Question

please show clear explanation of process thank you

4) Determine (a) the principal stresses and (b) the maximum in-plane shear stress and average normal stress. Specify the orientation of the element in each case. Include a diagram of Mohr's circle indicating the current state of stress and locations of principal and max shear stresses. 12 ksi 20 ksi 10 ksi

Answer 1

Mohr's circle method

1. Plot the co-ordiate axes $\sigma$ and $\tau$
2. Calculate the radius of circle
   + ਵ ਸ = =
   Here
   Orr = -20ksi, Oyy = 12ksi, Try = -10ksi
   substituting we get
   12)2 + -102 = 18.87 R= V2
   
3. Centre of Circle at (C,0)
   C = Per +Oyy _ -20+ 12
   Centre at (-4,0)
4. Plot vertical and horizontal face cordinates
   (Orr, Try), (yy: -Try)
   ie; (-20,-10) (12,10)   

  

(-20,-10) op2=-22.87 a (-4 op1=14.87 o → (12,10) I max =18.87

Note;

(To get the angle of rotation from mohrs circle, Half the angle got from the circle)

a)

From figure angle rotation to Principle Stress is angle a

$a=tan^{-1}(\frac{10}{16})=32^0$

Orientation of Principal stress

Angle rotation in clockwise =
a/2 = 32/2 = 16°
(half the angle)

Principal stresses

Opi = 14.87ksi

0p2 = -22.87ksi

Figure

22.87 ksi 7 14.87 14.87 ksi 16

b)

Orientation of Principal shear stress

From mohrs circle rotation to maximum shear stress is =

089 = TE-06 = -06

So orientation is

Anti clockwise (half the angle)

580/2 = 29

Principal shear stresse

Teymar = 18.87ksi

Figure

4 ksi 4 ksi 18.87 ksi

Average stress =

+ Oy = -4ksi