Chapter 5
Two-Dimensional Problems in Cartesian
Co-ordinate System
5.1 Introduction
5.1.1 Plane Stress Problems
In many instances, the stress situation is simpler than that shown in Fig. 2.6. An
example of practical interest is that of a thin plate which is being pulled by forces in
the plane of the plate. Figure 5.1 shows a plate of constant thickness, t subjected to
axial and shear stresses in the x-directions and y-directions only. These stresses are
assumed to be uniformly distributed over the thickness t. The surface normal to the
z-axis is stress free.
The state of stress at a given point will depend only on the three stress components
such as
σ x , σ y and τ xy = τ yx
(5.1)
in which the stress components are functions of only x and y. This combination of
stress components is called “plane stress” in the x y plane. The stress–strain relations
for plane stress are given by
ε x =
1
E
σ x − νσ y
ε y =
1
E
σ y − νσ x
γ xy =
τ xy
G
(5.2)
and
γ xz = γ yz = 0, ε z = −
v
E
σ x + σ y
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
T. G. Sitharam and L. Govindaraju, Theory of Elasticity,
https://doi.org/10.1007/978-981-33-4650-5_5
125