International Journal of Solids and Structures 222–223 (2021) 111003
Contents lists available at ScienceDirect
International Journal of Solids and Structures
journal homepage: www.elsevier.com/locate/ijsolstr
A unified approach to the Timoshenko 3D beam-column element
tangent stiffness matrix considering higher-order terms in the strain
tensor and large rotations
Marcos Antonio Campos Rodrigues a,⇑, Rodrigo Bird Burgos b, Luiz Fernando Martha c
a
b
c
Espírito Santo Federal University, Department of Civil Engineering, Avenida Fernando Ferrari, 514, Goiabeiras, Vitória, ES 29075-910, Brazil
State University of Rio de Janeiro, Department of Structures and Foundations, Rua São Francisco Xavier, 524, Maracanã, Rio de Janeiro, RJ 20550-900, Brazil
Pontifical 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 2020
Received in revised form 14 January 2021
Accepted 16 February 2021
Available online 04 March 2021
Keywords:
Tangent stiffness matrix
Analytical interpolation functions
Timoshenko beam theory
Nonlinear geometric analysis
a b s t r a c t
A structural geometric nonlinear analysis using the finite element method (FEM) depends on the consideration 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 numerical
solution, the structure discretization has a great influence on the analysis response. However, when
applying interpolation functions calculated from the homogenous solution of the differential equation
of the problem, a numerical solution closer to the analytical response of the structure is obtained, and
the level of discretization could be reduced, as in the case of linear analysis. Thus, to reduce this influence
and allow a minimal discretization of the structure for a geometric nonlinearity problem, this work uses
interpolation functions obtained directly from the solution of the equilibrium differential equation of a
deformed infinitesimal element, which includes the influence of axial forces. These shape functions are
used to develop a complete tangent stiffness matrix in an updated Lagrangian formulation, which also
integrates the Timoshenko beam theory, to consider shear deformation and higher-order terms in the
strain tensor. This formulation was implemented, and its results for minimal discretization were compared with those from conventional formulations, analytical solutions, and Mastan2 v3.5 software. The
results clearly show the efficiency of the developed formulation to predict the critical load of plane
and spatial structures using a minimum discretization.
Ó 2021 Elsevier Ltd. All rights reserved.
1. Introduction
The continuous (analytical) behavior of a solid can be approximated by a discrete solution. Usually, the discrete response is
obtained by nodal displacements, and an approximated continuous
solution can be found by means of interpolating (shape) functions.
However, the discrete solution using the FEM introduces simplifications in the mathematical idealization of the structure behavior
as the interpolation functions that define the deformed configuration of a structure are not compatible with the mathematical idealization of the response of a continuous medium (Martha, 2018).
In a linear elastic analysis of frame models with beam elements
with constant cross-sections, interpolating functions are obtained
from the homogeneous solution of the equilibrium differential
equation of an undeformed infinitesimal element, leading to the
⇑ Corresponding author at: Avenida Fernando Ferrari, 514, Nexem, Goiabeiras,
Vitória, ES 29075-910, Brazil.
E-mail address: rodriguesma.civil@gmail.com (M.A.C. Rodrigues).
https://doi.org/10.1016/j.ijsolstr.2021.02.014
0020-7683/Ó 2021 Elsevier Ltd. All rights reserved.
so-called cubic Hermitian interpolation functions (Rodrigues
et al., 2019). In this case, the formulation does not consider any
other approximation except those already covered in the analytical
idealization of the element behavior. This explains the fact that, in
linear elastic analysis, the structure response of this type of model
does not depend on the level of discretization.
However, for geometric nonlinear or second-order analysis, in
which equilibrium should be considered in the deformed configuration, Hermitian interpolation functions do not represent the
analytical 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 problem
and used additional degrees of freedom within the elements to
calculate the critical load by eigenvalue analysis.
Another way of improving a second-order analysis is to use stability functions (Chen and Lui, 1991; Aristizábal-Ochoa, 1997,
2007, 2008, 2012). Some authors have used the consistent field
Marcos Antonio Campos Rodrigues, Rodrigo Bird Burgos and Luiz Fernando Martha
International Journal of Solids and Structures 222–223 (2021) 111003
Nomenclature
Nh2 ; Nh5
General geometric parameters
x; 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.
fag; fbg 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=h
X
generic auxiliary parameter for characterization of bar
elements.
Nh3 ; Nh6
cross-section rotation interpolation functions of a bar
for a transverse displacement.
cross-section rotation interpolation functions of a bar
for a rotation displacement.
Strain and stress tensor parameters
exx
normal deformation.
cxy ; cxz shear distortion in the plane xy and xz.
c
shear distortion.
eijðtþDtÞ
Green-Lagrange strain tensor in an unknown configuration.
eðtÞ
Green-Lagrange strain tensor in a known configuration.
ij
Deij
incremental strain.
Deij
linear component of incremental strain.
Dexx
axial deformation from linear component of incremental strain.
Dexy
shear distortion from linear component of incremental
strain.
Dgij
nonlinear component of incremental strain.
Dgxx
axial deformation from nonlinear component of incremental strain.
Dgxy
shear distortion from nonlinear component of incremental strain.
sxx
normal stress.
sxy ; sxz shear stress.
ðtþDtÞ
second Piola-Kirchoff stress tensor.
Sij
sðtÞ
Cauchy stress tensor.
ij
Dsij
stress increment.
E
Young’s modulus.
G
modulus of rigidity (shear modulus).
m
Poisson’s ratio.
C ijkl
material constitutive tensor.
Displacement field parameters
u
axial displacement of a point inside a bar in the x direction.
v; w
transverse displacement of a point inside a bar in the y
or z direction.
u0 ; v 0 ; w0 axial and transversal displacements at the crosssection center of gravity.
v h ; v p homogeneous and particular parts of analytical solution
for displacements and rotations.
vg; vl
global and local solutions for displacements and rotations.
h
cross-section rotation in relation to an axis.
0
0
d1 ; d4
axial displacements at the initial and final nodes of a
bar.
0
0
d2 ; d5
transverse displacements at the initial and final nodes of
a bar.
0
0
d3 ; d6
cross-section rotation at the initial and final nodes of a
bar.
0
d
displacement vector of a bar.
axial displacement vector of a bar.
fug
fv g; fwg transverse displacements vectors of a bar.
constant of integration.
ci ; fC g
½X interpolation polynomials matrix of the displacement
field.
½H interpolation polynomials matrix of the displacement
field at nodes.
½N interpolation functions matrix.
axial displacement interpolation functions vector.
fNu g
fNv g
transverse displacement interpolation functions vector
on the local xy plane of a bar.
transverse displacement interpolation functions vector
fNw g
on the local xz plane of a bar.
fNhx g
cross-section rotation interpolation functions vector
around
the local axis x.
Nhy
cross-section rotation interpolation functions vector
around the local axis y.
fNhz g
cross-section rotation interpolation functions vector
around the local axis z.
Nu1 ; Nu4
axial displacement interpolation functions at the initial
and final nodes of a bar.
Nv2 ; N v5
transverse displacement interpolation functions of a bar
for a transverse displacement.
Nv3 ; N v6
transverse displacement interpolation functions of a bar
for a rotation displacement.
External loads and internal forces parameters
q
load rate of transverse force on the bar.
P
constant generic axial load acting on infinitesimal element.
M
generic bending moment acting on a bar element.
l; K
auxiliary parameters for the differential equation development of the deformed infinitesimal element equilibrium.
N
normal force.
V
vertical component of acting force in a cross-section.
Q y; Q z
shearing force in directions y and z.
Q
shearing force.
Mx
twisting moment.
My ; Mz
bending moment.
M y1 ; M y2 bending moment acting at the initial and final nodes of
a bar in y direction.
M z1 ; Mz2 bending moment acting at the initial and final nodes of
a bar in z direction.
Stiffness matrix parameters
virtual work due to external loading.
RðtþDtÞ
U; U NL
linear and nonlinear component of virtual work expression.
K g;Rotfin finite rotations influence on the geometric stiffness matrix of a generic plane element.
2
International Journal of Solids and Structures 222–223 (2021) 111003
Marcos Antonio Campos Rodrigues, Rodrigo Bird Burgos and Luiz Fernando Martha
placement relations. The matrix is constructed for a spatial element by an updated Lagrangian formulation considering higherorder terms in the strain tensor using interpolation functions
obtained directly from the equilibrium differential equation of a
deformed infinitesimal element, including shear deformation
according to the Timoshenko beam theory. In addition, the element
tangent stiffness matrix is adjusted to consider finite rotations. The
fifth important aspect of a nonlinear analysis is the incremental
solution scheme. This paper does not address this problem, and a
standard Newton-Raphson solution procedure was used.
Since this formulation involves trigonometric and hyperbolic
basic functions, the resulting stiffness matrix can be conveniently
rewritten using a Taylor series expansion with more terms than
the usual formulations. In this paper, tangent matrices with up to
3 and 4 terms were developed. This formulation was implemented
in Framoop (Martha and Parente Junior, 2002), the solver used in
Ftool (Martha, 1999) structural analysis program.
approach and transformed the shape interpolation functions (axial
and transverse displacements and rotation) into a power series
(Yunhua, 1998; Tang et al., 2015). The consistent field approach
eliminates ‘‘shear locking” and ‘‘membrane locking” effects that
usually appear in a geometric nonlinear analysis that employs just
one element per member.
Other works have a formulation based on the equilibrium equation of the infinitesimal element. Some include the axial effect considering the equilibrium in the deformed configuration (Davis
et al., 1972; Nukulchai et al., 1981; Goto and Chen, 1987; Chan
and Gu, 2000; Balling and Lyon, 2011). Others incorporate additional effects, such as elastic foundations (Areiza-Hurtado et al.,
2005; Aydogan, 1995; Burgos and Martha, 2013; Chiwanga and
Valsangkar, 1988; Eisenberger and Yankelevsky, 1985; Morfidis,
2007; Morfidis and Avramidis, 2002; Onu, 2000, 2008; Shirima
and 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 formulate the stiffness coefficients of the elements without explicitly
presenting expressions for interpolation functions based on the
differential equation of the problem.
In addition to the enrichment of interpolating the field variables
in an element, another important aspect to improve the response
of a structure when performing a geometric nonlinear analysis is
related to the bending theory. The most commonly used bending
solution for frame elements is the Euler-Bernoulli beam theory
(EBBT). However, the effects of shear deformation are important
when predicting the behavior of beam-columns with a moderate
slenderness ratio or with a small shear-to-bending ratio, and the
Timoshenko 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, many
works cited above also consider this effect, such as references
(Burgos and Martha, 2013; Onu, 2008).
Moreover, a kinematic description of motion is needed when
formulating an incremental geometrically nonlinear analysis.
Based on the choice of the reference configuration, the following
three kinematic descriptions are commonly used in structural
mechanics to formulate the nonlinear system of equations: Total
Lagrangian (TL), Updated Lagrangian (UL), and Corotational (CR).
When consistently developed, the total Lagrangian and the
updated 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 general, the Timoshenko theory is not considered, or in some cases,
the formulation yields additional degrees of freedom, which
requires static condensation to reduce the matrix order (Bathe
and Bolourchi, 1979; Aguiar et al., 2014). Additionally, most studies that consider shear deformation employ a corotational formulation, as proposed by (Battini, 2002; Crisfield, 1991; Pacoste and
Eriksson, 1995, 1997; Santana and Silveira, 2019; Silva et al., 2016).
Furthermore, the strain–displacement relation plays an important role when developing a geometric stiffness matrix. The consideration of higher-order terms in the strain tensor improves
the accuracy of the geometric nonlinear analysis (Chen, 1994;
Conci, 1988; Yang and Kuo, 1994; Yang and Leu, 1994). More
recently, the authors (Rodrigues et al., 2019) considered higherorder terms in the strain tensor in the formulation of a Timoshenko
element using Hermitian interpolation functions and an updated
Lagrangian formulation.
The main objective of this paper, which differentiates this work
from others found in the literature, is to formulate the tangent
stiffness matrix of a frame element integrating four important
aspects that improve geometric nonlinear analysis: interpolation
functions, beam theory, kinematic description, and strain–dis-
2. Differential equilibrium relationships in beam-columns
This section formulates the differential equations that define
the analytical behavior of a deformed infinitesimal beam element
considering both the Euler-Bernoulli and Timoshenko beam theories. Fig. 2.1presents a deformed infinitesimal element subjected
to a distributed transverse load q and a constant axial load P.
The equilibrium of the deformed infinitesimal beam element
leads to Eqs. (2.1) and (2.2)
X
X
F y ! dV þ qðxÞdx ¼ 0 !
dV ðxÞ
¼ qðxÞ
dx
M o ! dM ðV þ dV Þdx P:dv þ qðxÞ
dx2
¼0
2
ð2:1Þ
ð2:2Þ
where v ðxÞ is the infinitesimal element transverse displacement, qðxÞ is the transverse distributed load, V ðxÞ is the vertical
component of the force acting on the cross-section, P is the horizontal component, and MðxÞ is the bending moment.
Considering that the element has a constant cross-section, using
Eq. (2.2) and the approximate relation between bending moment
and curvature, M ðxÞ ¼ EI dh=dx, where hðxÞ corresponds to the
cross-section rotation, the differential equation of the problem
can be written according to expression (2.3).
d hðxÞ
d v ð xÞ
V ðxÞ P
dx2
dx
3
2
d hðxÞ dV ðxÞ
d v ð xÞ
¼ 0 ! EI
P
¼0
dx3
dx
dx2
2
EI
Fig. 2.1. Equilibrium of a deformed beam element.
3
ð2:3Þ
Marcos Antonio Campos Rodrigues, Rodrigo Bird Burgos and Luiz Fernando Martha
International Journal of Solids and Structures 222–223 (2021) 111003
Equation (2.9) can be written using the relation
Q ðxÞ ¼ dMðxÞ=dx in Eq. (2.8). According to the relation between
bending moment and curvature M ðxÞ ¼ EIdh=dx, Eq. (2.10), which
relates transverse displacement v ðx) with cross-section rotation
hðxÞ, is obtained.
The static fundamental relation in Eq. (2.1) shows that the gradient of the vertical component of the force acting on the crosssection is equal to the acting distributed load. Therefore, substituting this relation in Eq. (2.3), the differential equilibrium relation of
a deformed infinitesimal element can be found, as shown in
expression (2.4).
d h
d v ð xÞ
P
¼ qðxÞ
dx3
dx2
3
EI
dMðxÞ
dv ðxÞ
¼ vGA: hðxÞ dx
dx
2
ð2:4Þ
ð2:9Þ
2.1. Euler-Bernoulli beam theory
dv ðxÞ
EI d h
¼ hðxÞ dx
vGA dx2
The EBBT considers the rotation as the gradient of the transverse displacement (h ¼ dv =dx). Thus, the differential relation
developed in expression (2.4) can be written according to (2.5).
Finally, applying equation (2.10) in expression (2.4), the equilibrium differential relation of a deformed infinitesimal element, considering Timoshenko beam theory, can be written according to
expressions (2.11) or (2.12).
2
4
d
v ð xÞ
dx4
P d v ðxÞ qðxÞ
¼
EI dx2
EI
3
P
d h
dhðxÞ
¼ qðxÞ
EI 1 þ
P
dx
vGA dx3
2
ð2:5Þ
2.2. Timoshenko beam theory
3. Solutions of the differential equations
The solution of the equilibrium differential relations presented
before can be obtained by the composition of a homogeneous solution, v h ðxÞ, with a particular solution, v p ðxÞ, according to
v ðxÞ ¼ v h ðxÞ þ v p ðxÞ.
In the direct stiffness method, the solution is obtained by
the superposition of a global and a local solution, hence
v ðxÞ ¼ v g ðxÞ þ v l ðxÞ. The global solution is the one obtained by
using nodal displacements and rotations as coefficients for the
interpolating functions. The local solution is a fixed-end element solution, thus presenting null values for nodal displacements and notations. Martha (2018) used the scheme in
Fig. 3.1 to explain this superposition. Nodal displacements and
rotations of the final solution are obtained directly from the
global solution. Within an element, displacements are obtained
by superposition of the local and global solutions. In a classical
FEM analysis for continuous domains, generally, the local solution is not available, and the final solution is approximated
by the global solution.
If the right-side of the differential equation is null, then there is
no need for a particular solution and the homogeneous part is the
final solution, then v ðxÞ ¼ v h ðxÞ. Similarly, if there is no transverse
force within the element (q(x) = 0), the local solution is null, giving
v ðxÞ ¼ v g ðxÞ. Thus, the homogeneous and the global solutions are
equivalent: v h ðxÞ ¼ v g ðxÞ. Therefore, when adopting analytical
solutions for the behavior of the element, the displacements and
nodal rotations of the discrete model are exact, independently of
discretization. Moreover, in case a fixed-end local element solution
is available, nodal displacements and rotations of the final solution
are obtained directly from the global solution, also independently
of discretization.
These observations justify the use of interpolation functions calculated from the homogenous solution. When the differential
equation is calculated from the equilibrium of an undeformed
infinitesimal element, the so-called Hermitian cubic functions correspond to the homogeneous solution for EBBT, and because of
ð2:6Þ
Q ðxÞ ¼ Psenðh þ cÞ þ V ðxÞcosðh þ cÞ ! Q ðxÞ
¼P
dv ðxÞ
þ VðxÞ
dx
ð2:7Þ
Substituting Eqs. (2.7) into (2.2), the static fundamental relation
between the shear force and the bending moment is obtained:
Q ðxÞ ¼ dMðxÞ=dx. The shear force acting on the section is given
by Eq. (2.8)
dv ðxÞ
Q ðxÞ ¼ vGA:cðxÞ ! Q ðxÞ ¼ vGA: hðxÞ dx
ð2:12Þ
Equation (2.11), or alternatively Eq. (2.12), relates the crosssection rotation hðxÞ with the applied distributed transverse
loadqðx), considering the element axial force and the shear and
bending rigidity parameters. The next section shows the solution
of differential Eqs. (2.10) and (2.12) for an isolated beam element.
NðxÞ ¼ Pcosðh þ cÞ V ðxÞsenðh þ cÞ ! N ðxÞ
dv ðxÞ
dx
ð2:11Þ
3
d h
P
dhðxÞ
qðxÞ
¼
P
P
dx3
1 þ vGA
1 þ vGA
EI dx
EI
In TBT, shear distortion ðcÞ is constant for each cross-section (no
warping) and is considered an additional rotation of the section.
Therefore, the section rotation and the transverse displacement
are not associated, and both are considered independent variables,
as indicated in Fig. 2.2.
Therefore, for the deformed element, internal forces are calculated using the cross-sectional rotation ðdv =dx ¼ h þ cÞ shown in
Fig. 2.3 and expressed in Eqs. (2.6) and (2.7).
¼ P V ðxÞ
ð2:10Þ
ð2:8Þ
in which G is the material shear modulus, A is the cross-section
area, and v is the factor that defines the cross-section effective area
for shear.
Fig. 2.2. Shear deformation in the Timoshenko beam theory.
4
International Journal of Solids and Structures 222–223 (2021) 111003
Marcos Antonio Campos Rodrigues, Rodrigo Bird Burgos and Luiz Fernando Martha
Fig. 2.3. Internal forces, displacements, and rotations in the Timoshenko beam theory.
Fig. 3.1. Solution composition from the direct stiffness method.
4
that, linear analyses can be performed without discretization. Similar cubic functions are obtained from the undeformed infinitesimal equilibrium for TBT. The same applies to the functions that
represent the homogenous solution of the equations obtained from
the equilibrium of a deformed infinitesimal element.
Therefore, this section seeks the homogeneous solution of the
differential equations of equilibrium of a frame element in the
deformed configuration. The next sections indicates how the interpolation (shape) functions resulting from the solutions to these differential equations are obtained; and Section 5 shows how these
interpolation functions are used in the formulation of the tangent
stiffness matrix of a frame element in its local axis system. It
should be noted that the homogenous solution considers approximations inherent to the analytical model adopted for the behavior
of beam-column elements, such as the hypothesis of small rotations in the equilibrium of the infinitesimal element in the
deformed configuration.
d
v ð xÞ
dx4
pffiffiffiffiffiffiffiffiffiffi
v ðxÞ
¼ 0; l ¼ P=EI
dx2
2
l2
d
ð3:1Þ
v h ðxÞ ¼ c1 elx þ c2 elx þ c3 x þ c4
ð3:2Þ
where 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 ðxÞ ¼ c1 lelx c2 lelx þ c3
ð3:3Þ
The particular solution does not introduce any additional coefficients that need to be determined. Thus, the coefficients of the
exponential function can be calculated from the boundary
conditions.
For a tensile force, l is a real number, and the homogeneous
solution of the differential equation can be written by hyperbolic
functions according to (3.4) and (3.5). In the case of a compressive
force, l is a complex number, and the displacement can be written
by trigonometric functions (3.6) and (3.7).
3.1. Euler-Bernoulli beam theory
v h ðxÞ ¼ c1 sinhðlxÞ þ c2 coshðlxÞ þ c3 x þ c4 ; l ¼
Considering the EBBT and qðxÞ ¼ 0, the homogeneous solution
of the differential equation presented in (2.5) can be obtained from
relation (3.1) and written as given in (3.2),
hh ðxÞ ¼ c1 lcoshðlxÞ þ c2 lsinhðlxÞ þ c3
5
pffiffiffiffiffiffiffiffiffiffi
P=EI
ð3:4Þ
ð3:5Þ
Marcos Antonio Campos Rodrigues, Rodrigo Bird Burgos and Luiz Fernando Martha
pffiffiffiffiffiffiffiffiffiffiffiffiffi
v h ðxÞ ¼ c1 sinðlxÞ þ c2 cosðlxÞ þ c3 x þ c4 ; l ¼ P=EI
ð3:6Þ
hh ðxÞ ¼ c1 lcosðlxÞ c2 lsinðlxÞ þ c3
ð3:7Þ
International Journal of Solids and Structures 222–223 (2021) 111003
3.2. Timoshenko beam theory
In TBT, the cross-section rotation is not the gradient of the
transverse displacement, and it is not possible to write just one differential relation; thus, Eqs. (2.10) and (2.12) need to be solved.
The homogenous solution can be calculated considering qðxÞ ¼ 0;
thus, the differential equations are written as shown in (3.8) and
(3.9).
Fig. 4.1. Deformed configuration of an isolated element.
4. Interpolation functions
dv ðxÞ
EI d h
¼ hðxÞ dx
vGA dx2
2
ð3:8Þ
pffiffiffiffiffiffiffiffiffiffi
d h
dhðxÞ
l
¼ 0; K ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; l ¼ P=EI
K2
EI
dx3
dx
2
1þ
:l
The deformed configuration of an isolated element can be
described by interpolating nodal displacements according to
Fig. 4.1 and Eqs. (4.1) to (4.4).
3
ð3:9Þ
0
Using constantX, expression (3.10), introduced by Reddy
(Reddy, 1997) and used in (Burgos and Martha, 2013; Martha
and Burgos, 2015; Martha and Burgos, 2014), the differential equations can be rewritten as given in (3.11) and (3.12).
0
0
0
0
ð4:2Þ
hðxÞ ¼ N h2 ðxÞd2 þ N h3 ðxÞd3 þ N h5 ðxÞd5 þ N h6 ðxÞd6 ! hz ðxÞ ¼ fN hz ðxÞgfv g
ð3:10Þ
0
u0 ðxÞ ¼ fNu ðxÞgfug
dv ðxÞ
d h
¼ hðxÞ XL2 2
dx
dx
ð4:1Þ
v 0 ðxÞ ¼ Nv2 ðxÞd2 þ Nv3 ðxÞd3 þ Nv5 ðxÞd5 þ Nv6 ðxÞd6 ! v 0 ðxÞ ¼ fNv ðxÞgfv g
0
EI 1
X¼
vGA L2
0
u0 ðxÞ ¼ Nu1 ðxÞd1 þ Nu4 ðxÞd4 ! u0 ðxÞ ¼ fNu ðxÞgfug
vGA
0
0
v 0 ðxÞ ¼ fNv ðxÞgfv g
hz ðxÞ ¼ fNhz ðxÞgfv g
ð4:4Þ
2
ð3:11Þ
When interpolating functions are obtained from the homogeneous solution of the equilibrium differential equation of an
infinitesimal element, the analytical behavior of the element is
represented. In this work, the equilibrium of a deformed infinitesimal element is considered taking the axial force into account.
3
d h
dhðxÞ
l
K2
¼ 0; K ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dx3
dx
1 þ Xl2 L2
ð3:12Þ
Solving equation (3.12), the homogeneous solution for the
cross-section rotation is given by (3.13). This solution can be
applied in Eq. (3.11). Thus, the transverse displacement is written
according to (3.14).
hh ðxÞ ¼ K c1 e
v h ð xÞ ¼
Kx
Kx
c2 e
1 XL2 K2
þ c3
Kx
c1 e c2 eKx þ c3 x þ c4
4.1. Euler-Bernoulli beam theory
The interpolation functions can be calculated based on the
homogeneous solution for the EBBT presented in Eqs. (3.2) and
(3.3). First, the displacement is written in matrix form according
to Eq. (4.5).
ð3:13Þ
ð3:14Þ
v h ðxÞ ¼ c1 elx þ c2 elx þ c3 x þ c 4
Finally, for a tensile force, l is a real number, and the homogeneous solution of the differential equation can be written by the
hyperbolic functions in (3.15) and (3.16); for a compressive force,
l is a complex number, and the displacement is written by the
trigonometric functions in (3.17) and (3.18).
hh ðxÞ ¼ K½c1 coshðKxÞ þ c2 sinhðKxÞ þ c3 ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K ¼ l= 1 þ Xl2 L2
v 0 ðxÞ hðxÞ
lx
e
¼
lelx
elx
lelx
x
1
9
8
c
>
> 1 >
>
1 < c2 =
¼ ½X fC g
c3 >
>
0 >
>
;
:
c4
ð4:5Þ
The boundary conditions are obtained by evaluating the homogeneous solution of these displacements (v h ðxÞ and hh ðxÞ) at the
extreme nodes of the bar, as shown in expression (4.6).
8 0 9
2
3
9
8
>
1
1
0 1
> > v 0 ð0Þ >
> d20 >
= <
= n 0o 6
< >
n 0o >
7
l
l
1
0
d3
hð0Þ
7
¼
! d ¼6
d ¼
0
Ll
4 eLl
5
v
ð
L
Þ
e
L
1
>
>
>
>
0
d
;
>
:
5 >
>
>
Ll
Ll
: 0 ;
hðLÞ
l
e
l
e
1
0
d
ð3:15Þ
6
1 XL2 K2 ½c1 sinhðKxÞ þ c2 coshðKxÞ þ c3 x þ c4 ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K ¼ l= 1 þ Xl2 L2
ð3:16Þ
8
c1
>
>
>
<c
2
:
> c3
>
>
:
c4
ð3:17Þ
9
>
>
>
=
>
>
>
;
¼ ½H f C 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 by
Rodrigues (2019) for exponential, hyperbolic and trigonometric
functions and were implemented in file ShpFuncBeamEulerBernoulli
using MATLAB and C in an open source code that can be accessed
and used (Rodrigues et al., 2020, 2021).
v h ð xÞ ¼
1 þ XL2 K2 ½c1 sinðKxÞ þ c2 cosðKxÞ þ c3 x þ c4 ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K ¼ l= 1 Xl2 L2
!
hh ðxÞ ¼ c1 lelx c2 lelx þ c3
v h ð xÞ ¼
hh ðxÞ ¼ K½c1 cosðKxÞ c2 sinðKxÞ þ c3 ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K ¼ l= 1 Xl2 L2
ð4:3Þ
ð3:18Þ
6
International Journal of Solids and Structures 222–223 (2021) 111003
Marcos Antonio Campos Rodrigues, Rodrigo Bird Burgos and Luiz Fernando Martha
v 0 ð xÞ
hðxÞ
¼ ½X ½H1 fd0 g ) ½N ¼ ½X ½H1
5. Local stiffness matrix
ð4:7Þ
According to Fig. 5.1, the stiffness matrix development can be
performed by initially analyzing a two-dimensional structure (xy
and xz planes). Then, for spatial structures, as shown in Fig. 5.2,
the integration between the planes is coupled to the stiffness
matrix. Finally, to consider large rotations, the concept of finite
rotations was added to the matrix.
4.2. Timoshenko beam theory
The homogeneous solution for the TBT is presented in Eqs.
(3.13) and (3.14). Thus, the same development is performed to calculate the interpolation functions, according to expressions (4.8)
and (4.9).
v h ð xÞ ¼
1 XL2 K2
c1 eKx c2 eKx þ c3 x þ c 4
hh ðxÞ ¼ K c1 eKx þ c2 eKx þ c3
2
½X ¼ 4
1 XL2 K2 eKx
1 XL2 K2 eKx
KeKx
KeKx
8 0 9
8
>
d2 >
>
>
>
>
> >
> v 0 ð0Þ
>
>
= >
< hð0Þ
< d0 >
3
¼
fd0 g ¼
0
>
> >
> v 0 ðLÞ
> d5 >
>
>
> >
:
>
>
;
: 0 >
hðLÞ
d6
9
>
>
>
=
>
>
>
;
¼
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
!
v 0 ð xÞ hðxÞ
3
x
1
5
1 0
ð4:8Þ
Z
1 XL2 K2 ðc1 c2 Þ þ c4
Kðc1 þ c2 Þ þ c3
1 XL2 K2 c1 eKL c2 eKL þ c3 L þ c4
KL
KL
þ c3
K c1 e þ c2 e
ð1 XL2 K2 Þ
ðXL2 K2 1Þ
6
6K
K
½H ¼ 6
6 eLK ð1 XL2 K2 Þ
eLK ðXL2 K2 1Þ
4
LK
Ke
KeLK
0
1
L
1
The stiffness matrices are calculated considering the updated
Lagrangian formulation, and the steps shown in this work have
been 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 virtual work of the external forces, according to (5.1),
¼ ½X fC g
8
9
c1 >
>
>
< c >
=
2
fC g ¼
> c3 >
>
>
:
;
c4
2
5.1. Updated Lagrangian formulation
9
>
>
>
>
>
>
=
>
>
>
>
>
>
;
3
1
7
0 7
7
7
1 5
0
V
ðtþDt Þ
tensor, ij
ðtþDt Þ
e
u0 ðxÞ ¼
þ
x
¼1
L
Nu4 ðxÞ
ð5:1Þ
corresponds to the second Piola-Kirchoff stress
corresponds to the Green-Lagrange strain tensor,
corresponds to the virtual work due to external loading,
and R
which could include body, surface, inertia and damping forces
(Aguiar et al., 2014). It is important to note that the term d is
related to the application of a virtual displacement to consider
the virtual displacement principle, while D is related to a small displacement increment, changing the configuration.
However, the equilibrium equations of an unknown configuration t þ Dt must be written using a known reference configuration
t. Thus, the linearized incremental equation requires small displacement increments according to the equations in (5.2),
ð4:9Þ
ðtþDt Þ
Sij
eðijtþDtÞ ¼ etij þ Deij
¼ stij þ Dsij
ð5:2Þ
where stij corresponds to the Cauchy stress tensor, Dsij is the
stress increment and Deij is the deformation increment. In the
known reference configuration, there is no deformation so
detij = 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).
For the axial displacement and the cross-section rotation about
the x axis, in this research, a linear interpolation was adopted
according to Mcguire et al. (2000), which is given in expression
(4.10).
Nu1 ðxÞ
dV ¼ RðtþDtÞ
ðtþDt Þ
4.3. Axial interpolation functions
0
Nu4 ðxÞd4
ðtþDt Þ
deij
where Sij
¼ ½HfC g
Once again, equation (4.7) can be used to calculate the interpolation functions. The results are also presented in the work by
Rodrigues (2019) and implemented in file ShpFuncBeamTimoshenko; an open source code that can be accessed and used
((Rodrigues et al., 2020, 2021)).
0
Nu1 ðxÞd1
ðtþDt Þ
Sij
Z
V
ðtþDt Þ
Sij
ðtþDt Þ
deij
Z
dV ¼
ZV
Z
Dsij dDeij dV þ
¼
x
¼
L
stij þ Dsij d Deij dV
V
V
stij dDeij dV
ð5:3Þ
The Green-Lagrange strain tensor has a linear (Deij ) and a nonlinear part (Dgij ), according to expressions (5.4) to (5.6) in the xy
plane.
ð4:10Þ
Some authors, such as Tang et al. (2015) and Silva et al., (2016);
employed consistent interpolation functions for the axial displacement and not just linear interpolation.
Deij ¼ Deij þ Dgij
Fig. 5.1. Bending and torsion for a spatial element.
7
ð5:4Þ
Marcos Antonio Campos Rodrigues, Rodrigo Bird Burgos and Luiz Fernando Martha
International Journal of Solids and Structures 222–223 (2021) 111003
5.3. Local stiffness matrix equations
The development of the local stiffness matrix using cubic functions can be seen in Mcguire et al. (2000) considering EulerBernoulli beam theory and in Rodrigues et al. (2019) for a
Timoshenko beam element. In this research, the adopted form to
achieve the complete matrix is the same as that presented in the
cited 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 nonlinear parts of the Green-Lagrange strain tensor components in Eq.
(5.6), can be rewritten as expressions (5.15) and (5.16), respectively, for the Euler-Bernoulli beam theory.
Fig. 5.2. 3-D element (Mcguire et al., 2000).
Dexx ¼
@u
@x
Dexy ¼
"
1 @u
Dgxx ¼
2 @x
2
@v
þ
@x
@u @ v
þ
@y @x
2
ð5:5Þ
#
Dgxy ¼
@u @u @ v @ v
þ
@x @y @x @y
exx ¼
ð5:6Þ
Deij ¼ Deij
gxy ¼
Z
Z
C ijkl Dekl dDeij dV þ
V
V
stij dDeij dV þ
stij Dgij dV ¼ RðtþDtÞ
V
exx ¼
ð5:8Þ
The first integral leads to the elastic stiffness matrix, while the
third integral leads to the geometric stiffness matrix. The second
integral represents the virtual work of forces acting on the element
in configuration t and is usually represented at the right-hand side
of the expression. However, in this case, the rotation components
are approximated by displacement derivatives, and the finite rotation effect is not considered (Mcguire et al., 2000).
In the xy plane, the stress vector, the constitutive matrix, the
linear and nonlinear strain vectors are given according to equation
(5.9). Thus, relations (5.10)–(5.12) can be written.
s
s ¼ xx
sxy
C¼
E
0
(
e¼
0 G
exx
cxy
)
(
g¼
gxx
gxy
Z
C ijkl Dekl dDeij dV ¼
gxy ¼
V
V
V
stij dDeij dV ¼
sij Dgij dV ¼
sxx dexx dV þ
V
V
sxy dcxy dV
sxx dgxx dV þ
V
sxy dgxy dV
)
Z
Z
A
ð5:11Þ
dx
v ðx; yÞ ¼ v 0 ðxÞ
2
2
@u 2
z
þ @@xv þ y2 @h
@x
@x
¼ 12
@hz
y @u
@x @x
ð5:17Þ
ð5:18Þ
ð5:12Þ
cxy :Gdcxy dV
!
! !
@u
@2v
@u
@2v
y 2 E d
yd 2 dx dA
@x
@x
@x
@x
L
0
! dU ¼
RL
0
RL
R
2
d @@xv2 dx E A y2 dAþ
RL
R
2
@ 2 v @u
d dx E A ydA 0 @u
d @ v dx E A ydA
@x @x2
@x2 @x
@u @u
d dx
0 @x @x
E
R
R
A
RL
@2 v
0 @x2
dA þ
In the beam centroidal axis,
expression is reduced to (5.20).
@u @u
d dx EA þ
@x @x
L
0
Z
L
0
R
A
y2 dA ¼ Iz ;
R
A
ð5:19Þ
ydA ¼ 0, and the
!
@2v @2v
d
dx EIz
@x2 @x2
ð5:20Þ
From Eq. (4.4), the displacements can be written using the
interpolation functions resulting in (5.21). The notation
@=@x ¼ ð Þ’ is adopted for simplicity.
ð5:13Þ
Considering the Euler-Bernoulli beam theory, the rotation is the
derivative of the transverse displacement (h ¼ dv =dx); thus, the
displacement field can be rewritten according to (5.14).
:y
@ v @u @ v 0
¼
þ
hZ
@x @y
@x
V
¼
dU ¼
v 0 ð xÞ
ð5:16Þ
Z
Z
ð5:10Þ
The displacement field of a beam element is shown in Fig. 5.3
and defined by Eq. (5.13).
uðx; yÞ ¼ u0 ðxÞ cxy ¼
exx :Edexx dV þ
Z
v ðx; yÞ ¼ v 0 ðxÞ
2
@u @u @ v @ v
@hz
@u
þ
¼y
hz hz
@x @y @x @y
@x
@x
dU ¼
5.2. Displacement field
uðx; yÞ ¼ u0 ðxÞ hðxÞ:y
y @@xv2 @u
@x
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 equation, relation (5.10), can be written according to Eq. (5.19).
Z
V
2
@u 2
þ @@xv
@x
Z
Z
Z
V
V
Z
Z
@u @u0
@hZ
¼
y
@x
@x
@x
V
cxy :Gdcxy dV
2
2
2
@u 2
þ @@xv þ y2 @@xv2
@x
¼ 12
ð5:15Þ
5.3.1. Elastic matrix
Z
exx :Edexx dV þ
@ v @u
¼0
þ
@x @y
@u @u @ v @ v
@ 2 v @ v @ v @u
þ
¼y 2
@x @y @x @y
@x @x @x @x
gxx ¼ 12
ð5:9Þ
Z
cxy ¼
Meanwhile, for the Timoshenko beam theory, the linear and
nonlinear parts of the strain tensor components in Eq. (5.6) can
be written according to Eqs. (5.17) and (5.18), using the displacement field presented in (5.13).
ð5:7Þ
Based on equations (5.3) and (5.7), the virtual work equation
becomes (5.8).
Z
2
@u2
þ @@xv
@x
gxx ¼ 12
The stress increment is obtained from the material constitutive
relation. The consideration of the linear approximation for the
stress and strain increment leads to the expression in Eq. (5.7).
Dsij ¼ C ijkl Dekl ¼ C ijkl Dekl
@u @u0
@2v
¼
y 2
@x
@x
@x
dU ¼ fdugT
Z
ð5:14Þ
0
8
Z
0
L
L
T
EA N0u N0u dxfug þ fdv gT
T
EIz N00v N00v dxfv g
ð5:21Þ
International Journal of Solids and Structures 222–223 (2021) 111003
Marcos Antonio Campos Rodrigues, Rodrigo Bird Burgos and Luiz Fernando Martha
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 calculated in section 4.
The axial part was already used for the xy plane. Finally, writing
the expression using the interpolation functions, equation (5.26)
can be achieved:
Z
Plane xy (TBT)
L
dU ¼ fdwgT
0
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.
Z
dU ¼
V
cxy :Gdcxy dV ! dU
exx ¼
V
Z Z
L
@u
@hz
@u
@hz
¼
y
y
E
dx dA
@x
@x
@x
@x
A
0
Z Z L
@v
@v
þ
hZ G d
dhZ dx dA
@x
@x
0
A
Z
@u @u
d dx EA þ
@x @x
Z
@hZ @hZ
! dU ¼
d
dx EIz
@x @x
0
0
Z L
Z L
@v @v
þ
hZ dhZ dx GA
d
dx GA þ
@x
0 @x
0
Z L
Z L
@v
@v
hZ d
dx GA dhZ dx GA
@x
0
0 @x
L
0
L
RL
cxz ¼
RL
@hy
0 @x
EA þ
0
hy dhy dx GAþ
0
hy d @w
dx GA @x
RL
@w @u @w0
þ
¼
hy
@x @z
@x
d
@hy
@x
dx EIy þ
RL
@w
dhy dx
0 @x
RL
@w @w
d @x dx
0 @x
ð5:27Þ
GA
GA
ð5:28Þ
n on oT
R L T
EIy N 0hy N 0hy dxfwg þ fdwgT 0 GA N 0w N 0w dxfwg
R L T
R L T
þ fdwgT 0 GA N hy N hy dxfwg fdwgT 0 GA Nhy N0w dxfwgþ
R
T
L
fdwgT 0 GA N0w N hy dxfwg
dU ¼ fdwgT
ð5:22Þ
RL
0
ð5:29Þ
5.3.2. Geometric matrix
Plane xy (EBBT)
The third integral from the virtual work expression (5.8) generates the usual geometric stiffness matrix. Thus, the nonlinear part
of the virtual work principle, considering the complete strain tensor and higher order terms, is given by Eq. (5.30).
For the plane xz behavior, the linear part of the strain tensor can
be written just by switching the displacement v with w in equation
(5.24). Thus, the first integral of the virtual work equation is given
by expression (5.25).
dU ¼
@u @u
d dx
0 @x @x
L
Plane xz (EBBT)
Z
RL
þ
ð5:23Þ
@u @u0
@2w
¼
z 2
@x
@x
@x
@u @u0
@hy
¼
z
@x
@x
@x
dU ¼
R L T
R L T
dU ¼ fdugT 0 EA N0u N0u dxfug þ fdv gT 0 EIz N0hz N0hz dxfv g
R L T
RL
þ fdv gT 0 GA N0v N0v dxfv g þ fdv gT 0 GAfNhz gfNhz gT dxfv gþ
T
RL
RL fdv gT 0 GAfNhz g N0v dxfv g fdv gT 0 GA N0v fNhz gT dxfv g
exx ¼
ð5:26Þ
where N w corresponds to the same N v 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 elastic matrix are given by relations (5.28) and (5.29).
Z
exx :Edexx dV þ
T
EIy N00w N00w dxfwg
cxz ¼
@u @u
d dx EA þ
@x @x
Z
0
@w @u @w0 @w0
þ
¼
¼0
@x @y
@x
@x
L
Z
sxx dgxx dV þ
V
Z
ð5:24Þ
dU NL ¼
0
@
Z
þ
ð5:25Þ
A
9
sxy dgxy dV
V
A
!
@2w @2w
d
dx EIy
@x2 @x2
Z
dU NL ¼
Z
0 0
1
@u
t xx d@ @
2
@x
@2 v
þy
@x2
0
!
!
Z L
@ 2 v @ v @ v @u
dx dA
t xy d y: 2 :
:
@x @x @x @x
0
L
2
@v
þ
@x
2
2
1 1
!2 1
2
A y @ v @uAdxAdA
@x2 @x
Marcos Antonio Campos Rodrigues, Rodrigo Bird Burgos and Luiz Fernando Martha
! dU NL ¼
2
@u
dx
t xx dA
@x
A
0
!Z
Z L 2
1
@v
d
dx
txx dA
þ
2 0
@x
A
0
!2 1 Z
Z
1 @ L @2v
d
dxA y2 t xx dA
þ
@x2
2
A
0
!Z
Z L 2
@ v @u
d 2
txx ydA
dx
@x @x
0
A
!Z
Z L 2
@ v @v
d 2
t xy ydA
þ
dx
@x @x
A
0
Z L
Z
@ v @u
d
txy dA
dx
@x @x
0
A
Z
1
2
International Journal of Solids and Structures 222–223 (2021) 111003
!Z
L
d
Fig. 5.4. Frame element.
dU NL
ð5:30Þ
gxx ¼
ð5:33Þ
gxz ¼
L
txx d
dU NL ¼
!
y
2
@w
@x
@w
þ
@x
2
!
2
þy
2
!2 1
@2w A
@ 2 w @u
y 2
2
@x
@x @x
RL
0
RL
0
d
2
@u 2
dx
@x
d @@xw2
@u
dx
@x
R
t dA
A xx
þ 12
R
t zdA
A xx
þ
RL
0
RL
0
d
@w 2
dx
@x
2
d @@xw2
@w
dx
@x
Applying relations A txx dA ¼ P;
expression becomes (5.40).
sxy dgxy dV
2
þ
ð5:37Þ
ð5:38Þ
R
t dA
A xx
þ 12
R
t zdA
A xz
RL
0
RL
0
d
@2 w
@x2
d @w
@x
2
R
dx
@u
dx
@x
A
R
z2 txx dAþ
t dA
A xz
ð5:39Þ
V
Z
2
@u
@x
R
Z
2
2
1
@u
@v
@hz
þ
þ y2
2
@x
@x
@x
0
A
Z Z L
@hz
@u
dx dA
t xy d y
þ
hz hz
@x
@x
0
A
Z
ð5:35Þ
@u @u @w @w
@ 2 w @w @w @u
þ
¼y 2
@x @y @x @y
@x @x
@x @x
dU NL ¼ 12
When the Timoshenko beam theory is considered, the nonlinear
part of the virtual work principle in Eq. (5.8) is written by Eqs.
(5.34) and (5.35) applying the same relations presented for
Euler-Bernoulli beam theory.
dU NL ¼
dx
With the same development made in plane xy, the third integral
from the virtual work expression in Eq. (5.8) can be written as
(5.39) for bending in plane xz.
Plane xy (TBT)
V
1
2
0
1 @ @u
¼
2
@x
Substituting the complete shape functions and solving the integrals of the problem leads to the geometric stiffness matrix, considering the Euler-Bernoulli beam theory and higher-order terms
in the strain tensor.
sxx dgxx dV þ
!#
The nonlinear parts of the Green-Lagrange strain tensor considering bending in plane xz are written according to expressions
(5.37) and (5.38).
Considering a constant shear force, the bending moment and
the shear force equations of a planar frame, as shown in Fig. 5.4,
can be calculated by the expressions in (5.33).
Z
2
Pd
Plane xz (EBBT)
ð5:32Þ
ðMZ1 þ M Z2 Þ
L
"
L
ð5:36Þ
R L T
R L T
dU NL ¼ fdugT 0 P N0u N0u dxfug þ fdv gT 0 P N0v N0v dxfv gþ
T
RL
R L T
þfdv gT 0 P IAz N00v N00v dxfv g þ fdv gT 0 Mz N00v N 0u dxfugþ
R L T
R L T
þfdugT 0 Mz N0u N00v dxfv g fdv gT 0 Q y N0v N0u dxfug
R L T
fdugT 0 Q y N0u N0v dxfv g
dU NL ¼
@u
dx
@x
RL
Writing Eq. (5.35) with interpolation functions leads to Eq.
(5.36).
R L T
R L T
dU NL ¼ fdugT 0 P N0u N0u dxfug þ fdv gT 0 P N 0v N0v dxfv g
T
RL
R L T
þfdv gT 0 P IAz N0hz N0hz dxfv g þ fdv gT 0 M z N0hz N0u dxfug
T
R L T
RL
þfdugT 0 Mz N0u N0hz dxfv g fdv gT 0 Q y fNhz g N0u dxfug
RL fdugT 0 Q y N0u fNhz gT dxfv g
Once again, the displacements can be written using the interpolation functions in equation (4.4), resulting in expression (5.32).
Qy ¼ z
d @h
@x
2
2
RL
R
R
z
d @@xv dx A txx dA þ 12 0 d @h
dx A y2 txx dA
0
@x
R L @hz
R
RL
R
t ydA þ 0 d @x hz dx A txy ydA 0 dhz @u
dx A txy dA
A xx
@x
þ 12
t dA
A xx
R
!
2
2
@u
@v
Iz
@hz
þP d
þ
@x
@x
A
@x
0
Z L
@hz @u
@u
þ
Q y d hz
dx
Mz d
@x
@x @x
0
Z
1
2
dU NL ¼
0
!
!2 13
2
2
1 L4
@u
@v
I z @ @ 2 v A5
þP d
dx
¼
Pd
þ
2 0
@x
@x
A
@x2
!
#
Z L"
@ 2 v @u
@ v @u
Q yd
þ
dx
ð5:31Þ
Mz d
@x2 @x
@x @x
0
ðMZ1 þ M Z2 Þx
L
0
R
ð5:34Þ
2
M Z ¼ M Z1 þ
@u 2
dx
@x
d
0
RL
R
R
R
Applying relations, A txx dA ¼ P; A txx ydA ¼ Mz ; A txy dA ¼ Q y ;
the expression becomes (5.31).
Z
RL
! dU NL ¼ 12
! !
@hz @u
dx dA
@x @x
1
2
t zdA
A xx
¼ My ;
R
t dA
A xz
¼ Q z , the
0
!
!2 13
2
2
@w
I
@
w
y
4Pd
A5dx
þ
þ P d@
@x
@x2
A
0
!
#
Z L"
@ 2 w @u
@w @u
þ Q zd
dx
My d
@x2 @x
@x @x
0
Z
2
R
L
@u
@x
2
ð5:40Þ
10
International Journal of Solids and Structures 222–223 (2021) 111003
Marcos Antonio Campos Rodrigues, Rodrigo Bird Burgos and Luiz Fernando Martha
R L 0 0 T
R L I n on oT
P N w N w dxfwg þ fdwgT 0 P Ay N 0hy N0hy dxfwgþ
0
n o n oT
RL
RL
T
fdwgT 0 M y N 0hy N 0u dxfug fdugT 0 M y N 0u N 0hy dxfwgþ
R L T
R L T
fdwgT 0 Q z Nhy N 0u dxfug fdugT 0 Q z N 0u N hy dxfwg
The axial part was already used for the xy plane. The expression
using the interpolation functions is given according to (5.41):
dU NL ¼ fdwgT
R L T
R L I T
dU NL ¼ fdwgT 0 P N 0w N 0w dxfwg þ fdwgT 0 P Ay N00w N 00w dxfwgþ
T
T
RL
RL
fdwgT 0 M y N 00w N 0u dxfug fdugT 0 M y N0u N 00w dxfwgþ
0 0 T
T
T RL
T RL
fdwg 0 Q z N w N u dxfug fdug 0 Q z N 0u N 0w dxfwg
ð5:47Þ
ð5:41Þ
5.3.3. Combined torsion and axial force
The interaction between torsion and axial force, as shown in
Fig. 5.5, has an important influence on the geometric stiffness
matrix. The transverse displacements, v and w, need to take these
effects into account.
where N w corresponds to the same N v interpolation functions.
Additionally, considering a constant shear force, the bending
moment and the shear force equations of a planar frame in plane
xz can be calculated by the expressions in (5.42).
M y1 þ My2 x
L
M y ¼ My1 þ
My1 þ M y2
L
Qz ¼
ð5:42Þ
eBBT
Substituting the complete shape functions that were previously
calculated and solving the integrals of the problem leads to the
geometric stiffness matrix, which considers the Euler-Bernoulli
beam theory and higher-order terms in the strain tensor.
Considering the Euler-Bernoulli beam theory, the displacement
field is written according to expression (5.48)
u ¼ u0 z
Plane xz (TBT)
@u
@x
1
@u
¼
2
@x
gxz ¼
@w
@x
2
@w
þ
@x
2
1
2
2
!
v ¼ zhx
þ
2
þ z2
@hy
@x
2
!
@u @hy
z
@x @x
dU NL ¼ 12
RL
0
RL
0
d
d
@u 2
dx
@x
@hy @u
dx
@x @x
R
t dA
A xx
R
þ 12
t zdA
A xx
þ
RL
0
RL
0
d
d
@w 2
dx
@x
@hy
@x
hy dx
R
t dA
A xx
R
ð5:44Þ
þ 12
t zdA
A xz
RL
0
RL
0
d
@hy
@x
2
dx
dhy @u
dx
@x
R
R
A
gxy ¼
z2 txx dAþ
t dA
A xz
1
2
ð5:48Þ
ð5:49Þ
!
2
ð5:50Þ
@u @u @ v @ v @w @w
þ
þ
@x @y @x @y @x @y
@u @ v
@2w @v
@ 2 v @ v @w
@hx
hx þ y
þz 2
þy 2
þ
hx
@x @x
@x @x
@x @x @x
@x
@u @u @ v @ v @w @w
þ
þ
@x @z @x @z @x @z
2
@u @w
@ w @w
@ 2 v @w @ v
@hx
¼
þz 2
þy 2
hx þ z
hx
@x @x
@x @x
@x @x @x
@x
ð5:51Þ
gxz ¼
"
@w
@x
!
Iy
@hy
d
A
@x
0
Z L
@hy @u
@u
þ Q z d hy
dx
My d
@x
@x @x
0
w ¼ yhx
1
@hx
z
2
@x
¼
ð5:45Þ
dU NL ¼
w ¼ w0 þ yhx
2
@hx
@ v @hx
@w @hx
þy
z
þy
@x @x
@x
@x @x
!
@hx 2
1
@ v @hx
@w @hx
z
¼ z2 þ y 2
þy
2
@x @x
@x
@x @x
gxx ¼
ð5:43Þ
@u @u @w @w
@hy
@u
þ
¼z
hy hy
@x @z @x @z
@x
@x
v ¼ v 0 zhx
However, most terms of this equation have already been
employed when planes were analyzed independently. Therefore,
it is necessary to consider only terms that correspond to the rotation about the x axis, which is given in equation (5.49). Thus, the
nonlinear parts of the Green-Lagrange strain tensor can be written
according to expressions (5.50) to (5.52).
Meanwhile, for this beam theory, the nonlinear parts of the
strain tensor considering bending in plane xz are written by
expressions (5.43) and (5.44). Thus, the third integral from the virtual work expression is given according to (5.45) or with (5.46).
gxx ¼
@w
@v
y
@x
@x
Z
L
Pd
@u
@x
2
þ
2
þP
2
!#
dx
ð5:52Þ
The virtual work principle is then written as (5.53) and (5.54).
Z Z
dU NL ¼
ð5:46Þ
A
Z
¼
Equation (5.47) can be written by considering only terms that
have not been used and applying interpolation functions.
A
L
txx dgxx dx dA
"
Z L
@hx
1 2
z þ y2
t xx d
2
@x
0
0
2
!
z
# !
@ v @hx
@w @hx
þy
dx dA
@x @x
@x @x
ð5:53Þ
Fig. 5.5. Combined torsion and axial force (Mcguire et al., 2000).
11
Marcos Antonio Campos Rodrigues, Rodrigo Bird Burgos and Luiz Fernando Martha
Z Z
Z Z
L
dU NL ¼
0
A
RL
dU NL ¼
RL
þ
0
0
d
t xy dgxy dx dA þ
d
@2 w @v
@x2 @x
@ 2 v @w
@x2 @x
R
dx
R
dx
t zdA
A xy
t ydA
A xz
RL
0
@u @u @ v @ v @w @w
þ
þ
@x @y @x @y @x @y
@u
@hy
@hz
@w
@hx
¼ hz þ z
hx þ y
hz þ y
hz þ
hx
@x
@x
@x
@x
@x
L
gxy ¼
txz dgxz dx dA
RL
R
R
x
dx A txy dA þ 0 d @h
h dx A txy ydAþ
@x x
R
R
R
L
x
d @@xv hx dx A txz dA þ 0 d @h
h dx A t xz zdA
@x x
RL
þ
0
A
International Journal of Solids and Structures 222–223 (2021) 111003
0
@w
h
@x x
d
@u @u @ v @ v @w @w
þ
þ
@x @z @x @z @x @z
@u
@hy
@hz
@v
@hx
¼ hy þ z
hy þ y
hy hx þ z
hx
@x
@x
@x
@x
@x
gxz ¼
ð5:54Þ
Adopting the relations presented before for the bending
R
moment and A z2 þ y2 dA ¼ J p , equation (5.53) becomes (5.55).
Expression (5.56) is written using interpolation functions.
Z L 2
1 PJp L @hx
@ v @hx
dx
¼
d
dx My
d
2 A 0
@x
@x @x
0
Z L @w @hx
þ ðMz Þ
dx
d
@x @x
0
Z Z
ð5:55Þ
A
RL
dU NL ¼
0
þ
ZA
t xz dA ¼ Q z ;
t xz ydA ¼ aMx ;
A
A
t xy zdA ¼ ða 1ÞM x
RL
2
d @@xw2 @@xv dx ða 1ÞM x þ 0 d @w
h dx Q y
@x x
RL
RL
2
þ 0 d @@xv2 @w
dx aM x 0 d @@xv hx dx Q z
@x
RL
0
0
d
@hy
@x
RL
0
d
@hy
@x
ð5:65Þ
n oT T R L M x n 0 o
RL
dU NL ¼ dhy
N hy fNhz gT dxfhz g fdhz gT 0 M2x fN hz g N 0hy dx hy
0 2
R L T T R L M x 0 T
þfdhz gT 0 M2x N 0hz N hy dx hy þ dhy
N hy N hz dxfhz g
0 2
T
RL RL
þfdwgT 0 Q y N0w fN hx gT dxfhx g þ fdhx gT 0 Q y fNhx g N0w dxfwg
T
R
R
L
L
fdv gT 0 Q z N 0v fN hx gT dxfhx g fdhx gT 0 Q z fNhx g N0v dxfv g
ð5:57Þ
A
dU NL ¼
t xz dgxz dx dA
RL
hz dx ða 1ÞMx þ 0 d @w
h dx Q y þ
@x x
RL
RL
z
þ 0 d @h
h dx aM x 0 d @@xv hx dx Q z
@x y
dU NL ¼
Z
t xy dA ¼ Q y ;
0
Expression (5.64) is rewritten according to (5.65) or (5.66) using
interpolation functions with the same relations presented before in
(5.57).
For bisymmetric sections, the shear force and torsion moment
are defined by (5.57). Thus, expression (5.54) can be written
according to (5.58).
Z
A
L
ð5:64Þ
ð5:56Þ
Z
0
0
t xy dgxy dx dA þ
RL
RL
R
R
R
x
hz dx A txy zdA þ 0 d @w
h dx A t xy dA þ 0 d @h
h dx A t xy ydAþ
@x x
@x x
R
R
R
R
R
L
L
z
x
d @h
h dx A txz ydA 0 d @@xv hx dx A txz dA þ 0 d @h
h dx A txz zdA
@x y
@x x
RL
RL
Z Z
L
dU NL ¼
T
RL
PJ p 0 0 T
Nhx N hx dxfhx g fdv gT 0 M y N 0v N 0hx dxfhx g
A
T
R
R
T
L
L
fdhx gT 0 M y N 0hx N 0v dxfv g fdwgT 0 M z N 0w N 0hx dxfhx g
R
T
L
fdhx gT 0 M z N0hx N 0w dxfwg
dU NL ¼ fdhx gT
ð5:63Þ
Equation (5.61) is the same for both beam theories because it is
equal to (5.50). Finally, the virtual work principle for the
Timoshenko beam theory only changes in Eq. (5.64) due to the
nonlinear distortion.
Z
dU NL
ð5:62Þ
ð5:58Þ
ð5:66Þ
Considering Saint Venant pure torsion, M x ¼ Mx2 and a ¼ 1=2.
Then, Eq. (5.59) is written by rewriting (5.58) using interpolation
functions.
dU NL ¼ fdwgT
þfdv g
T
RL
0
RL
0
Mx
2
00 0 T
RL
N w N v dxfv g fdv gT 0
Mx
2
00
Nv
N0w
T
T
dxfwg þ fdwg
RL
0
Mx
2
Mx
2
N 0v
N0w
N 00w
T
00 T
Nv
5.3.4. Finite rotations
In spatial elements, the geometric stiffness matrix needs to consider finite rotations to satisfy the kinematic compatibility at the
joint of angled elements (Mcguire et al., 2000; Conci, 1988). Thus,
for a spatial vector, the rotation is given by equation (5.67), as
shown in Fig. 5.6.
dxfwg
dxfv g
T
RL RL
þfdwgT 0 Q y N 0w fN hx gT dxfhx g þ fdhx gT 0 Q y fN hx g N 0w dxfwg
fdv gT
RL
0
V 1 ¼ RðhÞV 0
T
RL
Q z N0v fN hx gT dxfhx g fdhx gT 0 Q z fN hx g N 0v dxfv g
ð5:67Þ
ð5:59Þ
The combined torsion and axial force contribution is found by
substituting the complete shape functions and solving the
integrals.
TBT
When the Timoshenko beam theory is considered, the displacement field cannot be written using the transverse displacement
derivative and instead should be given by Eq. (5.60). Thus, the nonlinear part of the strain tensor that is not used before is written
according to expressions (5.61) to (5.63).
v ¼ v 0 zhx
u ¼ u0 zhy yhz
gxx ¼
¼
1
@hx
z
2
@x
2
þy
@hx
1 2
z þ y2
2
@x
@hx
@x
2
z
2
!
z
w ¼ w0 þ yhx
ð5:60Þ
@ v @hx
@w @hx
þy
@x @x
@x @x
@ v @hx
@w @hx
þy
@x @x
@x @x
ð5:61Þ
Fig. 5.6. Spatial transformation between two vectors (Aguiar et al., 2014).
12
International Journal of Solids and Structures 222–223 (2021) 111003
Marcos Antonio Campos Rodrigues, Rodrigo Bird Burgos and Luiz Fernando Martha
Z Z
senðhÞ
1 cosðhÞ
W ðhÞ þ
W ðhÞ2
2
h
8 9
2
3h
0
hz hy
>
< hx >
=
6
7
0
hx 5; h ¼ hy
W ðhÞ ¼ 4 hz
>
: >
;
hy hx
0
hz
Z Z L
@u
@ v @u
dx dA þ
dx dA
txx d
t xy d
þ
@x
@x @y
0
0
A
A
Z Z L
@w @u
þ
þ
dx dA
txz d
@x @z
0
A
R¼Iþ
¼
Using a trigonometric series approximation given by expression
(5.68) and considering only terms up to the second order, the rotation matrix can be rewritten according to (5.69).
!
h3 h5 h7
þ þ 3! 5! 7!
2
h
h4 h6
cosðhÞ ¼ 1 þ þ 2! 4! 6!
L
t ij Deij ¼
V
senðhÞ ¼ h ð5:68Þ
2
1
RðhÞ ¼ I þ W ðhÞ þ W ðhÞ
2
L
R
Therefore, considering a bisymmetric section as shown in
Fig. 5.7, the position vector of a point, fag, and after rotation,fbg,
is written by equation (5.70).
V
R L z @ ðhx hy Þ
RL M
RL Q
tij Deij ¼ 0 M
d @x dx þ 0 2y d @ðh@xx hz Þ dx þ 0 2y d hx hy dx
2
RL
R L x @ ðhy hz Þ
RL
@ ðhy hz Þ
x
d @x dx þ 0 Q2z dðhx hz Þdx þ 0 aM
d @x dx
þ 0 ða1ÞM
2
2
ð5:74Þ
"
Expression (5.75) is obtained by adopting a ¼ 1=2 for Saint
Venant torsion and using integration by parts. Finally, the finite
rotation matrix is given in (5.76) and is independent of the beam
theory considered.
ð5:70Þ
The displacement can be written as: fbg fag or fug. However,
W ðhÞ has already been taken into account in the previous analysis
with the principle of virtual work. Thus, the displacement can be
L M
z
tij Deij ¼ M
d hx hy 0 þ 2y dðhx hz ÞjL0
2
t Deij ¼ 12 M z1 d hx1 hy1 þ M y1 dðhx1 hz1 Þ M z2 d hx2 hy2 þ M y2 dðhx2 hz2 Þ
V ij
R
RV
2
written only as a function of W ðhÞ , according to expressions
(5.71) and (5.72).
fug ¼
8 9
>
<u>
=
2
6
1
6
W ðhÞ2 fag ) v ¼ 6
>
2
: >
; 4
w
hy 2 þhz 2
2
hx hy
2
hx hy
2
hx
2
þhz 2
2
hy hz
2
hx hz
2
ð5:75Þ
3
8 9
>0>
=
7<
7
7 y
5>
: >
;
h 2 þh 2
z
x y
hx hz
2
hy hz
2
2
2
K g;Rotfin
ð5:71Þ
hx :hy :y hx :hz :z
þ
u¼
2
2
v
2
hy :hz :y
h2 hy
z x þ
w¼
2
2
2
!
hy :hz :z
h2 h2
y x þ z
¼
2
2
2
!
Z Z
tij Deij ¼
V
A
Z Z
0
t xy dexy dx dA
A
0
L
t xz dexz dx dA ¼
þ
A
M y1
0
0
0
0
0
0
0
0
0
0
0
0
0
M z1
0
0
Mz1
0
0
0
My2
0
0
3
hx1
0 7
h
7 y1
7
0 7 hz1
7
M y2 7
7 hx2
7
0 5 hy2
0
hz2
Therefore, considering these effects, the local complete tangent
stiffness matrix can be written by adding all the components for
both bending theories. The results are presented in the work by
Rodrigues (Rodrigues, 2019) and with an open source code available in the files StfBeamEulerBernoulliT (positive axial force) and
StfBeamEulerBernoulliC (negative axial force) considering EulerBernoulli beam theory. For the Timoshenko beam theory, the files
are StfBeamTimoshenkoT (positive axial force) and StfBeamTimoshenkoC (negative axial force) (Rodrigues et al., 2020, 2021).
To avoid numerical instability, the local tangent stiffness matrix
can be written using a Taylor series expansion, providing more
terms for the tangent stiffness matrix than the usual formulations.
These approximations provide greater precision in the results, and
in this work, tangent matrices with up to 3 and 4 terms were
developed. It is important to note that the 2-term approximation
provides the usual elastic and tangent stiffness matrices. These
matrices are also presented in the work by Rodrigues (Rodrigues,
2019) and implemented in (Rodrigues et al., 2020, 2021) in the files
GeoStfBeamEulerBernoulli and GeoStfBeamTimoshenko.
L
t xx dexx dx dA þ
M z1
ð5:76Þ
ð5:72Þ
Z Z
L
0
6 M z1
6
6
My1
16
¼ 6
26
6 0
6
4 0
0
To consider the finite rotations, the second integral of the virtual
work principle needs to be used according to expression (5.73).
Z
ð5:73Þ
Equation (5.74) can be written by employing the relations for
the bending moment, torsion, shear and axial force.
ð5:69Þ
#
2
1
fbg ¼ I þ W ðhÞ þ W ðhÞ fag
2
t xx @ hx hy
@ ðhx hz Þ
z dx dA
d
yþd
@x
2
@x
A
0
Z Z L
@ hy hz
t xy
þ
d hx hy þ d
z dx dA
2
@x
0
A
Z Z L
@ hy hz
t xz
y dx dA
dðhx hz Þ þ d
þ
@x
2
A
0
Z Z
Z
0
6. Numerical applications
In this section, numerical examples were developed with no
discretization to test the formulation from this work, the complete
formulation and the Taylor series expansion with 3 and 4 terms.
The elements can be identified based on the following
descriptions:
Fig. 5.7. Spatial transformation between two vectors (Mcguire et al., 2000).
13
Marcos Antonio Campos Rodrigues, Rodrigo Bird Burgos and Luiz Fernando Martha
International Journal of Solids and Structures 222–223 (2021) 111003
6.1. Isolated columns
EBBT_Large_Complete – Euler-Bernoulli beam theory, hyperbolic and geometric functions in the tangent stiffness matrix, and
higher order terms in the strain tensor.
EBBT_Large_4tr – Euler-Bernoulli beam theory, 4 terms in the
tangent stiffness matrix (1 elastic + 3 geometric), and higher order
terms in the strain tensor.
EBBT_Large_3tr – Euler-Bernoulli beam theory, 3 terms in the
tangent stiffness matrix (1 elastic + 2 geometric), and higher order
terms in the strain tensor.
EBBT_Large or Small_2tr – Euler-Bernoulli beam theory, 2
terms in the tangent stiffness matrix (1 elastic + 1 geometric),
and higher order terms in the strain tensor (Large) or (Small) if this
influence 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, hyperbolic
and geometric functions in the tangent stiffness matrix, and higher
order terms in the strain tensor.
TBT_Large_4tr – Timoshenko beam theory, 4 terms in the tangent stiffness matrix (1 elastic + 3 geometric), and higher order
terms in the strain tensor.
TBT_Large_3tr – Timoshenko beam theory, 3 terms in the tangent stiffness matrix (1 elastic + 2 geometric), and higher order
terms in strain tensor.
TBT_Large or Small_2tr – Timoshenko beam theory, 2 terms in
the tangent stiffness matrix (1 elastic + 1 geometric), and higher
order terms in the strain tensor (Large) or (Small) if this influence
is not considered (Rodrigues et al., 2019). This formulation is the
most conventional.
The results were compared with the results obtained with Mastan2 v3.5. The software is able to perform a geometric nonlinear
analysis considering both beam theories while considering cubic
interpolation functions and disregarding higher order terms in
the strain tensor. The element tangent stiffness matrix considered
in Mastan2 v3.5 is described in the work by McGuire et al.
(Mcguire et al., 2000). In this research, Mastan2 v3.5 elements
are labeled as follows:
EBBT_Mastan – Mastan2 v3.5 software Euler-Bernoulli beam
theory;
TBT_Mastan – Mastan2 v3.5 software Timoshenko beam
theory.
In cases that are no available analytical solution, the reference
response for comparison purposes was given by a numerical solution with discretized structure (EBBT or TBT_Large_2tr_ number of
elements in each member).
Euler Critical Load – Analytical Euler buckling load
Timoshenko and Gere – Analytical Timoshenko buckling load
(Timoshenko and Gere, 1963)
To verify the developed tangent stiffness matrix, the buckling
load of columns was studied as shown in Fig. 6.1, adopting just
one element per member. Additionally, a reduced slenderness ratio
(k ¼ L=h) was employed because of the influence of Timoshenko
beam theory. The columns have a length of L = 1 m, Young’s modulus of E = 107 kN/m2, section form factor of v ¼ 1 and null Poisson’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 simply
supported and fixed and simply supported columns, respectively,
with a slenderness ratio of k ¼ 10:0 for the Euler-Bernoulli beam
theory and k ¼ 4:0 for the Timoshenko beam theory. The numerical
results for the buckling loads for the columns are shown in Table 1.
Analyzing the equilibrium paths, it can be noted that the complete
formulation developed in this work, where the tangent stiffness
matrix was calculated with the complete interpolation functions
(EBBT_Large_complete and TBT_Large_complete), provides the correct prediction for the buckling loads of columns using just one element per member for both beam theories. With no discretization,
the complete formulation provides the best approximation for the
analytical response for both beam theories. In Table 1 the results
can be seen numerically and the efficiency of the complete formulation is observed when compared with literature values.
Additionally, it can be seen that geometric matrices obtained
from a Taylor series approximation (Large_3tr and Large_4tr) also
provide an accurate prediction for the buckling load. The approximation using 4 terms in some cases provides the same response
from the complete formulation, and the approximation with 3
terms follows those curves closely. Table 1 also shows the proximity of the complete formulation and the Taylor series expansion
and the differences with the usual formulation.
6.2. Continuous beam-column
The next example studies a continuous beam-column; this problem is presented in the work by (Timoshenko and Gere (1963) and
is shown in Fig. 6.5. The geometry of the structure, material and section proprieties are the same as in the first example with a length of
L = 1 m, Young’s modulus of E = 107 kN/m2, section form factor of
v ¼ 1 and null Poisson’s ratio (m ¼ 0). Fig. 6.6 shows the equilibrium
path considering a slenderness ratio of k ¼ 10:0 for the EulerBernoulli beam theory and k ¼ 4:0 for Timsohenko beam theory.
The structures were modeled with one element in each span.
Again, using just one element per member, the complete formulation and the Taylor series expansion with 4 terms provide the
Fig. 6.1. Analyzed columns (Silva et al., 2016).
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International Journal of Solids and Structures 222–223 (2021) 111003
Marcos Antonio Campos Rodrigues, Rodrigo Bird Burgos and Luiz Fernando Martha
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.
has a length of L = 1 m, Young’s modulus of E = 107 kN/m2, section form factor of v ¼ 5=6 and Poisson’s ratio of m ¼ 0:3. The
frame is loaded by vertical loads (P) and two small lateral disturbing loads (H ¼ 0:001P). The structure was modeled with one
element per member, and the response was compared with a discretized structure using the usual formulation. The equilibrium
paths can be seen in Fig. 6.8, considering k ¼ 10:0 for the EulerBernoulli beam theory and k ¼ 4:0 for the Timoshenko beam
theory.
Once again, the complete formulation and the Taylor series
expansion with 4 terms provide the buckling load of the spatial
frame. The expansion with 3 terms overlaps these results for the
correct buckling load of the structure. The expansion with just 3
terms gives approximate results. These results can be seen numerically in Table 2.
The conventional geometric stiffness matrix and the Mastan2
v3.5 software cannot predict the buckling load of the continuous
beam-column, providing differences on the order of 50% to the
expected result with a discretized structure.
6.3. Spatial frame
The example presented is the first to evaluate the developed
formulation in a spatial structure, according to Fig. 6.7. The frame
15
Marcos Antonio Campos Rodrigues, Rodrigo Bird Burgos and Luiz Fernando Martha
International Journal of Solids and Structures 222–223 (2021) 111003
Table 1
Numerical Buckling Loads for Columns.
Euler Bernoulli Beam Theory - EBBT
Discretization
1 Element
Large
2tr
Large
3tr
Euler Critical Load
EBBT- Element
Mastan
Small
2tr
Clamped col. (k = 10.0)
Simply supported col. (k = 10.0)
Fixed – simply sup. col. (k = 10.0)
2.496
12.255
31.651
2.491
12.312
31.630
TBT - Element
Mastan
Small
2tr
Large
2tr
Large
3tr
Large
4tr
Large
Complete
Clamped col. (k = 4.0)
Simply supported col. (k = 4.0)
Fixed - simply sup. col. (k = 4.0)
2.475
13.914
40.642
2.475
13.940
> 60
2.444
12.866
38.065
2.419
10.064
22.572
2.417
9.546
19.892
2.417
9.368
17.826
2.486
12.197
30.798
Timoshenko Beam
Discretization
Large
4tr
Large
Complete
2.469
2.468
10.348
10.047
23.583
21.779
Theory – TBT
2.468
9.95
20.541
1 Element
2.467
9.869
20.142
(Timoshenko and Gere, 1963)
2.408
8.949
16.401
Fig. 6.5. Beam-column (Timoshenko and Gere, 1963).
Fig. 6.6. Equilibrium paths for a beam-column.
6.4. Roorda spatial frame
Euler-Bernoulli beam theory and provides intermediate results for
the Timoshenko theory. The usual formulation and Mastan2 v3.5
software provide more elevated buckling loads. Table 3 presents
these conclusions numerically.
Using the same bar length, material and section proprieties as
the structure presented before, another spatial frame was studied
Table 2
Numerical buckling loads for the beam-columns.
Discretization
1 Element
(Timoshenko and Gere, 1963)
Element
Mastan
Small
2tr
Large
2tr
Large
3tr
Large
4tr
Large
complete
EBBT -k = 10.0
TBT - k = 4.0
12.240
13.989
12.310
13.843
12.240
12.867
10.453
10.074
10.145
9.553
9.953
9.368
16
9.869
8.949
International Journal of Solids and Structures 222–223 (2021) 111003
Marcos Antonio Campos Rodrigues, Rodrigo Bird Burgos and Luiz Fernando Martha
Fig. 6.9. Roorda spatial frame.
3 terms. These formulations give better approximations when
compared with the usual formulations with 2 terms, which present
relevant errors. Table 4 shows the numerical results.
When considering just one element per member, Table 4 shows
that the conventional formulation doubles the buckling load, while
the complete formulation maintains the error at a rate of 12% for
TBT elements and 8% for EBBT theory.
Fig. 6.7. Spatial frame.
(Fig. 6.9). The frame is loaded by a vertical load (P) and by two
small lateral disturbing loads (H ¼ 0:001P). The structure was
modeled with one element per member, and the response was
compared with a discretized structure using the usual formulation.
The equilibrium paths can be seen in Fig. 6.10 with k ¼ 10:0 for the
Euler-Bernoulli beam theory and k ¼ 4:0 for the Timoshenko beam
theory.
There is no available analytical solution for the critical load of
this frame. In this case, the reference solution for comparison purposes was the Large_2tr formulation with 4 segments in each
member. One may observe that, with no discretization, the complete formulation provides the best approximation for the reference response. The second-best solution is for the Taylor series
expansion with 4 terms and then the Taylor series expansion with
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 length
of L = 1 m, Young’s modulus of E = 107 kN/m2, section form factor of
v ¼ 5=6 and Poisson’s ratio of m ¼ 0:3.
The equilibrium paths found for the structure employing one
element per member and both beam theories, considering
k ¼ 4:0, are given in Fig. 6.12 and numerically presented in Table 5.
Fig. 6.8. Equilibrium paths for a spatial frame.
Table 3
Numerical buckling loads for a spatial frame.
Discretization
1 Element
4 Elements
Element
Mastan
Small
2tr
Large
2tr
Large
3tr
Large
4tr
Large
Complete
Large
2tr_4el
EBBT -k = 10.0
TBT-k = 4.0
5.577
5.068
5.569
5.115
5.554
5.030
5.507
4.977
5.501
4.837
5.500
4.934
5.488
4.927
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Marcos Antonio Campos Rodrigues, Rodrigo Bird Burgos and Luiz Fernando Martha
International Journal of Solids and Structures 222–223 (2021) 111003
Fig. 6.10. Equilibrium paths for a Roorda spatial frame.
Table 4
Numerical buckling loads for a Roorda spatial frame.
Discretization
1 Element
4 Elements
Element
Mastan
Small
2tr
Large
2tr
Large
3tr
Large
4tr
Large
Complete
Large
2tr_4el
EBBT -k = 10.0
TBT - k = 4.0
19.008
21.135
24.461
21.166
20.897
19.630
15.262
13.670
14.484
12.445
14.136
11.753
13.200
10.414
form factor of v ¼ 5=6 and Poisson’s ratio of m ¼ 0:3 and is loaded
by 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 illustrating the torsion experienced by the space frame with the scale
factor increased tenfold. Finally, the equilibrium path of the structure considering k ¼ 10 for the Euler-Bernoulli beam theory and
k ¼ 4:0 for the Timoshenko beam theory is illustrated.Fig. 6.15.
The solution obtained by the complete formulation and for the
Taylor series expansion provides the correct buckling load employing just one element per member for both beam theories, while the
usual 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 moderate
slenderness, k ¼ 6:6 for TBT, as shown in Fig. 6.16, loaded by two
vertical 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, section
form factor of v ¼ 5=6 and Poisson’s ratio of m ¼ 0:3. Fig. 6.17
shows the influence of torsion on this structure, and Fig. 6.18
shows the equilibrium paths.
Finally, the complete formulation and the Taylor series expansion provide the buckling load for the spatial frame, with the
curves overlapping. The usual formulation and Mastan software
overcome this load for both beam theories. Table 7 shows these
conclusions numerically.
Fig. 6.11. Spatial frame with inclined columns (adapted from Zugic et al. (Zugic
et al., 2016).
As concluded for the other examples, the complete formulation
calculates the exact buckling load of the structure. In this example,
the Taylor series expansion with 4 or even 3 terms reaches this
load as well, and their results overlap. Although the usual formulation gives a good approximation with just one element, employing
the formulation developed in this work improves the results. In
addition, consideration of Timoshenko beam theory reduces the
critical buckling load for structures with small slenderness.
7. Conclusions
This research developed a complete formulation of the tangent
stiffness matrices of a spatial frame element for the Euler-Bernoulli
and Timoshenko beam theories, considering interpolation functions obtained directly from the solution of the equilibrium differential equation of a deformed infinitesimal element, which
includes the influence of axial forces. Additionally, the formulation
uses higher-order terms in the strain tensor, and the geometric
6.6. Spatial frame with torsion
This example explores the influence of torsion. The frame also
has a length of L = 1 m, Young’s modulus of E = 107 kN/m2, section
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International Journal of Solids and Structures 222–223 (2021) 111003
Marcos Antonio Campos Rodrigues, Rodrigo Bird Burgos and Luiz Fernando Martha
Fig. 6.12. Equilibrium paths for the spatial frame with inclined columns.
stiffness matrix is corrected by employing finite rotations. The
complete formulation was rewritten using a Taylor series expansion considering 3 and 4 terms, improving the usual tangent stiffness matrices for both beam theories that involve only 2 terms (1
elastic and 1 geometric).
The formulation was evaluated in several numerical tests for
planar and spatial structures to verify the geometric nonlinear
response of the structure considering distinct bending theories
and the consideration of high-order terms for the strain. The first
set of examples employs the proposed element to analyze isolated
columns with different boundary conditions and a continuous
beam-column. Similarly, this formulation was evaluated for spatial
structures, and some examples explored the torsion effect, applying a load or geometric asymmetry.
The complete formulation presented precisely predicted results
for the critical load of the developed examples with a minimum
discretization of the structure, while the usual tangent stiffness
matrix would require a more refined mesh to predict this behavior.
Fig. 6.13. Spatial frame with asymmetric loads.
Table 5
Numerical buckling loads for the spatial frame with inclined columns.
Discretization
1 Element
4 Elements
Element
Mastan
Small
2tr
Large
2tr
Large
3tr
Large
4tr
Large
Complete
Large
2tr_4el
EBBT-k = 4.0
TBT- k = 4.0
7.070
6.182
7.179
6.371
7.157
6.227
6.952
6.155
6.942
6.155
6.942
6.082
6.919
6.082
Fig. 6.14. Deformed spatial frame with torsion.
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Marcos Antonio Campos Rodrigues, Rodrigo Bird Burgos and Luiz Fernando Martha
International Journal of Solids and Structures 222–223 (2021) 111003
Fig. 6.15. Equilibrium paths for the spatial frame with torsion.
Table 6
Numerical buckling loads for the spatial frame with torsion.
Discretization
1 Element
4 Elements
Element
Mastan
Small
2tr
Large
2tr
Large
3tr
Large
4tr
Large
Complete
Large
2tr_4el
EBBT- k = 10.0
TBT-k = 4.0
7.488
6.389
7.328
6.548
7.313
6.374
7.259
6.367
7.257
6.367
7.256
6.341
7.260
6.341
Fig. 6.17. Deformed asymmetric spatial frame.
Due to the numerical instability that can occur using the hyperbolic and trigonometric expressions of the complete formulation,
an alternative geometric stiffness matrix using a Taylor series
expansion with 4 terms is proposed to solve this problem with
accurate results. All necessary equations for this implementation
are available online in open source.
Although precise buckling loads are achieved using just one element if one is interested in following the actual equilibrium path
up to the critical load, a refined discretization should be used. This
is suggested because one element is not able to capture local
effects, such as the P-‘‘small delta”.
The continuation of this work will focus on monitoring the
equilibrium path in the postcritical stage. For this, more improved
solution schemes for incremental geometric nonlinear analysis
should be employed.
Fig. 6.16. Asymmetric spatial frame with inclined columns (adapted from Zugic
et al. (Zugic et al., 2016).
As expected, the examples have also shown that for structures with
small slenderness, the consideration of the Timoshenko beam theory leads to lower buckling loads, and the proposed formulation
illustrated this effect. The Timoshenko beam theory is also used
in cases of structures with a small shear-to-bending ratio, regardless of their slenderness.
The reported results clearly illustrate the efficiency of the complete formulation and the Taylor series expansion using 4 terms in
the solution of nonlinear geometric analyses using only one element for planar and spatial structures. Using 3 terms also provides
better results than considering the usual geometric stiffness
matrix, which overestimates the buckling load.
The presented formulation solves geometric nonlinear problems and can be implemented in any structural analysis software.
8. Funding sources
This work has been partially supported by Conselho Nacional de
Desenvolvimento Científico e Tecnológico (CNPq) and FAPERJ.
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International Journal of Solids and Structures 222–223 (2021) 111003
Marcos Antonio Campos Rodrigues, Rodrigo Bird Burgos and Luiz Fernando Martha
Fig. 6.18. Equilibrium paths for an asymmetric spatial frame.
Table 7
Numerical buckling loads for asymmetric spatial frames.
Discretization
1 Element
4 Elements
Element
Mastan
Small
2tr
Large
2tr
Large
3tr
Large
4tr
Large
Complete
Large
2tr_4el
EBBT- k = 10
TBT- k = 6.6
7.584
7.282
7.606
7.298
7.586
7.259
7.499
7.152
7.486
7.132
7.483
7.083
7.452
7.089
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Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared
to influence the work reported in this paper.
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21
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Marcos Antonio Campos Rodrigues. Adjunct professor in
the Department of Civil Engineering at Federal University of Espírito Santo (UFES). B.Sc. in Civil Engineering
from Federal University of Espírito Santo (UFES) in 2011.
M.Sc. in Aeronautical and Mechanical Engineering from
the Aeronautics Institute of Technology (ITA) in 2014
and Ph.D. in Civil Engineering from Pontifical Catholic
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Rodrigo Bird Burgos Associate Professor in the Department of Structures and Foundations at the State
University of Rio de Janeiro (UERJ). B.Sc. (2003), M.Sc.
(2005) and Ph.D. (2009) in Civil Engineering from Pontifical Catholic University of Rio de Janeiro (PUC-Rio).
Postdoctoral researcher at Pontifical Catholic University
of Rio de Janeiro from 2009 to 2012. He has experience
in computational modeling and advanced numerical
methods for solving differential equations. His main
research interests include instability and structural
dynamics, wavelet functions, interpolets, WaveletGalerkin Method, Finite Element Method.
Luiz Fernando Martha Associate professor of the
Department of Civil Engineering at Pontifical Catholic
University of Rio de Janeiro. B.Sc. (1977), M.Sc. (1980) in
Civil Engineering from Pontifical Catholic University of
Rio de Janeiro (PUC-Rio) and Ph.D. (1989) in Structural
Engineering from Cornell University. Postdoctoral
researcher at Superior Technical Institute from Technical University of Lisbon in 2012. Acts as coordinator of
research projects at Tecgraf / PUC-Rio (Tecgraf Institute
for Technical and Scientific Software Development). He
has experience in structural analysis and his main
research interests include Computer Graphics, Geometric Modeling, Numerical Methods Applied to Engineering and Geology Simulations, Computational Fracture Mechanics, and Educational Software for
Engineering Teaching.
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