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ScienceDirect
Available online at www.sciencedirect.com
Procedia Manufacturing 44 (2020) 489–496
2351-9789 © 2020 The Authors. Published by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Peer-review under responsibility of the scientific committee of the 1st International Conference on Optimization-Driven Architectural Design
10.1016/j.promfg.2020.02.265
10.1016/j.promfg.2020.02.265 2351-9789
© 2020 The Authors. Published by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Peer-review under responsibility of the scientic committee of the 1st International Conference on Optimization-Driven
Architectural Design
Available online at www.sciencedirect.com
ScienceDirect
Procedia Manufacturing 00 (2019) 000–000
www.elsevier.com/locate/procedia
2351-9789 © 2019 The Authors. Published by Elsevier Ltd.
This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)
Selection and peer-review under responsibility of the scientific committee of the International Conference on Optimization-Driven Architectural
Design (OPTARCH2019)
1st International Conference on Optimization-Driven Architectural Design (OPTARCH 2019)
Series solution of beams with variable cross-section
Valerio De Biagi1*, Bernardino Chiaia1, Giuseppe Carlo Marano1, Alessandra Fiore2, Rita
Greco3, Laura Sardone2, Raffaele Cucuzza1, Giuseppe L. Cascella4, M. Spinelli4, and
Nikos D. Lagaros5
1Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino (Italy)
*corresponding author: valerio.debiagi@polito.it
2Politecnico di Bari, DICAR, Via Edoardo Orabona, 4, 70126 Bari (Italy)
3Politecnico di Bari, DicaTech, Via Edoardo Orabona, 4, 70126 Bari (Italy)
4IDEA75 s.r.l., Via Brigata e Divisione Bari, 122 – 70123 Bari (Italy)
5Institute of Structural Analysis and Antiseismic Research, School of Civil Engineering, National Technical University Athens, Heroon
Polytechniou St. 9, Athens 15780, (Greece)
Abstract
In structural engineering beams with non-constant cross-section or beams with variable cross-section represent a
class of slender bodies, aim of practitioners’ interest due to the possibility of optimizing their geometry with respect
to specific needs. Despite the advantages that engineers can obtain from their applications, non-trivial difficulties
occurring in the non-prismatic beam modeling often lead to inaccurate predictions that vanish the gain of the
optimization process. As a consequence, an effective non-prismatic beam modeling still represents a branch of the
structural engineering of interest for the community, especially for advanced design applications in large spans
elements.
A straight beam of length l with variable inertia J(z) is provided in figure, subject to a generic live load condition
q(z). The vertical displacement y(z) can be obtained from the solution of the differential equation of the elastic line,
i.e., taking into consideration the inertia variability and neglecting, as first approximation, any shear contribution.
Even if this solution is an approximate one, it is able to deal with the problem in its basic formulation.
In this paper a solution for the problem stated is formulated using a series expansion of solutions, in a general
load and cross section variability condition. Solution is thus obtained for a generic rectangular cross section beam
with a variable height. Analytical solution is presented and evaluated using numerical evaluation of some cases of
practical interest.
Keywords: Structural Design, Structural Optimization, Parametric Design, Architectural Design
Available online at www.sciencedirect.com
ScienceDirect
Procedia Manufacturing 00 (2019) 000–000
www.elsevier.com/locate/procedia
2351-9789 © 2019 The Authors. Published by Elsevier Ltd.
This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)
Selection and peer-review under responsibility of the scientific committee of the International Conference on Optimization-Driven Architectural
Design (OPTARCH2019)
1st International Conference on Optimization-Driven Architectural Design (OPTARCH 2019)
Series solution of beams with variable cross-section
Valerio De Biagi1*, Bernardino Chiaia1, Giuseppe Carlo Marano1, Alessandra Fiore2, Rita
Greco3, Laura Sardone2, Raffaele Cucuzza1, Giuseppe L. Cascella4, M. Spinelli4, and
Nikos D. Lagaros5
1Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino (Italy)
*corresponding author: valerio.debiagi@polito.it
2Politecnico di Bari, DICAR, Via Edoardo Orabona, 4, 70126 Bari (Italy)
3Politecnico di Bari, DicaTech, Via Edoardo Orabona, 4, 70126 Bari (Italy)
4IDEA75 s.r.l., Via Brigata e Divisione Bari, 122 – 70123 Bari (Italy)
5Institute of Structural Analysis and Antiseismic Research, School of Civil Engineering, National Technical University Athens, Heroon
Polytechniou St. 9, Athens 15780, (Greece)
Abstract
In structural engineering beams with non-constant cross-section or beams with variable cross-section represent a
class of slender bodies, aim of practitioners’ interest due to the possibility of optimizing their geometry with respect
to specific needs. Despite the advantages that engineers can obtain from their applications, non-trivial difficulties
occurring in the non-prismatic beam modeling often lead to inaccurate predictions that vanish the gain of the
optimization process. As a consequence, an effective non-prismatic beam modeling still represents a branch of the
structural engineering of interest for the community, especially for advanced design applications in large spans
elements.
A straight beam of length l with variable inertia J(z) is provided in figure, subject to a generic live load condition
q(z). The vertical displacement y(z) can be obtained from the solution of the differential equation of the elastic line,
i.e., taking into consideration the inertia variability and neglecting, as first approximation, any shear contribution.
Even if this solution is an approximate one, it is able to deal with the problem in its basic formulation.
In this paper a solution for the problem stated is formulated using a series expansion of solutions, in a general
load and cross section variability condition. Solution is thus obtained for a generic rectangular cross section beam
with a variable height. Analytical solution is presented and evaluated using numerical evaluation of some cases of
practical interest.
Keywords: Structural Design, Structural Optimization, Parametric Design, Architectural Design
490 Valerio De Biagi et al. / Procedia Manufacturing 44 (2020) 489–496
2 Valerio De Biagi et al./ Procedia Manufacturing 00 (2019) 000–000
1. Introduction
In structural engineering beams with non-constant cross-section or beams of variable cross-section are represents
a class of slender bodies, object of practitioners’< interest due to the possibility of optimizing their geometry with
respect to specific needs. Beams with variable cross-section are widely adopted in many engineering fields, such as
large span structures (bridges)[1], protective structures [2] or mechanical components [3]. Interests in such type of
structures recently emerged in other disciplines, such as energy harvesting [4]. Evidences of variable cross-section
largely emerge in damaged structures [5],[6],[7]. The modeling of such elements is nontrivial since the variable
properties of the cross-section do not allow for straightforward solutions, event for simple geometries, loads and
boundary conditions [8],[9]. Therefore, numerical tools need to be considered for solving such problems. Recently, a
novel theory based on a simple definition of kinematics, equilibrium and constitutive equations was proposed for
non-prismatic beams by Balduzzi et al. [10]. They showed that in Timoshenko beams the stress distribution of
stresses due to the geometry play a relevant role in the response of the element. In addition, axial and shear-bending
problems are strictly coupled. The solution of their model requires advanced numerical techniques, as reported in
[1]. Other approaches can be found in the literature to solve the variable cross-section beams. Attarnejad et al.
adopted displacement functions coupled with structural matrices [11], Beltempo et al. derived the displacements
though the Hellinger-Reissner functional theory [12], Shooshtari and Khajavi proposed a procedure based on shape
functions and stiffness matrices [13].
Despite the real advantages that practitioners can have from the use of beams with variable cross-section (for
example, tapered beams), the nontrivial procedures and difficulties emerging in the solution vanish the gain of the
optimization process [14]. In this conference paper we propose a simple solution scheme for an Euler-Bernoulli
beam with variable cross section. Despite simple, the model offers interesting possibilities for future optimization
procedures.
The paper is organized as follows: Section 2 details the basic theoretical aspects, Section 3 proposes the solution
scheme based on power series. Section 4 illustrates the problem of applying boundary conditions to the solution and
Section 5 proposes an example.
2. Elastic line for beams with variable cross-section inertia
The elastic line formula for the transversal displacement of beams is a well-known mathematical expression that
originates from Euler-Bernoulli beam theory [15]. Consider a straight beam with variable cross-section inertia
(Fig.1). The length of the beam is denoted as and direction coincides with beam axis. Beam cross-section inertia
is denoted as . The beam is loaded with a variable load, denoted as . The displacement of the beam along
the transversal -axis is related to the other variables as:
where is the Young’s modulus of elasticity. Considering that the function representing the inertia belongs, at least,
to differentiability class, the previous expression can be rewritten as:
Valerio De Biagi et al. / Procedia Manufacturing 44 (2020) 489–496 491
Valerio De Biagi et al./ Procedia Manufacturing 00 (2018) 000–000 3
Fig. 1. Example of a beam with variable height, thus, variable stiffness. The distributed load is not constant along the length of the beam. For sake
of simplicity, the beam has fixed supports at both ends.
3. Power series solution
The partial differential equation describing the transversal displacement at each position z along beam axis can be
solved using various mathematical techniques. As described in the introduction, the solution of the equation can be
performed through numerical integration.
To overcome such problem, we propose a solution based on power series. The idea at the base of the method is
that the lateral displacement can be expressed as
where , with , is a set of constant terms to be determined on the bases of the boundary conditions. The
length of the beam is finite, thus a normalization of the position in the range [0;1] can be done. The power series
expansion of the lateral displacement, thus, becomes
Similarly, the variable inertia and the applied loads are expressed as
and
In case of constant load, is the only nonnull term of the series. The solution scheme consists in deriving the
previous terms and inserting them in a unique expression with the same summation index. Details on how to deal
with power series solutions of differential equations are provided in [16].
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The elastic line differential equation can be rewritten as
where
and I
I
From the previous equation, a general expression for the coefficient can be found:
The previous equation represents the general expression for computing the coefficients of the power series of the
lateral displacement of a straight beam with variable cross-section inertia and variable distributed load.
The coefficient can be evaluated starting from the previously computed terms and from the known values of
and . The values and have to be determined from the boundary conditions.
4. Boundary conditions and coefficients , , and
The boundary conditions allow to determine the values and . For a fourth-order differential equation,
4 boundary conditions have to be determined. In power series solutions adopted in physics, the initial conditions, i.e.
at , are provided. This allows to directly evaluate the unknown four coefficients. In structural mechanics, the
boundary conditions (b.c.) are usually provided for both ends of the beams. The conditions are directly related to the
supports of the beams. For example, the fixed end at can be described through two b.c.:
A simply supported end at can be mathematically described as
Focusing the attention on beams with fixed-fixed boundary conditions, it clearly results that and . The
terms and must be determined from the solution of the following system of equations
Valerio De Biagi et al. / Procedia Manufacturing 44 (2020) 489–496 493
Valerio De Biagi et al./ Procedia Manufacturing 00 (2018) 000–000 5
where are computed through the expression previously mentioned. To solve this problem an optimization
scheme is proposed in which dimensionless kinematic quantities expressing the displacement and the rotations are
minimized.
5. Example
As an example of the approach herein presented, a straight doubly fixed beam is proposed, as sketched in Fig. 2.
The beam has variable cross-section depth defined as a function of beam length
Fig.2. Example of a beam with variable depth and loading.
and a variable distributed load defined as
Dimensionless quantities are adopted following Buckingham’s -theorem. The length of the beam and material
Young’s modulus are chosen as repeating terms. Considering a dimensionless beam width , the
dimensionless inertia is
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In order to determine the coefficients
and
, the optimization procedure previously described was
implemented on Matlab considering a maximum summation index equal to 2000. An unconstrained nonlinear
minimization scheme was adopted to solve the optimization problem. The following parameter vector results
The stem plot of Fig. 3 reports the values of the coefficients for . Double colors are adopted for
marking positive and negative coefficients. Since the y-axis of the plot is in logarithmic scale, the values of the
coefficients decreases with increasing coefficient index, . As an example, the coefficient , referring to the
maximum computed index, is equal to 3.39 x 10-99. Fig. 4 shows the displacement of the beam under variable
distributed load. It is observed that the boundary conditions are satisfied since the displacement and the curvature at
and are null. The forces diagrams, i.e., bending moment and shear force, are reported in Fig. 5.
Fig. 3. Stem plot of the coefficients for . Negative values are marked in red and logarithmic y-axis is adopted.
Valerio De Biagi et al. / Procedia Manufacturing 44 (2020) 489–496 495
Valerio De Biagi et al./ Procedia Manufacturing 00 (2018) 000–000 7
Fig. 4. Plot of the vertical displacements of the beam under the variable distributed load. The maximum displacement /ℓ = 6.59 × 10-4 occurs at
/ℓ = 0.5675.
Fig. 5. Plot of the bending moment and shear force diagrams. The plotted forces are dimensionless.
6. Conclusions
Structures with variable cross-section elements are common in civil and mechanical engineering. The solution of
such elements, i.e., the evaluation of the end reactions, the displacement and the internal forces, is usually obtained
through numerical integration schemes or FEM modeling. We proposed a novel solution scheme based on power
series solution for estimating the end reactions and the displacement of such type of beams. The method integrates
the fourth-order ordinary elastic line differential equation considering the effects of variable inertia. The idea at the
base of the method is that the lateral displacement can be expressed as a summation of terms. Substituting the terms
into the original differential equations, a recurrence formula that allows to compute the coefficients of the power
series can be found. An optimization algorithm is proposed for estimating the initial terms, which depend on the
boundary conditions.
The proposed example, despite simple, clearly shows the capabilities of the method, with reference to the
possible implementations for studying the behavior of structures with known variable inertia [17], components
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subjected to corrosion with variable resisting area [18], or for the optimization of the structural behavior by varying
cross-section depth.
References
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