We were discussing basic
concept of bending stress , section modulus of the beam and derivation for beam bending equation in our previous session. We have also discussed assumptions
made in the theory of simple bending and expression
for bending stress in pure bending during our
last session.

Now we are going ahead to start new topic i.e. bending
stress of composite beam in the strength of material with the help of this
post.

Let us go ahead step by step for easy understanding,
however if there is any issue we can discuss it in comment box which is
provided below this post. So let us come to the main subject i.e. bending stress
of composite beam.

###
**Bending
stress of composite beam**

Composite beam is basically defined as
the beam made by two or more than
two beams of similar length but different materials and rigidly fixed with each
other in such a way that it behaves as one unit and strain together against
external load i.e. it behaves as single unit for compression and extension
against compressive and tensile stress.

We can say here from the definition of
composite beam that strain will be same for each beam of composite beam and
hence actual change in dimensions will be similar for each beam or we can say
that actual strain will be same for each beam of composite beam or flitched
beam and we will use this concept during bending stress analysis for composite
beams.

Let us consider that we have one
composite beam as displayed here in following figure, we can see here that wooden
beam or timber beam reinforced by steel plates and it is displayed here in
following figure.

Above arrangements of wooden beam or
timber beam with steel plates is termed as composite beam or flitched beam.

Concepts that we will have to remind
here for analysis of bending stress for composite beam or flitched beam is as
mentioned here.

Composite beam behaves as one unit and
strain together against external load i.e. actual strain will be same for each
beam of composite beam or flitched beam.

Total moment of resistance will be
equivalent to the sum of moments of resistance of the individual beam sections.

Bending stress at a point will be
directionally proportional to the distance of the point from the common neutral
axis of the composite beam or flitched beam.

Strain at a point will be directionally
proportional to the distance of the point from the common neutral axis of the
composite beam or flitched beam.

We have assumed following information
for above figure of composite beam or flitched beam

σ

_{1}= Stress developed in steel plate
σ

_{2}= Stress developed in wood
Ԑ

_{1}= Strain developed in steel plate at a distance y from the common neutral axis
Ԑ

_{2}= Strain developed in wood at a distance y from the common neutral axis
E

_{1}= Young’s Modulus of steel plate
E

_{2}= Young’s Modulus of wood
I

_{1}= Area moment of inertia of steel about the common neutral axis of the composite beam
I

_{2}= Area moment of inertia of wood about the common neutral axis of the composite beam
M

_{1}= Moment of resistance of steel plates
M

_{2}= Moment of resistance of wood
y = Distance from the common neutral
axis of the composite beam

Let us find the strain developed in
steel plate at a distance y from the common neutral axis of the composite beam

Ԑ

_{1}= Stress developed in steel plate / Young’s Modulus of steel plate

*Ԑ*_{1}= σ_{1}/ E_{1}
Strain developed in wood at a distance y
from the common neutral axis of the composite beam

Ԑ

_{2}= Stress developed in wood / Young’s Modulus of wood

*Ԑ*_{2}= σ_{2}/ E_{2}
As we have seen above during studying
the concepts that strain will be same for each beam of composite beam or
flitched beam.

Ԑ

_{1}= Ԑ_{2}
σ

_{1}/ E_{1}= σ_{2}/ E_{2}
σ

_{1}= σ_{2}x (E_{1}/ E_{2})
σ

_{1}= σ_{2}x m
Where, m= E

_{1}/ E_{2}i.e. modular ratio between wood and steel
Let us recall the flexural formula and
moment of resistance and we will use the following equation to determine the
moment of resistance for wood and steel and finally we will determine the total
moment of resistance of the composite beam or flitched beam.

M = (σ/y) x I

Moment of resistance for Steel plates

M

_{1}= (σ_{1}/y) x I_{1}
Moment of resistance for wood plates

M

_{2}= (σ_{2}/y) x I_{2}
Again we will see above mentioned
concepts that total moment of resistance will be equivalent to the sum of
moments of resistance of the individual beam sections

Total moment of resistance of the
composite beam, M = M

_{1}+ M_{2}
M = (σ

_{1}/y) x I_{1}+ (σ_{2}/y) x I_{2}
M = (m.σ

_{2}/y) x I_{1}+ (σ_{2}/y) x I_{2}
M = (σ

_{2}/y) x [m.I_{1}+ I_{2}]
Where, Equivalent moment of inertia of
the cross-section, I = m.I

_{1}+ I_{2}
M = (σ

_{2}/y) x I
Therefore, total moment of resistance of
the composite beam i.e. M will be written as mentioned here

*M = (σ*_{2}/y) x I

*Where,*

*I = m.I*_{1}+ I_{2}###
**Reference:**

Strength
of material, By R. K. Bansal

Image
Courtesy: Google

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