv.com.
Acknowledgements
The authors would like to thank dr.scient.Eivind Steen and dr.ing. Eirik Byklum, both atDNV, for initially suggesting the topic of thepaper, for their interest and discussions, and
additionally for making a computer subroutinefor stress calculations available for use in thisstudy.
References
[1] prEN 1993-1-5, Eurocode 3: Design of steelstructures. Part 1.5: Plated structural el-ements, CEN, European Committee forStandardisation, Brussels, 2005
[2] Det Norske Veritas, DNV Rules for classifi-cation of ships, Det Norske Veritas, Høvik,Norway, 2002
[3] Det Norske Veritas, Recommended prac-tice DNV-RP-C201, Buckling strength ofplated structures, Høvik, Norway, 2002
[4] E. Byklum and J. Amdahl, A simplifiedmethod for elastic large deflection analysisof plates and stiffened panels due to localbuckling, Thin-Walled Structures, 2000;40(11): 925–953
[5] E. Byklum and E. Steen and J. Amdahl, Asemi-analytical model for global bucklingand postbuckling analysis of stiffened pan-els, Thin-Walled Structures, 2004; 42(5):701–717
[6] E. Steen, Application of the perturbationmethod to plate buckling problems, Re-search Report in Mechanics, No. 98-1, Me-chanics Division, Dept. of Mathematics,University of Oslo, Norway, 1998, 60 pp.
[7] E. Steen, Buckling of stiffened plates us-ing a Shanley model approach, ResearchReport in Mechanics, No. 99-1, Mechan-ics Division, Dept. of Mathematics, Uni-versity of Oslo, Norway, 1998, 84 pp.
[8] L. Brubak and J. Hellesland and E. Steen,Semi-analytical buckling strength analysisof plates with arbitrary stiffener arrange-ments, Journal of Constructional Steel Re-search, 2007; 63(4): 532– 543
[9] E. Steen and E. Byklum and K.G. Vilmingand T.K. Østvold, Computerized bucklingmodels for ultimate strength assessments
15
of stiffened ship hull panels, Proceedingsof The Ninth International Symposium onPractical Design of Ships and other Float-ing Structures, Lubeck-Travemunde, Ger-many, Sept. 12-17, 2004; 235–242
[10] J.K. Paik and M.S. Lee, A Semi-analyticalmethod for the elastic-plastic large deflec-tion analysis of stiffened panels under com-bined biaxial compression/tension, biax-ial in-plate bending, edge shear, and lat-eral pressure loads, Thin-Walled Struc-tures, 2005; 43(3): 375–410
[11] M. Azhari and A.R. Shahidi and M.M.Saadatpour, Local and post local buck-ling of stepped and perforated thin plates,Applied Mathematical Modelling, 2005;29(7): 633–652
[12] Y. Xiang and G. W. Wei, Exact solu-tions for buckling and vibration of steppedrectangular Mindlin plates, InternationalJournal of Solids and Structures, 2004;41(1): 279–294
[13] S.P. Timoshenko and J.M. Gere, Theory ofelastic stability, McGraw-Hill Book Com-pany, second edition, 1963
[14] L. Brubak, Semi-analytical bucklingstrength analysis of plates with con-stant or varying thickness and arbitrarilyoriented stiffeners, Research Report inMechanics, No. 05-6, Mechanics Division,Dept. of Mathematics, University of Oslo,Norway, 2005, 65 pp.
[15] K. Marguerre, Zur theorie dergekrummten platte grosser formanderung,Proceedings of The 5th InternationalCongress for Applied Mechanics, 1938;93–101
[16] S. Levy, Bending of rectangular plates withlarge deflections, Report 737, NACA, 1942
[17] ANSYS Inc., ANSYS Documentation 9.0,Southpointe, Canonsburg, PA, 2004.
[18] L. Brubak and J. Hellesland, Computa-tional buckling strength analysis of stiff-ened plates with varying thickness, Pro-ceedings of The First International Confer-
ence on Computational Methods in MarineEngineering (MARINE 2005), Det NorskeVeritas, Høvik, Norway, 2005, 355–364
A Appendix
A.1 Coefficients in Airy’s stress function
The coefficients fij in Airy’s stress functionare defined by
fij =E
4(i2 bL
+ j2 Lb)2
M∑
r=1
N∑
s=1
M∑
p=1
N∑
q=1
crspq(arsapq
+ arsbpq + apqbrs)
(A.1)
where f00 is zero, ars and bpq are the amplitudesof w and w0, respectively, and crspq are integersgiven by
crspq = rspq + r2q2 (A.2)
if ±(r − p) = i and s + q = j, or r + p = i and±(s − q) = j, or
crspq = rspq − r2q2 (A.3)
if r + p = i and s + q = j, or ±(r − p) = i and±(s − q) = j, or
crspq = 0 (A.4)
for other cases. More details of the derivation ofthe coefficients fij can be found in the literature[4].
A.2 Definition of Gjl and Hjl
Gjl =∫ ys2
ys1
sin(πj
by)sin(
πl
by) dy (A.5)
and
Hjl =∫ ys1
ys1
cos(πj
by)cos(
πl
by) dy (A.6)
The results of these integrals are
16
Gjl =b
2π(j2 − l2)
(
(j + l)sin(π(j − l)
by)−
(j − l)sin(π(j + l)
by)
)
∣
∣
∣
∣
∣
∣
ys2
ys1
(A.7)
if j 6= l, and
Gjl =
(
y
2− b
2πjcos(
πj
by)sin(
πj
by)
)
∣
∣
∣
∣
∣
∣
ys2
ys1
(A.8)
if j = l, and
Hjl =b
2π(j2 − l2)
(
(j + l)sin(π(j − l)
by)+
(j − l)sin(π(j + l)
by)
)
∣
∣
∣
∣
∣
∣
ys2
ys1
(A.9)
if j 6= l, and
Hjl =
(
b
2πjcos(
πj
by)sin(
πj
by) +
y
2
)
∣
∣
∣
∣
∣
∣
ys2
ys1
(A.10)if j = l.
17