evolutionary topology optimization method – Global Homework Experts

322 ACI Structural Journal/March-April 2000
ACI Structural Journal, V. 97, No. 2, March-April 2000.
MS No. 99-032 received January 8, 1999, and reviewed under Institute publication
policies. Copyright
Ó 2000, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors.
Pertinent discussion will be published in the January-February 2001
ACI Structural
Journal
if received by September 1, 2000.
ACI STRUCTURAL JOURNAL TECHNICAL PAPER
This paper presents a performance based evolutionary topology
optimization method for automatically generating optimal strutand-tie models in reinforced concrete structures with displacement constraints. In the proposed approach, the element virtual
strain energy is calculated for element removal, while a performance index is used to monitor the evolutionary optimization
process. By systematically removing elements that have the
least contribution to the stiffness from the discretized concrete
member, the load transfer mechanism in the member is gradually characterized by the remaining elements. The optimal
topology of the strut-and-tie model is determined from the performance index history, based on the optimization criterion of
minimizing the weight of the structure while the constrained
displacements are within acceptable limits. Several examples
are provided to demonstrate the capability of the proposed
method in finding the actual load transfer mechanism in concrete
members. It is shown that the proposed optimization procedure
can produce optimal strut-and-tie models that are supported by
the existing analytical solutions and experimental evidence, and
can be used in practice, especially in the design of complex reinforced concrete members where no previous experience is available.
Keywords: corbels; deep beams; performance; reinforced concretes.
INTRODUCTION
The strut-and-tie model is considered the basic tool in the design and detailing of reinforced concrete members loaded in
bending, shear, and torsion. The structural idealization of a reinforced concrete member is to develop an analogous truss
model consisting of compressive struts and tensile ties that represent the actual load transfer mechanism in the concrete member for the applied loads and given support environments. The
strut-and-tie model developed is then employed to investigate
the equilibrium between the loads, the reactions, and the internal forces in the concrete struts and in the reinforcements
(Marti
1). The actual load carried by the strut-and-tie model is
treated as a lower-bound ultimate load for the reinforced concrete member based on the lower-bound theorem of plasticity.
This simple approach provides a clear understanding of the behavior of reinforced concrete members.
2 Moreover, it offers a
unified, intelligible, rational, and safe design framework for
structural concrete under combined load effects.
3
The truss models with 45-degree inclined compressive diagonals were originally developed by Ritter4 for the analysis and
design of reinforced concrete beams under shear in 1899. Later,
Mörsch
5 extended the truss models to the design of reinforced
concrete members under torsion. Recently, the consistent design approach proposed by Schlaich et al.
2,6 allows any part of
a concrete structure to be designed using the strut-and-tie modeling. The truss model approach is also recommended by the
ASCE-ACI Committee 445 on Shear and Torsion
3 for the shear
design of structural concrete. Based on this approach, developing an appropriate strut-and-tie model in the structural concrete
member is an important task for the structural designer. The
elastic stress distribution and load path methods have been used
to develop strut-and-tie models in structural concrete.
1,2 In the
elastic stress distribution approach, the strut-and-tie model is
formed by orientating the struts and ties to the principle stress
flows, which are obtained by performing a linear elastic finite
element analysis on the uncracked homogenization concrete
member. It is difficult, however, to find the correct models in
concrete members with complex loading and geometry conditions using these conventional methods that usually involve a
trial-and-adjustment process.
The layout optimization theory has been well-developed in
the past few decades (Rozvany et al.;
7 Topping8). Kumar9 and
Biondini et al.
10 used the truss topology optimization techniques to find optimal strut-and-tie models in structural concrete
members based on the ground structure approach. The continuum concrete member is represented by a ground structure that
consists of many truss members, and the linear programming
technique is employed to solve the truss topology optimization
problem. This method offers an automatic search for strut-andtie models in reinforced concrete members in an iterative process. Since the ground structure grid has a significant effect on the
optimal topology of the structure,
11 however, the chosen ground
structure may not adequately simulate the nature of a continuum
concrete member.
The topology optimization of continuum structures has received considerable attention in recent years. The homogenization method (Bendsøe and Kikuchi
12) has been proposed by
treating topology optimization as a material redistribution problem within a fixed continuum design domain. The solid isotropic microstructure with penalty (SIMP) method (Zhou and
Rozvany;
13 Rozvany et al.14) for intermediate densities can be
used to generate solid-empty type topologies for continuum
structures. The evolutionary structural optimization (ESO)
method (Xie and Steven;
15-17 Chu et al.18) is developed based
on material removal criteria. The performance indexes developed by Liang et al.
19-21 are used to monitor the optimization
process, and to measure the performance of structural topologies and shapes generated by different structural optimization
methods. The optimal topology of a plane stress continuum
structure produced by the continuum topology optimization is
often a truss-like structure. Therefore, it is appropriate to apply
Title no. 97-S36
Topology Optimization of Strut-and-Tie Models in
Reinforced Concrete Structures Using an Evolutionary
Procedure
by Qing Quan Liang, Yi Min Xie, and Grant Prentice Steven
ACI Structural Journal/March-April 2000 323
this technique in finding the best strut-and-tie models in reinforced concrete structures.
In this paper, a performance based evolutionary topology
optimization method based on the stiffness formulation is proposed and applied to automatically developing optimal strutand-tie models in reinforced concrete structures. The key features of the present topology optimization method are outlined.
Examples are provided to show the effectiveness and validity of
the proposed design optimization procedure in automatically
tracing the actual load transfer mechanism in concrete deep
beams, with and without web openings, slender beams, and a
corbel. The optimal strut-and-tie models obtained by the present
study are qualitatively compared with existing analytical solutions and experimental observations. The results and various related aspects are discussed.
RESEARCH SIGNIFICANCE
Although the potential of modern structural optimization
techniques has been realized by aeronautical, automotive, and
mechanical industries, they are still viewed by civil engineers as
academic exercises.
22 It should be esteemed, however, that
structural optimization can be much more than weight savings.
Structural optimization techniques can be used to find better topologies and shapes for the design of civil engineering structures. The present research exposes the capability of an
evolutionary topology optimization method for producing optimal strut-and-tie models in concrete members to civil engineers.
Extensive work has been undertaken on shear in structural concrete, but the shear behavior of a reinforced concrete member is still
difficult to understand due to its complicated nature. The load transfer mechanism of a reinforced concrete member is not the function
of a single variable, and it depends on the geometry, loading, and
support conditions of the member. It is time consuming and difficult for the structural designer to find appropriate strut-and-tie
models in complex members by using conventional methods.
Therefore, the proposed automatic design optimization procedure
is not only helpful for concrete researchers to understand the shear
resistance mechanism, but also a valuable design tool for concrete
designers in the design and detailing of reinforced concrete members using the strut-and-tie modeling.
PERFORMANCE BASED EVOLUTIONARY
OPTIMIZATION
Problem formulation
The topology optimization of a continuum structure is to find
the optimal geometry that minimizes the weight of the structure
under the applied loads while satisfying the requirement of constraints imposed on the structure. The optimization problem can
be stated as follows
(1)
(2)
where
W is the total weight of the structure; we is the weight of
the
e th element; te is the thickness of the e th element that is
treated as the design variables;
uj is the absolute value of the jth
constrained displacement;
uj* is the prescribed limit of uj; m is
the total number of displacement constraints; and
n is the total
number of elements within the structure.
It is known that some part of materials in the initial design domain is inefficient in carrying loads. The optimization task is to
identify the inefficient materials and to remove them from the
structure so that the objective of minimizing the weight can be
achieved while the constrained displacements are within the
prescribed limits. By means of systematically removing elements from the discretized concrete member, the actual load
paths within the concrete member can be gradually characterized by the remaining elements. In detail design, the concrete
usually remains in the structural member. Hence, the strut-andtie idealization of a reinforced concrete member offers a conservative design.
Element removal criteria
For structures subject to displacement constraints, it is desirable to eliminate elements that have a minimum effect on the
changes in the constrained displacements from the design. The
effect of element removal on the constrained displacements can
be evaluated by the element sensitivity numbers, which are calculated from the results of the finite element analysis. In the finite element analysis, the equilibrium equation for a static
structure is expressed by
(3)
in which [
K] is the stiffness matrix of the structure; {u} is the
nodal displacement vector; and {
P} is the nodal load vector.
When the
eth element is removed from a structure, the stiffness
and displacements are changed, and Eq. (3) can be rewritten as
(4)
where [
DK ] is the change of stiffness matrix, and {Du} is the
change of displacement vector. The change of the stiffness matrix due to the removal of the
eth element is
(5)
in which [
Kr] is the stiffness matrix of the resulting structure,
and [
ke] is the stiffness matrix of the e th element. The change of
displacement vector can be obtained by subtracting Eq. (3) from
Eq. (4) and neglecting the higher-order term as
(6)
To measure the change of the constrained displacement
uj due
to an element removal, a virtual unit load is applied to the position of the
j th constrained displacement. By multiplying Eq. (6)
with the virtual unit load vector {
Fj}T, the change of the constrained displacement is
(7)
minimize
W w
e( ) te
e
= 1
nå
=
subject to
uj uj* £ 0 j = 1 2 , , , ¼ m
[ ] K { } u = { } P
( ) [K ] D + [ ] K ( ) {u } D + { } u = { } P
[ ] DK = [ ] Kr [ ] K = –[ ] ke
{ } Du = –[ ] K –1[ ] DK { } u
Du
j = ( ) { } Fj T[ ] K –1[ ] DK { } u
ACI member Qing Quan Liang is a Research Associate in the School of Civil Engineering and Environment at University of Western Sydney, Australia. He graduated from the
Architectural Engineering Institute of Guangdong, China, in 1986, and received his ME
in civil engineering from the University of Wollongong, Australia, in 1998. His research
interests include structural concrete, stability and strength of composite steel-concrete
members, and structural optimization.
Yi Min Xie is an associate professor in the School of the Built Environment at Victoria University of Technology. He graduated from the Shanghai Jiao Tong University,
China, in 1984, and received his PhD from the University of Wales at Swansea, UK, in
1990. His research interests include computational mechanics and structural optimization.
Grant Prentice Steven is the Lawrence Hargrave Professor and Head of the Department of Aeronautical Engineering at the University of Sydney, Australia. He is also
the vice president of the Australian Association for Computational Mechanics. He
received his DPhil from Oxford University, UK, in 1970. His research interests

ACI Structural Journal/March-April 2000 324
include computational mechanics, finite element analysis, and structural optimization.
325
where {uj} and {uej} are the nodal displacement vectors of the
structure and the
eth element under the virtual unit load, respectively; and {ue} is the displacement vector of the eth element under the real loads. Equation (7) indicates the change in the
constrained displacement due to the removal of the
eth element,
and can be used as a measure of the element efficiency. Therefore,
the virtual strain energy for the
eth element in the design subject
to a displacement constraint is denoted by
(8)
For a structure under multiple displacement constraints, the
weighted average approach is used to calculate the virtual strain
energy for element removal. The virtual strain energy for the
eth
element for multiple displacement constraints is determined by
(9)
where the weighting parameter
lj is defined as uj/uj*. If the constrained displacement is far from the prescribed limit, it will be
less critical in the optimization process.
The elements with the lowest sensitivity numbers have little
effect on the changes in the constrained displacements, and can
be removed from the structure to obtain an efficient design. For
a structure under multiple load cases, only elements having the
lowest sensitivity numbers for all load cases are eliminated from
the design. This ensures that remaining materials in the structure
can safely carry all the loads. It is noted that minimizing the
changes in constrained displacements is equivalent to maximizing the stiffness of the structure. Since concrete permits only
limited plastic deformation, the best strut-and-tie model within
the concrete member is the one with the maximum stiffness or
minimum deflections, while its weight is the minimum.
2,9,23
Therefore, the optimization method based on displacement formulation is appropriate for finding optimal strut-and-tie models.
Performance index
In topology optimization, the cycle of finite element analysis
and element removal is repeated, and the quality of the resulting
structure is gradually improved. The performance of the resulting
topology at each iteration is evaluated by the performance index
that can be derived by the scaling design approach.
20
For a linear elastic plane stress continuum structure, the stiffness matrix is a linear function of the design variable such as the
element thickness, which has a significant effect on constrained
displacements and the weight of the final optimal design. To obtain the best feasible topology of a structure with the minimum
weight, the thickness of elements can be scaled at each iteration
in the optimization process so that the critical constrained displacement always reaches the prescribed limit.
24,25 By scaling
the initial design domain with a factor of
u0j/uj*, the scaled weight
of the initial design domain can be expressed by

order now
(10) sW0
u
j*
= è ø æ ö ——- W0

where W0 is the actual weight of the initial design domain, and
u0j is the absolute value of the j th constrained displacement that
is the most critical in the initial design under real loads. Similarly, by scaling the current design with respect to the most critical
u
= –{ } j T[ ] DK { } u = { } uej T[ ] ke { } ue
a
e
u
= { } ej T[ ] ke { } ue
a
e
l
j { } uej T[ ] ke { } ue
j
= 1
må
=
u
0j
displacement limit, the scaled weight of the current design at the
ith iteration is represented by
(11)
where
u
ij is the absolute value of the jth constrained displacement that is the most critical in the current design at the ith iteration under real loads, and Wi is the actual weight of the current
design at the
ith iteration.
The efficiency of material layout in a structure at the
ith iteration can be measured by the performance index, which is
defined as

(12) P I
sW0
——-
u
0jW0
= = ————–
sWi u
ijWi

It is seen from Eq. (12) that the performance index is a dimensionless number that measures the efficiency of material layout
in resisting the deflection and failure of a structure. The performance of a structural topology is improved by removing materials having the least contribution to the stiffness from the
structure. The objective of minimizing the weight of a structure
with displacement constraints can be achieved by maximizing
the performance index in an optimization process. The peak value of the performance index indicates that the best structure is
the one with the minimum weight and deflection, as pointed out
by Hemp.
23 The displacement limit uj* is eliminated from Eq.
(12). This means that the optimal topology does not depend on
the magnitude of the displacement limits. Therefore, displacement limits are set to large values in the optimization process in
the present study to obtain the optimal topology, which can then
be sized by changing the width of the member with respect to
the actual displacement limits.
It is worth noting that changing the element thickness has no
effect on the topology of the structure or on the performance index, but it has a significant influence on the weight of the structure and the constrained displacements. As a result of this, it is
not necessary to change the thickness of elements in the model
in the finite element analysis at each iteration. The performance
index can be employed to evaluate the efficiency of the resulting
topology at each iteration and to identify the optimum, which
can then be sized by changing the thickness of the structure to
satisfy the actual displacement limit.
Evolutionary optimization procedure
The design of a reinforced concrete member by using strutand-tie models usually involves the estimation of an initial size,
finding an appropriate strut-and-tie model, and dimensioning
struts, ties, and nodes. Developing an appropriate strut-and-tie
model for a complex concrete member is perhaps the most difficult task in the design process. Afterwards, dimensioning the
truss model is straightforward according to codes of practice,
26
and is not the objective of this paper. Interested readers should
refer to References 1 and 2 for details. The present topology optimization method can be used for developing the best strut-andtie models in concrete members in the design process. The optimization procedure is given as follows:
Step 1: Model the concrete member with fine finite elements;
Step 2: Analyze the concrete member for real loads and virtual unit loads;
Step 3: Calculate the performance index using Eq. (12);
Step 4: Calculate the virtual strain energy for each element
using Eq. (8) or (9);
Step 5: Delete a small number of elements with the lowest virtual strain energy; and
Wis
ui j
uj*
= è ø æ ö ——- Wi

ACI Structural Journal/March-April 2000 326
Step 6: Repeat Steps 2 to 5 until the performance index is less
than unity.
The performance index is used as an indicator of material efficiency and as the termination criterion in the previously mentioned iterative optimization process. Because the virtual strain
energy is calculated by neglecting the higher-order term in the
sensitivity analysis, only a small number of elements is removed
from the structure at each iteration to obtain a sound solution.
The element removal ratio (ERR) is defined as the ratio of the
number of elements to be removed to the total number of elements in the initial design domain and kept constant in the optimization process. It is obvious that the accuracy of the solution is
improved by adopting a smaller element removal ratio, but the
computational cost will be considerably increased. It has been
found that the ERR of 1 or 2% provides reasonable results for
use in engineering practice.
After extensive cracking of concrete, the loads applied to a reinforced concrete member are mainly carried by compressive
concrete struts and tensile steel reinforcements. The failure of a
reinforced concrete member cannot simply be explained by the
tensile stresses attaining the tensile strength of concrete; rather,
it is due to the breakdown of the load transfer mechanism at the
crack.
3 In current engineering practice, the behavior of reinforced concrete members is usually approximated by uncracked, cracked linear and limit analyses.27 Strut-and-tie
models are primarily used to predict the behavior of fully
cracked structural concrete members under the ultimate load
condition. Because the locations of tensile ties and the amounts
of steel reinforcement are not known in advance, the concrete
member is modeled using plane stress elements in the present
study. The linear elastic behavior of cracked concrete is assumed in the analysis as suggested by Schlaich et al.
2 Because
tensile ties in the strut-and-tie model obtained will be replaced
with steel reinforcements in a reinforced concrete member, the
effect of cracking due to stresses attaining the tensile strength of
concrete is not considered. The progressive cracking of a concrete member, however, is characterized by gradually removing
concrete from the member that is fully cracked at the optimum.
In nature, the loads are transmitted so that the associated strain
energy is a minimum. The topology optimization in this paper is
to find a strut-and-tie model as stiff as possible. The strength of
concrete struts, ties, and nodes can be treated when dimensioning the model.
EXAMPLE 1
The proposed procedure is used to find the best strut-and-tie
model in a simply supported concrete deep beam under two concentrated loads of
P
1 = 1200 kN placed at the bottom of the
beam, as shown in Fig. 1. The compressive cylinder strength of
concrete
fc¢ = 32 MPa; Young’s modulus of concrete E = 28,567
MPa; Poisson’s ratio
n = 0.15; and the initial width of the beam
b
0 = 250 mm are assumed in the study. The width of the beam
can be adjusted when dimensioning the strut-and-tie model obtained. The concrete beam is modeled using 50 mm square fournode plane stress elements. Two displacement constraints of the
same limit are imposed on the two loaded points in the vertical
direction. The ERR = 1% is adopted in the optimization process.
The performance index history of the deep beam loaded at the
bottom is presented in Fig. 2. While elements are systematically
removed from the deep beam, the performance index is gradually
increased from unity to the maximum value of 1.32, which corresponds to the optimal topology of the strut-and-tie model within
the concrete member. The topologies obtained at different iterations for this deep beam are given in Fig. 3. It can be observed
from Fig. 3 that when elements having the least contribution to the
structural stiffness are removed from the member, the actual load
transfer mechanism in the concrete member becomes clear, as
characterized by the remaining elements.
The optimal topology shown in Fig. 3(c) indicates the best
layout of the strut-and-tie model, in which the compressive arch
is formed, and the rest of the members are in tension. The strutand-tensile ties can be accurately located according to this optimal topology. The optimal topology shown in Fig. 3(c) is idealized as the strut-and-tie model shown in Fig. 3(d), which can be
used to determine the internal forces of the truss and reinforcement arrangements in the detail design. The vertical and inclined reinforcements should be provided to transfer the loads to
the compressive concrete arch. The dimension of the strut, ties,
and nodes should be undertaken according to codes of practice,
and is not discussed herein.
EXAMPLE 2
A simply supported deep beam with two web openings based
on the test specimen (O-O.3/3) presented by Kong and Sharp
28 is
illustrated in Fig. 4. Two concentrated loads of
P1 = 140 kN are
applied to the top of the deep beam. The compressive cylinder
strength of concrete
fc ¢= 35.5 MPa; Young’s modulus of concrete
E = 30088 MPa; Poisson’s ratio n = 0.15; and the width of the
beam
b = 100 are used in the analysis. The concrete beam is discretized into 25 mm square four-node plane stress elements. The
displacement constraints of the same limit are imposed on the two
loaded points in the vertical direction to obtain the optimal strutand-tie model with minimum deflections. The ERR = 1% is
adopted in the optimization process.
Figure 5 shows the performance index history of the deep
beam with web openings obtained by using the present procedure. The maximum performance index is 1.58, which indicates
that the resulting design represents the optimal topology of the
strut-and-tie model within the deep beam. The evolutionary topology optimization history is shown in Fig. 6(a) to (c), from
which it can be observed that the load transfer mechanism within the concrete deep beam are gradually manifested by the remaining elements. Ideally, the loads are transmitted along the
Fig. 1—Deep beam loaded at bottom.
Fig. 2—Performance index history of deep beam loaded at bot

327
shortest natural load paths between the loading and reaction
points. If the opening intercepts the natural load path, the load is
to be rerouted around the opening.
9 This is confirmed by the optimal strut-and-tie model shown in Fig. 6(d), which indicates
that the loads are transmitted to the supports by the upper and
lower struts around the opening. The presence of two inclined
tensile ties that connect the upper and lower struts around the
opening in Fig. 6(d) is supported by the experimental observations conducted by Kong and Sharp.
28
EXAMPLE 3
The best strut-and-tie model is needed to be found for a simply supported deep beam under the factored load P = 3000 kN
with a large hole, as shown in Fig. 7. This concrete deep beam
is based on the example given by Schlaich et al.
2 The compressive design strength of concrete fc = 17 MPa; Young’s modulus
of concrete
E = 20820 MPa; Poisson’s ratio n = 0.15; and the
initial width of the beam
b0 = 400 mm are used in this study. The
concrete beam is modeled using 100 mm square four-node plane
stress elements. A displacement constraint is imposed on the
loaded point in the vertical direction, and the ERR = 1% is
adopted in the optimization process.
Figure 8 demonstrates the performance index history of the
deep beam with a large hole. After reaching the peak value, the
performance index may drop sharply. This is because further element removal will cause large deflections. The maximum performance index is obtained as 1.65, which corresponds to the
optimal topology given in Fig. 9(c). The topologies obtained at
different iterations in the optimization process are shown in Fig.
9. It is seen that the load is to be rerouted around the opening,
even if the opening is very close to the support. The inclined tensile tie is developed across the upper right corner of the opening,
which tends to crack under the applied load. The optimal strutand-tie model obtained by the present study, as shown in Fig.
Fig. 3—Optimization history of strut-and-tie-model in deep
beam loaded at bottom: (a) topology at iteration 20; (b) topology at iteration 40; (c) optimal topology; and (d) optimal strutand-tie model (Note:
– – – = compressive strut; and —— = tenFig. 4—Deep beam with web openings.
Fig. 5—Performance index of deep beam with web openings.

ACI Structural Journal/March-April 2000 328
9(d), is similar to the strut-and-tie model given by Schlaich et
al.
2
EXAMPLE 4
This example is to investigate the effect of span-depth ratios
on optimal strut-and-tie models in simply supported concrete
beams under a concentrated load at the midspan of the beams, as
shown in Fig. 10. The depth of the beams
D is 1000 mm for all
cases, while the span-depth ratio for Cases (a) to (d) is 2, 3, 4,
and 5. The applied point load
P = 1200 kN, and the initial width
of the beam
b0 = 250 mm are assumed for all cases. It is noted
that the optimal topology of a linear elastic continuum structure
under the plane stress condition does not depend on the scale of
the point load and the width of the member. This can be seen
from Eq. (12). The value of the loading, however, affects the final dimensions of struts and ties. The width of the beams can be
changed to satisfy strength and stiffness requirements when dimensioning the truss models. The compressive cylinder strength
of concrete
fc ¢ = 32 MPa; Young’s modulus of concrete E =
28567 MPa; and Poisson’s ratio
n = 0.15 are used for all cases.
The concrete beams are modeled using 50 mm square four-node
plane stress elements, and the element removal ratio ERR = 1%
is employed for all cases.
The maximum performance indexes obtained for Cases (a) to
(d) are 1.88, 1.3, 1.23, and 1.21, respectively. The optimal topology and corresponding strut-and-tie idealization for each case
are presented in Fig. 11. It can be observed from Fig. 11 that the
truss model that ideally represents the load transfer mechanism
is changed from deep beams to slender beams. For beams with
a span-depth ratio
L/D ³ 3, inclined tensile ties connecting the
compressive concrete struts are necessary to form the truss model, as shown in Fig. 11(b) to (d). For very slender concrete
beams, optimal topologies obtained by the continuum topology
optimization method are continuum-like structures, in which
strut-and-tie actions are difficult to be identified, such as that
shown in Fig. 11(d). For such cases, the flexural beam theory
may be applied. These optimal strut-and-tie models indicate that
the angles between compressive concrete struts and longitudinal
ties are equal to or larger than 45 degrees. In detail design, some
of the bottom steel bars may be bent up to resist the inclined tensile stresses or the shear in the shear spans.
Fig. 6—Optimization history of strut-and-tie model in deep
beam with web openings: (a) topology at iteration 20; (b)
topology at iteration 40; (c) optimal topology; and (d) optimal
Fig. 7—Deep beam with large hole.
Fig. 8—Performance index history of deep beam with large
Fig. 9—Optimization history of strut-and-tie model in deep
beam with large hole: (a) topology at iteration 20; (b) topology
at iteration 40; (c) optimal topology; and (d) optimal strut-and

329
EXAMPLE 5
In this example, the corbel and column are considered as a whole
structure that is designed to support a point load of 500 kN, as illustrated in Fig. 12. The column is fixed at both ends. The compressive cylinder strength of concrete
fc ¢ = 32 MPa; Young’s
modulus of concrete
E = 28567 MPa; Poisson’s ratio n = 0.15; and
the width of the corbel and column
b = 300 mm are assumed. This
structure is modeled using 25 mm square four-node plane stress elements. A displacement constraint is imposed on the loaded point
in the vertical direction, and the element removal ratio ERR = 1%
is used in the optimization process.
Figure 13 shows the performance index history of the structure. The maximum performance index is 1.34, and the corresponding optimal strut-and-tie topology is shown in Fig. 14(c).
It can be observed from Fig. 14 that the applied load is transferred to the whole range of the structure along the paths of
compressive struts and tensile ties. This example shows that the
column and corbel should be treated as a whole structure in developing the best strut-and-tie model. The optimal strut-and-tie
model illustrated in Fig. 14(d) is supported by the solution obtained by the load path method.
2
DISCUSSIONS
Various examples given herein have shown that optimal strutand-tie models in concrete members can be generated by using
the proposed procedure. Although the present model considers
the elastic behavior of cracked structural concrete, it provides a
clear understanding of the nature of the load transfer mechanism
in reinforced concrete members. Moreover, the results obtained
by the present study confirm the findings of other researchers,
and are supported by experimental evidence. It should be noted
that there are no absolute optimal solutions. The objective of
shape finding is principally used as a vehicle to get a better design in terms of overall structural performance, and to free concrete designers from the time-consuming development of truss
models using conventional methods.
As mentioned previously, the load transfer mechanism in a reinforced concrete member depends on its geometry, loading, and
support condition. Without modification, the strut-and-tie model developed for a specific reinforced concrete member cannot be
Fig. 10—Simply supported beams with various span-depth
Fig. 11—Optimal topologies and truss models showing transition
from deep beams to slender beams: (a)
L/D = 2; (b) L/D = 3; (c)
L/D = 4; and (d) L/D = 5.
Fig. 12—Corbel jointed with column.
Fig. 13—Performance index history of corbel.

ACI Structural Journal/March-April 2000 330
used for different members. The initial size of a reinforced concrete member should be estimated based on the serviceability requirement. The proposed stiffness-based method produces the
optimal topology, which indicates only the locations of struts,
ties, and nodes of a strut-and-tie model in a structural concrete
member. Dimensioning the struts, ties, and nodes is left to the
designer. Since the width of a concrete member does not affect
the optimal topology, it can be adjusted to satisfy strength and
stiffness requirements when dimensioning the truss model obtained. It is common for continuum topology optimization
methods that the mesh size has a considerable effect on the result. The member geometry and computational time need to be
considered in choosing the mesh size. Since an optimal topology obtained is still a continuum structure, the strut-and-tie idealization based on the continuum topology may have redundant
members. It is suggested that the layout arrangement of steel reinforcement should follow the optimal strut-and-tie model as
closely as possible.
Conventional drawing board methods are especially not efficient in developing optimal strut-and-tie models in concrete
members under multiple load cases because it is difficult to superpose different models for different load cases. The present
computer-based topology optimization procedure, however, can
easily deal with multiple loading conditions, and it is not limited
to single and symmetry loading, although examples presented
herein consider only one load case. The proposed design optimization procedure can also be applied to finding optimal strutand-tie models in prestressed concrete structures and reinforced
concrete shearwalls.
FURTHER RESEARCH
Further theoretical research should be focused on studying
the effect of material property on the optimal strut-and-tie models and minimizing the effects of element mesh size and removal ratios on the results. Experimental work is also needed to
investigate the ultimate load capacity of reinforced concrete
members that are designed using optimal strut-and-tie models
generated by the present topology optimization technique. Test
results will be compared with current codes of practice.
CONCLUSIONS
A performance-based evolutionary topology optimization
method for automatically developing optimal strut-and-tie models in reinforced concrete structures has been presented in this
paper. Five examples that cover various types of reinforced concrete members have been provided to illustrate the effectiveness
of the proposed optimization procedure. It has been shown that
strut-and-tie models generated by the present optimization procedure are supported by existing analytical solutions and experimental observations. The method can also be applied to finding
optimal strut-and-tie models in prestressed concrete structures
and reinforced concrete shearwalls. Further theoretical and experimental work is needed to make topology optimization an integrated and friendly routine design tool for concrete designers.
Based on the present study, the following conclusions are drawn:
1. The proposed method in this paper is most appropriately
used for finding optimal strut-and-tie models in nonflexural
concrete members and in slender beams with
L/D £ 5;
2. For very slender concrete beams, the optimal topologies
obtained by the topology optimization method are continuumlike structures in which strut-and-tie actions are difficult to be
identified;
3. The present study shows that for a deep beam loaded at the
bottom, the vertical and inclined reinforcement should be provided to transfer the loads to the compressive arch with sufficient anchorage, but not necessarily to the top of the deep beam,
depending on the span-depth ratio of the beam;
4. When openings intercept the natural load paths, the load is
to be rerouted around the openings where inclined tensile ties
join the upper and lower struts. It is important to provide inclined reinforcement at the top and bottom of the opening. This
inclined reinforcement is efficient for crack control and for increasing the ultimate load capacity of the deep beam;
5. For reinforced concrete beams with
L/D ³ 3, inclined reinforcements bent up from bottom steel bars are most efficient in
resisting shear in the shear spans; and
6. In the structural idealization of corbels, the column that
joins the corbel should be considered together with the corbel in
developing the strut-and-tie model.
ACKNOWLEDGMENTS
This paper forms part of a research program into the evolutionary structural optimization as an efficient and reliable design tool. This program was
funded by the Australian Research Council under the Large Grants Scheme.
The first author is supported by an Australian Postgraduate Award and a
Faculty of Engineering and Science Scholarship. The first author wishes to
acknowledge helpful discussions with Prof. Lewis C. Schmidt in the Department of Civil, Mining, and Environmental Engineering at the University of
Wollongong, Australia.
Fig. 14—Optimization history of strut-and-tie model in corbel:
(a) topology at iteration 20; (b) topology at iteration 40; (c)
optimal topology; and (d) optimal strut-and-tie model.

ACI Structural Journal/March-April 2000 331
CONVERSION FACTORS
1 mm = 0.039 in.
1 kN = 0.2248 kips
1 MPa = 145 psi
NOTATIONS

b
b
0
D
E
=
=
=
=
width of member
initial width of beam
depth of beam
Young’s modulus of concrete
virtual unit load vector
compressive design strength of concrete
compressive cylinder strength of concrete
stiffness matrix of structure
stiffness matrix of resulting structure
stiffness matrix of
e th element
span of beam
total number of displacement constraints
total number of elements
load vector
performance index
thickness of
e th element
jth constrained displacement most critical in initial design
jth constrained displacement most critical in current design
absolute value of
jth constrained displacement
prescribed limit of
uj
nodal displacement vector
displacement vector of
eth element under real loads
displacement vector of
eth element under virtual unit load
displacement vector of structure under virtual unit load
total weight of structure
actual weight of initial design
scaled weight of initial design
actual weight of current design at
ith iteration
weight of
e th element
virtual strain energy of
eth element
change of stiffness due to element removal
change of displacement vector
{Fj} =
fc
fc ¢
=
=
[K ] =
[
Kr] =
[
ke] =
L
m
n
=
=
=
[P] =
PI
t
e
u0 j
uij
uj
u
j*
=
=
=
=
=
=
{u} =
{
ue} =
{
uej} =
{
uj} =
W
W
0
=
=
sW0
=
Wi
w
e
a
e
=
=
=
[DK] =
{
Du} =

l
j = weighting parameter
n = Poisson’s ratio
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