Strut-and-Tie Design Methodology for Three-Dimensional
Reinforced Concrete Structures
Liang-Jenq Leu1; Chang-Wei Huang2; Chuin-Shan Chen, M.ASCE3; and Ying-Po Liao4
Abstract: A strut-and-tie design methodology is presented for three-dimensional reinforced concrete structures. The unknown strut-and-
tie model is realized through the machinery of a refined evolutionary structural optimization method. Stiffness of struts and ties is
computed from an evolved topology of a finite element model to solve statically indeterminate strut-and-tie problems. In addition,
compressive strength for struts and nodal zones is evaluated using Ottosen’s four-parameter strength criterion. Numerical examples are
studied to demonstrate that the proposed design methodology is suitable for developing and analyzing three-dimensional strut-and-tie
models for reinforced concrete structures.
CE Database subject headings: Concrete, reinforced; Design; Finite elements; Optimization; Three-dimensional analysis; Struts;
In designing reinforced concrete structures, it is common practice
to classify portions of structures as either B or D regions. Most
design practices for B regions are well developed. On the other
hand, design for D regions, such as deep beams, corbels, joints,
and pile caps, is mostly based on heuristic methods and past ex-
perience ?Hsu 1993; MacGregor 1997?.
One approach to replace ad hoc D-region design practices is
the strut-and-tie method ?Schlaich et al. 1987?. In this method, the
complex flow of internal forces in D regions is transformed to a
truss-like structure carrying the imposed loading to adjacent B
regions or to its supports. A strut-and-tie model consists of struts,
ties, and nodes. Struts are compression members which represent
resultants of parallel or fan-shaped compressive stress fields. Ties
are tension members which mostly represent reinforcing steels,
but can occasionally represent prestressing tendons or tensile
stress fields. Nodes are the locations where the axes of the struts,
ties, and concentrated forces intersect. Nodal zones are thus sub-
ject to a multidirectional stress state. The strut-and-tie method
offers numerous advantages ?ASCE-ACI 1998? and has been
adopted in design code provisions recently ?FIP 1996; ACI 2002?.
Although the strut-and-tie method is conceptually simple, its
realization for complex D regions is not straightforward. The
major complexity involves how to transform a continuous de-
scription of a structural region to a discrete strut-and-tie model
?Liang et al. 2002?. Additional complexities include how to ac-
count for stiffness of struts and ties and how to evaluate concrete
effective strength ?Yun 2000; Tjhin and Kuchma 2002?. For the
strut-and-tie method to be reliably adopted in design practice, it is
imperative to manage these complexities in a unified and consis-
The advances in the field of structural topology optimization
open up new ways of resolving the implementation complexities
of the strut-and-tie method ?Liang et al. 2000, Liang et al. 2002;
Ali and White 2001; Biondini et al. 2001?. However, these studies
are mainly focused on two-dimensional structures. Thus, the main
objective of this study is to develop a strut-and-tie design meth-
odology for three-dimensional reinforced concrete structures. In
the following, the deficiencies of the conventional strut-and-tie
design methodology are addressed, in particular for tackling
three-dimensional problems. A design methodology aimed to re-
solve these deficiencies is then delineated. In the proposed meth-
odology, an appropriate strut-and-tie model is generated from
structural topology optimization. Member forces of a statically
indeterminate strut-and-tie model are calculated based on the
evolved topology. The bearing capacity of strut-and-tie models is
predicted by a concrete failure criterion. Finally, two numerical
examples are studied to demonstrate the applicability of this pro-
posed design methodology.
Challenges of Conventional Strut-and-Tie Design
The standard procedure of the conventional strut-and-tie design
methodology can be found in design code provisions ?ACI 2002?;
a conceptual flowchart is depicted in Fig. 1 for later comparison
in this study. Generally speaking, this procedure is a trial-and-
error iterative design process based mainly on designers’ intuition
1Professor, Dept. of Civil Engineering, National Taiwan Univ., Taipei
10617, Taiwan ?corresponding author?. E-mail: email@example.com
Management, St. John’s Univ., Taipei 25135, Taiwan. E-mail: cwhuang@
3Associate Professor, Dept. of Civil Engineering, National Taiwan
Univ., Taipei 10617, Taiwan. E-mail: firstname.lastname@example.org
4Research Associate, Dept. of Civil Engineering, National Taiwan
Univ., Taipei 10617, Taiwan. E-mail: email@example.com
Note. Associate Editor: Elisa D. Sotelino. Discussion open until
November 1, 2006. Separate discussions must be submitted for individual
papers. To extend the closing date by one month, a written request must
be filed with the ASCE Managing Editor. The manuscript for this paper
was submitted for review and possible publication on September 5, 2003;
approved on June 3, 2005. This paper is part of the Journal of Structural
Engineering, Vol. 132, No. 6, June 1, 2006. ©ASCE, ISSN 0733-9445/
of Industrial Engineeringand
JOURNAL OF STRUCTURAL ENGINEERING © ASCE / JUNE 2006 / 929
The conventional design methodology faces at least three
major challenges in design practices. The first challenge is that it
is not easy to generate an appropriate strut-and-tie model for a
structure, especially a three-dimensional one. In the literature,
most strut-and-tie models are developed by utilizing stress trajec-
tories from finite element analysis ?Schlaich et al. l987; Schlaich
and Schäfer 1991?. Directions of struts and ties are taken in ac-
cordance with those directions of principal compressive and ten-
sile stresses, respectively. This strategy, however, suffers from
two drawbacks. First, for a region with complex stress distribu-
tion, it is not a simple task to generate a corresponding strut-and-
tie model by inspection. Second, for a three-dimensional region, it
is very difficult for designers to visualize the stress trajectories in
the interior. Recently, structural topology optimization methods
have been employed to generate strut-and-tie models automati-
cally for two-dimensional reinforced concrete structures. For ex-
ample, Ali and White ?2001? and Biondini ?2001? used a linear
programming method to find an optimum strut-and-tie model. A
concrete structure was replaced by a ground structure consisting
of many truss members. However, the discrete ground discretiza-
tion may not be adequate to model the continuous structure. Con-
sequently, the resulting strut-and-tie model may not faithfully
represent the structure ?Liang et al. 2000?.
The second challenge faced by the conventional design meth-
odology is the issue concerned with stiffness determination of
each member in a statically indeterminate strut-and-tie model.
One way to handle the statically indeterminate case is to employ
the so-called plastic truss method. However, care must be taken
because of strain compatibility requirements and limited ductility
in concrete ?Tjhin and Kuchma 2002?. Schlaich and Schäfer
?1991? suggested that a statically indeterminate strut-and-tie
model could be decomposed into several statically determinate
ones. Each statically determinate truss model was then respon-
sible for sustaining an equal portion of external loading. How-
ever, it is not clear whether or not a statically indeterminate truss
model can always be decomposed. In addition, the assumption
that each statically determinate truss model takes the same exter-
nal loading may not be reasonable ?Ali 1997?. Recently, Yun
?2000? proposed an iterative approach to determine the relative
stiffness of statically indeterminate strut-and-tie members. How-
ever, its convergence property may not be assured.
The third challenge faced by the conventional design method-
ology is related to the issue of indirectly evaluating concrete bear-
ing capacity. The effective compressive strengths of the struts
and nodes are first obtained by looking up codes and guidelines
?e.g., Schlaich et al. 1987; FIP 1996; MacGregor 1997; ACI
2002?. The effective width of the strut is then computed from
dividing the member force by the effective strength. The shapes
and dimensions of the nodes are determined after the widths of
incoming struts are resolved. Finally, the truss model with finite
widths is evaluated to determine its suitability for the structure. If
the widths of struts are not suitable, the selected strut-and-tie
model has to be modified. For two dimensions, values for effec-
tive compressive strengths of struts and nodal zones are specified
in codes and guidelines. However, for three-dimensional strut-
and-tie models, no proven guidelines are yet available.
Proposed Strut-and-Tie Design Methodology
In this study, a strut-and-tie design methodology is proposed to
overcome the above-mentioned difficulties. The development is
centered upon the concept of transforming the design to a topol-
ogy optimization problem of a continuum structure ?Liang et al.
2002?. In addition, the proposed methodology utilizes information
passing from a continuum to determine effective stiffness and
strength properties needed in a strut-and-tie model. Doing so thus
allows us to reduce the ambiguous features encountered by the
Refined Evolutionary Structural Optimization Method
To generate a three-dimensional strut-and-tie model for a struc-
ture, the concept of topology optimization is employed herein.
Several methods have been proposed in the literature to solve the
topology optimization problem. Among them, the evolutionary
structural optimization ?ESO? method ?Xie and Steven 1993? is
one of the most popular methods. The ESO method is widely
reported and currently received intensive attention. One reason is
that the method is conceptually simple but the obtained results are
quite accurate compared with other methods. The other reason is
that it is easy to implement the ESO method in conjunction with
the finite element method.
Despite its popularity, the ESO method proposed by Xie and
Steven suffered some weaknesses ?Zhao et al. 1998?. In this
paper, the ESO is improved to find the optimal topology for a
three-dimensional continuum, from which the strut-and-tie model
can be derived. To distinguish the original ESO method from the
refined ESO method, the latter is denoted as the refined ESO
?RESO? method in this study. A brief introduction to the RESO
method is presented herein; details can be found in Huang ?2003?.
Fig. 1. Flowchart of conventional strut-and-tie design methodology
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The RESO method starts from a design domain constructed by
a finite element model with given loading and support conditions.
Through gradually removing ineffective materials, one can obtain
new topologies of the structure that are more effective than the
initial one. The material efficiency is determined from the strain
energy density of each element. Elements are deleted when their
strain energy densities are less than a rejection ratio ?RR? times
the average strain energy density of the structure ?wave?. The con-
dition can be expressed as
wi? RR ? wave
then the ith element is deleted
where wi= strain energy density of the ith element defined as
in which ?ui?=displacement vector; ?ki?=element stiffness matrix;
and Vi=volume of the ith element, respectively.
The finite element analysis and element elimination cycle are
repeated using the same RR until a steady state is reached;
this state represents null element deletion according to Eq. ?1?.
At this state, an evolution rate ?ER? is introduced and added to
RR= RR+ ER
in which ER is used as an increment of RR so that the elimination
criterion of elements is increased to the next level. It is worth
noting that the evolutionary process works properly only when
the number of elements removed at each evolution is small ?Hin-
ton and Sienz 1995?. If a large number of elements are removed
in a typical design evolution, it may result in missing the opti-
mum topology in the evolutionary process. To avoid this problem,
a maximum percentage of elements that can be removed in each
design evolution are prescribed. In this study, 1% of the elements
in the initial design domain are allowed to be removed in each
evolution. If the number of removed elements in each evolution
exceeds this maximum number, the current RR will be reduced
until the number of removed elements is smaller than the maxi-
The evolutionary process continues until a defined stop crite-
rion is reached. There are several ways to define the stop criterion
in the literature ?Xie and Steven 1993?. In this study, the perfor-
mance index ?PI? proposed by Liang and Steven ?2002? and Liang
et al. ?2002? is adopted to monitor whether the optimal topology
is reached. The index provides a measure of efficiency for differ-
ent design evolutions. A higher value of PI means a more efficient
topology for the structure.
The performance index, in terms of the total strain energies
and volumes at the initial and ith design evolutions, can be
where the superscripts 0 and i represent the initial and ith design
evolutions, respectively. In addition, W and V denote the total
strain energy and volume of the structure, respectively. The per-
formance index increases from unity to a maximum value when
the inefficient elements are removed gradually. However, further
element removal from the “optimal” topology structure that
has the highest value of the index will lead to decrease of the
stiffness of the structure. It then results in decreasing values of the
Sometimes, it is found that elements with restraints may be
removed without losing structural stability during the evolution-
ary process. However, removal of restraints leads to change of
design boundary conditions in the strut-and-tie design method.
Therefore, once any element with restraints is removed in a typi-
cal design evolution, the evolutionary process is also stopped.
The proposed RESO procedures can be summarized as
1.Assign initial values: RR=1%; ER=1%;
2.Discretize the design domain with a finite element mesh;
3.Perform linear elastic analysis for given loads and boundary
4. Calculate strain energy density values of all elements and
performance index of the structure;
5.Smooth element strain energy density values to avoid the
checkerboard pattern ?Li et al. 2001? and calculate the aver-
age strain energy density;
6. Remove inefficient elements with lower strain energy density
values according to Eq. ?1? with the constraint that the num-
ber of removed elements in each evolution shall be less than
1% of the number of elements of the initial structure. Other-
wise, a smaller value for RR shall be used; and
7.Repeat steps 3–6 until a stop criterion is reached.
Table 1. Values for Ottosen’s Four-Parameter Concrete Failure Criterion
Fig. 2. Illustration of region ?enclosed by dashed lines? for
two-dimensional truss element with cutoff distance r
Fig. 3. Illustration of second deviatoric stress invariant of applied
load and of material failure state
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In summary, the main differences between the RESO and ESO
methods are the removing algorithm in each evolutionary itera-
tion, the smooth process to prevent checkerboard patterns, and the
stop criterion by the performance index. Three remarks regarding
the practical implementation of the RESO method are listed in
order. First, removed elements are ignored in assembling the glo-
bal stiffness matrix in the RESO method. This technique has the
advantage in solving the equilibrium equations of finite element
analysis because the number of degrees of freedom becomes
less and less during the evolutionary process. Second, the removal
of elements may result in the case where one or more remained
elements do not have sufficient connectivities with other ele-
ments. For instance, if a quadrilateral element is connected to
the other elements by only one node, the structural stiffness ma-
trix will become singular. In this case, this element shall be re-
moved in order to have a nonsingular stiffness matrix. Finally,
two stop criteria are adopted in the RESO method. The evolution-
ary process is stopped when the performance index has reached
its maximum or elements with restraint are removed during the
Statically Indeterminate Strut-And-Tie Model Analysis
To compute relative stiffness of truss members, a simple stiffness
finding algorithm is developed. It is necessary because after an
appropriate strut-and-tie model is selected, member forces of the
model need to be determined. For statically determinate strut-and-
tie models, member forces can be obtained from equilibrium
equations. However, for statically indeterminate strut-and-tie
models, equilibrium equations and relative stiffness of members
are needed to compute member forces. The truss-like evolution-
ary topology offers a good estimate for the relative stiffness of
truss members. Each truss member within its reach should com-
prise some number of remaining finite elements. These finite el-
ements are naturally potential candidates to contribute stiffness
for the truss member.
Our stiffness finding algorithm can thus be established as fol-
lows. First, a cutoff distance r is chosen to allocate those finite
elements within the reach of a truss member. Fig. 2 illustrates the
region generated by the cutoff distance in two dimensions. In
Table 2. Member Forces in Anchorage Block Design Example from FIB
Fig. 4. Flowchart of proposed strut-and-tie design methodology
Fig. 5. Anchorage block design example of FIB ?load unit: kN?: ?a? configuration; ?b? strut-and-tie model
932 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / JUNE 2006
three dimensions, this cutoff distance yields a region with a cyl-
inder around the longitudinal axis and two hemispheres around
two nodes of this truss member. Second, remaining finite ele-
ments from the evolved topology located within this region are
identified. The effective area for the truss member is then ob-
tained from dividing the summing volume by the length of the
truss member. The proposed procedure can be summarized as
1. For the ith truss member connecting with the jth and
kth solid elements, calculate the length Lifor the truss
Li=??xj− xk?2+ ?yj− yk?2+ ?zj− zk?2
2. For the qth remaining element with the central point
?xq,yq,zq?, calculate the projected point ?xp,yp,zp? on the
longitudinal axis of the truss
zp= zj+ ?zk− zj?t
xp= xj+ ?xk− xj?t
yp= yj+ ?yk− yj?t
t =?xq− xj??xk− xj? + ?yq− yj??yk− yj? + ?zq− zj??zk− zj?
3. Calculate the distance dpqof all remaining elements to the
longitudinal axis of the ith truss. The volume of the ith truss
member Viis then the sum of the volumes of the solid ele-
ments within the cutoff distance r
dpq=??xq− xp?2+ ?yq− yp?2+ ?zq− zp?2
dpq? r:Vi= Vi+ Velem
in which Velem
Finally, the effective area of the ith truss member, Ai, can be
obtained by dividing the volume with the member length Li
=volume of the qth finite element.
We note that the evaluation of effective areas of truss members
depends naturally on the choice of cutoff distances. However,
thanks to the truss-like arrangement of the remaining finite ele-
ments, it is generally not very difficult to identify a suitable range
of cutoff distances for a given problem. In this range, member
forces of truss members are relatively insensitive to the choice of
After the effective areas of truss members are determined, the
elastic modulus of concrete ?Ec? is assigned to each truss member.
Member forces for the statically indeterminate strut-and-tie model
are obtained from the analysis. Given the force, the cross-
sectional area for the tension member is then calculated by
where As= required area for a tension tie. The T and fyrepresent
the tensile force and yield stress of the reinforcement. Care must
be taken as the real distribution of bars may need to be considered
when treating the nodes and checking for adequate anchorage.
Finally, consideration of secondary reinforcements may be
needed to control cracking.
Evaluation of Concrete Compressive Strength
To evaluate the effective compressive strength of the struts and
nodes, an analytical three-dimensional failure criterion for con-
Fig. 6. Optimal topology of anchorage block: ?a? evolutionary structural topology; ?b? connecting truss elements
Fig. 7. Performance index history of anchorage block
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crete is utilized. The strength of concrete under multiaxial stresses
is a function of the stress state and can only be properly deter-
mined by considering the interaction of various stress compo-
nents. The failure evaluation is only carried out for the initial
continuum structure, not during the evolutionary process. This is
because the evolutionary process is used to help engineers estab-
lish the corresponding strut-and-tie model. In real world, no ele-
ments are removed from the considered structure.
For two-dimensional cases, Yun ?2000? employed the biaxial
stress failure envelope proposed by Kupfer and Gerstle ?1973?
to evaluate the compressive strength of nodal zones from finite
element analyses. In this study, Ottosen’s four-parameter strength
criterion for concrete ?Ottosen 1977? is adopted. Ottosen’s
strength criterion is in good agreement with the experimental re-
sults over a wide range of stress states ?Chen 1982?. The four-
parameter failure function can be described as
?fc??2+ ??cos 3???J2
f?I1,J2,cos 3?? = a
cos 3? =3?3
where a and b=material parameters; I1=first invariant of the stress
tensor and J2and J3=second and third invariants of the deviatoric
stress tensor; fc?=uniaxial compressive strength of concrete; and
?= function of cos 3? and can be determined from the following
? = k1cos?1
? = k1cos??
3cos−1?− k2cos 3???
for cos 3? ? 0
for cos 3? ? 0
where k1and k2=material parameters. The values of a, b, k1, and
k2depend on the ratio of the uniaxial tensile strength ftto the
uniaxial compressive strength fc? ?Table 1?. For a given stress
state, if the value of the failure function, f?I1,J2,cos 3??, is
greater than zero, it means that concrete will fail under this state.
We note that only those elements under the compressive stress
state need to be checked by Ottosen’s four-parameter strength
criterion. A stress state that satisfies
is considered compressive. The tensile elements ?I1?0?, antici-
pated to fail in the cracking type, represent tension members of a
strut-and-tie model. Reinforcing steel and prestressing tendons
will be arranged for these tension members.
In addition to evaluating the compressive strength of concrete,
the strength criterion can be used to determine load-carrying
capacity of the structure. To this end, a stress ratio ? is defined as
where J2denotes the second deviatoric stress invariant of a stress
state due to a given load. The J2fdenotes the failure deviatoric
stress invariant by solving the quadratic equation defined by let-
ting Eq. ?10? be zero. For all the compressive elements satisfying
Eq. ?13? in an initial continuum, we can calculate the stress ratio
? according to Eq. ?14? and consequently find the maximum
value ?max. The load carrying capacity can then be determined by
dividing the given load by ?max. It is worth noting that the above-
obtained load-carrying capacity is in general rather conservative
because it assumes that the first occurrence of the limit state of an
element represents the limit state of a whole structure.
Flowchart for Proposed Design Methodology
Putting it all together, a flowchart for the proposed design meth-
odology is shown in Fig. 4. We first model the structure with solid
finite elements and evaluate the stress ratio ? for each compres-
sive element. An optimal topology is then generated using the
RESO method. Through the optimal topology, a truss model is
Table 3. Member Forces in Anchorage Block Design Example by RESO
Fig. 8. Effective area versus cutoff distance r for truss member CA3
Fig. 9. Strut-and-tie model of anchorage block resulting from RESO
934 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / JUNE 2006
developed interactively. The effective areas of the truss members
can be computed by the proposed stiffness finding method. Con-
sequently, the member forces of the truss model can be obtained.
Finally from the forces of the tension members, the required re-
inforcements can then be determined. Unlike the conventional
methodology illustrated in Fig. 1, most tasks in the proposed
methodology are performed automatically or semiautomatically.
The feasibility of the proposed strut-and-tie design methodology
for three-dimensional reinforced structures is demonstrated
below. In these examples, the eight-node brick element is used for
the finite element analysis ?Cook et al. 2002?. These examples are
conducted using an in-house object-oriented program, developed
to maximize flexibility and maintainability for future extension.
Additionally, interactive computer graphics are utilized to aid
modeling processes in three dimensions. Detailed features of the
program can be found in Huang ?2003?.
An anchorage block is considered in this example. The block is
supported by two rollers and one hinge at the corners of the lower
surface. The configuration and external loads of the anchorage
block are shown in Fig. 5?a?. The corresponding strut-and-tie
model suggested by FIB ?2002? is shown in Fig. 5?b?. In the
figure, the solid and dashed lines represent tension ties and com-
pressive struts, respectively. The member forces of the FIB model
are listed in Table 2.
The design domain is discretized with 2,250 equal-sized finite
elements. Each side of the element is equal to 4.5 cm. The elastic
modulus Ecand Poisson’s ratio ? of concrete are assumed to be
26,000 MPa and 0.2, respectively. Fig. 6?a? shows the optimal
topology developed by the RESO method with the highest value
of the performance index. Fig. 6?b? shows the truss model derived
from the evolved topology. The performance index history is
shown in Fig. 7; it takes 1,556 evolution steps to reach the opti-
From the truss model in the evolved topology ?Fig. 6?b??,
designers can select an appropriate cutoff distance r to calculate
the effective areas and consequently member forces of the truss.
Fig. 8 shows the relationship between cutoff distances r and com-
puted areas for the member CA3in Fig. 9. From Fig. 8, we can
observe a tendency that in a specific range of cutoff distances, the
increasing rate of effective areas is reduced. In this range, the
computed member forces are relatively insensitive to the choice
of cutoff distances. The member forces computed in this example
are listed in Table 3.
The strut-and-tie model suggested by the RESO method is
shown in Fig. 9. From Figs. 5?b? and 9, one can observe that the
strut-and-tie model developed by the RESO method is similar to
that of FIB ?2002?. The main difference occurs in the transmis-
sion of the upward load PZ8. In Fig. 5?b? ?FIB 2002?, the load PZ8
is carried through the tension member T7. In the RESO method
or Fig. 9, however, the load PZ8is carried by both the tension
member TA4and the compressive member CA5, which makes the
tension member T6of Fig. 5?b? become compressive member CA6
of Fig. 9.
There are two possible reasons to explain the difference be-
tween the two. First, the model suggested by the RESO method
depends mainly on the material efficiency in carrying external
loading. In general, the summing product of the tie length and the
tensile force can be used as a simple criterion for optimizing a
strut-and-tie model ?Schlaich et al. 1987?. In this example, the
proposed model has a smaller value than that of the FIB design
example; it is thus believed to be an optimum one. Second, the
model suggested in the FIB design examples may need to take the
construction process into account. Thus the FIB model leads to
the reinforcement arrangements being parallel to the edges of the
structure in order to simplify construction process.
Table 4. Summary of Reinforcements in Pile Cap Design Example
Area of steel
7-No. 10aat 45 mm
7-No. 10 at 45 mm
7-No. 10 at 45 mm
3-No. 10 at 45 mm
700 ?layer 1?
700 ?layer 2?
700 ?layer 3?
300 ?layer 1?
5-No. 10 at 45 mm
3-No. 10 at 45 mm
4-No. 10 at 70 mm
500 ?layer 2?
300 ?layer 3?
400 ?layer 1?
4-No. 10 at 70 mm
4-No. 10 at 70 mm
400 ?layer 2?
400 ?layer 3?
aNo. 10 is a 11.3 mm diam bar with a cross-sectional area of 100 mm2
Fig. 10. Pile cap design example ?unit: cm?: ?a? top view; ?b? side view
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The example is one of the six pile caps conducted experimentally
by Adebar et al. ?1990? to examine the suitability of three-
dimensional strut-and-tie models for the pile cap design. Details
of the test specimen are given in Fig. 10 and Table 4. The pile cap
is supported on six piles and designed using a strut-and-tie model
for an applied load of 3,000 kN acting on the column. The elastic
modulus Ecand Poisson’s ratio ? for concrete are assumed to be
26,000 MPa and 0.2, respectively. The concrete cylinder uniaxial
compressive and tensile strength are 27.1 and 3.7 MPa, respec-
tively. Consequently, the ratio of concrete tensile strength to com-
pressive strength is roughly taken as 0.12 in the four-parameter
failure criterion. In addition, the yield stress and nominal area for
Number 10 steel are 479 MPa and 100 mm2, respectively.
The design domain from Fig. 10 is discretized with 4,420
equal-sized finite elements. The length, width, and height for an
element are 10, 10, and 6 cm, respectively. The final evolutionary
topology is reached when the elements in conjunction with the
four outer piles are ready to be removed. The resulting topology is
shown in Fig. 11. One can observe that the two legs connected to
the middle piles comprise more elements than the other piles.
This load-carrying mechanism agrees well with the experimental
observations in which these two piles resist the majority of loads
?Adebar et al. 1990?.
From the final topology, designers can develop a strut-and-tie
model as shown in Fig. 12, where the solid and dashed lines
represent tension ties and compression struts, respectively. This
model is the same as that of Adebar et al. ?1990? which was
obtained from the designers’experiences. For this example, a cut-
off distance of 40 cm is employed. The calculated effective areas
and member forces are listed in Table 5. As one would expect, the
C2members in the middle have larger cross-sectional areas than
the other members. The required reinforcement areas of the strut-
and-tie model are shown in Table 6.
It is worth noting that designers have to take specifications
about temperature reinforcements ?ACI 2002? into consideration.
The minimum ratio for temperature reinforcements to the gross
concrete area is 0.0018. Consequently, the final reinforcement
arrangements for the strut-and-tie model are listed in Table 6.
One can observe that the longitudinal reinforcements ?T1? and
the outer transverse reinforcements ?T2? are increased to satisfy
It is interesting to compare the reinforcement requirements
computed from the optimal topology with those obtained empiri-
cally in Adebar et al. ?1990?. From Table 6, one can observe that
the derived strut-and-tie model by the RESO method uses a total
of 2,800 mm2reinforcements ?or 3,600 mm2with the tempera-
ture specification taken into account?. On the other hand, the
empirical arrangement in Adebar et al. ?1990? from Table 4 uses
a total of 4,400 mm2reinforcements. This is expected because
the optimal topology from the RESO method should produce a
more efficient load-carrying mechanism.
Finally, the ultimate load of the pile cap from the experimental
measurement is 2,892 kN ?Adebar et al.?. The external load used
in the finite element model is 2,980 kN, and the maximum stress
ratio ?maxis 1.053. In other words, the ultimate load for concrete
failure predicted by this study is 2,830 kN. The predicted ulti-
mate load is less than the experimental result by only 2.1%.
In this paper, a design methodology for developing a three-
dimensional strut-and-tie model is presented. A strut-and-tie
Table 5. Areas and Member Forces in Pile Cap Design Example from
Cross section area
Fig. 11. Pile cap design example by RESO method: ?a? topology from RESO ?top view?; ?b? topology from RESO ?side view?
Fig. 12. Strut-and-tie model suggested by RESO method for pile cap
936 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / JUNE 2006
model for the considered reinforced concrete structure is estab-
lished based on the RESO method. For a statically indeterminate
case, a simple stiffness finding algorithm is developed to estimate
the relative stiffness of the truss members from the evolved to-
pology. The concrete compressive strength evaluation is per-
formed by Ottosen’s four-parameter strength criterion.
Two design examples demonstrate the suitability of the pro-
posed methodology. In the anchorage block example, the derived
strut-and-tie model by the RESO method agrees with those from
design experience. A small difference between the two can mainly
contribute to the issue concerning with construction in practice. In
the pile cap example, we observe that the load-carrying mecha-
nism from the evolved topology agree well with the experimental
measurements. The load-carrying capacity of the structure is also
well predicted by Ottosen’s failure criterion.
Finally, we must emphasize that the proposed method is
mainly based on linear elastic analysis. The effects of concrete
crack are indirectly taken into account while evaluating concrete
compressive strength. However, the effects of the tension stiffen-
ing, the arrangement of steel reinforcements and the degree of
disruption of the nodal zone due to the incompatibility of tensile
strains in the ties are not considered. Furthermore, the effects of
nonlinear load-deformation responses and softening of concrete
are not taken into account. These limitations and uncertainties
need to be addressed in the future.
The following symbols are used in this paper:
Ai? relative area of ith truss member in strut-and-tie
As? cross-sectional area of reinforcement;
dpq? distance of pth remaining element to projected
point on ith truss member;
ER ? evolution rate;
fc? ? uniaxial compressive strength of concrete;
fy? yield stress of reinforcement;
I1? first invariant of stress tensor;
J2? second invariant of deviatoric stress tensor;
J3? third invariant of deviatoric stress tensor;
?ki? ? element stiffness matrix corresponding to ith
Li? length of ith truss element;
PI ? performance index;
RR ? rejection ratio;
T ? tension force in strut-and-tie models;
?ui? ? displacement vector corresponding to ith element;
V ? volume of structure;
Vi? volume of ith truss element;
W ? strain energy of structure;
wi? strain energy density of ith element;
wave? average strain energy density of structure;
?x,y,z? ? coordinates; and
? ? stress ratio of compression elements.
This research was supported by the National Science Council of
the Republic of China through Grant No. NSC 91-2211-E-002-
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