Information Technology Reference
In-Depth Information
1.8.2
Material Laws
The constitutive relations for the material of the beam should reflect the assumption that
the extension and contraction of longitudinal fibers are the dominant deformations. This
is equivalent to assuming that the material is rigid in the
z
direction. In terms of fibers
being deformed in the longitudinal
(
x
)
direction, this rigidity means that there will be no
contribution to the longitudinal strain
x
by stresses in the
z
direction. Thus,
x
of Eq. (1.32)
reduces to
x
=
(σ
x
−
νσ
y
)/
E
. Since the loading is in the
xz
plane, it is reasonable to assume
that
σ
y
=
0. Thus,
x
=
σ
x
/
E
or
σ
x
=
E
x
. Recall from Eq. (1.98) that for the beam
u
=
u
0
+
z
θ
,
so that
θ
dx
=
du
dx
=
du
0
dx
+
z
d
du
0
dx
+
=
κ
z
(1.105)
x
The stress distribution over the cross-section gives rise to the stress resultants or the net
internal forces:
The axial force
N
E
du
0
dA
EA
du
0
N
=
A
σ
x
dA
=
E
x
dA
=
dx
+
z
κ
=
dx
=
EA
(1.106a)
0
x
A
A
and the bending moment
M
E
du
0
zdA
E
z
2
M
=
A
σ
x
zdA
=
E
x
zdA
=
dx
+
z
κ
=
κ
dA
=
κ
EI
(1.106b)
A
A
A
where
I
is the moment of inertia about the
y
axis. The integral involving
z
, i.e.,
zdA
,is
zero if
z
is measured from a centroidal axis of the beam. In matrix notation, Eqs. (1.106a)
and (1.106b) appear as
N
M
EA
0
0
x
κ
=
(1.107)
0
EI
s
=
E
or
1
s
/
EA
0
01
=
(1.108)
/
EI
E
−
1
=
s
where
s
is chosen to be equal to [
NM
]
T
.
If shear deformation effects are to be taken into account, the constitutive equation relating
the shear strain and the net internal shear force is needed. Hooke's law for shear is
τ
xz
=
τ
=
=
τ
average
A
as a stress-force relationship where
V
is the shear force
and
A
is the cross-sectional area. If
G
γ
. Choose
V
τ
is the shear stress at the centroid of the cross-section,
then select
,
where
k
s
is a dimensionless
shear form
or
shear stiffness factor
that
depends on the cross-sectional shape. Later we will discuss techniques for finding
k
s
. The
reciprocal of
k
s
is the
shear correction factor
that is often tabulated in handbooks of formulas
for stress analysis. The desired material law relationship becomes
τ
average
=
k
s
τ
=
γ
V
k
s
GA
(1.109)
Search WWH ::
Custom Search