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Nonlinear programming approach to a shear-deformable hybrid beam element for large displacement analysis


Abstract and Figures

In the present work, a hybrid beam element based on exact kinematics is developed, accounting for arbitrarily large displacements and rotations, as well as shear deformable cross sections. At selected quadrature points, fiber discretization of the cross sections facilitates efficient computation of the stress resultants for any uniaxial material law. The numerical approximation is carried out through the lens of nonlinear programming, where the enengy functional of the system is treated as the objective function and the exact strain-displacement relations form the set of kinematic constraints. The only interpolated field is curvature, whereas the centerline axial and shear strains, along with the displacement measures at the element edges, are determined by enforcing compatibility through the use of any preferable constrained optimization algorithm. The solution satisfying the necessary optimality conditions is determined by the stationary point of the Lagrangian. A set of numerical examples demonstrates the accuracy and performance of the proposed element against analytical or approximate solutions available in the literature.
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Charilaos M. Lyritsakis, Charalampos P. Andriotis, Konstantinos G.
The Pennsylvania State University
University Park, PA, 16802
Keywords: geometrically exact beam theory, fiber elements, shear deformable beam , nonlinear
programming, hybrid finite element
Abstract. In the present work, a hybrid beam element based on exact kinematics is devel-
oped, accounting for arbitrarily large displacements and rotations, as well as shear deformable
cross sections. At selected quadrature points, fiber discretization of the cross sections facili-
tates efficient computation of the stress resultants for any uniaxial material law. The numerical
approximation is carried out through the lens of nonlinear programming, where the enengy
functional of the system is treated as the objective function and the exact strain-displacement
relations form the set of kinematic constraints. The only interpolated field is curvature, whereas
the centerline axial and shear strains, along with the displacement measures at the element
edges, are determined by enforcing compatibility through the use of any preferable constrained
optimization algorithm. The solution satisfying the necessary optimality conditions is deter-
mined by the stationary point of the Lagrangian. A set of numerical examples demonstrates the
accuracy and performance of the proposed element against analytical or approximate solutions
available in the literature.
7th ECCOMAS Thematic Conference on
Computational Methods in Structural Dynamics and Earthquake Engineering
M. Papadrakakis, M. Fragiadakis (eds.)
Crete, Greece, 24–26 June 2019
Available online at
Eccomas Proceedia COMPDYN (2019) 4707-4718
ISSN:2623-3347 © 2019 The Authors. Published by Eccomas Proceedia.
Peer-review under responsibility of the organizing committee of COMPDYN 2019.
doi: 10.7712/120119.7262.19709
Charilaos M. Lyritsakis, Charalampos P. Andriotis, Konstantinos G. Papakonstantinou
Problems frequently encountered by the engineering community over the last decades were
concerned with structural systems whose response involved large displacements, rotations and
strains on one hand, and inelastic behavior on the other. Such problems, in cases where the
subject matter involved rod-like structures, necessitated the advancement of the classical Euler-
Bernoulli theory, where displacements, rotations and strains were typically kept in the small
In the geometrically nonlinear case, the deformed geometry of the beam can be drastically
different from the undeformed one, resulting in a rather involved description of beam kinemat-
ics. In general, there are two approaches in describing the kinematics of beams. The first one is
the so-called continuum-based approach [1], which is employed for the derivation of the classi-
cal beam theory [2] and where all vectorial components are obtained from the three-dimensional
theory of solids, with additional assumptions imposed on cross-section kinematics. The second
approach, which is followed in this work, is concerned with the description of a material curve
- an assemblage of material points representing the beam centroid - embedded in E2(or E3in
the 3D case). The analysis of the curve by means of differential geometry of curves leads to
the notion of intrinsic parameterization or one-dimensional formulation of beams, where spa-
tial quantities can be expressed as functions of only one parameter. In the study of beams, this
parameter is taken to be the arc-length of the beam. This in turn leads to the so-called arc-
length parameterization of the beam with respect to a reference configuration. This approach
was adopted in early works by Reissner [3, 4] and, next, Simo [5], Simo and Vu-Quoc [6] and
can be traced back to Kirchhoff and his treatment of inextensible elastic rods. In these formu-
lations, often termed as geometrically exact, the thickness of the rod is taken into consideration
by attaching two vectors at each point on the material curve that would translate and rotate with
the points, thus defining the properties of curvature and torsion at these points.
Approaches where strain measures are recast as primary field variables are often termed
strain or deformation-based and were explored by Planinc et al. [7] for the planar case of
the geometrically exact beam in order to properly account for the effect of local instabilites
on the tangent stiffness matrix due to plastification, when global stability considerations are
also present. It was later extended to the 2D dynamic case by Gams et al. [8] and to the
three-dimensional case by Zupan and Saje [9] as a means to cope with the strain objectivity
issue arising from the interpolation of the rotation vector [10]. Interpolation of strain measures
is also encountered in mixed formulations, where more than one fields are interpolated and
the underlying functional is augmented by additional terms to be satisfied in the weak sense
[11]. Several other works with geometrically exact formulations in various settings can be
found in the literature. Approaches based on mixed, hybrid, flexibility- and displacement-based
considerations can be indicatively seen in [12, 13, 14, 15].
The present work is an extension of the geometrically exact hybrid formulation presented
in [16] in order to account for the effect of shear deformation at the section level. As opposed
to deriving the system equations from the Galerkin form after appropriate discretization, in the
aforementioned work the problem is originally recast in a nonlinear programming framework,
where the total potential energy functional (TPE) is augmented via Lagrange multipliers that
enforce satisfaction of the exact kinematic conditions. The resulting functional is then approx-
imated by employing a Gauss-Legendre quadrature rule, which yields the objective function
to be minimized. Another interesting feature of this particular approach is that the primary
variables in the element interior contributing to the elastic strain energy are the generalized
Charilaos M. Lyritsakis, Charalampos P. Andriotis, Konstantinos G. Papakonstantinou
strain measures of the centroid, which are sought at quadrature points. Displacement mea-
sures, namely, the translations along the coordinate axes and the rotation of the cross sections,
occur only at the nodes of the element and are associated with the external work. Kinematic
consistency between the rotational measures of displacement and strain is enforced by using a
Lagrange interpolation scheme to approximate the curvatures over the entire element domain.
We should note that quadrature points coincide with the Lagrange interpolation points. In the
remainder a succinct presentation of the formulation is presented, along with numerical investi-
gations based on benchmark nonlinear problems studied in the literature, verifying the accuracy
and efficiency of the suggested approach.
We now proceed with the derivation of the kinematic equations that serve as constraints for
the optimization problem. Let us consider two dimensional Euclidian space E2and a fixed
Cartesian coordinate system (X1, X2)with unit basis vectors {Ei},i= 1,2and a material
curve of length Lwhich represents the centroid of the beam embedded in that space. We
will assume that the beam is initially straight, aligned with the coordinate axis X1and in an
unstressed state. We also define the arc-length parameterization of the material curve sI=
[0, L]rR2, where sis the distance of a material point on the line of centroids in
the reference configuration. The position vector of any material point on the centroid in any
configuration can be written as:
r(s) = (s+u(s))E1+w(s)E2(1)
where uand ware the displacements of the material point along the coordinate axes X1and X2
Beam Kinematics
According to [3] the strain-displacement relations, assuming small axial strain of the beam
centroid, are:
u0= (1 + ) cos φγsin φ1(2)
w0= (1 + ) sin φ+γcos φ(3)
where , γ and κare the axial, shear and bending strains of the line of centroids respectively,
φis the tangent angle to the material curve in the current configuration and derivatives with
respect to the sare denoted by ( )0. Integrating (2)-(4) over Iyields:
u(L)u(0) = ZL
(1 + ) cos φγsin φ ds L
w(L)w(0) = ZL
(1 + ) sin φ+γcos φ ds
φ(L)φ(0) = ZL
κ ds
The integral form of the kinematic equations in (5) is utilized to impose the constraints on the
nonlinear program.
Charilaos M. Lyritsakis, Charalampos P. Andriotis, Konstantinos G. Papakonstantinou
Total Potential Energy
The total potential energy functional of the beam under a set of point loads P= [ P1P2]T, with
Pithe point loads at the element edge nodes, can be expressed in the reference configuration
Π(, γ, κ, d) = ZL
W(, γ, κ)ds PTd(6)
where Wis the strain energy per unit length of the beam centroid. The displacement degrees of
freedoms at the two edge nodes are collectively represented in vector das:
d1=u(0), d2=w(0), d3=φ(0),
d4=u(L), d5=w(L), d6=φ(L)
The stress resultants on a section are associated with the strain energy as follows:
∂ ,Q=W
∂γ ,M=W
∂κ (8)
The stress resultants defined in (8) can be numerically computed using appropriate fiber dis-
cretization at the cross section level. Thereby, any nonlinear constitutive law may be incorpo-
rated in beam element formulations to capture the effects of distributed elastoplastic behavior
or damage [17, 18, 19].
In this section we formulate the equations pertaining to the description of our hybrid element as
a nonlinear program.
Element Objective Function
The discrete form of the TPE in (6) is given by applying Gauss-Legendre quadrature to approx-
imate the integral for one element:
f(x) =
ciW(i, γi, κi)PTd(9)
where ciare the weights of the quadrature, nthe number of quadrature points and x,dand yi
are defined as:
2. . . yT
with yi=iγiκiT
Element Constrains
The first set of constraints derived from the approximation of kinematic relations (5) is given
Charilaos M. Lyritsakis, Charalampos P. Andriotis, Konstantinos G. Papakonstantinou
eq =
i=1 ci[(1 + i) cos φiγisin φi] + L
i=1 ci[(1 + i) sin φi+γicos φi]
i=1 ciκi
In accordance with [16] we then interpolate the curvature field with Lagrange polynomials in
order to obtain the rotations φiat the quadrature points:
Θij κj(12)
2. . . ξn
2. . . ξn
1ξ1. . . ξn1
1ξn. . . ξn1
where ξ=x
Land Gis the Vandermonde matrix.
Notice that the first two equations of (11) are nonlinear equality constraints, while the third,
along with the nequations of (12) are linear equality constraints. It is convenient to recast all
constraints in one vector as follows:
Ceq =
eq =
j=1 Θ1jκj
j=1 Θnj κj
Element Lagrangian Function
We now introduce a vector λof the Lagrange multipliers and augment the TPE (6), constructing
the Lagrangian of the optimization problem as:
f(x,λ) =
ciW(i, γi, κi)PTd+λTCeq (16)
Stationary points are provided by satisfying Karush-Kuhn-Tucker [20] optimality conditions for
the Lagrangian function of (16).
In the following examples we examine the performance of the proposed formulation and com-
pare it with other well-known works in the literature. We first explore the accuracy when the
shear stiffness is reduced and then we test against different loading cases. For each example,
only one element with five quadrature points is used.
Charilaos M. Lyritsakis, Charalampos P. Andriotis, Konstantinos G. Papakonstantinou
Table 1: Cantilever with constant transverse load at free end.
w u
GAsNumerical Analytical Present Numerical Analytical Present
5·1020 0.30172077 0.301720774 0.30172432 0.05643324 0.056433236 0.05643126
5·1020.31781387 0.317813874 0.31781567 0.06131566 0.061315658 0.06131317
5·1010.46541330 0.465413303 0.46541543 0.10328492 0.103284917 0.10328294
1·1021.16709588 1.167095878 1.16709542 0.25213661 0.252136606 0.25213357
5·1002.10408747 2.104087473 2.10409063 0.37612140 0.376121399 0.37612451
Problem data: EA = 1021, EI = 10, L = 1. Numerical data in [21], analytical in [22].
0 2 4 6 8 10 12
E=200 10 5
= 0.3
L = 1.0
F = 10
Figure 1: Timoshenko compared to Bernoulli solutions for different levels of slenderness.
Example 1 - Effect of shear deformation in cantilever deflection
In this example we explore the effect of shear flexibility on the tip deflection of a cantilever. A
constant point transverse force P= 10 units is applied at the free end and, then, several analyses
are performed by varying the shear stiffness coefficient, GAs. Numerical and analytical results
obtained by Batista in [21] and [22], respectively, are compared with the present formulation
and are illustrated in Table 1.
Fig. 1 demonstrates the effect shear deformations have when not neglected, compared to the
Bernoulli solutions, by varying the ratio L/h, with hbeing the height of the cross section and L
the length of the beam. The applied load is F,νis the Poisson’s ratio and wBis the deflection
when shear flexibility is neglected. As can be seen from the results, when L= 2h, transverse
Charilaos M. Lyritsakis, Charalampos P. Andriotis, Konstantinos G. Papakonstantinou
displacements due to shear deformation are increased by rougly 13%.
Example 2 - Cantilever with transverse point load at its tip
This problem has been analyzed in [23, 24], whereas Mattiasson [25] provided solutions by
solving the elliptic integrals of the problem of large deflections of beams. Moreover, the prob-
lem was also examined in [26] using a co-rotational transformation for the Timoshenko beam,
whereas Nanakorn [27] used 3 elements and a total tagrangian formulation.
0 0.2 0.4 0.6 0.8 1
Figure 2: Cantilever with point transverse load at its free end.
Table 2: Cantilever with transverse load at its free end.
w/L u/L
P L2/EI Mattiasson Present Mattiasson Present
2.0 0.49346 0.49347 0.16064 0.16063
4.0 0.66996 0.67001 0.32894 0.32892
6.0 0.74457 0.74465 0.43459 0.43457
8.0 0.78498 0.78509 0.50483 0.50481
10.0 0.81061 0.81073 0.55500 0.55498
Problem data: EI = 1000 lb/in2,L= 100in, P= 1lb, N= 20 steps.
Results in Mattiasson [25].
Charilaos M. Lyritsakis, Charalampos P. Andriotis, Konstantinos G. Papakonstantinou
Figure 2 illustrates the performance of the element when compared against the analytical and
numerical solutions - mentioned above - and Table 2 demonstrates the accuracy up to six deci-
mal points when compared to the analytical solutions for a sample of loading levels. The dotted
line indicates the linear response.
Example 3 - Cantilever with point moment at its free end
This example tests the capabilities of our developed model capturing the response of an inex-
tensional beam subjected to a point moment, forcing a curl into a complete circle. As men-
tioned previously, in all examples we only used one element with five quadrature points for our
analysis. Bathe and Bolourchi [28] using five and twenty elements and an updated lagrangian
procedure showed accuracy up to 90 degrees. In subsequent works, Simo & Vu-Quoc [6] (five
elements), Rankin & Brogan [29] (ten elements) and Crisfield [30] (five elements) duplicated
the exact solution. In the second and third works a corotational formulation was employed.
The mechanical and geometric properties for this problem are I= 0.01042in4,L= 100in,
A= 0.5in2,E= 1.2×104psi.
In Fig. 3 our solution is compared with the one using twenty elements. As mentioned earlier,
with the element proposed by Bathe & Bolourchi [28], which is based on large-displacement
and large-rotation assumptions, the response starts to diverge from the exact solution at an angle
of 90 degrees rotation, irrespectively of the mesh density. Our proposed formulation is instead
able to capture the response for the whole loading scenario (360 degrees), as can be seen from
the line that represents the normalized displacement φ/2π.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Bathe & Bolourchi
Bathe & Bolourchi
Figure 3: Cantilever with point moment at its free end.
Charilaos M. Lyritsakis, Charalampos P. Andriotis, Konstantinos G. Papakonstantinou
Example 4 - Cantilever beam with eccentric compressive point load
We consider a cantilever beam of length L= 100, cross section thickness b= 1 and elastic
modulus E= 12, with the load parameter λ=P/Pcr is increased up to 4.0. The critical load
for the cantilever is Pcr = 0.25π2EI/L2. Wood & Zienkiewicz [31] used five continuum-based
elements that allow for shear deformation and employed a total Lagrangian formulation. The
results are illustrated in Fig. 4. Analytical solutions to the problem, provided in [23, 32] where
it is assumed no axial or shear deformation occurs, show negligible discrepancy compared to
the ones proposed here and in [31]. It should be noted that the eccentricity is =b/2. The
configurations for each load step for Examples 2,3 and 4 are depicted in Fig. 5, from left to
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Zienkiewicz & Wood
Zienkiewicz & Wood
Figure 4: Cantilever with eccentric axial load at its free end.
Figure 5: Configurations at each step for examples 2, 3 and 4 (left to right).
Charilaos M. Lyritsakis, Charalampos P. Andriotis, Konstantinos G. Papakonstantinou
An extension to the geometrically exact hybrid element derived in [16] is presented herein that
accounts for shear deformations. The system of equilibrium equations is originally derived
within a nonlinear programming framework, where the total potential energy functional is dis-
cretized and then augmented by the exact kinematic constraints of the physical problem, also
in discretized form, and solved by determining the stationary point of the Lagrangian. The
suggested nonlinear programming hybrid formulation is capable of capturing the response of
the benchmark problems with accuracy, using only one element, which is a desirable feature
for framed structure applications. Ongoing work explores a variety of different approaches as
far as the optimization algorithms are concerned, which is something the proposed formulation
supports and enables, the extension of the material yield rule to account for the interaction of
shear and axial stresses, as well as the extension to spatial and dynamic formulations.
This material is based upon work supported by the National Science Foundation under Grant
No. 1634575. The authors gratefully acknowledge this support.
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This work presents a hybrid shear‐flexible beam‐element, capable of capturing arbitrarily large inelastic displacements and rotations of planar frame structures with just one element per member. Following Reissner’s geometrically‐exact theory, the finite element problem is herein formulated within nonlinear programming principles, where the total potential energy is treated as the objective function and the exact strain‐displacement relations are imposed as kinematic constraints. The approximation of integral expressions is conducted by an appropriate quadrature, and by introducing Lagrange multipliers, the Lagrangian of the minimization program is formed and solutions are sought based on the satisfaction of necessary optimality conditions. In addition to displacement degrees of freedom at the two element edge nodes, strain measures of the centroid act as unknown variables at the quadrature points, while only the curvature field is interpolated, to enforce compatibility throughout the element. Inelastic calculations are carried out by numerical integration of the material stress‐strain law at the cross‐section level. The locking‐free behavior of the element is presented and discussed, and its overall performance is demonstrated on a set of well‐known numerical examples. Results are compared with analytical solutions, where available, and outcomes based on flexibility‐based beam elements and quadrilateral elements, verifying the efficiency of the formulation.
Modern structural analysis necessitates numerical formulations with advanced nonlinear attributes. To that end, numerous finite elements have been proposed, spanning from classical to hybrid standpoints. In addition to their individual features, all formulations originally stem from an underlying variational principle, which can be deemed as a unified energy metric of the system. The corresponding equations of structural equilibrium define a stationary point of the assumed principle. Following this logic in this work, the total potential energy is directly treated as an objective function, subject to some kinematic compatibility constraints, within the conceptions of nonlinear programming. The only approximated internal field is curvature, whereas displacements occur solely as nodal entities and Lagrange multipliers serve compatibility. Thereby, a new nonlinear programming hybrid element formulation is derived, which uses exact kinematic fields, can incorporate nonlinear assumptions of any extent, and is amenable to various applicable nonlinear programming algorithms. The suggested nonlinear program is presented in detail herein, together with its consistent second-order iterative solution procedure. The results obtained in benchmark nonlinear structural problems are validated and compared with OpenSees flexibility-based elements, showcasing notable performance in terms of accuracy, mesh density discretization, computational speed, and robustness.
This paper extends the gradient inelastic (GI) beam theory, introduced by the authors to simulate material softening phenomena, to further account for geometric nonlinearities, and formulates a corresponding force‐based (FB) frame element computational formulation. Geometric nonlinearities are considered via a rigorously derived finite‐strain beam formulation, which is shown to coincide with Reissner's geometrically nonlinear beam formulation. The resulting finite‐strain GI beam theory: (i) accounts for large strains and rotations, unlike the majority of geometrically nonlinear beam formulations used in structural modeling that consider small strains and moderate rotations, (ii) ensures spatial continuity and boundedness of the finite section strain field during material softening via the gradient nonlocality relations, eliminating strain singularities in beams with softening materials, and (iii) decouples the gradient nonlocality relations from the constitutive relations, allowing use of any material model. On the basis of the proposed finite‐strain GI beam theory, an exact FB frame element formulation is derived, which is particularly novel in that, it: (a) expresses the compatibility relations in terms of total strains/displacements, as opposed to strain/displacement rates which introduce accumulated computational error during their numerical time integration, (b) directly integrates the strain–displacement equations via a composite two‐point integration method derived from a cubic Hermite interpolating polynomial to calculate the displacement field over the element length, and, thus, address the coupling between equilibrium and strain–displacement equations. This approach achieves high accuracy and mesh convergence rate, and avoids polynomial interpolations of individual section fields, which often lead to instabilities with mesh refinements. The FB formulation is then integrated into a Co‐rotational framework and is used to study the response of structures, simultaneously accounting for geometric nonlinearities and material softening. The FB formulation is further extended to capture member buckling triggered by minor perturbations of the initial member geometry.
In the article, we introduce an analytical solution for a large-deflection, finite-strain, and planar Reissner beam that is subject to an end force and a bending moment. The solution is provided in terms of Jacobi elliptic functions. The obtained analytical solution is enhanced with numerical examples. The buckling and post-buckling behavior of a beam under an axial compressive load applied at its end and subject to various boundary conditions is also discussed in detail. In particular, the buckling factor is derived for all boundary conditions.
In this work, a new smooth model for uniaxial concrete behavior that combines plasticity and damage considerations, together with unsymmetrical hysteresis for tension compression and nonlinear unloading, is presented. Softening and stiffness degradation phenomena are handled through a scalar damage-driving variable, which is a function of total strain. Smoothening of the incremental damage behavior is achieved, following similar steps as for Bouc-Wen modeling of classical plasticity, thus exploiting their common mathematical structure. The uniaxial model for concrete, together with the standard steel model exhibiting kinematic hardening, are employed to derive a fiber beam-column element that is used to assemble the numerical model of frame structures. Following the displacement-based approach, the solution of the entire system is established using a standard Newton-Raphson numerical scheme, which incorporates the evolution equations of all fibers elevated at the section, element, and structural level in the inner loop. Numerical results that compare well with existing experimental data are presented, demonstrating the accuracy and efficacy of the proposed formulation.
A nonlinear finite-element formulation for the static and dynamic behavior of flexible beams was developed by appropriately modifying and extending the three-dimensional (3D) finite-deformation beam model originally developed by Simo. By introducing energy dissipation in a physically consistent way through a linear viscoelastic constitutive equation, the main contribution in this paper lies in the derivation of a tangent stiffness operator that includes the effect of damping. Moreover, a solution to issues concerning the interpolation of total rotation vectors of magnitude greater than pi is proposed, along with an alternative approach for the update of curvatures based on total rotation vectors, taking advantage of special features of Lie groups and of the notion of right-trivialized derivative. Both two-dimensional (2D) and three-dimensional (3D) numerical examples are presented. In particular, static and dynamic analyses of an electrical conductor commonly used in power substations were performed. Energy balance calculations and the convergence rate of Newton's method illustrate the accuracy of the computed solutions. (C) 2014 American Society of Civil Engineers.
The classical problem of large displacements of thin curved beams is considered.