Civil Engineering Reference
In-Depth Information
P
iz
P
jz
M
jy
∆
iz
M
iy
EI
y
EI
y
Figure 4.29.
Example 4.16 Δ
iz
stiffness.
The tangential deviation of a point at the
i
i-end from a tangent to the curve
on the
j
-end is the implied deflection. This is equal to the moment of the
area of the
M
/
EI
diagram about the point at the
i
-end.
j
M
L
2
M
EI
xdx
L
P
22
L
EI
2
3
L
∫
iy
iz
t
== =−
∆
+
ij
iz
i
EI
y
y
i
Substituting the first equation for
P
iz
into the ∆
iz
equation results in one
of the stiffness terms. The second term is found by substituting the first
stiffness term back into the
P
iz
equation. Note that
M
iy
was assumed as
negative in the free-body diagram, so the sign must be switched.
M
L
EI
EI
L
2
iy
∆
=
iz
6
y
6
y
M
=−
∆
(4.17)
iy
iz
2
=
12
EI
L
y
P
iz
∆
(4.18)
iz
3
The four terms given in Equations 4.15 through 4.18 are the flexural stiff-
ness terms for the forces at the
i
i-end due to motions at the
i
i-end. This is
denoted as stiffness matrix [
K
ii
] in Equation 4.19. The stiffness equation
and matrix form of this are as follows:
[[]
=
[]
K
d
F
ii
i
i
