Topology Optimization for Conceptual Design of Reinforced Concrete Structures

Topology Optimization for Conceptual Designof Reinforced Concrete Structures

Oded Amir∗ and Michael Bogomolny∗∗

∗ Technical University of Denmark, Department of Mechanical Engineering∗∗ Plasan LTD, Research and Development Department

WCSMO-9, Shizuoka, Japan, June 14 2011

Topology Optimization for Conceptual Design of Reinforced Concrete Structures 1/18

Topology optimization as an architectural design tool

The facade of the Qatar convention center, designed by Arata Isozakiand Mutsuro Sasaki:

DTU Mechanical Engineering, Technical University of Denmark

Applications in Architecture/Design


Topology optimization as an architectural design toolProject UNIKABETON (2010):

DTU Mechanical Engineering, Technical University of Denmark

Applications in Architecture/Design

Stromberg et al. (2011):

176 L.L. Stromberg et al.

stress trajectories for the cantilever beam problem consid-ered (see Fig. 12a). One set of lines represent compressionlines while the other set represents tension lines. The trajec-tories are acting as streamlines such that the lateral windforce ‘‘enters’’ the continuum at a certain location alongthe height and flows through the trajectories to the foun-dation (this is due to the non-shear condition along theselines). Since the principal stress trajectories represent thenatural flow of forces in the structure, they offer an ana-lytical method to identify the optimal layout of structuralmaterial in a high-rise. The optimality comes from the ideaof understanding how the forces are ‘‘moving’’ through thestructure to the foundation and embrace this flow with thestructural members.

The principal stress trajectories in Fig. 12a show the fol-lowing important characteristics in relation to the behaviorof high-rise buildings:

• the tension and compression lines meet at a 45◦ angle atthe centerline-----in beam theory there is a state of pureshear stress at the centerline.

• the stresses at the beam edges are vertical because theproblem is purely axial. It can be noted how the linesbecome very dense toward the edges, emphasizing how

in a high rise the most efficient way to carry the over-turing moment is to put material as far away as possiblefrom the neutral axis.

• the trajectories tend to be more vertical toward thebase of the cantilever and closer to 45◦ bracing towardthe top. This is caused by the fact that at the topthere is mainly shear-type loading while the bottomof the cantilever is controlled by the overturningmoment.

The principal stress analysis conducted in two dimen-sions can be extended to three dimensions assuming acantilever beam with the cross section of a hollow tube (seeFig. 12b). In this case, when the wind is blowing orthogo-nally to one of the tube faces, the side of the tube parallel tothe wind directions are behaving similarly to the two dimen-sional problem while, in the sides orthogonal to the winddirection, the stress trajectories are mainly vertical. Thisresult emphasizes the typical behavior of a tubular highrise structure which behaves similarly to a simple I-beamsection. The faces of the tube orthogonal to the wind direc-tion are acting as flanges and mainly carry the overturningmoment, while the faces of the tube along the wind directionare carrying the shear force.

Fig. 13 Illustration of theconcept of pattern gradationalong the height of a building:a pattern gradation constraints;b topology optimization resultwith similarities to c JohnHancock Center in Chicago, IL(taken from en.wikipedia.org/wiki/John_Hancock_Center)

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70

90

110

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a b cTopology Optimization for Conceptual Design of Reinforced Concrete Structures 3/18

Topology optimization of reinforced concrete

Potential applications:Conceptual design phase - enhance collaboration betweenarchitects and engineers.Optimize the utilization of concrete - reduce weight -> materialconsumption -> CO2 emissions: reinforced/plain, heavy/light.Optimize the distribution of steel reinforcement in non-standardstructural elements.

The challengeIn the context of structural topology optimization:consider anisotropic nonlinear material behavior.


Elasto-plastic approachThe main components of the proposed procedure:

SIMP-type interpolation of nonlinear elasto-plastic properties,including interpolation of yield criteria.Adjoint sensitivity analysis with no simplifying assumptions.Hybrid approach to sensitivity analysis, enabling to optimizecompliance for distributed loads while using displacementcontrol.

Recent related work:Kato et al. 2009, Kato and Ramm 2010: optimization of fiberreinforced concrete.Liu and Qiao 2011: topology optimization with different tensileand compressive properties.Liang et al. 2000, Kwak and Noh 2006, Bruggi 2009: optimizedstrut-and-tie layouts (linear, elastic, isotropic modeling).


Distribution of elasto-plastic material

Steel is assumed to obey the von-Mises yield criterion;Concrete is assumed to obey the Drucker-Prager yield criterion;Intermediate topology optimization densities represent an artificialmixture.

the von Mises yield criterion which is widely used for metals (having equal strength intension and compression) can be seen as a particular case of the Drucker-Prager criterion.

(a) von Mises (b) Drucker-Prager

Figure 3: Yield surfaces in 2D principal stress space.

As a demonstrative case we focus throughout this article on the distribution of concreteand steel. In essence, the purpose of utilizing nonlinear modeling is to identify the failureof concrete when tension stresses appear and then redistribute material so that such failuredoes not occur. Other aspects of the elasto-plastic behavior, namely yielding of steel inboth stress states or yielding of concrete in compression, are categorized as less importantfor the purpose of this study. Therefore some simplifying assumptions are made in theformulation of the nonlinear material model, which would not be allowed if the purposewas accurate prediction of failure and damage in reinforced concrete structures.

In the following, we present the governing equations of the elasto-plastic model, leadingto the local constitutive problem to be solved on a Gauss-point level. We follow classicalrate-independent plasticity formulations, based on the textbooks by Simo and Hughes[25] and Zienkiewicz and Taylor [29]. The Drucker-Prager yield function can be expressedas

f(σ, κ) =√

3J2 + α(κ)I1 − σy(κ) ≤ 0

where J2 is the second invariant of the deviatoric stress tensor and I1 is the first invariant(trace) of the stress tensor. α is a material property and σy is the yield stress in uniaxialtension, both functions of the internal hardening parameter κ according to some hardeningfunctions. The expression

√3J2 is usually known as the von Mises stress or equivalent

stress. When α = 0, we obtain the von Mises yield criterion. We assume simple isotropichardening rules

α(κ) = constant (2)

σy(κ) = σ0y +HEκ (3)

where σ0y is the initial uniaxial yield stress, E is Young’s modulus and H is a constant,

typically in the order of 10−2. The assumptions (2) and (3) are not necessarily suitable foraccurate modeling of concrete but do not affect the ability to capture the most importantfailure in concrete, that is failure in tension. We assume an associative flow rule and a

5

Figure 4: Demonstrative example of the interpolation betweentwo yield surfaces, presented in 2D principal stress space. The“Hybrid” surface represents the behavior of an artificial mixture,corresponding to an intermediate density in topology optimiza-tion.

the interpolation of the yield surfaces is demonstrated, for two materials resembling steeland concrete.

In order to approach optimal strut-and-tie designs, we extend this interpolation so itaccommodates also void regions. Following [3], we add another design variable x for eachfinite element. Void regions are represented by x = 0 and solid regions are representedby x = 1. Within the solid regions, the value of ρ determines the distribution of the twocandidate materials. This leads to the following interpolation functions, replacing Eqs.(9), (10), (11), (12)

E(xe, ρe) = xpExe (Emin + (Emax − Emin)ρpEe ) (13)

f(σ, λ, ρe, xe) =√

3J2 + α(xe, ρe)I1 − σy(λ, xe, ρe) = 0 (14)

α(xe, ρe) = xpαxe (αmax − (αmax − αmin)ρpαe ) (15)

σy(λ, xe, ρe) = xpσxe (σ0y,min + (σ0

y,max − σ0y,min)ρ

pσye +HE(xe, ρe)λ) (16)

where pEx, pαx and pσx are penalization factors for x. In practice, one may choose to usethe same penalty factors for both design variables, x and ρ.

4.2 Optimization problem and sensitivity analysis

In this article, we focus mainly on one demonstrative class of objective functions. Theaim is to find the stiffest structural layouts given certain amounts of available material.When only linear elastic response is considered, the corresponding objective is the widelyused minimum compliance problem, presented above (1). When nonlinear response istaken into account, one may define several different objectives that are related to themaximization of the structural stiffness (see for example [28], [18], [7]). Since displacementcontrol is preferred in the nonlinear FE analysis, a possible equivalent to minimizingcompliance in linear elasticity is maximizing the end compliance for a given prescribed

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Distribution of elasto-plastic materialNonlinear material properties are a function of the density variable ρ.

Elastic modulus (Sigmund and Torquato 1997) :

E (ρe) = Emin + (Emax − Emin)ρpEe

Yield function:

f (σ, λ, ρe) =√3J2 + α(ρe)I1 − σy (λ, ρe)

Pressure-dependency of the strength:

α(ρe) = αmax − (αmax − αmin)ρpαe

Yield stress (Maute et al. 1998) :

σy (λ, ρe) = σ0y ,min + (σ0

y ,max − σ0y ,min)ρ

pσye + HE (ρe)λ


Problem formulation - maximize end-complianceDistributing steel within a solid concrete domain, point load:

minρ

c(ρ) = −θNp f puN

s.t.:Ne∑

e=1veρe ≤ V

0 ≤ ρe ≤ 1, e = 1, ...,Ne

with: Rn(vn, θn) = 0 n = 1, ...,NHn(un,un−1, vn, vn−1,ρ) = 0 n = 1, ...,N

For distributing concrete, steel and void: another element design variable xeand two volume fractions

∑Nee=1 vexe ≤ V1,

∑Nee=1 veρe ≤ V2.

For distributed loads, the objective −θN fT uN does not give “the stiffeststructure” - requires special treatment.


Adjoint sensitivity analysisThe augmented objective functional

c(ρ) = −θNp f puN −

N∑n=1

λTn Rn(vn, θn)

−N∑

n=1γT

n Hn(un,un−1, vn, vn−1,ρ)

The explicit design sensitivities

∂cexp∂ρe

= −N∑

n=1γT

n∂Hn∂ρe

Due to path dependency, a backwards-incremental procedure is requiredfor computing the adjoint variables (following the framework byMichaleris et al. 1994).


Adjoint sensitivity analysis

Typical increment: solve the coupled adjoint equations to determine nλ[−∂(nR)

∂(nv)∂(nH)

∂(nv)

−1 ∂(nH)

∂(nu)

]Tnλ =

∂c∂(nu)

T

−[∂c∂(nv)

∂(nH)

∂(nv)

−1 ∂(nH)

∂(nu)

]T

−[∂(n+1H)

∂(nu) − ∂(n+1H)

∂(nv)∂(nH)

∂(nv)

−1 ∂(nH)

∂(nu)

]Tn+1γ

∂(nR)

∂(nθ)

Tnλ =

∂c∂(nθ)

Compute the local adjoint vector nγ on a Gauss-point level

∂(nH)

∂(nv)

Tnγ = −∂(nR)

∂(nv)

Tnλ − ∂(n+1H)

∂(nv)

Tn+1γ +

∂c∂(nv)

T


Hybrid approach for displacement-controlled objectives

In a load-controlled setting:

Design 1

Optimized

design

u

f

Design k

updateupdatefp



In a displacement-controlled setting:

Design 1

Optimized

design

u

f

up

Design k

update

update



In a hybrid setting:

Design k-1

Optimized

design

u

f

up

Design k+1

update

Design kupdate

fk-1

fk


Implementation

Nonlinear finite element analysis:Global nonlinear equations solved by a Newton-Raphson procedurewith displacement control and automatic incrementation.Local nonlinear elasto-plastic problems solved implicitly afterassuming an elastic trial stress.Simplified flow and hardening rules.

Topology optimization:Regularization by a density filter (Bruns and Tortorelli 2001; Bourdin 2001) .Design update by MMA (Svanberg 1987) .


Example: ss-beam, 90% concrete, 10% steel

?

f

8

1

(a) Design domain and boundary conditions.

(b) Optimized layout after 150 design iterations with gradual refinement.

Figure 9: Maximum end-compliance of a simply supported beamsubject to a distributed load. Black = steel, white = concrete.Steel consists of 10% of the total volume.

?

f

7.2

1 ? ?

2.42.4



Figure 10: Maximum end-compliance of a cantilevered beam sub-ject to a distributed load. Black = steel, white = concrete. Steelconsists of 10% of the total volume.

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Example: cantilevered beam, 90% concrete, 10% steel

?

f

8

1



Figure 9: Maximum end-compliance of a simply supported beamsubject to a distributed load. Black = steel, white = concrete.Steel consists of 10% of the total volume.

?

f

7.2

1 ? ?

2.42.4



Figure 10: Maximum end-compliance of a cantilevered beam sub-ject to a distributed load. Black = steel, white = concrete. Steelconsists of 10% of the total volume.

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Example: short cantilever

?

δ{

f

1

1

(a) Design domain, boundary conditions and pre-scribed displacement.

(b) Optimized layout after 500 design iter-ations, 80% concrete, 20% steel. Black =steel, gray = concrete.

(c) Optimized layout after 200 design iter-ations, 30% concrete, 10% steel, 60% void.Black = steel, gray = concrete, white =void.

Figure 12: Maximum end-compliance of a short cantilever

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Example: clamped beam, 32.5% concrete, 7.5% steelprescribed displacement directed downwards at the middle of the bottom edge. The modelof the symmetric half is discretized with a 120 × 80 FE mesh. Further details regardingthe numerical implementation are given in Table 2.

(a) Design domain, boundary conditions and prescribed displacement.

(b) Optimized layout: 32.5% concrete, 7.5% steel, 60% void. Black =steel, gray = concrete, white = void.

Figure 9: Maximum end-compliance of a thick clamped beam

Examining the resulting design, it can be seen that the optimization process leads toa truss/frame layout where steel is used for members in tension and concrete for membersin compression. In this particular case, steel is used to transfer the hanging load to acompressed concrete arch. In essence, such layouts resemble strut-and-tie models that arewidely used in practical analysis and design of reinforced concrete. As in the previousexample, the support regions are also enforced with steel in compression. This is probablydue to the high elastic modulus of steel and the fact that the high cost of steel is notconsidered in the current study.

One of the difficulties encountered in this study is in obtaining a 0-1 distribution inregions where tension is transferred from steel to concrete. In the optimized layout of

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Example: ss-beam, 35% concrete, 5% steel

Figure 5: Maximum end-compliance of a simply supported beam: Design domain, boundary conditions andprescribed displacement.

The layouts obtained for the simply supported beam resemble actual design of reinforcedconcrete beams. Away from the supports, bending action is dominant so steel is necessaryin the bottom fibers where tension stresses appear. Near the supports, shear forces aredominant so concrete typically cracks in an angle of 45◦, corresponding to the direction ofthe principal stresses in pure shear. Consequently, the steel reinforcement should be bentin order to accommodate the tensile stresses due to shear.

Examining the results of the optimization procedure, it is evident that the distributionof material is strongly influenced by the choice of penalization factors. For standard pe-nalization (Figure 6(a)) steel is positioned mainly in the bottom fiber but also in the topfiber, where it is not absolutely necessary. This is because the objective is to maximizethe stiffness regardless of the cost. Near the supports, no clear layout is obtained but it isevident that the optimization drives the design towards including a bent steel bar. Raisingthe penalty factors (Figure 6(b)) results in a similar layout but with distinct steel barsin shear-dominated regions. Finally, raising the penalty factors once more (Figure 6(c))leads to the distribution of more steel for shear reinforcement while eliminating the steelin the top fiber. In any case, the importance of incorporating nonlinear material modelingis evident when comparing the designs to that obtained for the standard reinforcementproblem formulation using linear modeling (Figure 2).

The convergence of the objective values for the three cases is plotted in Figure 7. Itcan be seen that the higher penalty factors lead to higher local minima points. On theother hand, the best performing structure is not a 0-1 design from a topological point ofview. Further study of the procedure is required in order to suggest the most appropriatepenalization scheme so that a distinct topology is generated, which also represents the bestperforming structure.

When distributing concrete, steel and void we obtain a strut-and-tie model, wherethe beam is seen as a truss structure with concrete bars in compression and steel bars

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Towards realistic modelingLimitations of the approach:

Difficult to accommodate small volume fractions of the steel phase.Strain softening of the concrete is not considered - difficult toachieve optimized strut-and-tie layouts.

Current work in progress:Replace elasto-plasticity with continuum damage modeling.Consider anisotropic reinforced concrete: mixing theory,homogenization, other approaches.

3

2a 2

a 1

l

C1

C2

Cm

ε∗

ε∗

∂V

Figure 2: Representative volume element (RVE).

3.3 Micro-macro mappingThe local strain and stress fields within the RVE areaveraged over the total volume V of the RVE in orderto evaluate the homogenized values of the strains

< ε >V =1

V

∫

Vε(x)dV =

n∑

i=1

ci < ε >i (12)

and of the stresses

< σ >V =1

V

∫

Vσ(x)dV =

n∑

i=1

ci < σ >i . (13)

Since the local averaged field values are assumedto be constant within each phase (σi =< σ(x) >i

and εi =<ε(x)>i), they can be summed up accord-ing to the volume fraction ci = Vi/V , whereby Vi isthe total volume of the phase iwithin the RVE. Hence,the volume of the RVE of the considered three-phasecomposite is assumed to be filled completely by allphases i.e. c1 + c2 + cm = 1. According to the aver-age strain theorem, for any perfectly bonded hetero-geneous body the averaged strains < ε >V given byequation (12) can be identified as the macroscopicstrain tensor ε∗ applied on the RVE i.e. < ε >V = ε∗,which is independent of the considered constitutivelaws (Zohdi and Wriggers 2005). For a three-phasecomposite, a reformulation of equation (12) and (13)leads to

ε∗ =< ε >V = c1 ε1 + c2 ε2 + cm εm (14)

describing the homogeneous macroscopic strains ap-plied onto the composite material and to

σ∗ =< σ >V = c1 σ1 + c2 σ2 + cm σm (15)

for the macroscopic stresses representing the compos-ite stress field. The related local stress tensor σi(εi)is calculated according to each constitutive law givenby equations (6) and (8). The unknown local strainfields εi have to estimated by means of the forth-order

localization (concentration) tensor Ai which relatesthe homogenized macroscopic strains ε∗ to the localstrains within each phase

εi = Ai : ε∗, i = 1,2,m. (16)

The tensor Ai of each phase accounts for the mor-phology of the microstructure by considering the elas-ticity, the volume fraction, the aspect ratio, the orien-tation and the shape of each constituent. It should em-phasized that Ai relates micro and macro quantitiesand depends therefore on the theory chosen for themicromechanical model. If a three-phase compositeis considered only two concentration tensors have tobe known. The third one can be determined from theaverage value

<A >V = c1A1 + c2A2 + cmAm = 11, (17)

with 11 denoting the forth-order unit tensor. Dueto different orientation and shape of the inhomo-geneities, which is captured by Ai, the mechanicalresponse of the related homogenized stiffness tensorC∗ is in general anisotropic even if all constituentsare isotropic. As long as all constituents are in elas-tic regime, the mechanical constitutive relation for acomposite material is defined by

σ∗ = C∗ : ε∗, (18)

where C∗ can be derived from the localization tensorsof each phase

C∗ =< C : A >V =n∑

i=1

ciCi : Ai. (19)

In the post-cracking range of the matrix or in theyielding regime of the rebars, however, the macro-scopic tangent stiffness tensor of the composite C∗,tan

needs to be computed according to

C∗,tan = dσ∗/dε∗. (20)

Depending on the considered micromechanical modelthe macroscopic tangent tensor C∗,tan is obtainedfrom linearization of each constitutive law

Ctani = dσi/dεi i = 1,2,m. (21)

3.4 Three-phase MORI-TANAKA approachAn appropriate homogenization scheme to derive theeffective mechanical response of a RVE is providedby the widely used MORI-TANAKA approach (Moriand Tanaka 1973). This micromechanical model en-sures continuity of the matrix phase and accountsfor mechanical interactions between the inclusions inan average manner. According to this homogeniza-tion scheme, the reference material playing the pre-dominant morphological role of the composite is the

2 Chapter 1. Introduction

y

d

>d

x

y

z

z

composite

material

matrix:

concrete

fibers: steel

matrix

matrix

y

d

>d

x

y

z

z

composite

material

matrix:

concrete

fibers: steel

matrix

matrix

x

y

z

matrix:

concrete

fibers: steel

composite material:

reinforced concrete

x

y

x

y

x

y

zx

y

z

matrix:

concrete

fibers: steel

composite material:

reinforced concrete

x

y

x

y

Figure 1.1. Numerical modeling of reinforced concrete walls subjected to shear.

Figure 1.2. Numerical modeling of reinforced concrete beams.

In continuum mechanics, the phenomenon of strain localization caused by the appear-

ance of a crack can be represented by means of enriched kinematics describing a jump in the

strain field.

If the constitutive model of the composite material considers a common strain field

among the materials that form it, then the kinematics equipped to capture discontinuities in

simple materials can be applied in an equal way in the composite material, this constitutes an

important advantage which is used by the formulation presented here.

2 Chapter 1. Introduction

y

d

>d

x

y

z

z

composite

material

matrix:

concrete

fibers: steel

matrix

matrix

y

d

>d

x

y

z

z

composite

material

matrix:

concrete

fibers: steel

matrix

matrix

x

y

z

matrix:

concrete

fibers: steel

composite material:

reinforced concrete

x

y

x

y

x

y

zx

y

z

matrix:

concrete

fibers: steel

composite material:

reinforced concrete

x

y

x

y

Figure 1.1. Numerical modeling of reinforced concrete walls subjected to shear.

Figure 1.2. Numerical modeling of reinforced concrete beams.

In continuum mechanics, the phenomenon of strain localization caused by the appear-

ance of a crack can be represented by means of enriched kinematics describing a jump in the

strain field.

If the constitutive model of the composite material considers a common strain field

among the materials that form it, then the kinematics equipped to capture discontinuities in

simple materials can be applied in an equal way in the composite material, this constitutes an

important advantage which is used by the formulation presented here.

Images: Rumanus and Meschke 2007; Linero, Oliver and Huespe 2007


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Topology Optimization for Conceptual Design of Reinforced Concrete Structures