c4301 Unit 3

REINFORCED CONCRETE STRUCTURAL DESIGN C4301/UNIT3/

UNIT 3

DESIGN THEORY: LIMIT STATES AND BENDING

GENERAL OBJECTIVE

To understand the reinforced concrete design theory in Limit States and Bending

At the end of this unit, you will be able to: -

1. calculate the design strength for concrete.

2. calculate the design strength for steel reinforcement.

3. state the 3 modes of failure.

4. differentiate among the 3 modes of failure.

5. identify the behaviour of beams subjected to bending.

6. use the BS 8110 stress block.

7. calculate the depth to the neutral axis.

OBJECTIVES

SPECIFIC OBJECTIVES

1

Stress

0.67fcu m

0.0035Strain

Figure 3.1 Concrete Stress-strain relationships


3.1 Introduction

When load in imposed on a structural element, deformation occurs due to the

induced stress and strain in the element. It is of paramount importance for us to

understand the stress-strain relationship in order to analyze and design reinforced

concrete. Reinforced concrete is a composite material of concrete and steel.

Therefore we need to know the stress-strain relationship of both materials.

3.2 Concrete

The stress-strain relationship for concrete is shown in Figure 3.1 below: -

INPUT 1

2

C

B

A

Visible cracks

0.02 0.0035Strain

Stress

Figure 3.2: The stress-Strain Curve


The actual stress-strain curve depends on the grade of concrete that is used. For

normal concrete mixes, it can be concluded that:

i) The stress-strain curve can be assumed to be a straight line up to about

50% of the maximum stress.

ii) Maximum stress is achieved at about 0.02 strains.

iii) Cracks and the disintegration of concrete are visible when the strain is at a

value of 0.0035.

The actual stress-strain curve is shown in Figure 3.2 : -

For design purposes, the simplified curve BS 8110 is used as shown in Figure 3.1.

From the curve, it can be seen that, the maximum stress is equal to

and the concrete is assumed yield (fail) at an ultimate strain equal to 0.0035.

0.67fcu

m

3

Stress(N/mm2)

High yield steel(fy=460 N/mm2)

Mild steel(fy=250N/mm2)

Strain


3.3 Steel

The stress-strain relationship of steel reinforcement is shown in Figure 3.3 below:

The figure shows a typical stress-strain curve for steel reinforcement. This curve

can be used for both compression and tension conditions. For design purposes,

BS8110 is used as a simplified curve. Please refer to Figure 3.4.

Figure 3.3: The stress-strain relationship of steel reinforcement

4

200 kN/mm2

Strain

Tension

Stress

fym

fym

Figure 3.4 BS 8110 Design Curve


fy is the characteristic strength of steel which is similar to that, which is given by

BS 8110 (Table 3.1). m for steel is given as 1.15. Therefore, the design strength

of steel reinforcement is,

= 0.87 fy

Compression

fy

1.15

5


Fill in the blanks:

3.1 Figure 3.1, BS 8110 shows the stress-strain relationship for

________________________________.

3.2 Figure 3.2, BS 8110 shows the stress-strain relationship for

_________________________________.

3.3 The ultimate strain of concrete is equal to

________________________________.

3.4 The maximum stress of concrete in design is equal

to________________________________.

3.5 The maximum stress of steel reinforcement in design is equal to

__________________________________.

ACTIVITY 3a

6


ANSWERS:

1.1. Concrete

1.2. Steel

1.3. 0.0035

1.4. 0.67f cu

m

1.5. 0.87fy

FEEDBACK 3a

7

tension crack

load

tension

compression


1.6. The Behaviour of Beam in Bending

When load is applied on a reinforced concrete beam, it will bend as shown in

Figure 3.5 below: -

The intensity and distribution of the bending is depicted in the bending moment

diagram which is covered in the Theory of Structures. Because of the bending

effect, one face of the beam will be shortened due to compression force and the

other face will be elongated due to the tension force. The tension face will crack

because as explained earlier, concrete is weak in tension.

Figure 3.5: Bending in Beam

INPUT 2

8

S=0.9x

0.45fcu

0.87fyst

Neutral axis

b

d

cc


In order to counter this tensile force, steel reinforcement is provided as shown in

Figure 3.6: -

3.5 Stress And Strain Distribution

The stress and strain distribution for a beam of rectangular section is shown in

Figure 3.7: -

Above the neutral axis of the beam, the section experiences compression stress

while the area below the neutral axis experiences tension stress. Steel

reinforcement is provided in the tensile stress region because as we know concrete

is very weak in tension, i.e. steel reinforcing the concrete.

Steel reinforcement

Figure 3.6: Steel Reinforcement

Figure 3.7: The stress and strain distribution for a beam of rectangular section

9


The strain distribution shows that concrete reaches a maximum at cc (in

compression) and strain in steel is st (in tension). At a depth x from the

compression face, the stress is zero and the axis passing this point is called neutral

axis. (x is known as the depth to the neutral axis)

The x-value varies depending on the load and moment applied to the beam. An

increase in load or moment will increase the value of x. The stress-strain

relationship with respect to x can be explained as follows: -

st = cc

x (d-x)

so, (d-x) = cc

x st

d – 1 = cc

x st

d = 1 + cc

st

x = d 1 + cc

st

10


At the time when failure occurs at ultimate limit state, steel and concrete reach

their maximum stress and strain, i.e.,

Concrete strain, cc = 0.0035

Steel strain, st = stress Modulus of elasticity

= fy

m

Es

= fy

1.15

2.00 x 103

= 4.35 x 10 -6 f y

To determine the value of x, during failure, say for high yield steel, fy = 460

N/mm2,

st = 4.35 x 10-6 x 460

= 0.002

Therefore, x = d

1 + 0.002

0.0035

= 0.64d

11


The stress distribution is divided into three (3) phases. They are as follows: -

i) Triangular stress distribution whereby stress is directly proportional to

strain. This type of distribution occurs when a small load is applied on the

beam.

ii) Parabolic rectangular stress distribution occurs when concrete reaches

the maximum stress or strength and the ultimate limit state is reached

when this happens.

iii) Rectangular stress distribution. This type of distribution is the

simplified maximum stress distribution. The parabolic shape is simplified

into a rectangular shape. BS 8110 uses this stress distribution for design

purposes. The depth of the block, s = 0.9x. Please refer to Figure 3.3 BS

8110 for a clarification on this.

The following assumptions are made when analyzing reinforced concrete

sections: -

i) Stress in concrete and steel reinforcement is obtained assuming that plane

sections remain plane after applying the load.

ii) The concrete tensile strength is ignored.

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Now answer the following questions by filling in the blanks.

3.6 Concrete possesses considerable compressive strength but has very little

____________ strength.

3.7 The tensile forces resulting from bending are resisted by

_____________________________ placed near to the outermost fibres in

tension.

3.8 It is assumed that concrete doesn’t have ________________ strength.

3.9 BS 8110 uses the simplified _______________________ stress block for

design purposes.

3.10 x is called the depth to the _______________________ axis.

3.11 The depth of the simplified stress block is equal to

____________________.

3.12 For high yield steel, x is equal to _______________, where d is the

effective depth of the tension reinforcement.

ACTIVITY 3b

13


ANSWERS: -

3.6 tensile

3.7 steel reinforcement.

3.8 tensile

3.9 rectangular

3.10 neutral

3.11 0.9x

3.12 0.64d

FEEDBACK 3b

14


3.6 Failure Modes

There are three failure modes that may occur in a reinforced concrete beam due to

failure in bending. They are given below: -

a) Under reinforced.

When the area of reinforcement provided is relatively smaller compared to

the area of concrete section, this is termed as under reinforced. Under this

condition, the steel yields before the concrete crushes in compression.

Failure occurs because steel fails in tension. The failure of an under-

reinforced beam is characterized by large steel strains, and hence presence

of extensive cracking of the concrete and by substantial deflection. The

depth to the neutral axis: x < 0.64d.

b) Balance section.

This is achieved when the area of steel reinforcement provided is about

equal to the area of concrete section. The concrete and the steel strain

reach their maximum value simultaneously. The depth to neutral axis of a

balanced section, x = 0.64d.

INPUT 3

15


c) Over reinforced.

The area of steel provided is relatively bigger than the concrete area. The

concrete strain will reach the ultimate value before the steel strain reaches

the yield value. The failure is characterized by a small deflection and by

the absence of extensive cracking in the tension zone. The depth to neutral

axis depth of a over reinforced, x > 0.64d.

16


3.13 Match the depths to the neutral axis with the corresponding modes of

failure.

3.14 Read the statements and circle Y if it is true and N if it is false

a) In an under-reinforced beam, the steel yields after the concrete crushes.

b) In over-reinforced beam, the concrete crushes before the steel yields.

c) In a balance-section, the steel yields before the concrete crushes.

d) The failure of over-reinforced beam occurs with little warning.

x = 0.64d

x > 0.64d

x < 0.64d

Under reinforced

Balance section

Over reinforced

Yes/No

Yes/No

Yes/No

Yes/No

ACTIVITY 3c

a

b

c

17


ANSWERS: -

3.13 Check your answers below:-

3.14 a) No

b) Yes

c) No

d) Yes

Under reinforced

Balance section

Over reinforced

x = 0.64d

x > 0.64d

x < 0.64d

FEEDBACK 3c

a

b

c

18


Award one mark for every correct answer: Total 10 marks.

1. What is the maximum stress in a concrete grade 30?

A. 0.175 N/mm2

B. 1.75 N/mm2

C. 175.0 N/mm2

D. 17.5 N/mm2

2. The ultimate strain of concrete is equal to…

A. 35.0

B. 0.35

C. 0.0035

D. 0.035

3. The design strength of high yield steel reinforcement is equal to …

A. 4002.0 N/mm2

B. 400.2 N/mm2

C. 4.02 N/mm2

D. 40.02 N/mm2

SELF-ASSESSMENT

19


4. If depth to neutral axis, x = 250mm, what is the depth of simplified

rectangular stress block?

A. 225.0 mm

B. 22.5 mm

C. 2.25 mm

D. 0.225 mm

5. Given that Es = 200 x 103 N/mm2. What is the failure of steel strain if

grade 460 is used?

A. 2.0 x 10-3

B. 0.2 x 10-3

C. 20.01 x 10-3

D. 200.1 x 10-3

6. The modes of failure for beam subjected to bending are listed below

EXCEPT …

A. balance section

B. unbalance section

C. under-reinforced

D. over-reinforced

20


7. The depth to neutral axis if steel and concrete yield at the same time is…

A. 3.52 mm

B. 35.2 mm

C. 352.0 mm

D. 3520.0 mm

8. The BS 8110 stress block is the simplification of the following stress

block:

A. Triangular stress block

B. Parabolic stress block

C. Square stress block

D. Circular stress block

9. The design strength of grade 40 concrete is equal to 17.87 N/mm2. The

value of m used in calculating this value is equal to…

A. 1.15

B. 1.25

C. 1.25

D. 1.4

21


10. Which of the following statement is characteristic of under-reinforced

section?

A. Small deflection and absence of extensive cracking in the tension

zone.

B. Crushing of the concrete when compression reaches maximum.

C. Yielding of steel reinforcement in the tension zone.

D. Extensive cracking of the concrete and substantial deflection.

22


ANSWERS:

1. D

2. C

3. B

4. A

5. A

6. B

7. C

8. B

9. C

10. D

FEEDBACK ON SELF-ASSESSMENT

23


You should score 80% or more to pass this unit!

Proceed to the next unit if you have score 80% or more. Otherwise, go through this unit or part of this unit and redo the self-assessment until you score 80% or more

Now, you can proceed to the next unit.

END OF UNIT 3

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c4301 Unit 3

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