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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 scienti
fic
theory to support the practical idea developed by Ritter and Mörsch.
The scienti
fic 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 identi
fication 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
de
flection prevents large plastic deformations in the concrete. Maximizing stiffness correlates mathematically to minimizing reinforcing steels elastic strain energy. However, Schlaich acknowledges
that selecting the optimum truss model may be dif
ficult 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 Schlaichs 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 identi
fied 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 de
fined,
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 insigni
ficant 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 maxe 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 de
finiteness 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 er
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 ef
ficient 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 re
fined 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 de
fines 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 bene
fit of providing the
designer a tool for in
fluencing constructability, as requiring larger
features tends to produce simpler topologies. Minimum-length
scales can be imposed on a topology using an ef
ficient 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 ef
ficiency, 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 speci
fically enhance RC performance at
an ultimate limit state.
The primary advantage of the continuum approach is the freeform design evolution that identi
fies 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 identi
fies these locations and orientations in continuum topology optimization. Disadvantages are
that the tension regions are not de
fined 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 dif
ficult 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
con
fined 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 simpli
fies 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 quanti
fied 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 ef
ficiency, 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 ef
ficiency. This is in contrast to the truss approach, which restricts the design space by requiring loads to flow
in straight paths along prede
fined 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 identi
fied
shortcomings in the prevision section, speci
fically 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 prede
fined
(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. Youngs 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 Youngs 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
c
ðre 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 Poissons ratio n:

order now
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
de
fined 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, Poissons 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 Youngs modulus of the concrete in compression and tension, respectively, and ncc and nct 5 Poissons 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
s
sx þ2 sy2 þ t2 xy
u ¼ 1
2
tan
21sx22txysy
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
c
ðre 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
de
fined 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 ef
ficiently using gradientbased optimizers with adjoint sensitivity analysis, enabling the
identi
fication 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 ef
ficient, topologies. Conversely, the free-form nature
of continuum topology optimization typically enables discovery of
solutions with higher ef
ficiency. 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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