Strength of Materials - 1 - Mechanical Engineering MCQ

Maximum and minimum principal stresses:

Bending stress due to bending moment:

Shear stress due to twisting moment/ torque:

If the maximum bending stress equals to maximum shear stress developed:

In some applications the shaft are simultaneously subjected to bending moment M and Torque T.

From the simple bending theory equation:

If σ_b is the maximum bending stresses due to bending moment M on shaft:

Torsion equation:

The maximum shear stress developed on the surface of the shaft due to twisting moment T:

Equivalent Bending Moment:

Equivalent Torque:

Here R_A = R_B = W

Between A to C:

Mx=W×x⇒Ma=0,Mc=WaMx=W×x⇒Ma=0,Mc=Wa

Between C to D:

Mx=W×x−W(x−a)⇒MD=W×4a−W(4a−a)=WaMx=W×x−W(x−a)⇒MD=W×4a−W(4a−a)=Wa

Between D to B:

Mx=W×x−W(x−a)−W(x−4a)⇒MB=W×5a−W(5a−a)−W(5a−4a)=0

P_n = The component of force P, normal to section FG = P cos θ

P_t = The component of force P, along the surface of the section FG (tangential to the surface FG) = P sin θ

σ_n = Normal Stress across the section FG

σn=σcos2θσn=σcos2θ

σ_t = Tangential/Shear Stress across the section FG

• 
• So it is evident form the graph that the strain is proportional to stress or elongation is proportional to the load giving a straight line relationship. This law of proportionality is valid upto a point A. Point A is known as the limit of proportionality or the proportionality limit.
• For a short period beyond the point A, the material may still be elastic in the sense that the deformations are completely recovered when the load is removed. The limiting point B is termed as Elastic Limit.
• Beyond the elastic limit plastic deformation occurs and strains are not totally recoverable. There will be thus permanent deformation or permanent set when load is removed. These two points are termed as upper and lower yield points respectively. The stress at the yield point is called the yield strength.
• A further increase in the load will cause marked deformation in the whole volume of the metal. The maximum load which the specimen can with stand without failure is called the load at the ultimate strength. The highest point ‘E' of the diagram corresponds to the ultimate strength of a material.
• Beyond point E, the bar begins to form neck. The load falling from the maximum until fracture occurs at F.

In the case of bi-axial state of normal stresses:

Normal stress on an inclined plane:

Shear Stress on an inclined plane:

Normal stress on 45° plane:

Thermal strain ϵ_T is proportional to the temperature change ΔT

ϵT=αΔTϵT=αΔT

α is coefficient of thermal expansion.

When there is some restriction to the bar to expand, thermal stress will generate in the material:

σT=strain×E=α.ΔT.E

Change in length:

Young's is a property of the material of construction of a wire. It depends only on the nature of the material used for making the wire.

Young's modulus does not depend on the physical dimensions such as the length and diameter of wire. Thus there will be no change in Young's modulus.

As in thin cylinder shell:

Longitudinal stress = Half of circumferential stress

∴ circumferential stress ≤ Permissible stress

From the simple bending theory equation

If σ_b is the maximum bending stresses due to bending moment M on shaft:

For same bending moment:

Slenderness ratio is the ratio of the length of a column and the least radius of gyration of its cross section. It is used extensively for finding out the design load as well as in classifying various columns in short/intermediate/long.

Long column: buckling occurs elastically before the yield stress is reached.

Short column: material failure occurs inelastically beyond the yield stress.

The maximum load at which the column tends to have lateral displacement or tends to buckle is known as buckling or crippling load. Load columns can be analysed with the Euler’s column formulas can be given as

• For both end hinged, n = 1
• For one end fixed and other free, n = ½
• For both end fixed, n = 2
• For one end fixed and other hinged, n = √2
• 
• 

100 +10x - x² = 0

x² -10x - 100 = 0

x = 16 cm

The elastic strain energy stored in a member of length s (it may be curved or straight) due to axial force, bending moment, shear force and torsion is summarized below:

Strain energy due to combined effect of load P and torque T:

The ability of a material to absorb energy in the elastic region. This is given by the strain energy per unit volume which is the area of the elastic region.

 Ability to absorb energy in the plastic range. This is given by the total area under the stress-strain curve.