Lecture 11 Concrete

CIVL2230Introduction to Structural

Concepts and Design

Lecture 16Bending of Concrete Sections

Part 1

Dr Elisha HarrisA/Prof. Peter Ansourian

Concrete Properties• Concrete design in Australia follows AS 3600;

• Concrete behaviour varies depending on whether it is in tension or compression;

• The strength of concrete is given in terms of the characteristic strength;

• Characteristic strength is defined in AS 3600 as “the value of material strength, as assessed by standard test, which is exceeded by 95% of the material”;

• Characteristic strengths are denoted by the apostrophe;

• Standard characteristic compressive strengths are 20, 25, 32, 40 50 and 65 MPa.

Concrete Properties• The characteristic flexural strength can be

calculated using Equation 6.1.1.2, AS 3600:

f’cf = 0.6 (f’c)0.5

• Concrete strength is determined by testing a standard concrete cylinder in compression (cylinder is 200 mm high with a 100 mm diameter)

Concrete Properties

Strain

Stress

Typical stress-strain curve for a concrete cylinder in compression

Concrete Properties• Concrete stress-strain curve shows no definite

yield point;

• Concrete does not have the large plastic deformation capacity of structural steel in the stress-strain curve, and so does not display the same ductile behaviour;

• Concrete has a brittle failure.

Pre-cracking Behaviour of Concrete• When a concrete beam is sagging, the top of

the beam is in compression and the bottom of the beam is in tension;

• If the tension in the bottom of the beam is less than the tensile capacity of the concrete, the concrete is able to carry the load;

• At relatively low strains (e.g. when cracking begins), the stress-strain relationship is approximately linear.

Pre-cracking Behaviour of Concrete

< cr

< cr

M < Mcr M < Mcr

f < fcf

Strain

Stress

At relatively low strains, the stress-strain relationship is

approximately linear

fcf is the actual flexural tensile strength where f’cf

is the characteristic flexural tensile strength.

Pre-cracking Behaviour of Concrete• If the stress distribution is linear, the relationship

between bending moment and stress is given by:

f = M y / I

where: f = bending stress;

M = bending moment;

y = distance from neutral axis to point;

I = second moment of area of the section.

• The moment which causes the concrete in the tensile region to crack is called the cracking moment, Mcr.


• If the concrete has cracked, it can no longer take tensile load;

• The portion of the concrete in the tensile region which has cracked therefore has no contribution to the strength of the section.

• The solution adopted is to place reinforcing steel in the tensile region of the section to take the tensile loads.


• Individual bars are named with a letter and a number with the letter giving the type of bar while the number gives the diameter of the bar;

• In order to protect the reinforcing steel, the bars need a certain amount of ‘cover’;

• The amount of cover is specified in the concrete code, AS 3600, and depends on factors including concrete strength, location of the building, whether the member is interior or exterior etc;

• Cover is typically 20 – 50 mm.

Concrete Beam Dimensions

b

Ddkd

Neutralaxis

Analysis of Concrete Sections• So long as the stresses are low (elastic response), we

‘transform’ the entire section to concrete (these calculations were performed in Mechanics);

• Total compressive force = Total tensile force;

• Before cracking occurs the steel is taking tension, Ts, and so is the concrete, Tc;

• After cracking, only the steel is taking tension, Ts;

• All of the compression is taken by the concrete, Cc;

• The moment carried by the section can be found by finding the moment of these forces about the neutral axis.

EXAMPLE 1• For the cross-section shown in Figure 6:• (a) Determine the characteristic flexural tensile

strength of the concrete;• (b) Draw the transformed cross-section for the section

just below the cracking moment;• (c) Calculate the cracking moment for the cross-

section;• (d) Draw the stress distribution just below the cracking

moment;• (e) Draw the stress distribution just above the cracking

moment.300 mm200 mm250 mm2 N16 bars• f’c = 32 MPa• Ec = 30 000 MPa• Es = 200 000 MPa

300 mm

200 mm

250 mm

2 N16 bars

f’c = 32 MPaEc = 30 000 MPaEs = 200 000 MPa

SOLVE PROBLEM

Ultimate Behaviour of Concrete• As the strains increase, the relationship between stress and strain

is no longer linear;

• The strain distribution in the section is still assumed to be linear;

• The stress distribution will be non-linear.

Strain

Stress




• The stress distribution will be non-linear;

• The beam cross-section is assumed to be at its ultimate load when the concrete extreme compression fibre reaches a strain of 0.003.

Strain

Stress

c = 0.003

> cr

M = Muo M = Muo s

f’c

kud

Ultimate Behaviour of Concrete• In order to make analysis simpler, the compressive stress in the

concrete is approximated by a rectangular stress block;

• The depth of the rectangular stress block is kud where can be calculated by:

= 0.85 – 0.007(f’c – 28)

within the limits: 0.65 0.85

f’c

kud

0.85f’ckud

fs

Ultimate Behaviour of Concrete• The total compressive force in the concrete can now be

calculated as:

Force = Stress Area

Cc = 0.85f’ckudb

• This force acts in the middle of the rectangular stress distribution at a distance of kud / 2 from the top of the section;

• This gives a lever arm from the NA of

Lever arm from NA = kud – kud / 2

0.85f’cf’c

kud kud

fs

Cc

kud -kud / 2

Ultimate Behaviour of Concrete• The total tensile force in the steel can now be calculated as:

Force = Stress Area

Ts = fsAst

• This force acts at the level of the steel which gives a lever arm from the NA of:

Lever arm from NA = d – kud

f’c

kud

0.85f’ckud

fs

Cc

kud -kud / 2

Ts

d - kud

Ultimate Behaviour of Concrete• The ultimate moment taken by the section can now be calculated

by:

Moment = Force Lever arm

Muo = Cc(kud - kud / 2) + Ts(d – kud)

• Substituting values for Cc and Ts gives:

Muo = 0.85f’ckudb(kud - kud / 2) + fsAst (d – kud)

f’c

kud

0.85f’ckud

fs

Cc

kud -kud / 2

Ts

d - kud

Ultimate Behaviour of Concrete• As Cc = Ts:

Muo = Cc(kud - kud / 2) + Cc(d – kud)

Muo = Cc(d - kud / 2)

Muo = Ts(d - kud / 2)

f’c

kud

0.85f’ckud

fs

Cc

kud -kud / 2

Ts

d - kud

Types of Concrete Section• The calculation procedure varies depending on when the steel

yields;

• In under-reinforced sections the steel has already yielded when the concrete reaches its ultimate state with strains of 0.003 at the extreme compressive fibre;

• In balanced sections the steel yields just as the concrete reaches its ultimate state with strains of 0.003 at the extreme compressive fibre;

• In over-reinforced sections the steel has not yielded when the concrete reaches its ultimate state with strains of 0.003 at the extreme compressive fibre;

Types of Concrete Section• When the reinft yields, its deformations increase dramatically

without the steel actually breaking and this is ductile behaviour;

• Under-reinforced sections will display significant ductility as the steel yields before the concrete reaches its ultimate state;

• This ductile behaviour gives warning of imminent failure;

• Concrete behaviour is more brittle;

• In over-reinforced sections, the steel has not yielded and the concrete reaches its ultimate state without any significant deformation;

• Without any significant deformation, failure can occur suddenly and consequences may be catastrophic;

• It is desirable for sections to be under-reinforced;

• We will only look at under-reinforced sections.

AS 3600 requirements• Clause 8.1.3 of AS 3600 states that the design

strength of a section with ku = 0.4 is ΦMuo ;• From Table 2.3, Φ = 0.8;• The limit of ku = 0.4 is adopted to ensure that the

section is under-reinforced and hence demonstrates ductile behaviour;

• We will not consider the case of ku > 0.4;• AS 3600 also requires that the ultimate moment be at

least 20% higher than the cracking moment;• AS 3600 also places requirements on the spacings of

bars such as ensuring that the bars are far enough apart to ensure that the concrete can be properly placed and compacted.

Pre-cracking Behaviour of Concrete• When a concrete beam is sagging, the top of the beam is in

compression and the bottom of the beam is in tension;

• If the tension in the bottom of the beam is less than the tensile capacity of the concrete, the concrete is able to carry the load;

• Concrete tensile capacity is low, so at this point the strains in the cross-section are relatively low;

• At relatively low strains, the stress-strain relationship is approximately linear and so the relationship between applied moment and stress is given by:

f = My / I

• This equation can be used after first transforming the entire section to concrete;

• The moment which causes the concrete in the tensile region to crack is called the cracking moment.

Concrete Beam Dimensions

b

Ddkd

Neutralaxis




• The stress distribution will be non-linear;

• The beam cross-section is assumed to be at its ultimate load when the concrete extreme compression fibre reaches a strain of 0.003.

Strain

Stress

c = 0.003

> cr

M = Muo M = Muo s

f’c

kud

Ultimate Behaviour of Concrete• In order to make analysis simpler, the compressive stress in the

concrete is approximated by a rectangular stress block;

• The depth of the rectangular stress block is kud where can be calculated by:

= 0.85 – 0.007(f’c – 28)

within the limits: 0.65 0.85

f’c

kud

0.85f’ckud

fs

Types of Concrete Section• The calculation procedure varies depending on when the steel

yields;

• In under-reinforced sections the steel has already yielded when the concrete reaches its ultimate state with strains of 0.003 at the extreme compressive fibre;

• In balanced sections the steel yields just as the concrete reaches its ultimate state with strains of 0.003 at the extreme compressive fibre;

• In over-reinforced sections the steel has not yielded when the concrete reaches its ultimate state with strains of 0.003 at the extreme compressive fibre;

AS 3600 requirements• These basic AS 3600 requirements which we will consider are:

M* ≤ Muo

Muo ≥ 1.2Mcr

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Lecture 11 Concrete