Civil Engineering Reference
In-Depth Information
Equations (17.3) and (17.4) may be expressed in matrix form as follows
F
A
F
B
AE
/
L
w
A
w
B
−
AE
/
L
=
−
AE
/
L
/
L
or
F
A
F
B
1
w
A
w
B
AE
L
−
1
=
(17.5)
−
11
Equation (17.5) may be written in the general form
{
F
}=
[
K
AB
]
{
w
}
(17.6)
in which
are generalized force and displacement matrices and [
K
AB
]isthe
stiffness matrix
of the member AB.
{
F
}
and
{
w
}
Suppose now that we have two axially loaded members, AB and BC, in line and
connected at their common node B as shown in Fig. 17.2.
In Fig. 17.2 the force,
F
B
, comprises two components:
F
B,AB
due to the change in
length of AB, and
F
B,BC
due to the change in length of BC. Thus, using the results of
Eqs (17.3) and (17.4)
A
AB
E
AB
L
AB
F
A
=
(
w
A
−
w
B
)
(17.7)
A
AB
E
AB
L
AB
A
BC
E
BC
L
BC
F
B
=
F
B,AB
+
F
B,BC
=
(
w
B
−
w
A
)
+
(
w
B
−
w
C
)
(17.8)
A
BC
E
BC
L
BC
F
C
=
(
w
C
−
w
B
)
(17.9)
in which
A
AB
,
E
AB
and
L
AB
are the cross-sectional area, Young's modulus and length
of the member AB; similarly for the member BC. The term
AE
/
L
is a measure of the
stiffness of a member, this we shall designate by
k
. Thus, Eqs (17.7)-(17.9) become
F
A
=
k
AB
(
w
A
−
w
B
)
(17.10)
F
B
=−
k
AB
w
A
+
(
k
AB
+
k
BC
)
w
B
−
k
BC
w
C
(17.11)
F
C
=
k
BC
(
w
C
−
w
B
)
(17.12)
F
B
,
w
B
C
A
B
F
C
,
w
C
F
A
,
w
A
F
IGURE
17.2
Tw o
axially loaded
members in line
L
BC
node
L
AB
