Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

Real Moment M: Remove the virtual load and reapply the
real load w.
Again, draw a free-body diagram around end A of the
beam. Note that the same x coordinate used to derive the virtual
moment must be used to derive the real moment; therefore,
the origin of the x coordinate system must be placed at
A. From the free-body diagram, derive the following equation
for the real internal moment M:
A
x
w
L
w
B
M
M
2
wx
=− 0 ≤ x ≤ L
2
A
x
V
Virtual-Work Equation for Beam Deflection: From Equation (17.35), the beam deflection
at A can now be calculated:
1⋅D =
A
∫
0
L
⎛
m M ⎞
⎜ ⎟ dx =
⎝ EI ⎠
4
wL
∴D A = ↓
8EI
∫
0
L
2
(-1 x)( -wx
/2)
dx =
EI
w
2EI
∫
0
L
3
x dx
Ans.
Since the result is a positive value, the deflection occurs in the same direction as was
assumed for the unit load—that is, downward.
(b) Virtual Moment m for Calculating the Beam Slope:
To compute the angular rotation of the cantilever beam at A,
remove the real load w from the beam and apply a virtual
unit moment at A. The unit moment will be applied counterclockwise
in this instance because it is expected that the
beam will slope upward from A.
Again, draw a free-body diagram around end A of the
beam, placing the origin of the x coordinate system at A.
From the free-body diagram, derive the following equation
for the virtual internal moment m:
m =-1 0 ≤ x ≤ L
1
A
x
L
1 A
x
v
B
m
Real Moment M: The real-moment equation M is the same as was derived previously.
Virtual-Work Equation for Beam Slope: From Equation (17.36), the beam slope at A
can now be determined:
L ⎛ ⎞
1⋅ θ = ∫ m⎜
M A
⎟ dx =
⎝ ⎠
∫
0
EI
3
wL
∴ θ A = (CCW)
6EI
0
L
2
(-1)(-wx
/2)
dx =
EI
w
2EI
∫
0
L
2
x dx
Ans.
Since the result is a positive value, the angular rotation occurs in the same direction
as was assumed for the unit moment—that is, counterclockwise.
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