ON THE FINITE ELEMENT ANALYSIS OF THE …blogs.uakron.edu/cmg/files/2015/11/On-the-Finite-Element-Analysis...Spatial response of curved beams 641 Nodal section section Fig. 1. A typical

Com/wters d Struclwes Vol. 44, No. 3, pp. 639652, 1992 0045-7949192 S5.W + 0.00 Printed in Great Britain. Pergamon Press ttd

ON THE FINITE ELEMENT ANALYSIS OF THE SPATIAL RESPONSE OF CURVED BEAMS WITH ARBITRARY

THIN-WALLET SECTIONS

A. S. GENDY and A. F. SALEEB

Department of Civil Engineering, University of Akron, Akron, OH 443253905, U.S.A.

(Received 13 May 1991)

Abstract-A three-Dimensions, two-field, variational fo~ulation is empioyed to derive the differential equations governing the stretching, shearing, bending, twisting, as well as warping modes of defo~ations in a spatially curved beam subjected to general loading and boundary conditions. Cor~spoudingly, a simple two-noded finite element model was developed and utilized in a number of numerical simulations. In particular, attention was given to the significant curvature effects on the results in cases involving unsymmetric cross-section of the thin-walled type, in which any inconsistencies introduced when using the classical notion of two different reference lines. i.e., centroid and shear center, for sectional deformations may lead to large errors.

1. INTRODUCTION

Compared to their straight counterparts, the behavior exhibited by curved beams is far more complex. For example, a horizontally curved beam subjected to lateral loads undergoes not only vertical displacements but also twist defo~ations with respect to its longitudinal axis. Such flexure-torsion interactions depend on the girder curved geometry as well as its cross-sectional rigidities. In view of the mathematical complications involved, the early-stage developments in matrix analysis techniques [l, 2] have approximated the curved geometry by a series of straightline segments. More general treatments were made in recent years, both in the analytical solutions [3-q, as well as finite-element numerical simulations(8-211. However, in the majority of these works, only the case of doubly symmetric sections was considered. In addition, although the planar problem has been studied extensively, comparatively little has been done concerning the general three- dimensional, nonplanar, or coupled lateral-torsional, response of curved beams [8,9,22-25Ea subject of particular importance in lateral-buckling and linear/ nonlinear dynamics of these thin-walled curved components. A sampling of the representative works mentioned above is briefly reviewed in the sequel.

Following the early pion~~ng work of Saint- Venant [26], the next theoretical treatment of thin- walled curved beams may be traced back to the work of Gottfeld [27] who investigated a two-girder system interconnected by cross bracings and subjected to transverse loads to the plane of curvature. A more complete analysis for curved girders with doubly symmetric I-shaped cross-sections was given by Umanskii [28] who calculated the bimoments due to transverse loads. The governing differential equations for the coupling bending and restrained-warping

torsional response of curved beams with general, asymmetrical, open sections are given by Vlasov [3], These were later modified by Dabrowski [29], to account for the torsional deformations of closed, box-type, cross-sections. Unlike most of the curved beam theories available at the time, Konishi and Komatsu j30] have considered the case of very-sharp curved beams, where the elastic flexural stress and strain distributions over the beam section are nonlinear. The slope-deflection equation method has been employed to solve curved-girder systems under various loading and boundary conditions [4-6,31]. ~urthe~ore, some of these analytical results for strains and displacements were compared to labora- tory tests conducted on a curved girder by Heins and Spates [S].

The above curved-beam models have been formu- lated exclusively on the basis of technical beam theories in which the additional (intrinsic) kinematic constraints of vanishing flexural and warping- torsional shear strains are imposed, together with the conventional assumption of undeformed cross- section. That is, the Euler-Bernoulli hypothesis is used for flexural action and Wagner-Vlasov assumption for nonuniform torsional warping of thin-walled open sections [3,32]. Moreover, in dealing with general, unsymmetric, cross-sections, a conventional approach has been to select two different reference lines, i.e., the line of sectional centroids for stretch bending deformations, and the shear-center axis for torsion-warping kinematic parameters.

Despite its convenience in providing uncoupled forms for the corresponding governing differential equations for the two response components, thus facilitating their analytical soiutions for some ‘simple’ boundary and loading conditions, this approach has three distinct shortcomings. Firstly, it is strictly valid only for a cross-section having one axis of symmetry

639

640 A. S. GENDY and A. F. SALEEB

that lies in the plane of beam curvature; otherwise, ‘coupling terms’ still exist, e.g., in the strain- displacement relations. This must be accounted for properly in the general unsymmetric-section case; any inconsistencies introduced here, e.g., as in [23], may lead to significant errors as will be illustrated later in the applications part of this paper. Secondly, extensions to the geometrically nonlinear analysis problems are not straightforward. Finally, there are degenerated cases where the shear center is not defined (e.g., a thin-walled circular tube), or it may lose its significance as a ‘fixed’ geometric section property (e.g., due to inelasticity).

Alternative, more comprehensive, mathematical formulations for the linear and nonlinear analysis of curved beams have been developed in more recent years. Important contributions here include those by Antman [33], Reissner [34, 351, and Whitman and DeSilva [36], among many others. These are sufficiently general to allow for including such effects as shear deformations due to flexure as well as torsional warping, pretwist, finite deformations, etc. Also, in this context, the notion of selecting the shear center as an additional reference axis can be dispensed with. In fact, similar framework is utilized for the development reported here, i.e., a single reference axis (con- veniently taken as the line of centroids) is selected for defining all the deformation and (generalized) sectional forces (associated with stretching-transverse shear-bending-twisting-warping).

From the numerical standpoint, two general ap- proaches have been employed in the development of finite element equations for curved beams. The first, and more popular, approach is based on the principle of minimum potential energy, i.e., displacement models. The resulting elements are either of the C’ continuous type [lO-14, 191, when the Love- Kirchhoff approximations for thin beam/plate/shell theories are utilized, or the Co- or shear-flexible type based on Mindlin-Reissner hypotheses for thick beams/shells [1618,20]. An extensive body of literature currently exists on such formulations, in particular with regard to way to overcome the well-known phenomena of shear and/or membrane locking in the thin structure regimes [ll, 12, 1417, 19-21, 37,381. In the second approach, different forms of multifield variational principles [21, 38411 are utilized to derive mixed-hybrid types of curved beam models.

2. ORJECI-IVES AND OUTLINE

Our objective in this paper is to develop a simple finite element model (referred to here as HMCZ) for the analysis of spatial behavior of circular curved beams with arbitrary loadings and boundary conditions. The starting point in this development is an extension, to the present case of coupled extensional-flexural-warping response of spatial response, of our previous mixed formulations utilized in [38] for the straight-beam case, and in [21] for the restricted

problem of in-plane behavior of curved beams. In what follows, we delineate some of the basic features and important aspects of the proposed formulation.

The assumed kinematic description is based on a sufficiently comprehensive technical theory for generalized beam behavior. In particular, it accounts for shear deformations due to both flexural as well as restrained torsional warping effects [38], and it straightforwardly accommodates thin-walled beams with general cross-sections; e.g., of the open-, or closed-, or mixed-type [3, 321. The underlying mixed variational principle is of the Hellinger-Reissner type with two, independently assumed, fields displacements and generalized strains. The carefully selected basis (shape) functions for the latter play a crucial role in alleviating the shear/membrane locking phenomena [21,38].

The classical notion [3,23] of using two different reference axes (i.e., the line of sectional centroids and shear centers, respectively) plays no explicit role here; instead a single reference line (selected as the line of centroids for convenience) is employed in reducing all the sectional stress-strain resultants. This proves very convenient in treating both symmetric as well as unsymmetric section cases; the general unsymmetric section case simply amounts to the nonvanishing of a number of coupling terms in the generalized stress-strain matrix [see eqn (18) below]. Properly accounting for the significant curvature effects in the kinematic relations, this avoids any possible inconsistencies that may arise in utilizing the two reference lines formulations for sections with no axes of symmetry, e.g., refer to [23] and comparisons of its results with those in Sec. 6.5.

3. WEAK (VARIATIONAL) FORMULATION

We consider a circular beam element, with its initial radius of curvature denoted by R, and the line of centroids taken as the beam reference axis. Corre- sponding to the reference coordinate axes: x = Rq5, y,

and z, in the tangential, lateral, and radial directions, respectively, we thus define the orthonormal triad e, (i = 1,2,3), as shown in Fig. 1. The p,, p,, and p2 are the resultant distributed line load components (per unit arc length) in x-, y-, and z-directions, respectively. In this case, the employed two-field functional, nR, is written as [21, 381

nR=

s , [-fc;Cc,+a&]dx - W, (1)

where uR is generalized stress vector (superscript T indicates transposition)

uR = [F,, Fy, FL, M,., My> Mz, Ts,, T,.l* (2)

in which F, is the normal force; F, and F: are the shearing forces in they and z directions, respectively; M, is the bimoment; M, and M, are the bending

Spatial response of curved beams 641

Nodal section Nodal section

Fig. 1. A typical curved beam element.

moments about y and z directions, respectively; T, and T,,,--the two contributions to the total twist moment M,-are the Saint-Venant and warping, bishear, or ‘bitwist’, torsional moments, respectively. We again emphasize that all these stress resultants are referred to the section centroid ‘0’. In eqn (l), Lo and & are, respectively, the independent and geometric (derived from displacement) generalized strains

where $ is the axial stretch, 7, and p12 are the (average) transverse shear strains due to flexure; li-,,,, I$ and R,, are the warping and bending curvatures, respectively; and 9, and f,,, are the torsional shear strains associated with the Saint-Venant (uniform torsion) and warping (nonuniform torsion) response components, respectively. The C is the spatial elasticity tensor, i.e., section rigidities (or moduli) matrix.

Finally, W is the external work for distributed and end loads

w= s

(p,u +p,u +p,w)dx + i IQrql,=.~,, (4) I i=l

where (see Fig. 1)

9 = In, 0, w, 0x9 e,, 02, xl’ (5)

0 = F’x, F,, Fz, M,, My, M,, M,,lT (6)

define the components of nodal displacement and force vectors at the end points, respectively; and the overbar signifies a prescriped quantity. Note that the kinematic boundary conditions (b.c.) here correspond simply to specifying any of the components in q. Evidently, this is far more convenient for accommodating various supports, compared to other alternative formulations, e.g., see the coupled expressions for the y and t kinematic b.c. in

[231.

CAS 44,3--J

4. GOVERNING EQUATIONS FOR THIN-WALLED CURVED BEAMS

4.1. Kinematics

Referring to Fig. 1, the displacement field u at any point on the beam cross-section can be expressed in terms of the reference line translations uO, the rigid cross-sectional rotation vector 0, as well as a ‘super- posed’, local out-of-plane warping displacement whose amplitude is x and having a prescribed distribution over the cross-section, w(x,y), i.e., with u = (u, v, w), we have

^ u=u,+@r-aye,, (7)

where 0 - 8; 6,

e^ = skew(e) =

[ I

e, 0 -8, @aI -e, 8, 0

~o=~wvJ~I~~ e=[e,,e,.,e,r (8b)

r=[O,y,z]q e,=[l,O,O]r (gc)

w = w( y, z) = warping function. (gd)

In the above equation, u,,, vO, and w, denote the axial, lateral, and radial displacements, respectively; and 0,) f?,, and 0, are the rotations about x, y, and z axes, respectively.

Remark 4.1. Expressions of the generalized warping function include the contour, 6, and thickness, 5, warping contributions in thin-walled open sections [32,38], as well as the additional contribution from the ‘indeterminate’ shear flow distribution associated with the Saint-Venant in the cells of closed tubular sections; see [3,23] for more details.

Remark 4.2. With the centroid ‘0’ (not the shear center) selected as the pole [3,23] in constructing the sectorial are a diagram for torsion/warping properties, it is no longer the same principle sectorial area diagram [32], i.e., one generally has a JA wy d.4 # 0 and jA wz dA # 0, except in the special case of doubly


symmetric cross-sections. Refer to eqns (18x21) in which a prime indicates differentiation with respect below. However, the contour warping coordinate, 6, to (w.r.t.) the coordinate x, and a comma subscript can still be such that its integral over the cross- denotes differentiation w.r.t. the indicated coordinate sectional area, A, vanishes, i.e. y or z.

6dA =0, (9)

by selecting an appropriate ‘sectorial origin’ for calculating 6.

4.2. Linearized kinematics-the generalized strain- displacement relations

Only three strain components and the associated stresses are significant in the present one-dimensional curved beam model. The well-known expressions for strains in cylindrical coordinates [43] are given by

(104

(104

(lob)

Assuming (z/R) Q 1, the following Lure-type approximations may be made

( > 1,; -I

Z El--

R (11)

One can easily show that the linear strain components are now given by, after neglecting the effect of (z/R) in the shear strain equations (viewed here to be a higher-order effect by itself)

i,, = [lo - yk, + z$ - Ok”.] ( >

1 - ; (12a)

where the generalized strain vector, iR, the following ordered components

(12b)

WC)

is defined by

W)

(13b)

Remark 4.3. A possible source of difficulty in using the two-reference-line formulation for the general unsymmetric-section case becomes obvious upon closer examination of eqns (13). Here, the coupled kinematic relation for i. in eqn (13a) combines two terms; the first ah, is stretch-related and is taken to be at the centroid. On the other hand, the second bending-related term, w/R, would be measured with reference to the shear center. Similar observations can also be made for other strain components, e.g., r?, combines 0: at the centroid and 0, at the shear center. Inconsistencies may thus result, except for the special case of single-symmetric sections with the z-axis of symmetry in the plane of curvature.

4.3. Generalized stresses and constitutive equations

For an isotropic linear-elastic material, the stress components

u = diag.[E, G, G]c, (14)

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

where

Fi = s

Q,~ dA (for i = x, y, z) A

M = j

crJ--w, z, -ylrdA A

M, = T, + T,,,

Tm = s

[-Q~z + ~,,.I + a,(~ - o,,)l dA A

T, = s

axio,i dA (sum on i = y, z). A

(154

(15b)

(15c)

U5d)

(164

(16b)

(164

U6d)

(164

Using eqns (12), (13), and (15), one can finally arrive at the resultant-type constitutive equations


where the symmetric (8 x 8) matrix C is the spatial 4.4. Equilibrium equations elasticity tensor, i.e., section-rigidities (or moduli) Substituting eqns (13) and (15) into (1) and follow- matrix ing standard arguments in the calculus of variations,

c=

EA i

(symn-4

EI wz -EI, R R ’ ’ ’

GA, . . .

EI, - EI,, EIw,, .

4 -El,, .

EI, ’ .

GJ . ‘Wp - J)

where the following definitions are employed

A = j=“dA, IY=JAz2dA (19a)

I,= s

y2dA, I,;= yzdA (19b) A I A

t, = 02dA, I,, = yo dA s I

(19c) A A

I,,= zodA. I

(19d) A

A,, and A, are the flexural shear correction factors; and the following Saint-Venant and warping torsional rigidities are defined [3, 32,381

J = I

[(y - cQ2 + (z + cQ2] dA (2Oa) A

I,, - J = I

[(oQ2 + (w,,)~] dA. (2Ob) A

Using the conventional thin-walled assumption, i.e., neglecting the shear strain in the thickness direction, the expression of warping torsional rigidity, I,, can be written as

Ip= p2dA, s A

(21)

where p is the perpendicular distance from the centroid ‘0’ to the tangent at the middle surface of the considered section (see Fig. 2 and [3,32]).

Remark 4.4. In order to compare the HMC2 results with those obtained from conventional two- reference-lines treatments, some of displacement components, u, W, and 0,; as well as the shearing forces, bimoment, and torsions, must be transformed to another reference point (e.g., the shear center). These transformation rules between two reference sets of axes are well known; e.g., refer to [32,38].

9 (18)

one can finally obtain governing differential equations of equilibrium

F:+; +p,=o, (22a)

F;+p,=o, (22b)

F:-;+p;=O, (22c)

T:, + z + T,,. = 0, (22d)

M; - F, = 0, (2W

M; + t,, - f (T, + T,.) = 0, (220

M&. + T,. = 0. (22g)

Substituting from (17) and (18) into the above equations, the displacement-forms of the differential equations of a curved beam are obtained

+iGA,(w;+B,-:)+p*=O, (23a)

S . zAP n

P s

c 6ectorial origin)

Fig. 2.

0 (Pole) v

Sectional profile-coordinates and sign conventions.


(23b)

-++, +p:=O, (23~) 1

+G(I,,-J)(B:+;-x)=0 (23d)

-Cal(w;+S,--~)=O, (23e)

+G(I,-J)(B;+$)=O. Gw

Remark 4.5. In their present form, the coupled eqns (23) constitute the most comprehensive set governing the space behavior an initially circular beam with general cross-section. They should be contrasted to, for example the truncated forms in [23] for the no shear/two-reference-lines versions [i.e., q.YJ, f,,+O in eqns (13)]. Note, however, that, presently, the in-plane, eqns (22a, c, and e), and out-of-plane, eqns (22b, d, f, and g), responses will decouple for the doubly symmetric section case.

On the two ends of the element, we have the following boundary conditions

Remark 5.1. From the viewpoint of membrane/shear constraints in applications to thin-beam regimes, and following arguments in [21, 381, the case of the doubly symmetric section, and for in-plane loading of the present curved elements, appears to be most severe. Here, three kinematic degrees of freedom become operative (u, W, OJ), while only two constants per element are activated (one for the transverse shear, yXZ, and one for the extensional strain, eO, i.e., j, and &-+O, respectively), thus indi- cating that the constant index, CI defined for locking potential in the above references, has a favorable value of 1 (locking-free response).

By substituting (25) and (26) into (1) and grouping appropriate terms, the functional rcR can be written as follows, after eliminating the strain-parameter vector

B, e.g., ]21,38,421

xR = $lxKq - Q’s (28)

from which the stiffness matrix for the hybrid-mixed element is given by

Qi= Q, or (qi =gi and 6qi= O), (24)

where qi and Qi are, respectively, the ith component of the nodal degrees of freedom and forces which are defined in eqns (5) and (6); and the prefix 6 denotes a variation.

K = Grl--‘G, H = s

P’CPdx, I

G= PrCBdx, b =H-‘Gq. (29)

5. FINITE ELEMENT FORMULATION

In the above, B is the generalized strain-displacement defined using eqn (13). The stiffness matrix coefficient for the doubly symmetric section are given explicitly in the Appendix.

For the present model, the displacement, II, and For the purpose of comparing analysis results strain, eR, fields within the element are interpolated in later, the stiffness matrix of the displacement-based

terms of nodal displacement, q, and strain parameters, /I, respectively

u=Nq (25)

CR = PA (26)

where N and P are the matrices of interpolation functions. All displacement variables are interpolated using compatible, linear, shape functions of the fam- iliar Lagrange type [8,9]. That is, with r = natural coordinate goes from - 1 to 1, the interpolation functions are, for the HMC2 element

N, =f(l -r), N,=i(l +r). (27)

Unlike the displacement approximations, the assumption of strain functions, i.e., entries in the matrix P, is not as straightforward. Based on the discussion on the basic strategy used in selecting an appropriate set of strain parameters (see [21,38] for details), a constant field is assumed for components Lo, i.e., P is a (7 x 7) unit (identity) matrix.


counter-part for the present curved beam element with two nodes is also recorded here [8,9]

(30)

6. NUMERICAL EXAMPLES

In the present section, we examine the numerical performance of the HMC2 element developed. To this end, a number of test problems are considered, and results are compared with those obtained from theoretical, as well as other numerical, and experimental solutions reported in the literature. Since the in-plane version of the present curved beam has been discussed extensively in [21], we direct attention here to the out-of-plane response component.

6.1. Mesh convergence

To test the convergence characteristics, a cantilever circular beam (shown in Fig. 3) is analyzed, with an aspect ratio R/d N R/b = 24, where d and b are the depth and the flange width of the 10WF49 section, using HMC2, HMB2 (hybrid-mixed two-node straight element [38] and DC2 elements). The girder

+

1.25-

HMB2

5 -*- .- DC2

0.25 -

0 5 10 25 50 loo 200

Number of elements (log scale)

Fig. 3(a). Convergence study for tip lateral displacement of a cantilever curved beam.

1.50 * HMC2 1 -+-

1.25- HMBP

: -*- DC2

0.25 -

0 5 IO 26 60 100 200

Number of elements (log scale)

Fig. 3(b). Convergence study for tip rotation of a cantilever curved beam.

is subjected to two cases of loadings: (i) out-of-plane concentrated tip load and (ii) concentrated tip torsion. The tip deflection in the direction of load in the first case, and the tip rotation is the second, are calculated and normalized with respect to the con- verged values obtained from the HMC2 elements. These normalized values in (i) and (ii) are plotted versus mesh sizes in Figs 3(a) and (b).

Evidently, both HMC2 and HMB2 do not exhibit any sign of locking. In fact, the solutions using either an eight-element mesh of HMC2, or 16 HMB2 elements, give results which are less than 1% in error. On the other hand, the sever locking in case of DC2 element is clearly shown in Fig. 3, where the solution using even 200 elements is still in error by more than 40%.

6.2. Displacements and stresses predictions

A curved girder with a lOWF49 section, 15ft long, a 20-ft radius, and subjected to six various loadings and boundary conditions is considered here. The different cases of loadings and boundary conditions are listed in Table 1. The Young’s modulus and Poisson’s ratio are 30,000 ksi and 0.3, respectively. The results given by 12 HMC2 elements are compared with those obtained by solving the general Vlasov equations [3] using the sloping-deflection- equation method in [S]. The resulting deflection, Saint-Venant and warping torsions are given in Figs 4-9. The excellent agreement between results is evident.

6.3. Comparison with experimental results

A curved 7 I 15.3 girder with two fixed-end supports is considered. The girder is 27-ft long with a SO-ft radius and subjected to a concentrated unit torque moment at the midspan (in the negative direction of x-axis). The Young’s modulus and Poisson’s ratio are taken as 30,000 ksi and 0.3, respectively. The deflection and rotation along the girder length are reported in Figs 10(a) and (b) using 12 HMC2 elements. Both HMC2 results and those obtained from analytical solutions in [5] agree reasonably well with the test values, with less than 5% as maximum of difference.

6.4. Comparison with shell model

In contrast to the one-dimensional, beam-type solutions considered in the previous comparisons of Sets 6.2 and 6.3 above, a more accurate solution, of the general shell type, is used for the present vali- dation. That is, an unsymmetric L-shaped-section curved beam is solved first using the quadrilateral shell element HMSHS developed in [41]. The beam is fixed at the two ends and subjected to an out-of-plane concentrated load at the middle of the span. The geometric properties are shown in Fig. 11(a), while the material properties are taken as E = 30 x lo6 psi and v = 0.3. By making use of symmetry, the beam is idealized using eight HMC2 elements, whereas in


Table 1. Loading and boundary condition cases of study

:ASE

1

2

3

1

5

6

5. C

lURSIONAL MCMENY

M, r20 KIP- IN

m, :0.222 KIP IN / in ---c-e

Mx i20 KIP-IN 4z

m, z 0.222 KIP-IN/in

rcc---

m, =O-222 KIP-IN/in -L-c---

t20 KIP-IN

HINGED SUPPORT 9, :O

FIXED SUPPORY ex & X -0

the sheli analysis, a mesh of 24 HMSHS eIements

(two elements in the plane of the cross-section) are used. The results for the out-of-plane displacement component at the corner of the L-shaped cross- section are plotted in Fig. II(b), where excellent agreement is observed.

6.5. A co~#~~~~~s curved beam with u~~~~et~ic section

A two equai-span curved girder with an unsymmetric cross-section is considered next. Here, the effects

- Deflection - - - St. Venant Torsion -.-. Warping Torsion

1

IHMCZ)

0 Helm and spates (1970)

Fig. 4. Deflection and torsional stress resuftants, case 1.

LATERAL LOAD

Fr ~2 KIP

I Fy:t KIP

Py ~0,022 KIPI in

Py 5 0.022 KIP /in

t t t t it t tt

I Fu r2 KIP

.m

u,v &W =o

u,v,w,ey a ez =o

of the coupling terms in the moduli matrix, eqn (IQ and curvature terms in eqn (13) become very important. The girder has a radius of curvature R = 30 ft and the length of each span is 20 ft. It is subjected to a lateral distributed load pr = 0.05 kips/in, a concentrated load of 18 kips applied at the midpoint of each span, together with a horizontal (radial) distributed load pZ = 0.007 kips/in. Both the lateral and the radial loads are passing through the centroid of the cross- section. Pertinent data of this problem is given in Fig. 12. Values used for elastic moduii are E = 30,000 ksi and G = 12,000 ksi. The girder has pinned-end conditions with respect to both bending

W- Fig. 5. Deflection and torsional stress resultant, case 2.

Spatial response of curved beams

0.8

0.0 0.0 0.2 0.4 0.6 0.8 1.0

Fig. 6. Deflection and torsional stress resultants, case 3.

-30 0.0 0.2 0.4 0.6 0.8 1.0

WL

Fig, 7. Deflection and torsional stress resultants, case 4.

20 0.4

10

0

-10 0.2

-20

30 0.0 0.0 0.2 0.4 0.6 0.8 1.0

xn. Fig. 8. Deflection and torsional stress resultants, case 5.

0.0 0.2 0.4 0.6 0.8 1.0

Fig. 9. Deflection and torsional stress resultants, case 6.

Heins and Spates (1970)

w” IO(a). Comparison with experimental and analytical

results-deflection

2.5

10.01 I I I I 0.0 0.2 0.4 0.6 0.8

IO(b). Comparison with experimentai and analytical results-rotation.

648 A. S. GENDY and A. F. SALFEB

I 2.5’ I Y

1 Ii 1000.0 ,:.i-iI.3

I il 1 A67

4.167

Geometric data:

R = 360.0 in , L = 240.0 in , A = 3.75 in2

'Y = 1.95 in4, I: = 10.417in4, I, = 2.604 in4

J = 0.315 in4, I, = 3.905 in4, I, = 10.849 in6

Iw= -8.679 in5, I, = -4.34 id',

Fig. 1 t(a). The problem of unsymm~tric L-shaped girder.

0 0.2 0.4 0.6 0.8 1.0

WL

Fig. 1 l(b). Lateral deflection at the corner of L-shaped girder.

and torsion actions. In additions to the three translations, the axial rotation is also prevented at the middle support. The results using lZ-HMC2-element mesh for half the girder (by making use of symmetry condition at its midsection are reported together with those numerical results given in [23] in Figs 13-l 5). To facilitate the comparison with [34], the lateral and radial displacements as well as warping torsions and bimoments are transformed from the centroid to the shear center using the transformation scheme in [21,32]. Significant discrepancies exist between the present solution and the results given in [23]. As mentioned previously [see Sec. 2 and Remark 4.3), a major factor here is due to the inconsistent treatment of the coupled curvature terms in the strain expressions using two reference lines; this form in [23] is not valid for the present unsymmetric section. Also,

the value reported in [23] for the Z, coefficient appears to be in error. Although stated in [23] to be with reference to the shear center, the value given co- incided with one c~culation for I,,. with respect to the centroid. Furthermore, there is no indication in [23] that ‘equivalent’ coupling terms in the section-moduli matrix (e.g., I “,*, etc.) are accounted for.

6.6. Curved box girder bridge; mixed cross-section

Our final example concerns the analysis of a more complex structural configuration, i.e., a curved box girder as depicted in Fig. 16. The girder is simply supported at the two ends and is subjected to eccen- tric load Fy = 20 ib located at the exterior web. fn addition to the two end ~aphra~s, two intermedi- ate diaphragms at angles 15” and 35” from the line of support were installed to prevent the distortion of the


to 8.57 T

Geometric data:

R = 360.0 in. , L = 240.0 in., A = 17.50 in?

Iv = 132.5 in?, Iz = 1047.6 in!, I, = -142.57 in?

J = 1.458 inf , I,, = 725.88 in?, Iw = 26649.8 in?

Iw= 4527.9 in!, I,_ = 1190.8 in!,

Fig. 12. The problem of an unsymmetric continuous girder.

cross-section. By making use of symmetry, one half of the girder is idealized by 10 elements (70 DOF). The vertical deflection at the interior and exterior web and the tangential stresses at the midpoint of the bottom flange are presented in Figs 17 and 18, together with the experimental results, and analytical solutions (using she11 finite element approach), reported in [43]. The present HMC2 element per- formed surprisingly well in all the comparisons reported.

7. CONCLUSIONS

As a rational consequence of an appropriate three- dimensional, mixed variational formulation, we have established suitable, and sufficiently general, forms of the governing differential equations for spatial curved beam analysis. In addition to stretching, shearing, twisting, and bending modes of deformation, this analysis also accounts for the important curvature effects on warping modes in thin-walled beams with

u.wz5

0.m

-0.0025

0.0050

4.0075

-0.0100

- - HMCZ --- Yoo

3

-9.0125c. I I I I 0.0 0.2 0.4 0.6 0.8

? 5 >

0.0 0.2 0.4 0.8 0.5

m Fig. 13. Unsymmetric continuous girder-circumferential Fig. 14. Unsymmetric continuous girder-radial displace-

and lateral displacements. ment and torsional rotation.

- HMCZ O.Ol- --- Yoo

-O.o5r* I I I I 0.0 0.2 0.4 0.6 0.8 1

w


12C - HMC2

lo- --- Yoo

6-

6-

4-

2-

o-

-2 - __________________________I

-4 :. I I I I 0.0 0.2 0.4 0.6 0.6 1.0

1400

2000 /I

ii!!; -12OOr. I I I I- c

0.0 0.2 0.4 0.6 0.6 1.0

Fig. 15. Unsymmetric continuous girder-normal force and bimoment.

arbitrary cross-sections. To this end, a single reference line is utilized for all beam straining actions, thus facilitating the treatment of both symmetric-as well as unsymmetric-section cases.

For the purpose of numerical simulations, this same formulation was used to develop a simple, two-noded (with seven degrees-of-freedom per node) finite element. In particular, based on the results obtained using this latter model in an extensive number of test cases, it was clearly demonstrated that significant errors may result from inconsistencies

s 10lb !

\ !

0.200

0.175

0.150

0.125

0.100

0.075

0.060

0.025

0.000 0.0

Fam eta/. (1976

0.2 0.3

X/L Fig. 17. Deflection along the external and internal webs of

a curved box bridge.

1WL - HMC2

475- *Exp. Fam et al. (1976)

$

0.0 0.1 0.2 0.3 0.4 0

V-

Fig. 18. Tangential stresses along the midpoint of the lower flange of a curved box bridge.

introduced in alternative formulations using the classical notion of two different lines of reference (centroid and shear center) for reducing the stretching/ bending and the shearing-twisting-warping actions. The significance of this latter point will become even more critical in extensions to the geometrically nonlinear regimes; e.g., buckling and nonlinear vi- brations.

5

18.0 ” 1

Geometric data: R = 51.0 in., L E 80.11 in., A = 7.98 in?

‘Y = 191.46 in?, I, = 12.13 in?, I, = 0.00 in?

J = 31.37 in?, I, = 25.42 in?, I, = 169.01 in?

I, = 0.00 in!, I, = 99.39 in!,

Fig. 16. The problem of a curved box bridge-mixed section.


Acknowledgement-The work reported here is part of a research supported by NSF under grant No. EET-8714628.

REFERENCES

1. D. W. Funkhouser and C. P. Heins, Skew and elevated support effects on curved bridges. Civil Engineering Report No. 46, University of Maryland, College Park, MD (1972).

2. F. H. Lavalle and J. S. Boick, A program to analyze curved girder bridges. Engineering Bulletin No. 8, Uni- versity of Road Island, Kingston, RI (1965).

3. V. Z. Vlasov, Thin Walled Elastic Beams, 2nd Edn. National Science Foundation, Washington, DC (1961).

4. L. C. Bell and C. P. Heins, The solution of curved bridge system using the slope deflection method. Civil Engineering Department, University of Maryland, Col- lege Park, MD.

5. C. P. Heins and K. R. Spates, Behavior of single horizontally curved girder. J. S~ruct. Div., ASCE %, 1511-1524 (1970).

6. C. P. Heins, The presentation of the slope-deflection Fourier series method for the analysis of curved ortho-

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

tropic highway bridges. Civil Engineering Department, Universitv of Maryland. Collage Park. MD (1967). 34.

7. R. Dabrowski, Cuived Thin-walled Girders, Theory’and Analysis, (translated from German). Cement and Con- 35, Crete Association, London (1968).

8. K. J. Bathe, Finite Element Procedures in Engineering Analysis. Prentice-Hall, Englewood Cliffs, NJ (1982). 36.

9. R. D. Cook, Concepts and Applications of Finite Element Analysis, 2nd Edn. John Wiley, New York (1981).

10. D. G. Ashwell and A. B. Sabri, Limitations of certain 37.

curved finite elements when applied to arches. ht. J. Mech. Sci. 13, 133-139 (1971).

11. D. G. Ashwell, A. B. Sabri and T. M. Roberts, Further studies in the application of curved finite elements to 38.

circular arches. Int. J. Mech. Sci. 13, 505-517 (1971). 12. D. S. Dawe, Curved finite elements for the analysis of

shallow and deep arches. Comput. Struct. 4, 559-580 39.

(1974). 13. D. J. Dawe, Numerical studies using circular arch finite 4o

elements. Comput. Struct. 4, 7299740 (1976). 14. D. J. Dawe, Some high-order elements for arches and

shells. In Finite Elements for Thin Shells and Curved 41

Members, (Edited by D: G. Ashwell and R. H. Gallagher), pp. 131-154. John Wiley, London (1976).

15. I. Fried, Shape functions and accuracy of arch finite elements. AIAA Jnl 11, 287 (1973). 42

16. A. K. Noor, W. H. Greene and S. J. Hartley, Nonlinear “,

A. Rosen and H. Abromovich, Galerkin method as a tool to investigate the planar and non-planar behavior of curved beams. Comput. Struct. 18, 165-174 (1984). D. H. Hall, A personal correspondence. Bridge Soft- ware Department International Ltd (1986). _ A. E. H. Love, The Mathematical Theorv of Elasticitv. Dover, New York (1944).

- I

H. Gottfeld, The analysis of spatially curved steel bridges (in German). Die Baufechnik 715 (1932). A. A. Umanskii, Spatial Structures (in Russian). Moscow (1948). R. Dabrowski, Warping torsion of curved box girders of non-deformable cross-section. Der Stahlbau 34, 135-141 (1965). I. Konishi and S. Komatsu, Three dimensional analysis of curved girder with thin walled cross section. Int. Assoc. Bridge and Struct. Engineers, Zurich, Germany, Vol. 25, pp. 143-203 (1965). G. C. Brookhart, Circular-arc I-type girders. J. Struct. Div., ASCE 93, 133-159 (1967). A. Gjelsvik, The Theory of Thin-walled Bars. John Wiley, New York (1981). S. S. Antman, Ordinary differential equations of nonlinear elasticity-Part I. Arch. Rat. Mech. Anal. 61, 307-351 (1976). E. Reissener, On the finite deformations of space-curved beams. J. ADDI. Math. Phvs. 32. 734744 (1981). E. Reissener: A variational analysis of ‘small finite deformations of pretwisted elastic beams. Int. J. Solids Struct. 21, 773-779 (1985). A. B. Whitman and C. N. DeSilva, An exact solution in a nonlinear theory of rods. J. Elasticity 4, 265-280 (1974). T. J. R. Hughes, M. Cohen and M. Haroun, Reduced and selective integration techniques in the finite element analysis of plates. Nucl. Engng Design 46, 203-222 (1978). A. S. Gendy, A. F. Saleeb and T. Y. Chang, Generalized thin-walled beam models for flexural-torsional analysis. Comput. Struct., 42, 531-550 (1992). K. Washizu, Variational Methods in Elasticity and Plas- ticity, 3rd Edn. Pergamon Press, Oxford (1982). T. H. H. Pian and D. P. Chen, Alternative ways for formulation of hybrid stress elements. Int. J. Numer. Mefh. Engng 18, 167991684 (1982). A. F. Saleeb, T. Y. Chang, W. Graf and S. Yingyeun- yong, A hybrid/mixed model for nonlinear shell analysis and its applications to large-rotation problems. Int. J. Numer. Meth. Engng (1989). A. P. Boresi and K. P. Chong, Elasticity in Engineering Mechanics. Elsevier (1987).

finite element analysis of curved beams. Compur. Mech. Appl. Mech. Engng 12, 289-307 (1977).

17. A. K. Noor and J. M. Peters, Mixed models and reduced/selective integration displacement models for nonlinear analysis of curved beams. Int. J. Numer. Meth. Engng 17, 615631 (1981).

18. I. Sheinman, Large deflection of curved beam with shear deformation. J. Engng Mech. Div., AXE 108, 636647 (1982).

~2. A. R. M. Fam and C. Turkstra, Model study of horizontally curved box girder. J. Engng. Struct. Div., ASCE 102, 1097-I 108 (1976).

19. H. Stolarski and T. Belytschko, Membrane locking and reduced integration for curved elements. J. Appl. Mech. 49, 172-176 (1982).

20. H. Stolarski and T. Belytschko, Shear and membrane locking in curved Co-elements. Comput. Mefh. Appl. Mech. Engng 41, 279-298 (1983).

21. A. F. Saleeb and T. Y. Chang, On the hybrid-mixed formulation of Co curved beam elements. Comput. Meth. Appl. Mech. Engng 60, 95-121 (1987).

22. C. H. Yoo, D. R. Evick and C. P. Heins, Non-prismatic curved girder analysis. Comput. Struct. 4,675-698 (1974).

23. C. H. Yoo, Matrix formulation of curved girders. J. Engng Mech. Div., ASCE 105, 971-987 (1979).

APPENDIX: LINEAR STIFFNESS MATRIX FOR CURVED BEAM

The terms in the (14 x 14) symmetric linear stiffness matrix for the doubly symmetric section are given below; entries not shown are zeros, and ordering is for degrees of freedom in eqn (5) folr node 1 followed by those for node 2

K(1, 1) = K(8,8) = 7 (EA + $GA,~#J~)

K(1,3) = -K(E, 10) = ; (EA - GA,)

K(1,5) = K(1, 12) = K(5,8) = K(8, 12) = -$ GA/$

K(I,B)=;(-EA +fGA;42)


K(2,2) = - K(2,9) = K(9,9) = y

K(2,6) = K(2, 13) = - K(9, I 3) = - ~(6.9) = 4 GA,

Kfl, 10) = - K(3,8) = -$ @A +GA,)

Kt3,3f = K(I0, IO) = ‘1($ EAdrz + GA:)

K(3,5) = K(3,12) = -K(lO, 12) = -K(& 10) = -4 GA,

K(3, 10) = f ($ EAc$~ - GA,)

K(4,4)=K(11,11)=f($1~q52+G~p)

K(4,6)= -K(~,,l3~=~(~~:-G~~~

K(4,?)=K(4,14)= -K(il, 14)

= -K(7, lI)=+G(Z,-J)

K(4,11)=.~(:Ef;~z_GI,)

K(4, 13) = -K(6, I If = --$ (EI; + GI,)

K(S,S)=K(12, 12)=fEIy+,?GAzl

K(5, 12) = -; 5.1, + $ GA,1

K(6,7) = K(6, 14) = X(7,13)

=K(I3, 14)= -$-J)c#J

KU, 7) = K(14, 14) = ; Er,. + 1 Gl(r,, - J)

KU, 14) = -:; E& + fGl(I,, -J).

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ON THE FINITE ELEMENT ANALYSIS OF THE …blogs.uakron.edu/cmg/files/2015/11/On-the-Finite-Element-Analysis...Spatial response of curved beams 641 Nodal section section Fig. 1. A typical