elasto-plastic buckling of stiffener plates in beam-to-column flange connections

Compurers & S~nrc~ures Vol. 18, No. 2, pp. 201-213, 1984 004s7949/84 $3.00 + aI

Printed in Gnat Britain. 0 1984 Pergamon Press Ltd.

ELASTO-PLASTIC BUCKLING OF STIFFENER PLATES IN BEAM-TO-COLUMN FLANGE CONNECTIONS

H. A. EL-GHAZALY, R. N. DUBEY and A. N. SHERLUXJRNE~ University of Waterloo, Waterloo, Canada

Abstract-The problem of plastic buckling of steel plates is reviewed in relation to the load carrying capacity of stiffener plates in beam-to-column flange connections. Due to the non-uniformity of the stress distribution in these plates, the finite element method is used to compute the stresses in the elastic and plastic ranges. A bifurcation analysis is performed using both flow and deformation theory to evaluate the elast*plastic buckling of the stiffener. A scaled inverse iterative version of the power method is employed to evaluate the bifurcation load. A parametric study is conducted on stiffeners and design curves are obtained showing the relationship between the critical stress plate aspect ratios.

and the slenderness ratio for-different

NOTATION

element area long dimension of the plate matrix defining element strain-displacement re-

lationship short dimension of the plate inverse of [K,1 vector of element virtual curvature matrix defining moment-curvature relationship

in the elastic range elastic matrix moduli matrix moduli of flow or deformation theory matrix defining incremental moments curva-

tures relationship in the plastic range matrix defining the total stress-strain re-

lationship in a deformation theory infinitesimal stress and strain increments vec-

tors, respectively Young’s modulus of elasticity secant modulus tangent modulus total effective strain shear modulus in the plastic range intermediate value used in the power method refers to the ith load increment global bending stiffness matrix global geometric stiffness matrix Kbl + WI global elastic in-plane stiffness matrix element bending stiffness matrix in the global

coordinates element geometric stiffness matrix in the global

coordinates element elastic bending stiffness matrix in the

gIoba1 coordinates element plastic bending stiffness matrix in the

global coordinates in-plane element elastic stiffness matrix in-plane element plastic stiffness matrix lower triangular band matrix resulting from

factorizing [KS’] ratio of elastic stresses to total stress increment vector of internal moments stress resultants in the x,y directions, re-

spectively shear stress resultant element displacement-curvature matrix vector of global residual unbalanced forces vector of element residual unbalanced forces vector of arbitrary load refers to rth iteration in the initial stress method refers to rth iteration in the power method scaling factor

TPresent address: Michigan Technological University, Houghton, IM 49931, U.S.A.

Stiffener simply supported by column web, fixed to column flanges

stiffener simply supported to column web and column flanges

plate thickness displacement in X direction element volume displacement in Y direction element virtual slope in the X, Y directions,

respectively global rectangular coordinates location of a point along the Z axis

shear strain element displacement vector vector of infinitesimal internal moments infinitesimal displacement vector in the Z direc-

tion infinitesimal rotations about X, Y axes, re-

spectively load increment vector displacement increment vector elastic strain increment vecotr elastic stress increment vector stress vector to be supported by body forces element strain vector strain in X and Y directions. respectively uniaxial yield strain vectors needed for scaling in the power method E/E,) - 1 smallest eigenvalue second smallest eigenvalue eigenvector of the [C] matrix Poisson’s ratio buckling stress maximum stress material yield stress stresses in X and Y directions, respectively Ramberg-Osgood parameters uniaxial yield stress effective stress element stress vector shear stress

1. INTRODUCTION

Several studies on beam-to-column flange connections[l-31 have confirmed the importance of reinforcing the column web with horizontal stiffeners opposite the beam flanges. These stiffeners effectively increase the load carrying capacity of the column web and generally improve the overall rotational behaviour of the connection. Early attempts to establish limits on the slenderness ratio of stiffener plates[4] were based on the theory of plastic buckling of

201

202 H. A. EL-GHAZALY et al.

plates[5]. The current investigation aims at providing more precise design r&es for the stiffeners by conduc- ting a parametric study using the finite element method of analysis.

Plates stressed beyond the elastic limit under compression generally fail as a result of plastic buckling. TWO distinctive approaches have emerged to evaluate the maximum load for plates stressed into this range. The first approach was adopted by several investigators[6-91. In this approach, linear strain- displacement relationships are used; when equilibrium is studied in the out-of-plane configuration, which is in~nitesimally close to the plane state, the governing differential equation for plate buckling is obtained. The constitutive relationships could be based on either a flow or a deformation theory. The former was used by Handleman and Prager[7] and was more recently applied by Sherbourne and Murthy[2]. Among the several versions of deformation theory, those by IlyshinIB], Stowell[S] and Bijlaard[b] were successfufly employed to evaluate the buckling loads for plates in compression. Experiments on simply supported long plates[lO] confirmed the applicability of a deformation rather than a flow theory in estimating the buckling load. Shanley[l l] supported the previous findings and showed, on a qualitative basis, that the use of the secant modulus in defo~ation theory ac- counts for the shear strains caused by rotation of the principal axes. Incremental theory, on the other hand, fails to account fully for this effect. Geometric imperfections have been considered by Onat and Drucker[l2] in an effort to justify the use of incremental theory. Shanley [ 1 1] responded by showing that the geometric imperfections do not validate the correctness of incremental theory for inelastic buckling; on the contrary they lend support to deformation theory.

Sewell[l3] presented a general theory of elastic and inelastic plate buckling where it was shown that the bifurcation stress is markedly sensitive to the direction of the normal to the local yield surface. This casts doubt on the validity of any experiment that does not give attention to the precise determination of this parameter. Haaijer and Thurlimann [ 141 assumed an orthotropic plate model to predict the buckling loads of steel plates strained into the plastic range. They showed that plates of normal structural steel can be compressed beyond the yield stress into the strain hardening range, provided certain geometric conditions are met to prevent premature local buckling of the plate. Haaijer followed this paper with another publication[5] employing the same philosoply of orthotropic plate behaviour in the plastic range. He gave specific numerical values for the rigidity moduli for the orthotropic plate based on the flow theory of piasticity and good comparison was demonstrated between his theory and some experimental data.

In the previously mentioned plastic buckling investigations a uniform stress field was always assumed for mathematical simplicity. In situations where the stress distribution is not uniform, the finite element method successfully describes the non-uniform stress field in both the elastic and plastic ranges[l5, 161. Zienkiewicz et al. [17] proposed the use of the initial stress method as an effective procedure to yield the solution to the elastic-plastic analysis problem. The method proved to be numerically stable and com-

putationally efficient as compared with the tangent stiffness approachfl8j. Some years later Nayak and Zienkiewicz[l9] refined the previous initial stress technique and several practical problems were solved. In problems involving plastic buckling due to a non- uniform stress field, Isakson et u1.[20] conducted a discrete element analysis for the elastic-plastic buckling of plates using a deformation theory. Shrivastava[Zl] investigated shear effects in deter- mining the value of the bifurcation stress for plates under uniform compression. It was clearly shown that the deformation theory and flow theory yield similar results for small slenderness ratio. At higher ratios, the flow theory generally gives higher values for the bifurcation loads.

A second appraoch for evaluating the maximum capacity of plates under compression was done by performing a large deflection analysis with nonlinear strain~ispla~ment relationships including material nonlinearity due to plasticity. This appraoch necessi- tates integration over the plate volume (volume approach) or over the plate area (area approach) if the plate is thin. Crisfield[22,23] extensively discussed the two approaches and satisfactory restuls were obtained. Moxham presented a detailed theoretical[24] and experimental[25] study on plates under compression. Among the several findings it was concluded that for practical plates (e.g. components of fabricated members) it is reasonable to assume simple rather than fixed supports in design.

2. PROBLEM STATEMENT

Figure 1 shows a typical symme~~cal beam-to- column flange connection where stiffening plates are used. The effect of the stiffener area and eccentricity with respect to the beam flange on the overall capacity of the column web has been detailed in [2]. It has been reported in the literature[26,27] that the connection may fail as a result of the local buckling of the stiffener while the column web is still intact. In [26] test C-10 was concluded by buckling of the horizontal compression stiffener but the overall behaviour of the connection remained satisfactory. Also in [27] the failure of the stiffened connection was initiated by local buckling of the stiffener. It follows from the previous discussion, that the buckling of the compression stiffener is a possible mode of failure of the connection. Investigating the buckling of the compression stiffener

Fig. I. Typical beam-to-column flange condition.

Elastoplastic buckling of stiffener plates in beam-to-column flange connections 203

Fig. 2. In-plane boundary conditions. (Half stiffener used in analysis because of symmetry about Y axis.)

in beam-to-column flange connections is the prime goal of this study. The specific objective of this paper is to provide design rules governing the b/t ratio for the compression stiffener. Numerical experiments will be conducted using the finite element method to evaluate the buckling load for the stiffener.

The in-plane boundary conditions are shown in Fig. 2. The column web is assumed to provide full restraint to the stiffener (i.e. u = 1’ = 0) while the other three edges of the stiffener are free to move in the X-Y plane. The geometrical proportions of the stiffener suggest a plane stress analysis. The in-plane deformations are also assumed to be small relative to the plate dimensions. The forces acting on the stiffener are those from the beam compression flange and are assumed to act uniformly over the stiffener edge. Due to the complexity of the in-plane boundary conditions and the non-uniformity of the stress field, the finite

element method was selected to simulate the behaviour of the stiffener. The constant stress triangular finite element is used in the present analysis. The details of the elastic formulations for this element are available in [28].

In order to estimate the buckling load for the stiffener, the bending boundary conditions must be described. From experimental observations [26,27] the column flanges provide some degree of restraint to the stiffener. The degree of fixity depends on the relative rigidities of both the column flange and the stiffener. In the present analysis, the results will be obtained for both the hinged and fixed boundary conditions shown in Fig. 3. The column web is considered to provide very small rotational resistance to the stiffener since, at buckling, the column web rotates about its minor axis. The 9-degree of freedom bending triangular finite element is used for the stability analysis phase of this study. The element elastic bending and geometric stiffness matrices are shown in [28] and successful results were obtained when solving bending and elastic buckling problems.

The stress-strain model shown in Fig. 4 is based on two test coupons for structural steel G40.21-44W. The experimental stress-strain curves showed some yielding of the material which suggested the use of a trilinear mode1[29. 301. Haaijer clarified in [5] that, during the yielding process, the material is hetero- geneous and yielding takes place in slip bands. The strain actually jumps from its value at the elastic limit to that at the beginning of strain hardening. Therefore, during the yielding process, the element average

I .- , Y

I /

Fig. 3. Bending boundary conditions. (Full stiffener used in stability analysis since the buckling mode may not be symmetrical about Y axis.)

Fig. 4. Stress-strain relationship (G40.21~44W). (1 ksi = 6.9 MN/m2).


stress-strain modulus is less than E but larger than E,. When all the element material has been strained into strain hardening, the material again becomes homoge- neous and the stress-strain relationship of the different theories of plasticity could be applied. From the previous discussion it was decided to adopt a stress-strain model with constant E, once the stress reaches o4 as shown in Fig. 4. From the test coupons E, was found to take an average value of 500 ksi which is valid up to an applied stress of 60 ksi. Stresses beyond 60 ksi are not of practical interest in the current investigations. The modulus of elasticity E and the yield stress a,* were found to equal 29,000 ksi and 44 ksi, respectively.

3. FORMULATION

Buckling of the stiffener in the elastic or the plastic range is the mode that governs the failure of the stiffener. Buckling here is described as a bifurcation of equilibrium under increasing in-plane compressive stress. The formulation is divided into two parts. In the first section the formulation to establish the in- plane stress distribution in the elastic-plastic range is given. This is followed in the second section by the necessary formulation to study the stability problem.

(a) in -plane stress distribution The finite element method is the numerical tech-

nique adopted to delineate the stress distribution in the stiffener. The load is applied in small increments, since the plastic stress-strain relationship is incremental[31], and the stress level is monitored in each element to determine whether it remains elastic or is strained into the plastic range. If the element is in the elastic range its stiffness matrix takes the form [I 51

where [B] defines the element strain-displacement relationship

ic) = VW 1

In (l), [De] defines the elastic matrix moduli

{o) = [o&)3

where

The stiffness of the elements in the plastic range is calculated from the following expression [ 181

where [DC,,] defines the incremental const~tutive re-

lationship in the plastic range. Both the flow theory and the deformation theory for plastic deformations are employed in the present analysis. The matrix moduli [D,] takes a different form in the two theories.

(i) Matrix modui~ oj’jfow theor?. In [ 181 the explicit derivation of the matrix moduh according to flow theory is given. Zienkiewicz et al. in [I 71 gave the same matrix moduli following another approach. Since the incremental stress-strain relationship, according to flow theory. has been given in detail in both references, the derivation will not be repeated here. The final incremental co~stitutive relationship takes the form

where

{dc}‘= jdcr, da,. dr 1

{d<}r= {dc,dc,dyi

(ii) Matrix modu~i of deformution theory. In problems involving plastic buckling a comparison with experimental data[iO] shows that deformation theory is in better agreement than Row theory. This is obviously controversial since deformation theory contains fundamental inconsistencies not present in flow theory. Several opinions are available in the literature to justify the discrepancy between the two theories in problems involving plastic buckling]] l- 131. In a comparative study [ IO] to assess the different possible deformation theories, the theory by Stowell[9] was found to compare best with experimental data. It is therefore adopted in the present study.

Stowell assumed that a unique relationship exists between the effective stress, 5, and the total effective strain, efi which is given by

E, = C/e,

where E, is the secant modulus of the uniaxiai tensile stress-strain relationship

(5 = (a; + c; - (i,cr, + 322)“2 (4)

ep; ( L:+Lifc,Ly+;y2 ‘I2 1 (5)

Although the concept of a yield surface is not clearly defined in the Stowell theory, eqn (4) is considered to describe the yield surface for the stress vecotr (0).

The stress-strain relationship then takes the form

where

& I lv 0

n?pl - 1’ 1

0 0

i 0

l-v

2

Material incompressibility is assumed in the plastic

range, therefore

Elast+plastic buckling of stiffener plates in beam-to-column flange connections 205

unloads in the current rth iteration. Set {ps}l = {0} for that element and omit step 5.

2) = 0.5 (5) Elements that become plastic in the rth iteration are treated as follows

Equation (6) may seem similar to the elastic linear stress-strain relationship (2) but it is nonlinear due to the dependency of Es on the current stress level. Therefore an iterative solution is needed and it was decided to obtain the incremental version of (5) in the spirit of the flow theory of plasticity. Following [9], the incremental stress-strain relationship takes the

(i) Find the ratio m of the purly elastic stresses to the total stress increment {At};. For elements that were plastic at the end of the (r - 1)th iteration set m = 0.

m= %I -di_l

cTi-ci_, .

where

ID,1 = 4

(ii) Find ei (5). (7) (iii) Find Eli = Ci/ei;.

(iv) E, is taken constant and equal to 500 ksi in this

t;$;aluate [D’ ] using {ri};, 8 E’ E via (8). (vi) Calculate {&}I = [D,]; . {A:\:. $ -‘VI). (vii) Evaluate stresses that have to be equilibrated by body forces

(8) {ct}; = (6); - {At};.

Store current strains

and E, is the tangent modulus of the uniaxial stress-strain relationship of the material. Defining q in this form makes it possible to apply deformation theory to elastic-perfectly plastic materials (El = 0).

(iii) Method of solution. Equations (1) and (3) define the element stiffness matrices in the elastic and the plastic ranges, respectively, and are used in conjuction with the initial stress method[l7, 191 to find the in-plane elasteplastic stress distribution. The steps of calculation for the initial stress method using the flow theory are detailed in [17, 191 and will not be repeated here. When using deformation theory the procedure is generally similar to that described in [17] with some changes to reflect the characteristic of that theory. In the following, the steps of calculations using the initial stress method and a deformation theory are shown:

(1) Assemble the global elastic stiffness matrix [Kc’] and factorize it only once at the start of the analysis [@‘I = [L][LIT, where [L] is a lower triangular matrix.

(2) Appl;heload, increment {Ap},, .find through elimination part

{AS}; solution

[K$q{A6}j = {Ap}* Subscript “i” refers to the ith load increment while superscript “r” refers to the rth iteration during the ith load increment. Find {Ai}; = [B]{A6};, th en obtain the corresponding elastic stress increment {At?}; = [D,]{Ai}j.

{Aii j: = {A&}: + {b}; ty).

(xiii) Compute the nodal forces corresponding to the equilibrating body forces in (xii). For any element these forces are given by

{p,}; = j- [BIT{Aii}: d I’. Y

(3) Add (Ati): to the stresses existing at the start of the iteration {c+};-’ to obtain {ti};.

(4) Calculate 5: (4) and compare (1.005 5:) with the yield stress at the end of the i - 1 th iteration a,, _ , . If (1 00%:) is greater or equal to Q,,, _, then the element is plastic and proceed to step 5, while if 1.005 6; is found to be less than o,, _ I the element is still elastic or was elastic in the (r - 1)th iteration but

(6) Assemble the structural global load vector {pJ;.

(7) Find {AS}, ‘+ ’ through the elimination part of solution [K: ]{AS}:+’ = {P,}:, then find {Ai}‘+’ and IAtilY’ ‘. ’ (8)’ Repeat steps 3-7 until the following criterion is satisfied

. I III’~I’IIx 2 0.001 II{AP}iII2.

{At}); = {Ac+};(l -m) - {Au);.

(viii) Store current stresses

{i}; = {i}:-’ + {Ai};.

(ix) Re-calculate c?I using the current stress vector

(6): (x) Find the corrected yield stress oo, that corre- sponds to ei from the relationship

6, = 60. + (ei, -co,). E,

where 0 oO, co. are the uniaxial yield stress and strain, respectively. (xi) Scale the stresses (6); to the yield surface ai,

{b}:={ti}:.$ I

(xii) Scale the stresses {Ae}: as a result of the cor- rection in (xi)

206 H. A. EL-GHAZALY er al.

{S) Recalculate 6, for each element. If 1.005 6, is greater or equal to cr,,; 1 then the element is still plastic and loading, but If 1.005 0, is found less than G?, _ I then the element is considered elastic with a yield stress equal to a,,, _ ,. Steps l-9 establish the inplane stress distribution up to an externally applied load {P},+ , = {P)L__, + {AJJ}, using the deformation theory. The remainder of the steps to conclude plastic buckling will follow the formulation required for the piastic buckling analysis.

fb) Plustic buckling analysis Buckling, as considered herein and mentioned ear-

lier, is the condition for bifurcation of equilibrium. The criterion of bifur~tion commonty used is that the plate must be in equilibrium in the initial p1ane state as well as in a deflected state infinitesimally close to it[28]. All the necessary finite element formulations for the elastic buckling of plates are, explicitly, given in [28] for the 9-degree of freedom bending triangular element and will not be repeated here. Only the necessary assumptions and changes for elastic-plastic buckling are discussed in the fohowing.

In the elastic case the following results from equa- ting the internal and external virtual work when considering the equilibrium of the plate element in the infinitesimally close deflected configuration defined by &[28].

where { ?jr = {G,,V, G,, 2G,,} is a vector of curvature while d,,, 3,, are slopes with respect to X and Y axes respectively-that result from the application of an arbitrary virtuai displacement field defined by i. In the elastic range, M is linearly related to the vector of curvature C through the relationship

where

{Mf = - 1qq (10)

I

+ I12

PI = [L),]2 dz = ; [DJ (11) - 112

and [D,] is given in (2). From (lo), and since [D] is independent of SW

{6MJ = -[D](K). (12)

Therefore, in the elastic range, (9) reads

where [D] is given in (I 1). In the plastic range the [D] matrix (I 1) becomes

nonlinear because of the nonlinearity of the plastic moduh [D,] which complicates the problem. We may circumvent this difficulty by making use of Shanley’s model for column buckling[32]. This model assumed that the tangent modulus would apply over the entire

cross-section with no elastic untoading on the con- cave side of the cross-section. Therefore eqn ( 1 Z), in the plastic range, reads

where

J + r/2

r&J = r4J (14) - r/z

z’ d; = ; [D,].

De,, is the matrix moduli according to the flow theory[17] or the deformation theory (8) and is calculated using the stresses at the incipience of buckling. Equation (13) in the plastic range then reads

On applying (13) to the elastic elements and (15) to the plastic elements, following the procedure shown in [28], the following general matrix equation results for each element

[[k”] + [k”]]jSw f = 0 (16)

where [k*] is the element bending stiffness matrix defined as

WI = s

WtW’l d f’v P

for the elastic elements, and

V$l = J P’l’1~&‘1 df’, b for the plastic elements, and [D], [6J are defined in (11) and (14), respectively, [kg] is the element geometric stiffness matrix and its value depends on the geometrical dimensions of the element and the in- plane stress vector within the eiement.

The full derivation of [kt] and [kq is given in [28]. The assembly process eventually leads to an equation of the form

where

[KJ{Sw) = 0 (17)

[&I= Wl + WI 08)

and [K’] and [Kq are the global bending and geometric stiffness matrices of the structure. According to (17) instability is reached as a result of the combined effects of:

(a) Spread of plasticity, thus reducing [K’], since the plastic moduli [D,] is generally less than the elastic moduli [D].

(b) Increasing the compressive stresses, thus de- creasing the [KgJ matrix.

Mathematically bifurcation occurs when the [&] matrix becomes singular.

Elast*plastic buckling of stiffener plates in beam-to-column flange connections 207

(i) method of sorption. In the case of proportional loading, the elastic buckling load is obtained by directly solving the eigenvalue problem associated with (17) under any arbitrary load {P*}. The first buckling load will be 1, {P *} where 2, is the smallest eigenvalue[33]. This procedure cannot be applied directly in the case of plastic buckling since [KbJ is non-linearly dependent upon the in-plane stress distribution. Instead, the load is applied in small increments and a singularity test is performed on (17) at the end of each load increment. Previous investigators[2,3,34] used the determinant of [&I to check the singularity condition. The determinant does not reveal specific info~ation about the first eigen mode, but collects info~ation about al1 the eigenvector (modes) of the matrix. Gupta[35] proposed an automated digital computer procedure for the determination of the plastic bifurcation loads for plates in compression. This was achieved by a strum sequence method and employing a bisection strategy. In the present study a more convenient measure of the singularity of the [KS] is used, which involves evaluating the smallest numerical eigenvalue, kin, using a slightly modified version of the power method[36]. Buckling is then defined as the load level at which ai” becomes zero. The power method uses iterations to evaluate the largest numerical eigenvalue and the associated eigenvector for a given matrix. Some changes have been made to detect the smallest numerical eigenvalue and the corresponding eigenvector. The scaling technique[36] is also used to prevent numerical instability. The method, as applied in the present work, is shown by the following. Let

[Cl = K ‘I 119)

where [C] is an nth order real symmetric matrix and, therefore, has n linearly independent eigenvectors p(‘). The eigenvalues of [C] satisfy the relationship

/&I > I&l 2 /&I.. * 2 I&,/*

Define an arbitrary vector (Ylo as

(1) (Y), = a,{p,) + a&J + ( ” + a&>.

Multiply (20) by [C]

(20)

f ~“[q@“Jv (21)

and assume

(2) {r }, = [Cl{ q,- I’

Then (21) reads

{t;}, = WIIPJ + ml(h) + ’ * ’ + ~“[Cl{P”). (22)

Also, define the scaling relationship

(Y), J! SI

and

Since [C](pj = n{pf, therefore (22) reads

(4)

Repeating steps (l)-(4) r times gives

or,

Similarly,

Dividing the kth component of { Yfr by the kth component of {Y},_, yields,

(23)

but since I, > &, therefore ld,/R,l’-+O as r+o~ therefore

A, = ‘k-r 5 z.s’=*

In [36] L, could also be obtained as the fo~o~ng inner product

Also {,u,) will be directly proportional to {Yj, at convergence. The previous procedure predicts the largest eigenvalue &, of [C]. The ~lationship between 1,,, for [C] and A,,,,, for [KS] can be easily obtained as follows. Since,

[ClJfl) = 2 {II}. (24)

Pi-multiply (24) by [Cl-’

f IpO =PT’fPf (25)

therefore from (19)

IKl(P) =f 111). (26)

From (24,26) it follows that the largest eigenvalue of [C] is the inverse of the smallest eigenvalue of [KJ provided that the two matrices are positive definite.

CA.3 Vol. i8 No. 2-E


In the previous procedure since [C] is symmetric the rate of convergence will be twice that of the scheme explained before [36]. Moreover the rate of convergence is proportional to the natural logarithm of the ratio of the largest eigenvalue “A,” to the second largest eigenvalue “&” (23). As the buckling load is approached the ratio gets larger resulting in faster convergence. Usually 5-10 iterations are required to find A,,,,,. In (3-a-iii) steps l-9 are needed to find the resulting in-plane stress distribution at the end of load level (Pf,+,. In the following, the rest of the computational steps to check the stability of the plate at that load level are described:

(IO) For each element find the corresponding bending stiffness matrix [.@“I depending on whether the element is elastic or plastic.

(11) For each element find its geometric stiffness matrix [@] based on the current in-plane stress distribution found at the end of step 9.

(12) Assemble the global bending stiffness [!?‘I,+, and the global geometric [Kg],,. , matrices.

(13) Find [Kl, + I= [K’l, + 1 + [K”li + 1. (14) Factorize [K,], + , =,[L_lt: dLl,T, 1. (15) Set S, = 1 and {C,, - \l); for a typical rth

iteration proceed as follows: (i) {r), = {t),iK. & Fi$ $4 ‘ir L 1 from the elimination part of solution

rjtll jr+1 = I y),. (iii) Find i.,, j = (Y}: {4j,+,. (iv) Compare 4 + j = ( Y j t {t 1,+ 1. (iv) Compare &+ , and ;1, if ]A,+, -&I> 1 x 10m5 then return to step (i) to commence the r + lth iteration. (v) If I).,+, - A,1 5 1 x 10 - 5 then the procedure is terminated with, A,,,,, = l/A,+, and I{},+, is the corresponding eigen vector.

(16) The subroutine used to factorize [K,] gives an error message in case of a non positive definite matrix. Therefore if this error message is not issued (i.e. A,,, > 0), then apply the next load increment

{APL., and repe at steps l-15. (17) If an error message is issued because [KJ

ceases to be positive definite then omit step I5 and restart from load level (PI1 applying one-half of {Ap I1 and repeat steps l-15.

Incrementing and decrementing the load continues until numerical bounds are established on the buckling load.

4. NUMERICAL TESTING PROGRAM

The numerical testing program is divided into two phases. In the first phase the proposed method is checked against known solutions to verify the correctness of the computer program. Three cases are solved, the first involves loading a perforated plate in tension to failure due to excessive plasticity. The second case evaluates the plastic buckling load of simply supported long plates using both deformation and flow theories. The third example examines plastic buckling of simply supported and clamped uniaxially loaded square plates using both deformation and flow theories. The second phase of the numerical testing program is devoted to the evaluation of the buckling loads of stiffener plates with different aspect ratios (a/b) and different width to thickness ratios (b/t). AS mentioned previously the buckling load is evaluated

for the stiffener hinged at the column web and either fixed or hinged to the column flange.

(a) Verification (i) The perforated tension strip. Long aluminum

strips were tested by Theocaris and Marketos[37]. The strips were loaded in tension up to failure and the propagation of the elastic-plastic boundary was studied. The same example was solved by Zienkiewicz et al.[17] using the finite element method. The mechanical properties of the alunimum strips are given in [37, 171. The finite element mesh used in the present analysis is similar to the one used in [ 17J. The strip is symmetrica about the Xand Y axes hence only one quadrant is required to study plate behaviour. The displacement component normal to the axes of symmetry is set equal to zero. The case where the hole diameter to total width ratio equals 0.5 is selected to compare the experimental data and the solution using both flow and deformation theories. Figure 5 shows the variation of the strain in the direction of loading at the point of first yield with the applied mean stress (7, [ = applied loads/smallest cross sectional area]. The experimental strain values are obtained directly from test measurements[37] while the theoretical strains are evaluated as the average value of strain in the direction of loading of the elements meeting at the point of first yield. Both the flow and deformation theories gave identical results as can be seen in Fig. 5. The agreement is believed to be reasonable between the experimental and the anal- ytical results. Very slight difference could be noticed in the elastic-plastic boundary as predicted by the two theories but the boundary closely agrees with the plastic zone described in[17]. It may be mentioned that Ref. (171 did not give a specific criterion by which the strain at the point of first yield was calculated.

(ii) Plastic buckling of simply supported long plates. Pride and Heimerl[lO] experimentally evaluated the plastic buckling loads of simply supported aluminum plates. The condition of a simple support was simu- lated by testing seemless square tubes for which each of the four plates may be considered simply supported. To check the present computer program only case Fin Ref. [lo] is used since in this test buckling occurs well within the plastic range. The plate dimensions and properties are as follows:

a/b = 4.5; b/t = 20.8;

E = 10700 ksi; coo, = 50 ksi

dgi = 63.75 ksi; rrDBS = 60.3 ksi

E, co,,,,, b0.85 are the Ram~rg-Osgood parameters[38] and crO* is defined as the limit of initial linear behaviour in the stress-strain curve.

Although the boundary and loading conditions are symmetrical about the plate centoidai axes, the buckling mode may not be symmetrical. Therefore the full plate is used to evaluate the buckling load. The bending boundary conditions are imposed only at the edges. 576 elements and 333 nodes are used for the stability analysis of this plate. The experimental critical stress was found to be equal to 61.4 ksi. This load represents the initiation of out-of-plane displacements of the plates. The maximum stress, after which the

Elasto-plastic buckling of stiffener plates in beam-to-column flange connections 209

t

I IL 81 1.0 2 ” 4.0 L( 3.8

Fig. 5. Development of “c” in ‘* Y” direction at point of first yield (I in. = 25.4 mm).

plate unloads, was found equal to 61.6 ksi. The solution, using the deformation theory, estimates the critical stress as 63 ksi which is reasonably close to the experimental value. The flow theory, however, estimates the buckling stress at 73.2 ksi. The buckling mode according to both of the two theories consists of six half waves in the longitudinal direction and one half wave in the short direction. The experimental report[lO] did not give any information about the buckling mode. If the plate were to buckle elastically five half waves would describe the buckling mode in the longitudinal direction.

Shrivastava in [21], using an asymptotic approach, obtained a closed form expression for the buckling load of long simply supported plates including shear effects. The formulations were applied in conjunction with both the flow theory and Bijlaard’s deformation theory. According to flow theory the critical stress is equal to 70.5 ksi and 6.24 half waves describe the

buckling mode. Deformation theory estimates the buckling stress at 61.5 ksi with 5.55 half waves to describe the buckling mode. From this example it can be seen that the present formulation, when compared with the experimental results[lO] and to other anal- ytical solutions[21], showed satisfactory reliability in estimating both the buckling load and the associated mode.

(iii) Plastic buckling of square plates. The exact solution for the plastic buckling of uniaxially loaded simply supported square plates was obtained by Isakson et al. [20] and is shown in Table 1. The exact solution was obtained using Stowell’s deformation theory[9] and by imposing a buckled configuration taking the shape of a half sine wave in the two directions. Only one quadrant of the plate is used in the finite element analysis because of the complete symmetry of the problem. The results of the finite element analysis using 32 equal triangular elements is

Table 1. Plastic buckling of simply supported square plates

ocr(F.E.M.) Ceformation Ocr(F.E.M.l Flow Theory

b ocr(ksi)Exact theory

t [201 upper limit lower limit upper limit lower limit

ksi ksi ksi ksi

25.68 65 (elastic) 65.35 64.58 65.35 64.58

23.31 75 (plastic) 74.2 73.5 75.5 74.8

20.74 85 (plastic) 84.5 84. 88.3 88.9

17.85 95 (plastic) 94.5 93.7 111.9 111.

14.63 105 (plastic) 106.2 104.26 130.5* 129.*

11.32 115 (plastic) 114.5 113.15 130.5* 129.*

8.37 125 (plastic) 125.4 124.44 130.5. 129:


Table 2. Plastic buckling of clamped square plates

ocr(ksi)

66.414

81.712

91.234

97.549

r; P.E.M. Deformation Theory T upper limit

ksi

67.5

81.28

91.09

97.88

1

I shown in Table 1. The full plate dimensions and properties are as follows.

a = 20 in., b = 20 in.

uniaxially loaded was evaluated in [20] using a discrete element method and applying the deformation theory. Table 2 shows the comparison between the different solutions.

E = 10,000 ksi, cr,,, = 70 ksi, v = 0.5.

The Ramberg-Osgood parameters[38] are

The plate properties and dimensions are the same as the simply supported case discussed earlier. The limits shown in Table 2 are numerical bounds to bracket the buckling load.

u 07 = 100 ksi, croS5 = 90.612 ksi.

It must be emphasized that the limits shown in

(b) Buckling loads of stzxener plates

Table 1 are numerical bounds bracketing the estimated buckling stress at which the [K,] matrix (17)

becomes singular. The results of the finite element analysis using flow theory illustrates that the two theories give comparable results up to certain b/t ratio. For stiffer plates (b/t < 18) the flow theory gives consistently higher failure loads. The same conclusion was reached by Shrivastava[21] for long and short plates. The asterisk indicates that the initial stress solution did not converge in a limited number of iterations (300 in this case) while the [K,] matrix is mathematically still positive definite.

(i) General. Four different aspect ratios are considered in the present study, namely, a/b = 1, 1.5, 2 and 2.5. These ratios are believed to cover the practical range of stiffeners dimensions used in beam-to- column flange connections. For each aspect ratio the width to thickness ratio (b/t) takes on the values 40, 35, 30, 25, 20, 15, 12, 8 and 6 to cover the cases of non-compact to compact sections. Figures 6-9 show a non-dimensional plot of (a,,/~~,) vs (t/b) for each of the four selected aspect ratios. The critical stress (a,,) is evaluated by dividing the total applied load at buckling by the plate cross sectional area (b.t).

The finite element solution is based on a uniform symmetrical mesh of 392, 432,400 and 432 elements

The plastic buckling load for clamped square plates &th 14, 12, 10 and 9

.ower limit

ksi

66.875

80.2

90.2

96.9

I.E.?.!. Flow ‘Theory

qpper limit

ksi

67.5

83.4

96.37

111.3

I I

lower limit

ksi

66.875

82.3

95.5

110.3

1

equal divisions along

0 ‘).05 “.l 0.15 0.2

Fig. 6. Non-dimensional critical stress “u,,/u~,” vs slenderness ratio “t/b” for “a/b = 1”.


loaded edge for a/b equals to 1, 1.5, 2 and 2.5, respectively. The solution is obtained using both the flow and deformation theories and the elastic buckling solution is also shown for comparison. The solution by Haaijer[5] is also plotted and will be discussed subsequently. (ii) Discussion of results and conclusions. The following discussion stems from investigating the non- dimensional plots shown in Figs. 69.

(1) For the same aspect ratio, the effect of specifying the stiffener to be hinged or fixed to the column flanges is to increase or decrease, respectively, the

4

” “.05 0.1 Q.15 0.2

Fig. 7. Non-dimensional critical stress “o,,/u,,~” vs slenderness ratio “f/b” for “a/b = 1.5”.

range of the slenderness ratio at which buckling occurs in the elastic range. This is, obviously, due to the stiffening effect of the fixation.

(2) If buckling occurs in the elastic range the effect of increasing the aspect ratio is to decrease the value of the elastic buckling stress. This could also be concluded as a result of the invariance of the buckling mode with respect to the aspect ratio. The higher the aspect ratio of the plate the earlier it buckles.

(3) If the plate were to buckle in the plastic range the variation of the results due to specifying the stiffener as being fixed or hinged to the column

0 0.05 0.01 0.15 0.2

Fig. 8. Non-dimensional critical stress “~~,/a,,~” vs slenderness ratio “t/b” for “a/b = 2”.


Fig. 9. Non-dimensional critical stress “u~,/o,~” vs slenderness ratio “t/b” for “u/b = 2.5”

flanges is more sensitive when using a flow theory than when using a deformation theory than when using a deformation theory for the various aspect ratios and the various slenderness ratios.

(4) The difference between the results obtained using both of the flow and deformation theories is very sensitive to the slenderness ratio. Agreement between the two theories is limited to a relatively small range.

(5) The solution by Haaijer[5], which is then applied by Graham et al. [4] for a simply supported plate on three sides but free on one unloaded edge, is also plotted in Figs. 69. It can be seen that it over- estimates the plastic buckling stress in all cases as compared with the solution obtained using the deformation theory. The major source of error in Haaijer solution seems to be due to arbitrary choice of the plastic shear modulus G, as 2400 ksi.

(6) The buckling mode in all of the cases has the same shape and was found to be invariant with the a/b ratio. The buckling mode for each aspect ratio is shown at the inset of Figs. 69.

(7) The paper explains clearly the effectiveness of the power method to detect the bifurcation stress in problems involving plastic buckling. The direct evaluation of all the eigenvalues and the associated eigenvectors using conventional subroutines demands considerably more computation time and storage requirements.

(8) The use of the initial stress method to find the in-plane stress distribution when joined with the power method to detect the stability condition proved to be a computationally economical algorithm for problems involving elasto-plastic buckling resulting from non-uniform stress fields.

(9) The paper clarifies the big difference between the results of the flow and deformation theories in plastic buckling problems. On the other hand the two

theories may give similar answers in the analysis of stable structures.

(10) The elasto-plastic stability analysis when applied to the compression stiffener problem gave prac- tically accurate relationships between the buckling stress gc, and the stiffener slenderness ratio t/b for the different aspect ratios a/b.

It must be finally mentioned that the extension of the curves in Figs. 6-9 beyond (cTJu,,) equals to (1.363) is only shown for the academic interest rather than design purposes, since the assumption of a constant tangent modulus of value 500 ksi is imprac- tical in this range of high stresses.

5. DESIGN

The curves shown in Figs. 6-9 are of interest to designers and can be directly used to find the appropriate stiffener sizes. Assuming that premature buckling of the stiffener must be prevented until the framed beams in the connection attain the full plastic moment, M,, and undergo sufficient rotation for a mechanism to form, the maximum stress in the beam flanges is estimated (o,,,) based on the previous conditions. From the ratio (o,,,/a,J and the corresponding ratio (u/b), the appropriate (t/b) ratio can be found which defines the required thickness for the stiffener. The choice of bending boundary condition between stiffener and the column flange, as being hinged or fixed, is left to the designer based on the relative slenderness of both elements. However, based on the recommendations reported in [5,25] a hinged boundary condition is most likely to exist.

In the literature[ 10, 11,201 it has been emphasized that deformation theory gives more accurate and practical results than does the flow theory. This is also confirmed in a recent experimental study[27]. Therefore the authors recommend results based on deformation theory for use in design.


Acknowledgement-The authors are thankful to the lations including strain softening. In!. J. Num. Methoids NSERC for assistance received which made possible the Engng 5, 113-135 (1972). computational work presented in this paper. 20. G. Isakson. H. Armen. Jr. and A. Pifko. Discrete-

REFERENCES element methods for the plastic analysis of structures. CR-803, NASA (1967).

1. A. N. Sherboume, D. N. S. Murthy, J. C. Jofriet and P. 21. S. Varma, Behaviour and design of moment connections. Submitted to CSICC, Toronto (March 1975). 22.

2. A. N. Sherboume and D. N. S. Murthy, Plastic design of beam-column moment connections. Can. Sect. En- gng Conf., Montreal, 1976 23. A. N. Sherboume and D. N. S. Murthv, Computer 3.

4.

8.

9.

10.

Il.

12.

13.

14.

15.

16.

17.

18.

19.

simulation of column webs in structural steel-connections. Comput. Structures 8, 479490. 3. D. Graham. A. N. Sherboume. R. N. Khabbaz and C. D. Jensen, Welded Interior Beam-lo-Column Con- nections. AISC, New York (1959). G. Haaijer, Plate buckling in the strain-hardening range. Trans. AXE (1958). P. P. Bijlaard, Theory and tests on the plastic stability of plates and shells. J. Aero. Sic. 9, 529-541 (Sept. 1949). G. H. Handelmann and W. Prager, Plastic buckling of rectangular plates under edge thrusts. Tech. Note.-1 530, NASA (1948). A. A. Ilyushin, The elastic-plastic stability of plates. Tech. Note-l 188, NASA (1947). E. Z. Stowell, A unified theory of plastic buckling of columns and plates. Tech. Note-1556, NASA (1948). R. A. Pride and G. J. Heimeral, Plastic buckling of simply supported compressed plates. Tech. Note-l 8 17, NASA (1949). F. R. Shanley, On the inelastic buckling of plates. Symp. Plasticitri nella Scienza delle Construzioni-Tenutosi a Villa MonasterwVarenna, Sept. 1956. E. T. Onat and D. C. Drucker. Inelastic instability and incremental theories of plasticity. J. Aero. Sci. (1953). M. J. Sewell, A general theory of elastic and inelastic plate failure-II. J. Mech. Phys. Solids 12, 279-297 (1964). G. Haaijer and B. Thurlimann, On inelastic buckling in steel. J. EMD-Proc. ASCE (1958). 0. C. Zienkiewicz, The Finite Element Method in En- gineering Science. McGraw-Hill, New York (1971). D. R. J. Owen and E. Hinton, Finite Elements in Plus- ticity Theory and Practice. Pineridge Press, Swansea, Wales (1980). 0. C. Zienkiewicz, S. Valliappan and I. P. King. Elasto-plastic solutions of engineering problems “initial stress”, finite element approach. hr. J. Num. Methods Engng 1, 7>100 (1969). Y. Yamada and N. Yoshimura, Plastic stress-strain matrix and its application for the solution of elastic-plastic problems by the finite element method. Inr. J. Mech. Sci. 10. 343-354 (19681. G. C. Nayak and 0.6. Zienkiewicz, filast+plastic stress analysis. A generalization for various constitutive re-

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

S. C. Shrivastava, Inelastic buckling of plates including shear effects. Inr. J. Solids Structures 15567-575 (1979). M. A. Crisfield, Large deflection elastwplastic buckling analysis of plates using finite elements. Rep. LR 593 TRRL. U.K,. (1973). M. A. Crisfield, Large deflection elasto-plastic buckling analysis of eccentrically stiffened plates using finite elements. Rep. LR 725 TRRL, U.K. (1976). K. E. Moxham, Theoretical prediction of the strength of welded steel plates in compression. Univesity of Cam- bridge, Dept. of Engineering (1971). K. E. Moxham, Buckling tests on individual welded steel plates in copression. University of Cambridge, Dept. of Engineering (1971). J. S. Huang and W. F. Chen, Steel beam-to-column moment connections. Nat. Structural Engng Conf. ASCE, San Francisco, California, April 1973. A. N. Sherbourne and H. A. El-Ghazaly, Steel beam-to- columns connections. To be published. I. Holand and K. Bell, Finite Element Methods in Stress Analysis (Edited by I. Holand and K. Bell), Trondheim, Norway, TAPIR, Technical University of Norway. S. L. Lipson and M. I. Haque, Elasto-plastic analysis of single-angle bolted-welded connections using the finite element method. Cornput. Structures 9, 533-545 (1978). J. L. Dawe and G. L. Kulak, Local buckling of Wshapes used as columns beams, and beam-columns. Structural engineering, Rep. 95, Department of Civil Engineering, the University of Alberta, Edmonton, Albertaa, Canada (March 198 1). A. Mendelson, Plasticity Theory and Applicarions. (Mac- Millan, New York (1968). F. R. Shanley, The column paradox. J. Aero. Sci. 13( 12), 678 (Dec. 1946). J. S. Przemieniecki, Theory of Matrix SIructural Anal- ysis. McGraw-Hill, New York (1968). A. N. Sherboume, G. M. McNeice and S. K. Bose, Analysis and design of column webs in steel beam-to -column connections. Submitted to CISC, Toronto (March 1970). K. K. Gupta, On a numerical solution of the plastic buckling problems of structures. Int. Num.. Methods Engng 12, 941-947 (1978). E. Isaacson and H. B. Keller, Analysis of Numerical Methods. Wilev. New York (19661. P. S. Theokiiis and E. x Maiketos, Elastic-plastic analysis of perforated thin strips of strain-hardening material. J. Mech. Phys. Solids 12, 377-390 (1964). W. Ramberg and W. R. Osgood, Description of stress-strain curves by three parameters. TN 902, NASA (1943).

elasto-plastic buckling of stiffener plates in beam-to-column flange connections

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