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 P1
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
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 P1
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