Post on 19-May-2018
Topology Optimization of I-Section Beams with Web Openings
Karina Rocha¹, Anderson Pereira² and Rodrigo Bird Burgos¹
¹State University of Rio de Janeiro, Rio de Janeiro, Brazil
karina.mota.kmr@gmail.com, rburgos@eng.uerj.br
²Pontifical Catholic University of Rio de Janeiro, Brazil
anderson@tecgraf.puc-rio.br
Abstract
The placement of openings within the web of I-section beams has been employed in structural design for over 100 years to improve
their mass-to-stiffness ratio, enabling the use of longer spans and eliminating the probability of cutting holes in inappropriate locations
for electric, hydraulic and air conditioning installations. Castellated and cellular beams, with or without reinforcement, are the most
used types of perforated beams. Elliptical and sinusoidal openings have been recently studied. The fabrication of I-sections using the
plate assembly technique increases the number of options for positions and shapes of the openings. This “removal” of material that
creates web openings can be looked at from a topology optimization point of view. One of the goals of this work is to obtain different
web opening configurations using structural topology optimization. The bending stiffness reduction due to the openings is not very
significant, since the contribution of the web to the moment capacity is very small. Since transversal forces are usually resisted by the
web, the reduction in shear capacity at the opening can be significant. It is also necessary to check the possibility of the beam’s lateral
torsional buckling. In order to study the structural performance of the obtained configuration in comparison to a beam with circular web
openings (cellular), Finite Element Analyses were performed. The beam was optimized for two different boundary conditions and the
obtained configurations were subjected to a reference distributed loading. Results were compared for those configurations and the
cellular beam.
Keywords: Topology optimization, Castellated beams, Cellular beams, Web openings.
1. Introduction
A common type of a steel beam with web openings is the castellated beam where openings occur at regular intervals as shown in Fig. 1.
Such beams are formed from laminates of the I-profiles which are cut longitudinally in the web and pooled forming openings in
circular, hexagonal or square shapes. Such openings improve their mass-to-stiffness ratio, allowing the use of longer spans.
Figure 1. Examples of castellated beams [1,2]
The shapes of the web openings are usually regular, such as circular, rectangular or hexagonal. This shape of holes only takes into
consideration the manufacturing process regardless the stress distribution or the collapse behavior. Regions with high stress
concentrations indicate that they may be prone to failure, while regions with very low stresses indicate the existence of underutilized
material (Fig. 2). Thus, an efficient design of the web openings can be looked at from a topology optimization point of view where a
new format for the holes in the web of castellated beams can be achieved.
New fabrication techniques of I-section beams increase the options for the positions and shapes of the openings (Fig. 3). The plate
assembly technique for instance offers significantly increased design freedom in terms of the shape and layout of web openings, which
can be cut in various sizes and shapes. This "removal" of material that creates web openings can be looked at from topology
optimization a point of view.
Figure 2. Cellular beam subjected to bending
Figure 3. Laser beams cutting [3]
3. Topology optimization
Topology Optimization seeks to find the best layout for a structure by optimizing the material distribution in a predefined design
domain. There are a number of objectives functions used in topology optimization, but here we focus our attention in compliance
minimization problems. In such problems, the optimal solution provides the stiffest structure for a defined set of loads, and uses a
constraint on the volume on the structure. The discrete topology optimization problem can be written as follows:
min𝜌
𝑐(𝜌(𝒙), 𝒖) = 𝒇𝑇𝒖 (1)
s. t. 𝑉(𝜌(𝒙)) = ∫ 𝜌(𝒙)𝑑𝑉 ≤ 𝑉𝑆Ω
(2)
with 𝑲(𝜌)𝒖 = 𝒇, (3)
where 𝒇 and 𝒖 are the global force and displacement vectors, 𝑉 is the volume as a function of the densities, and 𝑉𝑆 is the prescribed
volume fraction. The material distribution 𝜌 for every point 𝒙 is defined as follows:
𝜌(𝒙) = {0, if void1, if structural member
(4)
A continuous density is desired in order to use gradient based optimization algorithms. The set of admissible densities is relaxed to
allow the appearance of intermediate values, ranging from 0 to 1, leading to some sort of “grey regions". A penalization technique is
used to ensure the solution is directed towards solid/void results. A commonly used approach in topology optimization is the Solid
Isotropic Material with Penalization (SIMP) model, first described by Zhou and Rozvany [4] and extensively analyzed by Bendsøe [5]
and references therein. In the SIMP method, the relationship between the density 𝜌(𝒙) and the material tensor 𝑪(𝒙) in the equilibrium
analysis is written as:
𝑪(𝒙) = [𝜌(𝒙)]𝑝𝑪0, (5)
where 𝑝 is the SIMP penalization factor (𝑝 ≥ 1) and 𝑪0 is the material tensor (constitutive matrix) for the solid phase, corresponding
to 𝜌 = 1. In this work, an “element-based" approach is used so every finite element in the domain is assigned to a constant design
variable.
Region of high stress
concentration
Region of low stress, underutilized
material
3.1. Numerical implementation
The implementation is done in MATLAB using as a starting point the educational code PolyTop (Talischi et al. [6,7] and Pereira et al.
[8]). In this work we expanded in order to consider three-dimensional problem and incorporate non design regions. Owing to the
modular structure of PolyTop, the extension for three dimensional problems involves changes that are limited mainly to the analysis
routine. The finite element formulation makes use of eight-node hexahedral elements so the basis function construction and element
integration routines are changed. The three-dimensional constitutive matrix for an isotropic element 𝑒 is given by
𝑪𝑒0 =
𝐸
(1 + 2𝜈)(1 − 2𝜈)
[ 1 − 𝜈 𝜈 𝜈 0 0 0
𝜈 1 − 𝜈 𝜈 0 0 0𝜈 𝜈 1 − 𝜈 0 0 0
0 0 01 − 2𝜈
20 0
0 0 0 01 − 2𝜈
20
0 0 0 0 01 − 2𝜈
2 ]
(6)
where 𝐸 is the Young’s modulus and 𝜈 is the Poisson’s ratio. The elastic element stiffness matrix is given by
𝒌𝑒 = ∫ ∫ ∫ 𝑩𝑇𝑪(𝒙)+1
−1
+1
−1
+1
−1𝑩𝑑𝜉1𝑑𝜉2𝑑𝜉3, (7)
where 𝜉𝑒 (𝑒 = 1,… ,3) are the natural coordinates and 𝑩 is the strain–displacement matrix. For a deeper discussion on the finite
element method, the reader is referred to [9,10].
4. Numerical Examples
Initially, a topology optimization study was performed on the web of a steel I-section beam with a 5 meters span, for two different
boundary conditions: simply supported and doubly clamped. Both were optimized for a distributed load case, as shown in Fig. 4. A
typical section was selected on the basis that it is a fairly common section to find in practice and mainly in building applications. The
target volume fraction was chosen so that the optimized beam would have the same volume as the cellular one. The structural behavior
of the optimized beam was then compared to a similar beam with circular web openings, by carrying out an elastic linear FE analysis.
The topology optimization studies were performed using MATLAB, obtaining the structures shown in Figs. 5 and 6. Then, the results
were interpreted using the AutoCAD software, where lines were drawn and all dimensions of the structure were measured, leading to
the structures shown in Figs. 7 and 8. The comparative FE analysis studies were performed using ABAQUS software, for a reference
total load of 100 kN. This example is similar to what is done in [11]. For each optimized beam, two boundary conditions were used:
simply supported and doubly clamped.
Figure 4. Boundary conditions, loads and cross section analyzed
Figure 5. Result of topology optimization for doubly clamped boundary condition
Figure 6. Result of topology optimization for simply supported boundary condition
Figure 7. Model of optimized beam in ABAQUS software (doubly clamped)
Figure 8. Model of optimized beam in ABAQUS software (simply supported)
In all cases, the cellular and optimized beams were subjected to a distributed load equivalent to 100 kN and discretized in a mesh of
approximately 30,000 quadratic elements. Material parameters are Young's modulus of 200 GPa and Poisson's ratio equal to 0.3. The
results obtained for the distribution of von Mises stress is shown in Figs. 9 and 10. It is evident the maximum stresses in the cellular
beams are higher than in the optimized ones. Additionally, after the finite element analysis the maximum displacements found for the
cellular beam also higher than the optimized ones. The exceptions were the cases in which the beam was subjected to different
boundary conditions than the ones they were optimized for. A summary of those results is shown in Tab. 1.
Figure 9. Results for beams subjected to simply supported boundary condition
Figure 10. Results for beams subjected to doubly clamped boundary condition
Table 1. Summary of results in terms of stresses and displacements
Analysis case Beam model Maximum von
Mises Stress
Maximum
displacement
Simply supported
Cellular 374.6 MPa 14.3 mm
Optimized as simply supported 301.2 MPa 12.5 mm
Optimized as doubly clamped 391.3 MPa 14.73 mm
Doubly clamped
Cellular 375.3 MPa 6.14 mm
Optimized as simply supported 214.0 MPa 4.15 mm
Optimized as doubly clamped 288.0 MPa 3.84 mm
5. Final Remarks
Topology optimization is a powerful tool allowing a design freedom. It is widely used in mechanical engineering industry and now
spreading to civil engineering. The optimized beams present fewer stress concentration regions in comparison with the cellular one
(with the same volume/weight), which makes it less susceptible to failure; Furthermore, the optimized beam also presents smaller
central displacement when compared to the cellular one (considering the same boundary condition it was optimized for).
It is clear how structural topology optimization can lead to strength gains just by finding the best placement for the material, with no
need to increase the structure total weight; There is still a wide range of studies to be done on this subject, among which: optimization
of steel beams for other boundary and loading conditions; evaluation of lateral and/or torsional buckling; use of shell elements instead
of polygonal; complete nonlinear analysis of the cellular and optimized beams, for comparisons regarding post critical behavior.
Fabricated beams where three plates are welded together to form an I-section could be also optimized.
6. References
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http://www.framestudio.cl/clientes/bming/newsletter-02/noticia-4.htm. Accessed 24 August 2016.
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http://techne.pini.com.br/engenharia-civil/164/artigo286765-1.aspx. Accessed 24 August 2016.
[3] CMM Laser. Available: http://www.tagliolaser.net/en/gallery-laser-services/drilling-coping-i-beams/index.html. Accessed 22
August 2016.
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Methods in Applied Mechanics and Engineering, vol. 89, pp. 309-336, 1991.
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