Strength of Compressed Rectangular Plates Subjected to Lateral Pressure

Journal of Constructional Steel Research 57 (2001) 491–516www.elsevier.com/locate/jcsr

Strength of compressed rectangular platessubjected to lateral pressure

A.P. Teixeira, C. Guedes Soares*

Unit of Marine Technology and Engineering, Technical University of Lisbon, Instituto SuperiorTecnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Received 7 August 2000; accepted 23 November 2000

Abstract

This paper presents the results of a parametric study to quantify the effect of lateral pressureon the collapse of square and rectangular steel plates under a predominantly compressive load.The load-shortening behaviour of square and rectangular plates under the combined effect oflongitudinal compression and lateral pressure were obtained using a general-purpose non-linearfinite element code for different breadth to thickness ratios. Finally design curves are proposedto predict the collapse strength of the compressed plates under lateral pressure. 2001 ElsevierScience Ltd. All rights reserved.

Keywords:Plate collapse; Lateral pressure; In-plane compression; Design curves

1. Introduction

The behaviour of plate elements under compressive loads has been studied formany years. Major developments occurred during the early 1970s with the develop-ment of numerical procedures based on finite differences and on finite-elements. Itthen became possible to study realistic cases of elasto-plastic collapse of plates withlarge deflections. Several parametric studies have been performed to describe theeffect of different parameters on the collapse strength, including the initial distortionsand residual stresses [5,11,16,17,8,21,14]. Most of the studies have dealt with uniax-ial loads, but some have considered the collapse resistance under biaxial loads asreviewed in Ref. [12].

* Corresponding author.

0143-974X/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved.PII: S0143 -974X(00)00033-X

492 A.P. Teixeira, C. Guedes Soares / Journal of Constructional Steel Research 57 (2001) 491–516

Although the main load component for the ship deck and bottom structures is theaxial and biaxial compression, the external bottom plating and the lower parts of theside shells can in addition be subjected to relatively high external lateral pressureand the inner bottom and inner longitudinal bulkheads to lateral pressure loads fromthe cargo.

In general, lateral pressure on panels causes out-of-plane displacements in a modethat is one half wave in both directions. These out-of-plane deflections will decreasethe plate strength whenever they coincide with the main buckling mode. However,when this is not the case the presence of lateral pressure can in fact increase theultimate strength of the plate. For example for long plates, which have a primarybuckling modem.1, the lateral load will force the lower mode (m=1) leading to anincrease of the plate strength. Although there are some circumstances that have theeffect of raising the buckling stress, they should be ignored when calculating theultimate strength of plates since they are not a permanent and reliable feature ofthe structure.

When dealing with this phenomenon one must distinguish between the two differ-ent cases. Under moderate in-plane compression, such that the compressive stress iswell below any value that would of itself cause any type of collapse, the role of thein-plane compression only magnifies the deflections and stresses caused by the lateralload. If the in-plane compression is large, then the analysis is concerned with thequestion of the actual buckling and/or collapse of the plate. In this case it is importantto determine the most critical or primary buckling mode, that is, the shape of bucklingdeflection that corresponds to the lowest value of critical stress.

There are only a few publications dealing with this complicated problem, althoughone is able to find some analytical, numerical and experimental results in the litera-ture. Steen and Valsgard [19] presented a design method for plates subjected tobiaxial compression and lateral pressure, which was based on deriving simplifiednon-linear elastic response curves for the in-plane and laterally loaded cases andcombining the local stresses obtained into an equivalent stress criterion. The elasticbuckling and the initial postbuckling behaviour of plates was described by the pertur-bation theory of Budiansky [4], which is based on the non-linear Von Karman equa-tions and includes the effect of geometrical imperfections. The non-linear elasticbehaviour of the plate under lateral load was then addressed. The stresses correspond-ing to both types of behaviour are assessed and combined in a Von Mises equivalentstress, which is used as a criterion for the initial yield and the ultimate collapse load.Interaction curves are provided for square plates of different slenderness and degreeof initial imperfection.

Dier and Dowling [7] conducted an extensive numerical study dealing with plateswith an aspect ratio of 3 and with square plates with simply supported and fullyclamped boundary conditions subjected to biaxial compression. The plates hadb/tratios of 40, 60 and 80 and different levels of initial imperfections and residualstresses. The results were obtained for different levels of lateral pressure and theywere shown in the form of interaction curves.

Experimental results are also scarce. Becker and Colao [2] conducted some testson the square tubes subjected to transverse load and internal pressure. Further experi-

493A.P. Teixeira, C. Guedes Soares / Journal of Constructional Steel Research 57 (2001) 491–516

mental results were presented by Yoshiki et al. [25], Yamamoto et al. [24] andOkada et al. [18], although all of them are of plates under an uniaxial load andlateral pressure.

Davidson et al. [6] formulated a design model for plate panels subjected to thecombined effect of biaxial compression and lateral pressure based on a simple largedeflection elastic analysis. Interaction curves for resistance at an average panel strainequal to yield strain and for maximum resistance were derived assuming that thecompressive strength of plates reduces linearly with increasing levels of lateral press-ure.

Recently, Guedes Soares and Gordo [12] have compared different proposals tomodel the collapse of plates under biaxial compression and also lateral pressure withexperimental results and with predictions of numerical codes. Strength assessmentformulas were derived based on these results that were mainly of plates subjectedto biaxial compression.

With the frequent longitudinal stiffening in ship structures, it has became importantto model situations of a structure such as a bottom under the effect of lateral pressure,which is then subjected to a longitudinal compressive load that results from thelongitudinal bending of the hull girder. In this situation, the collapse only occurs inthe longitudinal direction and the large out-of-plane deformations developed due tothe combined effect of lateral pressure and longitudinal compression will inducetensile forces in the transverse edge of the plates elements. This special state ofstress at collapse is not well represented by the usual design equations and interactionformulas used to predict the collapse strength of the plates.

The purpose of this paper is therefore to study the effect of lateral pressure onthe ultimate compressive strength of unstiffened square and rectangular plates withtransverse edges restrained to in-plane displacements subjected to longitudinal com-pression. Calculations are presented concerning the effect of different levels of lateralpressure on the strength of square and rectangular plates of different slendernesseswith initial imperfections. Interaction curves are also derived for plates that areinitially subjected to lateral pressure and later to in-plane longitudinal compression.

2. Effect of lateral pressure on the ultimate compressive strength of plates

Different proposals have been put forward to model the collapse of plates thatresult from the interaction between longitudinal and transverse stresses in a mannerthat is suitable for design in general and for code specifications in particular [10].The approach that has been generally adopted consists of predicting the longitudinaland the transverse stresses in a plate as a function of plate slenderness and of otherparameters like the aspect ratio and eventual initial defects. The equivalent stress inthe plate is determined from a combination of the stress components. Thus, interac-tion curves have been proposed to combine the non-dimensional longitudinal stressratio Rx=sx/sxu with the non-dimensional transverse stress ratioRy=sy/syu where thesubscript u indicates ultimate strength.

It should be noticed that different formulations ofsxu andsyu have been advanced


by the authors that have proposed the interaction curves. Thus each of the stressratiosRx andRy indicated hereafter should be referred to the original author’s formu-lations of longitudinal and of transverse strength, that is,Rx=Tx/fxu and Ry=Ty/fyu,whereTx=sx/so andTy=sy/so.

One expression that has been used in the Det norske Veritas (DnV) rules, Amer-ican Bureau of Shipping (ABS) rules and in the British Standard BS.5400 is thequadratic interaction:

R2x1R2

y51 (1)

which represents a circle in theRx2Ry plane. DnV proposed as normalising strengthsFaulkner’s formula [9] in the longitudinal direction and Valsgard’s formula [22] inthe transverse direction, which are respectively:

fxu52b

21b2 for b$1 andfxu51 for b#1 (2)

and:

fyu5fxu

a10.08S11

1b2D2S12

1aD (3)

wherefxu is given by Eq. (2),a is the plate aspect ratio andb its slenderness.The rules of the ABS use the same interaction formula but with different normalis-

ing strengths. They prefer to use a formulation based on the Bryan elastic bucklingstress combined with the Johnson–Ostenfeld approach to account for the effect ofplastic deformation. The buckling strengthscr of a plate is equal to the elastic buck-ling strengthse:

se

so

5p2

12(1−n2)Kb2 for se#0.5so (4)

when buckling occurs in the elastic range i.e. whense#0.5so. The Poisson’s ration is 0.3 for steel plates and the buckling coefficientK accounts for the type of loadingand of boundary conditions. For a wide plate with linearly varying transverse loadingit is given by Bleich [3]:

K5S111a2D2 2.1y+1.1

0#y#1 (5)

where the factor is such that when the stresses on one transverse edge of the plateare s on the other one they areys. Thus for plates under uniform compressivestressesy=1. For longitudinal loading with uniform applied stresses (y=1) Kbecomes equal to 4. It was shown by Guedes Soares and Gordo [13] that this formu-lation leads to unconservative designs, that is, the collapse stress is overestimated.

When the predicted strength is greater than half the yield stress, the collapsestrength is given by:

scr

so512

so

4sefor se.0.5so (6)


which implies an elasto-plastic collapse.Other authors have proposed parabolic curves, for example [10]:

Rx1R2y51. (7)

In this case the ultimate transverse strength should be computed by the expressionof the same author [10]:

fyu50.9b2 1

1.9baS12

0.9b2D (8)

Valsgard [23] generalised this expression by including cross terms and making theexponent ofRx a variableg:

Rgx2hRxRy1R2y#1 (9)

where h$0 is a constant. The proposed normalising equations are (2) and (3),respectively, for the longitudinal and transverse directions. On the basis of hisnumerical results on a plate with aspect ratio of 3, Valsgard proposed the followingdesign curve:

Rx20.25RxRy1R2y#1 (10)

which corresponds in fact tog=1 andh=0.25.Dier and Dowling [7] have considered a more comprehensive treatment, which

would also be applicable to cases in which one of the load components was tensile.This implies that they are considering the interaction curve not only in the firstquadrant of theRx2Ry plane (biaxial compression), but also in the others (biaxialtension). They proposed:

R2x10.45RxRy1R2

y51 (11)

which includes a positive contribution of the cross terms.In view of all the uncertainty of the results and the different interaction curves

available, Stonor et al. [20] proposed a lower bound curve to the existing data, whichturned out to be:

R1.5x 1R1.5

y 51 (12)

Very stocky plates should behave according to the von Mises equation, which wasgeneralised in terms of the ultimate stress in each direction instead of the yield stress:

R2x2RxRy1R2

y51 (13)

The choice of the adequated normalising equations,fux and fuy, should make Eq.(13) an upper bound curve.

Guedes Soares and Gordo [12] have compared the interaction curves just describedwith experimental results and with predictions of numerical codes, suggesting thatthe Von Mises curve [Eq. (13)] should be used for stocky plates (b<1) and thecircular interaction curve [Eq. (1)] for the other plates.

The effect of lateral pressure on plate collapse strength is usually accounted for


by including an additional termRQ to the interaction equation used for biaxial load.Different options have been considered and the one finally adopted has the form:

R2RQ50 (14)

whereR is obtained by the quadratic interaction [Eq. (1)].The effect of lateral pressure has been modelled by a regression equation, depen-

dent on the plate slendernessb and on a non-dimensional lateral pressure parameterQL. Different types of relations were tested and the one that showed best results wasof the form:

RQ5A2BQLb2 (15)

Dier and Dowling [7] have proposed that the non-dimensional lateral pressureparameter could be represented by:

QL5qoEs2

o(16)

where qo is the intensity of the lateral pressure. On the other hand, the Japaneseauthors [18,24,25] use an alternative formulation:

QLJ5qob4

Et45QLb4 (17)

which includes also some information on plate geometry. The first formulation wasadopted here because it is independent of the plate geometry.

Based on the regression study on data of various sources of ultimate strength ofplates subjected mainly to biaxial loads and lateral pressure, Guedes Soares andGordo [12] proposed the following equation

RQ51.020.116QLb2 (18)

3. Numerical results

The numerical calculations were performed usingasasnl software [1]. This is ageneral-purpose non-linear finite element code in which large displacement effectsare handled using an updated Lagrangian formulation with inclusion of geometricstiffness terms for plate elements. The calculation of the element stiffness can beeither elastic or elasto-plastic, depending whether plasticity defined by the Von Misesyield criteria has occurred at an integration point.

The calculations were conducted for several simply supported plates with aspectratio of 1 and 3, and slenderness (b/t) from 40 to 100. The lateral pressure load isapplied on the plate surface, keeping the boundary conditions of the plate restrained


to displacements in the plane. The stress level will rise but the collapse of the plateonly occurs by imposing a longitudinal displacement to the edge of the plate (Fig. 1).

An average level of initial geometric imperfections was considered in thepresent study

wmax

t50.10b2 (19)

whereb is the plate slenderness given by:

b5bt!so

E(20)

whereso and E are the yield stress and Young’s modulus, respectively.The shape of the initial imperfections is represented by a Fourier series:

w5Om

On

dmnsinmpx

asin

npyb

(21)

wherea andb are the plate dimensions andδmn is the amplitude of the components.In each calculation the initial distortion of the plate was represented by a shape

with only one component of this series. However, each type of plate was consideredtwice with a different initial distortion described by the order (m, n) of the Fouriercomponent of the initial distortions in order to quantify the sensitivity to this para-meter. Thus all plates were run with the pair (m=1; n=1) and some of them with thepair (m=a/b; n=1).

Three levels of pressure were considered:Po=0.1 MPa,Po=0.2 MPa andPo=0.4MPa and it was assumed that lateral pressure was applied first and remained constantduring the subsequent application of the compressive load.

Table 1 describes the finite element model used to derive the strength curves forplates subjected to lateral pressure.

4. Strength curves for plates subjected to lateral pressure

When lateral pressure and in-plane compression are applied together, each of theseloads can alter the effect, which the other would have if it were acting alone. If only

Fig. 1. Imposed displacements and boundary conditions for the plate model.


Table 1Geometric modelling

a (mm) b (mm) Mode of t (mm) b/t Mild steelimperfection

b wmax (mm)

1000 1000 m=1 25.0 40 1.34 4.5n=1 16.7 60 2.01 6.7

3000 m=1 and 3 12.5 80 2.68 9.0n=1 10.0 100 3.35 11.2

lateral pressure is applied, it will induce lateral deflection and consequently tensileforces on its restrained boundaries. Fig. 2 illustrates the curves of normalised stressas function of the level of lateral pressure obtained for this case. The figure clearlyshows that the plate can be loaded beyond its elastic limit before it fails in anysignificant way, or before the deformation becomes unacceptably large.

Considering now that in-plane longitudinal compressive loads are applied to aplate initially subjected to lateral pressure two different cases can occur. While mod-erate in-plane compression merely magnifies the deflections and the stresses causedby the lateral pressure, large in-plane compression assumes a primary importanceand cannot be regarded as simply a magnifying effect. In this case the ultimatecompressive strength of plates subjected to lateral pressure loads should be evaluated.

The procedure used to account for lateral pressure consists of initially applyingthe lateral pressure load to the plate and then imposing a longitudinal compressionin its plane so as to produce the load deflection curves. This loading sequence aimsat modelling a situation of a structure, such as a bottom structure of a ship, which

Fig. 2. Axial stress–lateral pressure curves for different plate slenderness.


is subjected to a longitudinal compressive load that results from longitudinal bendingof the hull girder.

In some situations a different load sequence can occur. For example, when plateelements subjected to an operational loading condition are then submitted to lateralpressure load that result from cargo or from an accidental situation. Fig. 3 comparesthe behaviour of a plate initially loaded with lateral pressure and then in-plane com-pression (PD) and four other cases in which the lateral pressure loading is appliedat four different levels of in-plane compression (curves DPD1 to DPD4). The figureclearly shows that after the lateral load has been applied the dotted line falls to thefull line and from that moment the same behaviour is obtained for all the cases.

4.1. Square plates

Before starting this study the effect of different boundary conditions on the resultswas analysed by considering a simply supported square plate subjected to longitudi-nal compression with the longitudinal edges restrained and then unrestrained againstany transverse contractions.

It is known that the effect of the boundary conditions on the collapse of strengthof plates cannot be considered of the same type for the whole range of variation ofparameters. It depends on plate slenderness, on the shape of initial imperfections aswell as on the boundary conditions as shown by Guedes Soares and Kmiecik [15].

This effect is illustrated in Fig. 4 and shows the longitudinal stress–displacementcurves of different plate slendernesses considering the longitudinal edges restrainedand unrestrained against transverse contraction. It is clear that the loss of longitudinalstrength for the unrestrained case increases for increasing plate slenderness.

However, restraining the longitudinal plate edge when applying the longitudinalcompression will induce stress levels on the transverse direction and therefore the

Fig. 3. Behaviour of square plate (b/t=60) under compression with lateral pressure of 0.2 MPa appliedat different levels of in-plane compression.


Fig. 4. Longitudinal normalised stress–displacement curves for restrained (—) and unrestrained (- - -)longitudinal plate edge.

biaxial state of stress must be analysed as illustrated in Fig. 5. It is interesting tosee from Fig. 5 that the stress developed in the transverse direction is negative whenb/t.60, which means that the out-of-plane deformations of slender plates inducestension on its transversal edge.

Considering now the plates subjected to lateral pressure, one can see in Fig. 6that the ultimate capacity of the plate ofb/t=40 is almost unaffected by changingits boundary condition even for increasing levels of lateral pressure. However, forslender plates the strength reduction when passing from the restrained to the unre-strained case is magnified by the level of lateral pressure.

Fig. 5. Longitudinal (—) and transversal (- - -) normalised stress–displacement curves for restrainedlongitudinal plate edge.


Fig. 6. Ultimate longitudinal strength of restrained (—) and unrestrained (- - -) square plates subjectedto lateral pressure.

Figs. 7 and 8 show the plate behaviour for a lateral pressure of 0.1 and 0.2 MPa(i.e. 10 and 20 m water depth), respectively. It can be observed that, initially theplates are in tension due to the effect of the lateral pressure but the increasing com-pression, tend to create a compression stress state in the plates. This effect is muchmore clear when high levels of lateral pressure are applied on slender plates as shownin Fig. 10.

For slender plates loaded with high levels of lateral pressure the compressive loadis compensated by the large out-of-plane deformations due to the pressure loading,which decrease the average compressive stress. In spite of these low levels ofstresses, square plates of high slenderness show very large out-of-plane deformationsat the collapse when the compression is associated with lateral pressure.

Fig. 7. Behaviour of square plates under compression with applied lateral pressure of 0.1 MPa.


Fig. 8. Behaviour of square plates under compression with applied lateral pressure of 0.2 MPa.

Figs. 9 and 10 show the effect of different levels of lateral pressure on the behav-iour of square plates with a breadth to thickness ratio of 40 and 80, respectively. Itcan be seen that for typical levels of lateral pressure that are between 10 and 20 mof water depth, the ultimate strength decreases in about 9 and 14% for a plate ofb/t=40 and 14 and 24% for a plate ofb/t=80.

One may conclude that the presence of lateral pressure do not change significantlythe form of the average stress–displacement curve for the square plate and, thus, thecurve with lateral pressure may be estimated from the ones without pressure byintroducing a correction factor calculated as a function of the level of pressure andalso of the plate slenderness.

Fig. 9. Behaviour of square plates under longitudinal compression with applied lateral pressure. Effectof increasing lateral pressure on plates ofb/t=40.


Fig. 10. Behaviour of square plates under longitudinal compression with applied lateral pressure. Effectof increasing lateral pressure on plates ofb/t=80.

4.2. Rectangular plates

The presence of lateral pressure will induce a plate deflection that corresponds tomodem=1 andn=1. For square plates this coincides with the primary buckling mode,but rectangular plates have a primary buckling modem.1 and then the lateral press-ure will increase its collapse strength.

The effect of different modes of initial distortions on the collapse strength ofrectangular plates subjected to lateral pressure is illustrated in Fig. 11. One mainconclusion may be inferred from the analysis of the figure. The lateral pressuredecreases the plate collapse strength but its effect is different depending on the shape

Fig. 11. Ultimate strength of rectangular plates with different mode of initial imperfections,m=1, n=1(—) and m=3, n=1 (- - -).


of the initial distortions. In fact the lateral pressure load increases the collapsestrength of the primary buckling mode (m=3, n=1) to values close to the onesobtained considering a shape of initial distortions ofm=1 andn=1 for the slenderplates and also when the high levels of lateral pressure are applied on plates ofb/t=40. Therefore, since a lower limit of the plate strength is obtained by consideringa shape of initial distortions ofm=3 andn=1, this was assumed for further calcu-lations presented in this paper.

Fig. 12 shows the longitudinal average stress–displacement curves for 0.2 MPaof lateral pressure. A reduction may be noticed in the maximum stress and the curvestend to be smoother when the plate slenderness increases. This behaviour also occursfor increasing levels of lateral pressure as shown from Fig. 13 that illustrates theeffect of different levels of lateral pressure on the behaviour of rectangular plateswith a breadth to thickness ratio of 60.

For high compression levels and high slenderness the lateral pressure is mainlysupported by tension stresses, that is, the very high deformations of the plate dueto the applied pressure compensates compression stresses induced by longitudinalcompression of the plate edge.

Fig. 14 compares the stress levels in the longitudinal and transverse directions fortwo rectangular plates of different slenderness subjected to a lateral pressure of 0.2MPa. It can be seen that the collapse is achieved in the transverse direction by theapplication of the lateral pressure load and the load carrying capacity in this directionremains almost constant during the compression stage. This means, once more, thatthe expansion in the transverse direction is converted in out-of-plane deformationshelped by the work done by the lateral pressure.

Fig. 12. Behaviour of rectangular plates (a/b=3) plates under compression with applied lateral pressureof 0.2 MPa.


Fig. 13. Longitudinal stress–displacement curves of rectangular plates (a/b=3). Effect of increasing lat-eral pressure on plates ofb/t=60.

Fig. 14. Longitudinal (—) and transversal (- - -) normalised stress–displacement curves of rectangularplates (a/b=3) with lateral pressure of 0.2 MPa.

5. Design curves for plate collapse

For design purposes including code specifications, several semi-empirical formulaehave been proposed to predict the collapse strength of the plate elements subjectedto predominantly compressive in-plane loads as already reviewed in this paper.

Since the longitudinal plate edge was considered to be restrained to transverse in-plane displacements, the applications of longitudinal compression will induce stresslevels on the transverse direction and therefore the biaxial state of stress must be


analysed. Fig. 15 shows the interaction ratios obtained for square plates by the circu-lar [Eq. (1)], Von Mises [Eq. (13)] and the interaction curve proposed by Dier andDowling [Eq. (11)] using the ultimate longitudinal and transversal strength given byFalkner’s [Eq. (2)]) and Valsgard’s [Eq. (3)] expressions, respectively.

As would be expected the interaction curve proposed by Dier and Dowling thatis applicable to the cases in which one of the components of the load was tensile,gives better predictions. It can also be seen that the interaction formulas are conserva-tive for slender plates (b/t$80). This is probably due to the stress levels developedin the transverse direction at the longitudinal collapse. For stocky plates when thelongitudinal collapse occurs the transverse collapse has already occurred but for slen-der plates the longitudinal collapse will induce tensile stress in the transverse direc-tion due to large out-of-plane deformations and therefore the usual theory of platecollapse for transverse direction cannot be used, instead the membrane theory shouldbe applied. This behaviour is also present and even aggravated when lateral pressureis initially applied to the plates as illustrated in Fig. 14.

Fig. 16 compares the interaction curves shown in Fig. 15 with the normalisedlongitudinal strength of the plates. It can be concluded that taking only the longitudi-nal strength of the plates constitutes a good design equation and therefore it will beused to predict the ultimate strength of plates subjected to lateral pressure.

Fig. 17 illustrates the longitudinal ultimate strength of plates subjected to lateralpressure normalised by the proposal of Faulkner. It is clear that the degradation ofstrength can be associated with the level of lateral pressure. Furthermore the degra-dation is almost identical for all range of slenderness of the plate. This suggests thatfor design purposes a linear dependence of only the non-dimensional pressure para-meterQL proposed by Dier and Dowling can be adopted.

In fact the best results are obtained by considering the effect of lateral pressuremodelled by an equation directly dependent onQL such as:

Fig. 15. Interaction ratios for restrained square plates.


Fig. 16. Interaction ratios for restrained square plates.

Fig. 17. Normalised longitudinal strength of restrained square plates subjected to lateral pressure.

RQ51

1+AQL(12BQLb2) (22)

The regression study for the case of square plates has led to regression coefficientsA andB of 0.36 and 0, respectively, showing the strong dependence of the reductionof plate ultimate strength with the level of lateral pressure for all range of plateslenderness. Therefore the longitudinal plate strength under lateral pressurefp

ux canbe easily calculated by:


fpux5fux

11+0.36QL

(23)

Fig. 18 and Table 2 illustrate the applicability of the proposed equation to predictthe longitudinal strength of square plates subjected to different levels of lateral press-ure. The mean value and the coefficient of variation are now reduced to the accept-able values of 1.0 and 7.9%, respectively.

Fig. 19 shows the lack of adequacy of Eq. (15) withA=1 andB=0.025 that isusually used to account for the effect of lateral pressure on the interaction equation.The fact that this equation has been derived mainly for a biaxial state of stress atcollapse can explain the deviation on the results in the case of uniaxial compression.

Considering now rectangular plates ofa/b=3 with initial distortions correspondingto a modem=3 andn=1, one can see from Fig. 20 that the degradation of the platesstrength is dependent on the level of lateral pressure but also on the plate slenderness.

Fig. 21 illustrates the normalised longitudinal strength of restrained rectangularplates taking into account the degradating effect of lateral pressure represented byEq. (24). It can be seen that the coefficientB equal to20.018 eliminates the depen-dence of the degradation of the plate strength on the plate slenderness and the mainvariation which is associated with the level of lateral pressure is then reduced by acoefficientA of 0.2. Using this approach a mean value and a coefficient of variationof 1.03 and 8%, respectively were obtained.

RQ5S 11+0.2QL

D(120.018QLb2) (24)

Fig. 18. Normalised longitudinal strength of restrained square plates taking into account the degradatingeffect of lateral pressure represented by Eq. (22).


Tab

le2

Nor

mal

ised

long

itudi

nal

stre

ngth

ofre

stra

ined

squa

repl

ates

taki

ngin

toac

coun

tth

ede

grad

atin

gef

fect

ofla

tera

lpr

essu

re

b/t

bf u

xf x

Rx=f x

/fp xu

P=0

Pa

P=0

.1M

Pa

P=0

.2M

Pa

P=0

.4M

Pa

P=0

Pa

P=0

.1M

Pa

P=0

.2M

Pa

P=0

.4M

Pa

401.

340.

940.

900.

820.

730.

580.

960.

991.

000.

9660

2.01

0.75

0.71

0.61

0.53

0.45

0.94

0.93

0.90

0.92

802.

680.

610.

630.

540.

480.

381.

041.

001.

000.

9610

03.

350.

510.

600.

500.

450.

311.

181.

121.

121.

03

Mea

nva

lue

1.00

4c.

o.v.

0.07

9


Fig. 19. Normalised longitudinal strength of restrained square plates taking into account the degradatingeffect of lateral pressure represented by Eq. (15).

Fig. 20. Normalised longitudinal strength of restrained rectangular plates subjected to lateral pressure(m=3 andn=1).

6. Conclusions

The strength of compressed square and rectangular plates subjected to lateral press-ure was investigated by a comprehensive series of numerical calculations that pro-vided the axial stress–displacement curves for several plates ranging fromb/t=20 to100 with levels of lateral pressure up to 0.4 MPa (i.e. 40 m water depth).

The effect of different boundary conditions was analysed by considering the longi-


Fig. 21. Normalised longitudinal strength of restrained rectangular plates taking into account the degrad-ing effect of lateral pressure represented by Eq. (24)

tudinal plate edges restrained and unrestrained to transverse displacements. It wasshown that the ultimate longitudinal strength decreases for unrestrained plates andthis reduction increases when increasing levels of lateral pressure are appliedespecially on slender plates.

The effect of different modes of initial distortions on the collapse strength ofrectangular plates was also studied showing that the lateral pressure increases thecollapse strength of the primary buckling mode (m=3, n=1) to values close to theones obtained considering the shape of initial distortions induced by lateral pressure(m=1, n=1).

Finally, design equations were derived to predict the reduction on the longitudinalstrength of plates due to lateral pressure. It was suggested that the design equationsshould be directly dependent on the non-dimensional pressure parameter, since thedegradation of the plate strength for a given level of lateral pressure is almost ident-ical for all ranges of plate slenderness especially for square plates.

Acknowledgements

The first author is grateful to Fundac¸ao para a Cieˆncia e a Tecnologia for havingfinanced his work under Contract No. PRAXIS XXI/BD/15930/98.


Appendix A. Strength of square plates subjected to lateral pressure(restrained)


Appendix B. Strength of square plates subjected to lateral pressure(unrestrained)


Appendix C. Strength of rectangular plates subjected to lateral pressure(restrained) a/b=3 (m=1; n=1)


Appendix D. Strength of rectangular plates subjected to lateral pressure(restrained) a/b=3 (m=3; n=1)


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Documents

Strength of Compressed Rectangular Plates Subjected to Lateral Pressure