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Beam Deflection (9.1-9.5)
MAE 314 – Solid Mechanics
Yun Jing
Beam Deflection
1
Introduction


Up to now, we have been primarily calculating normal and
shear stresses.
In Chapter 9, we will learn how to formulate the deflection
curve (also known as the elastic curve) of a beam.
Beam Deflection
2
Differential Equation of Deflection
 d   ds
d
ds

1

dy
ds
Recall from Ch. 4 that
1/ρ is the curvature of
the beam.
dy
θ
dx
dx
dx
y
Beam Deflection
 cos 
ds
dy
Slope of the deflection curve
 tan 
 sin 
ds
3
Assumptions

Assumption 1: θ is small.
1

1. ds  dx  d 

ds


2.
dy
dx
 tan    


d
dx
d
2

dx
d y
dx
2

1

Assumption 2: Beam is linearly elastic.
2

d y
dx
1


2

M
EI
Thus, the differential equation for the deflection curve is:
2
d y
dx
2

M
EI
Beam Deflection
4
Diff. Equations for M, V, and w

Recall from Ch. 5:
dV
 w
dM
dx

So we can write:
dx
2
EI
d y
dx

2
3
M
EI
d y
dx
3
4
V
EI
d y
dx
4
 w
Deflection curve can be found by integrating




V
Bending moment equation (2 constants of integration)
Shear-force equation (3 constants of integration)
Load equation (4 constants of integration)
Chosen method depends on which is more convenient.
Beam Deflection
5
Boundary Conditions



Sometimes a single equation is sufficient for the entire length of the
beam, sometimes it must be divided into sections.
Since we integrate twice there will be two constants of integration for
each section.
These can be solved using boundary conditions.


Deflections and slopes at supports
Known moment and shear conditions
V B  0 
M B  0 
M A
Beam Deflection
 0
M B
 0
6
Boundary Conditions


Section AC: yAC(x)
Continuity conditions:

Displacement continuity
y AC ( C )  y CB ( C )

Slope continuity
dy AC
dy CB
( C )   AC ( C ) 
( C )   CB ( C )
dx
dx
Symmetry conditions:
Section CB: yCB(x)
dy
0
dx
Beam Deflection
7
Example Problem
For the beam and loading shown, (a) express the magnitude and location
of the maximum deflection in terms of w0, L, E, and I, (b) Calculate the
value of the maximum deflection, assuming that beam AB is a W18 x 50
rolled shape and that w0 = 4.5 kips/ft, L = 18 ft, and E = 29 x 106 psi.
Beam Deflection
8
Statically Indeterminate Beams


When there are more reactions than can be solved using statics,
the beam is indeterminate.
Take advantage of boundary conditions to solve indeterminate
problems.
Problem:
Number of reactions: 3 (MA, Ay, By)
x=0, y=0
x=0, θ=0
x=L, y=0
Number of equations: 2 (Σ M = 0, Σ Fy = 0)
One too many reactions!
Additionally, if we solve for the deflection curve, we
will have two constants of integration, which adds
two more unknowns!
Solution: Boundary conditions
Beam Deflection
9
Statically Indeterminate Beams
Problem:
Number of reactions: 4 (MA, Ay, MB, By)
x=0, y=0
x=0, θ=0
x=L, y=0
x=0, θ=0
Number of equations: 2 (Σ M = 0, Σ Fy = 0)
+ 2 constants of integration
Solution: Boundary conditions
Beam Deflection
10
Example Problem
For the beam and loading shown, determine the reaction at the roller
support.
Beam Deflection
11
Beam Deflection: Method of
Superposition (9.7-9.8)
MAE 314 – Solid Mechanics
Yun Jing
Beam Deflection: Method of Superposition
12
Method of Superposition

Deflection and slope of a beam produced by multiple loads
acting simultaneously can be found by superposing the
deflections produced by the same loads acting separately.

Reference Appendix D (Beam Deflections and Slopes)

Method of superposition can be applied to statically
determinate and statically indeterminate beams.
Beam Deflection: Method of Superposition
13
Superposition

Consider sample problem 9.9 in text.

Find reactions at A and C.

Method 1: Choose MC and RC as
redundant.

Method 2: Choose MC and MA as redundant.
Beam Deflection: Method of Superposition
14
Example Problem
For the beam and loading shown, determine (a) the deflection at C, (b) the
slope at A
Beam Deflection: Method of Superposition
15
Example Problem
For the beam and loading shown, determine (a) the deflection at C, and (b)
the slope at end A.
Beam Deflection: Method of Superposition
16
Example Problem
For the beam shown, determine the reaction at B.
Beam Deflection: Method of Superposition
17
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