Beam Deflection 1 1

Beam Deflection
By Touhid Ahamed

Introduction
• In this chapter rigidity of the beam will be considered
• Design of beam (specially steel beam) base on strength consideration and deflection evaluation

Introduction
Different Techniques for determining beam deflection
• Double integration method
• Area moment method
• Conjugate-beam method
• Superposition method
• Virtual work method

Double Integration Method
The edge view of the neutral surface of a deflected beam is called the elastic curve

1 M ( x)


EI

ρ

Double Integration Method
• From elementary calculus, simplified for beam parameters, d2y 2
2
1 d y dx 

2
 
2 3 2 dx  dy 
1    
  dx  

• Substituting and integrating,
1
d2y
EI EI 2 M  x 

dx x dy
EI  EI
 M  x  dx  C1 dx 
0

x

x

EI y dx M  x  dx  C1x  C2
0

0

Double Integration Method
Boundary conditions for statically determinate beam

x

x

0

0

EI y dx M  x  dx  C1 x  C2

Solved Problem

SOLUTION:
• Develop an expression for
M(x) and derive differential equation for elastic curve.
W 14 68

I 723 in 4

P 50 kips L 15 ft

E 29 106 psi a 4 ft

For portion AB of the overhanging beam, (a) derive the equation for the elastic curve, (b) determine the maximum deflection,
(c) evaluate ymax.

• Integrate differential equation twice and apply boundary conditions to obtain elastic curve.
• Locate point of zero slope or point of maximum deflection. • Evaluate corresponding maximum deflection.

Solved Problem
SOLUTION:
• Develop an expression for M(x) and derive differential equation for elastic curve.
- Reactions
:
Pa
 a
RA 

L

RB P1   
 L



- From the free-body diagram for section AD, a M  P

L

x

 0  x  L

- The differential equation for the elastic curve,
EI

d2y

a


P
x
2
L dx Solved Problem
• Integrate differential equation twice and apply boundary conditions to obtain dy elastic
1 a 2 curve.
EI

dx



2

P

L

x  C1

1 a 3
P x  C1x  C2
6 L at x 0, y 0 : C2 0
EI y