Civil Engineering Reference
In-Depth Information
which, since
A
d
A
=
A
, reduces to
δ
w
i,
N
=
N
ε
v
δ
x
(15.9)
In other words, the virtual work done by
N
is the product of
N
and the virtual axial
displacement of the element of the member. For a member of length
L
, the virtual
work,
w
i,
N
, done during the arbitrary virtual strain is then
w
i,
N
=
N
ε
v
d
x
(15.10)
L
For a structure comprising a number of members, the total internal virtual work,
W
i
,
N
,
done by axial force is the sum of the virtual work of each of the members. Thus
w
i,
N
=
N
ε
v
d
x
(15.11)
L
Note that in the derivation of Eq. (15.11) we have made no assumption regarding the
material properties of the structure so that the relationship holds for non-elastic as
well as elastic materials. However, for a linearly elastic material, i.e. one that obeys
Hooke's law (Section 7.7), we can express the virtual strain in terms of an equivalent
virtual normal force. Thus
N
v
EA
Therefore, if we designate the
actual
normal force in a member by
N
A
, Eq. (15.11)
may be expressed in the form
σ
v
E
=
ε
v
=
N
A
N
v
EA
w
i,
N
=
d
x
(15.12)
L
Shear force
The shear force,
S
, acting on the member section in Fig. 15.5 produces a distribution
of vertical shear stress which, as we saw in Section 10.2, depends upon the geometry
of the cross section. However, since the element,
A
, is infinitesimally small, we may
regard the shear stress,
τ
, as constant over the element. The shear force,
δ
δ
S
, on the
element is then
δ
S
=
τ
δ
A
(15.13)
Suppose that the structure is given an arbitrary virtual displacement which produces a
virtual shear strain,
γ
v
, at the element. This shear strain represents the angular rotation
in a vertical plane of the element
x
relative to the longitudinal centroidal axis
of the member. The vertical displacement at the section being considered is therefore
γ
v
δ
δ
A
× δ
x
. The internal virtual work,
δ
w
i,
S
, done by the shear force,
S
, on the elemental
length of the member is given by
δ
w
i,
S
=
τ
d
A
γ
v
δ
x
A
