Shear Stresses Mohr Circle : 642865

Question:

For the elements in plane stress presented in the following pages:

  1. Identify Normal and Shear Stresses (5 points)
  2. Determine stresses acting on an element oriented at an angle πœƒ [Using Equations]

(15 points)Β 

  1. Show this stresses on a sketch of an element oriented at this angle (10 points)
  2. Determine the Principle stresses and Principle plane [Using Equations] (15 points)
  3. Determine average normal stress, maximum in-plane shear stress and maximum shear stress plane [Using Equations] (15 points)
  4. Draw the Mohr Circle diagram with the current state of stress in the Graph paper to scale (20 points)
  5. Identify the state of the stress for the element oriented at an angle πœƒ(5 points)
  6. Determine Principle stresses, Maximum in-plane Shear Stress, Principle plane and Maximum shear stress plane from the diagram [show working] (15 points)

Answer:

The given element is shown above:

  1. Identify Normal and Shear Stresses (5 points)

The normal stress are:

= 350 MPa
= 112 MPa

And, the shear stress is:

= -120 MPa

  1. Determine stresses acting on an element oriented at an angle πœƒ [Using Equations]

(15 points)

We know that,

= 350 MPa
= 112 MPa
= -120 MPa
= 15 clockwise = -15

=

=

= + (-120) (-0.5)

= – 60 = 274.09 MPa

=

=

= -59.5 -103.92 = -163.4 Mpa

=

= 350 + 112 – 274.09

= 187.7 Mpa

  1. Show this stresses on a sketch of an element oriented at this angle (10 points)

1

 

  1. Determine the Principle stresses and Principle plane [Using Equations] (15 points)

= 350 MPa
= 112 MPa
= -120 MPa

Principal Angles:

= = -22.61

= – 22.61 + 90 = 67.39

= = 231 -83.82 – 85.17 = 62.01 MPa

= = 231 + 83.82 + 85.17 = 399.9 MPa

 

  1. Determine average normal stress, maximum in-plane shear stress and maximum shear stress plane [Using Equations] (15 points)

Average Normal Stress: = = = 231 MPa

Max shearing stress is given by: = Β± = Β± 168.9 MPa

Maximum in-plane shear stress, = + = 231 + 168.9 = 399.9 MPa

 

 

  1. Draw the Mohr Circle diagram with the current state of stress in the Graph paper to scale (20 points)

The Mohr Diagram can be drawn as follows:

Now, substituting the values and drawing on the graph paper to scale, we can have the Mohr circle as drawn below:

  1. Identify the state of the stress for the element oriented at an angle πœƒ(5 points)

πœƒ = -15

2πœƒ = -30

R = = 350/2 = 175 MPa

Point C: = R = 175 MPa

Point D: = R + Rcos|2πœƒ| = 175 + 175 cos30 = 326.55 MPa

= R – Rcos|2πœƒ| = 175 – 175 cos30 = 23.44 MPa

-R sin (2πœƒ) = 87.5 MPa

  1. Determine Principle stresses, Maximum in-plane Shear Stress, Principle plane and Maximum shear stress plane from the diagram [show working] (15 points)

 

Question 2

 

 

 

 

 

 

 

 

The given element is shown above:

  1. Identify Normal and Shear Stresses (5 points)

The normal stress are:

= 30 MPa
= -20 MPa

And, the shear stress is:

= 80 MPa

  1. Determine stresses acting on an element oriented at an angle πœƒ [Using Equations]

(15 points)

We know that,

= 30 MPa
= -20 MPa
= 80 MPa
= 10 clockwise = -10

=

=

= (-0.34)

 

= 5 -27.36 = 1.14 MPa

=

=

= -8.5 + 75.2 = 66.7 Mpa

=

= 30 + (-20) – 1.14

= 8.86 Mpa

  1. Show this stresses on a sketch of an element oriented at this angle (10 points)

 

 

  1. Determine the Principle stresses and Principle plane [Using Equations] (15 points)

= 30 MPa
= -20 MPa
= 80 MPa

Principal Angles:

= = 36.32

= 36.32 + 90 = 126.32

= = 5 -7.45 – 76.35 = -78.8 MPa

= = 5 + 7.45 + 76.35 = 88.8 MPa

 

  1. Determine average normal stress, maximum in-plane shear stress and maximum shear stress plane [Using Equations] (15 points)

Average Normal Stress: = = = 10 MPa

Max shearing stress is given by: = Β± = Β± 167.6 MPa

Maximum in-plane shear stress, = + = 10 + 167.6 = 177.6 MPa

 

 

  1. Draw the Mohr Circle diagram with the current state of stress in the Graph paper to scale (20 points)

The Mohr Diagram can be drawn as follows:

Now, substituting the values and drawing on the graph paper to scale, we can have the Mohr circle as drawn below:

  1. Identify the state of the stress for the element oriented at an angle πœƒ(5 points)

πœƒ = -10

2πœƒ = -20

R = = 30/2 = 15 MPa

Point C: = R = 15 MPa

Point D: = R + Rcos|2πœƒ| = 15 + 15 cos20 = 29.09 MPa

= R – Rcos|2πœƒ| = 15 – 15 cos20 = 0.9 MPa

-R sin (2πœƒ) = -5.13 MPa

  1. Determine Principle stresses, Maximum in-plane Shear Stress, Principle plane and Maximum shear stress plane from the diagram [show working] (15 points)

 

Question 3

2

The given element is shown above:

  1. Identify Normal and Shear Stresses (5 points)

The normal stress are:

= 2100 MPa
= 300 MPa

And, the shear stress is:

= -560 MPa

  1. Determine stresses acting on an element oriented at an angle πœƒ [Using Equations]

(15 points)

We know that,

= 2100 MPa
= 300 MPa
= -560 MPa
= 65 clockwise = -65

=

=

= + 428.9 = 1050.4 MPa

=

=

= 689.4 +359.96= 1049.4 Mpa

=

= 2100 + 300 – 1050.4

= 1349.6 Mpa

  1. Show this stresses on a sketch of an element oriented at this angle (10 points)

 

 

  1. Determine the Principle stresses and Principle plane [Using Equations] (15 points)

= 2100 MPa
= 300 MPa
= -560 MPa

Principal Angles:

= = -15.94

= – 15.94 + 90 = 74.06

= = 1200 -773.06 – 295.09 = 139.8 MPa

= = 1200 + 764.24 + 295.75 = 2259.9 MPa

 

  1. Determine average normal stress, maximum in-plane shear stress and maximum shear stress plane [Using Equations] (15 points)

Average Normal Stress: = = = 1200 MPa

Max shearing stress is given by: = Β± = Β± 2128.19 MPa

Maximum in-plane shear stress, = + = 1200 + 2128.19 = 3328.19 MPa

 

 

  1. Draw the Mohr Circle diagram with the current state of stress in the Graph paper to scale (20 points)

The Mohr Diagram can be drawn as follows:

Now, substituting the values and drawing on the graph paper to scale, we can have the Mohr circle as drawn below:

  1. Identify the state of the stress for the element oriented at an angle πœƒ(5 points)

πœƒ = -65

2πœƒ = -130

R = = 2100/2 = 1050 MPa

Point C: = R = 1050 MPa

Point D: = R + Rcos|2πœƒ| = 1050 + 1050 cos130 = 375.07 MPa

= R – Rcos|2πœƒ| = 1050 – 1050 cos130 = 1724.9 MPa

-R sin (2πœƒ) = -428.9 MPa

  1. Determine Principle stresses, Maximum in-plane Shear Stress, Principle plane and Maximum shear stress plane from the diagram [show working] (15 points)

Question No 4

The given element is shown above:

  1. Identify Normal and Shear Stresses (5 points)

The normal stress are:

= -40 MPa
= 80 MPa

And, the shear stress is:

= 0 MPa

  1. Determine stresses acting on an element oriented at an angle πœƒ [Using Equations]

(15 points)

We know that,

= -40 MPa
= 80 MPa
= 0 MPa
= 55 clockwise = -55

=

=

= = 40.5 MPa

=

=

= 56.38 Mpa

=

= -40 + 80 – 40.5

= -0.5 Mpa

  1. Show this stresses on a sketch of an element oriented at this angle (10 points)

 

  1. Determine the Principle stresses and Principle plane [Using Equations] (15 points)

= -40 MPa
= 80 MPa
= 0 MPa

Principal Angles:

= = 0

= 0 + 90 = 90

= = 20 +60 = 80 MPa

= = 20 -60 = -40 MPa

 

  1. Determine average normal stress, maximum in-plane shear stress and maximum shear stress plane [Using Equations] (15 points)

Average Normal Stress: = = = 20 MPa

Max shearing stress is given by: = Β± = Β± 60 MPa

Maximum in-plane shear stress, = + = 60+20 = 80 MPa

 

 

  1. Draw the Mohr Circle diagram with the current state of stress in the Graph paper to scale (20 points)

The Mohr Diagram can be drawn as follows:

Now, substituting the values and drawing on the graph paper to scale, we can have the Mohr circle as drawn below:

  1. Identify the state of the stress for the element oriented at an angle πœƒ(5 points)

πœƒ = -55

2πœƒ = -110

R = = -40/2 = -20 MPa

Point C: = R = -20 MPa

Point D: = R + Rcos|2πœƒ| = -20 + (-20) cos110 = -13.2 MPa

= R – Rcos|2πœƒ| = -20 – 6.8 = -26.8 MPa

-R sin (2πœƒ) = 18.79 MPa

Determine Principle stresses, Maximum in-plane Shear Stress, Principle plane and Maximum shear stress plane from the diagram [show working] (15 points)

3