Civil Engineering Reference
In-Depth Information
Since both ends of the beam are fixed for translation, the tangential
deviation of a point at the
i
i-end from a tangent to the curve on the
j
i-end is
zero. This is equal to the moment of the area of the
M
/
EI
diagram about
the point at the
i
-end.
j
+
P
L
EI
2
=−
M
L
M
EI
xdx
L
2
3
L
∫
iy
t
==
0
iz
ij
i
EI
22
z
z
i
3
M
iz
L
P
=
iy
2
The change in rotation from the
i
-end to the
j
i-end is equal to the negative
of the implied rotation. This is the area under the
M
/
EI
diagram between
those points.
2
P
L
EI
M
L
iy
iz
∆
q
=− =− =−
0
q
q
+
ij
iz
iz
EI
2
z
z
Substituting the first equation for
P
iy
into the q
iz
equation results in one of
the stiffness terms. The second term is found by substitution of the first
stiffness term back into the
P
iy
equation.
M
L
iz
−=−
q
iz
EI
EI
L
4
z
4
M
=
z
q
(4.24)
iz
iz
=
6
EI
L
P
iy
z
q
(4.25)
iz
2
Example 4.18
∆
iy
stiffness
Derive the ∆
iy
stiffness using the conjugate beam method for a linear
member.
A free-body diagram is shown in Figure 4.31 with an imposed deflec-
tion of one unit on the
i
i-end of the member. The moments are assumed
in the positive
z
direction using the right-hand rule and the Cartesian
right-hand coordinate system. The forces are shown consistent with the
deformation. The moment diagram divided by
EI
is shown for the reaction
forces on the
i
i-end of the member.
