Strut-and-tie modeling technique is a simple and effective method which can be used as a
quick tool for analysis of discontinuous region (D-region) in reinforced and prestressed concrete
structures. It serves practicing engineers to grasp load transfer characteristics in order to provide
good details of reinforcement and to determine load carrying capacity of the members in very
effective way. Since the method is based on lower bound theorem of plasticity, it can be assured
to deliver safe designed structure. Using of strut-and-tie modeling demands clear understanding
of load path and good skill in visualization of stress field. Those understandings can be improved
by studying several model examples and practicing with various design problems.
This course material aims to provide fundamental knowledge of strut-and-tie modeling
starting from basic principles, basic of strut-and-tie modeling, review of ACI 318 Appendix A
provision for strut-and-tie modeling. The last two chapters are related to software used for strut-
and-tie modeling with worked examples. Many contents in this material are obtained from
several excellent textbooks and technical papers that the author found to be useful and
understandable. An interested reader may follow the list of references provided for further
reading. Finally, the author hopes that this material will be useful not only for students but also
practicing engineers to gain understanding in this topic and lead to improvement in their
1. Basic Principles
2. Basic Strut-and-Tie Modeling
3. ACI 318M-08 Appendix A Provisions
4. STM Software and Applications
5. Worked Examples with CAST
1.1 Discontinuous region and applications of STM
Discontinuous region (D-region) is the region where elementary beam theory (cf. Bernoulli’s
beam theory) ceases to be valid, i.e. plane section does not remain plane (due to nonlinear strain
distribution), beam thickness does change (where normal stress along thickness direction cannot
be neglected), and plane section does not remain normal to the neutral axis (shear strain may
become large). D-regions are usually located where loads are applied or adjacent to support.
They are also caused by load discontinuity due to openings or sudden change in model geometry
like dapped end beam or stepped beam. Fig. 1.1 identifies common B- and D-regions in
reinforced and prestressed concrete frame. To analyze complete stress and strain distribution in
D-region, we need to apply continuum mechanics approach by considering state of equilibrium,
compatibility and stress-strain relationship. Numerical methods like Finite Element Method
(FEM), or Boundary Element Method (BEM) are commonly applied to solve such problems for
arbitrary loads and model geometries.
Figure 1.1 Identification of B- and D- regions in typical concrete frame
Should neither complete elastic stress (or strain) nor displacement of the D-region is necessary
to be concerned as in typical reinforced concrete design problems, a simplified approach
satisfying equilibrium without considering compatibility and constitutive relationship can be
used. As proposed by Schlaich et al. (1987), Strut-and-Tie modeling (STM) can be used as an
alternative approach for analysis and design of D-region. STM has been evolved and included in
ACI 318 appendix A since ACI 318-02. A typical STM consists of struts, ties and nodal zones as
shown in Fig. 1.2. Those three elements form in a truss-like structure in order to transfer loads to
Figure 1.2 Typical strut-and-tie model (ACI 318M-08 Appendix A)
In STM methodology, factored axial loads in struts and ties are checked against their strength
multiplied by reduction factor as written in the following expression:
i i n
= service load;
corresponding load factors;
1.2 Lower bound theorem of theory of plasticity
STM relies on force equilibrium at nodes. The equilibrium equations applied at all nodes can
be written by
i = 1..ndim (at all nodes) (1.2)
It should be noted that no compatibility requirement is required in STM. Furthermore, the
influence of stiffness must be treated independently in statically indeterminate structure. Neither
deformation (strain) nor stress distribution can be evaluated by STM.
The lower bound theorem of plasticity
“A load system
, based on a statically admissible field which nowhere violates the yield
condition is a lower bound to the collapse load
” (Muttoni, Schwartz and Thuerlimann,
This implies that the solution obtained from the lower bound theorem (static method) usually lies
on safe side or in conservative sense. The static method does not consider kinematic condition
formed by plastic hinge mechanism. By neglecting plastic mechanism to be formed, STM will
always give the collapse load below the true collapse load estimated by plastic analysis and
hence safe design can be achieved. It should be noted that since different STMs also yield
different results, the one that give highest factor of safety against collapse load shall be adopted
as it is still less than the true collapse load. On the other hands, for a given set of loads, the model
having minimum strain energy or giving minimum rebar amount shall be the best utilized model.
Fig. 1.3 summarizes the solution by plastic theory. The true collapse load is the load where static
solution (lower bound) meets kinematic solution (upper bound).
Figure 1.3 Overview of plastic solution methods (Muttoni et al., 1997)
1.3 Topology optimization and applications to STM
Topology optimization is a class of optimization techniques (size, shape, and topology) to
reveal the most efficient material usage for a given set of loads and boundary conditions (Fig.
1.4). Topology optimization plays an important role in finding optimum STM and has become a
root of several techniques use for optimization of STM. Among other techniques, Evolutionary
Structural Optimization technique (ESO) (Xie & Steven, 1992; Xie and Steven, 1997; Huang &
Xie, 2010) have been widely used and applied to well-posed 2D and 3D problems.
Figure 1.4 Topology optimization of a 2D structure (Cristensen and Klarbring, 2008)
1.3.1 Evolutionary Structural Optimization Technique
Evolutionary structural optimization (ESO) technique (Xie & Steven, 1992; Xie and Steven,
1997; Huang & Xie, 2010) is based on the simple concept of gradually removing inefficient
material from a structure. Through this process, the resulting structure will evolve towards its
optimal shape and topology. Many criteria can be used to reject part of material that deems
inefficient such as level of stress, strain energy, and so on. One simple method to determine
element removal is to consider the level of element stress, for example, the von Mises stress
of the element with the maximum von Mises stress of the whole structure
. After each finite
element analysis, elements which satisfy the following condition are deleted from the model.
is the current rejection ratio (RR). Several cycles of finite element analysis are to be
performed until the topology is unchanged under the constant RR, which means that no more
element can be deleted using the current RR. By increasing RR by evolutionary ratio (ER) as
RR RR ER
After performing several cycles of analysis, more elements can be removed and the more
efficient topology can be obtained.
Fig 1.5 shows the process of ESO in obtaining optimal topology of a deep beam subjected to a
Figure 1.5 ESO topologies for a Mitchell type structure with two simple supports (Xie and
Steven, 1993) (a) RR = 5%; (b) RR = 10%; (c) RR = 15%; (d) RR = 20%; (e) RR = 25%
1.3.2 Sensitivity Number and Evolutionary Procedure
Considering stress level in removing elements that subject to low stress is not suitable to the
structure with limited plastic deformation like reinforced concrete. Sensitivity of the element to
structural stiffness can be good criterion in removing the elements that contribute less to the
overall stiffness of the structure.
Consider the mean compliance written in the form
is nodal force vector;
is displacement vector. In finite element analysis, static
equilibrium of the structure is expressed as
is the global stiffness matrix.
Varying Eq.(1.6) and assume that external nodal force does not change,
0 K u K u f
u K Ku
It is simple to prove that change of global stiffness is equal to negative of the stiffness matrix of
the elements that have been removed, so that
Using Eqs. (1.5), (1.8), and (1.9), the change in the mean compliance can be determined from
1 1 1
2 2 2
f u f K K u u K u
*Since the component of
does not contribute to strain energy to element
is nodal displacement vector of element ith. Thus the sensitivity number of element th
can be defined as
u K u
It can be noted that increase in the mean compliance as a result of the removal of the ith element
is equal to its elemental strain energy. To minimize the mean compliance (which is equivalent to
maximizing the stiffness) through the removal of elements, it is evident that the most effective
way is to eliminate the elements which have the lowest value of
so that the increase in
be minimal. The elemental removal ratio (ERR) as defined by the number of elements to be
removed to the total number of elements can be used to update the initial or current analysis
The evolutionary procedure for stiffness optimization (maximize stiffness) is given as follows:
Step 1: Discretize the structure using a fine mesh of finite elements;
Step 2: Carry out finite element analysis;
Step 3: Calculate the sensitivity number for each element using (1.11);
Step 4: Remove a number of elements with the lowest sensitivity numbers according to a
predefined element removal ratio ERR;
Step 5: Repeat steps 2 to 4 until the mean compliance (or the maximum displacement, etc.) of the
resulting structure reaches a prescribed limit.
1.3.3 Evolutionary Structural Optimization with Minimum Weight
The previously described evolutionary structural optimization techniques do require the analyst
to manually specify the portion of elements to be removed. Based on the sensitivity number
described above Liang et al. (1999) proposed the topology optimization problem of a continuum
structure which can be stated as follows:
W w t
is the total weight of the structure,
is the weight of the
is the thickness
element that is treated as the design variables,
is the magnitude of the jth
is the prescribed limit of jth displacement, m is the number of
displacement constraints and n is the total number of elements within structure.
To determine the sensitivity of the element to change in constrained displacement
, a unit
load is applied to the jth displacement. Sensitivity number of the element subjected to unit load
can be computed from
j ej e e
u u k u
is the nodal displacement vector of the eth element for the unit load and
displacement vector of the eth element under the applied loads. Eq. (1.14) indicates the change in
constrained displacement due to element removal, and can be used as a measure of the element
The element with low sensitivity number will make the whole structure subject to small change
when being removed. In other word, the elements that have the lowest sensitivity number do
have little contribution to the stiffness of the structure and can be removed from the structure to
achieve a more efficient design. Since concrete permits only limited plastic deformation, the
strut-and–tie model with maximum stiffness or minimum deflections is the best while its weight
is the minimum (Schlaich et al. 1987). Based on this fact, the ESO method for plane stress
continuum structures subject to displacement constraints is appropriate for the topology
optimization of strut-and-tie in non flexural reinforced concrete members. To minimize weight
of the structure, performance index (PI) is used to measure the material efficiency in resisting
deflection and failure of a plane stress structure. It can be defined as
is the initial weight of the structure;
is the weight of the structure at iteration ith;
is the magnitude of the jth nodal displacement in the initial design under the applied loads,
is the jth nodal displacement in the current design at the ith iteration under applied loads.
If the density is constant, performance index can be written by
is the volume of the initial design domain and
is the volume of the current design
at the ith iteration. Performance index varies reversely to the volume and nodal displacement of
the current design. For a typical solution, PI will increase when the iteration bring the structure to
be lighter. For a certain cycles of iteration when some effective elements are removed, the
displacement of structure become larger and performance index becomes lower. The iteration
shall be terminated when performance index become less than 1.0.
The optimization procedure is given as follows:
Step 1: Model the concrete member with a fine mesh of finite elements;
Step 2: Analyze the structure for the applied loads and unit loads;
Step 3: Calculate the performance index using Eq. (1.6)
Step 4: Calculate the sensitivity number for each element using Eq. (1.14) or (1.15)
Step 5: Remove elements with the lowest sensitivity numbers;
Step 6: Repeat step 2 to 5 until the performance index is less than unity.
Figure 1.7 Optimization of strut-and-tie model in RC deep beam with web opening (Liang et al.,
1.4 STM vs. FEM
Finite Element Method (FEM) is a powerful numerical method for solving partial differential
equations. Based on the principle of virtual work or the principle of minimum total potential
energy, the equilibrium equation can be formulated where displacement, stress, strain or all of
them can be used as independent variables. The FE solution must satisfy the following
conditions (Wilson, 2010):
• Force equilibrium (at node),
• Kinematic (strain-displacement) relationship, and
• Constitutive equations (at Gauss points)
Formulating a FE system requires four main steps (Bathe, 1996):
1. The structure or continuum is idealized as an assemblage of discrete elements
connected at nodes pertaining to the elements.
2. The externally applied forces (body forces, surface traction initial stresses,
concentrated loads) are lumped to these nodes using the virtual work principle to obtain
equivalent externally applied nodal point forces.
3. The equivalent externally applied nodal point forces (calculated in 2)are equilibrated
by the element nodal point forces that are equivalent (in the virtual work sense) to the
element internal stresses; i.e., we have
4. Compatibility and the stress-strain material relationship are satisfied exactly, but
instead of equilibrium on the differential level, only global equilibrium for the complete
structure, at the nodes, and of each element m under its nodal point force
Generally, the FE system equation is sparse and symmetric where most non-zero values are
located within the diagonal band of the system matrix (
). Specific solvers employing those
special characteristics like active column solver (Wilson, Bathe) can be used efficiently to solve
such system. The nodal solution can be obtained by solving a system of linear algebraic
equation. For displacement based FEM, nodal displacements are solved. Strain components at
integration points can be computed from the strain-displacement relationship. Stress components
at integration points are computed from constitutive relationship.
For assuring monotonic convergence of the FE solution, the finite elements used must also
satisfy the following conditions:
Completeness stands for the capability of the element to represent constant strain condition
including zero strain as in case of rigid body motion. The element that is incapable to represent
constant strain will never reach to the exact solution. Compatibility of the element inter-
connection is not stringent as long as the elements can pass patch test. Incompatible (non-
conforming) elements, if properly designed, can overcome locking problems as in case of shear
locking and volumetric locking.
For displacement-based FEM, the solution of nodal displacement always brings the model to
satisfy force equilibrium at nodes and at elements as shown in Fig. 1.7.
Figure 1.7 Nodal point and element equilibrium in a finite element analysis (Bathe, 1996)
However, continuous of stress or traction along inter-element boundary are not satisfied, except
at a specific state of constant strain (stress). Hence, the element is known to satisfy equilibrium
in ‘global’ sense. The accuracy of FE solution can be judged by considering smoothness of stress
along element boundary. A useful concept of pressure band visualization (Bathe, 1996) can be
used to evaluate stress discontinuities and accuracy of FE solutions as shown in Fig. 1.8.
Fig. 1.8 Pressure band for estimating stress discontinuities, with of bands = 5 Mpa
(a) Smooth pressure band (b) visible discontinuities (c) significant discontinuities (Bathe,
To simplify the problem, STM employs only equilibrium condition at nodes without
satisfying other two stringent conditions. It is therefore only applied to statically determinate
system without any specific material characterization. The solution of STM then only
contains element and nodal forces (reactions). It does not give any information about nodal
displacement and stress (strain). However, as we use STM to model the equilibrium
condition at limit state, elastic stress (strain) and deformation are not so relevant. Since both
STM and (elastic) FEM employ the equilibrium equation without considering material
yielding, they are expected to give the limit load less than theoretical collapse load as against
to plastic mechanism analysis.
The solution of FE depends strongly to mesh quality (performance, aspect ratio, etc.) and
mesh density. When the finer mesh is employed, the structure becomes less stiff and should
lead to a better solution. The coarse mesh leads to smaller displacement and smaller stresses.
Nonlinear finite element for plastic analysis is nothing related to the bound theorem of
plasticity. The method fully employs nonlinear stress-strain relationship where plastic region
can be determined by integrating stress over the domain as so-called stress return algorithm.
The solution is therefore unique, at least for monotonic stress-strain increment. For elastic
analysis, since equilibrium is applied, it is possible to use elastic FE solution to figure out an
optimal ST model. Kanok-Nukulchai & Anwar (1996) employ FEM to come across optimal
ST models of pilecaps with arbitrary pile arrangement. With aids of visualization technique,
Dammika & Anwar (2010) demonstrate general approach to extract 3D ST models from FE
solid element mesh for several complex structures such as pilecaps and bridge piers as shown
in Fig. 1.9.
Figure 1.9 Principal Compressive Stress Contours of Pier Head under Point Loading & Extracted
Strut and Tie Layout (Dammika & Anwar, 2010)
Basic Strut-and-Tie Modeling
Struts are compression elements used to transfer loads from nodes to another nodes. Struts
may be reinforced as columns or as typical walls. The influence of passing rebar must be
considered in specifying compressive strength of struts. According to ACI 318-08 Appendix A,
struts can be assigned as prismatic or bottle-shaped as shown in Fig. 2.1. Idealization of bottle-
shaped strut is shown in Fig. 2.1 (b). The neck struts are assumed to incline at 2:1 (longitudinal:
Figure 2.1 Bottle-shaped strut (a) Cracking of a bottle-shaped strut (b) strut-and-tie model of a
bottle-shaped strut (ACI 318-08 App. A)
Bottle-shaped strut is usually located in the region where lateral spreading of compression field
is possible; for example, at beam web. In practice, there is a rare need to model the idealized
bottle-shaped strut as shown in Fig. 2.1, especially when quick hand calculation is performed.
ACI 318 allows the bottle-shaped to strut to be modeled as prismatic or taper struts.
Ties are tension elements used to transfer tensile force to nodes. Ties can be prestressed or
non-prestressing rebar surrounded by a portion of concrete that is concentric with the axis of ties.
In design of tie element, care must be taken on providing sufficient anchorage length, so that full
tensile strength of ties can be developed. For design purpose, it is assume that concrete within the
ties does not carry any tensile force. The dimension of surrounding concrete is one factor to
determine nodal zone dimension.
2.3 Nodal zones
Nodal zones are finite region that struts and ties are coincided. Nodal zones may be treated as
a single region or may be subdivided into two smaller zones to equilibrate forces as shown in
Figure 2.2 Subdivision of nodal zone
Depending on the direction of strut and ties joining at node, nodal zone is classified in to CCC,
CCT, CTT and TTT nodes as shown in Fig. 2.3. Whatever nodal zones are defined, nodes are
assumed to be in compression as they are under hydrostatic pressure.
Figure 2.3 Classification of nodes
The hydrostatic pressure concept can be used to compute the dimensions of nodal zone. It can be
assumed that the pressures at nodal faces are equal where tie is assumed to be yield. Consider
CCC and CCT nodes in Fig. 2.4, the relationship between strut forces and node dimensions can
be written by
1 2 3
a a a
Figure 2.4 Nodal zone dimensions for a CCC and CCT node
Figure 2.5 Extended nodal zone
Figure 2.6 Nodal zones within the intersections of members
The use of hydrostatic elements sometimes does not represent the situation where nodes are
confined by two boundary prescribed dimension as shown in Fig. 2.5 (a). According to Fig. 2.6,
is determined from
is the area of vertical face. In contrast to
hydrostatic concept, the value of
can be different provided that
1. The resultants of the three forces coincide.
2. The stresses are within the limits as given by ACI 318 A.3.2, and A.5.2
3.The stress is constant on any one face.
Although, non-hydrostatic approach is not a standard method stipulated by ACI Code, it has been
widely used by many researchers.
2.4 STM methodology
The following strut-and-tie modeling procedure proposed by MacGregor (1996) can be
summarized as follows:
a. Define and isolate the D-regions.
b. Compute the internal stress on the boundaries of the element.
c. Subdivide the boundary and compute the force resultants on each sub-length.
d. Draw a truss to transmit forces from boundary to boundary of the D-region.
e. Check the stresses in the individual members of the truss.
2.5 Fundamental problems
Utilizing STM requires the designer to manually identify load path and nodal zone dimension,
for which, in particular cases, they are not easily defined. However, for many common
structures, STM models are available and well validated from experimental results. This section
presents some fundamental STMs which can be used as guidelines for constructing the more
complicated models. It should be mentioned that the best STM is not necessary the model with
minimum amount of steel as theoretically stated, but it is the model that behaves right with
compliance to good construction detailing and practice.
2.5.1 Transfer girder
Transfer girders are moderately deep beam with 3 < L/d < 5. They are generally used to
transfer load from the upper building stories down to the main columns below in order to enlarge
clear space for lobby or parking area (Fig, 2.7). As shown in Fig. 2.8, diagonal strut can be
idealized as prismatic or bottle-shaped strut. Although, bottle-shaped strut modeling is well
replicated to experimental results as demonstrated in Fig. 2.9 (b), its reinforcement detail is quite
complicated and found to be cumbersome with hand calculation. With aid of the computer
software, analysis using bottle-shaped strut does not make any different effort from that of
prismatic strut. Bottle-shaped strut shall be used when spreading of compression field is possible
as in case of diagonal strut in deep beam as shown in Fig. 2.8. Code section A.3.3 requires that a
minimum percentage of reinforcement be distributed across bottle-shaped struts to control
cracking along the axis of the strut. This reinforcement can be provided either in an orthogonal
mesh or as vertical-only or horizontal only reinforcement (Wight & Parra-Montesinos, 2003).
Prismatic strut, as simplified version of bottle-shaped strut, shall be used where spreading of
compression field is not possible as in case of horizontal strut in compression face of deep girder.
Minimum reinforcement as stated by ACI A3.3 is necessary for crack control due to lateral
spreading of compressive stress field.
Figure 2.7 Three-span deep beam, Brunwich building, Chicago. (JG MacGregor, 1997)
Figure 2.8 Strut-and-tie model of a deep girder to transfer load from single column
Figure 2.9 (a) Experiment speciment (b) failure of deep beam indicating spreading of crack
within compression strut region (Quintero-Febres et al., 2006)
2.5.2 Beam with openings
Openings in beam are sometimes required for conduits or pipes routing. Strut-and-tie
modeling can be used to evaluate the edge reinforcement around opening and to investigate the
adequate thickness of concrete close to the openings to prevent cracking by large compressive
field. Fig. 2.10 shows strut-and-tie models for beam with web openings.
Figure 2.10 Truss model for beam with openings (Muttoni et al., 1997)
2.5.3 Bracket or corbel
Bracket or corbels are reinforced or prestressed concrete element to support vertical and
horizontal load. They are typically projected from column face as shown in Fig. 2.11. Fig. 2.12
presents stress field for STM identification. It should be mentioned that ties must be properly
anchored to the boundary face of the corbel as required by code of practice. They can be hooked
with horizontal bar or welded to support plate. The latter detail is more preferred to prevent
corner or face spalling as might occur as shown in Fig. 2.13 (b) or (c).
Figure 2.11 Typical requirements for detailing concrete corbel
Figure 2.12 Strut-and-tie models for corbel (Muttoni et al., 1997)
Figure 2.13 Corbel details and possible failure modes (Wang & Salmon, 1992)
2.5.4 Dapped-end beam
Dapped-end beam is the beam with reduced depth at its end. Dapped beam is generally applied
for precast beams, bridge girders and so on. The main purpose is to reduce overall height of the
structure and to match with the geometry of corbel or bracket for load bearing support. Fig. 2.14
presents truss model for dapped end beam.
Figure 2.14 Truss model for dapped-end support (Muttoni et al.)
Attention must be paid to the location of vertical and horizontal ties as well as diagonal strut. The
horizontal tie must be properly anchored (usually by mechanical welding) to the bearing support
to prevent pull out force. The vertical tie acts as hanging bar to carry vertical load of the beam.
Diagonal strut usually control the depth of dapped end as well as beam thickness. An example of
reinforcement detail for dapped-end beam is shown in Fig. 2.15.
Figure 2.15 Strut and tie model and truss idealization for beams with dapped ends (Collins and
2.5.5 Pile caps
Pile caps or footings are reinforced concrete elements that transfer loads from columns or
walls to the foundation. They are usually thick. Strut-and-tie models of pile caps depend on pile
arrangement and load characteristics. Fig. 2.16 present strut-and-tie models of 3-pile footing
subjected to axial load and bending moment. The reinforcement details corresponding to strut-
and-tie model are also shown. Fig. 2.17 present strut-and-tie model for 4-pile footing with
suggested reinforcement details. It should be mentioned that ties must be properly anchored at
the boundary of footing in order to develop its yield strength as assumed in the analysis model.
The anchorage length is a key factor to dictate concrete covering
Figure 2.16 Strut-and-tie model of 3-pile footing (Muttoni et al.)
Figure 2.17 Strut-and-tie model of 4-pile footing (MacGregor, 1997)
184.108.40.206 Beam-column joint
Beam-column joints are subjected to bending moment and shear force from end of beam and
column members. Capacity of beam-column joint can be evaluated by using strut-and-tie
modeling as shown in Fig. 2.18
Figure 2.18 Strut-and-tie model for behavior of beam-column joint (Nilson et al., 2010)
220.127.116.11 Beam-to-girder joints
When secondary beam is joined to the main girder, vertical load may introduce diagonal crack
near to the joint. Fig. 2.19 presents truss model for beam-to-girder joint. It should be noted that
vertical force from the secondary beam is transferred through diagonal strut and vertical tie beam
acting as hanging bar. Hanging stirrups shall be provided for complete load path as shown in Fig.
Figure 2.19 Main girder supporting secondary beam (Nilson et al., 2010)
18.104.22.168 Corner joints
Various types of truss model and corresponding reinforcement details are given in Schalach et
al. (1987) as shown in Fig. 2.20. It should be mentioned that, those different truss models share
the same joint behavior.
Figure 2.20 Truss models of corner joint and corresponding reinforcement details (Schlaich et al.
ACI 318M-08 Appendix A Provisions
Guidance for determining size of struts, nodes and ties has been given in ACI 318M-08
Appendix A. The effective compressive strength of concrete
is adjusted by a modified
to account for the effects of cracks (caused by spreading compressive resultants) and
confining reinforcement in struts and the anchorage of ties in nodal zones.
3.1 Strength of compression struts
The effective strength of strut is computed from
ce s c
The value of
is dependent on shape of idealized strut as well as sufficiency of transverse
reinforcement. It accounts for the effects of cracking and confinement within the strut. The
are summarized in Table 3.1.
Compression steel may be added to increase the strength of a strut, so that
ns ce cs s s
F f A A f
is based on the strain in the concrete at peak stress. For grades 280 and 420 Mpa
may be assumed as
. If compression rebar is accounted for calculation of
strength, ACI code A3.5 requires that the compression bar must be properly anchored, oriented
parallel to the axis of strut, located within the strut, and enclosed by ties or spirals, as required
Where that lateral spreading of compression field is possible, ACI code A3.3 allows strut to be
idealized as bottle-shaped (Fig. 2.1). Spreading of compressive stress can cause crack due to
longitudinal splitting. Because of this, bottle-shaped struts are weaker than prismatic struts.
Transverse reinforcement shall be provided to control longitudinal splitting. For
the ACI code considers the transverse reinforcement requirement to be satisfied if the strut is
crossed by layers of reinforcement that satisfy
is the content of the rebar in layer i. The multiplication by
is to account for the
component of reinforcement transverse to the longitudinal axis of strut.
Figure 3.1 Reinforcement crossing at strut
3.2 Strength of nodal zones
Once nodal zone dimensions are defined, the nominal strength of nodal zone can be computed
from the following equation
nn ce nz
F f A
is the area of the face of nodal zone taken perpendicular to the line of action of the
force from strut or tie
is the effective compressive strength of the concrete in the nodal zone, given by
ce n c
The value of
is dependent on type of nodal zone as shown in Table 3.2. For CCC node, the
is maximized where the
value are penalized in CTT or TTT nodes.
3.3 Strength of ties
The nominal strength of ties
is contributed by the strengths of reinforcing steel and
prestressing steel within the tie. Concrete in tension does not contribute any strength to ties.
nt s y p pe p
F A f A f f
area of reinforcing steel
yield strength of reinforcing steel
= area of prestressing steel, if any
= effective stress in prestressing steel (initial stress)
= increase in prestressing steel stress due to factored load increment
must be less than the yield stress of the prestressing reinforcement
. The value of
may be found by analysis. ACI Code A.4.1 allows a value 420 Mpa to be used for bonded tendons
and 70 Mpa to be used for unbounded tendons.
Effective width of tie
depends on the distribution of tie reinforcement. It can be taken as the width of
anchor plate. The practical upper limit for tie width
is equal to the width corresponding to the width
of hydrostatic nodal zone, given as
is the effective nodal zone compressive stress and
is thickness of strut
Ties must be properly anchored before they leave the nodal zone at point defined by the centroid
of the bars in the tie and the extension of the outlines of either the strut or the bearing area as
shown in Fig. 2.5. If the combined lengths of nodal zone and extended nodal zone are inadequate
to provide for development of the reinforcement, additional anchorage may be obtained by
extending the reinforcement beyond the nodal zone , using 90o hooks, or by using mechanical
3.4 Shear requirements for deep beam
Deep beams are defined as the beam having clear spans to total member depth (
) less than
or equal to 4 (Fig. 3.2 (a)). Also, for the beam with concentrated load placed within twice of total
beam depth from support (Fig. 3.2 (b)) is considered as deep beam. ACI code 11.8.3 specifies
that the nominal shear in deep beam may not exceed
f b d
is the width of the
is effective depth. ACI Code 11.7.4 and 1.7.5 provide minimum steel requirements
for horizontal and vertical reinforcement within a deep beam. The minimum reinforcement
perpendicular to a span is
A b s
is the spacing of the reinforcement. The minimum reinforcement parallel to a span is
A b s
is the spacing of the reinforcement perpendicular to the longitudinal reinforcement.
may not exceed
or 300mm (whatever less). ACI Code 11.7.6 allows Eq.
(3.3) to be used instead of Eqs. (3.8) and (3.9). For strut-and-tie models,
is equal to the
thickness of the element
Figure 3.2 Deep beam D-regions
STM Software and Applications
4.1 Fachwerk & Fachwerk3D
Fachwerk and Fachwerk3D are Java-based applications developed by Adrian Vontobel (2010)
are software for 2D and 3D STM analysis, respectively. The programs base on the principle of
minimum complimentary strain energy (Castigliano Theorem), so it can be applied without
knowing information of element stiffness. They are under GNU open source project that means
they are freely distributed. Fachwerk and Fachwerk3D can be downloaded at
http://fachwerk.sourceforge.net/index_en.html. The main features of Fachwerk and Fachwerk3D
are listed as the following:
• Calculation of forces in strut and tie elements
• Indicate failure mechanism when the program detect any static inadmissible conditions
• Allow user to input specified member forces for indeterminate model, i.e. forces along
prestressed concrete tendon
• Allow user to specify force to any ties, i.e. yield force to predict limit load behavior
• The program employs only equilibrium condition without considering element stiffness
as required by STM methodology
• Allow node relocation to satisfy assumed target element forces. This is benefit to
sensitivity analysis of STM topology.
• Indicate degree of static indeterminacy at real time
• Interactive graphical user interface with edit, pan, and zoom capabilities
• Calculation report can be printed out
Some screenshots from Fachwerk and Fachwerk3D are shown in Fig. 4.1 and 4.2, respectively.
Figure 4.1 A screenshot from Fachwerk
Figure 4.2 A screenshot from Fachwerk3D
At the bottom line, Fachwerk and Facwerk3D neither check element strength nor provide nodal
zone dimensioning. The software aims to structural engineers who just want a portable modeling
and analysis tools for 2D and 3D STM. The software serves very well for such objective.
CAST stands for Computer aided for Strut-and-Tie design tools. It is only an existing
windows-based application for STM with rich of graphical user interfaces. CAST has been
developed by the University of Illinois since 2001 (Kuchma & Tjhin, 2001). It was used to serve
students and practicing engineers to grasp the concept and to try various idealized strut-and-tie
models with ease. Main features of CAST can are listed as shown in Fig. 4.3.
CAST components comprise of nodes and elements. Elements represent struts or ties meeting
at nodes. For particular cases, stabilizer is necessary to make the model numerically stable. The
stabilizer is a strut with zero force as shown in Fig. 4.4. Three types of boundary conditions can
be applied at exterior nodes: plate, point loads, and supports. Body force and support can also be
applied to interior nodes within D-region. The model must be restrained to prevent rigid body
movement. In CAST, positive value of forces is point outward from node and vice versa. Since
the direction of point loads on the boundary will follow the axis of the strut, no element
connectivity is allowed on boundary. The following procedure is necessary for performing strut-
and-tie analysis by CAST:
Figure 4.3 Current features of CAST (Version 0.9.11/ 01.26.2004)
1. Specify general properties of the STM (Fig. 4.4). They are D-region thickness, compressive
strength of concrete (
), tensile strength of concrete (
) and non-prestressed reinforcement
yield strength (
Figure 4.4 General properties window
2. Create the boundary of the D-region by using command. Interior boundary (voids) can
be constructed by using command. The boundary nodes can be relocated by using
3. Define nodes by using command and elements by using command. After
construction, nodes can be relocated by using command.
4. Assign boundary conditions at nodes by using command for bearing support, for
nodal forces or supports. Body forces or supports can be assigned to interior nodes by using
5. Define and assign struts types (ACI prismatic strut , ACI bottle-shaped strut with
reinforcement , ACI bottle-shaped strut without reinforcement , or ACI strut in tension
member ). Moreover, user-defined struts are allowed by using command. Tie can be
defined and assigned by using command.
6. Specify and assign relative stiffness and width of struts and ties by using command
Figure 4.5 Input window for assigning relative stiffnesses and widths of struts and ties
7. Perform analysis run by command. By default, the program determines nodal zone
dimension and perform checking of nodal stresses.
8. Review the results for forces in struts, tie, as well as utilization stress ratio. If the stress ratio
is less than 1.0, the result is acceptable. Otherwise, width & thickness of struts must be increased
or the amount of rebar in ties must be increased. The maximum thickness of struts and ties is
controlled by the thickness of D-region.
Some screen shots of CAST are shown in Figs. 4.6 to 4.8.
Figure 4.6 CAST working environment
Figure 4.7 Deep beam under two-point load
Figure 4.8 Non-slender wall with openings
Worked Examples with CAST
This chapter demonstrates the strut-and-tie method for solving two practical problems. The
first example is double corbel problem and the second is deep beam problem. Although, it is
possible to solve those problems by hand calculation, we use CAST in order to give an
impression of computer aided design tool for strut-and-tie modeling.
5.1 Double Corbel
This example is obtained from CAST’s tutorial page
. It presents analysis of double corbel
problem using strut-and-tie method. The geometry of the corbel is shown in Fig. 5.1. By this
example, CAST is used to construct the truss model, to analyse and to determine stresses in
struts, ties and nodal zones. The program needs user to input almost all important parameters,
such as material properties, width and thickness of strut and ties, number of rebar in tie, as well
as node classifications. The program automatically determines nodal zone dimension based on
hydrostatic concept or prescribed nodal zone dimension. For the given problem,
420 Mpa. The following procedure shall be followed to complete
Figure 5.1 The geometry and loadings of the corbel
Step 1: Construct the outer boundary
Figure 5.2 Outer boundary with guidelines
Step 2: Construct strut-and-tie model
Figure 5.3 Node and elements
Step 3: Apply loads and boundary supports
Figure 5.4 Model with loads and supports
At this stage, it should be mentioned that CAST can only accept resultant forces. The
element E18 and E19 must be inclined in to the direction of resultant forces which can be
is measure from the vertical axis.
Step 4: Define all struts as ACI prismatic strut except E4 and E5 as bottle-shaped struts. Tie E1,
E2 and E3 are defined as non-prestressed rebar (6-M25) as shown in Fig. 5.5
Figure 5.5 Properties of ties
Step 5: Assign width of struts and ties
In this example, width of ties E1, E2 and E3 are 200 mm, width of struts E7-E12 are 300 mm,
and width of the rest struts are 150 mm.
Figure 5.6 Strut-and-tie dimension
Step 6: Assign node types
The user must manually assign node type to each nodes in the model. In this example, we
know that N5 and N6 are CCC nodes; N1 and N4 are CCT nodes. It should be noted that only
nodes to be checked by the program are needed to be specified. The program will determine the
maximum stress from the defined node type.
Step 7: Run analysis and check results
Figure 5.7 Analysis results for forces in truss model and utilized stress ratio
According to the results, there are two elements that have no internal force. They are
called by the developer as stabilizer. If such elements, are removed before the analysis to be run,
the error message indicating model instability will occur and the analysis cannot be proceeded.
Step 8: Check nodal zone stress
Figure 5.8 Nodal zone stress check
At this stage, only nodes with pre-defined node type contain the results of nodal zone stress
check. By right clicking at node, the window as shown in Fig. 5.8 will appear. It should be noted
that, the program construct nodal zone dimension based on the known strut and tie dimensions. It
simply computes nodal stress from the computed forces in the corresponding struts and ties
coincident at node divided by their cross dimension area. For example, at node N1, the face
perpendicular to tie, the stress at node is computed as
1833.3 10 6.94
Stress limit for CCT node is
0.75 0.85 0.80 35 17.85
Utilized stress ratio is 6.94/17.85 = 0.388
ratio is 6.94/35.0 = 0.198
0.85 ' 0.75 0.85 35
Beta ratio is a measure of perpendicular nodal stress to allowable compressive stress.
5.2 Deep Beam
Another example is deep beam subjected to concentrated load. This example is adopted from a
paper of Wight and Parra-Montesinos (2003). The model of deep beam in the paper can be
shown in Fig. 5.9.
Figure 5.9 Deep beam model (Wight and Parra-Montesinos, 2003)
For this example, we set up the general properties of the problem as shown in Fig. 5.10.
Figure 5.10 General properties of the problem
The following procedure should be followed in order to complete this example:
Step 1: Construct the outer boundary
Figure 5.11 Outer boundary
Step 2: Construct truss model
At this stage it is assumed that the depth of truss model is equal to 1245 mm. The truss model
can be constructed as shown in Fig. 5.12. Nodal coordinates of the model is shown in Fig. 5.13.
Figure 5.12 Truss model geometry
Figure 5.13 Nodal coordinates
It should be mentioned that stabilizer element E10 is added to the model to retain numerical
stability and enable the program to solve the problem properly.
Step 3: Apply loads and boundary supports
Bearing supports (depth
width = 500
500 mm) are assigned to N1, N2 and N3, respectively.
Vertical load -2909 kN is applied at N2.
Figure 5.14 Load and boundary supports
Step 4: Define strut types and tie properties
Since lateral spreading of compression field is feasible, the inclined struts E1, E5, and E7 are
defined as bottled-shape strut, while strut E8 is defined as prismatic strut.
Step 5: Assign width of struts and ties
CAST has ability to compute the width of struts based on specified utilized stress ratio. We shall
demonstrate this feature in this example. Setting stress ratio as 0.90, we do not prescribe the
width of the struts at this stage, but ask the program to compute for them.
Figure 5.15 Let the program determine width of strut
The main horizontal tie is defined as shown in Fig. 5.16, while the vertical tie is defined as in
Fig. 5.17. It should be noticed that the width of both ties are calculated from the depth of outer
layers plus of extended effective tensile zone. Width of ties does not affect to its strength, but it
affects to stress in nodal zone. Modification of rebar arrangement may be necessary if stress in
nodal zone is greater than the allowable value.
Figure 5.16 Properties of the main tie
Figure 5.17 Properties of the transverse tie
Step 6: Assign node types
Assign ACI CCC node to N2 and ACI CCT node to N1, N3, and N7
Step 7: Run analysis and check results
Figure 5.18 Analysis results
As shown by the analysis results, all elements are safe. A little overstressed in the horizontal tie
can be noticed. Increasing the main rebar might be necessary.
Step 8: Check nodal zone stress
Stress at nodal zones must be checked to assure safety of the model. As shown in Fig. 5.19,
stresses ratio at N1 for three perpendicular faces are confirmed to be less than 1.0
Figure 5.19 Stress check at nodal zone N1
As indicated by the results, the face perpendicular to the main tie is subjected to high stress ratio
of 0.98. Rearrangement of rebar in tie may help to reduce the stress ratio, but it seems
unnecessary for this example.
Step 9: Check for adequate development length. Since this problem is deep beam, minimum
reinforcement according to ACI 10.6.7 skin reinforcement or ACI A3.3 must be confirmed.
Moreover, maximum shear stress as stipulated in ACI Code 11.8.3 shall be checked.
5.3 Ledge Beam
This example shows an application of ledge beam. The 200 mm thick D-region is shown in Fig.
5.20. The ledge is subjected to vertical load 25 kN applied at bearing plate.
Figure 5.20 Outer boundary of D-region and strut-and-tie model
The model consists of two tie elements (E5, E6) and four compression struts (E1, E2, E3 and
E4). A stabilizer (E7) is added to make the model stable. The analysis results are shown in Fig.
5.21. For a trial design, ties E5 and E6 are assigned as 2T12 (226 mm2). The effective width of
ties are 30 mm. Since lateral spreading of compression field is possible within the ledge, struts
E2, E3 are assigned as bottle-shaped strut without reinforcement satisfying ACI318 A.3.3 (
). Compression field is confined within the beam web, so that prismatic strut (
) is used for E4. The analysis result in Fig. 5.21 shows capacity ratio of the strut-and-tie
model. Fig. 5.22 shows detailed calculation for strut E2. Investigation of nodal stress can be
performed as similar to the previous examples.
Figure 5.21 Analysis results
Figure 5.22 Detailed calculation for strut E2
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