52
2 Analysis of Stress
8. The components of stress at a point are
σ x = 2 MPa σ y = 1.5 MPa τ xy = τ yz = 1 MPa τ zx = −1 MPa
Determine the normal and shearing stresses on the octahedral plane and the
direction of the shearing stress.
9. The state of stress at a point is given by the following array of terms in the xyz
co-ordinates system
τ i j =
⎡
⎣
10 15 20
15 25 15
20 15 30
⎤
⎦ MPa
If this system of axes is rotated by 30° about the z-axis in the anticlockwise
direction, determine the new stress tensor.
10. The stress components at a point are
σ x = −5 MPa σ y = 3 MPa σ z = 2 MPa
τ xy = −6 MPa τ yz = 4 MPa τ xz = 5 MPa
Determine the principal stresses and principal directions.
11. A metal bar is having cross Sect. 2.40 mm × 30 mm. It is subjected to an axial
tensile load of 240 KN. Calculate the normal, shear and resultant stresses on a
plane whose normal has the following direction cosines.
(i) l = m = n =
1
√
3
(ii) l = m =
1
√
2
and n = 0
12. The principal stresses at a point in a material are
σ 1 = 300 N/mm
2
σ 2 = 300 N/mm
2 and σ 3 = 400 N/mm
2
The direction cosines of the principal planes are l = m = n = 0.577. Determine
the resultant stress and the invariants.
13. The principal stresses at a point in a three-dimensional stress system are defined
as
σ 1 = 400 N/mm
2
σ 2 = 200 N/mm
2 and σ 3 = 100 N/mm
2
.
Find the normal and shearing stresses on the octahedral planes using Mohr’s
circle and check the values by theoretical equations.