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force-displacement relations in stiffness matrix form are
.
−
/
3
−
/
2
/
3
−
/
2
V
a
M
a
···
V
b
M
b
12
EI
6
EI
12
EI
6
EI
w
a
.
θ
2
2
−
6
EI
/
4
EI
/
6
EI
/
2
EI
/
a
···
w
b
θ
b
...
...
.
...
...
=
(5)
.
−
/
3
/
2
/
3
/
2
12
EI
6
EI
12
EI
6
EI
.
2
2
−
6
EI
/
2
EI
/
6
EI
/
4
EI
/
p
=
k
v
3.2.2
The Unit Load Method
This method is used to determine the displacement
V
k
at a given point in a given direction of
a structure for which the state of stress, and through the material law
al
so the state of strain
,
is known. To proceed, apply a virtua
l
force
δ
P
k
in the direction of
V
k
.
Then the external
W
e
=
complementary virtual work is
δ
V
k
δ
P
k
.
The virtual force corresponds to a system of
becomes
V
δ
σ
k
dV
W
i
virtual stresses
δ
σ
k
,
and the complementary virtual work
δ
.
From
the principle of complementary virtual work,
V
δ
σ
k
dV
δ
P
k
V
k
=
=
V
ij
(δσ
ij
)
k
dV
(3.14)
Since
δ
P
k
is arbitrary, for simplicity it can be set equal to unity. Thus,
V
ij
(δσ
ij
)
k
dV
V
δ
σ
k
dV
1
·
V
k
=
=
(3.15)
This is a statement of the
unit load
or
th
e
dummy load
method. The stresses
δ
σ
k
are due to a
unit force applied in the direction of
V
k
.
They can be chosen to be the same as for a similar
but statically determinate system as the work done by the unknowns of the corresponding
indeterminate system through the stresses of the real system is zero. If initial strains, e.g.,
thermal strains, are present, then
should include these. Equation (3.15) remains valid for
nonlinear stru
ct
ures.
Physically,
V
k
can be interpreted as the
flexibility
of the solid. The method is helpful in
calculating flexibility properties of structures for use in matrix methods. In Example 3.14,
the stiffness characteristics for a beam will be obtained from beam flexibilities found using
the unit load method.
For a truss or beam element, use can be made of Eq. (3.15), or an equivalent theorem in
more indigenous notation can be derived. In the case of a beam, the external complementary
virtual work done by the virtual (unit) load at location
k
moving through the real deflection
w
k
is
w
k
δ
P
k
=
w
k
(
1
)
Let
M
designate the internal bending moment generated by the virtual (unit) load. Real
forces cause a rotation of an element of a beam of
d
δ
Hence, the internal com-
plementary virtual work done on an element of a beam by the moment
θ
=
Mdx
/(
EI
).
δ
M
is
δ
MMdx
/(
EI
)
,
and the total internal complementary virtual work for a beam of length
L
is
L
M
EI
δ
Mdx
(3.16)
0
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