STRUT AND TIE MODEL FOR PILE CAP REPORT

Phương pháp thiết kế đài cọc bê tông cốt thép theo phương pháp giàn ảo (strut and tie model) theo tiêu chuẩn châu âu EC và Hoa Kỳ ACI. Tính toán cấu kiện bê tông cốt thép theo mô hình giàn ảo theo các tiêu chuẩn nước ngoài

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

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.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

V Factored shear force at section

z Overall height of strut and tie model

 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

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

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…”

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)

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

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

1.4 Design procedure using strut and tie model

The design procedure of strut and tie model can be summarized as the flowchart in

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Figure 1.4 – Design flowchart using STM

Provide reinforcement and anchorage Check struts and nodal zones Calculate reinforcement Determine the forces in members Determine a sufficient STM model

Determine the reaction forces and the boundary conditions Separate B and D-regions

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

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

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

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 specified For a hydrostatic CCC nodal zone, the condition 1 2 3

C

w  w  w must be satisfied thus w w w are proportioned Therefore no shear stresses are created at the 1, 2, 3node 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

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

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]

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

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

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

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



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 C are resolved into 2 C while strut 1 C and 3 C are combined into 4

Figure 1.21 – Resolve struts together

Dimension of nodal zone can be calculated as illustrated in Figure 1.22, l is the b

bearing width (column width or pile width), w is the size of bearing plate (used as t