Bars of a square and circular cross-section with 0.5 m length is made of a material with a shear strength of 20 MPa. The square bar cross-section dimension is 4 cm × 4 cm and the cylindrical bar cross-section diameter is 4 cm. The specimens are loaded as shown in the figure. Which specimen (s) will fail due to the applied load as per the maximum shear stress theory?

Tensile and compressive load specimens

Match the following related to theories of failureA. Max normal stress theory1. Vonmises theoryB. Max shear stress theory2. Haigh’s theoryC. Max strain energy theory3. Guest and Tresca theoryD. Max distortion energy theory4. Rankiness theory

A – 3, B – 4, C – 1, D - 2

A – 4, B – 3, C – 2, D -1

A – 3, B – 4, C – 1, D - 2

A – 4, B – 3, C – 1, D - 2

A – 3, B – 4, C – 2, D -1

According to Von-Mises’ distortion energy theory, the distortion energy under three dimensional stress state is represented by

\(\frac{1}{{3E}}\left[ {\sigma _1^2 + \sigma _2^2 + \sigma _3^2 - v\left( {\sigma _1\sigma _2 + \sigma _3\sigma _2 + \sigma _1\sigma _3} \right)} \right]\)

\(\frac{1}{{2E}}\left[ {\sigma _1^2 + \sigma _2^2 + \sigma _3^2 - 2v\left( {\sigma _1\sigma _2 + \sigma _3\sigma _2 + \sigma _1\sigma _3} \right)} \right]\)

\(\frac{{1 - 2v}}{{6E}}\left[ {\sigma _1^2 + \sigma _2^2 + \sigma _3^2 + 2v\left( {\sigma _1\sigma _2 + \sigma _3\sigma _2 + \sigma _1\sigma _3} \right)} \right]\)

\(\frac{{1 + v}}{{3E}}\left[ {\sigma _1^2 + \sigma _2^2 + \sigma _3^2 - \left( {\sigma _1\sigma _2 + \sigma _3\sigma _2 + \sigma _1\sigma _3} \right)} \right]\)

\(\frac{1}{{3E}}\left[ {\sigma _1^2 + \sigma _2^2 + \sigma _3^2 - v\left( {\sigma _1\sigma _2 + \sigma _3\sigma _2 + \sigma _1\sigma _3} \right)} \right]\)

Choose the correct relation between modulus of elasticity (E), modulus of rigidity (G) and bulk modulus (K) from the following options:

\(\frac{1}{E} = \frac{9}{G} + \frac{3}{K}\)

\(\frac{2}{E} = \frac{9}{G} + \frac{3}{K}\)

\(\frac{3}{E} = \frac{9}{G} + \frac{1}{K}\)

\(\frac{9}{E} = \frac{3}{G} + \frac{1}{K}\)

\(\frac{1}{E} = \frac{9}{G} + \frac{3}{K}\)

A machine element develops principal stresses of magnitudes 2P and P. What is the maximum magnitude of P before the material reaches the yield stress fy as per Distortion Shear Energy Theory?

\(\frac{{{{\rm{f}}_{\rm{y}}}}}{2}\)

\(\frac{{{{\rm{f}}_{\rm{y}}}}}{{2\sqrt 3 }}\)

\(\frac{{{{\rm{f}}_{\rm{y}}}}}{{\sqrt 3 }}\)

In case of biaxial stresses, the maximum value of shear stress is______

Half the difference of normal stresses

Half the sum of normal stresses

According to the distortion-energy theory, the yield strength in shear is

0.377 times the maximum shear stress

0.477 times the yield strength in tension

0.577 times the yield strength in tension

21 + Mcqs in Theory Of Failure in Civil Engineering Page 1 McqOptions

1.	A plane-strain compression (forging) of a block is shown in the figure. The strain in the z-direction is zero. The yield strength (Sy) in uniaxial tension/compression of the material of the block is 300 MPa and it follows the Tresca (maximum shear stress) criterion. Assume that the entire block has started yielding. At a point where σx = 40 MPa (compressive) τxy = 0, the stress component σy is ________
A.	340 MPa (compressive)
B.	340 MPa (tensile)
C.	260 MPa (compressive)
D.	260 MPa (tensile)
Answer» B. 340 MPa (tensile)

2.	A solid circular shaft of 40 mm diameter transmits a torque of 3200 N-m. The value of maximum stress developed is
A.	400/π
B.	800/π
C.	1600/π
D.	600/π
Answer» C. 1600/π

3.	A machine element is subjected to the following bi-axial state of stress; σx = 80MPa; σy = 20 MPa; τxy = 40 MPa. If the shear strength of the material is 100 MPa, the factor of safety as per Tresca‟s maximum shear stress theory is
A.	1
B.	2
C.	2.5
D.	3.3
Answer» C. 2.5

4.	Consider the following theories of failure:A. Maximum stress theoryB. Maximum strain theoryC. Maximum shear stress theoryD. Maximum energy or distortion theoryThe most suitable for ductile material is
A.	A and B
B.	A and C
C.	A and D
D.	C and D
Answer» E.

5.	Bars of a square and circular cross-section with 0.5 m length is made of a material with a shear strength of 20 MPa. The square bar cross-section dimension is 4 cm × 4 cm and the cylindrical bar cross-section diameter is 4 cm. The specimens are loaded as shown in the figure. Which specimen (s) will fail due to the applied load as per the maximum shear stress theory?
A.	Tensile and compressive load specimens
B.	Torsional load specimen
C.	Bending load specimen
D.	None of the specimens
Answer» B. Torsional load specimen

7.	At a certain point in a structural member, there are perpendicular stresses 80 N/mm2 and 20 N/mm2, both tensile. What is the equivalent stress in simple tension, according to the maximum principal strain theory? (Poisson’s ratio = 0.25)
A.	0 N/mm2
B.	20 N/mm2
C.	60 N/mm2
D.	75 N/mm2
Answer» E.

8.	A structural element is subjected to a two-dimensional stress system, wherein σ1 = 225 N/mm2 (tensile) with σ2 being compressive. The yield stress in both simple tension (σy)t and simple compression (σy)c is 250 N/mm2 and μ = 0.25. What is the value of σ2, according to Maximum Strain Theory?
A.	200 N/mm2
B.	150 N/mm2
C.	125 N/mm2
D.	100 N/mm2
Answer» E.

9.	For the prediction of ductile yielding, the theory of failure utilized is
A.	Maximum strain energy theory
B.	Distortion energy theory
C.	Maximum normal strain theory
D.	Mohr theory
Answer» C. Maximum normal strain theory

10.	An element in a strained body is subjected to only shear stress of intensity 50 MPa tending to rotate the body in a clockwise direction. What is the magnitude of principal stresses?
A.	± 50 MPa
B.	+ 50 MPa, -25 MPa
C.	+ 25 MPa, -50 MPa
D.	± 25 MPa
Answer» B. + 50 MPa, -25 MPa

11.	In a two-dimensional stress system, the principal stresses are σ1 = 200 N/mm2 (tensile) and σ2 (compressive). The yield stress in compression is 250 N/mm2, with μ = 0.25. What will be the value of σ2 according to the maximum normal strain theory?
A.	160 N/mm2
B.	100 N/mm2
C.	200 N/mm2
D.	250 N/mm2
Answer» D. 250 N/mm2

14.	Maximum shear stress in a solid shaft of 40 mm diameter, subject to twisting moment of 'π' Nm is
A.	25 MPa
B.	0.25 MPa
C.	250 MPa
D.	None of the above
Answer» C. 250 MPa

21.	A cylindrical pressure vessel is 1200 mm in diameter. It is made of rolled mild steel plate. The vessel is subjected to an internal pressure of 2 N/mm2. If the material yields at 200 N/mm2, what should be the minimum safe thickness of the plate, based on Maximum principal stress Theory?
A.	18 mm
B.	15 mm
C.	12 mm
D.	9 mm
Answer» E.

15.	In theories of failure, which theory states that no shearing stresses and shearing strains will be present anywhere in the block but only the volume changes?
A.	Guest theory
B.	St Venant theory
C.	Prandl's theory
D.	Distortion Energy theory
E.	Bohr's theory
Answer» E. Bohr's theory

17.	A cold rolled steel is designed on basis of maximum shear stress theory. The principal stresses induced at its critical section are 60 MPa and -60 MPa respectively. It yield stress for the shaft material is 360 MPa, FOS of design is
A.	2
B.	3
C.	4
D.	6
Answer» C. 4

18.	In case of biaxial stresses, the maximum value of shear stress is______
A.	Difference of normal stresses
B.	Half the difference of normal stresses
C.	Sum of normal stresses
D.	Half the sum of normal stresses
Answer» C. Sum of normal stresses

19.	A steel specimen is subjected to the following principal stresses: 120 MPa tensile, 60 MPa tensile and 30 MPa compressive. If the proportionality limit for the steel specimen is 250 MPa; the factor of safety as per maximum shear stress theory will be nearly
A.	1.3
B.	1.7
C.	2.3
D.	2.7
Answer» C. 2.3

20.	According to the distortion-energy theory, the yield strength in shear is
A.	0.277 times the yield stress
B.	0.377 times the maximum shear stress
C.	0.477 times the yield strength in tension
D.	0.577 times the yield strength in tension
Answer» E.

Explore topic-wise MCQs in Civil Engineering.

A solid circular shaft of 40 mm diameter transmits a torque of 3200 N-m. The value of maximum stress developed is

A machine element is subjected to the following bi-axial state of stress; σx = 80MPa; σy = 20 MPa; τxy = 40 MPa. If the shear strength of the material is 100 MPa, the factor of safety as per Tresca‟s maximum shear stress theory is

Consider the following theories of failure:A. Maximum stress theoryB. Maximum strain theoryC. Maximum shear stress theoryD. Maximum energy or distortion theoryThe most suitable for ductile material is

Match the following related to theories of failureA. Max normal stress theory1. Vonmises theoryB. Max shear stress theory2. Haigh’s theoryC. Max strain energy theory3. Guest and Tresca theoryD. Max distortion energy theory4. Rankiness theory

At a certain point in a structural member, there are perpendicular stresses 80 N/mm2 and 20 N/mm2, both tensile. What is the equivalent stress in simple tension, according to the maximum principal strain theory? (Poisson’s ratio = 0.25)

A structural element is subjected to a two-dimensional stress system, wherein σ1 = 225 N/mm2 (tensile) with σ2 being compressive. The yield stress in both simple tension (σy)t and simple compression (σy)c is 250 N/mm2 and μ = 0.25. What is the value of σ2, according to Maximum Strain Theory?

For the prediction of ductile yielding, the theory of failure utilized is

An element in a strained body is subjected to only shear stress of intensity 50 MPa tending to rotate the body in a clockwise direction. What is the magnitude of principal stresses?

In a two-dimensional stress system, the principal stresses are σ1 = 200 N/mm2 (tensile) and σ2 (compressive). The yield stress in compression is 250 N/mm2, with μ = 0.25. What will be the value of σ2 according to the maximum normal strain theory?

According to Von-Mises’ distortion energy theory, the distortion energy under three dimensional stress state is represented by

Choose the correct relation between modulus of elasticity (E), modulus of rigidity (G) and bulk modulus (K) from the following options:

Maximum shear stress in a solid shaft of 40 mm diameter, subject to twisting moment of 'π' Nm is

In theories of failure, which theory states that no shearing stresses and shearing strains will be present anywhere in the block but only the volume changes?

A machine element develops principal stresses of magnitudes 2P and P. What is the maximum magnitude of P before the material reaches the yield stress fy as per Distortion Shear Energy Theory?

A cold rolled steel is designed on basis of maximum shear stress theory. The principal stresses induced at its critical section are 60 MPa and -60 MPa respectively. It yield stress for the shaft material is 360 MPa, FOS of design is

In case of biaxial stresses, the maximum value of shear stress is______

A steel specimen is subjected to the following principal stresses: 120 MPa tensile, 60 MPa tensile and 30 MPa compressive. If the proportionality limit for the steel specimen is 250 MPa; the factor of safety as per maximum shear stress theory will be nearly

According to the distortion-energy theory, the yield strength in shear is

A cylindrical pressure vessel is 1200 mm in diameter. It is made of rolled mild steel plate. The vessel is subjected to an internal pressure of 2 N/mm2. If the material yields at 200 N/mm2, what should be the minimum safe thickness of the plate, based on Maximum principal stress Theory?