REINFORCED CONCRETE ANALYSIS AND DESIGN WITH

TRUSS TOPOLOGY OPTIMIZATION

Cristopher D. Moen, PhD, PE, Virginia Tech, Blacksburg, VA

James K. Guest, PhD, Johns Hopkins University, Baltimore, MD

ABSTRACT

The objective of this research is to develop a computational framework to

assist in the analysis and design of reinforced concrete members. The

framework extends and automates the useful idea that a truss can

approximate disturbed strain fields in concrete by utilizing truss topology

optimization, a free-form design methodology for optimizing material

distributions within a domain. The flow of forces through a cracked

reinforced concrete member with general loading and support conditions are

identified with the convex form of the minimum compliance (maximum

stiffness) truss topology algorithm. The algorithm identifies a truss that

minimizes the strain energy in the reinforcing steel, which is consistent with

current design guidelines for limiting plastic deformations in reinforced

concrete at an ultimate limit state. Results from a freely available open

source computer program demonstrate that the truss topology optimization

approach produces reinforcing layouts consistent with the principal tension

stress trajectories in a member, even for complex domains such as members

with holes. In some cases, force spreading cannot be explicitly captured with

the truss topology formulation. Ongoing work in continuum topology

optimization of reinforced concrete members is summarized, including

consideration of constructability in the optimized solution.

Keywords: Reinforced concrete, Strut-and-tie model, Truss model, Topology optimization

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INTRODUCTION

Reinforced concrete is a complex composite material, which to this day, is

challenging researchers who are attempting to describe its behavior with mechanics-based

models. In the late 1800’s, Wilhem Ritter and Emil Mörsch developed a rational engineering

approach to circumvent concrete’s complexities, where a cracked reinforced concrete beam

was assumed to behave as a truss. The truss analogy, known today as a strut-and-tie model,

provided a convenient visualization of the flow of forces and the specific locations of the

reinforcing steel which can be used to detail a structural 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 practically minded idea

developed by Ritter and Mörsch. The scientific support for cracked reinforced concrete truss

models came several decades later with research by Marti, who established for the first time a

scientific foundation for the truss model concept by relating the truss behavior to a lower

bound plasticity theory

1

. Marti and others concluded that optimum concrete truss models

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 ultimate limit state could be achieved with a stiffer truss. Nonetheless, the

engineering judgment required to obtain a truss model was specifically noted by Marti as a

drawback of the truss design analogy, and he recommended future research on computational

tools that could automate the design process.

The momentum from Marti’s work, in combination with experimental and analytical

work by Collins and Mitchell on truss models for shear and torsion

2

, led to useful guidelines

for truss models proposed by Jörg Schlaich and his colleagues at the University of Stuttgart

3

.

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. Large plastic deformations are avoided by minimizing the stretching of the

reinforcing steel, which correlates mathematically to minimizing the reinforcing steel’s total

strain energy. However, Schlaich admits that selecting the optimum truss model may be

difficult with this criterion, requiring “engineering intuition” that he blames on past failures.

Recent advances in optimization algorithms, and specifically the growth of the field

of topology optimization, has led to a new family of methods for identifying truss models

consistent with the rules outlined by Schlaich. Truss topology optimization begins with a

densely meshed design domain, referred to as a ground structure (Fig. 1). Cross-sectional

areas are then optimized and members having zero or near-zero area are removed, eventually

yielding an optimized topology with optimal cross-sectional areas (Figure 2b), see Ohsaki

and Swan

4

, and Bendsøe and Sigmund

5

for reviews. Following this approach, Biondini et

al.

6

and Ali et al.

7

solved minimum strain energy formulations using formal mathematical

programming to derive concrete truss models that were consistent with the elastic stress

trajectories in a general concrete domain. Ali also demonstrated with nonlinear finite

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element modeling to collapse of short reinforced concrete cantilevers that ultimate strength

increases as truss stiffness increases, an important conclusion that was later confirmed with

experiments on other types of concrete members by Kuchma

8

.

Fig. 1 Truss ground structure

The objective of this research is to deliver a clear, accessible, and automated analysis

framework to support the extension and proliferation of the truss model approach, which will

lead to safer, more durable reinforced concrete members with higher levels of material

efficiency and lower life cycle costs when compared to traditional designs. Reinforced

concrete design guidelines employing truss models were introduced into European practice in

1990

9

, followed by the Canadian Concrete Design Code

10

, the AASHTO LRFD bridge code

11

, and finally the ACI building code

12

. The method’s widespread use is currently stymied

though by a lack of formal mechanics-based tools for obtaining the truss shape that leads to

optimal performance in service. The cornerstone of the proposed framework is a

visualization tool utilizing topology optimization algorithms, allowing engineers to identify

the best performing, i.e. the stiffest truss, which describes the flow of forces through a

general concrete member with general loading and support conditions.

CONCRETE TRUSS MODELS WITH TOPOLOGY OPTIMIZATION

MOTIVATION

Consider the traditional truss model and reinforcing layout for a reinforced concrete

deep beam in Fig. 2a. (Note experimental results from Nagarajan and Pillai

13

are placed

behind the truss models in Fig. 2). The steel reinforcement is located near the bottom of the

deep beam, which is an appropriate location at midspan, but does not provide resistance at

the locations of principal tension near the supports, allowing wide diagonal cracks to develop

under load. Fig. 2b shows an alternative minimum compliance truss model derived with a

topology optimization algorithm. The term “minimum compliance” refers to the fact that the

truss topology results in the smallest possible external work for a set volume of material, i.e.

the topology produces the stiffest truss. The minimum compliance truss model locates the

steel reinforcement to bridge the principal tension cracks, with the added bonus of reducing

the volume of required steel reinforcement when compared to the traditional model. This

idea, that minimum compliance truss models produce superior reinforced concrete designs, is

consistent with Schlaich’s design guidelines

3

and with experimental and computational

results

7, 8

.

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Fig. 2 Compare (a) traditional concrete truss model and (b) minimum compliance truss

model derived with topology optimization. Black dashed lines represent compression carried

by the concrete, red solid lines represent tension carried by the reinforcing steel.

Experimental results provided in the background are taken from Nagarajan and Pillai

13

.

OPTIMIZATION THEORY

The reinforced concrete analysis framework is based on topology optimization, a

free-form methodology for optimizing material distribution within a design domain. The goal

is to identify optimal distributions of concrete and steel for a given domain geometry and set

of loads and boundary conditions by considering design optimization formulations that

maximize stiffness (minimize compliance) and thereby limit plastic deformations in the

concrete member.

The independent design variable for the topology optimization problem is the truss

element area, A

e

, and the nodal displacements d are the state (dependent) design variables. A

common approach to maximizing stiffness of a fixed-mass system is to minimize internal

strain energy and equivalently external work. The minimum compliance problem is given

for truss domains as

Ω

Ω

∈∀≤≤

∑

∈

e A,VLA, )(:to subject

)(min

e

e

ee

TT

0K

2

1

K

2

1

f=d

df=dd

e

e

d,

e

Α

Α

Α

(1)

where f are the nodal applied loads, L

e

is element length for truss elements, V is the available

volume of material, and the global stiffness matrix K is assembled (

e

A

)

from element

stiffness matrices k

e

:

eeee

e

A)(A , )A( )(

A

0

kkk ==

eee

K Α (2)

where

k

e

0

is the element stiffness matrix for a unit A

e

.

The minimum compliance formulation above is a continuous, nonconvex

optimization problem. Employing the principle of minimum potential energy and assuming

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linear elastic behavior, the nonconvex problem can be converted into the following convex

displacements-only formulation

14

:

Ωθ

θ

θ

∈∀≤

+−

e

L

:to subject

V min

ee

T

e

e

T

dd

d

d,

0

k

2

1

f

(3)

where A

e

has been removed (through substitution) and the constraint reveals that

θ

is

proportional to the maximum strain energy density in the structural system

Ω

. This problem

is one of maximizing external work while minimizing the maximum strain energy (maximum

strain energy density ∙ total volume). The problem is convex as the objective function is

linear and the constraints are quadratic in

d

, yielding a positive semi-definite Hessian matrix

resembling (but not identical to) the global stiffness matrix. Convexity facilitates fast and

stable convergence of the optimization algorithm and means any local minimum is a global

minimum.

The optimal structural response is solved with Eq. (3), given by displacement field

d

(and strain energy density

θ

). The truss design that yields this response is then extracted

from the optimal Lagrange multipliers associated with each of the strain energy density

constraints

15

. Optimality conditions guarantee that (i) design and response fields will be

consistent and (ii) the optimal truss structure will be uniformly stressed, meaning truss cross-

sectional areas may simply be scaled to satisfy stress constraints and thereby making the

choice of V arbitrary. This approach is mathematically equivalent to the minimum strain

energy guidelines proposed by Schlaich.

IMPLEMENTATION AND EXAMPLES

A freely-available computer program written in MATLAB

16

is available for

exploring minimum compliance concrete truss models with general concrete shapes,

loadings, and boundary conditions, including members with holes at

http://www.ce.jhu.edu/jguest/

. A user inputs a general concrete domain with loadings and

boundary conditions, and the optimized truss geometry (xy coordinates) and truss member

forces are output for use in design. The output can be used with code-based concrete truss

model programs such as CAST

17

to automate design exploration of reinforced concrete

members.

Examples of minimum compliance concrete truss models generated with the

computer program are provided in Fig. 3 through Fig. 5. The minimum compliance model

for a beam with a point load in Fig. 3 demonstrates that the truss model with the maximum

elastic stiffness (minimum compliance) can be realized by placing the reinforcing steel

orthogonal to the compressive stress trajectories, which is similar to the practice of providing

inclined shear stirrups to bridge diagonal cracks

18

. Fig. 4 demonstrates that a fanned steel

reinforcing pattern is stiffer than the traditional concrete cantilever reinforcement layout,

providing new ideas for reinforcement layouts that could be studied experimentally to

determine their efficacy for seismic design. Reinforced concrete designs can be readily

obtained with topology optimization even for complex domains, for example the deep beam

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with an opening in Fig. 5. The minimum compliance design in this case results in a

simplified reinforcing layout when compared to the traditional design, because stirrups are

not required in the confined space under the hole.

Fig. 3 Compare (a) traditional truss model to (b) a minimum compliance truss model for a

simply supported beam with a point load

Fig. 4 Compare (a) traditional truss model to (b) a minimum compliance truss model for a

cantilever loaded with a point load at its tip

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Fig. 5 Compare (a) traditional truss model to (b) a minimum compliance truss model for a

deep beam with a hole

LOOKING AHEAD TO CONTINUUM TOPOLOGY OPTIMIZATION

RECENT WORK

Several researchers are exploring the use of continuum topology optimization as a

tool for reinforced concrete analysis and design. Continuum topology optimization is a free-

form design algorithm capable of generating new design ideas, for example, the beam loaded

with a point load in Fig. 6. Design variables are steered towards 0-1 (void-solid)

distributions because the solid phase in the continuum model (

ρ

e

=1) indicates either

localized tension or compression zones, with identification of the respective zone (and

consequently location of steel) occurring as part of the post-processing. The void phase in the

continuum model (

ρ

e

=0) indicates locations of ‘background’ concrete that is not part of the

force model. Liang et al.

19

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 such as a deep beam and a corbel. Kwak and

Noh

20

and Leu et al.

21

employ similar ESO-based algorithms. Recently a more general

continuum topology optimization approach was used to guide strut-and-tie design and

thereby improve solution efficiency and optimality

22

. Bruggi considers 2D and 3D design

problems by relying on heuristic sensitivity filtering

23

to overcome well-known numerical

instabilities of checkerboard patterns and mesh dependency. Truss topology optimization

will facilitate discovery of new design solutions. However, to fully realize the free-form

design potential of topology optimization in reinforced concrete design, we must consider

continuum topology optimization representations.

Fig. 6. Continuum topology optimization of a beam with a point load

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PRESTRESSED CONCRETE AND FORCE SPREADING

The truss topology optimization approach is based on the assumption of a linear

elastic constitutive equation, which in some design settings, for example prestressed

concrete, may lead to invalid strut-only solutions that falsely indicate that steel reinforcement

is not needed. Fig. 7 illustrates this shortcoming for a compressive point load on a column

solved with truss (Fig. 7a) and continuum topology optimization (Fig. 7b). The designs

indicate compression-only zones and fail to capture load spreading that will induce tensile

stresses into the concrete phase as indicated by the principal stress plot (Fig. 7c). Research is

underway to overcome this drawback with a continuum-truss hybrid approach governed by

piecewise linear elastic material models, although for now users of topology optimization

techniques should be wary of strut-only solutions.

Fig. 7 (a) truss and (b) continuum topologies produce strut only designs and do not capture

the force spreading indicated by the (c) principal stresses for the plane stress model

CONSTRUCTABILITY

To date, constructability and feature length scale considerations have not been

incorporated into continuum strut-and-tie topology optimization approaches. Controlling

minimum length scale of the solid (load-carrying) phase provides a means for influencing

constructability. Reducing the allowable minimum length scale improves structure stiffness

but typically leads to thinner members and more complex designs

24

. In reinforced concrete

topologies, this means smaller diameter and more complex reinforcing steel geometries.

Restricting the void (background) phase provides a means for enforcing concrete cover and

tie spacing bonding constraints. It is common for topology optimized designs to contain

dominant structural members, or members having undesirably large length scales in single or

multiple directions. Fig. 8 displays a reinforced concrete pile cap design problem and

optimized topology. In this case the entire bottom plane serves as a tension tie for the pile

cap. A more desirable result for reinforced concrete design could be obtained by imposing a

maximum length scale, e.g., Guest

25

, on the solid phase resulting in a system of distinct tie

members.

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Fig. 8 (a) pile cap design problem and (b) optimized topology (right, solid phase shown).

The entire bottom plane serves as a tension tie. Alternatively, distinct tie members could be

obtained by prescribing a maximum length scale for the solid phase

CONCLUSIONS

Topology optimization provides a convenient methodology for obtaining a minimum

compliance concrete truss model, i.e. a truss model where the strain energy in the reinforcing

steel 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 reinforced concrete members over

traditional strut-and-tie designs. The convex form of the truss topology optimization

problem is conveniently solved to find a minimum compliance truss model resulting in steel

reinforcement placed in line with the principal tension elastic stress trajectories. The

topology optimization can produce truss models even for complex domains, including

members with holes. Users should be suspicious of strut only solutions, as force spreading

cannot be readily modeled with existing truss or continuum topology optimization

techniques.

Research continues on improving force visualization and design tools for reinforced

concrete, including the development of continuum topology optimization approaches which

can accommodate constructability and concrete cover constraints. Advances in the

continuum constitutive models are needed to accommodate truss models that simulate force

spreading. The truss and continuum topology optimization methodologies provide a means

for efficient exploration of a design space and the potential to discover new, better

performing designs in reinforced concrete and other engineering materials.

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REFERENCES

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Structural Engineers, ETH, 1980.

2. Collins M. P., Mitchell D., "Shear and torsion design of prestressed and non-prestressed

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Recent Advances in Optimal Structural Design, 2002, pp. 97-123.

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7. Ali M. A., White R. N., "Automatic generation of truss model for optimal design of

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and-tie method for complex regions," ACI Structural Journal, V. 105, No. 5, 2008, pp. 578-

89.

9. Comité Euro-International du Béton, "CEB-FIP Model Code 1990." London: Thomas

Telford Services, Ltd; 1993.

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11. AASHTO, "AASHTO LRFD Bridge Specifications." 1st ed. ed. Washington, D.C.:

American Association of State Highway and Transportation Officials, 1st. ed.; 1994.

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16. Mathworks, "Matlab 7.8.0 (R2009a)," Mathworks, Inc., www.mathworks.com, 2009.

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Englewood Cliffs: Prentice-Hall, Inc.; 1992.

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19. Liang Q. Q., Xie Y. M., Prentice Steven G., "Topology optimization of strut-and-tie

models in reinforced concrete structures using an evolutionary procedure," ACI Structural

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20. Kwak H.-G., Noh S.-H., "Determination of strut-and-tie models using evolutionary

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22. Bruggi M., "Generating strut-and-tie patterns for reinforced concrete structures using

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Multidisciplinary Optimization, V. 37, 2009, pp. 463-73.

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