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Interactive buckling analysis of box sections using dynamic relaxation
INTERACTIVE BUCKLING ANALYSIS OF BOX SECTIONS USING DYNAMIC RELAXATION P. A. Civil Engineeri~ (Received
FRIEZES
and P. J.
Deptiment,
1 February
1977;
DOWLINGS
Imperial College, London, England received
for publication
1 lune 1974)
Abstract-A procedure is presented for the “exact” analysis of plates forming box sections subject to generalised loading. The numerical method used in the formulation. dynamic relaxation, is described in some detail. Emphasis is placed on the modifications necessary to extend the method beyond the form already used for isolated panel analyses. The equilibrium and compatibility requirements for complete interaction at the adjoining edges of plates are presented and their inco~~tion into the present formulation is considered. The influence of mesh size is studied and comparisons are made with isolated panel results. The simplifying assumptions concerning edge interaction adopted by earlier workers are validated for axial loading and some results are presented for a square column under axial loading in which the effect of varying the flange plate thickness and of the mode and magnitude of plate initial deformations is considered.
1.1 Buckg~u~d
1. faction to probiem and
aim
of paper
Consideration of the post-buckled strength of plate components has been an important feature in the development of cold-formed sections. With the increase in use of hot-rolled plate in more slender welded structures. it has become necessary to examine interactive buckling in this cons~uction. The theoretical work reported here has been undertaken to study this problem. The necessity for studying interactive buckling in box sections is two-fold. Firstly, the design of the component plates has often been based on the behaviour of isolated plates-a procedure which must be of limited application. Secondly, the overall behaviour of box section cam-columns may be affected by local bucking so that the stiffness of the plates forming the cross section must be known with some degree of certainty. A number of mathematical models have been developed to investigate elastic interactive buckling between component plates of box sections loaded in compression or compression and flexure [ l-31. Particular attention has been focussed on the effect which locat post-buckling behaviour has on overall buckling. Because of limitations on computer capacities, previously formulated procedures have usually incorporated simplifying assumptions and have been confined in application to a limited range of problems. The aim of the present paper is to describe an “exact” procedure for the analysis of the elastic interaction between plates forming box columns which can be used to establish the validity of some of the simplifying assumptions adopted by previous workers. A further aim is to compare the results obtained for uniformly compressed cotumns with those obtained from isolated plate analyses[B S] to determine those areas where consideration of individua1 plate behaviour afone is adequate in describing plate response in interactive situations. I.2
Panicular
problem
may, in general, be considered as a ~am~oiumn, the top and bottom plates have been designated “flanges”, and the side plate “webs”, although the study in this paper is limited to direct compression. The geometry is thus defined (Fig. la) by the length o, the width of the flanges 6,. and the web 6, and the plate thickness r, and 1, of the flanges and web respectively. A local co-ordinate system has been adopted for each element so that a positive value of out-of-plane deflection is always inwards. Plate behaviour is determined using the thin plate large deflection equations allowing for initial out-of-plane deformations. The positive directions of the stress resultants and couples are shown in Fig. l(b). The loaded ends of the plates have been assumed to be free to rotate out-of-plane although fixed against out-of-plane movement, and tangentially unrestrained, while the loading is applied as a uniform in-plane displacement. Z AFPLKXTEONOF DYNAMIC RELAXATION @it) 2.1 Brief desc~prjon of DR DR is an iterative finite difference procedure for solving the differential equations for elastic systems. It is analogous to the step-by-step numerical integration in time of the dynamic response of a viscously damped system to the sudden application of static loading. The response to loading is controled by two iteration parameters, viscous damping and a simple funct~n of $ime increment and density At2/p. The choice of near optimal values for these parameters ensures the oscillations are rapidly damped out leaving the static solution, and that a minimum of computational steps is needed to follow the procedure in finite difference form. Each DR iteration follows a cyclic pattern. If equilibrium equations for the system are written in finite difference form, out-of-balance forces are generated by lack of equilibrium in these equations until the oscillations have been damped out. Equating these forces to the sum of viscous and mass-acceleration forces at any node gives equations of motion. If these latter forces are also represented in finite difference form in terms of the velocities at a node and a time increment At, the equations of motion may be rearranged to give an explicit form for the velocities at any node in terms of previous
considered
The structure is shown in Fig. 1. It is a short length of rectangular box section in which advantage can be taken of symmetry of geometry and foading. As the section iResearch Fellow. SReaderin Steel Structures. CAS Vol. 9. No. 5-A
431
432
P. A. FRIEZEand P. J,
DOWLINC
(at Fig. 1. (a) Geometry and local coordinate system for box column model.(b) Positive direction of stress resultants _J .I. ana couples. velocity values and the current steps in the iteration procedure
lack of ~quitibr~um. The are then as follows. Step 1. Catcutate stress couples and resultants (e.g. moments and in-plane forces) in terms of strain resuttants (i.e. curvatures and mean in-plane strains); Step 2. Apply stress boundary conditions; Step 3. Calculate new velocities from the equations described above; Step 4. Integrate the velocities using a first order expression to provide new displacements; Step 5. Apply displacement boundary conditions; Step 6. Calculate strain resultants from displacements; Steg 7. Go to Step 1 and repeat. A DR solution often starts from a condition of zero displacement and stress everywhere as the toads are suddenly applied. This is not essential, however, a solution can, as is the case here, start from the static result of a previous toad increment. The iterations are continued until there are suitabty small velocities everywhere, and the optimal achievement of this equilibrium is subject to a suitable systematic choice of the iteration parameters: these are discussed in detail in [4] and [7f. 2.2 E4~ilib~Mrnequations The governing large deflection plate equations are as follows:
where M,, M,,, MXYare the stress couples, N,. N,, N,, are the stress resultants. 4 is the intensity of lateral pressure and W, and w are the initial out-of-plane displacements and deflections under load respectively. Until ~u~ib~urn is achieved, eqns (tH3) are treated as out-of-balance forces in the equations of motion. Taking central finite differences with respect to time, where stress resultants and couples are calculated at time t and velocities at times Ad2 before and after t, a typical velocity equation would be 161
&
l-Q2 “m&b+
[out-of-balance At p&l + KJ2) ’ force, eqn W
(41
where pW and K, are the fictitious density and nondimens~on~ised damping factor associated with out-ofplane movement, and 16~ and G+, are the out-of-plane velocities at times 1 t Ad2 and t-At/2 respectively, By integrating the velocity, the displacement at time f t Ar can be found (5) and W, and 6~~become wb and Ir;b in the next cycle of calculations where forces and moments are determined using the current values of the displacements. The forces and moments are then substituted into the equilibrium equations at time t + Abtto give the next cycle of out-ofbalance forces. The procedure is continued untif the equilibrium equations are approximately zero.
2.3 Co~sti~~fi~e eq~ut~ons Assuming planes normal to the plane of the plate remain plane, an incremental form of the large deflection strain resultants is as foilows
Interactive
433
buckling analysis of box sections using dynamic relaxation
I
au a~ a~ a~ aw a~ au,_?!G+,_.2+_!__!____e__e ay ax ax ax ay +(t$!$?)~+(y$c)$
ay
ax ay
J
(6)
specifiying loading in the form of in-plane displacements, and shear nodes to describe the zero shear condition. The slope at the edge can be described in terms of the boundary and first internal w nodes using the fact that the moment along the edge is zero.
where u and v are the x and y in-plane displacements, and subscripts t and p indicate the current and previous values of total displacements respectively. Using Hooke’s Law, the incremental stress-strain equations are
(8) and
2.5 Iteration parameters The parameters, damping factor and fictitious density, used to control the iterative procedure have been discussed in detail in relation to plates in Ref. [6]. No modification to either was necessary for the present application to plate assemblages. However, the number of nodes necessary for the analysis of three plates is approximately eight times that needed for studying isolated symmetrical plates. Since, in the recently developed isolated plate fo~ulation[61, densities were determined at every iteration, consideration was given as to how often densities needed to be calculated with a view to reducing computer time. It was found that the following modified procedure gave virtually the same degree of convergence as obtained by calculating the fictitious densities every iteration. n
where
Frequencyof density calculation
s N/3 > N/3
every iteration 20n I every-Iterations N
where n is the ite~tion number up to the maximum N. The corresponding saving in computer time was approximately 20%
a
1
D, 0
1 u
D=D,D~O Et” 0 Dx, = 12(1- vZ) [0 E is the material elasticity and t the plate thickness.
modulus,
0
Y
vo I
0
01-y
2 I
Poisson’s ratio,
2.4 Applicafion of finite diifeeences and inierlacin~ meshes Central finite differences are used to approximate the partiaf derivatives. The first derivatives are more accurately represented by specifying displacements and dependent stresses on interlacing rather than non-interlacing grids. Figure 2 shows the mesh arrangement at the boundary between two plates adjacent to a corner adopted for this formulation. (The edges have been shown separated for clarity.) The use of fictitious nodes which lie outside the plate boundary is a feature of finite difference representations. Here they have been kept to a minimum in order to limit the array sizes and, consequently, the required computer storage. Thus at the ends of the plates only two fictitious nodes have been introduced. These are of nodes for
3. EDGE XNTERACTION 3.1 Equilibn’um and compatibility Four boundary conditions, two membrane and two flexural, must be specified along the edge of each plate. For each condition, a choice exists as to whether to specify the restraint as a displacement or as a stress resultant or couple, i.e. either compatibility or equilibrium may be invoked. However, if a displacement condition is adopted for one plate, then equilibrium must be used for the co~esponding condition on the adjoining plate. For the plates shown in Fig. 2, the following are the equations for complete interaction between the adjoining edges : Boundary conditions in terms of displacements, (i) normal to the flange, w, = v,
(9)
(ii) normal to the web, w, = - 0,
(10)
(iii) tangential to the edges, uW= rrf
(11)
(iv) rotational. 3
=2
P. A. FRIEZEand P. J. bWW
434
internal .
and boundary w, Nx 8 Ny,M,,My
_:
;xY,“xY
-+
v
Fictitious Web-
nodes
0
w
0
“Jxy
nodes
--uU -NV
Fig. 2. Finite differencemesh. Note: Boundarieshave been separated for clarity.
Boundary conditions in terms of forces,
exceed yield stress. The latter is also a large deftection method but which incorporates an elasto-plastic analysis (i) normal to the flange, on the basis that strain reversal of plastic material does not occur. In Ref. [2] interaction is discussed in some detail and N (13) expressions similar to those in eqns (9x16) are presenYW ted in the case of three plates connected along two boundaries. When relating the normal in-plane stress (ii) normal to the web, resultant to the reaction (eqns (28), Ref. [2]), KlSppel has also assumed that the angle between the in-plane stress Nyr= _ %X++‘” (14) resultants and the reactions is small (see eqns 13 and > ay 14). The order of magnitude of the reactions normal to (iii) tangential to the edges, NX,+= N,,, 115) the plane of the plate and the in-plane displacement normal to the edge are then considered to be small and (iv) rotational, M,, = MY,. (16) are ignored. Graves Smith[3] also makes the same assumptions so that, in both formulations, the right hand In deriving eqns (13) and (14), the angle between the sides of eqns (9). (lo), (13) and (14) are all zero. This implies that the plate elements analysed by them have directions of the reactions, normal to the plane of the boundaries rigid with respect to out-of-plane movement undeformed plate, and the membrane stress resultants, but are completely free from any normal in-plane resparallel to the plane of the deformed plate, has been traint. assumed to be small, i.e. cos B^- 1. As indicated above, a choice exists at a particular edge 3.3 Treatment of interaction conditions using DR between either the compatibility or the equilibrium conDuring the first few cycles of DR the magnitude of the dition for each of the four independent restraints. Inspection of eqns (13) and (14). however, shows that it resultants and couples generated through the sudden would be difficult to use the right hand side of these application of loading can be large. This arises from the nature of the numerical procedure in its use of inparticular equations as the independent part of the contegration in both space and time. However, these large dition. Consequently, as the stress normal to the edge will be the dependent variable in conditions (i) and (ii), values, which are normally of little consequence, can cause convergence problems in the case of dependent this implies that w rather than u will be the dependent boundary conditions. displacement variable in eqns (9) and (10). An example of a stress resultant dependent boundary restraint encountered in isolated plate analyses is that of 3.2 Interaction in existing mathematical models the “constrained” edge. This is an in-plane condition in Two of the most complete models for rectangular which the funny is kept straight al~ou~ the average sections composed of individual plates are those desnormal stress along the edge is zero. It is incorporated by cribed by Kliippel et af.[2] and Graves SmithD]. The keeping the edge stationary for one cycle, averaging the former is an elastic large deflection analysis which resulting normal in-plane stress resultant and then applydetermines the maximum capacity of box columns on the ing a correction to the normal displacement for the next basis that the maximum membrane stress, allowing for cycle. The correction is such that a stress equal and the loss of effectiveness due to buckling, should not
Interactive
buckling analysis of box sections using dynamic relaxation
opposite to that calculated should be imposed. However, if the correction is attempted within the first N/S cycles the solution usually diverges so that the correction is only applied after 20% of the cycles have been completed when the magnitude of the stress resultants and couples is of the order of their static values. As several of the interactive boundary conditions are similar in nature to that just described for the isolated plate, precautions used for the “constrained” edge condition were also incorporated within this analysis. Boundary conditions (9) and (10). From Fig. 2 it can be seen that u,,, is not determined exactly on the boundary and must be found from the average of the values at the fictitious nodes and first internal nodes. i.e. wfb
=
f(%a + t’wi, )
(17)
where subscripts b, ex and in refer to the boundary, external and internal nodes respectively. Best results were obtained by delaying the application of this condition until N/5 cycles had been completed after which it was satisfied every iteration. Equation (10) was handled in an identical manner. Boundary condition (11). The order in which equality of the tangential displacements is incorporated is completely dependent upon the order in which equality of the shear stress resultants, eqn (IS), is used. With interlacing meshes it is also necessary that a fictitious shear node exists so that the appropriate lack of equilibrium (eqn 2) can be determined for the independent displacement. In this case, therefore, the choice of uf as the independent variable necessitated the use of fictitious shear node along the edge of the flange adjoining the web (see Fig. 2), and, to satisfy eqn (15), its value was put equal to that at the first internal web node, i.e. Nxyfer
=
(18)
Nxyvin.
Both these conditions were incorporated from the first iterative cycle. Boundary conditions (12) and (16). The satisfaction of these dual conditions proved to be the most difficult of all the interaction relationships. Success was achieved, however, by combining the two conditions to produce an unique expression for the fictitious displacement at one of the edges. This was derived in the following manner (shown here in non-incremental terms) with reference to the plates in Fig. 2. Moment equilibrium, MY,= I$,,., in terms of curvatures is
Expressing the y-direction difference terms leads to
4 =
Equality
wfex-
zw,b
+
curvatures
in
finite
wfi.+ D,,%
(AY,)~ (19)
4w
of the slopes also in finite difference terms is Wfex -
DAY,
Wfin _
-
WWi”-
wwex
2Ayw
435
which for the flange fictitious node gives W,,, = w,i,
+h(wyi” - w,). AY,
Substituting eqn (20) into (19) and rearranging provides the required expression for the out-of-plane deflection at the fictitious web node
a2w, +D,,~+~~w,-D,,~ ax I/ AYE !++A_ ( AY, AY,AY,>’ 2
(21)
The procedure adopted thus was, after N/5 cycles, to solve first for the fictitious web node using eqn (21), followed immediately by that for the fictitious flange node using eqn (20). The latter was simpler to apply than the alternative, eqn (19) rearranged to provide an explicit expression for wrcX. Boundary conditions (13) and (14). Both equations were incorporated without modification into DR and could be used from the first iteration. 3.4 Some additional procedures for assisting convergence One obvious approach for hastening convergence of the solution is to, after the first increment, multiply all the displacements including the fictitious node values by the ratio of the present to the previous load increment. This is most useful for loadings which are outside the range where relatively rapid movement may occur, for instance, when any plate is near its elastic critical buckling load. Similarly, when equating edge out-of-plane displacement of the flange to in-plane displacement of the web (eqn 9), the entire line of nodes across the width of the flange is adjusted rather than just the edge node. This does not introduce any differential in the flange out-ofplane displacements with respect to the y-direction (Fig. 2) as the edge interaction is satisfied. The web is treated in a like manner and in the case of flexural loading involves rigid body rotation of each transverse line of nodes about the web longitudinal axis. 4. PRELMNARY
RFSULTS
4.1 Influence of mesh size In the case of isolated panels, it has been found that for simply supported plates in compression an 8Ax x 8Ay mesh for a square section of plate is adequate for achieving a good degree of accuracy[6]. For clamped plates a 12 x 12 mesh is required. One of the aims of the present investigation was to study the effect of increasing the flange thickness on the behaviour of the webs of a square box column. For web and flange plates of equal thickness, the web will behave more or less as a simply supported plate. As the flange thickness is increased, however, rotational edge restraints will develop inhibiting out-of-plane movement of the web until the web becomes effectively rotationally clamped along the edges adjoining the flanges. This suggested that for the web ‘some 12 divisions might be needed in the y-direction with only 8 in the x-direction since the ends of the column were simply supported. It was found that a 6 x6 mesh could be used to
P. A. FRIEZEand P. J.
436
same plate bounded by flanges of the same width but four times thicker. Only the web of the box section wab initially deformed into three half sine waves of amplitude b/IO0000 (&,/t = 0.0008). A 19 x I I mesh was used in the analysis of the isolated plate, and a 18 x 10 mesh for the web of the column. The elastic critical buckling mode was five half sine waves and the critical loads determined from the isolated panel and box column analyses were 202.5 and 200.3 N/mm2 respectively. The small difference between the results is probably due to the slightly more refined mesh used for the analysis of the isolated plate. Despite the small difference between the buckling loads and a close similarity in the stress resultant and couple distributions at loads in the vicinity of the critical values. differences were noted in the membrane stress distributions at loads equal to twice the critical loads. In Fig. 3(a) the edge and centre-line distributions of the longitudinal stress resultant N, are shown for both the isolated plate and the web of the column. Little difference can be seen between the distributions except
satisfactorily predict the elastic critical buckling load when a small initial bow was set into the web plate only of square columns with plates of uniform thickness. Subsequently the flange plate thickness was increased considerably so that the web behaved as a clamped plate. Use of a 6x 12 mesh led to satisfactory estimates of the critical buckling load for this case. 4.2 Comparison
with isolated plate results edges simply supported.
Comparisons were first undertaken on a square panel with a slenderness ratio (b/t) of 100 for which isolated panel results already existediS. 81. The box was formed from four such plates and the results including those in the postbuckling range for three different levels of initial single sinusoidal bows (alternately inwards and outwards in adjacent faces) were identical to those derived from the isolated plate analysis. A 9 X 9 mesh was used for all the plates. Longitudinal edges rotationally clamped. The response of an isolated plate with clamped longitudinal boundaries (a/b = 3, and b/t = 80) was compared with that of the Longitudinal
DOWLING
for that along the longitudinal edge. Here, in the case of the column, N, is almost uniform, whereas, in the case
Nx
(N/mm) 1000 N, (N/mm) 0
I(a 1 Longitudinal
stress
(N,
X
-
) distribution
along
Box bf=b,
edges
and
centre-lines
of plate
___
Box bf= b, Box tf=tw
-
(b)
Shear
06
stress
(Nxy
) distribution
along
edge
of plate
-
Box bf = b,,,
---
Box
---
PLate
lf =1,
I
I 8
I 9
I
10
11
12
13
14
cl +E(?,’
(c)
Secant
Fig. 3. Comparison
stiffness-avemge
of behaviour
of isolated
stress clamped plate o/b=3,b/r=l.
(nonwith
dimensionalised web of box column
1 bounded
by stocky
flanges,
Interactive bucklinganalysis of box sections using dynamic relaxation of the isolated plate, the distribution varies considerably with the peaks coinciding with the crests of the buckies. It was noted from the analysis that the magnitude of N, at the edge of the web approximated that of the isolated panel (NY= 0). The distribution of longitudinal stress along the edge of the column web has been influenced by the resistance IO differential straining of the flange plate. Another effect of this is to introduce shear stresses at the column corner in contrast to the zero shear stress condition assumed for the isolated plate. Figure 3(b) shows that the resulting distribution of shear is sinusoidal varying from zero at the nodes and antinodes of the buckles to a value numerically equal to some 4% of the practically uniform N, value. A further effect of the thicker flange is to increase the apparent axial stiffness of the web as illustrated in Fig. 3(c) where secant stiffness I(, has been plotted against (u,JE) (bit)‘. For high levels of compressive stress, the stiffness of the box column web is some 4% greater than that of the isolated plate. To investigate further this influence of flanges on the distribution of longitudinal stress in the web, the flange plate thickness was set equal to that of the web, but the width was determined using the parameter given by Bulson191 for studying interacting plates, viz. (t?b). it was expected that this combination of plates would result in web out-of-plane behaviour similar to that already found for the isolated .plate and the stocky flanged column, but that the distribution of in-plane stresses would lie intermed~te to the earlier results. The outcome for this present case was as expected as far as flexural behaviour and the distribution of shear stresses (Fig. 3b) were concerned. However, Fig. 3(a) shows that the magnitude of the N, variation has increased significantly, and from Fig. 3(c) it can be seen the apparent axial stiffness of the web now lies well below that of the previous results. 4.3 Comparison with existing solutions From the evidence presented so far from this “exact” analysis, it would appear that the simplifying assumptions concerning edge interaction (Section 3.2) made by Kliippel el al.121 and Graves Smith[3] are valid for box cotumn sections under predominantly axial loading. S.SQLJ.UtE BOXCOLUMNSlNCOMPRESSlON 5.1 Varying fiange thickness In Fig. 4. results are shown for three square box columns all with a web of slenderness 100 and flange plate thicknesses of 1 x t,,1: x r, and 5 x t,:the corresponding flange slenderness ratios are 100, 60 and 20. The initial deformations, all of magnitude ~/l~, were doubly sinusoidal in nature and were sympathetic to the probable critical buckling mode of the web. These modes. two half waves when the flange b/t = 100 and 60, and three half waves when the Range b/t = 20, were confirmed by the behaviour of the box columns under uniform axial shortening. The resuits in Fig. 4 correspond to an average longitudinal stress in the web (flrr.0, 1 of 2.2 times the web buckling stress (u,,) determined from the analysis. The values plotted in Figs. 4(a)-4(c) are from transverse sections corresponding to peaks in the buckles: Fig. 4(d) corresponds to the position of a node. Figure 4(a) shows the out-of-plane displacement which, in the case of the flange with r, = 5 x rv (t, = 1 mm), was negligible. This suggests that such a stocky flange effectively
437
rotationally clamps the b/t = 100 web, a fact which is confirmed by the distribution of transverse moments in Fig_ d(b). It can also be seen in Figs. 4(c) and (d) from the uniform dist~bution of the lon~tudinai stress in the flange that it has not buckled. Despite the significant differences in behaviour of the flanges, the response of the central portion of the web is reasonably consistent in all three examples. This is to be is similar for each of the webs expected since tr,,&,, although the actual level of applied strain at which this particular ratio occurs is different in each case. A comparison of the longitudinal web stiffnesses is given in Fig. 5, the average stress in the web being nondimensionalised with respect to its critical stress. It is apparent that stocky flanges have the effect of increasing the Ion~tudin~ stiffness of the web. In contrast, the web with the Range of inte~ediate slenderness (b/l = 60) appears less stiff for loads at the upper end of the loading range, but more stiff for loading just above critical. Here, buckling of the web is complicated by that of the flange which follows shortly thereafter and is the cause of the more rapid decline in secant stiffness which occurs at KS = 0.94. The average stress in this flange when it buckled was 143.5N/mm2, a value well below the simply supported critical buckling stress of 205.9 N/mm’ indicating the influence negative edge moments can have on the buckling stress of an interactive plate. 5.2 Initial deformations In the three box sections just described, the initial deformations had been selected to be of the same shape as the critical buckling mode of the web, but with a magnitude small enough so that the elastic critical load could be determined with some degree of accuracy. The effect of increasing the magnitude of the initial deformations in the column of uniform plate thickness results in stiffness curves practically identicai to the isolated plate ones. In columns with flanges and webs of unequal thickness, increasing the size of the initial out-of-flatness has a similar effect although the magnitudes are different: e.g. for b/f, = 100 and 60 the initiai web stiffness equals 0.88 and 0.90 respectively when S, = b/200. As an alternative to selecting the elastic criticaI buckling mode for the web when the Range bit was 20, a two half sine wave form was considered. When the magnitude of this initial bow was small, b/lOOOOO, the web still buckled into three half waves although at a load slightly greater than the earlier case, 132 N/mm’ compared with 122Nlmm’. The change in mode occurred more rapidly than the buckling which had taken place in the presence of three half waves, but beyond the critical load the resutts were practically identical. For larger levels of initial bow, however, the two half wave form remained throughout the full loading history and resulted in a much stiffer web than when the initial shape was in the criticiil mode, e.g. the initial. stiffness for an initial defo~ation of ma~itude b/200 was 0.93 and 0.87 for two and three half waves respectively. 6.CONCLUSIONS
(i) A method has been developed for the “exact” analysis of the interaction between assemblages of rectangular plates forming structural sections subjected to generaiised loading. (ii) Dynamic relaxation, based on finite differences, has been successfully adapted for this analysis of interacting plates.
438
P. A. FRIEZEand P. J.
JIOWLING
I
,
mm
---
tf=
+12jtw
--
tf=
Nmm/mm
(a)Out-of-plane
deflection
w (b)Transverse
T -w
-9.
moments MY
..
em----
(c) Long. stress ax/a
\\
‘,
rati 0
w cr (antinode)
(d)
Long stress o,Rl
ratio
w cr (node)
Fig. 4. Squarecolumn: effect of varying flange thickness, b,,./t, = 100, &,, = So, = b/MOOOO, ozwa, = 2.20,,,.
t, 51,
Interactive
buckling analysis of box sections using dynamic relaxation
439
a-w cr 0 w cr Fig. 5. Effect of varying flange plate thickness on web secant stiffness, b Jt, (iii) The method has been applied to the study of the elastic post-buckling behaviour of box sections with flanges and webs of equal and unequal thickness loaded in uniaxial compression. (iv) For box columns of uniform thickness, reasonable accuracy can be achieved using a 6x6 mesh for each square section of plate. In the case of sections of nonuniform thickness, 10 divisions are needed in the thinner plate in the direction normal to the interacting edges. (v) A reappraisal of the frequency of calculation of fictitious density, one of the iteration parameters governing convergence in the dynamic relaxation procedure, led to a 20% saving in computer time as compared with earlier formulations using this same numerical approach. (vi) When incorporating rotational equilibrium and compatibility between adjacent edges, a combination of the two conditions into a single expression led to the most rapidly convergent solution. Ah other equilibrium and compatibility requirements were treated more or less independently. (vii) The simplifying assumptions concerning interaction made by earlier workers have been shown to be acceptable for the range of problems studied to date. (viii) The addition of flanges to an isolated web can lead to either an increase or a decrease in web axial stiffness when compared with that of the isolated panel: the magnitude of the longitudinal stress variation along the edge of the web is similarly affected. (ix) In the case of square columns, provided the web average stress is non-dimensionalised with respect to its observed critical buckling stress, varying the flange thickness has little effect on the behaviour of the central region of the web, although that of the edge may vary considerably. Both the mode and the magnitude of the plate initial deformations significantly affected the web axial stiffness. (x) The buckling stress of a flange adjoining a slightly
= 100, a,, = so, = b/]OOoOO.
more slender web was found to be well be-low its simply supported critical value. Elastic interactive burtizder more complex loading conditions is presently being studied and it is intended to extend the formulation to include the effect of material non-linearity. With such a tool the problems of load-shedding and redistribution can be studied so as to provide a clearer understanding of these complex structural phenomena. REFERENCES I. P. P. Bijlaard and G. P. Fisher,. Column strength of H-sections and square tubes in the post-buckling range of the component plates. N.A.C.A. Tech. Note. 2994 (1953). 2. K. Klgpel, R. Schmied and J. Schubert, Die Traglast mittig und aussermittig gedrllckter dlinnwandiger Kastentrgger under Verwendung der nichtlinearen Bealtheorie. Dcr Sfuhibou gg(H.11). 321-337 (1966). 3. T. R. Graves Smith, The ultimate strength of locally buckled columns of arbitrary length. In Thin-walled Steel Structures. (Edited by K. C. Rockey and H. V. Hill) pp. 35-60, Crosby
Lockwood,London (1%9). 4. P. A. Frieze, P. J. Dowling and R. E. Hobbs, Steel box girders. Parametric study on plates in compression. CESLIC Reo. BG39. Imoerial Collene. London (1975). 5. P.-A. Frieze, ‘P. J. Dow& and R.. E. Hobbs, Steel box girders. A contribution to the design of stiffened compression flanges. CESUC Rep. BG42, Imperial College. London (1976). 6. P. A. Frieze, R. E. Hobbs and P. J. Dowling. Application of dynamic relaxation to the elasto-plastic large deflection analysis of plates. Comput. Strucfures (In print). 7. P. A. Frieze, Ultimate strength of steel box girders and their components. Ph.D. Thesis, University of London (Imperial College) (1975). P. A. Frieze, P. J. Dowling and R. E. Hobbs, Ultimate load behaviour of plates in compression. In Steel Plated Sfrucfures (Edited by P. J. Dowling. J. E. Harding and P. A. Frieze). Crosby Lockwood Staoles. London (1977). P. S. Bulson. The Stobiliry ofFlot Plates. Chatto & Windus. London (1970).
FRIEZES
and P. J.
Deptiment,
1 February
1977;
DOWLINGS
Imperial College, London, England received
for publication
1 lune 1974)
Abstract-A procedure is presented for the “exact” analysis of plates forming box sections subject to generalised loading. The numerical method used in the formulation. dynamic relaxation, is described in some detail. Emphasis is placed on the modifications necessary to extend the method beyond the form already used for isolated panel analyses. The equilibrium and compatibility requirements for complete interaction at the adjoining edges of plates are presented and their inco~~tion into the present formulation is considered. The influence of mesh size is studied and comparisons are made with isolated panel results. The simplifying assumptions concerning edge interaction adopted by earlier workers are validated for axial loading and some results are presented for a square column under axial loading in which the effect of varying the flange plate thickness and of the mode and magnitude of plate initial deformations is considered.
1.1 Buckg~u~d
1. faction to probiem and
aim
of paper
Consideration of the post-buckled strength of plate components has been an important feature in the development of cold-formed sections. With the increase in use of hot-rolled plate in more slender welded structures. it has become necessary to examine interactive buckling in this cons~uction. The theoretical work reported here has been undertaken to study this problem. The necessity for studying interactive buckling in box sections is two-fold. Firstly, the design of the component plates has often been based on the behaviour of isolated plates-a procedure which must be of limited application. Secondly, the overall behaviour of box section cam-columns may be affected by local bucking so that the stiffness of the plates forming the cross section must be known with some degree of certainty. A number of mathematical models have been developed to investigate elastic interactive buckling between component plates of box sections loaded in compression or compression and flexure [ l-31. Particular attention has been focussed on the effect which locat post-buckling behaviour has on overall buckling. Because of limitations on computer capacities, previously formulated procedures have usually incorporated simplifying assumptions and have been confined in application to a limited range of problems. The aim of the present paper is to describe an “exact” procedure for the analysis of the elastic interaction between plates forming box columns which can be used to establish the validity of some of the simplifying assumptions adopted by previous workers. A further aim is to compare the results obtained for uniformly compressed cotumns with those obtained from isolated plate analyses[B S] to determine those areas where consideration of individua1 plate behaviour afone is adequate in describing plate response in interactive situations. I.2
Panicular
problem
may, in general, be considered as a ~am~oiumn, the top and bottom plates have been designated “flanges”, and the side plate “webs”, although the study in this paper is limited to direct compression. The geometry is thus defined (Fig. la) by the length o, the width of the flanges 6,. and the web 6, and the plate thickness r, and 1, of the flanges and web respectively. A local co-ordinate system has been adopted for each element so that a positive value of out-of-plane deflection is always inwards. Plate behaviour is determined using the thin plate large deflection equations allowing for initial out-of-plane deformations. The positive directions of the stress resultants and couples are shown in Fig. l(b). The loaded ends of the plates have been assumed to be free to rotate out-of-plane although fixed against out-of-plane movement, and tangentially unrestrained, while the loading is applied as a uniform in-plane displacement. Z AFPLKXTEONOF DYNAMIC RELAXATION @it) 2.1 Brief desc~prjon of DR DR is an iterative finite difference procedure for solving the differential equations for elastic systems. It is analogous to the step-by-step numerical integration in time of the dynamic response of a viscously damped system to the sudden application of static loading. The response to loading is controled by two iteration parameters, viscous damping and a simple funct~n of $ime increment and density At2/p. The choice of near optimal values for these parameters ensures the oscillations are rapidly damped out leaving the static solution, and that a minimum of computational steps is needed to follow the procedure in finite difference form. Each DR iteration follows a cyclic pattern. If equilibrium equations for the system are written in finite difference form, out-of-balance forces are generated by lack of equilibrium in these equations until the oscillations have been damped out. Equating these forces to the sum of viscous and mass-acceleration forces at any node gives equations of motion. If these latter forces are also represented in finite difference form in terms of the velocities at a node and a time increment At, the equations of motion may be rearranged to give an explicit form for the velocities at any node in terms of previous
considered
The structure is shown in Fig. 1. It is a short length of rectangular box section in which advantage can be taken of symmetry of geometry and foading. As the section iResearch Fellow. SReaderin Steel Structures. CAS Vol. 9. No. 5-A
431
432
P. A. FRIEZEand P. J,
DOWLINC
(at Fig. 1. (a) Geometry and local coordinate system for box column model.(b) Positive direction of stress resultants _J .I. ana couples. velocity values and the current steps in the iteration procedure
lack of ~quitibr~um. The are then as follows. Step 1. Catcutate stress couples and resultants (e.g. moments and in-plane forces) in terms of strain resuttants (i.e. curvatures and mean in-plane strains); Step 2. Apply stress boundary conditions; Step 3. Calculate new velocities from the equations described above; Step 4. Integrate the velocities using a first order expression to provide new displacements; Step 5. Apply displacement boundary conditions; Step 6. Calculate strain resultants from displacements; Steg 7. Go to Step 1 and repeat. A DR solution often starts from a condition of zero displacement and stress everywhere as the toads are suddenly applied. This is not essential, however, a solution can, as is the case here, start from the static result of a previous toad increment. The iterations are continued until there are suitabty small velocities everywhere, and the optimal achievement of this equilibrium is subject to a suitable systematic choice of the iteration parameters: these are discussed in detail in [4] and [7f. 2.2 E4~ilib~Mrnequations The governing large deflection plate equations are as follows:
where M,, M,,, MXYare the stress couples, N,. N,, N,, are the stress resultants. 4 is the intensity of lateral pressure and W, and w are the initial out-of-plane displacements and deflections under load respectively. Until ~u~ib~urn is achieved, eqns (tH3) are treated as out-of-balance forces in the equations of motion. Taking central finite differences with respect to time, where stress resultants and couples are calculated at time t and velocities at times Ad2 before and after t, a typical velocity equation would be 161
&
l-Q2 “m&b+
[out-of-balance At p&l + KJ2) ’ force, eqn W
(41
where pW and K, are the fictitious density and nondimens~on~ised damping factor associated with out-ofplane movement, and 16~ and G+, are the out-of-plane velocities at times 1 t Ad2 and t-At/2 respectively, By integrating the velocity, the displacement at time f t Ar can be found (5) and W, and 6~~become wb and Ir;b in the next cycle of calculations where forces and moments are determined using the current values of the displacements. The forces and moments are then substituted into the equilibrium equations at time t + Abtto give the next cycle of out-ofbalance forces. The procedure is continued untif the equilibrium equations are approximately zero.
2.3 Co~sti~~fi~e eq~ut~ons Assuming planes normal to the plane of the plate remain plane, an incremental form of the large deflection strain resultants is as foilows
Interactive
433
buckling analysis of box sections using dynamic relaxation
I
au a~ a~ a~ aw a~ au,_?!G+,_.2+_!__!____e__e ay ax ax ax ay +(t$!$?)~+(y$c)$
ay
ax ay
J
(6)
specifiying loading in the form of in-plane displacements, and shear nodes to describe the zero shear condition. The slope at the edge can be described in terms of the boundary and first internal w nodes using the fact that the moment along the edge is zero.
where u and v are the x and y in-plane displacements, and subscripts t and p indicate the current and previous values of total displacements respectively. Using Hooke’s Law, the incremental stress-strain equations are
(8) and
2.5 Iteration parameters The parameters, damping factor and fictitious density, used to control the iterative procedure have been discussed in detail in relation to plates in Ref. [6]. No modification to either was necessary for the present application to plate assemblages. However, the number of nodes necessary for the analysis of three plates is approximately eight times that needed for studying isolated symmetrical plates. Since, in the recently developed isolated plate fo~ulation[61, densities were determined at every iteration, consideration was given as to how often densities needed to be calculated with a view to reducing computer time. It was found that the following modified procedure gave virtually the same degree of convergence as obtained by calculating the fictitious densities every iteration. n
where
Frequencyof density calculation
s N/3 > N/3
every iteration 20n I every-Iterations N
where n is the ite~tion number up to the maximum N. The corresponding saving in computer time was approximately 20%
a
1
D, 0
1 u
D=D,D~O Et” 0 Dx, = 12(1- vZ) [0 E is the material elasticity and t the plate thickness.
modulus,
0
Y
vo I
0
01-y
2 I
Poisson’s ratio,
2.4 Applicafion of finite diifeeences and inierlacin~ meshes Central finite differences are used to approximate the partiaf derivatives. The first derivatives are more accurately represented by specifying displacements and dependent stresses on interlacing rather than non-interlacing grids. Figure 2 shows the mesh arrangement at the boundary between two plates adjacent to a corner adopted for this formulation. (The edges have been shown separated for clarity.) The use of fictitious nodes which lie outside the plate boundary is a feature of finite difference representations. Here they have been kept to a minimum in order to limit the array sizes and, consequently, the required computer storage. Thus at the ends of the plates only two fictitious nodes have been introduced. These are of nodes for
3. EDGE XNTERACTION 3.1 Equilibn’um and compatibility Four boundary conditions, two membrane and two flexural, must be specified along the edge of each plate. For each condition, a choice exists as to whether to specify the restraint as a displacement or as a stress resultant or couple, i.e. either compatibility or equilibrium may be invoked. However, if a displacement condition is adopted for one plate, then equilibrium must be used for the co~esponding condition on the adjoining plate. For the plates shown in Fig. 2, the following are the equations for complete interaction between the adjoining edges : Boundary conditions in terms of displacements, (i) normal to the flange, w, = v,
(9)
(ii) normal to the web, w, = - 0,
(10)
(iii) tangential to the edges, uW= rrf
(11)
(iv) rotational. 3
=2
P. A. FRIEZEand P. J. bWW
434
internal .
and boundary w, Nx 8 Ny,M,,My
_:
;xY,“xY
-+
v
Fictitious Web-
nodes
0
w
0
“Jxy
nodes
--uU -NV
Fig. 2. Finite differencemesh. Note: Boundarieshave been separated for clarity.
Boundary conditions in terms of forces,
exceed yield stress. The latter is also a large deftection method but which incorporates an elasto-plastic analysis (i) normal to the flange, on the basis that strain reversal of plastic material does not occur. In Ref. [2] interaction is discussed in some detail and N (13) expressions similar to those in eqns (9x16) are presenYW ted in the case of three plates connected along two boundaries. When relating the normal in-plane stress (ii) normal to the web, resultant to the reaction (eqns (28), Ref. [2]), KlSppel has also assumed that the angle between the in-plane stress Nyr= _ %X++‘” (14) resultants and the reactions is small (see eqns 13 and > ay 14). The order of magnitude of the reactions normal to (iii) tangential to the edges, NX,+= N,,, 115) the plane of the plate and the in-plane displacement normal to the edge are then considered to be small and (iv) rotational, M,, = MY,. (16) are ignored. Graves Smith[3] also makes the same assumptions so that, in both formulations, the right hand In deriving eqns (13) and (14), the angle between the sides of eqns (9). (lo), (13) and (14) are all zero. This implies that the plate elements analysed by them have directions of the reactions, normal to the plane of the boundaries rigid with respect to out-of-plane movement undeformed plate, and the membrane stress resultants, but are completely free from any normal in-plane resparallel to the plane of the deformed plate, has been traint. assumed to be small, i.e. cos B^- 1. As indicated above, a choice exists at a particular edge 3.3 Treatment of interaction conditions using DR between either the compatibility or the equilibrium conDuring the first few cycles of DR the magnitude of the dition for each of the four independent restraints. Inspection of eqns (13) and (14). however, shows that it resultants and couples generated through the sudden would be difficult to use the right hand side of these application of loading can be large. This arises from the nature of the numerical procedure in its use of inparticular equations as the independent part of the contegration in both space and time. However, these large dition. Consequently, as the stress normal to the edge will be the dependent variable in conditions (i) and (ii), values, which are normally of little consequence, can cause convergence problems in the case of dependent this implies that w rather than u will be the dependent boundary conditions. displacement variable in eqns (9) and (10). An example of a stress resultant dependent boundary restraint encountered in isolated plate analyses is that of 3.2 Interaction in existing mathematical models the “constrained” edge. This is an in-plane condition in Two of the most complete models for rectangular which the funny is kept straight al~ou~ the average sections composed of individual plates are those desnormal stress along the edge is zero. It is incorporated by cribed by Kliippel et af.[2] and Graves SmithD]. The keeping the edge stationary for one cycle, averaging the former is an elastic large deflection analysis which resulting normal in-plane stress resultant and then applydetermines the maximum capacity of box columns on the ing a correction to the normal displacement for the next basis that the maximum membrane stress, allowing for cycle. The correction is such that a stress equal and the loss of effectiveness due to buckling, should not
Interactive
buckling analysis of box sections using dynamic relaxation
opposite to that calculated should be imposed. However, if the correction is attempted within the first N/S cycles the solution usually diverges so that the correction is only applied after 20% of the cycles have been completed when the magnitude of the stress resultants and couples is of the order of their static values. As several of the interactive boundary conditions are similar in nature to that just described for the isolated plate, precautions used for the “constrained” edge condition were also incorporated within this analysis. Boundary conditions (9) and (10). From Fig. 2 it can be seen that u,,, is not determined exactly on the boundary and must be found from the average of the values at the fictitious nodes and first internal nodes. i.e. wfb
=
f(%a + t’wi, )
(17)
where subscripts b, ex and in refer to the boundary, external and internal nodes respectively. Best results were obtained by delaying the application of this condition until N/5 cycles had been completed after which it was satisfied every iteration. Equation (10) was handled in an identical manner. Boundary condition (11). The order in which equality of the tangential displacements is incorporated is completely dependent upon the order in which equality of the shear stress resultants, eqn (IS), is used. With interlacing meshes it is also necessary that a fictitious shear node exists so that the appropriate lack of equilibrium (eqn 2) can be determined for the independent displacement. In this case, therefore, the choice of uf as the independent variable necessitated the use of fictitious shear node along the edge of the flange adjoining the web (see Fig. 2), and, to satisfy eqn (15), its value was put equal to that at the first internal web node, i.e. Nxyfer
=
(18)
Nxyvin.
Both these conditions were incorporated from the first iterative cycle. Boundary conditions (12) and (16). The satisfaction of these dual conditions proved to be the most difficult of all the interaction relationships. Success was achieved, however, by combining the two conditions to produce an unique expression for the fictitious displacement at one of the edges. This was derived in the following manner (shown here in non-incremental terms) with reference to the plates in Fig. 2. Moment equilibrium, MY,= I$,,., in terms of curvatures is
Expressing the y-direction difference terms leads to
4 =
Equality
wfex-
zw,b
+
curvatures
in
finite
wfi.+ D,,%
(AY,)~ (19)
4w
of the slopes also in finite difference terms is Wfex -
DAY,
Wfin _
-
WWi”-
wwex
2Ayw
435
which for the flange fictitious node gives W,,, = w,i,
+h(wyi” - w,). AY,
Substituting eqn (20) into (19) and rearranging provides the required expression for the out-of-plane deflection at the fictitious web node
a2w, +D,,~+~~w,-D,,~ ax I/ AYE !++A_ ( AY, AY,AY,>’ 2
(21)
The procedure adopted thus was, after N/5 cycles, to solve first for the fictitious web node using eqn (21), followed immediately by that for the fictitious flange node using eqn (20). The latter was simpler to apply than the alternative, eqn (19) rearranged to provide an explicit expression for wrcX. Boundary conditions (13) and (14). Both equations were incorporated without modification into DR and could be used from the first iteration. 3.4 Some additional procedures for assisting convergence One obvious approach for hastening convergence of the solution is to, after the first increment, multiply all the displacements including the fictitious node values by the ratio of the present to the previous load increment. This is most useful for loadings which are outside the range where relatively rapid movement may occur, for instance, when any plate is near its elastic critical buckling load. Similarly, when equating edge out-of-plane displacement of the flange to in-plane displacement of the web (eqn 9), the entire line of nodes across the width of the flange is adjusted rather than just the edge node. This does not introduce any differential in the flange out-ofplane displacements with respect to the y-direction (Fig. 2) as the edge interaction is satisfied. The web is treated in a like manner and in the case of flexural loading involves rigid body rotation of each transverse line of nodes about the web longitudinal axis. 4. PRELMNARY
RFSULTS
4.1 Influence of mesh size In the case of isolated panels, it has been found that for simply supported plates in compression an 8Ax x 8Ay mesh for a square section of plate is adequate for achieving a good degree of accuracy[6]. For clamped plates a 12 x 12 mesh is required. One of the aims of the present investigation was to study the effect of increasing the flange thickness on the behaviour of the webs of a square box column. For web and flange plates of equal thickness, the web will behave more or less as a simply supported plate. As the flange thickness is increased, however, rotational edge restraints will develop inhibiting out-of-plane movement of the web until the web becomes effectively rotationally clamped along the edges adjoining the flanges. This suggested that for the web ‘some 12 divisions might be needed in the y-direction with only 8 in the x-direction since the ends of the column were simply supported. It was found that a 6 x6 mesh could be used to
P. A. FRIEZEand P. J.
436
same plate bounded by flanges of the same width but four times thicker. Only the web of the box section wab initially deformed into three half sine waves of amplitude b/IO0000 (&,/t = 0.0008). A 19 x I I mesh was used in the analysis of the isolated plate, and a 18 x 10 mesh for the web of the column. The elastic critical buckling mode was five half sine waves and the critical loads determined from the isolated panel and box column analyses were 202.5 and 200.3 N/mm2 respectively. The small difference between the results is probably due to the slightly more refined mesh used for the analysis of the isolated plate. Despite the small difference between the buckling loads and a close similarity in the stress resultant and couple distributions at loads in the vicinity of the critical values. differences were noted in the membrane stress distributions at loads equal to twice the critical loads. In Fig. 3(a) the edge and centre-line distributions of the longitudinal stress resultant N, are shown for both the isolated plate and the web of the column. Little difference can be seen between the distributions except
satisfactorily predict the elastic critical buckling load when a small initial bow was set into the web plate only of square columns with plates of uniform thickness. Subsequently the flange plate thickness was increased considerably so that the web behaved as a clamped plate. Use of a 6x 12 mesh led to satisfactory estimates of the critical buckling load for this case. 4.2 Comparison
with isolated plate results edges simply supported.
Comparisons were first undertaken on a square panel with a slenderness ratio (b/t) of 100 for which isolated panel results already existediS. 81. The box was formed from four such plates and the results including those in the postbuckling range for three different levels of initial single sinusoidal bows (alternately inwards and outwards in adjacent faces) were identical to those derived from the isolated plate analysis. A 9 X 9 mesh was used for all the plates. Longitudinal edges rotationally clamped. The response of an isolated plate with clamped longitudinal boundaries (a/b = 3, and b/t = 80) was compared with that of the Longitudinal
DOWLING
for that along the longitudinal edge. Here, in the case of the column, N, is almost uniform, whereas, in the case
Nx
(N/mm) 1000 N, (N/mm) 0
I(a 1 Longitudinal
stress
(N,
X
-
) distribution
along
Box bf=b,
edges
and
centre-lines
of plate
___
Box bf= b, Box tf=tw
-
(b)
Shear
06
stress
(Nxy
) distribution
along
edge
of plate
-
Box bf = b,,,
---
Box
---
PLate
lf =1,
I
I 8
I 9
I
10
11
12
13
14
cl +E(?,’
(c)
Secant
Fig. 3. Comparison
stiffness-avemge
of behaviour
of isolated
stress clamped plate o/b=3,b/r=l.
(nonwith
dimensionalised web of box column
1 bounded
by stocky
flanges,
Interactive bucklinganalysis of box sections using dynamic relaxation of the isolated plate, the distribution varies considerably with the peaks coinciding with the crests of the buckies. It was noted from the analysis that the magnitude of N, at the edge of the web approximated that of the isolated panel (NY= 0). The distribution of longitudinal stress along the edge of the column web has been influenced by the resistance IO differential straining of the flange plate. Another effect of this is to introduce shear stresses at the column corner in contrast to the zero shear stress condition assumed for the isolated plate. Figure 3(b) shows that the resulting distribution of shear is sinusoidal varying from zero at the nodes and antinodes of the buckles to a value numerically equal to some 4% of the practically uniform N, value. A further effect of the thicker flange is to increase the apparent axial stiffness of the web as illustrated in Fig. 3(c) where secant stiffness I(, has been plotted against (u,JE) (bit)‘. For high levels of compressive stress, the stiffness of the box column web is some 4% greater than that of the isolated plate. To investigate further this influence of flanges on the distribution of longitudinal stress in the web, the flange plate thickness was set equal to that of the web, but the width was determined using the parameter given by Bulson191 for studying interacting plates, viz. (t?b). it was expected that this combination of plates would result in web out-of-plane behaviour similar to that already found for the isolated .plate and the stocky flanged column, but that the distribution of in-plane stresses would lie intermed~te to the earlier results. The outcome for this present case was as expected as far as flexural behaviour and the distribution of shear stresses (Fig. 3b) were concerned. However, Fig. 3(a) shows that the magnitude of the N, variation has increased significantly, and from Fig. 3(c) it can be seen the apparent axial stiffness of the web now lies well below that of the previous results. 4.3 Comparison with existing solutions From the evidence presented so far from this “exact” analysis, it would appear that the simplifying assumptions concerning edge interaction (Section 3.2) made by Kliippel el al.121 and Graves Smith[3] are valid for box cotumn sections under predominantly axial loading. S.SQLJ.UtE BOXCOLUMNSlNCOMPRESSlON 5.1 Varying fiange thickness In Fig. 4. results are shown for three square box columns all with a web of slenderness 100 and flange plate thicknesses of 1 x t,,1: x r, and 5 x t,:the corresponding flange slenderness ratios are 100, 60 and 20. The initial deformations, all of magnitude ~/l~, were doubly sinusoidal in nature and were sympathetic to the probable critical buckling mode of the web. These modes. two half waves when the flange b/t = 100 and 60, and three half waves when the Range b/t = 20, were confirmed by the behaviour of the box columns under uniform axial shortening. The resuits in Fig. 4 correspond to an average longitudinal stress in the web (flrr.0, 1 of 2.2 times the web buckling stress (u,,) determined from the analysis. The values plotted in Figs. 4(a)-4(c) are from transverse sections corresponding to peaks in the buckles: Fig. 4(d) corresponds to the position of a node. Figure 4(a) shows the out-of-plane displacement which, in the case of the flange with r, = 5 x rv (t, = 1 mm), was negligible. This suggests that such a stocky flange effectively
437
rotationally clamps the b/t = 100 web, a fact which is confirmed by the distribution of transverse moments in Fig_ d(b). It can also be seen in Figs. 4(c) and (d) from the uniform dist~bution of the lon~tudinai stress in the flange that it has not buckled. Despite the significant differences in behaviour of the flanges, the response of the central portion of the web is reasonably consistent in all three examples. This is to be is similar for each of the webs expected since tr,,&,, although the actual level of applied strain at which this particular ratio occurs is different in each case. A comparison of the longitudinal web stiffnesses is given in Fig. 5, the average stress in the web being nondimensionalised with respect to its critical stress. It is apparent that stocky flanges have the effect of increasing the Ion~tudin~ stiffness of the web. In contrast, the web with the Range of inte~ediate slenderness (b/l = 60) appears less stiff for loads at the upper end of the loading range, but more stiff for loading just above critical. Here, buckling of the web is complicated by that of the flange which follows shortly thereafter and is the cause of the more rapid decline in secant stiffness which occurs at KS = 0.94. The average stress in this flange when it buckled was 143.5N/mm2, a value well below the simply supported critical buckling stress of 205.9 N/mm’ indicating the influence negative edge moments can have on the buckling stress of an interactive plate. 5.2 Initial deformations In the three box sections just described, the initial deformations had been selected to be of the same shape as the critical buckling mode of the web, but with a magnitude small enough so that the elastic critical load could be determined with some degree of accuracy. The effect of increasing the magnitude of the initial deformations in the column of uniform plate thickness results in stiffness curves practically identicai to the isolated plate ones. In columns with flanges and webs of unequal thickness, increasing the size of the initial out-of-flatness has a similar effect although the magnitudes are different: e.g. for b/f, = 100 and 60 the initiai web stiffness equals 0.88 and 0.90 respectively when S, = b/200. As an alternative to selecting the elastic criticaI buckling mode for the web when the Range bit was 20, a two half sine wave form was considered. When the magnitude of this initial bow was small, b/lOOOOO, the web still buckled into three half waves although at a load slightly greater than the earlier case, 132 N/mm’ compared with 122Nlmm’. The change in mode occurred more rapidly than the buckling which had taken place in the presence of three half waves, but beyond the critical load the resutts were practically identical. For larger levels of initial bow, however, the two half wave form remained throughout the full loading history and resulted in a much stiffer web than when the initial shape was in the criticiil mode, e.g. the initial. stiffness for an initial defo~ation of ma~itude b/200 was 0.93 and 0.87 for two and three half waves respectively. 6.CONCLUSIONS
(i) A method has been developed for the “exact” analysis of the interaction between assemblages of rectangular plates forming structural sections subjected to generaiised loading. (ii) Dynamic relaxation, based on finite differences, has been successfully adapted for this analysis of interacting plates.
438
P. A. FRIEZEand P. J.
JIOWLING
I
,
mm
---
tf=
+12jtw
--
tf=
Nmm/mm
(a)Out-of-plane
deflection
w (b)Transverse
T -w
-9.
moments MY
..
em----
(c) Long. stress ax/a
\\
‘,
rati 0
w cr (antinode)
(d)
Long stress o,Rl
ratio
w cr (node)
Fig. 4. Squarecolumn: effect of varying flange thickness, b,,./t, = 100, &,, = So, = b/MOOOO, ozwa, = 2.20,,,.
t, 51,
Interactive
buckling analysis of box sections using dynamic relaxation
439
a-w cr 0 w cr Fig. 5. Effect of varying flange plate thickness on web secant stiffness, b Jt, (iii) The method has been applied to the study of the elastic post-buckling behaviour of box sections with flanges and webs of equal and unequal thickness loaded in uniaxial compression. (iv) For box columns of uniform thickness, reasonable accuracy can be achieved using a 6x6 mesh for each square section of plate. In the case of sections of nonuniform thickness, 10 divisions are needed in the thinner plate in the direction normal to the interacting edges. (v) A reappraisal of the frequency of calculation of fictitious density, one of the iteration parameters governing convergence in the dynamic relaxation procedure, led to a 20% saving in computer time as compared with earlier formulations using this same numerical approach. (vi) When incorporating rotational equilibrium and compatibility between adjacent edges, a combination of the two conditions into a single expression led to the most rapidly convergent solution. Ah other equilibrium and compatibility requirements were treated more or less independently. (vii) The simplifying assumptions concerning interaction made by earlier workers have been shown to be acceptable for the range of problems studied to date. (viii) The addition of flanges to an isolated web can lead to either an increase or a decrease in web axial stiffness when compared with that of the isolated panel: the magnitude of the longitudinal stress variation along the edge of the web is similarly affected. (ix) In the case of square columns, provided the web average stress is non-dimensionalised with respect to its observed critical buckling stress, varying the flange thickness has little effect on the behaviour of the central region of the web, although that of the edge may vary considerably. Both the mode and the magnitude of the plate initial deformations significantly affected the web axial stiffness. (x) The buckling stress of a flange adjoining a slightly
= 100, a,, = so, = b/]OOoOO.
more slender web was found to be well be-low its simply supported critical value. Elastic interactive burtizder more complex loading conditions is presently being studied and it is intended to extend the formulation to include the effect of material non-linearity. With such a tool the problems of load-shedding and redistribution can be studied so as to provide a clearer understanding of these complex structural phenomena. REFERENCES I. P. P. Bijlaard and G. P. Fisher,. Column strength of H-sections and square tubes in the post-buckling range of the component plates. N.A.C.A. Tech. Note. 2994 (1953). 2. K. Klgpel, R. Schmied and J. Schubert, Die Traglast mittig und aussermittig gedrllckter dlinnwandiger Kastentrgger under Verwendung der nichtlinearen Bealtheorie. Dcr Sfuhibou gg(H.11). 321-337 (1966). 3. T. R. Graves Smith, The ultimate strength of locally buckled columns of arbitrary length. In Thin-walled Steel Structures. (Edited by K. C. Rockey and H. V. Hill) pp. 35-60, Crosby
Lockwood,London (1%9). 4. P. A. Frieze, P. J. Dowling and R. E. Hobbs, Steel box girders. Parametric study on plates in compression. CESLIC Reo. BG39. Imoerial Collene. London (1975). 5. P.-A. Frieze, ‘P. J. Dow& and R.. E. Hobbs, Steel box girders. A contribution to the design of stiffened compression flanges. CESUC Rep. BG42, Imperial College. London (1976). 6. P. A. Frieze, R. E. Hobbs and P. J. Dowling. Application of dynamic relaxation to the elasto-plastic large deflection analysis of plates. Comput. Strucfures (In print). 7. P. A. Frieze, Ultimate strength of steel box girders and their components. Ph.D. Thesis, University of London (Imperial College) (1975). P. A. Frieze, P. J. Dowling and R. E. Hobbs, Ultimate load behaviour of plates in compression. In Steel Plated Sfrucfures (Edited by P. J. Dowling. J. E. Harding and P. A. Frieze). Crosby Lockwood Staoles. London (1977). P. S. Bulson. The Stobiliry ofFlot Plates. Chatto & Windus. London (1970).