Drawing Mohr Circle by Pole Method if Inclination Agle Given
Introduction
German physicist Otto Mohr adult a method to graphically interpret the general state of stress at a point. This method is named every bit Mohr'southward circumvolve and can be used to evaluate principal stress, maximum shear stresses and normal and tangential stresses at whatever plane.
Let u.s.a. take upwardly a simple second stress element to sympathise most the concept of stress transformation and how Mohr's circle can exist used to simplify the procedure.
Stress Chemical element
A stress element is a useful way to describe the stresses acting at a single point within the body. Let united states consider a simple example of the beam with centric load. now the stress state in the stress element will exist very simple. Information technology will merely have normal stress along 10-axis and zip normal stress along Y-centrality and cipher shear stress. If we rotate the stress element to get the stress land for a specific angle. Depending on how nosotros orient the stress element we will get unlike values for normal and shear stress components.
The normal and shear stress for the stress element oriented in a specific angle (or in other words an inclined plane) by using the Stress Transformation Equations.
Stress Transformation Equation
The following are the stress transformation equations,
The inputs for these equations are normal stresses σ10,σy and shear stress τxy at the starting element orientation and θ is the angle the stress element to be rotated or the inclination of a plane at which nosotros demand normal and shear stresses.
As the stress chemical element rotates, the normal and shear stress volition vary and this variation can be easily understood with an example.
From this instance, information technology tin be observed that once the element rotates 180 degree, the values of normal and shear stress comes to the original position. Think, the actual state of stress practical to the torso is not changing when the chemical element is rotated.
Observations from the graph
- At certain angles, the normal stress curve reaches a maximum and minimum values and these values are separated by an bending of xc degrees.
- This means that at a particular angle of rotation when we get maximum normal stress at one face (say X confront) then the minimum stress will exist perpendicular to that face (at Y face).
- Whenever the normal stress curve reaches maximum or minimum value, the shear stress at that angle volition exist zero.
We know that the plane at which the shear stresses are zero, that aeroplane is known every bit Primary Airplane and the corresponding normal is Chief Stresses (σ1, σii). The rotation angle of the principal airplane is denoted every bit θp.
Mohr'south Circle
It is a graphical method for easily determining normal and shear stresses "WITHOUT" using the stress transformation equations.
Mohr's circle is drawn on a 2D graph where normal stress is taken on the horizontal centrality and shear stress is taken on the vertical axis.
Sign Convention
If the shear stresses tend to rotate in anti-clockwise direction are taken every bit positive and shear stresses tends to rotate in clockwise direction are negative. That'south why the shear stresses in Y-face are taken every bit negative. So, positive shear stresses are plotted on downward direction and negative shear stresses on upwardly direction.
Normal stresses tensile in nature are positive and compressive in nature are negative.
Steps to Draw Mohr's Circumvolve
- Prepare the coordinate centrality (σ – X-axis, τ – Y-axis)
- Plot the points A (σten,τxy) and B (σy,-τxy).
- With A and B every bit bore and C every bit the heart, draw a circle.
Note that at point A, θ=0 caste and at B, θ=90 degree.
i) Double Bending Method
Stress Transformation using Mohr's Circumvolve
Let us presume that we need normal and shear stress on a airplane inclined at an angle θ. In Mohr's circle both σten and σy are marked in a same axis (i.e., 180 caste) which is non a existent instance as σx and σy are perpendicular to each other (i.e., 90 degrees).
Therefore, with CA equally reference locate the bespeak D on the circumvolve at an angle 2θ. Locate another point E diametrically opposite to D. The X coordinates of D and Due east represents the normal stresses on x and y faces respectively and the Y coordinate represents the shear stress on the new inclined airplane.
Principal Stresses
Principal stresses human activity on the principal surface/plane at which the shear stress is nil. The horizontal centrality of the Mohr'southward circle the value of shear stress will ever exist zero. Hence, the points at which the horizontal axis cuts the circle gives the Maximum and Minimum master stresses. The angle between the CA and horizontal centrality is 2θp from which the inclination of master plane (θp) can be calculated.
Maximum Shear Stress
Visually it tin can be observed that the extreme points of the circumvolve in vertical axis gives the maximum shear stress. Geometrically, maximum shear stress is zero merely the Radius of the Mohr'south circle.
The Angle between CA to the vertical is 2θs and θs is the inclination of a plane at which the maximum shear stress volition occur.
ii) Pole Method:
What is a pole?
Pole method is based on a unique point on the Mohr's circumvolve known as the Pole. This pole is unique, considering whatever strainght line drawn through the pole intersects the Mohr'southward circumvolve at a indicate representing the land of stress on a plane with the same orientation equally the line. Equally shown in the figure, a line drawn through the pole and some stress point (σ,τ) on the circle is exactly parallel to the plane with respective σ and τ, making this approach very intuitive.
How to locate Pole?
Let u.s. locate the pole in Mohr'southward circumvolve with an example. The figure shows the stress state on an element and its Mohr's circle.
To locate the pole, From point A draw a line parallel to Ten face (dark-green) and from B draw a line parallel to Y face (red). These two points will intersect at a indicate on the circle and this signal is called Pole.
Stress Transformation and Principal Stresses:
For example, let us calculate normal and shear stresses for an element rotated for an angle thirty deg in anti-clockwise management. To locate the stress country for the inclines airplane, draw two lines from pole with an angle 30 deg anti-clockwise from greenish and red lines. These lines volition be parallel to the 10 and Y faces of the rotated element respectively. The coordinates of the points at which these lines intersect the circle will requite the stress land of that plane.
Similarly, the angle of Principal plane tin can be calculated by connecting the pole with the σ1 and σii.
The following effigy shows how pole can exist used to summate normal and shear stress for unlike orientation of stress element.
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Source: https://thestructuralblog.com/stress-transformation-and-mohrs-circle/
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