Calculation of Principal stress is necessary to know about the maximum normal stress an object can withstand at a given condition.
To design any structure a designer always have a knowledge of the Principal Stresses associated with the principal planes. Theories of failure are also used to know the limit of the structure up to which it can withstand the stresses.
The planes which are considered within a material in such a way that the resultant stresses are acting normally on them and no any shearing stresses occur are known as Principal planes and the stresses acting on Principal plane are called Principal stresses.
How to Calculate Principal Stress from Principal Strain?
To calculate the level of stress on a structure, strain is measured.
The stress(in magnitude and direction) acting on a body can be determined by using the measured strain and certain properties of material like Modulus of Elasticity and Poisson’s Ratio.
In comparison to measure the normal strains on the surface of a body, it is quite difficult to measure normal and shear stresses acting on a body mostly at a point.
If we can measure the normal strains at a point, it is possible to find out the magnitude of principal strains as well as their directions. Now Principal stresses acting on a material which is obeying Hooke’s law can be determined from the measured principal strains.
The methods applied for strain measurement are as follows:
- Direct: Electrical type gauges are used which follow resistive, capacitive, inductive or photoelectric principles.
- Indirect: Optical methods are used for example holographic interferometry, photoelasticity etc.
A strain gauge is capable only of measuring strain in the direction in which gauge is oriented. 2. There is no direct way to measure the shear strain or to directly measure the principal strains as directions of principal planes are not generally known.
The main drawback of using a strain gauge is it can measure the strain only in the direction of its orientation. Since the direction of the principal planes are generally unknown, we cannot get a direct way to measure shear strain or principal strains.
To measure strain in three direction which is necessary for strain analysis in biaxial state we prefer strain rosettes where strain gauges are arranged in three directions. Different types of rosettes are available depending upon the arrangement of gauges.
A rectangular strain rosette consists of three strain gauges arranged as follows:-
If in stress measurement the directions of principal stress are unknown, a triaxial rosette gage is used to find out the strain values and using the following equations Principal stress can determined.
Consider ∈a→∈b→∈c as the forward direction, angleθ is angle of maximum principal strain to the ∈a axis when ∈c>∈b angle of minimum principal strain to the ∈a axis when ∈a<∈c. Comparison between ∈a and ∈c in magnitude includes plus and minus signs
Maximum principal strain,
Minimum principal strain
Direction of principal strain （from εa axis)
Maximum shearing strain
Now Minimum Principal Stress,
Maximum shearing stress
v=Poisson’s ratio E:Young’s modulus
How to Calculate Principal Stresses from Stress Tensor?
If we consider a point as an infinitely small cube. Each face of the cube is represented by three separate stress vectors which are nothing but the stresses acted on each face.
In this way the total number of stresses act on the whole cube can be expressed by nine stress vectors inside a matrix . This stress matrix with nine stress vectors is known as the Stress Tensor.
If the cube is in equilibrium, then it follows that
σ12 = σ21
σ13 = σ31
σ32 = σ 23
Six independent components are available in the stress tensor and it is symmetric in nature. If the cube is slanted keeping in mind that the major stress becomes normal to one of the planes and also no shear stresses are acting, in that condition the stress tensor can be represented as follows:
Here are known as Principal Stresses.
The mean stress is simply the average of three stresses.
σm = (σ1+σ2+σ3)/3
How to Calculate 1st Principal Stress?
Using 1st Principal stress we can get the value of stress that is normal to a plane where the value of shear stress is zero.
The idea of the maximum tensile stress experienced by a part under a loading condition can get from 1st Principal Stress.
The normal and shear stress acting on the right face of the plane make up one point, and the normal and shear stress on the top face of the plane make up the second point.
The largest value of of sigma is the first principal stress, and the smallest value of sigma is the second principal stress.
How to calculate Principal Stress in 3D?
In case of 3D , the x,y and z are the orthogonal directions and we can consider one normal stress and two shear stresses are there. We can define the stress state at a point in 3D as shown below:
We can also represent it with the help of a stress tensor
If we go for a definite orientation of xyz axis, denoted by the directions 1,2 and 3, then only normal stresses will act and shear stresses will disappear.
These normal stresses are called Principal Stresses S1,S2 and S3.
The values of the three principal normal stresses (S1, S2 & S3) can be found from the three real roots of S of the following cubic equation:
S1,S2 and S3 can be found from the three real roots of S
The values of S1, S2 & S3 should include the maximum and minimum normal stresses and S1, S2 & S3 could be positive, zero or negative.
How to Calculate Minor Principal Stresses?
The normal stresses acting on a principal plane is known as principal stresses: Major Principal Stress and Minor Principal Stress.
Minor principal stress gives the minimum value of principal stress and major principal stress gives the maximum value of the principal stress. Using Mohr’s circle method we can calculate the major and minor principal stresses.
Considering a stress system where figure ABCD represent a small element of a material.
Here σx, σy = Normal stresses (may be tensile or compressive)due to direct force or bending moment.
are complementary and
Assume that σn is the normal stress and τ is the shear stress on a plane at an angle T.
At an equilibrium condition,
Normal stress, σn = (σx + σy)/2 + (σx – σy)/2 cos2θ + τxysin2θ
And Shear stress, τ = (σx – σy)/2 sin2θ – τxycos2θ
Above equations are the transformation equations and they don’t depend on material properties and valid for both elastic and inelastic behaviour.
If we locate the position of principal planes, where shear stress is zero, we have
δσn/δθ = 0
Here σn = (σx + σy)/2 + (σx – σy)2cos2θ +τxySin2θ
tan2θp = 2τxy(σx – σy)
θp is the angle of Principal Plane
Now Principal Stresses are
Major Principal Stress,
Minor Principal Stress,