Skip to main content

Deflection of Beams

Strength of Materials -1, Unit -5
The deformation of a beam is usually expressed in terms of its deflection from its original unloaded position. The deflection is measured from the original neutral surface of the beam to the neutral surface of the deformed beam. The configuration assumed by the deformed neutral surface is known as the elastic curve of the beam.

The angle through which the cross-section rotates with respect to the original position is called the angular rotation of the section.

Methods of Determining Beam Deflections:

1. Double integration method

The double integration method is a powerful tool in solving deflection and slope of a beam at any point because we will be able to get the equation of the elastic curve.

In Cartesian co-ordinates, the radius of curvature(R) of a curve y = f(x) is given by 

  In the derivation of flexure formula, the radius of curvature of a beam is given as

 1/R = M /EI

ThusThe deferential equitation of elastic curve is EI dy2 / dx2  = MX
The product EI is called the flexural rigidity of the beam, If EI is constant,the General equation may be written as:

 EI dy2 / dx2 =  MX     ---  (1)

Note :The down ward deflection will consider here as -ve ,
where x and y are the coordinates shown in the figure of the elastic curve of the beam under load, y is the deflection of the beam at any distance x. E is the modulus of elasticity of the beam, I represent the moment of inertia about the neutral axis, and M represents the bending moment at a distance x from the end of the beam. 
Direct Integration Method or Double Integration Method
The first integration of equation of (1) yields the slope of the elastic curve and the second integration equation (1) gives the deflection of the beam at any distance x.
The resulting solution must contain two constants of integration since EI y" = M is of second order.
These two Integral constants must be evaluated from known conditions concerning the slope deflection at certain points of the beam.
1.A simply supported beam with rigid supports, 
at x = 0 and x = L, the deflection y = 0, and in locating the point of maximum deflection, we simply set the slope of the elastic curve y' to zero.
2.A Fixed support,
at fixed end, the deflection y is zero and slope dy/dx ( ø)is zero.
Macaulay's Method For Beam Deflections
For complex loading, specially where the span is partially loaded or loaded with number of concentrated loads,this Macaulay's method is more useful.
Note : 
1. While using the Macaulay's method, the section 'x' is to be taken in the last portion of the beam.
2. the quantity with in brackets ( ), should be integrated as a whole.
3.The expression for Bending Moment (Mx) equation can be used at any point, provided the term within the brackets becomes negative is omitted. 


Popular posts from this blog

Three Hinged Arches

The arch is one of the oldest structures. The Romans developed the semi circular true masonry arch.which they used extensively in both bridges and aqueducts.
An arch is a curved beam in  elevation. The horizontal moment at the supports is wholly or partially prevented in arches. Hence a horizontal thrust induced at the supports.
The supports must effectively arrest displacements in the vertical and horizontal directions. Only then there will be arch action.

An arch with a hinge at each support and third hing at the high point / crown or any where in the rib of arch is known as Three Hinged Arch. If no hinge exists at the apex, it will normally called as atwo-hinged arch.The horizontal distance between the lower hinges is called Span of the arch.

Eddy’s theorem:  Eddy’s theorem states that “ The bending moment at any section of an arch is proportional to the vertical intercept between the linear arch (or theoretical arch) and the centre line of the actual arch.” 

*Parabolic arches are prefer…


Analysis of multi-storey building frames involves lot of complications and tedious calculations by using conventional methods. To carry out exact analysis is a time consuming task. Substitute frame method for analysis of multistory frame can be handy in approximate and quick analysis. This method has been applied only for vertical loading conditions.
The method assumes that the moments in the beams of any floor are influenced by loading on that floor alone. The influence of loading on the lower or upper floors is ignored altogether. The process involves the division of multi-storied structure into smaller frames. These sub frames are known as equivalent frames or substitute frames. The sub frames are usually analyzed by the moment distribution method, using only Two cycle of distribution. It is only necessary to consider the loads on the two nearest spans on each side of the point . The substitute frames are formed by the beams at the floor level under consideration, together with the col…


The stresses produced due to constant Bending Moment (with zero Shear Force or pure bending) are know as Bending stresses.Assumptions in Theory of Bending : In deriving the relations between the bending moments and flexure (bending)stresses and between the Shear forces and Sharing stresses the following assumptions are made. 1.Transverse sections of the beam that were plane before bending remain so even after bending. 2.The material of the beam is isotropic and homogeneous and follows Hooke's law and has the same value of Young's Modulus in tension and compression. 3.The beam is subjected to Pure bending and therefore bends in an arc of a circle.
4.The radius of curvature is large compared to the dimension of the cross-section.
5.Each layer is independent to enlarge or contract.
6.The stresses are purely longitudinal and local effects of point loads are neglected.

Flexure Formula : Stresses caused by the bending moment are known as flexural or bending stresses. Consider a beam to be lo…