Friday, 28 April 2017

BENDING STRESS OF COMPOSITE BEAM

BENDING STRESS OF COMPOSITE BEAM

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

σ1= Stress developed in steel plate
σ2= Stress developed in wood
Ԑ1= Strain developed in steel plate at a distance y from the common neutral axis
Ԑ2= Strain developed in wood at a distance y from the common neutral axis
E1= Young’s Modulus of steel plate
E2= Young’s Modulus of wood
I1= Area moment of inertia of steel about the common neutral axis of the composite beam
I2= Area moment of inertia of wood about the common neutral axis of the composite beam
M1= Moment of resistance of steel plates
M2= 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

Ԑ1= Stress developed in steel plate / Young’s Modulus of steel plate
Ԑ1= σ1/ E1

Strain developed in wood at a distance y from the common neutral axis of the composite beam
Ԑ2= Stress developed in wood / Young’s Modulus of wood
Ԑ2= σ2/ E2

As we have seen above during studying the concepts that strain will be same for each beam of composite beam or flitched beam.
Ԑ1= Ԑ2
σ1/ E1= σ2/ E2
σ1= σ2 x (E1/ E2)
σ1= σ2 x m
Where, m= E1/ E2 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
M1 = (σ1/y) x I1

Moment of resistance for wood plates
M2 = (σ2/y) x I2

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 = M1 + M2
M = (σ1/y) x I1 + (σ2/y) x I2
M = (m.σ2/y) x I1 + (σ2/y) x I2
M = (σ2/y) x [m.I1 + I2]
Where, Equivalent moment of inertia of the cross-section, I = m.I1 + I2

M = (σ2/y) x I

Therefore, total moment of resistance of the composite beam i.e. M will be written as mentioned here
M = (σ2/y) x I
Where, I = m.I1 + I2

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

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