# Strut-and-tie design methodology for three-dimensional reinforced concrete structures

**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.

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**ABSTRACT:**The performance-based optimization (PBO) method has recently been developed for topology design of continuum structures with stress, displacement and mean compliance constraints. The PBO method incorporates the finite element analysis, modern structural optimization theory and performance-based design concepts into a single scheme to automatically generate optimal designs of continuum structures. Performance indices are used to monitor the performance optimization history of topologies while performance-based optimality criteria are employed to identify the optimum from the optimization process. The performance characteristics of a structure in an optimization process are fully captured. The PBO technique allows the designer to tailor the design to a specific performance level required by the owner. This paper reviews the state-of-the-art development of automated performance-based optimal design techniques for topology design of continuum structures. Several practical design examples are provided to demonstrate the effectiveness and validity of the PBO technology as an advanced design tool.Advances in Structural Engineering 12/2007; 10(6):739-753. · 0.60 Impact Factor - SourceAvailable from: Cristopher Dennis Moen[Show abstract] [Hide abstract]

**ABSTRACT:**Topology optimization techniques are employed to automate the design of reinforced concrete members. Truss models are derived with maximum stiffness (minimum total strain energy) from an initial ground structure defined over a general concrete member. The optimization routine, implemented with a freely available computer program, produces strut and tie geometries consistent with elastic tensile and compressive stress trajectories, resulting in steel reinforcement layouts with the potential to minimize crack widths and improve member performance over traditional strut and tie models. Ongoing work in continuum topology optimization of reinforced concrete members is summarized, including consideration of constructability in the optimized solution and the development of solutions with curved compressive struts which are more consistent with elastic stress trajectories than traditional strut-and-tie models derived by hand.05/2010; - [Show abstract] [Hide abstract]

**ABSTRACT:**This paper aims to develop a method that can automatically generate the optimal layout of reinforced concrete structures by incorporating concrete strength constraints into the two-material topology optimization formulation. The Drucker–Prager yield criterion is applied to predict the failure behavior of concrete. By using the power-law interpolation, the proposed optimization model is stated as a minimum compliance problem under the yield stress constraints on concrete elements and the material volume constraint of steel. The ε-relaxation technique is employed to prevent the stress singularity. A hybrid constraint-reduction strategy, in conjunction with the adjoint-variable sensitivity information, is integrated into a gradient-based optimization algorithm to overcome the numerical difficulties that arise from large-scale constraints. It can be concluded from numerical investigations that the proposed model is suitable for obtaining a reasonable layout which makes the best uses of the compressive strength of concrete and the tensile strength of steel. Numerical results also reveal that the hybrid constraint-reduction strategy is effective in solving the topology optimization problems involving a large number of constraints.Structural and Multidisciplinary Optimization 01/2013; 47(1). · 1.70 Impact Factor

Page 1

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.

DOI: 10.1061/?ASCE?0733-9445?2006?132:6?929?

CE Database subject headings: Concrete, reinforced; Design; Finite elements; Optimization; Three-dimensional analysis; Struts;

Trusses; Ties.

Introduction

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-

tent manner.

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

Methodology

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

and experience.

1Professor, Dept. of Civil Engineering, National Taiwan Univ., Taipei

10617, Taiwan ?corresponding author?. E-mail: ljleu@ntu.edu.tw

2AssistantProfessor,Dept.

Management, St. John’s Univ., Taipei 25135, Taiwan. E-mail: cwhuang@

mail.sju.edu.tw

3Associate Professor, Dept. of Civil Engineering, National Taiwan

Univ., Taipei 10617, Taiwan. E-mail: dchen@ntu.edu.tw

4Research Associate, Dept. of Civil Engineering, National Taiwan

Univ., Taipei 10617, Taiwan. E-mail: liao@caece.net

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/

2006/6-929–938/$25.00.

ofIndustrialEngineeringand

JOURNAL OF STRUCTURAL ENGINEERING © ASCE / JUNE 2006 / 929

Page 2

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

conventional method.

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

if

wi? RR ? wave

then the ith element is deleted

?1?

where wi= strain energy density of the ith element defined as

wi=

1

2Vi

?ui?T?ki??ui??2?

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= RR+ ER

?3?

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-

mum threshold.

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

defined as

PI=W0V0

WiVi

?4?

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

performance index.

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

follows.

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

conditions;

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

ft/fc?

abk1

k2

0.08

0.10

0.12

1.8076

1.2759

0.9218

4.0962

3.1962

2.5969

14.4863

11.7365

9.9110

0.9914

0.9801

0.9647

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

JOURNAL OF STRUCTURAL ENGINEERING © ASCE / JUNE 2006 / 931

Page 4

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

process.

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

Model

Members

Length

?cm?

Member

force

?kN?

Members

Length

?cm?

Member

force

?kN?

T1

T2

T3

T4

T5

T6

T7

40.0

67.5

60.0

67.5

40.0

67.5

40.0

580.7

3,171.2

1,618

2,800

1,660

2,800

1,660

C1

C2

C3

C4

C5

—

—

78.5

98.8

90.3

72.1

78.5

—

—

−1,139.1

−2,665.1

−3,761.2

−1,047.0

−3,256.1

—

—

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

Page 5

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

follows:

1.For the ith truss member connecting with the jth and

kth solid elements, calculate the length Lifor the truss

member

Li=??xj− xk?2+ ?yj− yk?2+ ?zj− zk?2

?5?

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

?6a?

t =?xq− xj??xk− xj? + ?yq− yj??yk− yj? + ?zq− zj??zk− zj?

?Li?2

?6b?

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

?7a?

if

dpq? r:Vi= Vi+ Velem

q

?7b?

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

q

=volume of the qth finite element.

4.

Ai=Vi

Li

?8?

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

cutoff distances.

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

As=T

fy

?9?

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

JOURNAL OF STRUCTURAL ENGINEERING © ASCE / JUNE 2006 / 933

Page 6

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

J2

fc?

+ bI1

fc?− 1

?10?

cos 3? =3?3

2

J3

J2

3/2

?11?

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???

3cos−1?− k2cos 3???

for cos 3? ? 0

?12a?

3−1

for cos 3? ? 0

?12b?

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

I1? 0

?13?

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

?Fig. 3?

? =?J2

?J2f

?14?

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

Method

Members

Length

?cm?

Member

force

?kN?

Members

Length

?cm?

Member

force

?kN?

TA1

TA2

TA3

TA4

—

—

40.0

67.5

60.0

78.5

—

—

580.7

3,171.2

1,618

3,267.8

—

—

CA1

CA2

CA3

CA4

CA5

CA6

78.5

98.8

90.3

72.1

40.0

67.5

−1,139.1

−2,665.1

−3,761.2

−1,047.0

−5.9

−9.52

Fig. 8. Effective area versus cutoff distance r for truss member CA3

?unit: cm?

Fig. 9. Strut-and-tie model of anchorage block resulting from RESO

method

934 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / JUNE 2006

Page 7

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.

Numerical Examples

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?.

Anchorage Block

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-

mal topology.

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

Members

Steel

reinforcement

Area of steel

?mm2?

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?

T1

T2

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?

T3

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

JOURNAL OF STRUCTURAL ENGINEERING © ASCE / JUNE 2006 / 935

Page 8

Pile Cap

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

the specification.

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%.

Conclusions

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

RESO Method

Members

Cross section area

?cm2?

Member force

?kN?

T1

T2

T3

C1

C2

1,618

442.5

4,200

2,423

4,634

166.54

83.27

1,183.46

−207.92

−1,769.09

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

design example

936 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / JUNE 2006

Page 9

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.

Notation

The following symbols are used in this paper:

Ai? relative area of ith truss member in strut-and-tie

models;

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

elements;

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.

Acknowledgment

This research was supported by the National Science Council of

the Republic of China through Grant No. NSC 91-2211-E-002-

078.

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Table 6. Summary of Reinforcements in Pile Cap Design Example from RESO Method

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