Civil Engineering Reference
In-Depth Information
and boundary restraint. The complementary energy of the reactions at A and B is
therefore zero since both of their corresponding displacements are zero.
In Eq. (v) the term
∂
F
j
/∂
P
2
represents the rate of change of the actual forces in the
members of the truss with
P
2
. This may be found, as described in the non-linear case, by
calculating the forces,
F
j
, in the members in terms of
P
2
and then differentiating these
expressions with respect to
P
2
. Subsequently the actual value of
P
2
would be substituted
in the expressions for
F
j
and thus, using Eq. (v),
2
obtained. This approach is rather
clumsy. A simpler alternative would be to calculate the forces,
F
j
, in the members
produced by the applied loads including
P
2
, then remove all the loads and apply
P
2
only as an unknown force and recalculate the forces
F
j
as functions of
P
2
;
∂
F
j
/∂
P
2
is
then obtained by differentiating these functions.
This procedure indicates a method for calculating the displacement of a point in the
truss in a direction not coincident with the line of action of a load or, in fact, of a point
such as C which carries no load at all. Initially the forces
F
j
in the members due to
P
1
,
P
2
,
...
,
P
k
,
...
,
P
r
are calculated. These loads are then removed and a
dummy
or
fictitious
load,
P
f
, applied at the point and in the direction of the required displacement.
A new set of forces,
F
j
, are calculated in terms of the dummy load,
P
f
, and thus
∂
F
j
/∂
P
f
is obtained. The required displacement, say
C
of C, is then given by
n
F
j
L
j
E
j
A
j
∂
F
j
∂
P
f
C
=
(vi)
j
=
1
The simplification may be taken a stage further. The force
F
j
in a member due to
the dummy load may be expressed, since the system is linearly elastic, in terms of the
dummy load as
∂
F
j
∂
P
f
F
j
=
P
f
(vii)
Suppose now that
P
f
=
1, i.e. a
unit load
. Equation (vii) then becomes
∂
F
j
∂
P
f
F
j
=
1
so that
∂
F
j
/∂
P
f
F
1,
j
, the load in the
j
th member due to a unit load applied at
the point and in the direction of the required displacement. Thus, Eq. (vi) may be
written as
=
n
F
j
F
1,
j
L
j
E
j
A
j
C
=
(viii)
j
=
1
in which a unit load has been applied at C in the direction of the required displacement.
Note that Eq. (viii) is identical in form to Eq. (ii) of Ex. 15.6.
In the above we have concentrated on members subjected to axial loads. The argu-
ments apply in cases where structural members carry bending moments that produce
rotations, shear loads that cause shear deflections and torques that produce angles of
