STRUT AND TIE MODEL FOR PILE CAP REPORT

UNIVERSITY OF TRANSPORT

Strut and tie model for design of
reinforced concrete pile caps
Nguyen Thanh Trung
Pham Hong Thai

Faculty of Civil Engineering
HCMC UNIVERSITY OF TRANSPORT
Ho Chi Minh City, August 2016


MINISTRY OF TRANSPORT
HO CHI MINH CITY UNIVERSITY OF TRANSPORT
FACULTY OF CIVIL ENGINEERING

Strut and tie model for design of
reinforced concrete pile caps

Supervisor:
Research student:

Pham Tien Cuong, Ph.D
Nguyen Thanh Trung
Pham Hong Thai

Ho Chi Minh City, August 2016


Abstract

During country modernlizing and developing time, construction industry has developed
maturely. Many impressive structures have built among the country, as a result, the
theory of design of concrete structures has a lot of changes in order to gain a more
accurate and effective design. TCVN 5574-2012-the structural concrete design standard
is the main guide for engineers in Viet Nam, in which the calculation is mostly based on
sectional approach. This approach is derived from Bernoulli hypothesis that is suitable
only for B-region. Normally, engineers will apply this design approach for the whole
structure which includes both B and D-region. In this case, the calculation may not
appropriate to the members behaviors. Stocky pile caps with high thickness under
concentrated load are kind of members that are entirely in D-region which is not well
treated by design code.
International building codes such as Eurocode 2 (EC2), American building code (ACI
318), etc, now present an exact method for design of reinforced concrete in D-region
called strut and tie model. The model is based on the lower bound theorem of plasticity
that ensures the structure is safe under ultimate loading and it also gives the designer a
better look of structural behavior. Bridge engineers in Viet Nam are also familiar to this
method through 22TCN 272-02 but the application of this method in building-pile caps
design is still limited.
Therefore, the main objective of this work is to make a guide of using strut and tie model
for design of pile cap with instruction from ACI 318-11 which may take more
advantages than traditional bending theory.

Key words: pile caps, strut and tie model, nodal zone


Table of Contents
1....................................................................................................................................... 2
1.1 History of strut and tie model and specifications ................................................. 2
1.2 Definition of B and D-regions .............................................................................. 2
1.3 Lower bound theorem of plasticity ....................................................................... 4

1.4 Design procedure using strut and tie model.......................................................... 5
1.5 Development of strut and tie model ...................................................................... 6
1.5.1 Strut ................................................................................................................ 6
1.5.2 Tie .................................................................................................................. 7
1.5.3 Node and nodal zone...................................................................................... 7
1.6 Constructing strut and tie model ......................................................................... 11
1.6.1 Elastic analysis approach ............................................................................. 12
1.6.2 Load path approach ...................................................................................... 12
1.6.3 Standard model ............................................................................................ 13
1.6.4 Notices on modelling strut and tie model .................................................... 13
1.7 Analyzing model and ACI provisions ................................................................ 16
1.7.1 Analyzing model .......................................................................................... 16
1.7.2 ACI provisions for strut and tie model ........................................................ 18
2..................................................................................................................................... 21
2.1 Working mechanism ........................................................................................... 21
2.1.1 Direct arch action ......................................................................................... 22
2.1.2 Truss action .................................................................................................. 22
2.2 Traditional approach ........................................................................................... 22
2.3 Strut and tie approach ......................................................................................... 23
2.3.1 Boundary conditions .................................................................................... 23
2.3.2 Development of strut and tie model for pile caps ........................................ 25
2.3.3 Reinforcement and anchorage ..................................................................... 27
2.4 Comparison ......................................................................................................... 28
A .................................................................................................................................... 30


A.1 Synopsis ............................................................................................................. 30
A.2 Example 1: Two piles – pile cap ........................................................................ 30
A.2.1 Pile cap geometry ........................................................................................ 30
A.2.2 Design calculations ..................................................................................... 31

A.3 Example 2: Four piles – pile cap........................................................................ 35
A.3.1 Pile cap geometry ........................................................................................ 35
A.3.2 Design calculations ..................................................................................... 35


List of figures

Figure 1.1 – St.Venant’s principle (Brown et al, 2006). ................................................ 3
Figure 1.2 – Bernoulli’s hypothesis. .............................................................................. 3
Figure 1.3: B and D-regions on structures. .................................................................... 4
Figure 1.4 – Design flowchart using STM..................................................................... 5
Figure 1.5 – Prismatic strut. ........................................................................................... 6
Figure 1.6 – Bottle shape strut. ...................................................................................... 7
Figure 1.7 – Fan shape strut. .......................................................................................... 7
Figure 1.8 – CCC node. ................................................................................................. 8
Figure 1.9 – CCC node in pile caps. .............................................................................. 8
Figure 1.10 – CCT node. ................................................................................................ 9
Figure 1.11 – CTT node. ................................................................................................ 9
Figure 1.12 – Hydrostatic nodal zone. ......................................................................... 10
Figure 1.13 – Non-hydrostatic nodal zone. .................................................................. 10
Figure 1.14 – Extended nodal zone at a CCT node. .................................................... 11
Figure 1.15 – Good and poor STM model for pile caps. ............................................. 11
Figure 1.16 – Stress trajectories in deep element. ....................................................... 12
Figure 1.17 – Load path approach on deep beam [7]. ................................................. 13
Figure 1.18 – Singular and general model of a 16 piles – pile cap [5] ........................ 13
Figure 1.19 – Angle between strut and tie. .................................................................. 14
Figure 1.20 – Idealized prismatic strut. ....................................................................... 15
Figure 1.21 – Resolve struts together. ......................................................................... 15
Figure 1.22 – Nodal zone geometry. ............................................................................ 16
Figure 1.23 – Optimizing the height of model. ............................................................ 16

Figure 1.24 – Loading condition on the interface of B and D-regions. ....................... 17
Figure 1.25 – Statically determinate and statically indeterminate model. ................... 17
Figure 1.26 – Internally statically indeterminate model. ............................................. 18
Figure 1.27 – Development of anchored length........................................................... 20
Figure 2.1 – Direct arch and truss action mechanism of shear transfer ....................... 21
Figure 2.2 – Provisions of ACI code for strain, stress and rectangular stress diagram 22
Figure 2.3 – Simple modelling of pile cap ................................................................... 22
Figure 2.4 – Moment and shear diagram (neglecting pile cap self-weight) ................ 23
Figure 2.5 – Comparison of simplified analysis and FE analysis ................................ 24


Figure 2.6 – Position of applied forces by elastic distribution. ................................... 24
Figure 2.7 – Position of applied forces at ultimate limit state. .................................... 25
Figure 2.8 – Vertical position of ties and nodes at the bottom and top face................ 26
Figure 2.9 – Reinforcement layouts in pile caps.......................................................... 27
Figure 2.10 – Combination of square bunched and grid reinforcement layout ........... 28
Figure A.1 – Plan view of the pile cap......................................................................... 31
Figure A.2 – Analyzing the pile cap for ultimate limit state ....................................... 32
Figure A.3 – Stress distribution from finite element analysis ..................................... 32
Figure A.4 – Stress distribution from finite element analysis ..................................... 35
Figure A.5 – Analyzing the pile cap for ultimate limit state ....................................... 37


List of tables
Table 1.1 – Values of  s for strut strength .................................................................. 19
Table 1.2 – Values of  n for nodal zone strength........................................................ 19
Table 2.1 – Comparison of FE analysis and simplified analysis ................................. 24
Table A.1 – Piles reaction force ................................................................................... 35

List of notations

a

Depth of equivalent rectangular stress block

Acs

Cross sectional area at one end of the strut

Anz

The smaller of section through the nodal zone

As'

Area of compression reinforcement

Ats

Area of non-prestress reinforcement

f ce

Effective compressive strength of the concrete strut or nodal zone

f c

Specific compressive strength of concrete

f se

Effective stress in prestressing steel (after allowance for all prestress
losses)

f py

Specified yield strength of prestressing steel

f p

Increase in stress in prestressing steel

Fu

Factored force acting in strut, tie, bearing area, or nodal zone in a strut and
tie model;

Fn

Nominal capacity of strut, tie, or nodal zone;

Fns

Nominal strength of concrete strut.

Fnn

Nominal strength of nodal zone

Fnt

Nominal strength of ties

Mn

Nominal flexural strength at section

Mu

Factored moment at section

Vn

Nominal shear strength

Vu

Factored shear force at section

z

Overall height of strut and tie model


n

Factor to account for the effect of the anchorage or ties on the effective
compressive strength of a nodal zone

s

Factor to account for the effect of cracking and confining reinforcement
of the effective compressive strength of the concrete in a strut

1

Factor relating depth of equivalent rectangular compressive stress block to
neutral axis depth



Strength reduction factor.


INTRODUCTION

Reinforced concrete is a composite and anisotropic material, its mechanical behavior is
very complicated and not understood completely. Therefore, in order to make it possible
to practice, structures are mostly designed by empirical sectional approach. This
methodology is predicated on traditional bending theory which proposed the assumption
of plane section remains plane after bending. The assumption is not appropriate for
application of region (D stands for disturbed or discontinuity) due to the nonlinear strain
distribution. Because of the reasons above, the design of D-regions have to be carried
by a regional, rather than a sectional approach.
Strut and tie model (STM) is a conservative design method based on the lower bound
theorem, its applications are commonly used in the design of D-region such as brackets,
column-beam joints, pile caps, etc. Since it was accepted by codes, the application has
been widely increased over decade.
Pile caps are elements with the function of transmitting the load from superstructure to

concrete piles. Due to its discontinuities geometry, pile caps are considered as D-region
of the whole element. Therefore, applying empirical formulas for section under flexure
is questioned.
Another design method for disturbed regions like pile caps is STM which is
recommended by researchers and codes as well.

Aims and limitations
This work is done with the main aims of comparing the differences between traditional
and STM design method for the design of pile caps and also making a guide for designers
when they want to apply STM for pile caps.
Due to limitation of time and knowledge, the pile caps modeled in this paper work are
designed isolated from the structure. If designers can consider pile caps and the
superstructure in a single model, they could obtain a more reliable result.

HCMC University of Transport

1


CHAPTER

1

STRUT AND TIE MODEL

1.1 History of strut and tie model and specifications
In 1890’s, German engineer named Wilhelm Ritter introduced the ideal of design
concrete beam using truss analogy where reinforcing steel bars would carry tensile
forces and concrete would carry compressive forces. The ideas was later taken by Emil
Morsch in 1990’s who used the truss analogy to determine the amount of reinforcing

steel in beam. Since then, this method had been extensively used.
However, the research of STM was not be expanded until Schlaich et al (1987) who
gave a way to design the whole beams and structures with STM in the article “Toward
a Consistent Design of Structural Concrete”. The article suggests using STM could lead
to an efficient design based on the actual knowledge of mechanics, rather than test
results and experiences.
Since STM was introduced in the Canadian Concrete Code (CSA, 1984), American
Bridge and Highway standard (AASHTO, 1994) and American Building code (ACI
318, 2002), its applications have been highly increased. These codes’s requirements are
very similar to those proposed by Schlaich et al (1987). The specifications in this paper
work are mainly taken from ACI 318-11.

1.2 Definition of B and D-regions
Structures normally are divided into two kinds of regions due to the abrupt changes in
geometry and loading according to St.Venant’s principle.
“The localized effects caused by any load acting on the body will dissipate
or smooth out within regions that sufficiently away from the location of the
load…”

HCMC University of Transport

2


Chapter 1: Strut and tie model
The distance mentioned in St.Venant’s principle is approximately equal to the overall
height of the member, h, away from the discontinuities. This principle allows elasticians
replace complicated stress distribution into ones that are easier to solve. This one is
called B-region where Bernoulli’s hypothesis is valid.
Bernoulli’s hypothesis: “Plane section remain plane after bending…”

Bernoulli’s hypothesis facilitates the flexural design of reinforced concrete structures
by allowing a liner strain distribution for all loading stages, including ultimate flexural
capacity. This is the basic assumption adopted by codes when applying sectional
approach for the design.

Figure 1.1 – St.Venant’s principle (Brown et al, 2006).

c

fcu

x
M

M
As

c

fs

Figure 1.2 – Bernoulli’s hypothesis.
In contrast, D-regions are the regions of discontinuities resulting in nonlinear strain
distribution and thus, the assumption of codes is not applicable anymore. D-regions are
assumed to extend on both side a distance, h from the discontinuities. At geometric
discontinuities, a D-region may have different dimension on either side of discontinuity,
as show in Figure 1.3 below.
According to the definition, pile caps are usually in range of dimension where bending
theory is not applicable in any section and the entire pile caps are composed of D-region.
Therefore, design based on the procedure given by codes which mainly rely on sectional

approach is not appropriate.

HCMC University of Transport

3


Chapter 1: Strut and tie model

Figure 1.3: B and D-regions on structures.
Strut and tie model approach based on the lower bound theorem of plasticity is an
alternative design approach for D-region which strongly recommended by ACI 318, AS
3600 and EC2.
“Current design procedure for pile caps do not provide engineers with a
clear understanding of the physical behavior of these element. STM can
provide this understanding and hence offer the possibility of improving
current design practice” say Kuchma and Collins [4].

1.3 Lower bound theorem of plasticity
The lower bound theorem is based on the behavior of ideal rigid-plastic systems. It can
be summarized as below:
“A stress field that satisfies equilibrium and does not violate yield criteria
at any point provides a lower bound estimate of capacity of elastic-perfectly
plastic materials” or “each load for which any statically admissible stress
state can be given is either the collapse load or a lower bound of the collapse
load”
The statement above can be simply explain that if a designer can figure out a way for a
structure to carry a set of design ultimate loads such that equilibrium is satisfied, yield
criteria are not violated and ductile response is ensured, the structure will be able to
carry the load.

For the theorem to be true in the application on reinforced concrete structure, the
yielding of reinforcement has to be occur before the crushing of the compressive
concrete. Moreover, reinforced concrete is not an elastic-perfectly plastic material and
its plastic deformation capacity is limited in order to prevent brittle failure. Therefore,
good judgement is required when the choice of statically stress state is made. This
challenges the design of concrete structure for strength using strut and tie model.
HCMC University of Transport

4


Chapter 1: Strut and tie model

1.4 Design procedure using strut and tie model
The design procedure of strut and tie model can be summarized as the flowchart in
Error! Reference source not found. below.
Separate B and D-regions

Determine the reaction forces
and the boundary conditions

Determine a sufficient STM model

Determine the forces in members

Calculate reinforcement

Check struts and nodal zones

Provide reinforcement and anchorage

Figure 1.4 – Design flowchart using STM.

HCMC University of Transport

5


Chapter 1: Strut and tie model

1.5 Development of strut and tie model
Strut and tie model are apply within D-regions. It is a conceptual framework where the
stress distribution in the region is idealized as a truss. The truss consists of three main
components, struts and ties connected at nodal zones that are capable of transferring
loads to the support or adjacent B-regions.
In traditional expression, a continuous line indicates a tie while a strut is represented by
a dashed line. This convention is also used in this work hereafter.
1.5.1 Strut
A strut is an internal compression member in which the compressive stress is
transferred. Compression struts fulfill two functions that are serving as the compression
chord of the truss mechanism which resists moment and serving as the diagonal strut
which transfer shear to the supports. Diagonal struts are generally placed parallel to the
axis of cracking.
Depending on the stresses field in the vicinity, strut can be classified into three kinds.
This state of stresses also affects the design strength of struts.
1.5.1.1 Prismatic strut
If there is no space along the strut, the compression stress field could not spread out
laterally and the stress would be fairly uniform. In this case, the section of the strut
would remain constant and a prismatic or straight-sided strut is defined. This kind of
strut is often located along the compression flange of a beam and used in the design as

an idealized strut.

Figure 1.5 – Prismatic strut.
1.5.1.2 Bottle shape strut
Unless a strut is parallel to and immediately adjacent to a free surface, the stresses will
diverge laterally, transverse tension arises. In other words, the strut expands or contracts
along its length. The transverse tensile stress in bottle shape strut could lead to
unexpected cracking along the strut axis in the bottle zone. Bursting reinforcement may
be required to control this effect.
HCMC University of Transport

6


Chapter 1: Strut and tie model

Figure 1.6 – Bottle shape strut.
1.5.1.3 Fan shape strut
At some node, where an array of struts with varying inclination meet each other. This
type of struts is called fan-shape strut.

Figure 1.7 – Fan shape strut.
1.5.2 Tie
A tie is an internal member under tension within a strut and tie model. Ties may consist
of reinforcement, a portion of the concrete that is concentric with and surrounds the axis
of the tie and any special detail reinforcement.
The tie area is defined by the surrounding concrete. Although the tensile capacity of the
concrete is ignored for design purposes, it still contributes to decrease tie deformation
under service load stage.
The detailing of anchorage is a very important part and is often the critical consideration

in design using STM. If designers do not provide an adequate anchorage for the
reinforcement, brittle failure would be likely happened at anchorage when the external
load does not reach the expected ultimate bearing capacity.
1.5.3 Node and nodal zone
The distinction has to be made between the two terms above. Nodes correspond to the
points of intersection between the axes of struts and ties. All concurring forces at node
must satisfy equilibrium which is used to calculate the members forces. While nodal
HCMC University of Transport

7


Chapter 1: Strut and tie model
zones correspond to the concrete blocks around the nodes which transfer strut and tie
forces through the nodes. Forces in this region acting in different directions, meet and
balance (Schafer, 1999).
1.5.3.1 Nodes classification
The classification is made due to the difference in members connecting to nodes, three
common cases of nodes in two-dimensional model and their nodal zones are listed
below.
a) CCC node
In order to keep forces at node in equilibrium, there must be three forces act on a node
in different directions. A CCC node has three compressive struts intersect each other.

Figure 1.8 – CCC node.
In pile caps, CCC node often located at the interface of column and pile caps, where the
axial load is started transferring from column to piles.

CCC node

Figure 1.9 – CCC node in pile caps.
b) CCT node
When forces from two compressive struts and one tensile tie are resisted at a node, a
CCT node is defined.
HCMC University of Transport

8


Chapter 1: Strut and tie model

Figure 1.10 – CCT node.
c) CTT node
CTT note is the intersection of two tensile forces and one compressive force, it is often
located “inside” the STM and classified as smear node.

Figure 1.11 – CTT node.
1.5.3.2 Hydrostatic and non-hydrostatic nodal zone
In order to build-up a strut and tie model and proceed the design, the geometries and
forces of each components must be defined. Nodal zone is one of them and it can be
proportioned in two ways: hydrostatic nodal zone and non-hydrostatic nodal zone
a) Hydrostatic nodal zone
When forces acting on the node in directions produce equal stresses on all the loaded
faces which perpendicular to the axes of the struts and ties, a hydrostatic nodal zone is
C
C C
specified. For a hydrostatic CCC nodal zone, the condition 1  2  3 must be
w1 w2 w3
satisfied thus w1 , w2 , w3 are proportioned. Therefore no shear stresses are created at the
node. However, this is also the reason that make managing to have a geometry assures

hydrostatic nodal zone mostly impossible and unrealistic. Arrangement of
reinforcement layout also affect this choice which may lead to an impractical solution.

HCMC University of Transport

9


Chapter 1: Strut and tie model

Figure 1.12 – Hydrostatic nodal zone.
A CCT nodal zone can also be represent as a hydrostatic nodal zone if the tie is assumed
to be a compression strut acts on the opposite face of the nodal zone. For this to be true,
the tie has to extend through the node and be anchored by a bearing plate result in a
bearing stress that equal to the stresses produced by struts.
b) Non-hydrostatic nodal zone
For the reason of difficulties in applying hydrostatic nodal zones in STM, some
international codes define non-hydrostatic nodal zone, in which the stresses acting on
loaded faces must not be equal. In his article, Schlaich recommended to keep stress ratio
on adjacent edges of nodal zone below 2. If not, the non-uniform distribution of stresses
could make an unconservative strength check at node (Schlaich et al, 1987). The
advantage of non-hydrostatic nodal is not only facilitation in calculating dimensions of
nodal zones but also reflect the actual stress concentrations at nodal regions.

Figure 1.13 – Non-hydrostatic nodal zone.
Non-hydrostatic nodal zone is usually defined by extended nodal zone, which is a
portion of a member bounded by the intersection of the strut width and tie width as
show in Figure 1.14.

HCMC University of Transport

10


Chapter 1: Strut and tie model

Figure 1.14 – Extended nodal zone at a CCT node.

1.6 Constructing strut and tie model
According to the lower bound theorem, any statically determinate stress field that
satisfies equilibrium is shown, it can ensure that the structure is safe. This is not only
the advantage of the method but also the challenge in design. It can be shown that there
is no right or wrong model, but a good model could help creating an effective design
while a poor model could lead to an inappropriate result and cost ineffectively.

Figure 1.15 – Good and poor STM model for pile caps.
At ultimate limit state, when the concrete is cracked and the external load is very close
to the collapse load, plastic deformation has occurred, then it is possible that the chosen
force distribution would happen. However, in order to take account the limitation of
plastic deformation of reinforced concrete and the admissible performance at service
load stage, the chosen stress state should thus reflect the way the structure naturally
carries loads. This is a special requirement in the design of pile caps, which are members
with a low ability to plastic redistribution to prevent brittle failure of the compression
concrete [5].

HCMC University of Transport

11

Chapter 1: Strut and tie model
Therefore, a conclusion may be made, the structure tries to carry load as effectively as
possible with at least amount of deformation. Since the tensile force in reinforcement
contributes much more deformation than those of compression force in concrete strut,
the most effective model would be one which has shortest tensile tie and least tie forces.
As illustrated in Figure 1.15, the poor model requires large deformation of concrete in
order to make the tie reach its yield strength, this obviously violates the limitation of
plastic deformation of concrete.
There are some methods that engineers can use to formulate a proper strut and tie model.
1.6.1 Elastic analysis approach
Elastic analysis is based on the stress trajectories, the placement of struts and tie in the
model represent and follow the general pattern of compressive and tensile stress fields
within the structural component. A linear finite element analysis can be used to
determine direction and intensity of principal stresses for modelling STM. This method
of modelling also gives designers a better view of structural members behaviors.

Figure 1.16 – Stress trajectories in deep element.
Although building a model that represent exactly the elastic flow of stress is not strictly
required, developing the model which best follow the natural elastic stress distribution
would reduce the possibility of service crack. Any deviation would increase the risk of
cracking.
1.6.2 Load path approach
Another way to set up a strut and tie model is to place struts, ties in the model that
follow the visualized flow of forces (Schlaich [7]). In order to do this, the designer
must use the locations of applied external loads and forces on the boundary of D-region
to develop a logical and suitable load path. This intuitive method requires experience
as well as a good understanding of member behaviors under loading.

HCMC University of Transport

12


Chapter 1: Strut and tie model

Figure 1.17 – Load path approach on deep beam [7].
1.6.3 Standard model
This method is to use a general model that covers every load cases of a concrete member
could be subjected. It results in a much more complicated model which could be solved
only with enormous efforts. However, since its advantage is the suitability to all cases
without considering the state of loading, it is very proper to apply this method for
computer program where users can find the amount of reinforcement of every design
load combinations rapidly and conveniently without effort of applying try and error
process. Figure 1.18 gives an example of a singular model of a 16 piles – pile cap
subjected to axial load and its general model when considering moment about two axes.

(a)

(b)

Figure 1.18 – Singular and general model of a 16 piles – pile cap [5]
1.6.4 Notices on modelling strut and tie model
When building a strut and tie model, there are some notices that designer should keep
in mind.
1.6.4.1 Angle limitation
When an inclined compression strut intersects with a tensile tie, two problems could
arise. If the angle between the strut and the tie is too large, the requirement of plastic
redistribution may excessively high and strain in stressed and unstressed regions may

HCMC University of Transport

13


Chapter 1: Strut and tie model
have compatibility problems. In contrast, if this angle is too small, the strain
compatibility problems could also happen.
The recommendation of angle limitation by codes and authors are different. For
example, the maximum value given by the ACI 318 [3] is 65o while AS 3600 requires
a smaller value of 60o. The minimum angle can be calculated as min  90o  max .



Figure 1.19 – Angle between strut and tie.
1.6.4.2 Positions and sizes of components
Struts and ties in model have to comply some other rules. Except from angle limitation,
developing a strut and tie model should ensure that no struts are overlapping and cross
each other outside the node regions. Actually, struts are checked according to the
concrete effective strength given in ACI 318 [3], violating this rule could lead the
overlapping regions to yielding (Reineck [6]). However, no problem occur if ties cross
struts or other ties.
The width of strut can be determine depending on its internal force and nodal zones
geometries. If the strut is prismatic, it would be easily to specify its dimensions.
However if the strut is bottle shape or fan shape strut, the width at the middle where
tensile stress arises which normally happen in pile caps is not easily estimated.
Therefore, prismatic strut is often used in design.

HCMC University of Transport

14

Chapter 1: Strut and tie model

Figure 1.20 – Idealized prismatic strut.
External forces should concentrated at nodes, struts and ties should cross each other at
nodes. It can be say that determining nodal zone geometry is the most difficult problem
in design procedure using STM. The difficulties increase excessively when the node is
the intersection of many struts and ties. Due to its complicated geometry, number of
struts and ties is often minimized by resolving adjacent strut together. In Figure 1.3
below, strut C1 and C2 are resolved into C1 while strut C3 and C4 are combined into

C3 by vectors summation.
C1  C1  C2 and C1  C12  C2 2  2C1C2 cos 
C3
C2




C1

C'3

C4
C'1

T

T

A

A

Figure 1.21 – Resolve struts together.
Dimension of nodal zone can be calculated as illustrated in Figure 1.22, lb is the
bearing width (column width or pile width), wt is the size of bearing plate (used as
anchorage)

HCMC University of Transport

15