2.8 Two-Dimensional Stress at a Point
13
Fig. 2.6 Thin body
subjected to stresses in
xy plane
Fig. 2.7 Stress components
acting on faces of a small
wedge cut from body of
Fig. 2.6
Considering σ x and τ x y as positive and area of side MN as unity, the sides MP
and PN have areas cos θ and sin θ, respectively.
Equilibrium of the forces in the x- and y-directions requires that
T x = σ x cos θ + τ xy sin θ
T y = τ xx cos θ + σ y sin θ
(2.10)
where T x and T y are the components of stress resultant acting on MN in the x- and
y-directions, respectively. The normal and shear stresses on the x
plane (MN plane)
are obtained by projecting T x and T y in the x
- and y
-directions.
σ x = T x cos θ + T y sin θ
τ x y = T y cos θ − T x sin θ
(2.11)