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Reinforced Concrete Force

Visualization and Design Using

Bilinear Truss-Continuum Topology

Optimization

Cristopher Moen

Journal of Structural Engineering

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Reinforced Concrete Force Visualization and Design Using

Bilinear Truss-Continuum Topology Optimization

Andrew T. Gaynor, S.M.ASCE1; James K. Guest, A.M.ASCE2; and Cristopher D. Moen, M.ASCE3

Abstract: A new force visualization and design tool employing hybrid topology optimization is introduced for RC and prestressed concrete

structural members. The optimization scheme couples a minimum compliance (maximum stiffness) objective function with a hybrid trusscontinuum ground structure that can generate a strut-and-tie model for any general concrete member, loading, and set of boundary conditions.

The truss ground structure represents discrete steel reinforcing bars (tensile load paths) that can be sized based on axial forces output directly by

the optimization routine, whereas the continuum elements simulate concrete compression struts. This separation of compressive and tensile

load-carrying elements is achieved through bilinear elastic models with an orthotropic constitutive relationship for the continuum. Examples

are provided demonstrating the potential value of the optimization tool to RC design. Reinforcing layouts that can minimize cracking and reduce

steel quantities when compared with traditional designs are provided for a prismatic beam, a hammerhead pier, a stepped beam with a cutout,

and the local anchorage zone of a prestressed concrete block. A minimum length scale constraint is employed to control complexity of the strutand-tie topology, accommodating design solutions that balance material savings, structural performance, and constructability. DOI: 10.1061/

(ASCE)ST.1943-541X.0000692. © 2013 American Society of Civil Engineers.

CE Database subject headings: Optimization; Struts; Reinforced concrete; Prestressed concrete; Structural design.

Author keywords: Structural optimization; Topology optimization; Strut-and-tie model; Reinforced concrete; Prestressed concrete; Force

visualization.

Introduction

RC is a complex composite material that continues to challenge

those researchers attempting to describe its behavior with mechanicsbased models. In the late 1800s, Wilhem Ritter and Emil Mörsch

developed a rational engineering approach to circumvent these

analysis complexities (Ritter 1899; Mörsch 1909). The idea was

to assume a cracked, RC beam behaves like a truss. This truss

analogy, known today as a strut-and-tie model, provides a convenient visualization of force flow and identifies required-reinforcing

steel locations that can be used to design and detail a concrete

member.

A drawback of early concrete truss models was the arbitrary nature with which they could be formulated and the lack of scientific

theory to support the practical idea developed by Ritter and Mörsch.

The scientific support for cracked, RC truss models came several

decades later with research by Marti, who established a technical

foundation for the truss model concept by relating truss behavior to

a lower-bound plasticity theory (Marti 1980). Marti and others

concluded that an optimum concrete truss model could be achieved

by locating the compressive struts and tension ties coincident with

the elastic stress trajectories in a member, and that higher ductility

and improved structural performance at an ultimate limit state could

be achieved with a stiffer truss. The engineering judgment required

to obtain an accurate truss model was viewed as a drawback of

the design approach, and Marti recommended future research

on computational tools that could automate the identification of

viable strut-and-tie geometries.

The momentum from Marti’s work, in combination with experimental and analytical work by Collins and Mitchell (1980) on

truss models for shear and torsion, led to a groundbreaking set of

design guidelines for truss models proposed by Jörg Schlaich and

his colleagues at the University of Stuttgart (Schlaich et al. 1987).

Schlaich states that the stiffest truss model is the one that will

produce the safest load-deformation response because limiting truss

deflection prevents large plastic deformations in the concrete. Maximizing stiffness correlates mathematically to minimizing reinforcing steel’s elastic strain energy. However, Schlaich acknowledges

that selecting the optimum truss model may be difficult with the

energy criterion, requiring engineering intuition that has contributed to past structural failures.

RC design guidelines employing strut-and-tie models were introduced into the Canadian Concrete Design Code in 1984 [Canadian Standards Association (CSA) Technical Committee A23.3

1984], followed by European practice (Comité Euro-International

du Béton 1993), the AASHTO LRFD bridge code (AASHTO 1994),

and finally the American Concrete Institute (ACI) (2002) building

code. However, the method’s widespread use is currently stymied by

a lack of mechanics-based tools for identifying the force flow and

visualizing the truss shape needed in design.

It is the goal of the research described herein to create a new

automated tool for visualizing the flow of forces in RC and prestressed concrete structural members. The approach couples truss

and continuum topology optimization methodologies to create

a hybrid routine that leads to strut-and-tie solutions consistent

1Graduate Student, Dept. of Civil Engineering, Johns Hopkins Univ.,

Baltimore, MD 21218. E-mail: [email protected]

2Associate Professor, Dept. of Civil Engineering, Johns Hopkins Univ.,

Baltimore, MD 21218 (corresponding author). Email: [email protected]

3Assistant Professor, The Charles E. Via, Jr. Dept. of Civil and Environmental Engineering, Virginia Tech, Blacksburg, VA 24061. E-mail:

[email protected]

Note. This manuscript was submitted on December 20, 2011; approved

on July 26, 2012; published online on August 11, 2012. Discussion period

open until September 1, 2013; separate discussions must be submitted for

individual papers. This paper is part of the Journal of Structural Engineering, Vol. 139, No. 4, April 1, 2013. ©ASCE, ISSN 0733-9445/2013/

4-607–618/$25.00.

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with Schlaich’s hypothesis that placing reinforcing steel consistent

with the stiffest truss results in superior structural performance over

traditional designs, i.e., reduced crack widths and improved capacity. Force paths (topologies) can be identified and studied for

any general concrete domain and with any loading and boundary

conditions, and the tensile forces in the reinforcing steel are readily

available to a designer for sizing. With the force paths defined,

a designer may then apply existing code-based strut-and-tie design

provisions to evaluate ductility and ultimate strength. An introduction to truss and continuum topology optimization from the

perspective of RC design is provided in the subsequent section,

followed by a step-by-step implementation of the hybrid trusscontinuum topology optimization approach. Included in the discussion are examples of force flow topologies for a concrete beam,

a hammerhead pier, a deep beam with a cutout, and a prestressed

concrete block.

RC and Topology Optimization

Topology Optimization Background and Formulation

Recent advances in optimization algorithms, and specifically growth

in the field of topology optimization, have led to a new family of

methods for identifying RC truss models consistent with the rules

outlined by Schlaich for optimal performance in service and at an

ultimate limit state. In topology optimization, the design domain

(structural component) is discretized with structural elements, typically truss or continuum (solid) elements, with the goal of identifying the concentration of material in each element (Fig. 1).

Elements receiving little or no material by the optimizer at convergence are deemed structurally insignificant and removed from

the structural domain in postprocessing (Ohsaki and Swan 2002;

Bendsøe and Sigmund 2003).

Following the guidelines of Schlaich, the objective is to design

a truss topology with maximal stiffness. This may be equivalently

formulated as a minimum compliance problem, in which the goal is

to minimize the external work done by the applied loads (and strain

energy stored in the structure), for a limited volume of material,

expressed in general as follows:

min

r

f r ¼ FTd

subject to: KðrÞ d ¼ F

P

e 2 V

re ve # V

0 # re # re max” e 2 V

ð1Þ

where design variable vector r 5 encoding of the material concentration (the structural design), re 5 material concentration in

element e (e.g., the cross-sectional area of truss element e), F 5

applied nodal loads, d 5 nodal displacements, ve 5 element volume

for unit re (element length for truss structures), V 5 available

volume of material in the design domain V, and re max 5 design

variable upper bound. The global stiffness matrix, K, is assembled

( A

e 2 V

) from element stiffness matrices Ke as follows:

KðrÞ ¼ A

e 2 V

KeðreÞ, KeðreÞ ¼ ½ðreÞh þ re minKe 0 ð2Þ

where Ke

0 5 element stiffness matrix for unit re, re min 5 small

positive number to maintain positive definiteness of the global

stiffness matrix, and the exponent parameter, h $ 1 5 optional

penalty term that may be used to drive solutions to the design

variable bounds (Bendsøe 1989). This penalization approach is

known as the solid isotropic material with penalization (SIMP)

method and is widely used in the topology optimization community.

The optimization problem in Eq. (1) is solved using gradientbased optimizers, chosen as the method of moving asymptotes

(MMA) (Svanberg 1987) in this work. Such optimizers are guided

by design sensitivities, or derivatives with respect to the design

variables. Minimum compliance sensitivities may be found using the

adjoint method or direct differentiation, and take the well-known

form of scaled-elemental strain energies

∂f e∂r

¼ 2 hðreÞh21deTKe 0de ð3Þ

where de 5 elemental displacement vector of element e. The reader

is referred to Arora (1997) and Bendsøe and Sigmund (2003) for

sensitivity analysis background information.

Truss Topology Optimization

Truss topology optimization begins with a densely meshed domain,

referred to as ground structure [Fig. 1(a)], and cross-sectional areas

are then optimized. Following convergence, members having (near-)

zero area are removed to identify the final optimal topology and

corresponding distribution of cross-sectional areas (e.g., Bendsøe

et al. 1994). Following this approach, Biondini et al. (1999) and Ali

and White (2001) solved minimum compliance formulations using

mathematical programming to develop concrete truss models consistent with the elastic stress trajectories in a general concrete

domain. Ali and White demonstrated with nonlinear finite-element

modeling to the collapse of short, RC cantilevers that ultimate

strength increases as truss stiffness increases, an important result

supporting Schlaich’s hypothesis that was subsequently confirmed

with experimental results by Kuchma et al. (2008).

In typical truss topology optimization, the cross-sectional areas

re are considered unpenalized continuous variables (h 5 1) with

Fig. 1. Topology optimization design domain with hole discretized using (a) truss elements and (b) four-node quadrilateral elements

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relaxed upper bound re max. Under these conditions, it can be shown

that minimum compliance optimization yields a topology of uniform

strain-energy density and thus a uniformly stressed design (Bendsøe

et al. 1994). Thismeans the target volume V specified by the designer

is arbitrary and cross-sectional areas may be uniformly scaled to

satisfy a stress constraint, such as the reinforcing steel-yield stress.

This may be extended to the case in which truss members have

different properties in both tension and compression (Achtziger

1996; Rozvany 1996).

The minimum compliance topology-optimization truss is illustrated for a RC deep beam shown in Fig. 2. The topologies are

overlaid on experimental testing data indicating crack paths, and

therefore principal tension trajectories for this beam. The traditional

truss model and reinforcing layout places steel near the bottom of the

deep beam [Fig. 2(a)], which is an optimal location at midspan;

however, it is less efficient at providing resistance to principal

tension near the supports where wide diagonal cracks may develop

under a load as shown. The minimum compliance truss model

[Fig. 2(b)] helps the designer understand where the cracks will form,

in this case showing that inclined-steel reinforcement [tension ties in

Fig. 2(b)] should be provided to bridge to better resist the principal

tension cracks.

One of the drawbacks of truss topology optimization is that

solutions are dependent on the ground structure chosen by the designer. This includes nodal locations and element connectivity, as

the designer has essentially restricted potential force paths a priori.

Using a very fine mesh with small nodal spacing and extensive

element connectivity [e.g., connecting every node together as shown

in Fig. 1(a)] offers the most design freedom and allows topologies to

approximate curved trajectories [Fig. 2(b)]. By using a coarser mesh

and/or simpler connectivity, the designer restricts the design space

and subsequently global optima will underperform those found with

more refined ground structures, i.e., they will have higher compliance

and hence lower stiffness. The advantage of using coarse meshes,

however,isthatoptimal topologiesaretypicallylesscomplex andthus

easier to construct. This tradeoff between stiffness and constructability will be revisited in the examples section, Examples of Topology Optimization for RC.

Continuum Topology Optimization

Continuum topology optimization offers an alternative free-form

approach to visual force flow. The domain is discretized with finite elements [four-node quadrilateral elements shown in Fig. 1(b)]

and the goal is to determine whether an element contains material,

i.e., is a solid (re 5 re max 5 1) or a void (re 5 0). The resulting

connectivity of the solid elements defines the optimized structure. In

the application to RC-force visualization, the solid phase represents

load-carrying material (concrete or steel), while the void phase in

the continuum model indicates locations of background concrete

that are not part of the force model.

To enable its use with gradient-based optimizers, the binary

(solid-void) condition on re in is relaxed and solutions are steered

toward 0–1 distributions using the SIMP penalty term h . 1 in

Eq. (2) (e.g., h 5 3). It is well known that this approach leads to

numerical instabilities of checkerboard patterns and solution mesh

dependency if the design space is not restricted to prevent them [see

Sigmund and Peterson (1998) for review]. These issues are circumvented herein by imposing a minimum-length scale (minimum

thickness) on load-carrying members. This not only numerically

stabilizes the formulation, it has the added benefit of providing the

designer a tool for influencing constructability, as requiring larger

features tends to produce simpler topologies. Minimum-length

scales can be imposed on a topology using an efficient projectionbased algorithm (Guest et al. 2004; Guest 2009), in which an

auxiliary variable field f serves as the independent optimization

variable and is mapped onto the finite element space to determine the

topology, meaning finite element variables r are a function of f.

This mapping is rigorously constructed such that the minimumlength scale of the designed topological features is naturally controlled at a negligible added computational cost. The reader is

referred to Guest et al. (2011) for details on numerical implementation of the algorithm used herein.

Several researchers have explored the use of continuum topology

optimization as a tool for RC analysis and design. Liang et al.

(2000b) implemented a heuristic plane-stress topology optimization

approach, commonly referred to as evolutionary structural optimization (ESO), to derive concrete truss model shapes for common

cases including a deep beam and a corbel. Kwak and Noh (2006) and

Leu et al. (2006) employed similar ESO-based algorithms. Bruggi

(2009) solved two-dimensional (2D) and three-dimensional (3D)

strut-and-tie design problems using a gradient-based topology optimization algorithm with heuristic sensitivity filtering to improve

solution efficiency, while Victoria et al. (2011) used a heuristic

optimality criterion-updating scheme allowing various moduli for

tension and compression phases. More recently, Amir and Bogomolny

(2011) and Bogomolny and Amir (2012) used material-dependent

elastoplastic models to specifically enhance RC performance at

an ultimate limit state.

The primary advantage of the continuum approach is the freeform design evolution that identifies high-performance topologies

consistent with the force path in a structural component. Unlike

truss topology optimization, in which the designer selects node

locations and element orientations of the force flow model a priori,

it is the optimizer itself that identifies these locations and orientations in continuum topology optimization. Disadvantages are

that the tension regions are not defined as discrete bars, requiring

postprocessing of the continuum results to produce truss representations to size concrete reinforcement. Continuum topologies,

as they are generated in a free-form manner, are also typically more

complex and therefore may be more difficult to construct than

those found directly using truss topology optimization. As will

be shown, the geometric restriction methods (minimum-length

scale) discussed previously provide a means for controlling this

complexity.

Examples of Topology Optimization for RC

A traditional linear elastic topology optimization approach is demonstrated for several RC design examples. These examples will

Fig. 2. (a) Traditional concrete truss model and (b) minimum compliance truss model derived with topology optimization; dashed lines 5

compression carried by concrete; solid lines 5 represent tension carried

by the reinforcing steel (Nagarajan and Pillai 2008 with permission from

Multi-Science Publishing Co. Ltd.)

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provide baseline solutions that can be compared with the hybrid

model results discussed in the manuscript. For the truss topologies,

solid lines represent tension (steel) ties and dashed lines represent

the compressive struts (as in Fig. 2). Line thickness is proportional to

axial force and therefore the required cross-sectional area for the

steel tension ties. In the continuum representations, the solid black

features represent the force flow topology, i.e., the load-carrying

concrete and steel ties. A single isotropic, linear elastic material

model is assumed for both the concrete and the steel. Continuum

examples use four-node quadrilateral plane-stress elements. Domain

dimensions and loads are given in relative (unitless) measures.

Simply Supported Beam

The design domain for a RC beam with a point load is shown in

Fig. 3(a) along with a typical strut-and-tie model in Fig. 3(b). The

topology-optimized truss and continuum models are shown in

Figs. 3(c and d), respectively, with the truss solution achieving

a uniformly stressed state as expected. These solutions illustrate that

the maximum elastic stiffness (minimum compliance) can be achieved by placing the reinforcing steel orthogonal to the compressive

stress trajectories. This design philosophy is similar to the practice

of providing inclined shear stirrups to bridge diagonal cracks

(MacGregor 1992).

As previously mentioned, one of the disadvantages of the truss

approach is that solutions are mesh dependent. In selecting a ground

structure, the designer limits the potential force flow paths a priori.

Fig. 4, for example, shows three different ground structures containing a number of nodes ranging from 10 (coarse) to 85 (fine) in

a lattice format. The optimized topology found using the fine ground

structure closely resembles the principal stress trajectories, and

consequently offers a 22% improvement in stiffness and reduces

required steel quantities by an estimated 14% over the solution found

using the coarse mesh. The trade off, however, is constructability, as

the simpler topology is likely easier to construct. Ultimately, the

decision is left to the designer to balance the cost of material and

labor, while topology optimization offers a tool for exploring this

trade-off.

Deep Beam with a Cutout

RC designs can be readily obtained with topology optimization for

complex domains such as the deep beam with openings example

shown in Fig. 5(a). The minimum compliance design in this case

results in a reinforcing layout that does not require stirrups in the

confined space under the hole, simplifying construction. Also,

Figs. 5(c and d) show that there is tension in the lower-left corner of

the beam, below the cutout, which could result in splitting cracks

from the corner of the hole to the edge of the beam. A designer

may miss this potentially detrimental behavior with a traditional

strut-and-tie solution [Fig. 5(b)]. A drawback of the truss solution

[Fig. 5(c)] is the lack of reinforcement over the left support where

tensile stresses may develop as a result of bearing. This will be

revisited with the hybrid model.

Hammerhead Bridge Pier

Hammerhead bridge piers are widely used and typically designed

with the truss model shown in Fig. 6(b). Vertical shear stirrups are

spaced evenly across the pier cap with a top mat of reinforcing steel

to control cracking at the girder bearing line. The minimum compliance truss and continuum models in Figs. 6(c and d) demonstrate

that for the loading case considered, the shear stirrups do not coincide with the internal tensile-force trajectories; instead, draped

reinforcing steel or posttensioning would be a more appropriate

design solution.

Fig. 7 illustrates the potential benefits of imposing minimumlength scale on the continuum structural members. Increasing the

required minimum strut-and-tie thicknesses simplifies the topology

and reduces the number of designed steel ties from four draped

[Fig. 7(a)] to three (nearly) straight [Fig. 7(b)] to two straight [Fig.

7(c)] ties in each half of the pier. With direct control over the length

Fig. 3. Force visualization for RC simply supported beam: (a) design domain; (b) traditional truss model; (c) topology-optimized truss model;

(d) topology-optimized continuum model; (b) and (c) solid lines indicate tension (steel) members; dashed lines compression members, with line

thickness indicating relative axial force

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Fig. 4. Trusssolutions using various groundstructures having normalized compliances of(a) 1:000 for a 5 3 2 node mesh;(b) 0:792 for a 9 3 3 node mesh;

(c) 0:779 for a 17 3 5 node mesh

Fig. 5. Deep beam with cutout via topology optimization: (a) design domain; (b) traditional truss model; (c) topology-optimized truss model;

(d) topology-optimized continuum model

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scale, a designer may generate a suite of solutions in which structural

performance and constructability are balanced.

Motivation for a Hybrid Truss-Continuum Topology

Model

The results presented in the previous section are consistent with

those reported in the literature. Optimized topologies largely follow

the principal stress trajectories even for complex domains treated in

Biondini et al. (1999), Ali and White (2001), and Bruggi (2009).

While solutions found using truss and continuum topology optimization follow these general trends, there are distinct differences in

terms of solution stiffness, as quantified by the objective function,

and constructability between the two approaches.

The free-form nature of continuum topology optimization is

evident as the presented force trajectories may take any shape, have

varying thickness, and/or connect with other members at any angle.

In this sense, the optimizer selects both the locations of the nodes

of the force-transfer topology and also the corresponding flow paths.

This design freedom enhances solution efficiency, however, it may

produce solutions that are less practical from a construction point of

view (even with length-scale control), potentially negating any costsaving from solution efficiency. This is in contrast to the truss approach, which restricts the design space by requiring loads to flow

in straight paths along predefined candidate orientations. Truss

models, therefore, underperform continuum solutions; however,

they likely improve constructability as steel rebar and strands may

be placed in straight segments.

A key limitation of both topology optimization approaches as

presented is the assumption of isotropic, linear elastic constitutive

models. This assumption means that traditional topology optimization approaches to RC design may miss transverse tensile stresses

that develop in the concrete phase as a result of load spreading,

an outcome that is observed even when algorithms are implemented that use various moduli for the tension and compressive

materials (Victoria et al. 2011). In some design settings, for example,

Fig. 6. Hammerhead pier supporting four girder lines with topology optimization: (a) design domain; (b) traditional truss model; (c) topologyoptimized truss model; (d) topology-optimized continuum model

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prestressed concrete, this missing treatment of orthotropy, may lead

to invalid strut-only solutions that falsely indicate that steel reinforcement is not needed. Fig. 8 illustrates this shortcoming for

a concrete block subjected to a compressive load, which is representative of a column or prestressed anchorage zone. Minimum

compliance solutions found using truss and continuum topology

optimization are shown in Figs. 8(b and c), respectively. The topologies indicate a strictly compressive load path, failing to capture

load spreading that will create transverse tensile stresses in the

concrete phase as shown by the principal stress plot in Fig. 8(d)

(only major stresses are shown, as all minor stresses are compressive). A similar instance can be seen over the left support in the

topology-optimized truss solution for the deep beam with cutout

[Fig. 5(c)].

Overcoming this limitation requires breaking from traditional

linear elastic topology optimization methodologies. We propose

herein a bilinear hybrid approach. The idea is that tension members

are implemented as truss elements in the optimization formulation,

resulting in reinforcing steel design that is straight, simply placed,

and easily sized. Continuum elements form force paths consistent

with the elastic stress trajectories and couple with the tensile truss

members to carry compression in the concrete. This separation of the

compressive and tensile load-carrying elements allows various

moduli to be used for the various materials, and more importantly, is

shown to capture force-spreading that results in tensile stresses orthogonal to compression struts, i.e., splitting stresses near a prestressing steel anchorage. The details of this hybrid approach are

presented in the subsequent section.

Hybrid Truss-Continuum Strut-and-Tie Models

A new force visualization approach is proposed that utilizes a

hybrid truss-continuum design domain to address the identified

shortcomings in the prevision section, specifically the inability of the

topology solutions to simulate force-spreading and the cumbersome

postprocessing required to size reinforcing steel with continuum

solutions. In the hybrid approach, first postulated in Moen and Guest

Fig. 7. Hammerhead pier example solved using continuum topology optimization with minimum prescribed length scales (diameter 5 dmin) of

(a) dmin 5 0.007L (small dmin); (b) dmin 5 0.015L (medium dmin); and (c) dmin 5 0.030L (large dmin); increasing length scale decreases efficiency

but also complexity

Fig. 8. Compression block illustrating strut-only solutions: (a) load and boundary conditions, (b) truss optimization producing three vertical struts,

(c) continuum optimization producing single large strut; (d) strut-only solutions fail to capture tensile stresses because of force spreading, the maximum principal stress plot for solution (c)

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(2010), the steel phase is modeled using truss elements with high

tensile stiffness and zero compressive stiffness, while the concrete

phase is modeled using continuum elements with high compressive

stiffness and low tensile stiffness. The hybrid formulation therefore

requires tension load paths be carried with steel truss members

and compressive load paths to be carried with continuum concrete

members. This not only captures force-spreading as will be shown, it

also leverages the desirable properties of both topology optimization

approaches: load-carrying concrete continuum may take any topology, as it need not be constructed, while steel reinforcement is

placed in straight segments.

The idea of combining discrete structural elements with continuum elements is not a new idea. Several authors have proposed

using continuum topology optimization to reinforce predefined

(nonoptimized) frame systems, most notably for designing topologies of lateral bracing systems (Mijar et al. 1998; Liang et al.

2000a, Stromberg et al. 2012). A key difference here is that topology optimization simultaneously optimizes the discrete element

(truss) and continuum domains and leverages separation of these

domains to distinguish compression and tension load paths.

Hybrid Mesh

The hybrid mesh is achieved by embedding a truss ground structure

into a continuum finite element mesh. The design domain V is

discretized with a lattice mesh of nodes and the continuum mesh

Vc uses every node, while the truss mesh Vt is more sparse, with

members connected every few nodes to reduce the complexity of

the final steel configuration. This is seen in Fig. 9, where there are

12 continuum elements in each direction; however, the truss elements are connected every four nodes (gray 5 continuum elements,

black 5 truss elements). Force transfer between the meshes occurs at

the shared nodes and a nonslip condition for the steel reinforcement

is assumed. This relative node spacing of four to one was selected to

match approximately the truss and continuum ground structures in

the preceding examples. As previously discussed, using a densertruss ground structure (skipping fewer continuum nodes) would

likely lead to more complex reinforcement patterns, while coarsertruss ground structures would likely produce simpler patterns. It is

not recommended, however, that the truss node spacing be fewer

than the continuum node spacing, as this would require compression

truss elements to have nonzero stiffness.

Material Models

The bilinear stress-strain relationships for the steel and concrete are

shown in Fig. 10. Young’s moduli for the steel are assumed 200 GPa

(29,000 ksi) in tension and zero in compression, while moduli for

the concrete are assumed 24.9 GPa (3,600 ksi) in compression and

2.0 GPa (290 ksi) in tension. A nonzero tensile stiffness is used for

the concrete to prevent singularities in the global stiffness matrix.

Such singularities would otherwise arise at nodes that are not connected to truss elements and that are located in regions achieving

a state of tensile stress.

As truss members carry only axial forces, the bilinear constitutive

steel model is straightforward to implement. Denoting the truss elemental design variables (cross-sectional areas) as rt, the element

stiffness matrix of a truss element Ke

t is now stated as follows:

Ke

t ðre t , se t Þ ¼ ðre t Þht Ke 0,tEtðse t Þ ð4Þ

where Et 5 Young’s modulus of truss element and dependent on the

sign of the elemental (axial) stress se t (Fig. 10), and Ke 0,t 5 truss

element stiffness matrix for unit re t .

For the concrete continuum elements, an orthotropic constitutive

model is adapted from the bilinear elastic portion of a model proposed by Darwin and Pecknold (1977). The model combines the

various Young’s moduli as a function of the principal normal

stresses and orientation of the principal stress plane. This rotational

dependence is key to capturing force-spreading and gives preference

over existing isotropic stress-dependent stiffness tensors proposed in

the literature [e.g., Cai (2011)].

Denoting the continuum elemental design variables (volume

fractions) as rc, the element stiffness matrix of a continuum element

Ke

c is now stated as

Ke

cðre c, se cÞ ¼ ½ðre cÞhc þ re min

Ke

0,c½Dðse cÞ ¼ ½ðre cÞhc þ re min ð BeT Dðse cÞ Be dV ð5Þ

where D 5 (stress-dependent) constitutive stiffness tensor relating

continuum stresses sc, and strains ɛ, and Be 5 elemental component

of the standard strain-displacement tensor ðɛ 5 BdÞ. Constitutive

stiffness tensor D is defined as follows for an isotropic material with

Young’s modulus E and Poisson’s ratio n:

Diso ¼ E 1 2 n2 |
3 |

5 | ð6Þ |

24

1 n 0

n 1 0

0 0 ð1 2 nÞ=2

The orthotropic material model of Darwin and Pecknold (1977)

is stress-dependent and uses the following approximation for the

stiffness tensor, denoted as Dp with the subscript, p, indicating it is

defined in the principal stress coordinate system:

D

p ¼

264

D11 | veffD11D22 | 0 |

veffD11D22 0 |
D22 | 0 |

0 | 0:25D11 þ D22 2 2veffpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D11D22 |

375

ð7Þ

Fig. 9. Interaction between continuum (four-node quadrilaterals) and

truss domains

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where neff 5 effective, or smeared, Poisson’s ratio, and Dij terms are

dependent on the principal normal stresses, sci, for i 5 1, 2 as

follows:

Dii ¼ Ect, ni ¼ nct if sci . 0

Dii ¼ Ecc, ni ¼ ncc if sci , 0

D12 ¼ D21 ¼ neff

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pD11D22

neff ¼

ffiffiffiffiffiffiffiffiffi

pn1n2

ð8Þ

where Ecc and Ect 5 Young’s modulus of the concrete in compression and tension, respectively, and ncc and nct 5 Poisson’s ratio

of the concrete in compression and tension, respectively. The

compression Poisson’s ratio of ncc 5 0:2 is used to compute the

tensile Poisson’s ratio from the following equation, which is required to achieve symmetry of Dp (Darwin and Pecknold 1977):

nct ¼ vccEct=Ecc ð9Þ

This relationship is deemed acceptable as the stiffness, and therefore

load-carrying potential, of the concrete continuum system in tension

is negligible.

The principal stresses si in Eq. (8) and the orientation u of the

principal plane are computed in the standard manner

s1,2 ¼

sx þ sy

2

6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ssx þ2 sy2 þ t2 xy

u ¼ 1

2

tan21sx22txysy

ð10Þ

where normal stresses sx and sy and shear stress txy are all defined

in the global coordinate system.

The constitutive stiffness tensor in Eq. (7), defined in the principal coordinate system, is then transformed to the global coordinate

system using

D ¼ QTDpQ ð11Þ

where the transformation tensor Q is defined as

Q ¼

264

cos2u sin2u |
sin2u 2 cosðuÞsinðuÞ cos2u 22 cosðuÞsinðuÞ |

2cosðuÞsinðuÞ cosðuÞsinðuÞ cos2u 2 sin2u

375 ð

12Þ

Optimization Formulation and Solution Algorithm

The hybrid minimum-compliance problem can now be expressed as

min

rt, rc

f rt, rc ¼ FTd

subject to: Kðrt, rc, st, scÞd ¼ F

P

e 2 Vt

re t ve t þ P

e 2 Vc

re c ve c # V

0 # re t ” e 2 Vt

0 # re c # 1 ” e 2 Vc

ð13Þ

where ve

t and ve c 5 truss element lengths and continuum element

volumes, respectively (as before), and the global stiffness matrix is

assembled in the standard manner

Kðrt, rc, st, scÞ ¼ A

e 2 Vt

Ke

t ðre t , se t Þ þ A

e 2 Vc

Ke

cðre c, se cÞ ð14Þ

The standard projection scheme (Guest et al. 2011) is again used for

the continuum elements, and thus rc remain a closed-form function

of f as before. This detail is omitted for brevity.

One of the key aspects of this hybrid technique is that the concrete and steel pull from the same prescribed volume of material.

Using this approach, if a structure sees only tension forces, the

optimization process will produce a truss (steel)-only structure.

Likewise, if only compressive forces are present, a continuum

Fig. 10. Stress-strain relationships for continuum concrete and truss steel models

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(concrete)-only structure will result, although this is unlikely because of the force-spreading effect.

The equilibrium conditions are now governed by a nonlinear

material model and thus require an iterative numerical solution.

However, they are bilinear elastic, meaning they are not loadmagnitude dependent and do not require load-stepping as in topology optimization for fully nonlinear material models, such as

in Swan and Kosaka (1997), Maute et al. (1998), and Bogomolny

and Amir (2012). The approach taken here is to initialize the analysis iterative process with all elements being active and isotropic

(no stress dependency) and updating the stiffnesses using the material models described previously.

The optimization algorithm is summarized as follows:

1. Initialize truss and continuum design-variable fields rt and rc

with a uniform distribution of material (or random or educated

guess).

2. Finite element analysis:

a. Solve linear elastic (stress-independent) problem

Kðrt, rcÞ d 5 F with truss elements having Et 5

200 GPa and all continuum elements being isotropic with

Ec 5 24:9 GPa (i.e., all elements at full stiffness).

b. Update truss element stiffness matrices according to

Eq. (4).

c. Update continuum-element stiffness matrices according

to Eqs. (5) and (7)–(12).

d. Solve Kðrt, rc, st, scÞ d 5 F with updated element

stiffnesses.

e. If analysis converged, go to Step 3; else, go to Step 2b.

3. Compute sensitivities using Eq. (3) and converged displacements and element stiffnesses from Step 2.

4. Update the independent design variables using a gradientbased optimizer.

5. Check optimization convergence. If converged, stop; else go

to Step 2;

Convergence of the finite element analysis (Step 2e) is herein

defined as fewer than 0.1% of truss elements changing between

tension and compression states and the average change in orientation

of the principal plane u in nonvoid elements is ,0.01. Convergence

was typically achieved in fewer than 10 finite element iterations in

the presented examples. It should also be noted that the Young’s

moduli shown in Fig. 10 for each phase are piecewise linear and thus

exhibit C0, but not C1, continuity. In gradient-based optimization,

this typically requires interpolation between the piecewise states

to achieve C1 continuity. Interestingly, oscillatory behavior was not

observed in either the analysis or optimization steps. This may be

owing to the result that the design sensitivities [Eq. (3)] are always

negative, indicating that adding material always improves stiffness.

Interpolation of Young’s moduli, however, may be required for

more challenging design problems in which sensitivities may be

positive or negative.

Hybrid Topology Optimization Results

The same examples presented previously are solved with the new

hybrid truss-continuum topology optimization approach using the

embedded mesh scheme shown in Fig. 9. The compression block

example (Fig. 8) highlights the capability of the model to capture

force-spreading. The optimal solution found using the hybrid topology optimization approach is shown in Fig. 11, where the white

regions indicate non-load-carrying concrete, the black region

indicates compressive load-carrying concrete, and lines indicate

the steel ties. Under the compressive load [Fig. 11(a)], a single

(large) compression strut is designed as before, however, it is now

reinforced with horizontal steel truss elements to capture principal

tensile stresses that develop because of force-spreading. This resembles splitting reinforcement that would be detailed in the local

anchorage zone of a prestressing strand anchorage. It is also worth

emphasizing that the volume constraint V is shared between the truss

and continuum topologies in Eq. (14). Fig. 11(b) highlights this idea:

When the same structure is subjected to a tensile load, the optimizer

concentrates all available material in vertical steel ties, as the concrete does not play a role in force transfer.

Fig. 12 contains solutions to the previously explored examples

found using the hybrid topology optimization algorithm. The compressive and tensile load paths are indicated by the continuum and

truss topologies. The compressive load paths may take any angle,

vary in thickness, and connect at any location, leveraging the

free-form nature of continuum topology and the idea that these

members are not explicitly constructed, but rather represent an

idealized load path. The tension load paths are straight and thus

more accurately represent rebar and its placement, although their

placement is still dependent on the truss ground structure as

previously discussed. As truss members, they also allow for direct

extraction of axial force and the calculation of required crosssectional areas in design.

The simply supported beam [Fig. 12(a)] and hammerhead pier

[Fig. 12(c)] solutions resemble a combination of the previously

presented continuum-only and truss-only topology-optimized solutions. The tension and compression load paths are orthogonal, with

load transfer occurring at the end of the truss members, typically in

the interior of compressive struts. An interesting highlight of the

deep beam with cutout solution [Fig. 12(b)] is the use of steel near

the supports. In the truss-only topology optimization solution [Fig.

5(c)], unbraced compressive struts transfer load to the supports. In

the hybrid model, lateral truss members are present in the region of

the support to pick up the tensile stresses that will develop because of

force-spreading at bearing. This more closely resembles the continuum solution [Fig. 5(b)], although an extra tie has been added

to the lower-right side of the domain to pick up tensile stresses

Fig. 11. (a) Compression block solution using new hybrid topology

optimization algorithm; horizontal truss (steel) elements carry tensile

stresses because of force spreading seen in Fig. 8(d); (b) under tensileapplied load the algorithm produces tie-only solution, illustrating that

the hybrid scheme allocates material to tension (steel) and compression

(concrete) constituents as needed

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developing as a result of the change in the thickness of the compression strut in this region.

Concluding Remarks

Topology optimization provides an effective and convenient

methodology for obtaining a minimum compliance concrete truss

model, i.e., a truss model in which the strain energy is minimized,

a generally agreed-upon design guideline, which is intended to

minimize plastic deformation at an ultimate limit state. Experiments

and nonlinear finite-element modeling have confirmed that a minimum compliance concrete truss model can increase peak load and

improve the load-deformation response of RC members over traditional strut-and-tie designs. However, as only elastic stiffness is

considered rigorously in the optimization formulation, a designer

must still check ductility and strength requirements of the selected

strut-and-tie model with existing code provisions. The minimum

compliance design problem is solved efficiently using gradientbased optimizers with adjoint sensitivity analysis, enabling the

identification of models for complex domains, including members

with holes. Results are shown to produce steel reinforcement

patterns that are in line with the principal tension elastic stress

trajectories.

The advantages of truss topology optimization are that steel ties

are discrete and straight, yielding realistic and constructible reinforcement patterns, and that axial force demand and therefore

required steel cross-sectional areas come directly out of the analysis.

The disadvantages are that the optimality is limited by the initial

choice of ground structure, with simpler ground structures yielding

simpler, less efficient, topologies. Conversely, the free-form nature

of continuum topology optimization typically enables discovery of

solutions with higher efficiency. These solutions, however, tend to

be more complex, potentially requiring curved rebar or rebar with

varying thickness. The number of required ties, however, may be

indirectly limited through the use of projection schemes for controlling minimum-length scale as shown in the hammerhead pier

design problem. Continuum solutions must also be postprocessed to

determine required steel areas. Neither truss nor isotropic continuum

topology optimization is capable of accounting for transverse tensile

stresses that may develop in compression members caused by forcespreading.

The key contribution of this work is the development of a hybrid truss-continuum topology optimization methodology that can

help designers understand the flow of forces in a RC member and

provide demand forces that can be used to size reinforcing steel.

Bilinear material models are used to find tensile load paths represented by truss (steel) elements and compressive load paths by

continuum (load-carrying concrete) elements. Rebar is therefore

kept discrete and placed in straight segments, with axial-force

demand directly output by the model, while the concrete load

path is free to take any shape. An orthotropic bilinear material

model is assumed for the concrete, requiring tensile principal

stresses to be carried by the steel truss members. This is shown to

capture force-spreading phenomena that traditional topology optimization approaches miss, particularly at locations of concentrated forces, such as applied loads, bearing at supports, and

prestressing anchorages.

The ability to algorithmically focus natural tension and compression force flow in separate structural elements enables consideration of more complex design objectives beyond elastic stiffness,

potentially enabling direct optimization for serviceability and

strength in RC. Very recently, Bogomolny and Amir (2012) have

provided an excellent step in this direction by optimizing for strength

with elastoplastic continuum-damage models. As shown, the proposed hybrid approach allows various constitutive models to be used

for the steel and concrete, which could include various yield and

hardening behaviors, and potentially be extended to design for

ductility and ultimate strength. The proposed approach may also be

extended to other composite material systems with constituents

having different properties in tension and compression.

Acknowledgments

This work was supported in part by the National Science Foundation (NSF) IGERT Program (DGE-0801471) and Grant No.

CMMI-0928613. Their support is gratefully acknowledged. The

authors also thank Krister Svanberg for providing the MMA optimizer code.

Fig. 12. Optimized topologies using new hybrid optimization algorithm: (a) simply supported beam; (b) deep beam with cutout; (c) hammerhead pier

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