Deflection of Beams

Chapter 9

Introduction:

In this chapter we learn how to determine the deflection of beams (the maximum deflection) under given load .

A prismatic beam subjected to pure Bending is bent into an arc of a circle in the elastic range ,the curvature of the neutral surface expressed as :

1/ρ = M/EI

Where , M is the bending Moment , E is the modulus of elasticity , I is the moment of inertia of the cross

section about it’s neutral axis .

Both the Moment “M” & the Curvature of neutral axis “1/ρ” will vary denoting by the distance of the section from the left of the beam “x” .

We write

1/ρ = M(x)/EI

Deflection

We notice that the deflection at the ends

y = 0 due to supports

dy/dx = 0 at A,B ,also dy/dx = 0 at the max.

deflection

To determine the slope and the deflection of the beam at any given point ,we first drive the following second order differential equations which governs the elastic curve

So the deflection (y) can be obtained through the boundary conditions .

In the next fig. two differential equations are required due to the effective force (p) at point (D) .

One for the portion (AD) ,the other one for the portion (DB) .

P

D

• The first eq. holds Q1 ,y1

• The second eq. holds Q2 ,y2

So ,we have four constants C1 ,C2 ,C3 ,C4 due to the integration process .

Two will be determined through that deflection (y=0) at A,B .

The other constants can be determined through that portions of beam AD and DB have the same slope and the same deflection at D

If we took an example

M(X) = -Px

We notice that the radius

of curvature “ρ” =

∞ ,so that M = 0

P

P

B

B A

A

L

Also we can conclude from the next example , P1>P2

We notice that +ve M so that the elastic curve is concaved downward .

P1 P2

AB

CD

-ve M

+ve M

Q(x,y)

Elastic curve

Equation of elastic curve:We know the curvature of a plane curve at point Q(x,y) is expressed as

Where , are the 1st & 2nd derivatives of a function y(x) represented by a curve ,the slopeis so small and it’s square is negligible ,so we

get that

Is the governing differential equation for the

elastic curve .

N.B.: ``EI`` is known as the flexural rigidity .

In case of prismatic beams (EI) is constant .