DEFLECTION OF BEAMS

DEFLECTION :

In engineering,deflection is the displacement of the member from original position under the influence of external forces. In beams generally the external forces causing the deflection are transverse loads.

But in general we cannot see the deflection in our structures, as the beams are designed to resist loading and deflection even if there is deflection it will be of small magnitude which we can't see with our naked eyes.

DEFORMATION :

On earth every thing has a form (i.e, its shape and size ), when these things are subjected to external factors they undergo some permanent changes in their shape, and size and the changes are known as Deformations.

What is difference between Deflection and Deformation?

Generally, there is a misconception in students that Deflection and Deformation are basically same but that is not totally true.Deflection is a type of deformation but with non permanent changes in form.

There are many types of deformation such as extension,shrinking etc,. and but they are permanent whereas deflection is temporary.

DEFLECTION IN PURE BENDING :

In pure bending , the deflection of the beam axis is in the shape of a circular arc. Generally this curved axis of the beam is called elastic curve or deflection curve.

CURVATURE OF THE AXIS = M/EI

where M = Moment of resistance(bending moment)

E = Modulus of elasticity

I = Moment of inertia of the beam section

CONDITIONS FOR A GOOD BEAM :

STRENGTH
STIFFNESS

STRENGTH :

This is the property of a material to resist induced stress. The beams should be designed strong enough as to resist the induced bending stress (stress due to transverse loads) and the stress in the beam should be under permissible limits.

STIFFNESS :

This is the property of a material to resist deformations and deflections. The beams should be designed such that the deflection in the beam should be minimal and within the limits.

A beam is said to be a good beam when the strength and stiffness conditions are satisfied.

DIFFERENTIAL EQUATION OF FLEXURE :

Differential equation of flexure is a widely used equation to calculate the slope and deflection at any point on the beam. It gives the relationship between the bending moment at a particular cross-section and the cartesian co-ordinates of the point in the bent beam.

where, M = Moment of resistence/B.M

EI = Flexural rigidity

METHODS OF DETERMINING SLOPE AND DEFLECTION :

Double integration method
Macaulay's method
Moment area method -Mohr's theorems
Conjugate beam method.

DOUBLE INTEGRATION METHOD:

The name itself is self explanatory of the above method . In this method to find the slope, integration of the Differential equation of flexure is done and to find deflection, Differential equation of flexure is integrated once again ,that's the reason the method is named as double integration method.

MACAULAY'S METHOD :

It is similar to the Double integration method and extended version of it.In this method a single bending moment equation is used for the whole beam,whereas in the Double integration method multiple bending moment equations are required.

MOMENT AREA METHOD - MOHR'S THEOREM'S :

This method is especially suitable to find the slope or deflection at only one point of the beam instead of the complete equation of the deflection curve.There are two Mohr's Theorems and they are :

Mohr's Theorem-1 :

The angle between tangents drawn at any two points on elastic curve will be equal to the area of M/EI diagram.

SLOPE = A/EI (A =Area of BMD )

Mohr's Theorem-2 :

The deflection at any point relative to another point on a beam subjected to bending is equal to the moment of M/EI diagram between the two points about the vertical line.

DEFLECTION = Ax/EI

where, A=Area of BMD

x = distance of C.G of BMD and the where deflection is calculated.

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