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NONLINEAR PROGRAMMING APPROACH TO A

SHEAR-DEFORMABLE HYBRID BEAM ELEMENT FOR

LARGE DISPLACEMENT ANALYSIS

Charilaos M. Lyritsakis, Charalampos P. Andriotis, Konstantinos G.

Papakonstantinou

The Pennsylvania State University

University Park, PA, 16802

e-mail: czl279@psu.edu

{cxa5246,kpapakon}@psu.edu

Keywords: geometrically exact beam theory, ﬁber elements, shear deformable beam , nonlinear

programming, hybrid ﬁnite element

Abstract. In the present work, a hybrid beam element based on exact kinematics is devel-

oped, accounting for arbitrarily large displacements and rotations, as well as shear deformable

cross sections. At selected quadrature points, ﬁber discretization of the cross sections facili-

tates efﬁcient computation of the stress resultants for any uniaxial material law. The numerical

approximation is carried out through the lens of nonlinear programming, where the enengy

functional of the system is treated as the objective function and the exact strain-displacement

relations form the set of kinematic constraints. The only interpolated ﬁeld is curvature, whereas

the centerline axial and shear strains, along with the displacement measures at the element

edges, are determined by enforcing compatibility through the use of any preferable constrained

optimization algorithm. The solution satisfying the necessary optimality conditions is deter-

mined by the stationary point of the Lagrangian. A set of numerical examples demonstrates the

accuracy and performance of the proposed element against analytical or approximate solutions

available in the literature.

4707

COMPDYN 2019

7th ECCOMAS Thematic Conference on

Computational Methods in Structural Dynamics and Earthquake Engineering

M. Papadrakakis, M. Fragiadakis (eds.)

Crete, Greece, 24–26 June 2019

Available online at www.eccomasproceedia.org

Eccomas Proceedia COMPDYN (2019) 4707-4718

ISSN:2623-3347 © 2019 The Authors. Published by Eccomas Proceedia.

Peer-review under responsibility of the organizing committee of COMPDYN 2019.

doi: 10.7712/120119.7262.19709

Charilaos M. Lyritsakis, Charalampos P. Andriotis, Konstantinos G. Papakonstantinou

1 INTRODUCTION

Problems frequently encountered by the engineering community over the last decades were

concerned with structural systems whose response involved large displacements, rotations and

strains on one hand, and inelastic behavior on the other. Such problems, in cases where the

subject matter involved rod-like structures, necessitated the advancement of the classical Euler-

Bernoulli theory, where displacements, rotations and strains were typically kept in the small

range.

In the geometrically nonlinear case, the deformed geometry of the beam can be drastically

different from the undeformed one, resulting in a rather involved description of beam kinemat-

ics. In general, there are two approaches in describing the kinematics of beams. The ﬁrst one is

the so-called continuum-based approach [1], which is employed for the derivation of the classi-

cal beam theory [2] and where all vectorial components are obtained from the three-dimensional

theory of solids, with additional assumptions imposed on cross-section kinematics. The second

approach, which is followed in this work, is concerned with the description of a material curve

- an assemblage of material points representing the beam centroid - embedded in E2(or E3in

the 3D case). The analysis of the curve by means of differential geometry of curves leads to

the notion of intrinsic parameterization or one-dimensional formulation of beams, where spa-

tial quantities can be expressed as functions of only one parameter. In the study of beams, this

parameter is taken to be the arc-length of the beam. This in turn leads to the so-called arc-

length parameterization of the beam with respect to a reference conﬁguration. This approach

was adopted in early works by Reissner [3, 4] and, next, Simo [5], Simo and Vu-Quoc [6] and

can be traced back to Kirchhoff and his treatment of inextensible elastic rods. In these formu-

lations, often termed as geometrically exact, the thickness of the rod is taken into consideration

by attaching two vectors at each point on the material curve that would translate and rotate with

the points, thus deﬁning the properties of curvature and torsion at these points.

Approaches where strain measures are recast as primary ﬁeld variables are often termed

strain or deformation-based and were explored by Planinc et al. [7] for the planar case of

the geometrically exact beam in order to properly account for the effect of local instabilites

on the tangent stiffness matrix due to plastiﬁcation, when global stability considerations are

also present. It was later extended to the 2D dynamic case by Gams et al. [8] and to the

three-dimensional case by Zupan and Saje [9] as a means to cope with the strain objectivity

issue arising from the interpolation of the rotation vector [10]. Interpolation of strain measures

is also encountered in mixed formulations, where more than one ﬁelds are interpolated and

the underlying functional is augmented by additional terms to be satisﬁed in the weak sense

[11]. Several other works with geometrically exact formulations in various settings can be

found in the literature. Approaches based on mixed, hybrid, ﬂexibility- and displacement-based

considerations can be indicatively seen in [12, 13, 14, 15].

The present work is an extension of the geometrically exact hybrid formulation presented

in [16] in order to account for the effect of shear deformation at the section level. As opposed

to deriving the system equations from the Galerkin form after appropriate discretization, in the

aforementioned work the problem is originally recast in a nonlinear programming framework,

where the total potential energy functional (TPE) is augmented via Lagrange multipliers that

enforce satisfaction of the exact kinematic conditions. The resulting functional is then approx-

imated by employing a Gauss-Legendre quadrature rule, which yields the objective function

to be minimized. Another interesting feature of this particular approach is that the primary

variables in the element interior contributing to the elastic strain energy are the generalized

4708

Charilaos M. Lyritsakis, Charalampos P. Andriotis, Konstantinos G. Papakonstantinou

strain measures of the centroid, which are sought at quadrature points. Displacement mea-

sures, namely, the translations along the coordinate axes and the rotation of the cross sections,

occur only at the nodes of the element and are associated with the external work. Kinematic

consistency between the rotational measures of displacement and strain is enforced by using a

Lagrange interpolation scheme to approximate the curvatures over the entire element domain.

We should note that quadrature points coincide with the Lagrange interpolation points. In the

remainder a succinct presentation of the formulation is presented, along with numerical investi-

gations based on benchmark nonlinear problems studied in the literature, verifying the accuracy

and efﬁciency of the suggested approach.

2 KINEMATICS AND TOTAL POTENTIAL ENERGY

We now proceed with the derivation of the kinematic equations that serve as constraints for

the optimization problem. Let us consider two dimensional Euclidian space E2and a ﬁxed

Cartesian coordinate system (X1, X2)with unit basis vectors {Ei},i= 1,2and a material

curve of length Lwhich represents the centroid of the beam embedded in that space. We

will assume that the beam is initially straight, aligned with the coordinate axis X1and in an

unstressed state. We also deﬁne the arc-length parameterization of the material curve s∈I=

[0, L]→r∈R2, where sis the distance of a material point on the line of centroids in

the reference conﬁguration. The position vector of any material point on the centroid in any

conﬁguration can be written as:

r(s) = (s+u(s))E1+w(s)E2(1)

where uand ware the displacements of the material point along the coordinate axes X1and X2

respectively.

Beam Kinematics

According to [3] the strain-displacement relations, assuming small axial strain of the beam

centroid, are:

u0= (1 + ) cos φ−γsin φ−1(2)

w0= (1 + ) sin φ+γcos φ(3)

φ0=κ(4)

where , γ and κare the axial, shear and bending strains of the line of centroids respectively,

φis the tangent angle to the material curve in the current conﬁguration and derivatives with

respect to the sare denoted by ( )0. Integrating (2)-(4) over Iyields:

u(L)−u(0) = ZL

0

(1 + ) cos φ−γsin φ ds −L

w(L)−w(0) = ZL

0

(1 + ) sin φ+γcos φ ds

φ(L)−φ(0) = ZL

0

κ ds

(5)

The integral form of the kinematic equations in (5) is utilized to impose the constraints on the

nonlinear program.

4709

Charilaos M. Lyritsakis, Charalampos P. Andriotis, Konstantinos G. Papakonstantinou

Total Potential Energy

The total potential energy functional of the beam under a set of point loads P= [ P1P2]T, with

Pithe point loads at the element edge nodes, can be expressed in the reference conﬁguration

as:

Π(, γ, κ, d) = ZL

0

W(, γ, κ)ds −PTd(6)

where Wis the strain energy per unit length of the beam centroid. The displacement degrees of

freedoms at the two edge nodes are collectively represented in vector das:

d=d1d2d3d4d5d6T(7)

with:

d1=u(0), d2=w(0), d3=φ(0),

d4=u(L), d5=w(L), d6=φ(L)

The stress resultants on a section are associated with the strain energy as follows:

N=∂W

∂ ,Q=∂W

∂γ ,M=∂W

∂κ (8)

The stress resultants deﬁned in (8) can be numerically computed using appropriate ﬁber dis-

cretization at the cross section level. Thereby, any nonlinear constitutive law may be incorpo-

rated in beam element formulations to capture the effects of distributed elastoplastic behavior

or damage [17, 18, 19].

3 NONLINEAR PROGRAMMING PROBLEM STATEMENT

In this section we formulate the equations pertaining to the description of our hybrid element as

a nonlinear program.

Element Objective Function

The discrete form of the TPE in (6) is given by applying Gauss-Legendre quadrature to approx-

imate the integral for one element:

f(x) =

n

X

i=1

ciW(i, γi, κi)−PTd(9)

where ciare the weights of the quadrature, nthe number of quadrature points and x,dand yi

are deﬁned as:

x=yT

1yT

2. . . yT

ndTT(10)

with yi=iγiκiT

Element Constrains

The ﬁrst set of constraints derived from the approximation of kinematic relations (5) is given

as:

4710

Charilaos M. Lyritsakis, Charalampos P. Andriotis, Konstantinos G. Papakonstantinou

CA

eq =

d4−d1−Pn

i=1 ci[(1 + i) cos φi−γisin φi] + L

d5−d2−Pn

i=1 ci[(1 + i) sin φi+γicos φi]

d6−d3−Pn

i=1 ciκi

=0(11)

In accordance with [16] we then interpolate the curvature ﬁeld with Lagrange polynomials in

order to obtain the rotations φiat the quadrature points:

φi=d3+

n

X

j=1

Θij κj(12)

Θ=L

ξ1

ξ2

1

2. . . ξn

1

n

.

.

..

.

.....

.

.

ξn

ξ2

n

2. . . ξn

n

n

G−1,G=

1ξ1. . . ξn−1

1

.

.

..

.

.....

.

.

1ξn. . . ξn−1

n

(13)

where ξ=x

Land Gis the Vandermonde matrix.

Notice that the ﬁrst two equations of (11) are nonlinear equality constraints, while the third,

along with the nequations of (12) are linear equality constraints. It is convenient to recast all

constraints in one vector as follows:

Ceq =

CA

eq

CB

eq

=0(14)

where

CB

eq =

φ1−d3−Pn

j=1 Θ1jκj

.

.

.

φn−d3−Pn

j=1 Θnj κj

(15)

Element Lagrangian Function

We now introduce a vector λof the Lagrange multipliers and augment the TPE (6), constructing

the Lagrangian of the optimization problem as:

f(x,λ) =

n

X

i=1

ciW(i, γi, κi)−PTd+λTCeq (16)

Stationary points are provided by satisfying Karush-Kuhn-Tucker [20] optimality conditions for

the Lagrangian function of (16).

4 NUMERICAL EXAMPLES

In the following examples we examine the performance of the proposed formulation and com-

pare it with other well-known works in the literature. We ﬁrst explore the accuracy when the

shear stiffness is reduced and then we test against different loading cases. For each example,

only one element with ﬁve quadrature points is used.

4711

Charilaos M. Lyritsakis, Charalampos P. Andriotis, Konstantinos G. Papakonstantinou

Table 1: Cantilever with constant transverse load at free end.

w u

GAsNumerical Analytical Present Numerical Analytical Present

5·1020 0.30172077 0.301720774 0.30172432 0.05643324 0.056433236 0.05643126

5·1020.31781387 0.317813874 0.31781567 0.06131566 0.061315658 0.06131317

5·1010.46541330 0.465413303 0.46541543 0.10328492 0.103284917 0.10328294

1·1021.16709588 1.167095878 1.16709542 0.25213661 0.252136606 0.25213357

5·1002.10408747 2.104087473 2.10409063 0.37612140 0.376121399 0.37612451

Problem data: EA = 1021, EI = 10, L = 1. Numerical data in [21], analytical in [22].

0 2 4 6 8 10 12

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Bernoulli

Timoshenko

E=200 10 5

= 0.3

L = 1.0

F = 10

Figure 1: Timoshenko compared to Bernoulli solutions for different levels of slenderness.

Example 1 - Effect of shear deformation in cantilever deﬂection

In this example we explore the effect of shear ﬂexibility on the tip deﬂection of a cantilever. A

constant point transverse force P= 10 units is applied at the free end and, then, several analyses

are performed by varying the shear stiffness coefﬁcient, GAs. Numerical and analytical results

obtained by Batista in [21] and [22], respectively, are compared with the present formulation

and are illustrated in Table 1.

Fig. 1 demonstrates the effect shear deformations have when not neglected, compared to the

Bernoulli solutions, by varying the ratio L/h, with hbeing the height of the cross section and L

the length of the beam. The applied load is F,νis the Poisson’s ratio and wBis the deﬂection

when shear ﬂexibility is neglected. As can be seen from the results, when L= 2h, transverse

4712

Charilaos M. Lyritsakis, Charalampos P. Andriotis, Konstantinos G. Papakonstantinou

displacements due to shear deformation are increased by rougly 13%.

Example 2 - Cantilever with transverse point load at its tip

This problem has been analyzed in [23, 24], whereas Mattiasson [25] provided solutions by

solving the elliptic integrals of the problem of large deﬂections of beams. Moreover, the prob-

lem was also examined in [26] using a co-rotational transformation for the Timoshenko beam,

whereas Nanakorn [27] used 3 elements and a total tagrangian formulation.

0 0.2 0.4 0.6 0.8 1

0

2

4

6

8

10

Mattiasson

Nanakorn

Present

Mattiasson

Nanakorn

Present

Linear

Figure 2: Cantilever with point transverse load at its free end.

Table 2: Cantilever with transverse load at its free end.

w/L u/L

P L2/EI Mattiasson Present Mattiasson Present

2.0 0.49346 0.49347 0.16064 0.16063

4.0 0.66996 0.67001 0.32894 0.32892

6.0 0.74457 0.74465 0.43459 0.43457

8.0 0.78498 0.78509 0.50483 0.50481

10.0 0.81061 0.81073 0.55500 0.55498

Problem data: EI = 1000 lb/in2,L= 100in, P= 1lb, N= 20 steps.

Results in Mattiasson [25].

4713

Charilaos M. Lyritsakis, Charalampos P. Andriotis, Konstantinos G. Papakonstantinou

Figure 2 illustrates the performance of the element when compared against the analytical and

numerical solutions - mentioned above - and Table 2 demonstrates the accuracy up to six deci-

mal points when compared to the analytical solutions for a sample of loading levels. The dotted

line indicates the linear response.

Example 3 - Cantilever with point moment at its free end

This example tests the capabilities of our developed model capturing the response of an inex-

tensional beam subjected to a point moment, forcing a curl into a complete circle. As men-

tioned previously, in all examples we only used one element with ﬁve quadrature points for our

analysis. Bathe and Bolourchi [28] using ﬁve and twenty elements and an updated lagrangian

procedure showed accuracy up to 90 degrees. In subsequent works, Simo & Vu-Quoc [6] (ﬁve

elements), Rankin & Brogan [29] (ten elements) and Crisﬁeld [30] (ﬁve elements) duplicated

the exact solution. In the second and third works a corotational formulation was employed.

The mechanical and geometric properties for this problem are I= 0.01042in4,L= 100in,

A= 0.5in2,E= 1.2×104psi.

In Fig. 3 our solution is compared with the one using twenty elements. As mentioned earlier,

with the element proposed by Bathe & Bolourchi [28], which is based on large-displacement

and large-rotation assumptions, the response starts to diverge from the exact solution at an angle

of 90 degrees rotation, irrespectively of the mesh density. Our proposed formulation is instead

able to capture the response for the whole loading scenario (360 degrees), as can be seen from

the line that represents the normalized displacement φ/2π.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Bathe & Bolourchi

Present

/2

Bathe & Bolourchi

Present

Figure 3: Cantilever with point moment at its free end.

4714

Charilaos M. Lyritsakis, Charalampos P. Andriotis, Konstantinos G. Papakonstantinou

Example 4 - Cantilever beam with eccentric compressive point load

We consider a cantilever beam of length L= 100, cross section thickness b= 1 and elastic

modulus E= 12, with the load parameter λ=P/Pcr is increased up to 4.0. The critical load

for the cantilever is Pcr = 0.25π2EI/L2. Wood & Zienkiewicz [31] used ﬁve continuum-based

elements that allow for shear deformation and employed a total Lagrangian formulation. The

results are illustrated in Fig. 4. Analytical solutions to the problem, provided in [23, 32] where

it is assumed no axial or shear deformation occurs, show negligible discrepancy compared to

the ones proposed here and in [31]. It should be noted that the eccentricity is =b/2. The

conﬁgurations for each load step for Examples 2,3 and 4 are depicted in Fig. 5, from left to

right.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0

0.5

1

1.5

2

2.5

3

3.5

4

Present

Zienkiewicz & Wood

Analytic

Present

Zienkiewicz & Wood

Analytical

Figure 4: Cantilever with eccentric axial load at its free end.

Figure 5: Conﬁgurations at each step for examples 2, 3 and 4 (left to right).

4715

Charilaos M. Lyritsakis, Charalampos P. Andriotis, Konstantinos G. Papakonstantinou

5 CONCLUSIONS

An extension to the geometrically exact hybrid element derived in [16] is presented herein that

accounts for shear deformations. The system of equilibrium equations is originally derived

within a nonlinear programming framework, where the total potential energy functional is dis-

cretized and then augmented by the exact kinematic constraints of the physical problem, also

in discretized form, and solved by determining the stationary point of the Lagrangian. The

suggested nonlinear programming hybrid formulation is capable of capturing the response of

the benchmark problems with accuracy, using only one element, which is a desirable feature

for framed structure applications. Ongoing work explores a variety of different approaches as

far as the optimization algorithms are concerned, which is something the proposed formulation

supports and enables, the extension of the material yield rule to account for the interaction of

shear and axial stresses, as well as the extension to spatial and dynamic formulations.

Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant

No. 1634575. The authors gratefully acknowledge this support.

REFERENCES

[1] T. Belytschko, L. Schwer, and M.J. Klein. Large displacement, transient analysis of space

frames. International Journal for Numerical Methods in Engineering,11(1):65–84, 1977.

[2] T.J.R. Hughes. The ﬁnite element method: linear static and dynamic ﬁnite element analy-

sis. Dover, 2000.

[3] E. Reissner. On one-dimensional ﬁnite-strain beam theory: The plane problem. Zeitschrift

f¨

ur Angewandte Mathematik und Physik ZAMP,23(5):795–804, 1972.

[4] E. Reissner. On one-dimensional large-displacement ﬁnite-strain beam theory. Studies in

applied mathematics,52(2):87–95, 1973.

[5] J.C. Simo. A ﬁnite strain beam formulation. The three-dimensional dynamic problem.

Part I. Computer Methods in Applied Mechanics and Engineering,49(1):55–70, 1985.

[6] J.C. Simo and L. Vu-Quoc. A three-dimensional ﬁnite-strain rod model. Part II: Computa-

tional aspects. Computer Methods in Applied Mechanics and Engineering, 58(1):79–116,

1986.

[7] I. Planinc, M. Saje, and B. Cas. On the local stability condition in the planar beam ﬁnite

element. Structural Engineering and Mechanics,12(5):507–526, 2001.

[8] M. Gams, M. Saje, S. Srpˇ

ciˇ

c, and I. Planinc. Finite element dynamic analysis of geomet-

rically exact planar beams. Computers & Structures,85(17-18):1409–1419, 2007.

[9] D. Zupan and M. Saje. Finite-element formulation of geometrically exact three-

dimensional beam theories based on interpolation of strain measures. Computer Methods

in Applied Mechanics and Engineering,192(49-50):5209–5248, 2003.

4716

Charilaos M. Lyritsakis, Charalampos P. Andriotis, Konstantinos G. Papakonstantinou

[10] M.A. Crisﬁeld and G. Jeleni´

c. Objectivity of strain measures in the geometrically exact

three-dimensional beam theory and its ﬁnite-element implementation. Proceedings of the

Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences,

455(1983):1125–1147, 1999.

[11] C.A. Felippa. A survey of parametrized variational principles and applications to com-

putational mechanics. Computer Methods in Applied Mechanics and Engineering,113(1-

2):109–139, 1994.

[12] M.V. Sivaselvan and A.M. Reinhorn. Collapse analysis: large inelastic deformations anal-

ysis of planar frames. Journal of Structural Engineering,128(12):1575–1583, 2002.

[13] H.A.F.A. Santos, P.M. Pimenta, and J.P.M. Almeida. A hybrid-mixed ﬁnite element

formulation for the geometrically exact analysis of three-dimensional framed structures.

Computational Mechanics,48(5):591, 2011.

[14] N.D. Oliveto and M.V. Sivaselvan. 3D ﬁnite-deformation beam model with viscous

damping: Computational aspects and applications. Journal of Engineering Mechanics,

141(1):04014103, 2014.

[15] M. Salehi and P. Sideris. A ﬁnite-strain gradient-inelastic beam theory and a corresponding

force-based frame element formulation. International Journal for Numerical Methods in

Engineering,116(6):380–411, 2018.

[16] C.P. Andriotis, K.G. Papakonstantinou, and V.K. Koumousis. Nonlinear programming hy-

brid beam-column element formulation for large-displacement elastic and inelastic analy-

sis. Journal of Engineering Mechanics,144(10):04018096, 2018.

[17] E. Spacone, F.C. Filippou, and F.F. Taucer. Fibre beam–column model for non-linear anal-

ysis of R/C frames: Part I. Formulation. Earthquake Engineering & Structural Dynamics,

25(7):711–725, 1996.

[18] P. Uriz, F.C. Filippou, and S.A. Mahin. Model for cyclic inelastic buckling of steel braces.

Journal of Structural Engineering,134(4):619–628, 2008.

[19] C. Andriotis, I. Gkimousis, and V. Koumousis. Modeling reinforced concrete struc-

tures using smooth plasticity and damage models. Journal of Structural Engineering,

142(2):04015105, 2015.

[20] J. Nocedal and S. Wright. Numerical Optimization. Springer Science & Business Media,

2006.

[21] M. Batista and F. Kosel. Cantilever beam equilibrium conﬁgurations. International Jour-

nal of Solids and Structures,42(16-17):4663–4672, 2005.

[22] M. Batista. A closed-form solution for Reissner planar ﬁnite-strain beam using Jacobi

elliptic functions. International Journal of Solids and Structures,87:153–166, 2016.

[23] R. Frisch-Fay. Flexible bars. Butterworths, 1962.

[24] K.E. Bisshopp and D.C. Drucker. Large deﬂection of cantilever beams. Quarterly of

Applied Mathematics,3(3):272–275, 1945.

4717

Charilaos M. Lyritsakis, Charalampos P. Andriotis, Konstantinos G. Papakonstantinou

[25] K. Mattiasson. Numerical results from large deﬂection beam and frame problems analysed

by means of elliptic integrals. International Journal for Numerical Methods in Engineer-

ing,17(1):145–153, 1981.

[26] N.D. Kien. A Timoshenko beam element for large displacement analysis of planar beams

and frames. International Journal of Structural Stability and Dynamics,12(06):1250048,

2012.

[27] P. Nanakorn and L.N. Vu. A 2D ﬁeld-consistent beam element for large displacement

analysis using the total Lagrangian formulation. Finite Elements in Analysis and Design,

42(14-15):1240–1247, 2006.

[28] Klaus-J¨

urgen Bathe and S. Bolourchi. Large displacement analysis of three-dimensional

beam structures. International Journal for Numerical Methods in Engineering,14(7):961–

986, 1979.

[29] C.C. Rankin and F.A. Brogan. An element independent corotational procedure for the

treatment of large rotations. Journal of Pressure Vessel Technology,108(2):165–174,

1986.

[30] M.A. Crisﬁeld. A consistent co-rotational formulation for non-linear, three-dimensional,

beam-elements. Computer Methods in Applied Mechanics and Engineering,81(2):131–

150, 1990.

[31] R.D. Wood and O.C. Zienkiewicz. Geometrically nonlinear ﬁnite element analysis of

beams, frames, arches and axisymmetric shells. Computers & Structures,7(6):725–735,

1977.

[32] S.P. Timoshenko and J.M. Gere. Theory of elastic stability. Dover, 2009.

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