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5 Two-Dimensional Problems in Cartesian Co-ordinate System
Fig. 5.1 General case of
plane stress
The constitutive relation for plane stress problems is given by
⎧
⎨
⎩
σ x
σ y
τ xy
⎫
⎬
⎭
=
E
1 − ν 2
⎡
⎣
1 ν 0
ν 1 0
0 0
1−ν
2
⎤
⎦
⎧
⎨
⎩
ε x
ε y
γ xy
⎫
⎬
⎭
(5.3)
5.1.2 Plane Strain Problems
Problems involving long bodies whose geometry and loading do not vary significantly
in the longitudinal direction are referred to as plane strain problems. Some examples
of practical importance, shown in Fig. 5.2, are a loaded semi-infinite half-space such
as a strip footing on a soil mass, a long cylinder; a tunnel; culvert; a laterally loaded
retaining wall; and a long earth dam. In these problems, the dependent variables can
be assumed to be functions of only the x and y-co-ordinates, provided a cross-section
is considered some distance away from the ends.
Hence the strain components will be
ε x =
∂u
∂ x
, ε y =
∂v
∂ y
, γ xy =
∂u
∂ y
+
∂v
∂ x
(5.4)
ε z =
∂w
∂z
= 0, γ xz =
∂w
∂ x
+
∂u
∂z
= 0, γ yz =
∂w
∂ y
+
∂v
∂z
= 0
(5.5)
Moreover, from the vanishing of ε z , the stress σ z can be expressed in terms of σ x
and σ y as
σ z = ν
σ x + σ y
(5.6)
The constitutive law for elastic, isotropic material for plane strain problems is
given by