PROBLEMS in COMBINED STRESS

1. Derive for a thin cylindrical pressure vessel expressions symbolically for determining the safe values of:

a) minimum thickness t given p, F, T and Db) maximum internal pressure p given t, F, T and Dc) maximum axial load F given t, p, T and Dd) maximum twisting moment T given t, p, F and De) maximum diameter D given t, F, T and D (this may end up in a

cubical equation in D2 to solve. One of the 3 values will be the right one)

2. Using data given in Example 4, find numerical solutions for problems 1 (b) to (e), for t = 4 mm.

3. In Example 1, if the shaft is hollow with outer dia 100 mm & inner dia 50 mm, solve it.

4. A solid circular shaft of diameter d is subjected to a bending moment of 15000 N-m, a twisting moment of 25000 N-m and an axial thrust (compressive) load of 500000 N. Find the minimum safe diameter d, for each of the five theories of failure, if the factor of safety required is 1.25, the yield strength of the material is 500 MPa (equal for tension and compression) and the Poisson’s ratio is 0.25.

5. A circular hollow shaft of outer diameter 100 mm and inner diameter 40 mm is subjected to a bending moment of 15000 N-m, a twisting moment Mt N-m and an axial tensile load of 500000 N. Find the maximum safe value of twisting moment Mt, for each of the five theories of failure, if the factor of safety required is 1.25, the yield strength of the material is 500 MPa and the Poisson’s ratio is 0.25.

6. A solid circular shaft of diameter 100 mm is subjected to a bending moment M N-m, a twisting moment of 25000 N-m and an axial tensile load of 500000 N. Find the maximum safe value of bending moment M, for each of the five theories of failure, if the factor of safety required is 1.2, the yield strength of the material is 500 MPa and the Poisson’s ratio is 0.25.

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7. A circular hollow shaft of outer diameter 100 mm and inner diameter 60 mm, is subjected to a bending moment of 15000 N-m, a twisting moment of 25000 N-m and an axial tensile load P N. Find the maximum safe value of tensile load P for each of the five theories of failure, if the factor of safety required is 1.25, the yield strength of the material is 500 MPa and the Poisson’s ratio is 0.25.

8. A cantilever beam of length L and of rectangular cross section of

B(breadth) and D (depth), having a tip load P. Derive expressions for the save value of the maximum load P, for five theories of failure, in terms of beam geometry, yield strength SY, Poison’s ratio ν, Modulus of elasticity E.

(a) A thin cylindrical vessel closed at ends, subjected to an internal pressure p, has the internal diameter D and thickness t. It is further subjected to an axial tensile force P and a twisting moment Mt. Set up the expression for the principal stresses in the following form. mms1, s2 = (3X + Y)/2 ± √[(Y – X)/2]2 + (Z) 2] where, X= pD/4t; Y = P/pDt; Z = 2Mt/pD2 t

9 (a) A thin cylindrical vessel closed at ends, subjected to an internal pressure p, has the internal diameter D and thickness t. It is further subjected to an axial tensile force P and a twisting moment Mt. Set up the expression for the maximum shear stress theory of failure in the following form. Sw = √[(Y – X)2 + 4(Z) 2] where, Sw = Working Stress,

X= pD/4t; Y = P/pDt; Z = 2Mt/pD2 t

b) If D = 100 mm, t = 4 mm, P= 20 kN, Mt = 5 kN-m, yield stress = 600 MPa, for a factor of safety of 2, find the maximum value of p that can be applied, using the maximum shear stress theory of strength

10 (a) A thin cylindrical vessel closed at ends, subjected to an internal pressure p, has the internal diameter D and thickness t. It is further subjected to an axial tensile force P and a twisting moment Mt. Set up the expression for the maximum strain energy theory of failure in the following form.

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Sw2 = [9 – 4(1+ν)]X2 + Y2 + [6 – 4(1+ν)]XY + 2(1+ν)Z 2

where, Sw = Working Stress,

X= pD/4t; Y = P/pDt; Z = 2Mt/pD2 t

Note: If we put ν = 0.5, then we get the distortion energy case as

Sw2 = 3X2 + Y2 + 3Z 2 , because in distortion, there is no volume change.

b) If D = 100 mm, t = 4 mm, p = 10 MPa, P= 20 kN, yield stress = 600 MPa, for a factor of safety of 2, find the maximum value of Mt that can be applied, using the maximum strain energy theory of strength (5 marks)

(11) a) Prove that for small strains, the change in volume per unit volume dV/V is the sum of three principal strains.

b) Prove that for small strains, if volume constancy is assumed, Poison’s ratio ν = 0.5

12) A hub is shrink-fit on a circular shaft. In a cubical element on the outer fiber of the shaft the the stresses caused fy shrink fit are: σy = - 75 MPa and σz = -75 MPa. The shaft is subjected to bending causing σx = ± 90 MPa at the extreme surface of the shaft. The transverse shear is neglected. Hence all these stresses can be considered as principal stresses. Consider modulus of elasticity E = 200 GPa and Poison’s ratio = 0.3. Determine

a) three principal strainsb) the maximum shear stressc) the strain energyd) the distortion energye) the volumetric strain dV/V (see problem 11)

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