45

CHAPTER 3

MODELLING OF GFRP REINFORCED DEEP BEAMS

USING “STRUT AND TIE” METHOD

3.1 INTRODUCTION

3.1.1 About Strut-and-Tie Method

The Strut-and-Tie Method (STM) is an analytical modelling

method has become a popular technique of designing due to its flexibility.

The idea of the strut-and-tie method originated from the truss analogy

method. The design basis of this method is a truss model which idealizes the

flow of force in a cracked concrete beam.

The Strut-and-Tie model has become one of the most useful design

methods for structures which are subject to shear critical load conditions.

They are also preferred to be used at disturbed regions in the concrete

structure where the stress variation across the section is non-linear.

The Strut-and-Tie models are formulated as a combination of strut

elements and tie elements to form an idealized truss member, capable of

resisting the complex flow of stresses in a structural member. The struts are

designed to resist the compressive stress, while the ties are designed as

members to resist the tensile stress. These idealized members offer to resist

the applied load by axial forces. The junction at which Strut-and-Tie members

meet is referred to as a nodal point. At each nodal point, a third element,

46

named as the nodal element is assumed to be present along with struts and

ties.

Strut-and-Tie Modelling of steel reinforced concrete deep beams

has been extensively carried out during the last few decades. This method of

design has been approved and adopted as a design method in the code of

practice in many countries.

3.1.2 D and B Regions

Ordinary reinforced concrete beams, which are designed to resist

shear and flexural forces, are modelled based on the assumption that the

variation of strain across its depth is linear. This follows the definition of a

beam under ‘Bernoulli hypothesis’ or ‘beam theory’, under the assumption

that plane sections remain plane after bending. The internal stresses are

ascertained from the forces in the B-region or the Bernoulli region either

before or after the concrete cracks. The B and D regions in a concrete beam

structure are shown in Figure 3.1.

On the other hand, in the D-region or Discontinuity region, the

variation of the strain is nonlinear and the assumptions used in the beam

theory are no longer valid. The reason for nonlinearity in the strain is

associated with either the changes made in loading or due to any change in the

geometry of the section under consideration and may be also due to the severe

load close to the point of loading or support.

The region close to the loading and support points for a distance

equal to the depth of the section is considered as D-region. The remaining

portion of the member is the linear stress distribution region also known as B

region. For a member to have a B and D region, the depth of the structure

should be comparable to the span of the structure.

47

Figure 3.1 B and D region in a concrete beam structure

3.1.3 Need for modelling using STM

The STM method of designing is one of the simple methods

presently used by many researchers due to its flexible nature. The results

obtained by using the STM modelling are generally lower bound and

conservative. This has been established by many early researchers. Research

work on STM method of design for deep beams has been carried out

extensively and some design codes have also been published in the recent

years. It has been shown by earlier researchers that, for deep beam models

with a ‘shear span to effective depth’ ratio (a/d) of less than 2.5, when

designed by using STM modelling were found to give reasonably good

results. Since this method was found to give better results for deep beams

with a low a/d ratio, it was decided to make use of Strut-and-Tie method for

designing and validating the experimental results of this study.

3.2 MODELLING OF GFRP REINFORCED DEEP BEAMS

USING STM

Till date there is no design code available for modelling FRP

reinforced concrete structures using STM. Hence, in this study, the deep

beams reinforced with GFRP web reinforcement were modelled using the

code ACI-318-05 meant for steel RC structures. Finally, a comparative study

between the experimental and STM results was done to evaluate the code’s

compatibility with FRP reinforcement. All the beams tested in this work were

48

modelled individually as the web reinforcements differed from one another.

The entire beam is considered to be a disturbed region or ‘D-region’ which

has a shear span to depth ratio of 0.72.

Due to a shorter shear span and a greater depth of the beams in this

study, there were constraints in modelling them. Since the strut angle was

restricted to be between 25o

and 65o as prescribed by the ACI 318-05 code,

the beams were modelled to have the simplest combination of struts and ties

that can be adopted for a simply supported beam. Each beam was modelled

with a combination of two struts and two ties. This was advantageous from

the point of view that the simplest combination of struts and ties was expected

to give the best result.

The elements of the modelled beams in Series-I were arranged in

such a manner that when were connected, they finally form a trapezium

shaped Strut-and-Tie arrangement as shown in Figure 3.2. The deep beam

model was designed to have only four elements i.e. three struts and one tie

element. The tie member placed at bottom of the beam represents the bottom

main reinforcement which is subjected to tension. The remaining three

elements were modelled as strut elements. At the junction, where the elements

meet, a node is provided. This node is used to connect all the elements

meeting at a point in the model to smoothly transfer forces and also to

maintain compatibility.

The size of the bearing plates plays a crucial role in deciding the

size of node along the loading and bearing faces. The top and bottom of each

beam tested, having loading and support points respectively, was provided

with 150mm x 160mm x10 mm size M.S. steel bearing plates. The size of the

bearing plate was checked to confirm its strength against crushing by

calculating the crushing strength at the individual bearing area of each node.

49

The compressive strength of concrete played an important role in checking

the bearing strength in each beam.

Figure 3.2 Strut-and-Tie Model (SERIES-I)

The nodes were shaped in the form of a triangle and their

dimensions varied depending on the angle of the strut or tie joining the node

on the faces of the triangle. The nodes were also positioned based on the

loading and support points. Based on this and also on the magnitude of the

force acting on the face of a strut or a tie, the dimensions of individual nodes

and the angles of inclination of them were finalised. Various classifications of

commonly used nodes in STM modelling are illustrated in Figure 3.3.

Figure 3.3 Classification of node

50

After finalizing the dimensions of the nodes and STM elements, the

nodes were classified as C-C-C, C-C-T, C-T-T and T-T-T node based on the

type of load that they encounter at each of their faces. Only two types of

nodes were used for modelling the GFRP deep beam specimens tested in this

work. The top nodal points namely ‘B’ node and ‘C’ node were designed as

C-C-C type of node, while the bottom nodes namely ‘A’ and ‘D’ node were

designed as C-C-T node.

By equating the force of tension to that of compression taking place

at the bottom and top main reinforcement of the beam, the widths of both the

strut and the tie members were calculated and subsequently the forces acting

on the faces of both the bottom and the top nodes were obtained. Once the

forces in the elements and their dimensions were obtained, the model was

checked for its capacity. The details of the design calculations of GFRP

reinforced deep beams with web reinforcement using STM has been shown in

Appendix 3.

The elements of the modelled beam for Series-II were arranged in

such that they formed a triangle as shown in Figure 3.4. Since the beams in

Series-II were tested under three point loading, the arrangement of elements

was made accordingly.

Figure 3.4 Strut-and-Tie Model (SERIES-II)

51

3.3 STM RESULTS AND ANALYSIS OF MODELLED BEAMS

All the thirteen beams were modelled based on the test results

obtained by experimentation. The details of the experimental results are

shown in Table 5.1 and Table 5.2 of chapter 5. Each of the modelled deep

beams was subjected to the ultimate load obtained from experimental results

to study and evaluate its capacity. The forces in the strut and tie members of

the modelled beams were calculated using the design equation of ACI 318 -05

Code of Practice for design. A typical model designed for beam GFRDB-1 is

shown in Figure 3.5.

Figure 3.5 STM Model of Beam GFRDB-1with internal forces

The internal forces in the strut and tie members were calculated

based on the individual Strut-and-Tie models and the results are as shown in

Figure 3.6. Since modelled beams have both struts AB and CD which were

placed symmetrically with respect to the centre line of the beam, they were

assumed to carry equal loads. Also, the top and bottom chord members BC

and AD respectively were assumed to carry equal forces while equating the

force of tension to the force of compression.

52

Figure 3.6 Internal Forces in the Strut and the Tie Members

Due to a smaller shear span, the force in the strut was found to be

relatively higher than the force in the tie members. After determining the

forces in the members, the stress on each face of the node was calculated.

Since each face of the node was assumed to have equal stress, similar to a

hydrostatic condition, the forces on the nodes were calculated so as to satisfy

this condition. For compatibility of the model, the face of the node in contact

with the strut or the tie should also be of the same width as the strut or the tie.

The dimensions of the strut and the tie members were altered to match the

new dimensions of the node’s side face according to the hydrostatic condition.

The Figure 3.7 shows a comparison of required and designed capacities of the

modelled struts. The applied load in each strut was calculated using simple

geometric relations and was compared with the designed capacity of the strut.

It was observed that the design capacity was higher than the required capacity

in all beams except in beams GFRDB-8 and GFRDB-9. This can be distinctly

seen in Figure 3.7.The variation in the ultimate capacity is shown in Figure

3.8. A detailed design calculation showing the difference in strut capacity in

53

the beam GFRDB-9 has been illustrated as an example in Appendix-3 of this

thesis.

Figure 3.7 Comparison of strut’s required and designed capacity

Figure 3.8 Comparison of experimental and STM ultimate load

54

Based on this study it was found that the factor s, accounting for

the effect of cracking and confining reinforcement on the effective

compressive strength of the concrete in a strut as adopted in the ACI 318-05

code in calculating the designed capacity, needs to be modified in order to

account for the high confinement by GFRP web reinforcement. The value of

s adopted for the steel reinforced members is 0.75. If this value is increased,

then the ACI 318-05 design procedure may be adopted for GFRP reinforced

deep beam members to check the struts capacity. This increase in the value of

s factor can be justified due to an increase in the confinement of concrete in

the strut region by high GFRP web reinforcement. It should be noted this

modification was needed only to check the design capacity of the beam. From

the comparative chart shown in Figure 3.8, it can be concluded that STM

results were greater than the experimental results. This higher design capacity

in STM models can be attributed to the reason that the modelling of GFRP

deep beams was carried out using the equations of ACI 318-05 code which is

intended for the design of steel reinforced concrete structures.

Figure 3.9 Width of Strut used for calculating the capacity

55

The calculated values of the top and bottom widths of the strut after resolving

the forces as per hydrostatic condition are as shown in Figure 3.9.

The values of the various design coefficients adopted and used for

calculation has been tabulated in Table 3.1. The dimensional details related to

the modelled struts and their inclinations have also been shown in the same

table. It can be observed that the widths of the strut and the tie in each

modelled beam increases as the amount of web reinforcement increases. This

increase in the sizes of the strut and the ties in turn increases its load carrying

capacity and vice-versa. As the width of the strut and the tie dimension

increases, the size of the nodes also increases and as a result the lever arm

distance (jd) reduces.

The variation of lever arm distance for the modelled beams in

Series-I is shown in Figure 3.10. Due to the decrease in the lever arm

distance, the angle of inclination of the strut was reduced by nearly 8%.

Figure 3.10 Variation of lever arm distance (jd)

56

The maximum deviation of the results for the modelled beams from

the experimental results was found to be 12% in case of Series-I beams and

19% for in Series-II beams as could be deduced from Tables 3.1 and 3.2

respectively.

Table 3.1 The Design coefficients and the dimensional details of STM

models (SERIES-I)

Sl.

No.Beam n (at

Node B)n (at

Node A)

Strength

reduction

Factor

S

( ForChecking

the Strut

Capacity )

Ws Wto P STM /

PEXP

1 GFRDB-1 1. 0 0. 8 0.75 0.60 30.5 38.1 54O

46'

1.09

2 GFRDB-2 1. 0 0. 8 0.75 0.75 44 55 53O

55'

1.12

3 GFRDB-3 1. 0 0. 8 0.75 0.75 40.7 50.9 54O

8'

1.114

4 GFRDB-4 1. 0 0. 8 0.75 0.75 35.5 44.5 54O

27'

1.106

5 GFRDB-5 1. 0 0. 8 0.75 0.75 56 70 53O

8'

1.12

6 GFRDB-6 1. 0 0. 8 0.75 0.75 50.8 63.5 53O

29'

1.121

7 GFRDB-7 1. 0 0. 8 0.75 0.75 70.4 88 52O

10 1.117

8 GFRDB-8 1. 0 0. 8 0.75 0.75 78 97.5 51O

38'

1.109

9 GFRDB-9 1. 0 0. 8 0.75 0.75 90 112.5 50O46

'1.09

Table 3.2 The Design coefficients and the dimensional details of STM

models (Series-II)

Sl.

No.Beam n (at

Node B)n (at

Node A)

Strength

reduction

Factor

s ( For

Checking

the Strut

Capacity )

Ws Wt

oP STM /

PEXP

1 GFRDB-1(a) 1.0 0.8 0.75 0.60 31 39 47O

43' 1.14

2 GFRDB-3(a) 1.0 0.8 0.75 0.75 44 55 46O

52' 1.17

3 GFRDB-5(a) 1.0 0.8 0.75 0.75 52 65 46O

20'

0.799

4 GFRDB-9(a) 1.0 0.8 0.75 0.75 95 119 44O

14' 1.193

57

In case of Series-II beams, the designed capacity of the strut in

beam GFRDB-9(a) was found to be lower than required as shown in Figure

3.11. However, the difference in strut capacity in GFRDB-9(a) was found to

be negligible compared to beam GFRDB-9.

Figure 3.11 Comparison of required and designed capacity of strut

(Series-II)

As can be seen in Figure 3.12, the ultimate load capacity of STM

modelled deep beams for most of the beams in Series-II were found to be

greater than the experimental results. This can be deduced by comparing the

values of the ultimate loads obtained from experimental tests and STM model

analysis.

58

Figure 3.12 Comparison of experimental and STM modelled Ultimate

capacity results (Series-II)

3.4 SUMMARY

The Strut-and-Tie method of modelling GFRP reinforced deep

beams which was developed using the AC1-318 -05 code was found to be

higher compared to the experimental values of the tested deep beams.

Although the STM results were found to be greater, the STM method of

modelling can be adopted for GFRP reinforced deep beams by suitable

modification to minimise the gap between the experimental and modelled

results.