generalized thin-walled beam models for flexural-torsional...

Computera & Smctwes Vol. 42, No. 4, pp. 531-550, 1992 m4s-‘IJw9/92 s5.00 + 0.00 Printed in Great Britain. psrslmon~plc

GENERALIZED THIN-WALLED BEAM MODELS FOR FLEXURAL-TORSIONAL ANALYSIS

A. S. GEIWY, A. F. SALEBB and T. Y. P. CLUNG Department of Civil Engineering, The University of Akron, Akron, OH 44325-3905, U.S.A.

(Received 3 December 1990)

Ahatrae-With nonuniform warping being an important mode of deformation, supplementary to the other six modes of stretching, shearing, twisting, and bending, we utilii a fairly comprehensive one-dimensional beam theory for the development of a simple finite element model for the analysis of arbitrary thin-walled beams under general loadings and boundary conditions. The formulation is valid for both open- and closed-type sections, and this is accomplished by using a kinematical description accounting for both flexural and warping torsional effects. To eliminate the shear/warping locking in this Co-element, a generalized mixed variational principle is utilized, in which both displacement and strain fields are approximated separately. In this, the strain parameters are of the interelement-independent type, and are therefore eliminated on the element level by applying the relevant stationarity conditions of the employed ‘mediAed Hellinger-Beissner functional, thus leading to the standard form of stltIneas equations for implementation. A rather extensive set of numerical simulations are given to demonstrate the versatility of the models in practical applications involving usage of such components in their stand-alone forms as well as in plate/shell stiffening.

1. INTRODUCTION

The thin-walled, beam-type, structural components and assemblages are widely used in many civil, mech- anical, and aerospace engineering applications, both in their stand-alone forms and as stiffeners for plate/shell structures. Intensive research works have thus been made over the years to develop finite element models that can accurately represent the complex extensional-flexural-torsional-warping de- formational response of these structures (e.g. [l-6,26-31,35,37]).

Traditionally, fluxural behavior has been modeled on the basis of classical Euler-Bernoulli beam theory for undeformable cross-sections, resulting in the so- called C-continuous ‘thin-beam’ elements, with cubic (Hermit-type) interpolation (shape) functions for transverse displacements. However, more recent developments [l, 2,461 emphasized the use of shear- flexible modeling approaches; i.e., CO-type elements, thus extending the range of applicability to thick and anisotropic composite beams where flexural shear deformations become significant, and allowing use of simpler, low-order, shape functions. A major effort here was made to avoid potential shear and/or membrane locking problems in straight and curved beam formulations; e.g., using both displacement- based elements [l, 6,7] as well as hybrid/mixed models [4,5,23].

Turning now to the torsional response component, we note that, in contrast with the uniform (St Venant) torsion of solid-section beams [32], both longitudinal (normal) and transverse shear stresses arise when a thin-walled beam is twisted and restrained from warping [32,39]. These latter warping stresses; i.e., l&moments and bi-twists, and the associated defor-

mations (bi-curvature and warping shear) must be accounted for in the tinite case of thin-walled open sections, the most widely-used approach is based on the so-called sectorial-area theory of Vlasov[39] neglecting the warping shear deformations; i.e., the Cl-type interpolations used in [26-311. For instance, the torsional stiffness matrix in [26-311. For instance, the torsional stiffness matrix in [25,26] was calculated using the closed-form solution for the governing differential equations, while cubic shape functions are utilized in [27,28] for interpolating the twist-rotational degree of freedom. The effect of shear deformations is accounted for a posteriori in [29,30] by means of a number of interactive procedures. An alternative formulation was employed in [31] based on the hybrid-stress method.

On the other hand, the effects of these shear strains are significant in case of thin-walled sections of the closed- or mixed-type, and a limited number of specialized beam models of this CO-type have been reported [21,22,40]. Note, however, that, from the theoretical standpoint, a fairly comprehensive frame- work for developing generalized one-dimensional beam models of general cross-section has been given in [41]. In fact, this same approach is utilixed as a basis for the development in the present paper,

Considering the analysis of thin-walled beams with general unsymmetric cross-sections, the prevailing trend in most of the finite element works has followed the classical notion of selecting two reference lines to uncouple the goveming equations for torsion and flexure; i.e., the centroidal axis for stretching/bending components, and the line of shear centers for shear/twisting/warping actions [26-311. However, de- spite its effectiveness in the linear analysis of 8traQht

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532 A. S. GENDY et al.

thin-walled beams, the approach has a number of disadvantages from the viewpoint of general applicability and extensions to nonlinear problems. For example, there are a number of ‘degenerate’ cases where the shear center is not definable; e.g., a thin- walled circular tube due to ‘section-experiencing- zero-warping’, or cases when it loses its significance as a geometric section property such as in inelastic problems. Also, treatments of the coupled extensional/bending formations in the case of curved beams become extremely difficult when using two reference axes for the unsymmetric-section case [39,42,43]. Indeed, incorrect results may even be obtained if simplified, inconsistent, treatments are introduced in this latter case (e.g. [43]).

Finally, in addition to the one-dimensional models alluded to in the above, several more general developments have also been reported dealing with flex- Ural-torsional analysis of thin-walled beams. A representative of these [37] makes use of specialized, three-dimensional, degenerated-shell theories to account for complex out-of-plane warping patterns of general sections. From the standpoint of solution economy, this is far more expensive than the conventional beam models, although it certainly still leads to much lesser degrees of freedom as compared to a truly three-dimensional shell analysis, depending on the number of thin segments comprising the cross- section.

Our main objective in this paper is to report on the utilization of the hybrid-mixed approach [l l-13,23,34] to formulate two simple one- dimensional CO-straight beam models of the linear and quadratic type, designated here for convenience as HMB2 and HMB3, respectively. In this, we employ the Hellinger-Reissner variational principle with independent spatial discretizations for the displacement and generalized strain fields, together with a fairly comprehensive beam theory accounting for stretching/shear/bending/twisting/warping deformation modes. The conventional assumption of undistorted cross-sections is utilized, but the notion of shear center does not enter the formulation explic- itly; i.e., only one arbitrary reference axis (e.g., chosen here to be the line of centroids for convenience) is utilized for all sectional actions. A fairly extensive number of numerical simulations are given to demonstrate the models effectiveness in practical applications. In light of these latter results, these models appear to be viable candidates for further extensions; e.g., to nonlinear dynamics, inelastic analyses, interactive buckling and post-buckling of stiffened plate/shell structures. These will be topics of our future work.

2. VARIATIONAL PRINCIPLE

2.1, The two-jeld form for geometrically and physically linear continuum

The formulation is restricted to linear elastic ma- terials with small deformations. Here, we utilize a

modified form of the Hellinger-Reissner (using strain instead of stress), two-field, variational principle which for the general continuum case takes the following form [lo], referring to a rectangular cartesian coordinate system x (i = 1,2,3)

I&= 5

[-+rCe +a*i -fru]dV "

where L is the vector of independently assumed strains; u the element stresses; C the material stiffness; u the element displacements; T the boundary tractions, f the body force; 0 a prescribed quantity); ( . )T the transpose of ( . ); P = iii = $uij -I- u,,~) the so-called ‘geometric’ strain-field (the comma subscript is used to indicate differentiation with respect to the spatial coordinate following); V the element volume; and S, , S, are portions of the total element boundary surface area S over which tractions and displacements are specified, respectively, (such that &US, = S and S,,nS, = 0). Note that the last term in the above equation will vanish in the case of a compatible displacement field as assumed here.

Invoking the stationarity of IIR with respect to the independent variations L and II, the following well- known governing equations (i.e., constitutive equations, equilibrium and boundary conditions) are recovered

2 =C’a in V (2.2a)

diva +f=O in I’ (2.2b)

T=T on& (2.2c)

u=ii on S,, (2.2d)

where the divergence operator div u = aUj and sum- mation convention is used for repeated indices.

2.2. Beam functional

Consider a typical thin-walled beam whose longitudinal axis is x and y and z are principal centroidal axes, as shown in Fig. 1, where point 0 represents the centroid of the cross-section. The conventions used for the positive directions of both displacement and stress resultants are also illustrated in Fig. 1. In this case the functional I& for the beam element can be rewritten in the form:

I&= s [+;C6,+u;&Jdx - w, (2.3) L

where L is the element length, and the stress-resultant vector can be conveniently written in terms of extensional/bending and torsion/warping actions as

Analysis of generalized thin-walled beams 533

Fig. 1. A typically thin-walled beam model-generalized forces and kinematic degrees of freedom.

follows:

uR = b,, %,lT w4

u~=LFy3z,~y,MIT (2.4b)

@RI = Wfw, Tm TAT, (2.44

where F, is the normal force; F, and F, the shear forces in the y and z directions, respectively; MY and M, the bending moments; M, the bi-moment (com- prised of a set of self-equilibrating direct or normal stresses); and T,,, and T, the two contributions to the total twist moment M,; the St Venant and warping (or ‘bi-twist’) torsional moments, respectively. It is noted that all of these stress-resultants are defined here with reference to a single axis through points 0.

The independently-assumed generalized strains cR and geometric strains gR (derived from displacements) are defined similarly to eqns (2.4); e.g.

ER ” = [&,lRJT (2Sa)

* tRb=[~~,~~~,rSxy,K~,$lT (2Sb)

~RRI = La r^ r^ ] w, NY w (2Sc)

where Z, is the axial stretch; % and R, the bending curvatures; ?v and FXz the (average) transverse shear strains due to flexure; & the warping curvature; jam and P, are the torsional shear strains associated with the St Venant (uniform torsion) and warping (nonuniform torsion) response components, respectively. Finally, the term Win eqn (1) denotes collectively the work of prescribed external forces and moments.

3. GOVERNING EQUATIONS FOR THIN-WALLED BEAM

3. I. Kinematics

In order to establish a relationship between the displacement components of a point on the beam cross-section to those for its reference line, the assumption of ‘no cross-sectional distortions’ is invoked. That is, II can now be expressed in terms of three displacement components ~b for reference line translations, the rigid cross-sectional rotation vector 8, and the ‘superimposed’ local warping displacement x

u=u,+Br-wxe,, (3.la)

where

(3.lb)

~o=~w~,wJ~; ~=FL,~,,~ (3.Ic)

r = LO, Y, zlT; 9 = [I, 0, 01

0 = w(y, z) = warping function. (3. Id)

In the above equation, r denotes a position vector in the plane of the cross-section, and ei is the unit vector along the beam reference line of centroids.

3.2. Sectorial area properties

Expression for the generalized warping function Q) giving the predefmed pattern of warping displaee- ments over typical cross-sections of different shapes


S \ zAP

9 P

P 5

c (sectorial origin)

0 (pole) v Fig. 2. Sectional profile-coordinates and sign conventions.

are available in the literature. These include contour and thickness warping contributions in open sections 1321 as well as the additional contribution from the ‘indeterminate’ shear flow distribution associated with St Venant torsion of closed sections. In particular, the inclusion of thickness warping is useful in two main respects. Firstly, even for such degenerated cases as open T- or L-sections for which the contour warping vanishes (when calculated with reference to the shear center as in the conventional treatment), it still provides a non-zero warping stiffness, thus no modifications in the formulation are needed. Sec- ondly, it obviates, the ad hoc introduction of St Venant torsional stiffness (e.g. [26-28, 351).

As shown in Fig. 2 and following the terminology of [32], the reference centroidal point 0 is utilized as a pole for constructing the sectorial-area diagrams and subsequent calculation of warping-torsion sectional properties. A right-handed curvilinear coordinate system (x, n, s) is placed on the section profile with n normal to a following contour coodinate s which has its origin at point C, ‘sectorial origin’. A generic point P with coordinate s on the section contour has p and h as its coordinates with reference to the pole 0 in the n and s directions, respectively. The sign convention is that p is positive if the radius vector OP rotates in the positive x-direction when P moves in the positive s-direction, and h is positive if OP rotates in the negative x-direction when P moves in the positive n-direction. The warping function w can then be written as

(3.2a)

(3.2b)

with

s h

13= dn n

(3.2~)

(3.2d)

The iirst term in eqn (3.2b) is the contour warping, and the second term is the indeterminate St Venant shear flow S, in the ith cell of a closed section. For

a single closed cell, for example, this term will be reduced to [ 16,321

(3W

where r is the area enclosed by the profile, t the wall thickness, C(S) the segment of the contour lying between the sectorial origin and an arbitrary point, and the integration is carried out over the entire section profile. On the other hand, b is the thickness warping function. For some sections, e.g. I-sections, UT, is much larger than b, and the thickness warping can often be neglected compared to the contour warping in calculating the warping properties.

Conventionally, the contour warping coordinate cij is normalized such that its integral over the cross- sectional area, A, vanishes, i.e.

s &dA=O (3.3)

A

by selecting an appropriate ‘sectorial’ origin for calculating c.G [32]. This is, in fact, assumed to be the case here.

3.3. Generalized strain -displacement relations

Three strain components and the associated stresses are significant in the present one-dimensional beam model. That is, it can be easily shown that, for the present linear-analysis case, the geometric strain components are given by

1 ~XX=E&--yy$+zi+& (3.4a)

1 * %y = YXU - (WY - z)li, + O,PW (3.4b)

GZ = 9, + 0, - cJs)liN + WS3W (3.4c)

where the generalized strain-displacement relations

1 Eo=ub; fXY = (0; - 6,); f, = (w; + 6,) (3.5a)

r+e;; &=e:; gW=x’ (3.5b)

fm=e:; r^,=(e:-X) (3.5c)

in which the prime indicates differentiation with respect to the axial coordinate x and a comma subscript denotes differentiation with respect to the indicated coordinate y or z.

3.4. Generalized stresses and constitutiue equations

For isotropi&inear-elastic material, the stress components

a = [ exxr axy, axzlr (3.6)


are expressed in terms of strain t as follows:

6 = diag[E, G, G]c, (3.7)

where diag indicates a diagonal matrix, and E and G are the Young’s and shear moduli, respectively. This can be used to define stress resultants as

F, = I

u, dA = EAQ, (3.8a) A

FY = s

ux,, dA = GAlyls;y (3.8b) A

F, = s

a,, dA = GA,y= (3.8~) A

MY= s

c,zdA =EI,K,-EI,,K, (3.8d) A

i&f,= - I

u,y dA = EI, K, + EI,, Kw (3.8e) A

I&=

s

- u,,w dA

A

= EZw K, - EZ= KY + Er,, K, (3.8f)

Tm= 5

[-u,,(z +~,)+d~ -~,)I64 A

= GJY, (3.W

I,-J- s

A [(Q2 + (?#I dA (3W

and where A,, = k,A, with k,, for i = y, z, being the flexural shear correction factors. Using the conventional thin-walled assumption, i.e., neglecting the shear strain in the thickness direction, the expression of torsional warping rigidity, ZP, can be written as

zP= prdA, I

(3.9f) A

where p is the perpendicular distance from the pole 0 to the tangent to the sectional profile at the considered point (see Fig. 2).

Including the effect of thickness warping in the definition of the warping function leads to the familiar expression for the St Venant torsional constants for both open and closed section, i.e.

J = fXb,r f for open section (3.1Oa)

J = g for a one-cell closed section, (3.lOb)

Y T

where b, and 1, are the width and the thickness of the ith plate segment, respectively.

Using eqns (3.5) and (3.8), one can fmally arrive at the ‘resultant-type’ constitutive equation

c-

(3.11a)

(3.llb)

Tw = s

[uxy~y + %~_A dA = G&a - J)Y,, (3.8h) A

where the following definitions are employed for various sectional properties

jAdA =A; JAy2dA =I, (3.9a)

I zZdA = ZY;

i 02dA =Z, (3.9b)

A A

Z Ivy = s

oy 64 ; zwz = s

oz dA (3.9c) A A

J- s

[(Y - a,)* + (z - q)‘1 dA (3.W A

where the dots in the above array indicate xero entries.

Remark 3.1. For a rectangular cross-section (xero warping over cross-section), the following approach can be used for its finite element treatment. Artifi- cially very large ‘penalty’ number is used for the diagonal term G(Z, - J) of eqn (3.1 lb), compared to GJ, thus imposing the physical requirement of xero warping shear strain [see the second equation of (3.5c)] applicable in this case. Simultaneously, a very small coefficient is assumed for EZw (e.g., of the order of wall thickness squared times the smaller of the two bending moduli to reflect the requirement of xero effective warping stiffness. This requires no mod& cations in the basic formulation outlined here.


Zl con tour

“*c* (Y;‘,z;)

2’ ‘i’ L

S

c(Yc

o* (Y;.Z;) Y*

0 Y

Fig. 3. Sectorial poles and origins-coordinate transfonn- ations.

3.5. Displacement and force transformations

Choosing the centroid as a pole for warping function leads to reduced all the displacement and force components are reduced to this reference point. In order to facilitate comparisons of the present finite element results with those obtained by the more conventional, two-reference lines treatments, some of these force and displacements need to be transformed to another point, e.g., shear center. Here, general transformation equations between two parallel sets of Cartesian axes (x, Y, z) and (x*, Y*, z*) are considered [32]. Two contour coordinate systems s and s+, each of them is referred to one of the Cartesian axes, are considered as shown in Fig. 3. The s has its origin ‘C’ with coordinate Y,, z, and a pole 0 with coordinate (0, 0), while s* has its origin at ‘C*’ with coordinate y:, zr and a pole at 0 with coordinates Y$, z$ . The coordinate of C* in the s system is sr, and the 6: is the value of B at C* which is given by

(J)*= c I

. SC d6. (3.12) c

The transformation equations between the displacements of the unstarred and starred systems in the direction of the Cartesian coordinates are

(3.13a)

(3.13b)

wQ=w,-y:e, (3.13c)

0*=0. (3.13d)

The force corresponding to the two coordinate systems can be related by noting that the work must be the same when calculated either in starred or unstarred system (see [32] for more details)

F+=F; F=[F,,F,,F,]r (3.14a)

M*=M+&F; M=[M,,M,,T$ (3.14b)

A4,+=A4,+y,:lu,+zQM,

+ F&o: + yrz: - z:yo’1 (3.14c)

T: = T, (3.14d)

(3.14e)

4. FINITE ELEMENT FORMULATION

In the finite element approximation, the displacements II and strains cR within an element are interp- olated in terms of nodal displacements, q, and strain parameters, /I, as follows:

u=Nq, (4.1)

&R = p,t (4.2)

where N and P are the interpolation matrices for element displacements and strains, respectively; they are polynomial functions in the axial coordinate x. Some comments on the rational for the particular choice of the interpolation functions for strains are given in the following sections.

We note that the strain parameters may be chosen to satisfy the homogeneous part of the stress equilibrium equations within the element. The resulting formulation is then referenced to as an ‘equilibrium’ hybrid element [ll, 141. However, in the present study, and following the recommendations made in recent works [13,23,34], the stress equilibrium conditions are not enforced initially, but we note that these are actually brought in, in an ‘approximate’ (integral or variational) sense, through the use of the functional form in eqn (2.3) itself.

By substituting (4.1), (4.2) into eqn (2.3) and following standard arguments [10-l 3,23,34], we can show that

nR = &=I - Q's (4.3)

where

H = PrCPdx; G = PrCBdx (4.4) s L s L

and the generalized strain-displacement matrix B is defined using eqn (3.5); i.e., iR = Bq. The Q is the equivalent nodal force vector, and q is the nodal displacement vector, defined at any of the nodes in the following ordering

q = ]a, a, W, e,, e,, e,, XIT. (4.5)

The stiffness matrix K for the hybrid/mixed element in eqn (4.3) is given by

K = G7’H-‘G (4.6)

and we also have for the independent strain parameters

Analysis of generalixed thin-walled beams 537

fi = H-‘Gq (4.7)

thus allowing for the elimination of the strain parameters on the element level.

5. APPROXIMATIONS FOR DISPLACEMENTS AND STRAINS

The general hybrid-mixed formulation outlined in Sec. 4 is employed in conjunction with the two beam elements considered herein, namely, the two-model (HMBZ) element and the three-noded (HMB3) element. At each node, seven degrees of freedom are considered. Thus, the total number of kinematic degrees of freedom per element is equal to 14 and 21 for HMB2 and HMB3, respectively.

the difference between the total numbet of kinematic degrees of freedom of the element minus the number of rigid-body modes. Pian and Chen [ll] have pro- posed a simple scheme whereby the total number of strain parameters are minimized while simultaneously suppressing all kinematic modes. In this, we basically choose one #?-term for each corresponding term in strain expressions obtained from strain-displacement relations. In the context of the present beam formulation this can be very easily achieved, and the following specific forms of P in eqn (4.2) are arrived at; i.e., for the HMB2 element

P = I, (5.3)

where I is a diagonal (8 x 8) unit matrix; and for the HMB3 element

1 r 0 0 0 0 0 0 0 0 0 0 0 0 0 0’ OOlrOOOOOOOOOOOO OOOOlrOOOOOOOOOO

p= OOOOOOlrOOOOOOOO OOOOOOOOlrOOOOOO OOOOOOOOOOlrOOOO OOOOOOOOOOOOlrOO 00OOOOOOOOOOOOlr,

(5.4)

5.1. Displacement

Here, only compatible displacement functions are used and they are the familiar lagrangian polynomial interpolation function for one-dimensional element [6-8]. That is, with natural coordinate

2x r= --I, ( > L

the interpolation functions are, for HMB2 element:

N,=$l -r); N,=# +r)

and for HMB3 element

(5.1)

N, =$(l -r); N,=i(l +r) (5.2a)

N, = (1 - r2). (5.2b)

5.2. Strains

In general, there are two important considerations that must be taken in the selection of strain parameters: (1) all the zero-energy kinematic modes must be avoided and (2) the ability of resulting element to handle applications to the constrained problems. Each of these is discussed separately below.

Suppression of kinematic deformation modes is perhaps the most important required in the hybrid- mixed formulation. A necessary condition for the stiffness matrix to be of su3lcient rank is that the number of parameters should be greater or equal to

The second consideration is to check the element behavior so that any potential locking problem is precalculated for its applications to ‘thin’ structures. A convenient way to examine this behavior is to use the method of constraint counting, often termed the ‘constraint index’ as suggested by Hughes et al. [7-g]. See also [23,34] for the utilization of this same concept in the context of mixed-type variational formulations. The constraint index, C1, is defined as the difference between the number of kinematic degrees of freedom brought by an element, when added to an existing finite element mesh, and the number of independent constraints per element when it is used in limiting case (e.g., thin beam). A favorable value of CZ is equal to or greater than one which implies that the element is locking free.

As mentioned above, it is necessary to choose at least one term in the strain polynomial corresponding to each of the basic deformation mode involved in the assumed displacement field in order to suppress all kinematic deformation mode. Which strain component the aforementioned b-term is assigned to is, however, immaterial. It is precisely this freedom in assigning the B-term to various strain components which enables us to control the locking phenomenon. By exercising this freedom, as for the present case of eqns (5.3) and (5.4), the number of element constraints is minimixed and the ‘constraint index’ is thus improved. Ultimately, it leads to better numerical performance of the element in question.

For illustration, consider the HMB2 element used to represent the response of a general-section beam


Fig. 4. A torsionally loaded cantilever beam.

under bending and torsional actions (uncoupled extensional-flexural/torsional behavior still exists with the centroid being selected as the reference point). In the limit, as the beam thickness approaches to zero, both flexural and torsional shear deformations should vanish. Correspondingly, the three strain parameters /I*, &, and /I, associated with shear strains due to flexure and warping approach to zero for such a general-section case. Referring to eqn (4.7), this will then impose three constraint conditions. Thus, when an HMB2 element is added to an existing mesh, the resulting CI is (6 - 3) = 3 (with six pertinent kinematic degrees of freedom associated with flexural/torsional response), hence no locking is expected in this case.

Turning now to the case of a three-noded mixed element HMB3, the CI = (12 - 6) = 6, thus indicat- ing an improved performance as far as nonlocking response for thin beams is concerned. However, a more critical situation arises when we consider the uncoupled-flexure-torsion behavior for bisymmetric- section beam, i.e., considering warping/torsion response, we now have the reduced values C1=(2-l)=l, and CZ=(4-2)=2, for HMB2 and HMB3, respectively. On the other hand, the corresponding calculation for the counterpart displacement-based models, with exact integration for the associated component stiffness arrays, will indicate that, for the example of the two-noded element with a general cross-section (that is using two Gauss integration points), CZ is equal to (6 - 6) = 0, thus indicates complete failure due to locking in the thin beam limit. Note, however, that the quadratic displacement model (that is three Gauss integration points) gives CI = (12 - 9) = 3 in the above calculation. Although this indicates no failure due to locking, its accuracy has been actually reduced, i.e., it is only comparable to the accuracy of the linear HMB2 element. This conclusion has indeed been supported by numerical results given later in Sec. 7.

6. ELEMENT STIFFNESS MATRICES

Once selection of displacement and strain interpolation functions, as defined before, is made, the stiffness matrices for HMB2 and HMB3 can be obtained from (4.6), in which the matrices H and G are found by carrying out the appropriate integration indicated in eqn (4.4). Also, the ‘consistent’ nodal load vector Q can be found from the appropriate

terms, e.g., corresponding to specified load components, in W expression of (2.3). The stiffness coefficients corresponding to the HMB2 elements is given in Appendix A.

For the purpose of comparing analysis results, we also obtained the stiffness matrix for a displacement- based beam element with two nodes, designated here as DB2 element. In this case, the stiffness matrix is obtained from the ‘familar’ expression (e.g. [1,2])

K= BrCBdx, s L

(6.1)

where B is the generalized strain-displacement matrix. The stiffness coefficients corresponding to this element is given in Appendix B.

7. NUMERICAL SIMULATIONS

In order to determine the numerical performance of the hybrid-mixed elements developed, a number of test problems are considered in this section. These include a variety of thin-walled beams under different flexural-torsional loading and support conditions, both in their stand-alone forms as well as in their usage as stiffeners for plates. All calculations were performed in double precision on the IBM 3090 Computer at the University of Akron.

7.1. The warping locking phenomenon

The effect of torsional warping shear locking is investigated first by considering an I-section cantilever beam under tip torsional moment M,, as shown in Fig. 4. The beam is modeled by a single element. In this case, using the stiffness matrices for HMB2 element, yields the following expression for tip rotation 0,

e,=$(;+;); CI =(@, (7.1)

where, for simplicity, we neglected the effect of the St Venant torsional rigidity, J, compared with IP in eqn (7.1), (which is an acceptable assumption for the present thin-walled, open-type, I-section). On the other hand, for DB2 element, one obtains

( > 1+; OX=%&-.

w 1+; ( >

(7.2)

30 LO 50

Number of Elrmmnts

--MM2

10 20 30 LO 50 Numbw of Ekmmnts

Fig. 5. Convergence study for tip rotation of a cantilever beam. (a) L/b = 10, (b) L/b = 0, (c) L/b = 40.

The expression (7.2) clearly illustrates the effect of warping-shear locking of the DB2 element. That is, for the limiting case of a thin beam, t( becomes very large; i.e., proportional to (Z./b)2 for the present I-section, where b is the flange width, and conse- quently the predicted value of 0, approaches to zero, obviously a worthless result contradicting the physical solution considered.

On the other hand, the expression obtained from the hybrid mixed element HMB2 is valid for the entire range of aspect ratio, L/b, of the I-beam. In fact, the normalized tip rotations, normalized with respect to analytical solution [16], using only one element and with L/b = 10, are 0.945 and 0.055 for HMB2 and DB2, respectively. The corresponding values in the case of a thin beam L/b 3: 40, are 1.077 and 0.030, respectively. Note that the analytical solution in [16] does not account for the effect of shear deformations due to either bending or warping.

7.2. Mesh convergence

To test the convergence characteristics of the elements developed, the cantilever beam shown in

Fig. 4 is analyzed again for three different values of aspect ratio (L/b = 10,20,40), using different meshes of HMBZ, HMB3, and DB2 elements. The normalized tip rotations versus mesh sizes are plotted in Fig. 5.

As shown from Fig. 5, both HMB2 and HMB3 elements do not exhibit any sign of locking. In fact, for all values of L/b considered, the solution using four HMB2 elements gives the results with less than 1% error, while using four HMB3 elements yielded results with less than 0.2% error. On the other hand, the convergence is very slow using DB2. For instance, for L/b = 40 (see Fig. 5c), and even with a mesh of 50 DB2 elements, the solution still exhibits an error of approximately 2%.

The curves for normalized warping torsional moment at the 6xed support versus mesh sizes are plotted in Fig. 6. Here, due to the severe locking exhibited by DB2 element for the thin beam, L/b = 40, the solution using 50 elements is in error by approximately 900% as shown in Fig. 6(c). On the other hand, very accurate results using 16 HMB2 elements and eight HMB3 elements are obtained in all cases.

A. S. GENOY et at.

(a) 12

Number of Elements

(b) 16

6

10 20 30 40 50

Number ot Elements

Numba of Elwncnts

Fig. 6. Convergence study for warping-torsional moment of a cantilever beam. (a) L/b = 10, (b) L/b = 20, (c) L/b = 40.

7.3. Stress predictions

Two additional problems are considered here to further evaluate the stress-prediction capability of the HMB2 elements. The numerical results are compared with those obtained by other investigators.

7.3.1. Torsionally loaded cantilever beam. Once again, we consider the analysis of a cantilever beam with an I-section having the following properties in

torsion: 1, = 0.1667(10)‘” mm6; J = 0.1333(10)6 mm4; Ip = 20(10)6 mm4 and L = 2000 mm, and with L/b ratio chosen to be equal to 20, and E = 200 GPa and G = 77 GPa, respectively. The beam is subjected to unit tip torsional moment. The distributions of bimoments along the beam length by using 16 HMB2 elements are compared to the analytical solution [16] in Fig. 7. The calculated stresses at the mid-points of each HMB2 element are fitted by the continuous curve shown in Fig. 7, evidently leading to quite


--- - Analytical Method

025

Fig. 7. Bimoment distribution for a torsionally loaded cantilever beam problem.

accurate results, compared to the fairly complex distribution given by the exact solution along the beam

iU, = O.O57[tanh( 1.755L)cosh( 1.755x)

- sinh( 1.755x)],

where x is the distance from the fixed support as shown in Fig. 7.

7.3.2. Continuous beam under concentrated torsional moments. A four-span continuous crane girder is considered in Fig. 8. The only torsional moment occurs in the first span AB and is due to horizontal forces H = 6.7 kips (29.8 15 kN) exerted by the wheels at the top of the crane rail. The location of the wheels in the span corresponds approximately to that which causes the largest warping moments in the beam [15]. The resulting torque at each wheel location is 115.78 kipin (6.33 kNm) referred to the centroid. The assumed value of material properties of steel are E = 29,000 kip/in2 (199,810 MPa), and G = 11,154 kip/in2 (76,851 MPa). In addition to conventional bending constraints, the simply supported boundary conditions correspond to no-twisting-but- free-warping at all ends and intermediate supports (from A to E).

This girder is idealized by nine HMB2 elements for each span. The results of bimoment, warping/twisting moment (both of these moments are transformed to the shear center), and St Venant torsion are depicted

in Fig. 9 together with the analytical results reported in [ 151. As is evident from the plot, HMB2 predictions are almost identical to those given by Stefan in [15]; the maximum deviation is about 1%.

7.4. An end-loaded perforated core

Considering the torsional analysis of beams with closed-type sections, two perforated core problems are solved here. To prevent the incompatibility in the warping function at the intersection of closed and open sections, a common approach has been to replace the connecting beams at each floor level by a continuous distribution of a connecting medium which is assumed to be with uniform properties throughout the height of the structure [21]. This same scheme is also utilized in this study and the results of HMB2 elements are compared with analytical and experimental results available [17-221.

Care 1: The analysis is applied to a 20-story model perforated core structure. Plan dimensions of core are: a = 6.25 in (158.75 mm) b = 4.75 in (120.65 mm); thickness of wall and connecting lintels t = 0.244 in (6.197 mm). The story height is 2.45 in (62.23 mm) and the depth of lintel beam is 3/8 in (9.5 mm). The model is subjected to an applied unit torque at the top and is fixed at the base (i.e., 0, and x are prevented). The properties of material used for this model are E = 4.3 x lo5 psi (29.67 N/mmt) and G = 1.6 x 105psi (11.04N/mm2). The comparison between results given by 40 HMB2 elements and those given by analytical and experimental


A M, M, B C E -- I 1 I

WV? A A A? 1273 10.60’ 6 ’

I 75’

I ’ 1

JO’ 1 30’ / JO’ 1 30’ I

I 1 1 1

F=6.7 kips 100.71 +

131.11

9

I w- Diagram

96.64

. Geometric Data . A = 46.75 in2 I,= 6222.9 in4 :

I,, = -7696.1 in5 ,

I,= 7620.2 in4 , I*= 863.7 in4 J = 11.11 in4 , I”= 120987.3 in6 I wz = 0.0

Fig. 8. The problem of a continuous beam under concentrated torques.

_ 2000 N.

f

.$

zi -2000 - St&n (1965

-4000

-60

20

;i IO

:0 J

2 -10

-20 1 I I I I

A B C D E

Fig. 9. Torsional-warping stress resultants of a continuous beam.

Analysis of generalized thin-walled beams

Jmkinr k Harriron (1

05 1.0 1.5 2.0 2.5

q/M.(lO)-‘(rad/N.m)

b)

(c) 20

1s

LO z

5

-0.25 0.0 0.25 0.50 0.75 1.00 1.25

M./M. (N.m’/N.m)

Fig. 10. A perforated core under end torque. (a) Rotation of open-section, (b) rotation of mixed section, (c) bimoments.

methoda [ 17-211 for rotation O,, and bimoment M,, compare favorably with those in [I81 in the open-sec. along the height of the structure for two cases: (i) tion case (Fig. lOa), and they are very cloa to the open croaa+ection (no lintel beams); and (ii) lintel analytical and experimental results of [21] when the beam (d - 9.5 mm) are considered at each floor, as lintel beams are considered (Fig. lob). Again, from shown in Fig. 1qa-c). The results given by HMB2 the plot in Fig. IO(c), the HMB2 curvea provide a

A. S. GENDY et al.

(b)

a)

!

i i

:

- HMt32

i _-____ Method 1

i Method 2

i -Method 3

i P Bredt

\

L

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.1 C.2 0.3 0.4 0.5 0 6 0.7 0.8

d/lb d./Cb

Fig. Il. A perforated core with lintel beam. (a) Rectangular cross-section, (b) square cross-section.

-HMBZ

- Method 1

- Method 2

.... .....‘.‘. Method 2 + Michael’s Correction

- - -.- Eqwvelont Frame _____ Fintte Element

- Bredt

good fit of the analytical bimoment distribution along the length of the model.

Case 2: In order to examine the performance of the HMB2 to handle the effect of shear deformations, two IO-story perforated core structures with different cross-sections are considered; see Fig. 11. The story height is 3.5 m. The depth of lintel beam to its span (d/l,), ranges between 0.1-0.7. The properties of material are E = 3.0 x lo6 t/m*, and v = 0.2. The structures are subjected to unit torque at the top and are fixed at the base. The torsional rotations along the core versus (d/l,) ratios are shown in Fig. 1 l(a, b) for HMB2 and some of the analytical and numerical

1’

:

20’ fl”I S,lO w5’

Geometric Dab

-+

‘50

c

methods given in [22]. Several solutions are included for comparisons in Fig. 11; e.g., the so-called method 1 signifies an analysis neglecting the shear deformation effects (open-section model), while method 2 is based on Omanshy-Benscoter [32] theory which considers the shear deformations, a finite element model and wide-column frame analogy given in [24], Berdt’s formula [22] for an equivalent continuum replacing the lintel beams, and finally the alternating open-section/closed-box model which is designated as method 3 in Fig. 11(b).

The HMB2 gives results which are in close agreement with those given by method 2, especially when

85.72

/jL+=+

I’

A = 70 ft2 5253.3 ft4 I,= 4192.9 ft4 :

I,=

23.3 ft4 I,= 6775.6 ft*

J = , I,= 2549634.4 fP I WY = 0.0 I,, = 103296.3 ft’

342.86

W _ Diagram

Fig. 12. The problem of a laterally loaded unsymmetric tall building.

Analysis of gencrdizcd thin-walled beams 545

(a)

::I l-X---t ---- Smith k Taranoth (1966)

16.0

6.0

0.00 ) I I

0.00 0.30 0.60 0.90 1.20 1.50

Angle of Twist x lo-’

0.00 I I I 1 I 0.00 1.00 2.00 3.00 4.00 5.00

Bimoment (Ib.ft’)

Fig. 13. Unsymmetric tall building. (a) Rotation, (b) bimoment.

the d/l, ratio gets bigger (i.e., with significant shear deformations).

7.5. A laterally loaded unsymmetric tall core building

An unsymmetric core-supported structure is considered next. Figure 12 gives the assumed dimensions and loading. This structure consists of 15 stories each of 12.5 ft. The Young’s modulus and Poisson’s ratio are E = 576,000 kip/ft’ and v = 0.2, respectively. This core is modeled by 15 HMBZ. The distribution of twist rotation and bimoment (referred to the shear center) along the core height are shown in Fig. 13, where they are compared with those given in [33].

7.6. A torsionally loadednon-prismatic cantilever beam

A cantilever beam with a variable cross-section and subjected to concentrated torsional moment at the free end is considered in Fig. 14. The depth of the cross-section varies linearly from d at the fixed end to 0.4d at the free end, corresponding to the ‘most flexible’ case as considered in [35]. The beam is idealized using 10 HMB2 elements whose sectional properties of the integration points (one point for each element in this case) are obtained by interpolation of the cross-section dimensions at the nodal points. The elastic material constants are E = 65347.1 x lo* N/cm* and v = 0.237. The angles of twist along the beam span are compared with those reported in [35] as shown in Fig. 14.

7.7. Two-span cable deck bridge

( b) 16 I 1 I 1

- HUB2 - Smith k Taranath (1966)

HMB2 elements are used to model a two-span cable deck bridge subjected to eccentric load as shown in Fig. 15. The girder is idealized using 25 elements, while one element is used for each of the cables or the tower members. The pertinent material and geometric data are given in Fig. 15. Since the girder is stiikned by diagrams, the undistorted cross-section assump tion employed in the beam model is valid in this case. The vertical deflections along the line of centroids of

Fig. 14. A torsionally loaded non-ptkmtic cantilever . As an example of more practical significance, the

0 42 a4 ae aa I.0 X/L

Ram.

546 A. S. @3NDY et al.

2

6.33 7.75 7.75 6.18

i&i ;5.40 mm

6.52 6.52

ial and Geometric Datsx

&&

u = 0.3768 I,= 151.07 Cm’ , I,= 1326.76 Cm+ J = 288.07 Cm4 , I,= 4642.59 Cm6

I, = 0.0

E - l.O5S(lCf kg/Cm2 , A = 0.0!767 cm=

&&.$&

E - 2.1(lOf kg/Cm’ 4 A = 1.0066 Cm*

Fig. 15. The problem of a two-span cable deck bridge.

0.010 - HMB2 1

- Timoshenko 0 HM82

ob I I I i 0 2 4

El /we 8 K)

Fig. 16. Weotion along the.~~od ?f.“f” upper 5ange of a two-span caoLc aecK onage.

Fig. 17. Deflection at the center of a stiffcasd, uniformly loaded. aauare olpte. .I .

Analysis of grwalbd thin-walled beams 547

0 2 4 6 6 K)

El /DL

Fig. 18. Moment M, at the center of a stiffened, uniformly loaded square plate.

the upper flange are plotted in Fig. 16 which agree remarkably well with the experimental results of [38], as well as with those reported in [371 using finiteel- ement shell-type analysis.

7.8. A tmt~ormly loaded sttrened square plate

Our linal test case concerns the utilization of HMB2 elements as stiffeners for plated structures. Here we consider the case of a square plate simply supported on two opposite edges and stiffened on the two other sides by two thin rectangular-section beam members (see Fig. 17). The plate is subjected to uniform pressure loading. Due to symmetry, only one quarter of the plate is idealized using a 4 x 4 mesh of the mixed plate/shell element HMSHS developed previously in [34], together with four HMB2 elements for the stiffener.

The solutions for the plate center deflection and center plate moments M, were obtained for a variety of stiffener-to-plate flexural rigidity ratios. These are plotted in Figs 17 and 18 and compared with the classical results of Timoshenko and Woinowsky- Kreiger [44], where the very good agreement of results in every case is noted.

8. CONCLUSION

Based on the ‘modified’ Hellinger-Reissner variational principle, two Co straight beam elements with linear and quadratic displacement assumptions, respectively, were developed. Their effectiveness were subsequently demonstrated in a fairly comprehensive set of numerical examples, including several compli- cated structures, such as perforated shear walls, nonprismatic members, cable-stayed box girder bridge, stiffened plates, etc. The models were shown to be free from flexural- as wall as warping-shear locking, and exhibiting fairly accurate stress-predic-

tion capabilities. Fkrallel studies concerning their extensions for the analysis of curved beams, spatial lateral-torsional buckling, nonlinear dynamics, etc. a in progress and the associated zesults will be reported in the near future.

Acknow&&evnent-This work is supported by NSF Grant No. EET-8714628.

REFERENCES

1. K. J. Bathe. Finite Element Procedures in &tgker& AM/ysi.r. PrenticeHall, Englewecd ClitTs, NJ-(1%2).-

2. R. D. Cook. htCCDt8 and zidcat~ of I&irr lZi&etU Analysis, 2nd Edn: John Wiiey, New Yerk (1982).

3. 0. C. Zienkiewi~ The &rite E&meat MetM 3rd Edn. McGraw-Hill, London (1977).

4. A. K. Nocr, W. H. Greene and S. J. Hartley, Nonlinear finite element analysis of curved baams. Comput. Meth. Appl Mech. Engng 12, 289-307 (1977).

5. A. K. Noor and J. M. Peters, Mixed model and reduc&elective integratien dirplrosnrent models for nonlinear analysis of curved beams. Int. .r. Namer. Meth. J3tgng 17, 615-631 (1981).

6. H. St&r&i and T. Belytacbko, Sbesr and meatbrute leekin in 0 curved C elements. Compnt. Meth. Appl. Mech. Engns 41, 172-176 (1983).

7. T. J. R. Hughes, M. Coben and M. Haroua, Redti and selective integration teclmiqws in the finite element analysis of plates. Nucl. Engng DeJign 46, 303-322 (1978).

8. T. J. R. Hughes and W. K. Liu, Nonlinear finite element analysis of sbellsz part I. -onal &ells. Comput. Meth. Appl. Mech. Engng 26,331-362 (1981).

9. D. S. Malkus and T. J. R. Hugbe+, Mixed &rite element method-reduced and selective integration tecbniqw: A unification of concepts. Comput. Meth. Appl. Mech. hgng 15, 63-81 (1978).

10. K. Wasbixu, Variational Method in Ebticity and Pb ticity, 3rd Edn. Per8amon Press, Oxford (1982).

11. T. H. H. Pian and D. P. Cben, Alternative wavs for formulation of hybrid stress elements. ht. J. h&w. Meth. Enpnp lg. 16791684 11982).

12. T. H. H. &I, Recent advances in hybrid/mixed finite elements. Proceehgs of the International Confm of Finite Element Methoak, pp. 1-19. Sbangbai, china (1982).

13. T. H. H. Pian, Finite element based on consisteatly assumed stresses and displacements. J. Icynite Ekmmts Anal. Des. 1, 131-140 (1985).

14. T. H. H. Pian and P. Tong, Basis of finite ahant methods for solid continua. I&. J. Numer. Meths hhgng 18, 3-28 (1959).

15. S.. J. M&twadowski, Warpins moment distribution. J. Struct. Engng III, 453-455 (1985).

16. R. D. Cook and W. C. Young, A&aced Mec~ of Mater@. MacmiUan, New York (1985).

17. T. C. Liauw and K. Wai Leung, Torsional analysis of core wall structures by transfer matrix metbocl. m Structurd Engineer 53, 187-194 (1975).

18. W. M. Jenkins and T. Harrisen, Analysis of tall building with shear wall under bending andtorsion. &UC. of Symposium on Tall Bui&g& Pergunon Prras (1967).

19. D. Miebael, Torsion co@ing of ecme wags in tall buildings. Fhe Sttwctural Brsfmt 47.2 (1969).

20. W. K. Tse and J. K. Biswas, Aaslyds of mre wall structum subjected to applied torque. B&i. sel. IJ, 251-252 (1973).

21. A. V. R&b&a and W. K. Tso. Torsional u&sis of perforated cm-structure. J. St&t. Dfu., AS& 101, 539-550 (1975).

548 A. S. Gir~nv et al.

22. A. Rutenberg and M. Eisenberger, Torsional analysis method for perforated cores. J. Srruct. &gag 112, 1207-1227 (1986).

23. A. F. Saleeb and T. Y. P. Chang, On the hybrid-mixed formulation of Co curved beam elements. Compur. &f&r. A&. Me& figrzg 60, 95-121 (1987).

24, I. A. MacLeod and H. M. Hosny, Frame analysis of shear wall cores. J. Sfruct. Div., ASCE 103,2037-2047 (1977).

25. J. L. Krahula, Analysis of bent and twisted bars using the finite element method. AIAA Jl5, 1194-I 197 (1967).

26. A. B. Chandhary, Generalized stiffness matrix for thin walled beams. J. Struct. Div.. ASCE 108. 559-577 (1982).

27. R. S. Barsoum and R. H. Gallagher, Finite element analysis of torsional and torsional flexural stability problems. Inr. .I. Namer. Meth. Engng 2, 335-353 (1970).

28. J. L. Meek and P. Swannel, Stiffness matrices for beam members including warping torsion effect. J. Engng Me& Dir., ASCE 193-197 (1976).

29. A. Pittaluga, Recent developments in the theory of thin walled beams. Comput. Strut. 3, 69-79 (1978).

30. J. Vasquez and R. Riddel, Thin walled core element for multistory building. J. Sfrucr. Div., ASCE 1021-1034 (1984).

31. J. L. Meek and P. T. S. Ho, A simple finite element for warping torsion problem. Cornput. h4eth. Appl. Meek. F&rig 37, 25-36-(1983). -

_.

32. A. Gielsvik. The Theorv of Thin Walled Barz. John Wiley, New’York (1981). _

33. B. S. Smith and B. S. Taranath, The analysis of tall core-supported structures subjected to torsion. Proc. Inst. Ciuil Engrs, fart 2 53, 173-187 (1972).

34. A. F. Saleeb and T. Y. Chang, An efficient quadrilateral

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

element for plate bending analysis. Znt. J. Numer. Meth. .Ehgng 24, 1123-l 155 (1987). J. W. Wekexer, Blastic torsion of thin walled bars of variable cross sections. Comput. Stnrct. 19, 401-407 (1984). S. ~tipu~~ and N. T&air, Elastic stability of tapered I beams, J. Struct. Div., ASCE 98, 713-721 (1972). W. Kanok-Nukulchai and M. Sivakumar, Degenerate elements for combined flexural and torsional analysis of thin-walled structures. J. Strut. Div.. ASCE 114, 657-674 (1988). S. Lankar, Bending torsion of thin-walled straight and curved beams with variable open cross section. Thesis presented to the Swiss Federal Institute of Technology, Zurich, Swltxerland, in partial fulfillment of the requirements of the degree of Doctor of Technical Sciences, V. Z. Vlasov, Thin-Walled Elastic Beams, 2nd Bdn. Israeli Program for Scientific Translation, Jerusalem (1961). S. Krenk and 0. Gunneskov, Statics of thin-walled pretwisted beams. Inr. J. Numer. Meth. Engng 17, 1407-1426 (1981). E. Reissner, On a variational analysis of finite deformations of prismatical beams and on the effect of warping stiffness on buckling loads. Z. Muth. Phys. 3!$ 247-251 (1984). C. H. Yoo, Matrix fo~~~on of curved girders. J. Engng Mech. Div., ASCE laS, 971-987 (1979). A. Gendy and A. F. Saleeb, On the finite element modeling of curved beams with arbitrary thin-walled sections. Submitted (1990). S. P. Timoshenko and S. Woinowsky-Kreiger, Theory oj Plates and She& McGraw-Hill, New York (1959).

APPENDIX A: STIFFNESS MATRIX FOR THE HME2 ELEMENT

The coefficients of the stiffness matrix in the special case of a doubly symmetric section for the HMB2 element, eqn (4.6). are given below. The nodal diligent degrees of freedom am ordered as in eqn (4.5). For ~rnpl~~, the stiffness matrix can be written in the partitioned form

K= &I K,, [ 1 &I Kzz ’

where the (7 x 7) submatrices, K# (i,j = 1,2) are given by (dots indicate zero entries)

K,, =

4 - 1

Gk,A . GkyA L 2

Gk,A -Gk,A __ 1

L 2

crp G(Z, -3) 2

(A-1)

(44.2)

KIZ =

GA,A L

-GkyA

2

EA

t .

CkyA L

Analysis of gcnmakd thin-walled beams

Gk,A - . L

-GI,

Gk,A 2

Gk,A

L

L

GkyA 2

-Gk,A

2

GU, - J)

(yy) . 1

. (s+Eg) *

EI,

-r+

G(I, - J)L

4

-Gk,,A

2

Gk,A 2

-G(I,- J)

)_

549

64.3)

(A.4)

APPENDIX B: STIFFNESS MATRIX FOR DB2 ELEMENT

Using the analogous expression to cqn (A.l), the submatrica of K in this case [see cqn (6.111 arc given by

K,, =

GkyA 2

Gk,A . -Gk,A

L 2

crp G(I, - J) L 2

(sm.) (s+yL) ,

El,,

L+

G(Z, - J)L’

3 I

550 A. S. WENDY et al.

Kn =

EA

t

L

Kn =

-Gk,A

L

-Gk,,A

2

I-

-Gk,A

L

+Gk,A

2

GkyA . L

Gk,A

L

-c’p L

-G(I,,-J)

2

G’I, L

fGk, A

-Gk,A

2

G&-J) 2

(%+““a3

. (s+E$!k)

b

-GkyA

2

+Gk,A

2

-G(I, - J)

2

(symm.) e + y) .

(B.2)

(B.3)

generalized thin-walled beam models for flexural-torsional...

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