A unified approach to the Timoshenko 3D beam-column

International Journal of Solids and Structures 222–223 (2021) 111003

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International Journal of Solids and Structures

journal homepage: www.elsevier .com/locate / i jsols t r

A unified approach to the Timoshenko 3D beam-column elementtangent stiffness matrix considering higher-order terms in the straintensor and large rotations

https://doi.org/10.1016/j.ijsolstr.2021.02.0140020-7683/� 2021 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Avenida Fernando Ferrari, 514, Nexem, Goiabeiras,Vitória, ES 29075-910, Brazil.

E-mail address: [email protected] (M.A.C. Rodrigues).

Marcos Antonio Campos Rodrigues a,⇑, Rodrigo Bird Burgos b, Luiz Fernando Martha c

a Espírito Santo Federal University, Department of Civil Engineering, Avenida Fernando Ferrari, 514, Goiabeiras, Vitória, ES 29075-910, Brazilb State University of Rio de Janeiro, Department of Structures and Foundations, Rua São Francisco Xavier, 524, Maracanã, Rio de Janeiro, RJ 20550-900, BrazilcPontifical Catholic University of Rio de Janeiro, Department of Civil Engineering, Rua Marques de São Vicente, 225, Gávea, Rio de Janeiro, RJ 22451-900, Brazil

a r t i c l e i n f o

Article history:Received 10 July 2020Received in revised form 14 January 2021Accepted 16 February 2021Available online 04 March 2021

Keywords:Tangent stiffness matrixAnalytical interpolation functionsTimoshenko beam theoryNonlinear geometric analysis

a b s t r a c t

A structural geometric nonlinear analysis using the finite element method (FEM) depends on the consid-eration of five aspects: the interpolation (shape) functions, the bending theory, the kinematic description,the strain–displacement relations, and the nonlinear solution scheme. As the FEM provides a numericalsolution, the structure discretization has a great influence on the analysis response. However, whenapplying interpolation functions calculated from the homogenous solution of the differential equationof the problem, a numerical solution closer to the analytical response of the structure is obtained, andthe level of discretization could be reduced, as in the case of linear analysis. Thus, to reduce this influenceand allow a minimal discretization of the structure for a geometric nonlinearity problem, this work usesinterpolation functions obtained directly from the solution of the equilibrium differential equation of adeformed infinitesimal element, which includes the influence of axial forces. These shape functions areused to develop a complete tangent stiffness matrix in an updated Lagrangian formulation, which alsointegrates the Timoshenko beam theory, to consider shear deformation and higher-order terms in thestrain tensor. This formulation was implemented, and its results for minimal discretization were com-pared with those from conventional formulations, analytical solutions, and Mastan2 v3.5 software. Theresults clearly show the efficiency of the developed formulation to predict the critical load of planeand spatial structures using a minimum discretization.

� 2021 Elsevier Ltd. All rights reserved.

1. Introduction

The continuous (analytical) behavior of a solid can be approxi-mated by a discrete solution. Usually, the discrete response isobtained by nodal displacements, and an approximated continuoussolution can be found by means of interpolating (shape) functions.However, the discrete solution using the FEM introduces simplifi-cations in the mathematical idealization of the structure behavioras the interpolation functions that define the deformed configura-tion of a structure are not compatible with the mathematical ide-alization of the response of a continuous medium (Martha, 2018).

In a linear elastic analysis of frame models with beam elementswith constant cross-sections, interpolating functions are obtainedfrom the homogeneous solution of the equilibrium differentialequation of an undeformed infinitesimal element, leading to the

so-called cubic Hermitian interpolation functions (Rodrigueset al., 2019). In this case, the formulation does not consider anyother approximation except those already covered in the analyticalidealization of the element behavior. This explains the fact that, inlinear elastic analysis, the structure response of this type of modeldoes not depend on the level of discretization.

However, for geometric nonlinear or second-order analysis, inwhich equilibrium should be considered in the deformed configu-ration, Hermitian interpolation functions do not represent theanalytical response of the structure. To cope with this problem,high-order finite elements can be used (So and Chan, 1991;Zheng and Dong, 2011; Rodrigues et al., 2016). Burgos et al.,(2005) employed classic linearization from the stability problemand used additional degrees of freedom within the elements tocalculate the critical load by eigenvalue analysis.

Another way of improving a second-order analysis is to use sta-bility functions (Chen and Lui, 1991; Aristizábal-Ochoa, 1997,2007, 2008, 2012). Some authors have used the consistent field

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Nomenclature

General geometric parametersx; y; z bar local axis in longitudinal (x) and transversal (y; z)

directions.dx; dy;ds infinitesimal increment in the (x,y) directions or in arc

length.l; L bar length.V spatial vector position of a point in space.R spatial transformation matrix.af g; bf g point in a cross-section.h generic cross-section height.A cross-section area.v form factor that defines the effective shear area.I cross-section moment of inertia in relation to an axis.Jp cross-section polar moment of inertia.k element slenderness given by:L=hX generic auxiliary parameter for characterization of bar

elements.

Displacement field parametersu axial displacement of a point inside a bar in the x direc-

tion.v;w transverse displacement of a point inside a bar in the y

or z direction.u0;v0;w0 axial and transversal displacements at the cross-

section center of gravity.vh; vp homogeneous and particular parts of analytical solution

for displacements and rotations.vg ; v l global and local solutions for displacements and rota-

tions.h cross-section rotation in relation to an axis.d01;d

04 axial displacements at the initial and final nodes of a

bar.d02;d

05 transverse displacements at the initial and final nodes of

a bar.d03;d

06 cross-section rotation at the initial and final nodes of a

bar.d0� �

displacement vector of a bar.uf g axial displacement vector of a bar.vf g; wf g transverse displacements vectors of a bar.ci; Cf g constant of integration.X½ � interpolation polynomials matrix of the displacement

field.H½ � interpolation polynomials matrix of the displacement

field at nodes.N½ � interpolation functions matrix.Nuf g axial displacement interpolation functions vector.Nvf g transverse displacement interpolation functions vector

on the local xy plane of a bar.Nwf g transverse displacement interpolation functions vector

on the local xz plane of a bar.Nhxf g cross-section rotation interpolation functions vector

around the local axis x.Nhy� �

cross-section rotation interpolation functions vectoraround the local axis y.

Nhzf g cross-section rotation interpolation functions vectoraround the local axis z.

Nu1;N

u4 axial displacement interpolation functions at the initial

and final nodes of a bar.Nv

2 ;Nv5 transverse displacement interpolation functions of a bar

for a transverse displacement.Nv

3 ;Nv6 transverse displacement interpolation functions of a bar

for a rotation displacement.

Nh2;N

h5 cross-section rotation interpolation functions of a bar

for a transverse displacement.Nh

3;Nh6 cross-section rotation interpolation functions of a bar

for a rotation displacement.

Strain and stress tensor parametersexx normal deformation.cxy; cxz shear distortion in the plane xy and xz.c shear distortion.eðtþDtÞij Green-Lagrange strain tensor in an unknown configura-

tion.eðtÞij Green-Lagrange strain tensor in a known configuration.Deij incremental strain.Deij linear component of incremental strain.Dexx axial deformation from linear component of incremen-

tal strain.Dexy shear distortion from linear component of incremental

strain.Dgij nonlinear component of incremental strain.Dgxx axial deformation from nonlinear component of incre-

mental strain.Dgxy shear distortion from nonlinear component of incre-

mental strain.sxx normal stress.sxy; sxz shear stress.SðtþDtÞij second Piola-Kirchoff stress tensor.sðtÞij Cauchy stress tensor.Dsij stress increment.E Young’s modulus.G modulus of rigidity (shear modulus).m Poisson’s ratio.Cijkl material constitutive tensor.

External loads and internal forces parametersq load rate of transverse force on the bar.P constant generic axial load acting on infinitesimal ele-

ment.M generic bending moment acting on a bar element.l;K auxiliary parameters for the differential equation devel-

opment of the deformed infinitesimal element equilib-rium.

N normal force.V vertical component of acting force in a cross-section.Qy;Qz shearing force in directions y and z.Q shearing force.Mx twisting moment.My;Mz bending moment.My1;My2 bending moment acting at the initial and final nodes of

a bar in y direction.Mz1;Mz2 bending moment acting at the initial and final nodes of

a bar in z direction.

Stiffness matrix parametersRðtþDtÞ virtual work due to external loading.U;UNL linear and nonlinear component of virtual work expres-

sion.Kg;Rotfin� �

finite rotations influence on the geometric stiffness ma-trix of a generic plane element.

Marcos Antonio Campos Rodrigues, Rodrigo Bird Burgos and Luiz Fernando Martha International Journal of Solids and Structures 222–223 (2021) 111003

2

Fig. 2.1. Equilibrium of a deformed beam element.


approach and transformed the shape interpolation functions (axialand transverse displacements and rotation) into a power series(Yunhua, 1998; Tang et al., 2015). The consistent field approacheliminates ‘‘shear locking” and ‘‘membrane locking” effects thatusually appear in a geometric nonlinear analysis that employs justone element per member.

Other works have a formulation based on the equilibrium equa-tion of the infinitesimal element. Some include the axial effect con-sidering the equilibrium in the deformed configuration (Daviset al., 1972; Nukulchai et al., 1981; Goto and Chen, 1987; Chanand Gu, 2000; Balling and Lyon, 2011). Others incorporate addi-tional effects, such as elastic foundations (Areiza-Hurtado et al.,2005; Aydogan, 1995; Burgos and Martha, 2013; Chiwanga andValsangkar, 1988; Eisenberger and Yankelevsky, 1985; Morfidis,2007; Morfidis and Avramidis, 2002; Onu, 2000, 2008; Shirimaand Giger, 1992; Ting and Mockry, 1984; Zhaohua and Cook,1983) and interlayer slip for sandwich beams (Ha, 1993;Girhammar and Gopu, 1993). However, these studies directly for-mulate the stiffness coefficients of the elements without explicitlypresenting expressions for interpolation functions based on thedifferential equation of the problem.

In addition to the enrichment of interpolating the field variablesin an element, another important aspect to improve the responseof a structure when performing a geometric nonlinear analysis isrelated to the bending theory. The most commonly used bendingsolution for frame elements is the Euler-Bernoulli beam theory(EBBT). However, the effects of shear deformation are importantwhen predicting the behavior of beam-columns with a moderateslenderness ratio or with a small shear-to-bending ratio, and theTimoshenko beam theory (TBT) provides good results(Timoshenko and Gere, 1963; Friedman and Kosmatka, 1993;Pilkey et al., 1995; Schramm et al., 1994). For this reason, manyworks cited above also consider this effect, such as references(Burgos and Martha, 2013; Onu, 2008).

Moreover, a kinematic description of motion is needed whenformulating an incremental geometrically nonlinear analysis.Based on the choice of the reference configuration, the followingthree kinematic descriptions are commonly used in structuralmechanics to formulate the nonlinear system of equations: TotalLagrangian (TL), Updated Lagrangian (UL), and Corotational (CR).When consistently developed, the total Lagrangian and theupdated formulation produce the same results (Mcguire et al.,2000). These formulations are developed in (Mcguire et al., 2000;Bathe and Bolourchi, 1979; Conci, 1988; Yang and Leu, 1994;Yang and Kuo, 1994; Chen, 1994; Bathe, 1996). However, in gen-eral, the Timoshenko theory is not considered, or in some cases,the formulation yields additional degrees of freedom, whichrequires static condensation to reduce the matrix order (Batheand Bolourchi, 1979; Aguiar et al., 2014). Additionally, most stud-ies that consider shear deformation employ a corotational formu-lation, as proposed by (Battini, 2002; Crisfield, 1991; Pacoste andEriksson, 1995, 1997; Santana and Silveira, 2019; Silva et al., 2016).

Furthermore, the strain–displacement relation plays an impor-tant role when developing a geometric stiffness matrix. The con-sideration of higher-order terms in the strain tensor improvesthe accuracy of the geometric nonlinear analysis (Chen, 1994;Conci, 1988; Yang and Kuo, 1994; Yang and Leu, 1994). Morerecently, the authors (Rodrigues et al., 2019) considered higher-order terms in the strain tensor in the formulation of a Timoshenkoelement using Hermitian interpolation functions and an updatedLagrangian formulation.

The main objective of this paper, which differentiates this workfrom others found in the literature, is to formulate the tangentstiffness matrix of a frame element integrating four importantaspects that improve geometric nonlinear analysis: interpolationfunctions, beam theory, kinematic description, and strain–dis-

3

placement relations. The matrix is constructed for a spatial ele-ment by an updated Lagrangian formulation considering higher-order terms in the strain tensor using interpolation functionsobtained directly from the equilibrium differential equation of adeformed infinitesimal element, including shear deformationaccording to the Timoshenko beam theory. In addition, the elementtangent stiffness matrix is adjusted to consider finite rotations. Thefifth important aspect of a nonlinear analysis is the incrementalsolution scheme. This paper does not address this problem, and astandard Newton-Raphson solution procedure was used.

Since this formulation involves trigonometric and hyperbolicbasic functions, the resulting stiffness matrix can be convenientlyrewritten using a Taylor series expansion with more terms thanthe usual formulations. In this paper, tangent matrices with up to3 and 4 terms were developed. This formulation was implementedin Framoop (Martha and Parente Junior, 2002), the solver used inFtool (Martha, 1999) structural analysis program.

2. Differential equilibrium relationships in beam-columns

This section formulates the differential equations that definethe analytical behavior of a deformed infinitesimal beam elementconsidering both the Euler-Bernoulli and Timoshenko beam theo-ries. Fig. 2.1presents a deformed infinitesimal element subjectedto a distributed transverse load q and a constant axial load P.

The equilibrium of the deformed infinitesimal beam elementleads to Eqs. (2.1) and (2.2)

XFy ! �dV þ q xð Þdx ¼ 0 ! dV xð Þ

dx¼ qðxÞ ð2:1Þ

XMo ! dM � V þ dVð Þdx� P:dv þ q xð Þ dx

2

2¼ 0 ð2:2Þ

where v xð Þ is the infinitesimal element transverse displace-ment, q xð Þ is the transverse distributed load, V xð Þ is the verticalcomponent of the force acting on the cross-section, P is the hori-zontal component, and MðxÞ is the bending moment.

Considering that the element has a constant cross-section, usingEq. (2.2) and the approximate relation between bending momentand curvature, M xð Þ ¼ EI dh=dx, where h xð Þ corresponds to thecross-section rotation, the differential equation of the problemcan be written according to expression (2.3).

EId2h xð Þdx2

� V xð Þ � Pdv xð Þdx

¼ 0 ! EId3h xð Þdx3

� dV xð Þdx

� Pd2v xð Þdx2

¼ 0 ð2:3Þ


The static fundamental relation in Eq. (2.1) shows that the gra-dient of the vertical component of the force acting on the cross-section is equal to the acting distributed load. Therefore, substitut-ing this relation in Eq. (2.3), the differential equilibrium relation ofa deformed infinitesimal element can be found, as shown inexpression (2.4).

EId3hdx3

� Pd2v xð Þdx2

¼ qðxÞ ð2:4Þ

2.1. Euler-Bernoulli beam theory

The EBBT considers the rotation as the gradient of the trans-verse displacement (h ¼ dv=dx). Thus, the differential relationdeveloped in expression (2.4) can be written according to (2.5).

d4v xð Þdx4

� PEI

d2v xð Þdx2

¼ qðxÞEI

ð2:5Þ

2.2. Timoshenko beam theory

In TBT, shear distortion cð Þ is constant for each cross-section (nowarping) and is considered an additional rotation of the section.Therefore, the section rotation and the transverse displacementare not associated, and both are considered independent variables,as indicated in Fig. 2.2.

Therefore, for the deformed element, internal forces are calcu-lated using the cross-sectional rotation dv=dx ¼ hþ cð Þ shown inFig. 2.3 and expressed in Eqs. (2.6) and (2.7).

N xð Þ ¼ Pcos hþ cð Þ � V xð Þsen hþ cð Þ ! N xð Þ

¼ P � V xð Þ dv xð Þdx

ð2:6Þ

Q xð Þ ¼ Psen hþ cð Þ þ V xð Þcos hþ cð Þ ! Q xð Þ

¼ PdvðxÞdx

þ VðxÞ ð2:7Þ

Substituting Eqs. (2.7) into (2.2), the static fundamental relationbetween the shear force and the bending moment is obtained:Q xð Þ ¼ dMðxÞ=dx. The shear force acting on the section is givenby Eq. (2.8)

Q xð Þ ¼ �vGA:cðxÞ ! Q xð Þ ¼ vGA: hðxÞ � dvðxÞdx

� �ð2:8Þ

in which G is the material shear modulus, A is the cross-sectionarea, and v is the factor that defines the cross-section effective areafor shear.

Fig. 2.2. Shear deformation in the Timoshenko beam theory.

4

Equation (2.9) can be written using the relationQ xð Þ ¼ dMðxÞ=dx in Eq. (2.8). According to the relation betweenbending moment and curvature M xð Þ ¼ EIdh=dx, Eq. (2.10), whichrelates transverse displacement vðx) with cross-section rotationhðxÞ, is obtained.

dM xð Þdx

¼ vGA: hðxÞ � dvðxÞdx

� �ð2:9Þ

dvðxÞdx

¼ h xð Þ � EIvGA

d2hdx2

ð2:10Þ

Finally, applying equation (2.10) in expression (2.4), the equilib-rium differential relation of a deformed infinitesimal element, con-sidering Timoshenko beam theory, can be written according toexpressions (2.11) or (2.12).

EI 1þ PvGA

� d3hdx3

� PdhðxÞdx

¼ qðxÞ ð2:11Þ

d3hdx3

� P

1þ PvGA

�EI

dh xð Þdx

¼ q xð Þ1þ P

vGA

�EI

ð2:12Þ

Equation (2.11), or alternatively Eq. (2.12), relates the cross-section rotation hðxÞ with the applied distributed transverseloadqðx), considering the element axial force and the shear andbending rigidity parameters. The next section shows the solutionof differential Eqs. (2.10) and (2.12) for an isolated beam element.

3. Solutions of the differential equations

The solution of the equilibrium differential relations presentedbefore can be obtained by the composition of a homogeneous solu-tion, vh xð Þ, with a particular solution, vp xð Þ, according tov xð Þ ¼ vh xð Þ þ vp xð Þ.

In the direct stiffness method, the solution is obtained bythe superposition of a global and a local solution, hencev xð Þ ¼ vg xð Þ þ v l xð Þ. The global solution is the one obtained byusing nodal displacements and rotations as coefficients for theinterpolating functions. The local solution is a fixed-end ele-ment solution, thus presenting null values for nodal displace-ments and notations. Martha (2018) used the scheme inFig. 3.1 to explain this superposition. Nodal displacements androtations of the final solution are obtained directly from theglobal solution. Within an element, displacements are obtainedby superposition of the local and global solutions. In a classicalFEM analysis for continuous domains, generally, the local solu-tion is not available, and the final solution is approximatedby the global solution.

If the right-side of the differential equation is null, then there isno need for a particular solution and the homogeneous part is thefinal solution, then v xð Þ ¼ vh xð Þ. Similarly, if there is no transverseforce within the element (q(x) = 0), the local solution is null, givingv xð Þ ¼ vg xð Þ. Thus, the homogeneous and the global solutions areequivalent: vh xð Þ ¼ vg xð Þ. Therefore, when adopting analyticalsolutions for the behavior of the element, the displacements andnodal rotations of the discrete model are exact, independently ofdiscretization. Moreover, in case a fixed-end local element solutionis available, nodal displacements and rotations of the final solutionare obtained directly from the global solution, also independentlyof discretization.

These observations justify the use of interpolation functions cal-culated from the homogenous solution. When the differentialequation is calculated from the equilibrium of an undeformedinfinitesimal element, the so-called Hermitian cubic functions cor-respond to the homogeneous solution for EBBT, and because of

Fig. 2.3. Internal forces, displacements, and rotations in the Timoshenko beam theory.

Fig. 3.1. Solution composition from the direct stiffness method.


that, linear analyses can be performed without discretization. Sim-ilar cubic functions are obtained from the undeformed infinitesi-mal equilibrium for TBT. The same applies to the functions thatrepresent the homogenous solution of the equations obtained fromthe equilibrium of a deformed infinitesimal element.

Therefore, this section seeks the homogeneous solution of thedifferential equations of equilibrium of a frame element in thedeformed configuration. The next sections indicates how the inter-polation (shape) functions resulting from the solutions to these dif-ferential equations are obtained; and Section 5 shows how theseinterpolation functions are used in the formulation of the tangentstiffness matrix of a frame element in its local axis system. Itshould be noted that the homogenous solution considers approxi-mations inherent to the analytical model adopted for the behaviorof beam-column elements, such as the hypothesis of small rota-tions in the equilibrium of the infinitesimal element in thedeformed configuration.


Considering the EBBT and q xð Þ ¼ 0, the homogeneous solutionof the differential equation presented in (2.5) can be obtained fromrelation (3.1) and written as given in (3.2),

5

d4v xð Þdx4

� l2 d2v xð Þdx2

¼ 0;l ¼ffiffiffiffiffiffiffiffiffiffiP=EI

pð3:1Þ

vh xð Þ ¼ c1elx þ c2e�lx þ c3xþ c4 ð3:2Þwhere c1, c2,c3 and c4 are the coefficients of an exponential

function.In EBBT, h ¼ dv=dx, and the cross-section rotation is obtained by

equation (3.3).

hh xð Þ ¼ c1lelx � c2le�lx þ c3 ð3:3ÞThe particular solution does not introduce any additional coef-

ficients that need to be determined. Thus, the coefficients of theexponential function can be calculated from the boundaryconditions.

For a tensile force, l is a real number, and the homogeneoussolution of the differential equation can be written by hyperbolicfunctions according to (3.4) and (3.5). In the case of a compressiveforce, l is a complex number, and the displacement can be writtenby trigonometric functions (3.6) and (3.7).

vh xð Þ ¼ c1sinh lxð Þ þ c2cosh lxð Þ þ c3xþ c4;l ¼ffiffiffiffiffiffiffiffiffiffiP=EI

pð3:4Þ

hh xð Þ ¼ c1lcosh lxð Þ þ c2lsinh lxð Þ þ c3 ð3:5Þ


vh xð Þ ¼ c1sin lxð Þ þ c2cos lxð Þ þ c3xþ c4;l ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi�P=EI

pð3:6Þ

hh xð Þ ¼ c1lcos lxð Þ � c2lsin lxð Þ þ c3 ð3:7Þ

Fig. 4.1. Deformed configuration of an isolated element.


In TBT, the cross-section rotation is not the gradient of thetransverse displacement, and it is not possible to write just one dif-ferential relation; thus, Eqs. (2.10) and (2.12) need to be solved.The homogenous solution can be calculated considering q xð Þ ¼ 0;thus, the differential equations are written as shown in (3.8) and(3.9).

dvðxÞdx

¼ h xð Þ � EIvGA

d2hdx2

ð3:8Þ

d3hdx3

�K2 dh xð Þdx

¼ 0;K ¼ lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ EI

vGA :l2q ;l ¼

ffiffiffiffiffiffiffiffiffiffiP=EI

pð3:9Þ

Using constantX, expression (3.10), introduced by Reddy(Reddy, 1997) and used in (Burgos and Martha, 2013; Marthaand Burgos, 2015; Martha and Burgos, 2014), the differential equa-tions can be rewritten as given in (3.11) and (3.12).

X ¼ EIvGA

1L2

ð3:10Þ

dvðxÞdx

¼ h xð Þ �XL2d2hdx2

ð3:11Þ

d3hdx3

�K2 dh xð Þdx

¼ 0;K ¼ lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þXl2L2

q ð3:12Þ

Solving equation (3.12), the homogeneous solution for thecross-section rotation is given by (3.13). This solution can beapplied in Eq. (3.11). Thus, the transverse displacement is writtenaccording to (3.14).

hh xð Þ ¼ K c1eKx � c2e�Kx �þ c3 ð3:13Þ

vh xð Þ ¼ 1�XL2K2 �

c1eKx � c2e�Kx� �þ c3xþ c4 ð3:14Þ

Finally, for a tensile force, l is a real number, and the homoge-neous solution of the differential equation can be written by thehyperbolic functions in (3.15) and (3.16); for a compressive force,l is a complex number, and the displacement is written by thetrigonometric functions in (3.17) and (3.18).

hh xð Þ ¼ K c1cosh Kxð Þ þ c2sinh Kxð Þ½ � þ c3 ;

K ¼ l=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þXl2L2

qð3:15Þ


c1sinh Kxð Þ þ c2cosh Kxð Þ½ � þ c3xþ c4 ;

K ¼ l=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þXl2L2

qð3:16Þ

hh xð Þ ¼ K c1cos Kxð Þ � c2sin Kxð Þ½ � þ c3 ;

K ¼ l=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�Xl2L2

qð3:17Þ

vh xð Þ ¼ 1þXL2K2 �

c1sin Kxð Þ þ c2cos Kxð Þ½ � þ c3xþ c4 ;

K ¼ l=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�Xl2L2

qð3:18Þ

6

4. Interpolation functions

The deformed configuration of an isolated element can bedescribed by interpolating nodal displacements according toFig. 4.1 and Eqs. (4.1) to (4.4).

u0 xð Þ ¼ Nu1 xð Þd0

1 þ Nu4 xð Þd0

4 ! u0 xð Þ ¼ Nu xð Þf g uf g ð4:1Þ

v0 xð Þ¼Nv2 xð Þd0

2þNv3 xð Þd0

3þNv5 xð Þd0

5þNv6 xð Þd0

6!v0 xð Þ¼ Nv xð Þf g vf gð4:2Þ

h xð Þ¼Nh2 xð Þd0

2þNh3 xð Þd0

3þNh5 xð Þd0

5þNh6 xð Þd0

6 ! hz xð Þ ¼ Nhz xð Þf g vf g ð4:3Þ

u0 xð Þ ¼ Nu xð Þf g uf g v0 xð Þ ¼ Nv xð Þf g vf g hz xð Þ ¼ Nhz xð Þf g vf gð4:4Þ

When interpolating functions are obtained from the homoge-neous solution of the equilibrium differential equation of aninfinitesimal element, the analytical behavior of the element isrepresented. In this work, the equilibrium of a deformed infinites-imal element is considered taking the axial force into account.


The interpolation functions can be calculated based on thehomogeneous solution for the EBBT presented in Eqs. (3.2) and(3.3). First, the displacement is written in matrix form accordingto Eq. (4.5).

vh xð Þ ¼ c1elx þ c2e�lx þ c3xþ c 4

hh xð Þ ¼ c1lelx � c2le�lx þ c3

! v0 xð Þh xð Þ

� �¼ elx e�lx x 1

lelx � le�lx 1 0

� � c1c2c3c4

8>><>>:

9>>=>>; ¼ X½ � Cf g

ð4:5ÞThe boundary conditions are obtained by evaluating the homo-

geneous solution of these displacements (vh xð Þ and hh xð Þ) at theextreme nodes of the bar, as shown in expression (4.6).

d0n o

¼d

02

d03

d05

d06

8>>><>>>:

9>>>=>>>;

¼v0 0ð Þh 0ð Þv0 Lð Þh Lð Þ

8><>:

9>=>;! d

0n o¼

1 1 0 1l �l 1 0eLl e�Ll L 1leLl �le�Ll 1 0

2664

3775

:

c1c2c3c4

8>>><>>>:

9>>>=>>>;

¼ H½ � Cf g ð4:6Þ

Finally, interpolation functions are calculated using Eqs. (4.4)–(4.6),resulting in relation (4.7). The results are presented in the work byRodrigues (2019) for exponential, hyperbolic and trigonometricfunctions and were implemented in file ShpFuncBeamEulerBernoulliusing MATLAB and C in an open source code that can be accessedand used (Rodrigues et al., 2020, 2021).


v0 xð Þh xð Þ

� �¼ X½ � H½ ��1 d0f g ) N½ � ¼ X½ � H½ ��1 ð4:7Þ


The homogeneous solution for the TBT is presented in Eqs.(3.13) and (3.14). Thus, the same development is performed to cal-culate the interpolation functions, according to expressions (4.8)and (4.9).


c1eKx � c2e�Kx� �þ c3xþ c 4

hh xð Þ ¼ K c1eKx þ c2e�Kx �þ c3

! v0 xð Þh xð Þ

� �¼ X½ � Cf g

X½ � ¼ 1�XL2K2 �

eKx 1�XL2K2 �

e�Kx x 1

KeKx Ke�Kx 1 0

24

35 Cf g ¼

c1c2c3c4

8>><>>:

9>>=>>; ð4:8Þ

d0f g ¼

d02

d03

d05

d06

8>>>>><>>>>>:

9>>>>>=>>>>>;

¼v0 0ð Þh 0ð Þv0 Lð Þh Lð Þ

8>>><>>>:

9>>>=>>>;

¼

1�XL2K2 �

c1 � c2ð Þ þ c4

K c1 þ c2ð Þ þ c3

1�XL2K2 �

c1eKL � c2e�KL� �þ c3Lþ c4

K c1eKL þ c2e�KL �þ c3

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

¼ H½ � Cf g

½H� ¼

ð1�XL2K2Þ ðXL2K2 � 1Þ 0 1K K 1 0

eLKð1�XL2K2Þ e�LKðXL2K2 � 1Þ L 1KeLK Ke�LK 1 0

266664

377775 ð4:9Þ

Once again, equation (4.7) can be used to calculate the interpo-lation functions. The results are also presented in the work byRodrigues (2019) and implemented in file ShpFuncBeamTi-moshenko; an open source code that can be accessed and used((Rodrigues et al., 2020, 2021)).

4.3. Axial interpolation functions

For the axial displacement and the cross-section rotation aboutthe x axis, in this research, a linear interpolation was adoptedaccording to Mcguire et al. (2000), which is given in expression(4.10).

u0 xð Þ ¼ Nu1 xð Þd0

1 þ Nu4 xð Þd0

4 Nu1 xð Þ ¼ 1� x

LNu

4 xð Þ ¼ xL

ð4:10ÞSome authors, such as Tang et al. (2015) and Silva et al., (2016);

employed consistent interpolation functions for the axial displace-ment and not just linear interpolation.

Fig. 5.1. Bending and torsio

7

5. Local stiffness matrix

According to Fig. 5.1, the stiffness matrix development can beperformed by initially analyzing a two-dimensional structure (xyand xz planes). Then, for spatial structures, as shown in Fig. 5.2,the integration between the planes is coupled to the stiffnessmatrix. Finally, to consider large rotations, the concept of finiterotations was added to the matrix.

5.1. Updated Lagrangian formulation

The stiffness matrices are calculated considering the updatedLagrangian formulation, and the steps shown in this work havebeen presented in the work by Mcguire et al. (2000), Chen(1994), Bathe (1996), Aguiar et al. (2014) and Rodrigues et al.(2019).

The virtual work of the internal forces must be equal to the vir-tual work of the external forces, according to (5.1),ZV

S tþDtð Þij de tþDtð Þ

ij dV ¼ R tþDtð Þ ð5:1Þ

where S tþDtð Þij corresponds to the second Piola-Kirchoff stress

tensor, e tþDtð Þij corresponds to the Green-Lagrange strain tensor,

and R tþDtð Þ corresponds to the virtual work due to external loading,which could include body, surface, inertia and damping forces(Aguiar et al., 2014). It is important to note that the term d isrelated to the application of a virtual displacement to considerthe virtual displacement principle, while D is related to a small dis-placement increment, changing the configuration.

However, the equilibrium equations of an unknown configura-tion t þ Dt must be written using a known reference configurationt. Thus, the linearized incremental equation requires small dis-placement increments according to the equations in (5.2),

S tþDtð Þij ¼ stij þ Dsij e tþDtð Þ

ij ¼ etij þ Deij ð5:2Þwhere stij corresponds to the Cauchy stress tensor, Dsij is the

stress increment and Deij is the deformation increment. In theknown reference configuration, there is no deformation sodetij = 0, and the element is only subject to rigid body motion. Thus,the left side of Eq. (5.1) can be rewritten as in (5.3).ZV

S tþDtð Þij de tþDtð Þ

ij dV ¼ZV

stij þ Dsij �

d Deij �

dV

¼ZV

DsijdDeijdV þZVstijdDeijdV ð5:3Þ

The Green-Lagrange strain tensor has a linear (Deij) and a non-linear part (Dgij), according to expressions (5.4) to (5.6) in the xyplane.

Deij ¼ Deij þ Dgij ð5:4Þ

n for a spatial element.

Fig. 5.2. 3-D element (Mcguire et al., 2000).


Dexx ¼ @u@x

Dexy ¼ @u@y

þ @v@x

ð5:5Þ

Dgxx ¼12

@u@x

� 2

þ @v@x

� 2" #

Dgxy ¼@u@x

@u@y

þ @v@x

@v@y

ð5:6Þ

The stress increment is obtained from the material constitutiverelation. The consideration of the linear approximation for thestress and strain increment leads to the expression in Eq. (5.7).

Dsij ¼ CijklDekl ¼ CijklDekl Deij ¼ Deij ð5:7ÞBased on equations (5.3) and (5.7), the virtual work equation

becomes (5.8).ZV

CijklDekldDeijdV þZVstijdDeijdV þ

ZVstijDgijdV ¼ R tþDtð Þ ð5:8Þ

The first integral leads to the elastic stiffness matrix, while thethird integral leads to the geometric stiffness matrix. The secondintegral represents the virtual work of forces acting on the elementin configuration t and is usually represented at the right-hand sideof the expression. However, in this case, the rotation componentsare approximated by displacement derivatives, and the finite rota-tion effect is not considered (Mcguire et al., 2000).

In the xy plane, the stress vector, the constitutive matrix, thelinear and nonlinear strain vectors are given according to equation(5.9). Thus, relations (5.10)–(5.12) can be written.

s ¼ sxxsxy

� �C ¼ E 0

0 G

� �e ¼ exx

cxy

( )g ¼ gxx

gxy

( )

ð5:9ÞZV

CijklDekldDeijdV ¼ZVexx:EdexxdV þ

ZVcxy:GdcxydV ð5:10Þ

ZVstijdDeijdV ¼

ZVsxxdexxdV þ

ZVsxydcxydV ð5:11Þ

ZVsijDgijdV ¼

ZVsxxdgxxdV þ

ZVsxydgxydV ð5:12Þ

5.2. Displacement field

The displacement field of a beam element is shown in Fig. 5.3and defined by Eq. (5.13).

u x; yð Þ ¼ u0 xð Þ � h xð Þ:y v x; yð Þ ¼ v0 xð Þ ð5:13ÞConsidering the Euler-Bernoulli beam theory, the rotation is the

derivative of the transverse displacement (h ¼ dv=dx); thus, thedisplacement field can be rewritten according to (5.14).

u x; yð Þ ¼ u0 xð Þ � v0 xð Þdx

:y v x; yð Þ ¼ v0 xð Þ ð5:14Þ

8

5.3. Local stiffness matrix equations

The development of the local stiffness matrix using cubic func-tions can be seen in Mcguire et al. (2000) considering Euler-Bernoulli beam theory and in Rodrigues et al. (2019) for aTimoshenko beam element. In this research, the adopted form toachieve the complete matrix is the same as that presented in thecited literature; however, the interpolation function is complete,i.e., the axial force is considered.

From the displacement field in Eq. (5.14), the linear and nonlin-ear parts of the Green-Lagrange strain tensor components in Eq.(5.6), can be rewritten as expressions (5.15) and (5.16), respec-tively, for the Euler-Bernoulli beam theory.

exx ¼ @u@x

¼ @u0

@x� y

@2v@x2

cxy ¼@v@x

þ @u@y

¼ 0 ð5:15Þ

gxx ¼ 12

@u@x

2þ @v@x

2 �

¼ 12

@u@x

�2þ @v@x

�2þy2 @2v@x2

�2� �y@2v

@x2@u@x

gxy ¼@u@x

@u@y

þ@v@x

@v@y

¼ y@2v@x2

@v@x

�@v@x

@u@x

ð5:16Þ

Meanwhile, for the Timoshenko beam theory, the linear andnonlinear parts of the strain tensor components in Eq. (5.6) canbe written according to Eqs. (5.17) and (5.18), using the displace-ment field presented in (5.13).

exx ¼ @u@x

¼ @u0

@x� y

@hZ@x

cxy ¼@v@x

þ @u@y

¼ @v0

@x� hZ ð5:17Þ

gxx ¼ 12

@u@x

�2þ @v@x

�2 �¼ 1

2@u@x

�2þ @v@x

�2þy2 @hz@x

�2 ��y@u

@x@hz@x

gxy ¼@u@x

@u@y

þ@v@x

@v@y

¼ y@hz@x

hz�hz@u@x

ð5:18Þ

5.3.1. Elastic matrix

� Plane xy (EBBT)From the displacement field in (5.14) and the linear part of the

strain tensor in (5.15), the first integral of the virtual work equa-tion, relation (5.10), can be written according to Eq. (5.19).

dU ¼ZV

exx:EdexxdV þZV

cxy:GdcxydV

¼ZA

Z L

0

@u@x

� y@2v@x2

!E d

@u@x

� yd@2v@x2

!dx

!dA

! dU ¼ R L0

@u@x d

@u@x dx

�ERA dAþ R L

0@2v@x2 d

@2v@x2 dx

�ERA y

2dAþ

� R L0

@2v@x2 d

@u@x dx

�ERA ydA� R L

0@u@x d

@2v@x2 dx

�ERA ydA

ð5:19Þ

In the beam centroidal axis,RA y

2dA ¼ Iz;RA ydA ¼ 0, and the

expression is reduced to (5.20).

dU ¼Z L

0

@u@x

d@u@x

dx�

EAþZ L

0

@2v@x2

d@2v@x2

dx

!EIz ð5:20Þ

From Eq. (4.4), the displacements can be written using theinterpolation functions resulting in (5.21). The notation@=@x ¼ ð Þ’ is adopted for simplicity.

dU ¼ duf gTZ L

0EA N0

u

� �N0

u

� �Tdx uf g þ dvf gT

�Z L

0EIz N00

v� �

N00v

� �Tdx vf g ð5:21Þ

Fig. 5.3. Beam displacement field.


The elastic matrix is formed when cubic functions are used.However, in this work, the interpolation functions are those calcu-lated in section 4.

� Plane xy (TBT)

The same idea can be used for the Timoshenko beam theory,however, considering now the displacement field presented in(5.13). Thus, the first integral of the virtual work equation, relation(5.10), is written by Eq. (5.22) or according to expression (5.23)when interpolation functions are used.

dU ¼ZV

exx:EdexxdV þZV

cxy:GdcxydV ! dU

¼ZA

Z L

0

@u@x

� y@hz@x

� E

@u@x

� y@hz@x

� dx

� dA

þZA

Z L

0

@v@x

� hZ

� G d

@v@x

� dhZ

� dx

� dA

! dU ¼Z L

0

@u@x

d@u@x

dx�

EAþZ L

0

@hZ@x

d@hZ@x

dx�

EIz

þZ L

0

@v@x

d@v@x

dx�

GAþZ L

0hZdhZdx

� GA

�Z L

0hZd

@v@x

dx�

GA�Z L

0

@v@x

dhZdx�

GA ð5:22Þ

dU ¼ duf gT R L0 EA N0

u

� �N0

u

� �Tdx uf g þ dvf gT R L0 EIz N0

hz

� �N0

hz

� �Tdx vf gþ dvf gT R L

0 GA N0v

� �N0v

� �Tdx vf g þ dvf gT R L0 GA Nhzf g Nhzf gTdx vf gþ

� dvf gT R L0 GA Nhzf g N0

v� �Tdx vf g � dvf gT R L

0 GA N0v

� �Nhzf gTdx vf g

ð5:23Þ

� Plane xz (EBBT)

For the plane xz behavior, the linear part of the strain tensor canbe written just by switching the displacement vwith w in equation(5.24). Thus, the first integral of the virtual work equation is givenby expression (5.25).

exx ¼ @u@x

¼ @u0

@x� z

@2w@x2

cxz ¼@w@x

þ @u@y

¼ @w0

@x� @w0

@x¼ 0 ð5:24Þ

dU ¼Z L

0

@u@x

d@u@x

dx�

EAþZ L

0

@2w@x2

d@2w@x2

dx

!EIy ð5:25Þ

9

The axial part was already used for the xy plane. Finally, writingthe expression using the interpolation functions, equation (5.26)can be achieved:

dU ¼ dwf gTZ L

0EIy N00

w

� �N00

w

� �Tdx wf g ð5:26Þ

where Nw corresponds to the same Nv interpolation functions.Plane xz (TBT)For the Timoshenko beam theory considering the xz plane, the

displacement field becomes (5.27), and the equations for the elas-tic matrix are given by relations (5.28) and (5.29).

exx ¼ @u@x

¼ @u0

@x� z

@hy@x

cxz ¼@w@x

þ @u@z

¼ @w0

@x� hy ð5:27Þ

dU ¼ R L0

@u@x d

@u@x dx

�EAþ R L

0@hy@x d

@hy@x dx

�EIy þ

R L0

@w@x d

@w@x dx

�GA

þ R L0 hydhydx

�GAþ

� R L0 hyd

@w@x dx

�GA� R L

0@w@x dhydx

�GA

ð5:28Þ

dU ¼ dwf gT R L0 EIy N0

hy

n oN0

hy

n oTdx wf g þ dwf gT R L

0 GA N0w

� �N0

w

� �Tdx wf gþ dwf gT R L

0 GA Nhy� �

Nhy� �Tdx wf g � dwf gT R L

0 GA Nhy� �

N0w

� �Tdx wf gþ� dwf gT R L

0 GA N0w

� �Nhy� �Tdx wf g

ð5:29Þ

5.3.2. Geometric matrix

� Plane xy (EBBT)

The third integral from the virtual work expression (5.8) gener-ates the usual geometric stiffness matrix. Thus, the nonlinear partof the virtual work principle, considering the complete strain ten-sor and higher order terms, is given by Eq. (5.30).

dUNL ¼ZV

sxxdgxxdV þZV

sxydgxydV

dUNL ¼ZA

Z L

0txxd

12

@u@x

� 2

þ @v@x

� 2

þ y2@2v@x2

!20@

1A� y

@2v@x2

@u@x

0@

1Adx

0@

1AdA

þZA

Z L

0txyd y:

@2v@x2

:@v@x

� @v@x

:@u@x

!dx

!dA

Fig. 5.4. Frame element.


! dUNL ¼ 12

Z L

0d

@u@x

� 2

dx

!ZAtxxdA

þ 12

Z L

0d

@v@x

� 2

dx

!ZAtxxdA

þ 12

Z L

0d

@2v@x2

!2

dx

0@

1AZ

Ay2txxdA

�Z L

0d@2v@x2

@u@x

dx

!ZAtxxydA

þZ L

0d@2v@x2

@v@x

dx

!ZAtxyydA

�Z L

0d@v@x

@u@x

dx� Z

AtxydA ð5:30Þ

Applying relations,RA txxdA ¼ P;

RA txxydA ¼ �Mz;

RA txydA ¼ Qy;

the expression becomes (5.31).

dUNL ¼ 12

Z L

0Pd

@u@x

� 2

þ @v@x

� 2 !

þ PIzAd

@2v@x2

!20@

1A

24

35dx

þZ L

0Mzd

@2v@x2

@u@x

!� Qyd

@v@x

@u@x

� " #dx ð5:31Þ

Once again, the displacements can be written using the interpo-lation functions in equation (4.4), resulting in expression (5.32).

dUNL ¼ duf gT R L0 P N0

u

� �N0

u

� �Tdx uf g þ dvf gT R L0 P N0

v� �

N0v

� �Tdx vf gþþ dvf gT R L

0 PIzA N00

v� �

N00v

� �Tdx vf g þ dvf gT R L0 Mz N00

v� �

N0u

� �Tdx uf gþ

þ duf gT R L0 Mz N0

u

� �N00v

� �Tdx vf g � dvf gT R L0 Qy N0

v� �

N0u

� �Tdx uf g�

� duf gT R L0 Qy N0

u

� �N0v

� �Tdx vf gð5:32Þ

Considering a constant shear force, the bending moment andthe shear force equations of a planar frame, as shown in Fig. 5.4,can be calculated by the expressions in (5.33).

MZ ¼ �MZ1 þ MZ1 þMZ2ð ÞxL

Qy ¼ � MZ1 þMZ2ð ÞL

ð5:33Þ

Substituting the complete shape functions and solving the inte-grals of the problem leads to the geometric stiffness matrix, con-sidering the Euler-Bernoulli beam theory and higher-order termsin the strain tensor.

� Plane xy (TBT)

When the Timoshenko beam theory is considered, the nonlinearpart of the virtual work principle in Eq. (5.8) is written by Eqs.(5.34) and (5.35) applying the same relations presented forEuler-Bernoulli beam theory.

dUNL ¼ZV

sxxdgxxdV þZV

sxydgxydV

dUNL ¼ZA

Z L

0txxd

12

@u@x

� 2

þ @v@x

� 2

þ y2@hz@x

� 2 !

� y@hz@x

@u@x

!dx

!dA

þZA

Z L

0txyd y

@hz@x

hz � hz@u@x

� dx

� dA

10

! dUNL ¼ 12

R L0 d

@u@x

�2dx � RA txxdAþ 1

2

R L0 d @v

@x


2

R L0 d

@hz@x

�2dx

� RA y

2txxdA

� R L0 d

@hz@x

@u@x dx

�RA txxydAþ R L

0 d @hz@x hzdx

� RA txyydA� R L

0 dhz @u@x dx

� RA txydA

ð5:34Þ

dUNL ¼ 12

Z L

0Pd

@u@x

� 2

þ @v@x

� 2 !

þ PIzAd

@hz@x

� 2 !" #

dx

þZ L

0Mzd

@hz@x

@u@x

� � Qyd hz

@u@x

� � �dx ð5:35Þ

Writing Eq. (5.35) with interpolation functions leads to Eq.(5.36).

dUNL ¼ duf gT R L0 P N0

u

� �N0

u

� �Tdx uf g þ dvf gT R L0 P N0

v� �

N0v

� �Tdx vf gþ dvf gT R L

0 PIzA N0

hz

� �N0

hz

� �Tdx vf g þ dvf gT R L0 Mz N0

hz

� �N0

u

� �Tdx uf gþ duf gT R L

0 Mz N0u

� �N0

hz

� �Tdx vf g � dvf gT R L0 Qy Nhzf g N0

u

� �Tdx uf g� duf gT R L

0 Qy N0u

� �Nhzf gTdx vf g

ð5:36Þ

� Plane xz (EBBT)

The nonlinear parts of the Green-Lagrange strain tensor consid-ering bending in plane xz are written according to expressions(5.37) and (5.38).

gxx ¼12

@u@x

� 2

þ @w@x

� 2 !

¼ 12

@u@x

� 2

þ @w@x

� 2

þ y2@2w@x2

!20@

1A� y

@2w@x2

@u@x

ð5:37Þ

gxz ¼@u@x

@u@y

þ @w@x

@w@y

¼ y@2w@x2

@w@x

� @w@x

@u@x

ð5:38Þ

With the same development made in plane xy, the third integralfrom the virtual work expression in Eq. (5.8) can be written as(5.39) for bending in plane xz.

dUNL ¼ 12

R L0 d

@u@x


2

R L0 d

@w@x


2

R L0 d

@2w@x2

�2dx

� RA z

2txxdAþ

� R L0 d @2w

@x2@u@x dx

� RA txxzdAþ R L

0 d@2w@x2

@w@x dx

� RA txzzdA� R L

0 d @w@x

@u@x dx

� RA txzdA

ð5:39Þ

Applying relationsRA txxdA ¼ P;

RA txxzdA ¼ My;

RA txzdA ¼ Qz, the

expression becomes (5.40).

dUNL ¼ 12

�Z L

0Pd

@u@x

� 2

þ @w@x

� 2 !

þ PIyAd

@2w@x2

!20@

1A

24

35dx

�Z L

0Myd

@2w@x2

@u@x

!þ Qzd

@w@x

@u@x

� " #dx

ð5:40Þ


The axial part was already used for the xy plane. The expressionusing the interpolation functions is given according to (5.41):

dUNL ¼ dwf gT R L0 P N0

w

� �N0

w

� �Tdx wf g þ dwf gT R L0 P

IyA N00

w

� �N00

w


0 My N00w

� �N0

u

� �Tdx uf g � duf gT R L0 My N0

u

� �N00

w


0 Qz N0w

� �N0

u

� �Tdx uf g � duf gT R L0 Qz N0

u

� �N0

w

� �Tdx wf gð5:41Þ

where Nw corresponds to the same Nv interpolation functions.Additionally, considering a constant shear force, the bendingmoment and the shear force equations of a planar frame in planexz can be calculated by the expressions in (5.42).

My ¼ �My1 þMy1 þMy2 �

xL

Qz ¼My1 þMy2 �

Lð5:42Þ

Substituting the complete shape functions that were previouslycalculated and solving the integrals of the problem leads to thegeometric stiffness matrix, which considers the Euler-Bernoullibeam theory and higher-order terms in the strain tensor.

� Plane xz (TBT)

Meanwhile, for this beam theory, the nonlinear parts of thestrain tensor considering bending in plane xz are written byexpressions (5.43) and (5.44). Thus, the third integral from the vir-tual work expression is given according to (5.45) or with (5.46).

gxx ¼12

@u@x

� 2

þ @w@x

� 2 !

¼ 12

@u@x

� 2

þ @w@x

� 2

þ z2@hy@x

� 2 !

� z@u@x

@hy@x

ð5:43Þ

gxz ¼@u@x

@u@z

þ @w@x

@w@z

¼ z@hy@x

hy � hy@u@x

ð5:44Þ

dUNL ¼ 12

R L0 d @u

@x

�2dx �RA txxdAþ 1

2

R L0 d

@w@x

�2dx �RA txxdAþ 1

2

R L0 d

@hy@x

�2dx

� RA z

2txxdAþ

� R L0 d

@hy@x

@u@x dx

� RA txxzdAþ R L

0 d @hy@x hydx

�RA txzzdA� R L

0 dhy @u@x dx

� RA txzdA

ð5:45Þ

dUNL ¼ 12

�Z L

0Pd

@u@x

� 2

þ @w@x

� 2 !

þ PIyAd

@hy@x

� 2 !" #

dx

�Z L

0Myd

@hy@x

@u@x

� þ Qzd hy

@u@x

� � �dx

ð5:46ÞEquation (5.47) can be written by considering only terms that

have not been used and applying interpolation functions.

Fig. 5.5. Combined torsion and axi

11

dUNL ¼ dwf gT R L0 P N0

w

� �N0

w

� �Tdx wf g þ dwf gT R L0 P

IyA N0

hy

n oN0

hy

n oTdx wf gþ

� dwf gT R L0 My N0

hy

n oN0

u

� �Tdx uf g � duf gT R L0 My N0

u

� �N0

hy

n oTdx wf gþ

� dwf gT R L0 Qz Nhy

� �N0

u

� �Tdx uf g � duf gT R L0 Qz N0

u

� �Nhy� �Tdx wf g

ð5:47Þ

5.3.3. Combined torsion and axial forceThe interaction between torsion and axial force, as shown in

Fig. 5.5, has an important influence on the geometric stiffnessmatrix. The transverse displacements, v and w, need to take theseeffects into account.

� eBBT

Considering the Euler-Bernoulli beam theory, the displacementfield is written according to expression (5.48)

u ¼ u0 � z@w@x

� y@v@x

v ¼ v0 � zhx w ¼ w0 þ yhx ð5:48Þ

However, most terms of this equation have already beenemployed when planes were analyzed independently. Therefore,it is necessary to consider only terms that correspond to the rota-tion about the x axis, which is given in equation (5.49). Thus, thenonlinear parts of the Green-Lagrange strain tensor can be writtenaccording to expressions (5.50) to (5.52).

v ¼ �zhx w ¼ yhx ð5:49Þ

gxx ¼12

z@hx@x

� 2

þ y@hx@x

� 2 !

� z@v@x

@hx@x

þ y@w@x

@hx@x

¼ 12

z2 þ y2 � @hx

@x

� 2 !

� z@v@x

@hx@x

þ y@w@x

@hx@x

ð5:50Þ

gxy ¼@u@x

@u@y

þ @v@x

@v@y

þ @w@x

@w@y

¼ � @u@x

@v@x

þ z@2w@x2

@v@x

þ y@2v@x2

@v@x

þ @w@x

hx þ y@hx@x

hx ð5:51Þ

gxz ¼@u@x

@u@z

þ @v@x

@v@z

þ @w@x

@w@z

¼ � @u@x

@w@x

þ z@2w@x2

@w@x

þ y@2v@x2

@w@x

� @v@x

hx þ z@hx@x

hx ð5:52Þ

The virtual work principle is then written as (5.53) and (5.54).

dUNL ¼ZA

Z L

0txxdgxxdx

� dA

¼ZA

Z L

0txxd

12

z2 þ y2 � @hx

@x

� 2 !

� z@v@x

@hx@x

þ y@w@x

@hx@x

" #dx

!dA

ð5:53Þ

al force (Mcguire et al., 2000).

Fig. 5.6. Spatial transformation between two vectors (Aguiar et al., 2014).


dUNL ¼ZA

Z L

0txydgxydx

� dAþ

ZA

Z L

0txzdgxzdx

� dA

dUNL ¼R L0 d @2w

@x2@v@x

�dx

�RA txyzdAþ R L

0 d @w@x hx �

dx � R

A txydAþ R L0 d @hx

@x hx �

dx � R

A txyydAþ

þ R L0 d @2v

@x2@w@x

�dx

�RA txzydA� R L

0 d @v@x hx �

dx � R

A txzdAþ R L0 d

@hx@x hx �

dx �R

A txzzdA

ð5:54Þ

Adopting the relations presented before for the bendingmoment and

RA z2 þ y2 �

dA ¼ Jp, equation (5.53) becomes (5.55).Expression (5.56) is written using interpolation functions.

dUNL ¼ 12PJpA

Z L

0d

@hx@x

� 2

dx�My

Z L

0d

@v@x

@hx@x

� dx

þ �Mzð ÞZ L

0d

@w@x

@hx@x

� dx ð5:55Þ

dUNL ¼ dhxf gT R L0

PJpA N0

hx

� �N0

hx

� �Tdx hxf g � dvf gT R L0 My N0

v� �

N0hx

� �Tdx hxf g� dhxf gT R L

0 My N0hx

� �N0v

� �Tdx vf g � dwf gT R L0 Mz N0

w

� �N0

hx

� �Tdx hxf g� dhxf gT R L

0 Mz N0hx

� �N0

w

� �Tdx wf gð5:56Þ

For bisymmetric sections, the shear force and torsion momentare defined by (5.57). Thus, expression (5.54) can be writtenaccording to (5.58).

ZAtxydA ¼ Qy;

ZAtxzdA ¼ Qz;

ZAtxzydA ¼ aMx;Z

AtxyzdA ¼ ða� 1ÞMx ð5:57Þ

dUNL ¼R L0 d

@2w@x2

@v@x dx

�ða� 1ÞMx þ

R L0 d

@w@x hx �

dx �

Qy

þ R L0 d

@2v@x2

@w@x

�dx

�aMx �

R L0 d

@v@x hx �

dx �

Qz

ð5:58Þ

Considering Saint Venant pure torsion, Mx ¼ Mx2 and a ¼ 1=2.Then, Eq. (5.59) is written by rewriting (5.58) using interpolationfunctions.

dUNL ¼ � dwf gT R L0

Mx2 N00

w

� �N0v

� �Tdx vf g � dvf gT R L0

Mx2 N0

v� �

N00w

� �Tdx wf g

þ dvf gT R L0

Mx2 N00

v� �

N0w

� �Tdx wf g þ dwf gT R L0

Mx2 N0

w

� �N00v

� �Tdx vf g

þ dwf gT R L0 Qy N0

w

� �Nhxf gTdx hxf g þ dhxf gT R L

0 Qy Nhxf g N0w

� �Tdx wf g

� dvf gT R L0 Qz N0

v� �

Nhxf gTdx hxf g � dhxf gT R L0 Qz Nhxf g N0

v� �Tdx vf g

ð5:59ÞThe combined torsion and axial force contribution is found by

substituting the complete shape functions and solving theintegrals.

� TBT

When the Timoshenko beam theory is considered, the displace-ment field cannot be written using the transverse displacementderivative and instead should be given by Eq. (5.60). Thus, the non-linear part of the strain tensor that is not used before is writtenaccording to expressions (5.61) to (5.63).

u ¼ u0 � zhy � yhz v ¼ v0 � zhx w ¼ w0 þ yhx ð5:60Þ

gxx ¼12

z@hx@x

� 2

þ y@hx@x

� 2 !

� z@v@x

@hx@x

þ y@w@x

@hx@x

¼ 12

z2 þ y2 � @hx

@x

� 2

� z@v@x

@hx@x

þ y@w@x

@hx@x

ð5:61Þ

12

gxy ¼@u@x

@u@y

þ @v@x

@v@y

þ @w@x

@w@y

¼ � @u@x

hz þ z@hy@x

hz þ y@hz@x

hz þ @w@x

hx þ y@hx@x

hx ð5:62Þ

gxz ¼@u@x

@u@z

þ @v@x

@v@z

þ @w@x

@w@z

¼ � @u@x

hy þ z@hy@x

hy þ y@hz@x

hy � @v@x

hx þ z@hx@x

hx ð5:63Þ

Equation (5.61) is the same for both beam theories because it isequal to (5.50). Finally, the virtual work principle for theTimoshenko beam theory only changes in Eq. (5.64) due to thenonlinear distortion.

dUNL ¼ZA

Z L

0txydgxydx

� dAþ

ZA

Z L

0txzdgxzdx

� dA

dUNL ¼R L0 d

@hy@x hz �

dx � R

A txyzdAþ R L0 d

@w@x hx �

dx � R

A txydAþ R L0 d

@hx@x hx �

dx � R

A txyydAþ

þ R L0 d @hz

@x hy �

dx � R

A txzydA� R L0 d

@v@x hx �

dx � R

A txzdAþ R L0 d @hx

@x hx �

dx � R

A txzzdA

ð5:64Þ

Expression (5.64) is rewritten according to (5.65) or (5.66) usinginterpolation functions with the same relations presented before in(5.57).

dUNL ¼R L0 d

@hy@x hzdx

�ða� 1ÞMx þ

R L0 d

@w@x hx �

dx �

Qyþ

þ R L0 d

@hz@x hy �

dx �

aMx �R L0 d

@v@x hx �

dx �

Qz

ð5:65Þ

dUNL ¼ � dhy� �T R L

0Mx2 N0

hy

n oNhzf gTdx hzf g � dhzf gT R L

0Mx2 Nhzf g N0

hy

n oTdx hy� �

þ dhzf gT R L0

Mx2 N0

hz

� �Nhy� �Tdx hy

� �þ dhy� �T R L

0Mx2 Nhy� �

N0hz

� �Tdx hzf gþ dwf gT R L

0 Qy N0w

� �Nhxf gTdx hxf g þ dhxf gT R L

0 Qy Nhxf g N0w

� �Tdx wf g� dvf gT R L

0 Qz N0v

� �Nhxf gTdx hxf g � dhxf gT R L

0 Qz Nhxf g N0v

� �Tdx vf gð5:66Þ

5.3.4. Finite rotationsIn spatial elements, the geometric stiffness matrix needs to con-

sider finite rotations to satisfy the kinematic compatibility at thejoint of angled elements (Mcguire et al., 2000; Conci, 1988). Thus,for a spatial vector, the rotation is given by equation (5.67), asshown in Fig. 5.6.

V1 ¼ RðhÞV0 ð5:67Þ


R ¼ I þ senðhÞh

W hð Þ þ 1� cosðhÞh2

W hð Þ2

W hð Þ ¼0 �hz hyhz 0 �hx�hy hx 0

264

375; h ¼

hxhyhz

8><>:

9>=>;

Using a trigonometric series approximation given by expression(5.68) and considering only terms up to the second order, the rota-tion matrix can be rewritten according to (5.69).

sen hð Þ ¼ h� h3

3!þ h5

5!� h7

7!þ � � �

cos hð Þ ¼ 1� h2

2!þ h4

4!� h6

6!þ � � � ð5:68Þ

R hð Þ ¼ I þW hð Þ þ 12W hð Þ

2

ð5:69Þ

Therefore, considering a bisymmetric section as shown inFig. 5.7, the position vector of a point, af g, and after rotation, bf g,is written by equation (5.70).

bf g ¼ I þW hð Þ þ 12W hð Þ

2" #

af g ð5:70Þ

The displacement can be written as: bf g � af g or uf g. However,W hð Þ has already been taken into account in the previous analysiswith the principle of virtual work. Thus, the displacement can be

written only as a function of W hð Þ2, according to expressions(5.71) and (5.72).

uf g¼ 12W hð Þ2

� �af g)

u

vw

8><>:

9>=>;¼

� hy2þhz

2

2hxhy2

hxhz2

hxhy2 � hx

2þhz2

2hyhz2

hxhz2

hyhz2 � hx

2þhy2

2

26664

37775

0y

z

8><>:

9>=>;

ð5:71Þ

u ¼ hx:hy:y2

þ hx:hz:z2

v ¼ hy:hz:z2

� yh2x2þ h2z

2

!

w ¼ hy:hz:y2

� zh2x2þ h2y

2

!ð5:72Þ

To consider the finite rotations, the second integral of the virtualwork principle needs to be used according to expression (5.73).Z

VtijDeij ¼

ZA

Z L

0txxdexxdx

� dAþ

ZA

Z L

0txydexydx

� dA

þZA

Z L

0txzdexzdx

� dA ¼

Fig. 5.7. Spatial transformation between two vectors (Mcguire et al., 2000).

13

¼ZA

Z L

0txxd

@u@x

� dx

� �dAþ

ZA

Z L

0txyd

@v@x

þ @u@y

� dx

� �dA

þZA

Z L

0txzd

@w@x

þ @u@z

� dx

� �dA

!ZVtijDeij ¼

ZA

Z L

0

txx2

d@ hxhy �@x

yþ d@ hxhzð Þ

@xz

� �dx

� �dA

þZA

Z L

0

txy2

d hxhy �þ d

@ hyhz �@x

z� �

dx� �

dA

þZA

Z L

0

txz2

d hxhzð Þ þ d@ hyhz �@x

y� �

dx� �

dA ð5:73Þ

Equation (5.74) can be written by employing the relations forthe bending moment, torsion, shear and axial force.

RV tijDeij ¼

R L0

�Mz2 d

@ hxhyð Þ@x dxþ R L

0My

2 d @ hxhzð Þ@x dxþ R L

0Qy

2 d hxhy �

dx

þ R L0

ða�1ÞMx2 d

@ hyhzð Þ@x dxþ R L

0Qz2 d hxhzð Þdxþ R L

0aMx2 d

@ hyhzð Þ@x dx

ð5:74ÞExpression (5.75) is obtained by adopting a ¼ 1=2 for Saint

Venant torsion and using integration by parts. Finally, the finiterotation matrix is given in (5.76) and is independent of the beamtheory considered.RV tijDeij ¼ �Mz

2 d hxhy ��L

0 þMy

2 d hxhzð ÞjL0RV tijDeij ¼ 1

2 �Mz1d hx1hy1 �þMy1d hx1hz1ð Þ �Mz2d hx2hy2

�þMy2d hx2hz2ð Þ� �ð5:75Þ

Kg;Rotfin ¼ 12

0 �Mz1 My1 0 0 0�Mz1 0 0 0 0 0My1 0 0 0 0 00 0 0 0 �Mz1 My2

0 0 0 �Mz1 0 00 0 0 My2 0 0

2666666664

3777777775

hx1hy1hz1hx2hy2hz2

ð5:76ÞTherefore, considering these effects, the local complete tangent

stiffness matrix can be written by adding all the components forboth bending theories. The results are presented in the work byRodrigues (Rodrigues, 2019) and with an open source code avail-able in the files StfBeamEulerBernoulliT (positive axial force) andStfBeamEulerBernoulliC (negative axial force) considering Euler-Bernoulli beam theory. For the Timoshenko beam theory, the filesare StfBeamTimoshenkoT (positive axial force) and StfBeamTi-moshenkoC (negative axial force) (Rodrigues et al., 2020, 2021).

To avoid numerical instability, the local tangent stiffness matrixcan be written using a Taylor series expansion, providing moreterms for the tangent stiffness matrix than the usual formulations.These approximations provide greater precision in the results, andin this work, tangent matrices with up to 3 and 4 terms weredeveloped. It is important to note that the 2-term approximationprovides the usual elastic and tangent stiffness matrices. Thesematrices are also presented in the work by Rodrigues (Rodrigues,2019) and implemented in (Rodrigues et al., 2020, 2021) in the filesGeoStfBeamEulerBernoulli and GeoStfBeamTimoshenko.

6. Numerical applications

In this section, numerical examples were developed with nodiscretization to test the formulation from this work, the completeformulation and the Taylor series expansion with 3 and 4 terms.The elements can be identified based on the followingdescriptions:


EBBT_Large_Complete – Euler-Bernoulli beam theory, hyper-bolic and geometric functions in the tangent stiffness matrix, andhigher order terms in the strain tensor.

EBBT_Large_4tr – Euler-Bernoulli beam theory, 4 terms in thetangent stiffness matrix (1 elastic + 3 geometric), and higher orderterms in the strain tensor.

EBBT_Large_3tr – Euler-Bernoulli beam theory, 3 terms in thetangent stiffness matrix (1 elastic + 2 geometric), and higher orderterms in the strain tensor.

EBBT_Large or Small_2tr – Euler-Bernoulli beam theory, 2terms in the tangent stiffness matrix (1 elastic + 1 geometric),and higher order terms in the strain tensor (Large) or (Small) if thisinfluence is not considered (Yang and Leu, 1994; Yang and Kuo,1994; Chen, 1994). This formulation is the most conventional.

TBT_Large_Complete – Timoshenko beam theory, hyperbolicand geometric functions in the tangent stiffness matrix, and higherorder terms in the strain tensor.

TBT_Large_4tr – Timoshenko beam theory, 4 terms in the tan-gent stiffness matrix (1 elastic + 3 geometric), and higher orderterms in the strain tensor.

TBT_Large_3tr – Timoshenko beam theory, 3 terms in the tan-gent stiffness matrix (1 elastic + 2 geometric), and higher orderterms in strain tensor.

TBT_Large or Small_2tr – Timoshenko beam theory, 2 terms inthe tangent stiffness matrix (1 elastic + 1 geometric), and higherorder terms in the strain tensor (Large) or (Small) if this influenceis not considered (Rodrigues et al., 2019). This formulation is themost conventional.

The results were compared with the results obtained with Mas-tan2 v3.5. The software is able to perform a geometric nonlinearanalysis considering both beam theories while considering cubicinterpolation functions and disregarding higher order terms inthe strain tensor. The element tangent stiffness matrix consideredin Mastan2 v3.5 is described in the work by McGuire et al.(Mcguire et al., 2000). In this research, Mastan2 v3.5 elementsare labeled as follows:

EBBT_Mastan – Mastan2 v3.5 software Euler-Bernoulli beamtheory;

TBT_Mastan – Mastan2 v3.5 software Timoshenko beamtheory.

In cases that are no available analytical solution, the referenceresponse for comparison purposes was given by a numerical solu-tion with discretized structure (EBBT or TBT_Large_2tr_ number ofelements in each member).

Euler Critical Load – Analytical Euler buckling loadTimoshenko and Gere – Analytical Timoshenko buckling load

(Timoshenko and Gere, 1963)

Fig. 6.1. Analyzed column

14

6.1. Isolated columns

To verify the developed tangent stiffness matrix, the bucklingload of columns was studied as shown in Fig. 6.1, adopting justone element per member. Additionally, a reduced slenderness ratio(k ¼ L=h) was employed because of the influence of Timoshenkobeam theory. The columns have a length of L = 1 m, Young’s mod-ulus of E = 107 kN/m2, section form factor of v ¼ 1 and null Pois-son’s ratio (m ¼ 0).

Fig. 6.2 shows the equilibrium paths for the clamped column,while Figs. 6.3 and 6.4 represent equilibrium paths for the simplysupported and fixed and simply supported columns, respectively,with a slenderness ratio of k ¼ 10:0 for the Euler-Bernoulli beamtheory and k ¼ 4:0 for the Timoshenko beam theory. The numericalresults for the buckling loads for the columns are shown in Table 1.

Analyzingtheequilibriumpaths, it canbenotedthat thecompleteformulation developed in this work, where the tangent stiffnessmatrix was calculated with the complete interpolation functions(EBBT_Large_complete and TBT_Large_complete), provides the cor-rect prediction for the buckling loads of columns using just one ele-ment per member for both beam theories. With no discretization,the complete formulation provides the best approximation for theanalytical response for both beam theories. In Table 1 the resultscan be seen numerically and the efficiency of the complete formula-tion is observedwhen comparedwith literature values.

Additionally, it can be seen that geometric matrices obtainedfrom a Taylor series approximation (Large_3tr and Large_4tr) alsoprovide an accurate prediction for the buckling load. The approxi-mation using 4 terms in some cases provides the same responsefrom the complete formulation, and the approximation with 3terms follows those curves closely. Table 1 also shows the proxim-ity of the complete formulation and the Taylor series expansionand the differences with the usual formulation.

6.2. Continuous beam-column

The next example studies a continuous beam-column; this prob-lem is presented in the work by (Timoshenko and Gere (1963) andis shown in Fig. 6.5. The geometry of the structure, material and sec-tion proprieties are the same as in the first example with a length ofL = 1 m, Young’s modulus of E = 107 kN/m2, section form factor ofv ¼ 1 and null Poisson’s ratio (m ¼ 0). Fig. 6.6 shows the equilibriumpath considering a slenderness ratio of k ¼ 10:0 for the Euler-Bernoulli beam theory and k ¼ 4:0 for Timsohenko beam theory.The structuresweremodeledwith one element in each span.

Again, using just one element per member, the complete formu-lation and the Taylor series expansion with 4 terms provide the

s (Silva et al., 2016).

Fig. 6.2. Equilibrium paths for a clamped column.

Fig. 6.3. Equilibrium paths for a simply supported column.

Fig. 6.4. Equilibrium paths for a fixed and simply supported column.


correct buckling load of the structure. The expansion with just 3terms gives approximate results. These results can be seen numer-ically in Table 2.

The conventional geometric stiffness matrix and the Mastan2v3.5 software cannot predict the buckling load of the continuousbeam-column, providing differences on the order of 50% to theexpected result with a discretized structure.

6.3. Spatial frame

The example presented is the first to evaluate the developedformulation in a spatial structure, according to Fig. 6.7. The frame

15

has a length of L = 1 m, Young’s modulus of E = 107 kN/m2, sec-tion form factor of v ¼ 5=6 and Poisson’s ratio of m ¼ 0:3. Theframe is loaded by vertical loads (P) and two small lateral dis-turbing loads (H ¼ 0:001P). The structure was modeled with oneelement per member, and the response was compared with a dis-cretized structure using the usual formulation. The equilibriumpaths can be seen in Fig. 6.8, considering k ¼ 10:0 for the Euler-Bernoulli beam theory and k ¼ 4:0 for the Timoshenko beamtheory.

Once again, the complete formulation and the Taylor seriesexpansion with 4 terms provide the buckling load of the spatialframe. The expansion with 3 terms overlaps these results for the

Fig. 6.5. Beam-column (Timoshenko and Gere, 1963).

Fig. 6.6. Equilibrium paths for a beam-column.

Table 1Numerical Buckling Loads for Columns.

Euler Bernoulli Beam Theory - EBBT

Discretization 1 Element Euler Critical Load

EBBT- Element Mastan Small Large Large Large Large2tr 2tr 3tr 4tr Complete

Clamped col. (k = 10.0) 2.496 2.491 2.486 2.469 2.468 2.468 2.467Simply supported col. (k = 10.0) 12.255 12.312 12.197 10.348 10.047 9.95 9.869Fixed – simply sup. col. (k = 10.0) 31.651 31.630 30.798 23.583 21.779 20.541 20.142

Timoshenko Beam Theory – TBT

Discretization 1 Element (Timoshenko and Gere, 1963)

TBT - Element Mastan Small Large Large Large Large2tr 2tr 3tr 4tr Complete

Clamped col. (k = 4.0) 2.475 2.475 2.444 2.419 2.417 2.417 2.408Simply supported col. (k = 4.0) 13.914 13.940 12.866 10.064 9.546 9.368 8.949Fixed - simply sup. col. (k = 4.0) 40.642 > 60 38.065 22.572 19.892 17.826 16.401


Euler-Bernoulli beam theory and provides intermediate results forthe Timoshenko theory. The usual formulation and Mastan2 v3.5software provide more elevated buckling loads. Table 3 presentsthese conclusions numerically.

Table 2Numerical buckling loads for the beam-columns.

Discretization 1 Element

Element Mastan Small Large Large2tr 2tr 3tr

EBBT -k = 10.0 12.240 12.310 12.240 10.45TBT - k = 4.0 13.989 13.843 12.867 10.07

16

6.4. Roorda spatial frame

Using the same bar length, material and section proprieties asthe structure presented before, another spatial frame was studied

(Timoshenko and Gere, 1963)

Large Large4tr complete

3 10.145 9.953 9.8694 9.553 9.368 8.949

Fig. 6.7. Spatial frame.

Fig. 6.9. Roorda spatial frame.


(Fig. 6.9). The frame is loaded by a vertical load (P) and by twosmall lateral disturbing loads (H ¼ 0:001P). The structure wasmodeled with one element per member, and the response wascompared with a discretized structure using the usual formulation.The equilibrium paths can be seen in Fig. 6.10 with k ¼ 10:0 for theEuler-Bernoulli beam theory and k ¼ 4:0 for the Timoshenko beamtheory.

There is no available analytical solution for the critical load ofthis frame. In this case, the reference solution for comparison pur-poses was the Large_2tr formulation with 4 segments in eachmember. One may observe that, with no discretization, the com-plete formulation provides the best approximation for the refer-ence response. The second-best solution is for the Taylor seriesexpansion with 4 terms and then the Taylor series expansion with

Fig. 6.8. Equilibrium path

Table 3Numerical buckling loads for a spatial frame.

Discretization 1 Elem

Element Mastan Small Large2tr 2tr

EBBT -k = 10.0 5.577 5.569 5.554TBT-k = 4.0 5.068 5.115 5.030

17

3 terms. These formulations give better approximations whencompared with the usual formulations with 2 terms, which presentrelevant errors. Table 4 shows the numerical results.

When considering just one element per member, Table 4 showsthat the conventional formulation doubles the buckling load, whilethe complete formulation maintains the error at a rate of 12% forTBT elements and 8% for EBBT theory.

6.5. Spatial frame with inclined columns

The frame presented in Fig. 6.11 is loaded by two vertical loads(P) and by small lateral disturbing loads (H ¼ 0:001P) with a lengthof L = 1 m, Young’s modulus of E = 107 kN/m2, section form factor ofv ¼ 5=6 and Poisson’s ratio of m ¼ 0:3.

The equilibrium paths found for the structure employing oneelement per member and both beam theories, consideringk ¼ 4:0, are given in Fig. 6.12 and numerically presented in Table 5.

s for a spatial frame.

ent 4 Elements

Large Large Large Large3tr 4tr Complete 2tr_4el

5.507 5.501 5.500 5.4884.977 4.837 4.934 4.927

Fig. 6.10. Equilibrium paths for a Roorda spatial frame.

Fig. 6.11. Spatial frame with inclined columns (adapted from Zugic et al. (Zugicet al., 2016).

Table 4Numerical buckling loads for a Roorda spatial frame.

Discretization 1 Element 4 Elements

Element Mastan Small Large Large Large Large Large2tr 2tr 3tr 4tr Complete 2tr_4el

EBBT -k = 10.0 19.008 24.461 20.897 15.262 14.484 14.136 13.200TBT - k = 4.0 21.135 21.166 19.630 13.670 12.445 11.753 10.414


As concluded for the other examples, the complete formulationcalculates the exact buckling load of the structure. In this example,the Taylor series expansion with 4 or even 3 terms reaches thisload as well, and their results overlap. Although the usual formula-tion gives a good approximation with just one element, employingthe formulation developed in this work improves the results. Inaddition, consideration of Timoshenko beam theory reduces thecritical buckling load for structures with small slenderness.

6.6. Spatial frame with torsion

This example explores the influence of torsion. The frame alsohas a length of L = 1 m, Young’s modulus of E = 107 kN/m2, section

18

form factor of v ¼ 5=6 and Poisson’s ratio of m ¼ 0:3 and is loadedby four vertical loads (P) and by two lateral loads (H ¼ 0:01P)according to Fig. 6.13. Fig. 6.14 shows the deformed structure illus-trating the torsion experienced by the space frame with the scalefactor increased tenfold. Finally, the equilibrium path of the struc-ture considering k ¼ 10 for the Euler-Bernoulli beam theory andk ¼ 4:0 for the Timoshenko beam theory is illustrated.Fig. 6.15.

The solution obtained by the complete formulation and for theTaylor series expansion provides the correct buckling load employ-ing just one element per member for both beam theories, while theusual formulation proceeds to higher values for the critical load.This is clearly illustrated numerically in Table 6.

6.7. Asymmetric spatial frame

The last example studies an asymmetric frame with moderateslenderness, k ¼ 6:6 for TBT, as shown in Fig. 6.16, loaded by twovertical loads (P) and small lateral disturbing loads (H ¼ 0:001P),with a length of L = 1 m, Young’s modulus of E = 107 kN/m2, sectionform factor of v ¼ 5=6 and Poisson’s ratio of m ¼ 0:3. Fig. 6.17shows the influence of torsion on this structure, and Fig. 6.18shows the equilibrium paths.

Finally, the complete formulation and the Taylor series expan-sion provide the buckling load for the spatial frame, with thecurves overlapping. The usual formulation and Mastan softwareovercome this load for both beam theories. Table 7 shows theseconclusions numerically.

7. Conclusions

This research developed a complete formulation of the tangentstiffness matrices of a spatial frame element for the Euler-Bernoulliand Timoshenko beam theories, considering interpolation func-tions obtained directly from the solution of the equilibrium differ-ential equation of a deformed infinitesimal element, whichincludes the influence of axial forces. Additionally, the formulationuses higher-order terms in the strain tensor, and the geometric

Table 5Numerical buckling loads for the spatial frame with inclined columns.

Discretization 1 Elem

Element Mastan Small Large2tr 2tr

EBBT-k = 4.0 7.070 7.179 7.157TBT- k = 4.0 6.182 6.371 6.227

Fig. 6.12. Equilibrium paths for the spatial frame with inclined columns.

Fig. 6.13. Spatial frame with asymmetric loads.

Fig. 6.14. Deformed spatia


19

stiffness matrix is corrected by employing finite rotations. Thecomplete formulation was rewritten using a Taylor series expan-sion considering 3 and 4 terms, improving the usual tangent stiff-ness matrices for both beam theories that involve only 2 terms (1elastic and 1 geometric).

The formulation was evaluated in several numerical tests forplanar and spatial structures to verify the geometric nonlinearresponse of the structure considering distinct bending theoriesand the consideration of high-order terms for the strain. The firstset of examples employs the proposed element to analyze isolatedcolumns with different boundary conditions and a continuousbeam-column. Similarly, this formulation was evaluated for spatialstructures, and some examples explored the torsion effect, apply-ing a load or geometric asymmetry.

The complete formulation presented precisely predicted resultsfor the critical load of the developed examples with a minimumdiscretization of the structure, while the usual tangent stiffnessmatrix would require a more refined mesh to predict this behavior.

ent 4 Elements

Large Large Large Large3tr 4tr Complete 2tr_4el

6.952 6.942 6.942 6.9196.155 6.155 6.082 6.082

l frame with torsion.

Table 6Numerical buckling loads for the spatial frame with torsion.



EBBT- k = 10.0 7.488 7.328 7.313 7.259 7.257 7.256 7.260TBT-k = 4.0 6.389 6.548 6.374 6.367 6.367 6.341 6.341

Fig. 6.16. Asymmetric spatial frame with inclined columns (adapted from Zugicet al. (Zugic et al., 2016).

Fig. 6.17. Deformed asymmetric spatial frame.

Fig. 6.15. Equilibrium paths for the spatial frame with torsion.


As expected, the examples have also shown that for structures withsmall slenderness, the consideration of the Timoshenko beam the-ory leads to lower buckling loads, and the proposed formulationillustrated this effect. The Timoshenko beam theory is also usedin cases of structures with a small shear-to-bending ratio, regard-less of their slenderness.

The reported results clearly illustrate the efficiency of the com-plete formulation and the Taylor series expansion using 4 terms inthe solution of nonlinear geometric analyses using only one ele-ment for planar and spatial structures. Using 3 terms also providesbetter results than considering the usual geometric stiffnessmatrix, which overestimates the buckling load.

The presented formulation solves geometric nonlinear prob-lems and can be implemented in any structural analysis software.

20

Due to the numerical instability that can occur using the hyper-bolic and trigonometric expressions of the complete formulation,an alternative geometric stiffness matrix using a Taylor seriesexpansion with 4 terms is proposed to solve this problem withaccurate results. All necessary equations for this implementationare available online in open source.

Although precise buckling loads are achieved using just one ele-ment if one is interested in following the actual equilibrium pathup to the critical load, a refined discretization should be used. Thisis suggested because one element is not able to capture localeffects, such as the P-‘‘small delta”.

The continuation of this work will focus on monitoring theequilibrium path in the postcritical stage. For this, more improvedsolution schemes for incremental geometric nonlinear analysisshould be employed.

8. Funding sources

This work has been partially supported by Conselho Nacional deDesenvolvimento Científico e Tecnológico (CNPq) and FAPERJ.

Table 7Numerical buckling loads for asymmetric spatial frames.



EBBT- k = 10 7.584 7.606 7.586 7.499 7.486 7.483 7.452TBT- k = 6.6 7.282 7.298 7.259 7.152 7.132 7.083 7.089

Fig. 6.18. Equilibrium paths for an asymmetric spatial frame.


Declaration of Competing Interest

The authors declare that they have no known competing finan-cial interests or personal relationships that could have appearedto influence the work reported in this paper.

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Marcos Antonio Campos Rodrigues. Adjunct professor inthe Department of Civil Engineering at Federal Univer-sity of Espírito Santo (UFES). B.Sc. in Civil Engineeringfrom Federal University of Espírito Santo (UFES) in 2011.M.Sc. in Aeronautical and Mechanical Engineering fromthe Aeronautics Institute of Technology (ITA) in 2014and Ph.D. in Civil Engineering from Pontifical CatholicUniversity of Rio de Janeiro (PUC-Rio) in 2019. He hasexperience in computational mechanics and structuraldesign. His main research interests include Finite Ele-ment Method, Nonlinear Analysis, Design of ReinforcedConcrete, Prestressed and Steel Structures.

Rodrigo Bird Burgos Associate Professor in the Depart-ment of Structures and Foundations at the StateUniversity of Rio de Janeiro (UERJ). B.Sc. (2003), M.Sc.(2005) and Ph.D. (2009) in Civil Engineering from Pon-tifical Catholic University of Rio de Janeiro (PUC-Rio).Postdoctoral researcher at Pontifical Catholic Universityof Rio de Janeiro from 2009 to 2012. He has experiencein computational modeling and advanced numericalmethods for solving differential equations. His mainresearch interests include instability and structuraldynamics, wavelet functions, interpolets, Wavelet-Galerkin Method, Finite Element Method.

Luiz Fernando Martha Associate professor of theDepartment of Civil Engineering at Pontifical CatholicUniversity of Rio de Janeiro. B.Sc. (1977), M.Sc. (1980) inCivil Engineering from Pontifical Catholic University ofRio de Janeiro (PUC-Rio) and Ph.D. (1989) in StructuralEngineering from Cornell University. Postdoctoralresearcher at Superior Technical Institute from Techni-cal University of Lisbon in 2012. Acts as coordinator ofresearch projects at Tecgraf / PUC-Rio (Tecgraf Institutefor Technical and Scientific Software Development). Hehas experience in structural analysis and his mainresearch interests include Computer Graphics, Geo-

metric Modeling, Numerical Methods Applied to Engineering and Geology Simu-lations, Computational Fracture Mechanics, and Educational Software forEngineering Teaching.










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