Civil Engineering Reference
In-Depth Information
2
wx
(
)
−
−
1
x
L
L
L
mM
EI
2
wx
EI
dx
3
wL
EI
4
∫
∫
∫
∆
B
=
dx
=
dx
=
=
EI
2
8
0
0
0
4.6.1
VISUAL INTEGRATION
The virtual work method can also be done using the moment diagrams of
the real and virtual loads. This is known as
visual integration
. The moment
diagrams for the real (Q) and virtual (q) loads are constructed. The integral
is equal to the area of the
M
/
EI
diagram multiplied by the value of the m
diagram taken at the centroid of the
M
/
EI
diagram.
Example 4.9
Visual Integration
Determine the deflection at C in inches for the simply supported beam
shown in Figure 4.18 using the virtual work method and visual integra-
tion. Use
E
= 2000 ksi and
I
= 2000 in
4
.
8k
40 k-ft
A
C
B
8ft
16ft
Figure 4.18.
Example 4.9 Visual integration.
Solve for the reactions due to the applied loads. The free-body dia-
gram is shown in Figure 4.19.
==
( )
−
( )
+−∴=
== +−∴=
Σ
Σ
M
0816
k
ft Aft
24
40
k
ft Ak
7
B
y
y
F
0
A Bk Bk
8
1
y
y
y
y
The shear and moment diagrams are constructed in Figure 4.19. The
moment diagram is broken into unique areas that will be continuous over
the area of the virtual moment diagrams shown in Figure 4.20.
Apply a virtual force
dQ
= 1 at point C as shown in Figure 4.20. Solve
for the reactions similar to the real loads. Draw the shear and moment
diagram locating the centroid of the real moment diagrams on the virtual
moment diagram.
