Number of forces are acting simultaneously almost on each body. Thus, it is complex loading condition. When these forces increase, a body fails by a simple principal stress or by maximum shear stress.
COMPLEX STRESS SYSTEM
When normal and shear stresses act simultaneously on an area, it is a complex stress system. Complex stress is there when an inclined force acts on an area. Resolve this inclined force into normal and tangential components. Normal component causes normal stress and tangential component causes the shear stress. Hence complex stress is there.
Practical Applications of Complex stresses
(iv) Chimneys etc.
FOR A PRINCIPAL STRESS
Only normal stress (with zero shear stress on an area) is principal stress.
There are at the most three principal stresses.
These act at right angles to each other.
σ1 , σ2 , and σ3 are principal stresses.
The units are N/mm2.
Principal stresses act on principal planes.
It is two dimensional.
Area is 2-Dimensional.
Rectangle, square, circle, triangle, ellipse or T-section are planes.
The area on which there is only normal stress with zero shear stress. P
θpp represents a principal plane.
DOUBLE SUBSCRIPT NOTATION ON STRESSES
First suffix represents direction of axis normal to the area considered.
Second suffix represents direction of axis inthedirection of the stress considered.
The normal stress on vertical plane BC will be σxx. The shear stress on the vertical plane BC will be zxy.
There is a double suffix notation for 3-D stress system.
Use single suffix for sigma stress
Use no suffix for shear stresses in 2-D stress system
Firstly Write σxx as σx
Secondly write σyy as σy
Thirdly write zyx and zxy as z
since zyx. = zxy (numerically equal)
These are at right angles to each other.
These are complementary of each other.
Thus one finds the followings from the given complex stress system of σx, σy and z
(i) Principal planes
(ii) Angle between principal planes
(iii) Principal stresses
(iv) Planes of maximum shear stress
(v) Angle between planes of maximum shear
(vi) Maximum shear stress
(vii) Angle between a principal plane and a plane of maximum shear
Find Principal stresses. When a body under number of forces fail in a simple manner. The failure is because of any one reason given below:
(a) by tension alone
(b) by compression alone
( c) by shear alone
(d) By maximum principal strain
(e) maximum principal strain energy
(f) maximum shear strain energy
PRINCIPAL STRESSES-ANALYTICAL METHOD
In this, it is required to determine
(i) principal planes
(ii) angle between principal planes
(iii) principal stresses
(iv) planes of maximum shear
(v) angle between planes of maximum shear
(vi) maximum shear stress
(vii) angle between a principal plane and a plane of maximum shear.
There are two methods to determine all the above quantities.
(i) Analytical Method
(ii) Graphical Method or Mohr’s Stress Circle Method
Fig. Complex, Principal stresses, Principal Planes,
maximum shear stress & Planes of Maximum Shear Stress
(i) Given a vertical plane AB.
(ii) Given complex stresses on this vertical plane
(a) Given a tensile stress (+ ve) σx in the x direction on the given vertical plane. (assumed)
(b) Given a counterclockwise (+ve) shear stress τ on the given vertical plane. (assumed)
(iii) Given a horizontal plane BC.
Given stresses on this horizontal plane.
(a) Given a tensile stress (+ve) σy in the y direction on the given horizontal plane. (assumed)
(b) Given a clockwise shear stress (-ve) τ on the given horizontal plane. (assumed)
USE ACTUAL NATURE OF STRESSES IN THE NUMERICAL PROBLEMS.
STRESSES ON ANY INCLINED PLANE WITHIN THE BODY UNDER COMPLEX STRESSES
Inclined plane AD is at an angle ‘θ’ with the given vertical plane AB. Stresses on this inclined plane (IP) will also be complex.