C4301 Unit 3

1 REINFORCED CONCRETE STRUCTURAL DESIGN

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UNIT 3 DESIGN THEORY: LIMIT STATES AND BENDING

OBJECTIVES 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.

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INPUT 1

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

Figure 3.1 Concrete Stress-strain relationships Stress Strain 0.67f 0.0035 cu

γm

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

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.

Strain V A B C Stress isible 0.0035 0.02 Figure 3.2: The stress-Strain Curve cracks

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

γFor design purposes, the simplified curve BS 8110 is used as shown in Figure 0.67fcu 3.1. From the curve, it can be seen that, the maximum stress is equal to γm and the concrete is assumed yield (fail) at an ultimate strain equal to 0.0035.

3.3

Steel

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

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High Mild steel yield steel Strain Stress 22 ((f =250N/mm =460 2 fy y (N/mmN/mm ) ))

Figure 3.3: The stress-strain relationship of steel reinforcement

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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.

200 kN/mm Stress Tension Strain f 2 y

γm

Figure 3.4 BS 8110 Design Curve

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γγ

Compression

fy is the characteristic strength of steel which is similar to that, which is given by BS 8110 (Table 3.1). γm for steel is given as 1.15. Therefore, the design strength of steel reinforcement is, fy 1.15 = 0.87 fy

8 REINFORCED CONCRETE STRUCTURAL DESIGN

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ACTIVITY 3a

Fill in the blanks: 3.1 Figure

3.1,

BS

8110

shows

the

stress-strain

relationship

for

stress-strain

relationship

for

________________________________. 3.2 Figure

3.2,

BS

8110

shows

the

_________________________________. 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 __________________________________.

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FEEDBACK 3a

ANSWERS: 1.1.

Concrete

1.2.

Steel

1.3.

0.0035

1.4.

0.67fcu γm

1.5.

0.87fy

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INPUT 2

1.6. The Behaviour of Beam in Bending tension compression tension loadcrack

When load is applied on a reinforced concrete beam, it will bend as shown in Figure 3.5 below: -

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Figure 3.5: Bending in Beam

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.

In order to counter this tensile force, steel reinforcement is provided as shown in Figure 3.6: Steel reinforcement

Figure 3.6: Steel Reinforcement

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3.5 Stress And Strain Distribution 0.45f 0.87f Neutral dε bycu axis S=0.9x st cc

The stress and strain distribution for a beam of rectangular section is shown in Figure 3.7: -

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

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. The strain distribution shows that concrete reaches a maximum at εcc (in compression) and strain in steel is εst (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: -

εst

=

x so,

(d-x) x

εcc (d-x)

=

εcc εst

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d–1

=

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εcc εst

x d

=

1 + εcc

εst x

=

d 1 + εcc

εst

At the time when failure occurs at ultimate limit state, steel and concrete reach their maximum stress and strain, i.e.,

Concrete strain, εcc = 0.0035 Steel strain,

εst =

stress Modulus of elasticity

=

fy γm Es

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=

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fy 1.15 2.00 x 103

= 4.35 x 10-6fy To determine the value of x, during failure, say for high yield steel, fy = 460 N/mm2,

εst = 4.35 x 10-6 x 460 = 0.002 Therefore, x

=

d 1 + 0.002 0.0035

=

0.64d

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.

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iii)

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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.

ACTIVITY 3b

Now answer the following questions by filling in the blanks. 3.6 Concrete possesses considerable compressive strength but has very little ____________ strength.

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3.7 The

tensile

forces

resulting

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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. 3.10x is called the depth to the _______________________ axis. 3.11The depth of the simplified stress block is equal to ____________________. 3.12For high yield steel, x is equal to _______________, where d is the effective depth of the tension reinforcement.

FEEDBACK 3b

ANSWERS: 3.6 tensile 3.7 steel reinforcement. 3.8 tensile

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3.9 rectangular 3.10neutral 3.110.9x 3.120.64d

INPUT 3

3.6

Failure Modes

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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 underreinforced 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.

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.

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ACTIVITY 3c

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

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a

b

3.14Read 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.

Yes/No

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

Yes/No

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

Yes/No

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

Yes/No

FEEDBACK 3c

ANSWERS: 3.13Check your answers below:-

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a

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x = 0.64d

Under reinforced

x > 0.64d

Balance section

x < 0.64d

Over reinforced

b

3.14a)

No

b)

Yes

c)

No

d)

Yes

SELF-ASSESSMENT

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

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2.

3.

4.

B.

1.75 N/mm2

C.

175.0 N/mm2

D.

17.5 N/mm2

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The ultimate strain of concrete is equal to… A.

35.0

B.

0.35

C.

0.0035

D.

0.035

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

If depth to neutral axis, x = 250mm, what is the depth of simplified rectangular stress block?

5.

A.

225.0 mm

B.

22.5 mm

C.

2.25 mm

D.

0.225 mm

Given that Es = 200 x 103 N/mm2. What is the failure of steel strain if grade 460 is used?

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6.

A.

2.0 x 10-3

B.

0.2 x 10-3

C.

20.01 x 10-3

D.

200.1 x 10-3

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The modes of failure for beam subjected to bending are listed below EXCEPT …

7.

8.

A.

balance section

B.

unbalance section

C.

under-reinforced

D.

over-reinforced

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

The BS 8110 stress block is the simplification of the following stress block: A.

Triangular stress block

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9.

B.

Parabolic stress block

C.

Square stress block

D.

Circular stress block

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The design strength of grade 40 concrete is equal to 17.87 N/mm2. The value of γm used in calculating this value is equal to…

10.

A.

1.15

B.

1.25

C.

1.25

D.

1.4

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.

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FEEDBACK ON SELF-ASSESSMENT

ANSWERS: 1.

D

2.

C

3.

B

4.

A

5.

A

6.

B

7.

C

8.

B

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9.

C

10.

D

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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.

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END OF UNIT 3