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 freeform 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, Strutandtie 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 mechanicsbased
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 strutandtie 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). Crosssectional
areas are then optimized and members having zero or nearzero area are removed, eventually
yielding an optimized topology with optimal crosssectional 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 mechanicsbased 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
freeform 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 fixedmass 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
displacementsonly 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 semidefinite 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 freelyavailable 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 codebased 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 01 (voidsolid)
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 postprocessing. 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 ESObased algorithms. Recently a more general
continuum topology optimization approach was used to guide strutandtie 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 wellknown numerical
instabilities of checkerboard patterns and mesh dependency. Truss topology optimization
will facilitate discovery of new design solutions. However, to fully realize the freeform
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 strutonly 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 compressiononly 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 continuumtruss hybrid approach governed by
piecewise linear elastic material models, although for now users of topology optimization
techniques should be wary of strutonly 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 strutandtie topology optimization approaches. Controlling
minimum length scale of the solid (loadcarrying) 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 loaddeformation response of reinforced concrete members over
traditional strutandtie 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.
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Rexdale, Ontario: Canadian Standards Association; 1984.
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16. Mathworks, "Matlab 7.8.0 (R2009a)," Mathworks, Inc., www.mathworks.com, 2009.
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Englewood Cliffs: PrenticeHall, Inc.; 1992.
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19. Liang Q. Q., Xie Y. M., Prentice Steven G., "Topology optimization of strutandtie
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Multidisciplinary Optimization, V. 37, 2009, pp. 46373.
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