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Stresses, strains and modulus Principal stresses and strains Shear force and bending moment diagrams Bending stresses in beams

Shear stresses in beams Slope and deflection in beams Torsional stresses in shafts Buckling stresses in columns Stresses and strain in thin shells

Principal stresses: At each point say at x, there exist

three mutually perpendicular directions along which

THERE ARE NO SHEAR STRESSES. These directions

are called the principal stress directions, and the

corresponding normal stresses are called the

principal stresses.

Principal Plane: The area on which principal stress acts

is a principal plane. There at the three mutually

perpendicular planes. There will always be planes on

which the shear stress components are zero. The

normal stresses acting on these planes are called the

principal stresses. Planes on which the principal stresses act are called the principal planes.

Compound Stress/complex stress system: A body is under a compound stress if normal and shear stresses

simultaneously act on an area (plane).

Plane: a two dimensional area may be a rectangle, square, circle, triangle, ellipse or T-section.

Plane State of Stress: 2-Dimensional stress system: stresses are in the x-y plane only and the stresses in z plane are

zero.

Refer figure below: Given two dimensional stress system: Material subjected to combined direct and shear

stresses:

A tensile stress (+ ve) xin the x direction (assumed)

A tensile stress (+ve) yin the y direction, (assumed)

A counterclockwise (-ve) shear stressxy

on the x surface (BC) and a balancing clockwise shear stress (+ve) shear

stressxyon the y surface (AB) (assumed)

NOTE: put the actual nature of stresses in the final equations developed

As per the double subscript notation the shear stress on the face BC should be notified as xy.

First suffix represents direction of axis normal to the area being considered.

Principal stresses

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Page 1 of 9Principal stresses, Mohr circle

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Second suffix representation of the direction of the stress being considered. Complementary shear stresses

generated are equal such that yx = xy.

Given complex stresses x, y and xy in a stressed element. Now find the principal stresses, principal planes,

maximum shear stress and planes of maximum shear stress. These are determined because irrespective of any

number of forces in any combination may be acting on a body it will fail either by tension alone or by compression

alone or by shear alone.

There are two methods to determine principal stresses, principal planes, maximum shear stress and theplanes of maximum shear i.e. 1, 2,p p1

,p p2

, max

and ms1, ms2

(i) Analytical method (ii) Mohrs graphical method

ANALYTICAL METHOD:

Normal and shear stresses on any Plane CP inclined at an angle to the REFERENCE PLANE BC containing x

and xy

They do not depend on material proportions and are equally valid for elastic and inelastic behavior. This equation

involves 2whereas is given for the element. Square the above equations and add these to get the equation of a

circle. Therefore circle involves 2and the element involves i.e. on element is2on Mohrs Circle.

If we put shear stress

equal to zero, we get the same equation as above. Therefore maximum or minimum normal

stresses are also principal stresses. It is to remind that principal stresses are normal stresses with zero shear

stress. Standard symbols for principal stresses are 1,2, and 3.

Draw the triangle and find

Substituting the values of cos2and sin2 in equation of mentioned below

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This shows that the values of shear stresses are zero on the pr incipal planes containing principal stresses.

The angle now becomes p for the principal plane

Equation is now written as

The above condition gives two values of 2p that differ by 1800 because tan2p is positive in the first and third

quadrants and is negative in second and fourth quadrants. Hence the planes on which principal stresses occur are at

900 apart on the element since the inclination considered was only on element and not 2.Thus the two principal

stresses occur on mutually perpendicular planes. Principal planes and principal stresses are shown in the figure

below. Reference vertical plane BC of x

is also highlighted in the figure.

Maximum shear stresses and the planes of maximum shear stress




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tan2p.tan2ms = -1

This shows that the product of two slopes is 1. Therefore 2p and 2ms planes are at 900. Therefore angle between

p and ms is 450 i.e. principal plane is inclined to the plane of maximum shear at 450.

In order to find the expression for maximum shear stress, draw the triangle and we get

The above value is maximum shear stress.

GRAPHICAL METHOD: MOHRS STRESS CIRCLE

The transformation equations for plane stress can be represented in a graphical format known as Mohrs circle. If we

square the above equations and add these we get the equation of a circle. Therefore circle involves 2 and the eleme

involves i.e. on element is2 on Mohrs Circle.




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Given the stresses x,

y, and

yx, then calculate the principal stresses

1,

2, the principal plane

p, the maximum

shear stressmax

and its plane of maximum shear ms

using Mohr's circle.

Construction of Mohrs circle

1. Identify the stressesx, y, and xyand list them with the proper sign i.e.

xyis -.

2. Draw a set of - . Co-ordinate axes with along x-axis and along y-axis being positive to the right and

being positive in the upward direction. Choose an appropriate same scale for each axis.

3. Sign conventions for drawing of Mohrs Circle

Normal Stresses along x-axis (horizontal axis): tensile (+) along right of pole 0

: Compressive (-) along left of pole

Shear Stresses along y-axis (Vertical Axis) : + For Clockwise Shear

: - For Counterclockwise Shear

4. X axis is the axis i.e. x, y,z1, 2, 3, i.e. all sigma values are along x-axis.

5. Y axis is theaxis i.e. xy, yz, zx, ,andmaxi.e. all shear stresses are along y axis.

6. Plot the given stresses (x and xy) on the x face (vertical plane) of the element and name this point V(x, -xy) .

It will lie on the circle.

7. Plot the stresses (y and + yx) on the y face (horizontal plane) of the element and name it point H (x,yx). It will

also lie on Mohrs Circle.

8. Draw a line between the two points V and H. The point where this line crosses the axis establishes the center of

the circle as point P (x+ y)/2.

9. Draw the complete circle with centre as point P and radius as PV or PH

10. Radial line PV represents the plane containing stresses (x and xy).

11. Radial line PH represents the plane containing stresses (y and + yx)

12. The line PV from the center of the circle to point V identifies reference axis or reference plane BC of stressed

element for angle measurements(i.e. = 0).

13. Remember every radial line is a plane in the stressed body.

14. All the stresses are measured from the pole (ORIGIN).

15. on stressed body becomes 2 on Mohrs circle and vice versa. Refer transformation equations. Any value of

angle read from Mohrs circle becomes half with respect to the stressed element.

16. All the data is given as per stressed element and all the answers are to be given with respect to the stressed

element VIA Mohrs circle.




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17. Measure two stresses from the horizontal diameter of Mohrs circle giving the principal stresses 1 =OMand

2=OL.

18. Measure angle between PV and PM i.e. 2p1.Measure angle PVML=180+2p1= 2p2

p2

= p1

+900i.e. principal planes are at right angles to each other.

19. Measure PL =max = the radius of the circle.

20. Measure angle PVMJ1

= 2ms1.

21. 2ms2=2ms1 + 1800 OR ms2=ms1 + 90

0 therefore planesof max shear are at right angles to each other.

22. Angle between PM and P is 900 on Mohrs circle therefore angle between a principal plane and a plane of

maximum shear is 450.

23. PV= plane of x

and xy, PL=PM= principal plane, CJ1=CJ2=Plane of maximum shear

24. 1,2 =avg R= normal stress along with max maximum shear stress =OPPM

Here we can define the average stress, avg (normal stress along with max)

and a "radius" R(which is just equal to the maximum shear stress).

Additional sketch of Mohrs Circle for better understanding

Additional sketch of Mohrs Circle for better understanding showing the magnitudes of principal stresses and the

maximum shear stress (all measured from the pole)

Note: The angle between the reference axis and the axis is equal to 2p say= 340 then answer becomes 170 with

respect to the element.

Important Observations:

1. Principal stresses occur on mutually perpendicular planes.

2. Principal planes are at right angles to each other and carry no shear stress.




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3. Planes of maximum shear stress are at right angles to each other.

4. Shear stresses are zero on pr incipal planes.

5. The angle between a plane of maximum shear and the principal plane is 450.

6. The maximum shear stress is equal to one half the differences of the principal stresses.

7. All angular measurements are made from the reference line PV.

8. All stress measurements are made from the pole (ORIGIN).

9. Every radius of Mohrs Circle represents a plane of the stressed body.

10. on element is 2 on Mohrs circle and vice-versa.

Mohr's circles representing different stress regimes are shown below:

(case of like equal stresses)

PROBLEM:

The state of stress in the wall of a cylinder is expressed as follows:

(a) 85 MN/m2 tensile,

(b) 25 MN/m2 tensile at right angles to 85 MN/m2

(c) Shear stresses of 60 MN/m2 on the planes on which the stresses (a) and (b) act;

(d) The shear couple acting on planes carrying the 25 MN/m2 stress is clockwise.

Calculate the principal stresses and the principal planes, maximum shear stress and the planes of maximum shear.

Solution: The problem is being solved both analytically as well as graphically.

Analytical solution

The principal stresses are given by the formula




8/9

For finding out the planes on which the principal stresses act, use the equation

The solution of this equation will yield two values of p i.e. they p1 and p2 giving p1= 31071' & p2= 121

071'

anticlockwise from PV.

max

=( 122 - (-12))/2 =67MN/m2

tan2ms= (x- y)/2xy

ms =p 450

ms1 = 76071' anticlockwise from PV

ms2= 166

071' anticlockwise from PV

Maximum shear stress occurs on planes oriented 450 to the principal planes.

Thus, the two principle stresses acting on the two mutually perpendicular planes along with the reference plane BC is

shown below:

In the second case of loading we are measure the angles (negative) in the opposite direction to the reference plane

BC.

The stressed body containing the principal planes, principal stresses and reference plane is shown below.




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GRAPHICAL SOLUTION:

Mohr's Circle solution: The same solution can be obtained using the graphical Mohr's stress circle, for the first

loading, the block diagram becomes

Construct the graphical construction as per the steps given earlier.

Taking the measurements from thepole 0 of Mohr's stress circle, the various quantities are

1

= 120 MN/m2 tensile , 2

= 10 MN/m2 compressive

p1 = 340 counter clockwise from BC , p2 = 34

0 + 90 = 1240 counter clockwise from BC

max= 65MN/m

2

ms1

= 110 CW from BC

ms2=1010 CW from BC

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