analysis of flexural members - جامعة نزوى · the compressive strain and stress in concrete...
TRANSCRIPT
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.
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.
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
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”.
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:
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
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