Th. Of Failure.docx

Theories of failure-Assignment 1. The force acting on a bolt consists of two components – an axial pull of 12 KN and a transverse shear force of 6 kN. The bolt is made of steel FeE 310 (Syt = 310 N/mm2) and the factor of safety is 2.5. Determine the diameter of the bolt using the maximum shear stress theory of failure. (Ans: 13.2 mm)

2. The principal stresses induced at a point in a machine component made of steel 50C4 (Syt = 460 N/mm2) are as follows; 𝜎1 = 200 N/mm2; 𝜎2 = 150 N/mm2; 𝜎3 = 0. Calculate the factor of safety by (a) the maximum normal stress theory (b) the maximum shear stress theory and (c) the distortion energy theory. (Ans: 2.3, 2.3, 2.55)

3.

A cylindrical. shell made of mild-steel plate yields at 200 N/mm2 in uniaxial tension. The diameter of the shell is 2 m and its thickness is 20 mm. Find the pressure at which the failure occurs according to (a) maximum shear stress theory and (b) distortion energy theory. (Ans. (a) 8 N/mm2 (b) 4.62 N/mm2).

4. For a biaxial stress system, σx = 80 N/mm2 and; σy = 40 N/mm2. Find the equivalent stress at the elastic limit assuming that failure occurs due to maximum principal strain theory. Take poisson's ratio to be 0.25.

(Ans. 70 N/mm2).

5. A pressure vessel of inside radius 300 mm is subjected to a pressure of 2.0 N/mm2. Find the thickness of the vessel according to total strain energy theory. The factor of safety is 3 and the yielding occurs in simple tension at 240 N/mm2. Take poisson's ratio to be 0.3. (Ans. 7. 31mm)

6.

A cylindrical tube of outside diameter 120 mm and thickness 3 mm is subjected to a torque of 2 x 10 4 N-m. The stress at the elastic limit in simple tension is 1200 N/mm2. Calculate the factor of safety according to (a) maximum shear stress theory and (b) distortion energy theory. (Ans: (a) 1.89, (b) 2.18)

7. A shaft of diameter 50 mm is subjected to a torque of 300 N.m and an axial thrust. For a factor of safety of 3, find the maximum value of the thrust according to (a) maximum shear stress theory and (b) distortion energy theory. The failure stress is 100 N/mm2 at the elastic limit. (Ans. (a) 44.51kN; (b) 50.56 kN).

8. A mild steel shaft of diameter 50 mm is subjected to a bending moment of 2 kN.m and a Torque T. If the yield point of steel in tension is 200 MPa, find the maximum value of the torque without causing yielding of the shaft material according to (i) maximum principal stress (ii) maximum shear stress and (iii) distortion energy theories. (Ans. 2112 N.m, 1422.6 N.m, 1642 N.m)

9. A thick spherical pressure vessel of inner radius 150 mm is subjected to an internal pressure of 80 MPa. Calculate its wall thickness based upon (a) maximum principal stress theory and (b) total strain energy theory. Poisson's ratio = 0.30, yield strength = 300 MPa. (Ans. 40 mm, 38.99 mm)

10. Find the maximum principal stress developed in a cylindrical shaft 80 mm in diameter and subjected to a bending moment of 2.5 kN.m and a twisting moment of 4.2 kN.m. If the yield stress of the shaft material is 300 MPa, find the

factor of safety of the shaft according to the maximum shearing stress theory of failure. (Ans. 73.49 N/mm2, 3.085)

11. A component in an aircraft flap actuator can be adequately modeled as a cylindrical bar subjected to an axial force of 8 kN, a bending moment of 55 N.m and torsional moment of 30 N.m. A 20 mm diameter solid bar of 7075-T6 aluminium having σu = 591 MPa, σyt = 542 MPa and τy = 271 MPa is recommended for its use. Determine the factor of safety available as per maximum principal stress theory, maximum shear stress theory. (Ans. 5.96, 5.27)

12. A rod of circular cross section is to sustain a torsional moment of 3 kN-m and a bending moment of 2 kN-m. Selecting C45 steal (t σytt = 353 MPa) and assuming factor of safety as 3, determine the diameter of the rod as per (a) Maximum normal stress theory, (b) Distortion energy theory and (c) Maximum shear stress theory. (d) Maximum strain theory (e) Maximum strain energy theory. Poisson’s ratio = 0.25. (Ans: 62.44 mm, 65.7 mm, 67.8 mm, 63.8 mm, 64.5 mm)

13. A critical section in a shaft is subjected to a bending stress of 50 MPa and torsional stress of 31.5 MPa simultaneously. Determine the factor of safety as per (a) Maximum normal stress theory (b) Maximum shear stress theory (c) Distortion energy theory (d) Maximum strain theory (e) Maximum strain energy theory. Proportionality limit in tension is 284 MPa. Poisson’s ratio is 0.25. (Ans: 4.35, 7.06, 3.85, 4.11, 4.02)

14. A rod of 50 mm diameter is subjected to a compressive load of 20 kN together with a twisting moment of 1.5 kN-m. It is made of C40 steal σytt = 328.6 MPa. Determine the factor of safety according to: (i) Maximum normal stress theory, (ii) Maximum shear stress theory (c) Distortion energy theory (d) Maximum strain theory (e) Maximum strain energy theory (Ans: 4.95, 2.68, 3.09, 3.94, 3.32)

Additional problems

15. A material has a yield strength of 630 MPa. Find the factor of safety for each of the states of stress using: (i) Maximum normal stress theory, (ii) Maximum shear stress theory, (iii) Distortion energy theory.

(a)

σ1 = 200 MPa,

σ2 = 0,

σ3 = -90 MPa

(b)

σ1 = 200 MPa,

σ2 = 90 MPa,

σ3 = 0

(c)

σ1 = 200 MPa,

σ2 = 200 MPa,

σ3 = 200 MPa

(d)

σ1 = 90 MPa,

σ2 = -90 MPa

σ3 = -200 MPa.