tension stiffening model for reinforced concrete …

The Pennsylvania State University

The Graduate School

College of Engineering

TENSION STIFFENING MODEL FOR REINFORCED CONCRETE

BASED ON BOND STRESS SLIP RELATION

A Thesis in

Civil Engineering

by

Yun Lin

© 2010 Yun Lin

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Master of Science

August 2010

ii

The thesis of Yun Lin was reviewed and approved* by the following:

Andrew Scanlon

Professor of Civil Engineering

Thesis Adviser

Gordon Warn

Assistant Professor of Civil Engineering

Swagata Banerjee

Assistant Professor of Civil Engineering

William Burgos

Professor of Environmental Engineering and Professor in Charge of Graduate Programs

*Signatures are on file in the Graduate School

iii

Abstract

Tension stiffening is a structural property of reinforced concrete that refers to the

contribution of concrete between cracks to the overall stiffness of the member. In assessing

the strength of reinforced concrete sections the tension in the concrete is usually ignored

because it contributes little to member strength. However it provides an important

contribution to the performance of members at service loads. Tension stiffening can be

modeled at the stress-strain level by use of a post-peak degrading modulus of elasticity as

first proposed by the first author in 1971. This paper proposes and evaluates a method to

evaluate the post-peak parameters of the tension stiffening model based on bond

characteristics between concrete and steel.

Table of contents

Chapter 1 INTRODUCTION……………………………………………………………….……...1

1.1 Background…………………………………………………………………………1

1.2 Objective and Scope………………………………………………………………..2

Chapter 2 LITERATURE REVIEW……………………………………………………………….3

2.1 Tension Stiffening Mechanism…………………………………...………………….3

2.2 Steel-concrete Bond…………………………………………………………………6

2.3 Tension Stiffening Models…………………………………………………………..6

2.4 Finite Element Models………………………………………………………………8

2.5 Harajli’s Bond Stress-slip Model……………………………………………………9

2.6 Summary…………………………………………………………………………..10

Chapter 3 DEVELOPMENT OF ANALYTICAL MODELS……………………………………11

3.1 Introduction………………………………………………………………………...11

3.2 Algorithm…………………………………………………………………………..11

3.3 Two-dimensional Simplification…………………………………………………...20

3.4 Bond Link Element………………………………………………………………...21

3.5 Mesh and Cracks…………………………………………………………...………25

3.6 Example Application of the Model………………………………………………...27

Chapter 4 BEEBY AND SCOTT TEST SPECIMEN………………………………………........38

4.1 Develop Tension Stiffening Model………………………………………………....38

4.2 Strain Distribution along the Steel Bar……………………….……………………..49

Chapter 5 APPLICATION OF THE MODEL………………………………………………...…53

5.1 Introduction……………………………………………….………………..……….53

5.2 Comparison with CEB Model……………………………………………………....53

5.3 Effect of Varying Reinforcement Ratio…………………………………………….57

Chapter 6 SUMMARY CONLUSIONS AND RECOMMENDATIONS……………………….57

6.1 Summary……………………………………………………………………………57

6.2 Conclusion………………………………………………………………………….57

6.3 Recommendations…………………………………………………………………..58

References……………………………………………………………………………………….59

Appendix Bond-slip Relation Chart……………………………………………………………...61

1

Chapter 1

Introduction

1.1 Background

Reinforced concrete members have been widely use for structural purposes. Tension stiffening

refers to tension carrying ability of concrete between cracks, contributing to the stiffness of a

reinforced concrete member before the reinforcement yields. If concrete is assumed to carry

tension between the cracks only, the reinforcement carries the entire axial load at the crack

location. The rigidity of the reinforced member affects the performance of a reinforced member in

terms of deflection and crack control.

Concrete cracks when the tensile stress limit is exceeded. Cracking causes a softening behavior

in plain concrete. As cracking progresses, concrete loses its stiffness at a relatively high rate.

However, this softening behavior is counteracted by the steel reinforcing bars in the tension zone

of concrete. The tensile stress in concrete gradually decreases as cracking develops.

The propagation of cracks is a complicated phenomenon that depends on the interaction

between concrete and reinforcement and plays an important role in the analysis of concrete

structures. This thesis investigates the development of a tension stiffening model based on bond

stress-slip relationships that for can be used in smeared cracking finite element analysis.

1.2 Objective and Scope

The objective and scope of this study is to investigate the behavior of concrete between cracks

considering bond stress-slip relationships to develop tension stiffening models for reinforced

concrete. Tension stiffening models are needed to simulate post-cracking behavior of reinforced

concrete which is important for evaluating serviceability and in particular, deflection control.

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This objective will be achieved within the scope of the following tasks:

1. Literature review

2. Development of analytical models to trace the development of cracking in axially loaded

prisms and the effect on overall stiffness.

3. Evaluation of the analytical models using available experimental data

4. Comparison with the CEB tension stiffening model and study of the effect of reinforcement

ratio

5. Development of conclusions and recommendations.

3

Chapter 2

LITERATURE REVIEWS

2.1 Tension Stiffening Mechanism

The tension stiffening mechanism is illustrated in Fig 2.1 which shows a prism loaded in

tension. As the tensile force P increases cracks form at intervals long the prism.

Figure 2.1(a) shows the stress distribution of concrete and steel along the bar length. At

individual cracks, the entire force is transferred across the crack by the steel bar. Between cracks,

force transfers from steel to concrete through bond stress and the tensile force is carried by both

steel and concrete. The total applied force is equal to the sum of the force in the steel and the force

in the concrete at any section. At the cracked locations, stress in concrete is assumed to be zero

and the entire axial load is concentrated at reinforcement bar.

Figure 2.2 shows the stress distribution of the concrete in a section perpendicular to the bar. At

a section near to the end of the bar or a cracked location, stress in the concrete is higher closer to

the reinforcement bar and lower further away. The distribution gradually becomes uniform at

sections away from the crack location.

4

Fig. 2.1 Stress distribution of concrete and steel along the prism (MacGregor 2009)

5

Fig. 2.2 Variation of longitudinal stress distribution of concrete in vertical direction

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2.2 Steel-Concrete Bond

Figure 2.3 shows a picture of a steel-concrete bond. The bond between steel and concrete has

three main components: chemical adhesion, friction and mechanical interlock (Wang and Liu

2003). Chemical adhesion is the original bond developed between concrete and steel before any

slip occurs. The effect of chemical adhesion is very small and it does not allow any slip. As the

steel bar is loaded up to a certain level, chemical adhesion bond cannot provide sufficient bond

force and breaks down. As soon as the chemical bond fails, relative movement can occur between

concrete and steel. As one of the components of bond force, friction comes into play. The radial

forces around the steel bar can create a certain amount of friction forces counteracting the slip

effect. Also, mechanical interlock, which is created by the ribs on the bar embedded in the

concrete, becomes the most important component as illustrated in Figure 2.3. As the load

continuous to increase, the steel bar is elongated more significantly. Poisson’s ratio’s effect causes

the cross section to decrease. The radial forces are significantly reduced due to that effect, so

friction becomes negligible at this stage and leaving the bearing of concrete becomes the primary

force transfer mechanism. Cracks begin to form adjacent to steel rebar.

2.3 Tension Stiffening Models

Since the early 1970’s a number of models have been proposed in the literature to represent the

stiffening effect of concrete between cracks for use in smeared crack finite element analyses. (e.g.

Scanlon (1971), Scanlon and Murray (1974), Nayall and Rashid, 2006). Figure 2.4 shows the

original tension stiffening model developed by Scanlon (1971). In most cases these models,

typically referred to as “tension stiffening models”, have been developed on an empirical basis and

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validated by comparing computed load – deformation response against available laboratory test

data. In this paper a methodology is proposed to develop such a model based on consideration of

bond characteristics between concrete and steel and progressive cracking under increasing load

Fig. 2.3 Steel-concrete Bond

Figure 2.4 Tensile Stresses in Reinforced Concrete (Scanlon 1971)

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Figure 2.5 shows the bond force distribution along the prism. Bond force is larger closer to crack

location.

Fig. 2.5 Bond force distribution along the prism

2.4 Finite element model

Finite element models for analysis of reinforced concrete have been under development since

the 1960’s. Ngo and Scordelis (1967) proposed a discrete crack analysis finite element model for

reinforced concrete beams. A unit width of the beam was modeled using plane stress elements. At

predefined crack locations, separate nodal points on either side of the crack were defined. Linear

bond link elements were used to model bond between the reinforcement and the concrete.

Nilson (1971) extended the model incorporating a nonlinear bond-slip relationship with

nonlinear material property to increase the accuracy of the model. A nonlinear incremental method

was used in his study. De Groot et al. (1981) developed a bond-zone element to distribute bond

stress. Keuser and Mehlhorn (1987) introduced a contact element to provide continuous interaction

9

between concrete and steel. Yankelevsky (1985) proposed a linear bond stress-slip law. G. Chen

and G. Baker (2002) use a single spring model to account for bond stress and slippage. Rots and

Invernizzi(2002) developed a saw-tooth tension stiffening model with sequential cracks. Lowes et

al (2004) created a concrete steel bond model under cyclic loading.

2.5 Harajli’s bond stress-slip model

Harajli (2002) generated a monotonic envelope bond stress-slip relationship using regression

analysis of test data. Using both analytical model and experimental results, he proposed an

equation for maximum bond stress and the corresponding slippage. Figure 2.5 shows the local

bond stress and slip relationship for axially loaded concrete prisms reinforced with steel bars. The

vertical axis U is the local bond stress and horizontal axis is local slip distance. Local bond stress

U is increasing in a descending rate instead of linear relationship with slip distance. Equation 2.4.1

shows the relationship between local bond stress and slip distance. The maximum bond stress 𝑈𝑚

and the corresponding slip distance 𝑠1 are defined in equation 2.4.2 and 2.4.3. The splitting stress

𝑈𝑠𝑝 is required stress for concrete splitting failure and it is not used in this study. The bond stress

slip relationship is used to determine the stiffness for linkage elements in later chapters.

𝑈 = 𝑈𝑚(𝑠

𝑠1)𝛼 (2.4.1)

𝑈𝑚𝑎𝑥 = 31.0 𝑓𝑐′ (psi) (2.4.2)

𝑠1= 0.15𝑐𝑜 (2.4.3)

𝑐𝑜 : Clear distance between lugs

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2.6 Summary

In most cases tension stiffening models have been developed on an empirical basis and verified

in terms of overall structural behavior. Models have been developed in this chapter to simulate the

load interaction between reinforcement and concrete in tensile zones. In the following chapter,

bond stress-slip elements will be used to provide the basis for an overall tension stiffening model

based on degrading concrete stiffness under sequential cracking.

Fig. 2.5 Monotonic envelope model (Harajli 2002)

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Chapter 3

DEVELOPMENT OF ANALYTICAL MODEL

3.1 Introduction

In this chapter, a finite element model is developed to simulate the response of an axially

loaded tensile prism under increasing load. Using a proposed bond stress-slip model, stresses in the

prism are monitored and cracks are inserted as the cracking stress is exceeded under increasing

load. The model is used to develop a saw-tooth type of tension stiffening model for use in smeared

crack finite element analysis.

3.2 Algorithm

An increasing tensile stress is applied to the prism. Whenever the maximum stress in the

concrete reaches the tensile stress limit, a new series of cracks are inserted to the prisms, crack

spacing is shortened and equivalent stiffness of the prism is decreased.

3.2.1 Step 1

The prism is initially assumed to be uncracked and under a uniform strain 𝜀𝑐𝑜𝑚𝑝 1

corresponding to a total axial force 𝑃𝑇 causing a stress fc in the concrete and fs in the steel. A

uniform strain is considered to be under plane stress with area of concrete Ac and area of steel As .

Figure 3.1 shows the stress and strain distribution in axially loaded the prism.

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a) Stress

b) Strain

Fig. 3.1 Stress and strain distribution in uncracked tensile prism

13

As the load P is increased, the stress in the concrete increases until the maximum tensile stress

of the concrete is reached 𝑓𝑡′ . The strain 𝜀𝑐𝑜𝑚𝑝 1 at any cross section is uniform. While increasing

the applied load, the tensile stress of concrete will eventually reach the tensile stress limit 𝑓𝑡′ .

𝑆𝑡𝑟𝑒𝑠𝑠 𝑖𝑛 𝑡𝑕𝑒 𝑐𝑜𝑛𝑐𝑟𝑒𝑡𝑒 = 𝑓𝑡1= 𝑓𝑡

′

𝜀𝑐𝑜𝑚𝑝 1 = 𝑓𝑡

′

𝐸𝑐

𝑇𝑜𝑡𝑎𝑙 𝑎𝑝𝑝𝑙𝑖𝑒𝑑 𝑓𝑜𝑟𝑐𝑒 = 𝑃𝑇 = 𝑓𝑐𝐴𝑐 + 𝑓𝑠𝐴𝑠 = 𝐸𝑐𝜀𝑐𝑜𝑚𝑝 1𝐴𝑐 + 𝐸𝑠𝜀𝑐𝑜𝑚𝑝 1𝐴𝑠

Figure 3.2 shows the plot of stress vs. strain in concrete up to the first cracking.

Fig.3.2 Tensile stress vs. Strain in concrete up to the first cracking

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3.2.2 Step 2

At first cracking, the concrete has reached its tensile strength 𝑓𝑡′ . It is assumed that at this stage

a series of cracks form at a spacing L along the prism, the force is concentrated in the steel bar.

Therefore, apply force P to the steel bar embedded in a prism with a length of L (Fig. 3.3). The

right end represents the section of the prism mid-way between adjacent cracks. Tensile stress

distribution in vertical direction closer to the force end is no longer uniform, and gradually

changing to uniform as moving to the middle of the prism (Fig.3.4). In any horizontal path,

concrete closer to the steel bar has higher tensile stress and the maximum tensile stress can exceed

the tensile stress limit 𝑓𝑡′ as the load P increases. New cracks are inserted when maximum stress of

concrete reaches 𝑓𝑡′and new crack spacing becomes L/2. Every time new cracks are inserted to the

prism, the equivalent overall prism stiffness is reduced to a new value which means the slope of

stress vs. stain diagram is reduced.

Fig. 3.3 Load P applied to reinforcing bar after formation of first series of cracks at spacing

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Fig. 3.4 Tensile Stress distribution in vertical paths

Apply force P in the prism at 1st crack to the steel. Obtain the stress distribution in the steel bar

from the finite element results. Figure 3.5 shows the tensile stress distribution along the steel bar

between two cracks. 𝑆11𝑆𝑡𝑒𝑒𝑙 is the average stress in the steel. The average total force in the steel

𝑃𝑠 is equal to the product of the average stress in the steel 𝑆11𝑆𝑡𝑒𝑒𝑙 and the area of steel 𝐴𝑠. Total

force in the concrete at any section is equal to the difference of total applied force 𝑃𝑇 and force in

the steel 𝑃𝑠.

Fig. 3.5 Tensile Stress distribution in steel between Cracks

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Force in the steel = 𝑃𝑠 = 𝑆11𝑆𝑡𝑒𝑒𝑙 × 𝐴𝑠

𝑆11𝑆𝑡𝑒𝑒𝑙 = Average stress in the steel can be obtained from ABAQUS model.

𝑃𝑇 = 𝑃𝑐 + 𝑃𝑠

𝑃𝑐 = 𝑃𝑇 − 𝑃𝑆

Equivalent modulus of elasticity of concrete is reduced to 𝐸𝑒2

𝐸𝑒2=

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑆𝑡𝑟𝑒𝑠𝑠 𝑖𝑛 𝑡𝑕𝑒 𝑐𝑜𝑛𝑐𝑟𝑡𝑒

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑆𝑡𝑟𝑎𝑖𝑛=

𝑃𝐶𝐴𝐶

𝜀𝑐𝑜𝑚𝑝 2

𝜀𝑐𝑜𝑚𝑝 𝑎𝑡 𝑎𝑛𝑦 𝑠𝑡𝑎𝑔𝑒𝑠 𝑐𝑎𝑛 𝑏𝑒 𝑜𝑏𝑡𝑎𝑖𝑛𝑒𝑑 𝑓𝑟𝑜𝑚 𝐴𝐵𝐴𝑄𝑈𝑆 𝑚𝑜𝑑𝑒𝑙.

𝜀𝑐𝑜𝑚𝑝 2 =∆𝑡𝑜𝑡𝑎𝑙 2

𝐿 𝜀𝑐𝑜𝑚𝑝 3 =

∆𝑡𝑜𝑡𝑎𝑙 3

𝐿

∆𝑡𝑜𝑡𝑎𝑙 : 𝑇𝑜𝑡𝑎𝑙 𝑒𝑙𝑜𝑛𝑔𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡𝑕𝑒 𝑝𝑟𝑖𝑠𝑚.

L: Total length of the prism.

In figure 3.7, point 2 shows the stress in the concrete when the original total load in step 1 is

applied at the steel bar. The new slope is reduced to 𝐸𝑒2due to the stiffness loss caused by cracks.

𝑓𝑡2= 𝜀𝑐𝑜𝑚𝑝 2 × 𝐸𝑒2

Continue increasing applied load to the ABAQUS model to make the maximum stress of

concrete reach limit again (point 3). Figure 3.4 shows in any vertical path, the maximum stress in

the concrete is always at the layer closest to the steel layer. Figure 3.6 shows the tensile stress

distribution of the horizontal path in the concrete closest to steel layer between two cracks.

Applied load P is increased until the maximum stress 𝑆𝑡 ,𝑐(𝑚𝑎𝑥 ) reaches the tensile stress limit 𝑓𝑡′ .

Due to the variation of concrete property, crack can take place within the length of 𝑆𝑐 , shown in

figure 3.6.

17

Fig. 3.6 Tensile stress distribution of the horizontal path in the concrete closest to steel layer

between two cracks

At point 3:


Figure 3.7 shows shows the plot of average stress of concrete vs. Strain of the composite up to

this stage and a new series of cracks is ready to be inserted in step 3.

fi

Fig. 3.7 Average tensile stress vs. Strain up to the second cracking

18

3.2.3 Step 3

Insert new series of cracks to the model. Crack spacing is decreased to L/4. Repeat the procedures

in step 2: Increase applied load (𝑃𝑇) so that maximum stress of concrete matches 𝑓𝑡′

𝑃𝑠 = 𝑆11𝑆𝑡𝑒𝑒𝑙× 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑠𝑡𝑒𝑒𝑙

Average stress in steel (𝑆11𝑆𝑡𝑒𝑒𝑙) can be obtained from ABAQUS model.

𝑃𝐶 = 𝑃𝑇 − 𝑃𝑠


𝐿

Equivalent modulus of elasticity of concrete = 𝐸𝑒3=

𝑃𝐶𝐴𝐶

𝜀𝑐𝑜𝑚𝑝 4

𝑓𝑡4= 𝐸𝑒3

× 𝜀𝑐𝑜𝑚𝑝4

Fig. 3.8 Average tensile stress vs. Strain

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3.2.4 Step 4

Repeat step number 3 until stress in the steel can no longer be increased or crack spacing is too

small. A stepped (saw-tooth) stress-strain diagram is completed once all the steps are properly

performed until reinforcement bar is yielded. Figure shows a complete saw tooth stress and strain

diagram.

Fig. 3.9 Stepped (saw-tooth) stress-strain diagram

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3.3 Two-dimensional simplification

Figure 3.10 shows a simplified two-dimensional Prism model. The model has a 16 inches long

prism with 5 x 5 inches cross-section with a number 8 steel rebar though the middle.

All parts have a unit width of 1 inch. A realistic model contains a circular cross-sectional steel

bar and a block of concrete. For simplicity, Scordelis’s two-dimensional model with unit width is

used in the modeling. A thin layer of steel is used to model the steel bar so that the 1 unit width

model has the same reinforcement ratio. The thickness of the thin layer is the product of the

vertical length of concrete and reinforcement ratio.

𝑇𝑕𝑖𝑐𝑘𝑛𝑒𝑠𝑠 𝑜𝑓 𝑡𝑕𝑖𝑛 𝑙𝑎𝑦𝑒𝑟 = 𝑕 × 𝜌

h: height of concrete block in two dimensional model [fig. 1]

ρ: reinforcement ratio

Fig. 3.10 Prism Dimensions

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3.4 Bond link element

Figure 3.11 shows a sample ABAQUS model of a cracked reinforced concrete beam with two

point loads on one third of the beam and its bond stress distribution along the horizontal distance.

A fully bonded condition was assumed. In reality, the bond stress curve does not have sudden

jumps at the locations of crack. A bond link element which can account for slip effect is needed.

Fig. 3.11 – Bond stress distribution for perfect bond condition

A series of spring elements will be added to provide the bond force and a slip distance

depending on the local bond force. A stiffness coefficient and the tributary width of each spring

gives a combination of bond force and slip distance.

3.4.1 Spring element setup

Figure 3.12 shows the spring element setup for the analytical models. Spring elements are

spaced out evenly along the prism member. Each spring pair provides the bond force over the

distance ∆L.

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Fig. 3.12 Spring element layout

3.4.2 Stiffness Coefficient of Spring Element

Initially, a linear relationship between bond slip and bond stress was assumed which means the

bond slip is proportional to bond stress. A single spring was used in each location. Cracking due to

a “press down effect” has not yet occurred. It is reasonable to believe that the vertical movement of

steel is very small and negligible. Thus, only horizontal elongations of spring elements are allowed

in this model.

Load is directly applied to steel. Part of the stress is transferred to the concrete due to bond

stress. The horizontal stiffness coefficient of spring element affects the rate of stress transfer

between concrete and steel. However, the amount of stress transfer is not directly proportional to

the stiffness coefficient K. The force in the spring is the product of stiffness coefficient K and

elongation S. Any increment in K value decreases the elongation S. To know how K value is

affecting the stress left in the steel in middle of the prism, parametric study of stiffness coefficient

K is performed

A stress of 6452 psi is applied to the bar at the end of the prism. Perfect bond between concrete

and steel is used. In the result, at middle of the prism, the steel has a residual horizontal stress of

1511 psi. Then, fully bonded condition is replaced by a number of evenly distributed spring

23

elements. Figure 3.13 shows the relationship between spring stiffness coefficient K and residual

stress in the steel at middle of prism.

Fig. 3.13 Residual stress distribution in the steel bar at the middle of the prism

When the stress reached its minimum value, in this case it is 1665.81 psi, any increment of K

has no effect on residual stress. The minimum residual stress is still larger than the case in perfect

bond condition. It is because spring element can only provide discrete resistance instead of

continuous resistance.

Stiffness coefficient K is affecting the amount of stress transferred from steel to concrete. The

variation of stress distribution shows exactly how sensitive the solution is to the selected value K.

Figure 3.14 shows how stress distribution from the end to the middle of the prism in the concrete

varies with different K value. It is assumed that from 50000 to 300000 lb/in is the reasonable range

24

of K in this model because within the range the distribution of the concrete is increasing from zero

and being constant when the maximum stress level is reached.

Fig. 3.14 Tensile stress distribution in concrete

In reality, bond slip relationship is not exactly linear as represented by a spring with constant

stiffness. Bond slip relationships are different at different locations, even at the same location the

relationship changes under different applied loads. Harajli’s model (chapter 2) is used to determine

variable bond link stiffness for Beeby and Scott’s model discussed in chapter 4. Figure 3.14 shows

the variation of bond link stiffness as a function of slip. The bond link stiffness is relatively greater

when the slip is small. The derivation will be presented in chapter 5. More slippage means more

damage due to the relative movement of steel and concrete.

0

100

200

300

400

500

600

0 2 4 6 8 10 12 14 16

K = 1E^7

K = 1E^6

K = 5E^5

K = 3E^5

K = 2E^5

K = 1.5E^5

K = 1E^5

K = 80000

K = 50000

K = 10000

K = 1000

x (in)

S11

(psi)

25

Fig. 3.14 Tensile stress distribution in concrete

3.5 Mesh and cracks

Figure 3.15 shows the mesh of the prism problem. Rectangular mesh is used for finite element

calculations. Smaller mesh is used in steel, because the steel thin layer has a small thickness. Nodal

points are needed at the middle of the layer to obtain the stress level at each position. Larger mesh

is used in concrete, but the layer closer to the steel has smaller mesh size. In general, smaller mesh

size is used at the place where more accurate result is needed.

For simplicity, straight through cracks are used in the model as is shown in figure 3.16. A

series of cracks will be generated where stress level in concrete passes the limit. Cracks are placed

slightly left of the spring element. A very small crack width of 0.0001 inch is used so that it does

not affect the mesh layout.

26

Fig. 3.15 Mesh

Fig. 3.16 Bond link element, mesh and Crack

27

3.6 Example application of the model

An arbitrary reinforced concrete prism is used as the first analytical model using spring linkage

element with constant stiffness. Fig.3.17 shows an axially loaded 32’ long prism with a 5” by 5”

cross-section reinforced with a number eight steel bar though the middle. Load is applied at each

end of the steel bar.

Fig. 3.17 Example Model

Assumptions:

Linear bond-slip relation (Constant bond link stiffness)

Constant spring stiffness K is assumed to be 1.5 × 105𝑙𝑏/𝑖𝑛.

Material property of concrete is uniform.

Material property:

Material property:

𝐶𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑣𝑒 𝑠𝑡𝑟𝑒𝑛𝑔𝑡𝑕 𝑜𝑓 𝑐𝑜𝑛𝑐𝑟𝑒𝑡𝑒 = 𝑓𝑐′ = 4000 𝑝𝑠𝑖

𝑇𝑒𝑛𝑠𝑖𝑙𝑒 𝑠𝑡𝑟𝑒𝑛𝑔𝑡𝑕 𝑜𝑓 𝑐𝑜𝑛𝑐𝑟𝑒𝑡𝑒 = 7.5 𝑓𝑐′ = 474 𝑝𝑠𝑖

𝑌𝑒𝑖𝑙𝑑 𝑠𝑡𝑟𝑒𝑠𝑠 𝑜𝑓 𝑠𝑡𝑒𝑒𝑙 𝑏𝑎𝑟 = 𝑓𝑦 = 60 𝑘𝑠𝑖

𝑀𝑜𝑑𝑢𝑙𝑢𝑠 𝑜𝑓 𝑒𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑡𝑦 𝑜𝑓 𝑐𝑜𝑛𝑐𝑟𝑒𝑡𝑒 = 𝐸𝑐 = 57000 𝑓𝑐′ = 3605000 𝑝𝑠𝑖

𝑀𝑜𝑑𝑢𝑙𝑢𝑠 𝑜𝑓 𝑒𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑡𝑦 𝑜𝑓 𝑠𝑡𝑒𝑒𝑙 = 𝐸𝑠 = 29000000 𝑝𝑠𝑖

28

Dimensions:

Thickness = 1 in

Cross-sectional area of Concrete = 𝐴𝑐 = 5 𝑖𝑛2 − 0.155 𝑖𝑛2 = 4.845 𝑖𝑛2

Cross-sectional area of Steel = 𝐴𝑠 = 0.155 𝑖𝑛2

Length = 32 in

Perform the steps mentioned in chapter 3.2.

Step 1

Applied a uniform strain to an uncracked prism and increase the uniform strain to 𝜀𝑐𝑜𝑚𝑝 1 when

tensile stress in concrete is reached 𝑓𝑡′ , 474 psi.

𝑇𝑜𝑡𝑎𝑙 𝑎𝑝𝑝𝑙𝑖𝑒𝑑 𝑓𝑜𝑟𝑐𝑒 = 𝑃𝑇 = 𝑓𝑐 × 𝐴𝑐 + 𝑓𝑠 × 𝐴𝑠

𝑓𝑐 = 474 𝑝𝑠𝑖

𝑓𝑠 = 𝐸𝑠 × 𝜀𝑐𝑜𝑚𝑝 1 = 29000000 𝑝𝑠𝑖 × 0.000131 = 3799 𝑝𝑠𝑖

𝑃𝑇 = 474 𝑝𝑠𝑖 × 4.845 𝑖𝑛2 + 3799 𝑝𝑠𝑖 × 0.155 𝑖𝑛2 = 2885 𝑙𝑏

𝑓𝑡1 = 𝑓𝑡′ = 474 𝑝𝑠𝑖

𝜀𝑐𝑜𝑚𝑝 1 =𝑓𝑡

′

𝐸𝑐=

474 𝑝𝑠𝑖

3605000 𝑝𝑠𝑖= 0.000131

Step 2

All the concrete has reached its maximum tensile stress capacity. Cracks can happen at

anywhere along the uncracked prism. Insert cracks and make a crack spacing 32”. Now the same

load 2885 lb is applied at both ends of the steel bar, the stress level in the concrete is reduced due

to the stress relief at cracks. Increase the load to 2945 lb, so that the maximum stress in the

concrete is reached to 𝑓𝑡′ again. Figure 3.18 shows the stress contour from ABAQUS result.

29

Fig. 3.18 Tensile Stress Contour for 32” crack spacing

Figure 3.19 shows the tensile stress distribution in the steel and average stress in the steel 𝑆11𝑆𝑡𝑒𝑒𝑙 .

Fig. 3.19 Tensile Stress distribution in the steel

𝑆11𝑆𝑡𝑒𝑒𝑙 = 5939 𝑝𝑠𝑖

Force in the steel = 𝑃𝑠 = 𝑆11𝑆𝑡𝑒𝑒𝑙 × 𝐴𝑠 = 5939 𝑝𝑠𝑖 × 0.155 𝑖𝑛2 = 920.5 𝑙𝑏

𝑃𝑇 = 𝑃𝑐 + 𝑃𝑠 = 2945 𝑙𝑏

𝑃𝑐 = 𝑃𝑇 − 𝑃𝑆 = 2945 𝑙𝑏 − 920.5 𝑙𝑏 = 2024.5 𝑙𝑏

∆𝑡𝑜𝑡𝑎𝑙 2= 0.00745 𝑖𝑛

30


𝐿=

0.00745 𝑖𝑛

32 𝑖𝑛= 0.000233


𝐸𝑒2=



𝑃𝐶𝐴𝐶

𝜀𝑐𝑜𝑚𝑝 2=

2024.5 𝑙𝑏4.845 𝑖𝑛2

0.000233= 1793362 𝑝𝑠𝑖


= 0.000233 × 1793362 𝑝𝑠𝑖 = 412.5 𝑝𝑠𝑖

In figure 3.20, maximum stress of the concrete was reached approximately from 4 in to 28 in of

the prism. Figure only shows half of the results due to symmetry. At zero horizontal distance, the

stress in concrete is not zero, because there are spring elements placed at both ends of the prism

causing an immediate stress transfer at both ends.


between two cracks

31

Step 3

Additional cracks are assumed to be at 8”, 16” and 24” of the prism. New crack spacing is 8”.

Load is increased to 3131 lb in ABAQUS model to make maximum stress in concrete reach 𝑓𝑡′ .

Stress contour is shown in figure 3.21.



Fig. 3.22Tensile stress distribution of steel


Force in the steel = 𝑃𝑠 = 𝑆11𝑆𝑡𝑒𝑒𝑙 × 𝐴𝑠 = 12871 𝑝𝑠𝑖 × 0.155 𝑖𝑛2 = 1995 𝑙𝑏

𝑃𝑇 = 𝑃𝑐 + 𝑃𝑠 = 3131 𝑙𝑏

32

𝑃𝑐 = 𝑃𝑇 − 𝑃𝑆 = 3131 𝑙𝑏 − 1995 𝑙𝑏 = 1136 𝑙𝑏

∆𝑡𝑜𝑡𝑎𝑙 3= 0.0138𝑖𝑛 (𝑓𝑟𝑜𝑚 𝑎𝑏𝑎𝑞𝑢𝑠 𝑟𝑒𝑠𝑢𝑙𝑡)


𝐿=

0.0138 𝑖𝑛

32 𝑖𝑛= 0.000431


𝐸𝑒3=



𝑃𝐶𝐴𝐶


1136 𝑙𝑏4.845 𝑖𝑛2

0.000431= 544010 𝑝𝑠𝑖


= 0.000431 × 544010 𝑝𝑠𝑖 = 234.5 𝑝𝑠𝑖

In figure 23, maximum stress of the concrete was reached approximately at 4”, 12”, 20” and

28” in of the prism. Figure only shows half of the results due to symmetry.


between two cracks

Step 4

Additional cracks are inserted at 4”, 12”, 20” and 28”. New crack spacing becomes 4”. Load is

increased to 3720 lb to make maximum stress in concrete reach 𝑓𝑡′ . Stress contour is shown in

figure 3.24.

33



Fig. 3.25 Tensile stress distribution of steel


Force in the steel = 𝑃𝑠 = 𝑆11𝑆𝑡𝑒𝑒𝑙 × 𝐴𝑠 = 20335 𝑝𝑠𝑖 × 0.155 𝑖𝑛2 = 3152 𝑙𝑏

𝑃𝑇 = 𝑃𝑐 + 𝑃𝑠 = 3720 𝑙𝑏

𝑃𝑐 = 𝑃𝑇 − 𝑃𝑆 = 3720 𝑙𝑏 − 3152 𝑙𝑏 = 568 𝑙𝑏

∆𝑡𝑜𝑡𝑎𝑙 4= 0.0212𝑛


𝐿=

0.0212 𝑖𝑛

32 𝑖𝑛= 0.000663

34


𝐸𝑒4=



𝑃𝐶𝐴𝐶


568 𝑙𝑏4.845 𝑖𝑛2

0.000663= 176824 𝑝𝑠𝑖

𝑓𝑡4= 𝜀𝑐𝑜𝑚𝑝 4 × 𝐸𝑒4 = 0.000663 × 176824 𝑝𝑠𝑖 = 117.2 𝑝𝑠𝑖


between two cracks

In step 1, figure 3.4(d) shows that it takes about 4” for the stress in concrete to rise from

minimum to maximum. This distance does not change on step 2 and 3. The required load

increment is small to form another series of cracks if the crack spacing is greater than 8”, which is

twice the distance of 4”. Once the crack spacing is less than 8”, it takes a lot more additional load

for the maximum stress in concrete to reach 𝑓𝑡′ . In this model, crack spacing is still able to reduce

to 2”, if enough additional load is provided, because spring stiffness is assumed to be constant. In

reality, bond link stiffness will be reduced when slippage is increased. When slippage is too large,

bond link elements might not be able to provide needed force transfer within a distance shorter

than 4”. It is suggested that minimum cracks spacing to be taken as 4”. A variable stiffness for

35

spring element will be adopted in Beeby and Scott’s model. Figure 3.27 shows the stepped (saw-

tooth) Stress strain diagram for this model.

Fig. 3.27 Stepped (saw-tooth) Stress strain diagram for example model

36

Comparison of bond link elements with different constant stiffness

For analytical model with constant spring stiffness, 50000 to 300000lb/in is the reasonable

range for stiffness coefficient (K) of spring elements (Fig. 3.14). In the previous calculation,

150000lb/in is used for K value. In the part of study, different k values will be used to determine

the sensitivity of K.

Figure 3.28 shows that how bond link stiffness K affect the cracking load and displacement.

Generally speaking, a prism with higher K value cracks in a faster rate. The faster rate is reflected

by a smaller initial cracking load and smaller crack spacing.

The saw-tooth form can also be converted to a gradually descending post-peak form by curve

fitting and an “equal energy” criterion, i.e. equal area under the post-peak portion of the stress-

strain diagram as shown in Figure 3.29.

Fig. 3.28 Effect of assumed bond element stiffness on stress-strain diagram

0

50

100

150

200

250

300

350

400

450

500

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016

K= 50000

K=150000

K=250000

ft

(psi)

37

Fig. 3.29 Smoothed post-peak stress-strain diagram

0

50

100

150

200

250

300

350

400

450

500

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016

K=150000

K=50000

K=250000

ft

(psi)

38

Chapter 4

BEEBY AND SCOTT TEST SPECIMEN

4.1 Introduction

A simplified two-dimensional model was made to compare with the Beeby and Scott’s test

specimen (Beeby and Scott 2002). A tension stiffening model was created based on the analytical

finite element results. The strain distribution along the steel bar was compared with experimental

data to confirm the accuracy of the analytical assumptions and results.

4.2 Development of tension stiffening model

Fig.4.1 shows an analytical model for a axially loaded 1600mm long prism with a 120 by

120mm cross-section reinforced with a number five steel bar though the middle. Load is applied at

each end of the steel bar. The intention of building this model is to match up the experimental

data with experiment T16B1. Variable Bond link stiffness is used in this analytical model.

Fig. 4.1 Beeby and Scott’s test specimen

Assumptions:

Bond link stiffness K is variable corresponding to the slip distance. Details are presented later

in this chapter.

Strength of concrete is increasing for each cracking stage.

39

Material property:

Material property: 𝑓𝑐′ = 3091 𝑝𝑠𝑖; 𝑓𝑦 = 60 𝑘𝑠𝑖

Modulus of Elasticity of Concrete = 𝐸𝑐 = 57000 𝑓𝑐′ = 3170000 psi = 21856 Mpa

Modulus of Elasticity of steel bar = 𝐸𝑠 = 200 𝐺𝑝𝑎

Dimensions:

Thickness = 1 mm

Cross-sectional area of Concrete = 14400 𝑚𝑚2

Cross-sectional area of Steel = 200 𝑚𝑚2

Specimen Length = 1600mmn

Element type:

2-D, 4 Nodes, Shell element

Load stages:

1. 8 KN

2. 20 KN

3. 29 KN

4. 44 KN

Bond link springs:

Total number of springs: 130 (65 pairs)

Spacing = 2.5 cm

40

Derivation of bond link stiffness coefficient

The variable bond link stiffness coefficient will be derived based on the bond stress-slip

relationship in Harajli(2002). K is the spring stiffness coefficient equal to the bond force divided

by slip distance S. Bond force is equal to the product of bond stress 𝜏 and tributary area at each

spring element. Equation (2) can be used to obtain local bond stress to get the bond force needed in

equation (3).

𝜏 ∶ 𝐿𝑜𝑐𝑎𝑙 𝐵𝑜𝑛𝑑 𝑆𝑡𝑟𝑒𝑠𝑠

S : Local slip distance

𝜏𝑚𝑎𝑥 ∶ 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑏𝑜𝑛𝑑 𝑆𝑡𝑟𝑒𝑠𝑠

𝑆1 ∶ 𝑆𝑙𝑖𝑝 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑐𝑜𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑡𝑜 𝜏𝑚𝑎𝑥

T.A. : Tributary area of each bond link element

K : Stiffness coefficient of bond link element

𝜏𝑚𝑎𝑥 = 2 𝑓𝑐′ (Mpa) (Harajli 2002) (1)

𝜏 = 𝜏𝑚𝑎𝑥 (𝑠

𝑠1)𝛼 (Harajli 2002) (2)

𝛼 = 0.4

𝐾 = 𝐵𝑜𝑛𝑑 𝑓𝑜𝑟𝑐𝑒

𝑆=

𝜏(𝑇.𝐴.)

𝑆=

𝜏𝑚𝑎𝑥 (𝑠

𝑠1)𝛼 (𝑇.𝐴)

𝑆= {𝜏𝑚𝑎𝑥 𝑆1

−𝛼(𝑇. 𝐴. )}𝑆(𝛼−1) (3)

All the parameters in {𝜏𝑚𝑎𝑥 𝑆1−𝛼(𝑇. 𝐴. )} are constant. The only variable for K is slip distance.

The value of bond link stiffness can be calculated if slip is known. All the value of bond link

stiffness is tabulated in appendix A from a 0.01 mm to 1.8 mm, where 1.8 mm is 15% of clear lug

distance of a number five steel rebar. Table 4.1 shows the different K value for slip distance

between 0 and 𝑆1 .

41

In the experimental (Beeby and Scott 2002), deflections of the bar at each location along the

bar have been recorded. Elongation in the steel bar is normally 10 times larger than the elongation

in the concrete at the same horizontal location. A series of slip distances can be generated

according to the local deflection of the steel bar. The corresponding series of stiffness K can be

obtained from the slip distance and the tabulated stiffness in Appendix A. All stiffness K values

have to be regenerated once the applied load is changed which means there are 4 different series of

K values at 4 different load stages based on equation (3).

The tensile stress limit 𝑓𝑡′ is assumed to be increasing in the sequential crack stages. For

example in step 1 where the first series of crack is about to form, 𝑓𝑡′ is assumed to be 1.6 Mpa, but

in step 2, 𝑓𝑡′ is assumed to be 1.7 Mpa. In reality, concrete property is not perfectly uniform. Crack

will always from at the location with weaker tensile strength first.

Step 1

Assume minimum tensile strength 𝑓𝑡1′ of concrete is 1.6 MPa.

𝜀𝑐𝑜𝑚𝑝 1 = 𝑓𝑟𝐸𝑐

=1.6𝑀𝑝𝑎

21856𝑀𝑝𝑎= 0.000073

𝑇𝑜𝑡𝑎𝑙 𝑎𝑝𝑝𝑙𝑖𝑒𝑑 𝑓𝑜𝑟𝑐𝑒 = 𝑃𝑇 = fc × Ac + fs × As

= 0.000073 200Gpa 200mm2

+ 0.000073 21856Mpa 14400mm2 − 200mm2 = 25.5 KN

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑠𝑡𝑟𝑒𝑠𝑠 𝑖𝑛 𝑡𝑕𝑒 𝑐𝑜𝑛𝑐𝑟𝑒𝑡𝑒 = 𝑓𝑡1= 𝑓𝑡1

′ = 1.6𝑀𝑝𝑎

42

Spring Stiffness(N/mm) Spring Stiffness(N/mm)

K1 381 K33 3051

K2 396 K34 3051

K3 413 K35 3051

K4 432 K36 3051

K5 453 K37 3051

K6 477 K38 3051

K7 506 K39 3051

K8 539 K40 3051

K9 578 K41 3051

K10 626 K42 3051

K11 687 K43 3051

K12 766 K44 3051

K13 879 K45 3051

K14 1041 K46 3051

K15 1328 K47 3051

K16 2013 K48 3051

K17 3051 K49 2013

K18 3051 K50 1328

K19 3051 K51 1041

K20 3051 K52 876

K21 3051 K53 766

K22 3051 K54 687

K23 3051 K55 626

K24 3051 K56 578

K25 3051 K57 539

K26 3051 K58 506

K27 3051 K59 477

K28 3051 K60 453

K29 3051 K61 432

K30 3051 K62 413

K31 3051 K63 396

K32 3051 K64 381

Table 4.1 Stiffness coefficients for spring elements for Beeby and Scott’s specimen

43

Step 2

Cracking spacing = 1600 mm, Load = 26 KN

Assume the minimum tensile strength of concrete increased to 1.7 MPa.

Figure 3.5(b) shows the tensile stress distribution along the steel bar and the average stress in

steel 𝑆11𝑆𝑡𝑒𝑒𝑙 .


Figure 4.2 shows the tensile stress distribution along the steel bar and the average stress in


𝑆11𝑆𝑡𝑒𝑒𝑙 = 46.8 𝑀𝑃𝑎

Force in the steel = 𝑃𝑠 = 𝑆11𝑆𝑡𝑒𝑒𝑙 × 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑠𝑡𝑒𝑒𝑙 = 46.8𝑀𝑝𝑎 × 200𝑚𝑚2 = 9260 𝑁


𝑃𝑐 = 𝑃𝑇 − 𝑃𝑆 = 26000 𝑁 − 9260 𝑁 =16740 N


∆𝑡𝑜𝑡𝑎𝑙 2= 0.3408 𝑚𝑚


𝐿=

0.3408 𝑚𝑚

1600 𝑚𝑚= 0.000213

𝜀𝑐𝑜𝑚𝑝 = 0.000213

44

𝐸𝑒2=



𝑃𝐶𝐴𝐶


16740𝑁(14400𝑚𝑚2 − 200𝑚𝑚2)

0.000213

= 5500 𝑀𝑝𝑎


= 5500 𝑀𝑝𝑎 × 0.000213 = 1.17 𝑀𝑝𝑎

In figure 4.3 maximum stress of the concrete was reached approximately from 500 mm to 1100

mm of the prism. According to the experimental results, the next crack will be inserted at 800 mm.


between two cracks

Step 3

Crack spacing = 800 mm

Load = 29 KN

Assume minimum tensile strength of concrete is 1.9 MPa.




Force in the steel = 𝑃𝑠 = 𝑆11𝑆𝑡𝑒𝑒𝑙 × 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑠𝑡𝑒𝑒𝑙 = 84.5 𝑀𝑃𝑎 × 200𝑚𝑚2 = 16900 𝑁


𝑃𝑐 = 𝑃𝑇 − 𝑃𝑆 = 29000 𝑁 − 16900 𝑁 = 12100 𝑁

45


𝜀𝑐𝑜𝑚𝑝 3 = 0.0004


𝐸𝑒2=



𝑃𝐶𝐴𝐶


12100𝑁 14400𝑚𝑚2 − 200𝑚𝑚2

0.0004

= 2130 𝑀𝑝𝑎


= 2130 𝑀𝑝𝑎 × 0.0004 = 0.85 𝑀𝑃𝑎

In figure 4.4 maximum stress of the concrete was reached approximately from 100 mm to 600

mm and 900mm to 1500mmof the prism. According to the experimental results, the next series of

cracks will be inserted at 220 mm, 550mm, 1100mm and 1400mm.

46


between two cracks

Step 4

Cracks are modeled at all same locations with experiment T16B1 (Beeby and Scott 2002).

Load = 44 KN




Force in the steel = 𝑃𝑠 = 𝑆11𝑆𝑡𝑒𝑒𝑙 × 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑠𝑡𝑒𝑒𝑙 = 183.6 𝑀𝑝𝑎 × 200𝑚𝑚2 = 36720 𝑁



𝑃𝑐 = 𝑃𝑇 − 𝑃𝑆 = 44000 𝑁 − 36720 𝑁 = 7280 𝑁

47


𝜀𝑐𝑜𝑚𝑝 4 = 0.000881

𝐸𝑒4=



𝑃𝐶𝐴𝐶


7280𝑁(14400𝑚𝑚2 − 200𝑚𝑚2)

0.000881

= 580 𝑀𝑝𝑎


= 580 𝑀𝑝𝑎 × 0.000881 = 0.51 𝑀𝑝𝑎

Figure 4.6 shows the 𝑓𝑡 𝑣𝑠 𝜀𝑐𝑜𝑚𝑝 diagram using the result from step1-4.

Fig. 4.6 𝑓𝑡 𝑣𝑠 𝜀𝑐𝑜𝑚𝑝 for Beeby and Scott specimen

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.001

ft (mpa)

Strain

48

Figure 4.7 shows the load vs. deflection curve of Beeby and Scott specimen.

Fig. 4.7 Load vs. Deflection for Beeby and Scott specimen

49

4.2 Strain distribution along the steel bar

Two sets of analysis were performed. Initially, an arbitrary model was analyzed using a

constant element stiffness, K = 15000 lb/in. Comparing with experimental results (Fig. 4.8), the

strain distribution pattern is quite similar except the strain in the steel with no crack developed

(Fig. 4.9). In the experimental model, the strain in the mid-section is close to uniform which means

the stress level is uniform. The real experiment, the bond link stiffness K varies from node to node.

K at the middle tends to be smaller than the ones on the sides. Therefore, most of the force transfer

occurs at the sides closer to applied force but not in mid-section. In analytical model, a linear

relationship was assumed for bond stress and slip distance, same K values has been used at every

bond link element. A relatively smaller K allows more force transfer in mid-section.

To compare with the results done with constant K, figure 4.10 shows Beeby and Scott’s prism

with variable bond link stiffness used in the model. The strain diagram looks very close to the

experimental result. A more accurate result has been achieved by using a more realistic bond slip

model.

50

Fig. 4.8 Strain distribution along the reinforcement at four stages during loading (Beeby and Scott,

2002)

51

Fig. 4.9 Strain distribution along the reinforcement at four stages during loading (Constant K)

0

100

200

300

400

500

600

700

800

900

0 2 4 6 8 10 12 14 16 18

distance (in)

rein

forc

emen

t S

trai

n x

10

⁶

3689 lb

3046 lb

1550 lb

2325 lb

52

Fig. 4.10 Strain distribution comparison of Analytical and experimental results (Variable K)

53

Chapter 5

APPLICATION OF THE MODEL

5.1 Introduction

In this chapter, comparison of the analytical load vs. Strain plot with two different

reinforcement ratio and CEB method are presented.

5.2 Comparison with CEB Method

CEB method uses a tension stiffening factor 𝛽2 to reduce the overall stiffness EA and β is the

ratio of cracking load 𝑁𝑐𝑟 and applied load N (Equation 5.1). Figure 5.1 shows a tensile load vs.

strain diagram of finite element analytical results, predicted results using CEB method and steel

bar only for Beeby and Scott specimen. (𝐸𝐴)𝑢𝑐 is the uncracked member stiffness. (𝐸𝐴)𝑒 is the

effective member stiffness at different stages. (𝐸𝐴)𝑐𝑟 is the remaining member stiffness when

concrete is fully cracked. It is equivalent to the stiffness of the steel bar only.

𝐸𝐴𝑒 = 𝐸𝐴𝑢𝑐 (𝑁𝑐𝑟

𝑁)2 (Equation 5.1)

The area between the analytical RC curve or CEB curve and steel bar only is the total

contribution of concrete to the stiffness.

The diagram shows finite element approach is more conservative than CEB approach. That is

due to the algorithm of analytical approach. Cracks take place one series at one time instead of

propagating steadily. Strain jump occurs when a new crack series takes place without any load

increase. The major different occurs when the first series of cracks occurs. A significant stiffness

loss due to first series of cracks occurs while the CEB method shows a smooth transition. With

increasing load beyond the first series of cracking the two curves are seem to be approximately

parallel.

54

Fig. 5.1 Tensile load vs. strain diagram

55

5.3 Effect of varying reinforcement ratio

The number 5 steel reinforcing bar was replaced by a number 3 bar with the same Beeby and

Scott concrete prism.

In comparison, concrete cracks in a faster rate for the member with larger reinforcement ratio.

Larger steel bar has a large contact area with concrete and larger ribs. The bond force is relatively

higher than the member with smaller reinforcing bars. The larger bond force causes concrete to

crack faster and more completely. Therefore, concrete block should contribute more to the whole

member due to smaller damage. As mentioned in previous sections, the area between the curve and

the straight line represents the total contribution of pure concrete. In figure 5.2, the area is larger

for the member with smaller reinforcement ratio.

There are also some disadvantages for the member with smaller reinforcement ratio. First is

lower tension capacity. They cannot carry as much tension force as the one with larger

reinforcement. Second is lower bond force. If the reinforcement ratio is too low, pull-out failure

can occur before crack is initiated. A total bond link failure is same as total concrete failure, which

means no contribution from concrete.

56

Fig. 5.2 Tensile load vs. strain diagram for models with different reinforcement ratio

57

Chapter 6

SUMMARY CONCLUSIONS AND RECOMMANDATIONS

6.1 Summary

This research investigated the development of a tension stiffening model based on bond stress-

slip relationships. Based on Harajli’s(2002) bond stress-slip relationship, single spring bond link

elements with variable stiffness were implemented to provide the basis for an overall tension

stiffening model based on degrading concrete stiffness under sequential cracking. Numerical

results provided by ABAQUS, a finite element program, were compared with experimental results

to establish the validity of the simplifications and assumptions made for the two dimensional

axially loaded prism model.

The method of constructing an analytical tension stiffening model for an axially loaded

reinforced concrete prism with degrading stiffness and sequential cracking algorithm has been

described. Models with constant and variable bond link stiffness have been presented for

illustration. A tension stiffening saw-tooth stress-strain diagram has been generated in the form

proposed by Scanlon and Murray (1974). Development of bond link element is the key factor

which has significant influence on the stress distribution of the entire model. Comparisons with

experimental results show that using variable bond link stiffness improves the accuracy of

modeling.

6.2 Conclusions

The finite element approach developed in this thesis can be used to construct a tension

stiffening model for axially loaded prism. Both constant stiffness and variable bond link stiffness

models were applied. The variable bond link stiffness model was found to produce better

correlation with available experimental results. The proposed methodology provides a rational

approach for the development of saw-tooth type tension stiffening models.

58

6.3 Recommendations

The algorithm can be improved by extending the modeling approach to three dimensional

stress states that can be used to evaluate the effects of additional factors such as bar spacing and

concrete cover, and other reinforcement materials such as FRP (fiber reinforced polymer

materials). Further refinement is also possible in the modeling of bond between concrete and steel.

The probabilistic nature of the problem can be addressed by considering spatial variation of tensile

strength and modulus of elasticity. The approach could also be extended to flexural members.

59

References:

Beeby, A.W., Scott,R. H. (2002). “Tension stiffening of concrete,behaviour of tension zones in

reinforced concrete including time dependent effects.” Technical Report 59, Supplementary

Information,The Concrete Society, Camberley

D’Ambrisi, A. and Filippou, F. C. (1997). “Correction studies on an RC frame shaking-table

specimen.” Earthquake Eng. Struct. Dyn., V. 26, 1021-1024.

De Groot, A. K., Kusters, G. M. A., and Monnier, T. (1981). “Numerical modeling of bond-slip

behavior.” Heron, Concrete mechanics, V. 26(1B), 1-90.

Gilbert, R., and Waner, R. (1978). “Tension stiffening in reinforced concrete slabs.”ASCE J.

Struct. Div., V. 104(12), 1885-1990.

Harajli, M. H., Hout, M. and Jalkh, W. (1995). “Local bond stress-slip behavior of reinforcing bars

embedded in plain and fiber concrete” ACI Materials Journal, V. 92, 343-353.

Harajli, M. H., Hamad, B. and Kram, K. (2002). “Bond-slip response of reinforcing bars embedded

in plain and fiber concrete” J. Mat. in Civ. Engrg., V. 14(6), 503-511.

Keuser, M., and Mehlhorn, G. (1987). “Finite element models for bond problems.” J. Struct. Eng.,

V.113(10), 2160-2173.

Lin, C. S., and Scordelis, A. C. (1975). “Nonlinear analysis of RC shells of general form.” J.Struct.

Div., V. 101(3), 523-538.

Lowes, L. N., Moehle, J. P. and Govindjee, S. (2004) “ Concrete-Steel bond model for use in finite

element modeling of reinforced concrete structures” ACI Structure Journal, V. 101, No.4. 501-

511.

Nayal, R., and Rasheed, H. A. (2006). “Tension stiffening model for concrete beams reinforced

with steel and FRP bars.” J. Mat. in Civ. Engrg., V. 19(11) , 1014-1015

Ngo, D. and Scordelis, A.C. (1967) “Finite element analysis of reinforced concrete beams”, ACI

Journal, V. 64,152-163

Nilson, A. H. (1971) “Internal measurement of bond slip.” J. AM. Concr. Inst., V. 69(7), 439-441.

Scanlon, A., and Murray, D. W. (1974). “Time dependent deflections of reinforced concrete slab

deflections.” ASCE J. Struct. Div., V. 100(9), 1911-1924.

Vebo, A., and Ghali, A. (1977). “Moment curvature relation of reinforced concrete slabs.”ASCE J.

Struct. Div., V. 103(3), 515-531.

60

Yankelevsky, D. Z. (1985). “New finite element for bond-slip analysis.” ASCE J. Struct. Eng., V.

111(7), 1533-1542.

Wang, X., Liu, L. (2003). “A strain-softening model for steel–concrete bond” Cement and

Concrete Research 33, 1669-1673.

61

Appendix

STIFFNESS COEFFICIENT WITH CORRESPONDING SLIP DISTANCE FOR BEEBY AND

SCOTT SPECIMEN

Bond-Slip Chart for Model#2 with #5 bar

Slip(mm) Bond Stress(MPa) Spring Stiffness(N/mm)

0.001 0.49 12145.93

1 0.01 1.22 3050.92

2 0.02 1.61 2012.86

3 0.03 1.89 1578.19

4 0.04 2.12 1327.99

5 0.05 2.32 1161.58

6 0.06 2.50 1041.21

7 0.07 2.66 949.23

8 0.08 2.80 876.15

9 0.09 2.94 816.37

10 0.1 3.07 766.36

11 0.11 3.18 723.76

12 0.12 3.30 686.95

13 0.13 3.40 654.73

14 0.14 3.51 626.26

15 0.15 3.61 600.86

16 0.16 3.70 578.04

17 0.17 3.79 557.39

18 0.18 3.88 538.60

19 0.19 3.96 521.41

20 0.2 4.04 505.61

21 0.21 4.12 491.02

22 0.22 4.20 477.50

23 0.23 4.28 464.94

24 0.24 4.35 453.21

25 0.25 4.42 442.25

26 0.26 4.49 431.96

27 0.27 4.56 422.29

28 0.28 4.63 413.18

29 0.29 4.69 404.57

30 0.3 4.76 396.42

62

Table A - Continued


31 0.31 4.82 388.70

32 0.32 4.88 381.36

33 0.33 4.94 374.39

34 0.34 5.00 367.74

35 0.35 5.06 361.40

36 0.36 5.12 355.34

37 0.37 5.17 349.55

38 0.38 5.23 344.00

39 0.39 5.28 338.68

40 0.4 5.34 333.58

41 0.41 5.39 328.67

42 0.42 5.44 323.95

43 0.43 5.49 319.41

44 0.44 5.54 315.04

45 0.45 5.59 310.82

46 0.46 5.64 306.74

47 0.47 5.69 302.81

48 0.48 5.74 299.01

49 0.49 5.79 295.33

50 0.5 5.84 291.78

51 0.51 5.88 288.33

52 0.52 5.93 284.99

53 0.53 5.97 281.75

54 0.54 6.02 278.61

55 0.55 6.06 275.56

56 0.56 6.11 272.60

57 0.57 6.15 269.72

58 0.58 6.19 266.92

59 0.59 6.23 264.19

60 0.6 6.28 261.54

61 0.61 6.32 258.96

62 0.62 6.36 256.45

63 0.63 6.40 254.00

64 0.64 6.44 251.61

65 0.65 6.48 249.28

63

Table A - Continued


66 0.66 6.52 247.00

67 0.67 6.56 244.79

68 0.68 6.60 242.62

69 0.69 6.64 240.50

70 0.7 6.68 238.44

71 0.71 6.71 236.42

72 0.72 6.75 234.44

73 0.73 6.79 232.51

74 0.74 6.83 230.62

75 0.75 6.86 228.77

76 0.76 6.90 226.96

77 0.77 6.94 225.18

78 0.78 6.97 223.45

79 0.79 7.01 221.75

80 0.8 7.04 220.08

81 0.81 7.08 218.44

82 0.82 7.11 216.84

83 0.83 7.15 215.27

84 0.84 7.18 213.73

85 0.85 7.22 212.22

86 0.86 7.25 210.73

87 0.87 7.28 209.28

88 0.88 7.32 207.85

89 0.89 7.35 206.44

90 0.9 7.38 205.06

91 0.91 7.41 203.71

92 0.92 7.45 202.38

93 0.93 7.48 201.07

94 0.94 7.51 199.78

95 0.95 7.54 198.52

96 0.96 7.58 197.27

97 0.97 7.61 196.05

98 0.98 7.64 194.85

99 0.99 7.67 193.66

100 1 7.70 192.50

64

Table A - Continued


101 1.01 7.73 191.35

102 1.02 7.76 190.23

103 1.03 7.79 189.12

104 1.04 7.82 188.02

105 1.05 7.85 186.95

106 1.06 7.88 185.89

107 1.07 7.91 184.84

108 1.08 7.94 183.81

109 1.09 7.97 182.80

110 1.1 8.00 181.80

111 1.11 8.03 180.82

112 1.12 8.06 179.85

113 1.13 8.09 178.89

114 1.14 8.11 177.95

115 1.15 8.14 177.02

116 1.16 8.17 176.10

117 1.17 8.20 175.19

118 1.18 8.23 174.30

119 1.19 8.25 173.42

120 1.2 8.28 172.55

121 1.21 8.31 171.70

122 1.22 8.34 170.85

123 1.23 8.36 170.02

124 1.24 8.39 169.19

125 1.25 8.42 168.38

126 1.26 8.45 167.57

127 1.27 8.47 166.78

128 1.28 8.50 166.00

129 1.29 8.53 165.23

130 1.3 8.55 164.46

131 1.31 8.58 163.71

132 1.32 8.60 162.96

133 1.33 8.63 162.23

134 1.34 8.66 161.50

135 1.35 8.68 160.78

65

Table A - Continued


101 1.01 7.73 191.35

102 1.02 7.76 190.23

103 1.03 7.79 189.12

104 1.04 7.82 188.02

105 1.05 7.85 186.95

106 1.06 7.88 185.89

107 1.07 7.91 184.84

108 1.08 7.94 183.81

109 1.09 7.97 182.80

110 1.1 8.00 181.80

111 1.11 8.03 180.82

112 1.12 8.06 179.85

113 1.13 8.09 178.89

114 1.14 8.11 177.95

115 1.15 8.14 177.02

116 1.16 8.17 176.10

117 1.17 8.20 175.19

118 1.18 8.23 174.30

119 1.19 8.25 173.42

120 1.2 8.28 172.55

121 1.21 8.31 171.70

122 1.22 8.34 170.85

123 1.23 8.36 170.02

124 1.24 8.39 169.19

125 1.25 8.42 168.38

126 1.26 8.45 167.57

127 1.27 8.47 166.78

128 1.28 8.50 166.00

129 1.29 8.53 165.23

130 1.3 8.55 164.46

131 1.31 8.58 163.71

132 1.32 8.60 162.96

133 1.33 8.63 162.23

134 1.34 8.66 161.50

135 1.35 8.68 160.78

136 1.36 8.71 160.07

137 1.37 8.73 159.37

138 1.38 8.76 158.67

66

Table A - Continued


139 1.39 8.78 157.99

140 1.4 8.81 157.31

141 1.41 8.83 156.64

142 1.42 8.86 155.98

143 1.43 8.88 155.32

144 1.44 8.91 154.67

145 1.45 8.93 154.03

146 1.46 8.96 153.40

147 1.47 8.98 152.77

148 1.48 9.01 152.15

149 1.49 9.03 151.54

150 1.5 9.06 150.93

151 1.51 9.08 150.33

152 1.52 9.10 149.74

153 1.53 9.13 149.15

154 1.54 9.15 148.57

155 1.55 9.18 147.99

156 1.56 9.20 147.42

157 1.57 9.22 146.86

158 1.58 9.25 146.30

159 1.59 9.27 145.74

160 1.6 9.29 145.20

161 1.61 9.32 144.66

162 1.62 9.34 144.12

163 1.63 9.36 143.59

164 1.64 9.38 143.06

165 1.65 9.41 142.54

166 1.66 9.43 142.03

167 1.67 9.45 141.51

168 1.68 9.48 141.01

169 1.69 9.50 140.51

170 1.7 9.52 140.01

171 1.71 9.54 139.52

172 1.72 9.57 139.03

173 1.73 9.59 138.55

174 1.74 9.61 138.07

175 1.75 9.63 137.60

67

Table A - Continued


176 1.76 9.65 137.13

177 1.77 9.68 136.66

178 1.78 9.70 136.20

179 1.79 9.72 135.74

180 1.8 9.74 135.29

tension stiffening model for reinforced concrete …

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