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://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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