Analysis of Flexural Members

Flexural members, such as beams and slabs subjected to transverse loads develop bending moment and shear force along their span.

The bending moment will induce a tensile and compressive stresses in the member. Concrete will carry the compression force since it has excellent compressive properties.

Reinforcing bars are placed in concrete to carry the tension

The moment caused by the load is resisted by the internal couple formed by the compression in concrete and Tension in steel. For equilibrium to be satisfied:

l

Steel Reinforcement As

w kN/m

Shear Force

Bending Moment

C

T

z d

As

b

h

Fx = 0 or C = T

M = 0 or M = T. z = C. z

Analysis

In the analysis of r.c section, the dimensions of the section, the steel reinforcement and the material strength are given and it is required to calculate the moment capacity of the r.c section.

Design

In design, the moment due to the applied load is given and it is required to calculate the dimension of the r.c section and steel reinforcement required to resist the applied moment safely.

When reinforcing bars are subjected to tension, they stretch. The concrete around the reinforcing bars is consequently subject to tension and stretches. When tension in excess of tensile strength of concrete is reached, transverse cracks may appear near the reinforcing bars.

It is well known that the concrete under compression posses a brittle mode of failure i.e. sudden failure without any warning.

While the steel reinforcement under tension shows a ductile failure mode i.e. shows large deformation and provide warning before failure.

When both materials are combined in a r.c section, the type of failure of the section will be dominated either by concrete brittle failure or steel ductile failure.

Since it is very important that the failure at r.c member is to be in a ductile way to provide sufficient warning to evacuate the people, the design of the section has to be carried out in such away to initiate the failure in steel before concrete.

Therefore the failure mechanism play important role in the analysis and design of the r.c section.

When both concrete and steel reach their ultimate strength at the same time, the failure of the r.c section is a called “balanced failure”. The balanced type of failure shows some ductility before collapse due to the factor of safety imposed by the different design code.

As

b

h

C

T

z

d N.A x

Strain-Stress distribution along the depth of r.c.

section

The distance “x” is the depth of the neutral axis from the top of the section. The shape of concrete compression stress is non linear and is similar to the stress – strain curve of the concrete.

When both material reach their ultimate capacity, the ultimate strain in concrete under compression as per BS 8110 Code is cu = 0.0035 and the yielding strain in reinforcing steel y= 0.002.

From similarity of triangles of the strain diagram

At balanced condition, the N.A. depth

dxycu

cubal

dxbal002.00035.0

0035.0

dxbal 636.0

To simplify the computation BS 8110, suggests using equivalent rectangular distribution of stress

the depth of the rectangular stress block “a” is equal to “0.9x”.

As

b

h

C

T

z

d N.A x a

N.A

d

a

b

x As

Z

The compressive force C = Area compressive stress in concrete

=

Taking

then the compressive stress in concrete will be:

m

cufba

67.0..

5.1m

cucucu ff

f45.0446.0

5.1

67.0

bafC cu ..45.0

The compression force will be therefore

This force is acting at a distance of “a/2” from

the fibre of the concrete section. Similarly the

tensile force in steel reinforcement will be:

sys AfAfy

T 87.0.15.1

zTzCM

The resisting moment

zCM .

2

adZ

245.0

adbafcuM

287.0

adAsfyM

ZTM .

When the steel reinforcement reach its yielding tensile stress fy before crushing of concrete, the failure will be controlled by the steel reinforcement and will be in ductile mode of failure.

This type of failure is called “Tension failure”.

As the steel yields, the stress in the steel will be constant and with any increase in the applied load, the steel reinforcement undergoes long deformation while the tensile force in the steel remains constant. This is due to the stress-strain behaviour of steel reinforcement

Constitutive relation for reinforcing steel

Stress

Strain εy=0.00219

Es

fy

1

As

h d

N.A

x1 x2

x3 cu

cucu

002.0yy y

C1

C2

C3

T=Asfy T T

N.A

N.A N.A

x1 x2

x3

z1

z2

z3

cuf

cuf

cuf

To maintain equilibrium the strain in top compression fibre of concrete will be increased and the N.A will be shifted up

Since the force in reinforcing steel is constant (The steel stress reach its ultimate capacity) the compressive forces in concrete C1 = C2 = C3 = C to maintain equilibrium.

balxx

To balance the moment due to extra applied load, the lever arm will be increased (z3 z2 z1), leading to shifting the N.A. up.

This in turn will result in decreasing the compression area.

To maintain equal compressive forces, the compressive stress in concrete will be increased as C = Area * compressive stress.

The compressive strain and stress in concrete will be increased gradually due to shifting of the N.A upwards, and the concrete area in compression is reduced.

The final collapse will be occurred when the concrete is crushed (the strain reaches 0.0035).

The process of tension failure indicates that there will be sufficient warning before the crushing of concrete and collapse of the beam. The warning is in the form of large displacement and cracking in the concrete at the tension side.

To ensure this mode of failure, the depth of N.A must be less than xbal hence, the BS 8110 limit the depth of N.A to

Xlimit = 0.5d

to ensure this type of ductile failure.

dxbal 61.0

In case where the concrete reach its ultimate capacity cu = 0.0035 before the steel attains its yielding strength, the failure will be characterized with Brittle failure due to the sudden crushing of concrete at the top.

In this case, with the increase of load, the N.A will be shifted down to increase the area of concrete under compression. This will lead to an increase in the steel strain and stress. This type of failure is called compression failure and is completely prohibited in any design codes since it will not give any warning before failure.

Considering the xmax = 0.5d as the maximum depth of N.A allowed in design as any value less than xmax will produce more ductile failure (tension failure ) and any value more than this limit is not allowed since it produce compression sudden failure.

Maximum Condition

Tension Failure Condition (T.F)

Compression Failure Condition (C.F)

Balance Condition

dxbal 61.0

dx 5.0max

As

b

h

C

T

z

d N.A

x a

Fig. 8 Stress-strain distribution at maximum limit condition

m

cuf

67.0

0035.0cu

00219.0y

dx 5.0max

dxa 45.09.0 maxmax

2

maxmax

adz

dz 775.0max

maxmax .zCM

dbafcuM 775.0.45.0 maxmax

cufbdM 2

max 156.0

cufbdkM 2

maxmax

156.02

maxmax

fcubd

Mk

zTM

dAsfyM 775.095.0max

Introducing to represent the steel ratio in the section

which is defined as the ratio between the steel area to

the effective area of the section:

s

100xbd

Ass 100/2bdA ss

100/)(95.0.45.0 maxmax bdfbaf ycu

cu

y

f

f3.21max

represent the maximum ratio of steel reinforcement in the concrete section which leads to a ductile failure in case of the increase of the applied load.

For the steel ratio lower than , the failure become more ductile (T.F) and the reinforced concrete section is called under-reinforced section.

While when is grater than the failure will brittle (C.F) and hence the section is called over reinforced section.

max

s max

maxs

T.F verses C.F

Maximum

Condition

(T.F)

(C.F)

,

maxaa

max

maxxx

maxkk

maxzz

dx 5.0max da 45.0max 156.0max k

y

cu

f

f%3.21max dz 775.0max

maxxx

maxaa

maxkk

max

maxzz