ADVANCED STRENGTH OF MATERIALS UNIT :1

1) Define Strain energy
When an elastic body is loaded with in an elastic limit, it deforms and some work is done which
is stored with in the body in the form of internal energy. This stored energy in the deformed body is
known as Strain energy.
2) Define Proof Resilience
Proof Resilience is the maximum energy stored in the body within the elastic limit.
3) Define Strain energy Density (Resilience)
The ability of the material to regain its original shape on removal of the applied load is known
as Strain energy Density (Resilience).
4) Define Modulus of Resilience
Proof Resilience per unit volume is known as Modulus of Resilience.
5) Write the formula to calculate the strain energy due to axial loads.
U=∫ P^2 dx limit 0 to L
2AE
Where,
P=Applied tensile load
L=length of the member
A=Area of the member
E=Young’s Modulus
6) Write the formula to calculate the strain energy due to bending
U=∫M^2 dx limit 0 to L
2EI
Where,
M=Bending moment due to applied loads
E=Young’s Modulus
I=moment of inertia
7) Write the formula to calculate the strain energy due to torsion in a solid shaft
U= V* (fs)^2
4N
Where,
Fs= maximum shear stress developed in the outermost layer.
V=volume of shaft
N= Modulus of rigidity
8) Write the formula to calculate the strain energy due to torsion in a Hollow shaft
U= fs^2(D^2+d^2)
4ND^2
Where,
Fs= maximum shear stress developed in the outermost layer.
D=outer diameter of the shaft
d=inner diameter of the shaft
N= Modulus of rigidity
9) Write the formula to calculate the strain energy if the moment value is given.
U=M^2
2EI
Where,
M=Bending moment due to applied loads
E=Young’s Modulus
I=moment of inertia
10) Write the formula to calculate the strain energy if the applied load value is given
U=P^2L
2AE
Where,
P=Applied tensile load
L=length of the member
A=Area of the member
E=Young’s Modulus
11) State Castigliano’s theorem
Castigliano’s theorem states that” If a body is acted upon by forces f1, f2, f3…..fn and U is the
strain energy stored in the body the partial derivative of the strain energy with respect to a force system
fi gives the displacement of the body in the direction of fi.
δi= ∂U
∂fi
12) What are the uses of Castigliano’s theorem?
i) To determine the deflection of complicated structures.
ii) To determine the deflection of curved beams and springs.
13) Define unit load method
The external loads are removed and the unit load is applied at places where deflection has to
be found out is known as unit load method.
14) Define Maxwell’s Reciprocal theorem
In any beam or truss the deflection at any point ‘A’ due to a load ‘W’ at any other point ‘C’ is
the same as the deflection at ’C’ due to the same load at ‘A’.
δA=δC
15) Compare the unit load method and Castigliano’s first theorem
In the unit load method one has to analyze the frame to find the load and deflection while
in the latter method, only one analysis is needed.
16) What is Williot Mohr’s diagram?
Williot Mohr’s diagram is a graphical method to find the deflection of the beam.
17) Write the formula for finding deflection of a fixed beam carrying a load w at the free end of length L
δ =wL^3
3EI
18) State the principal of virtual work
Direct use of deflection and strain energy for determining deflection of beam breaks down due
to several deflections. Hence an extraordinary device meant for solving this problem i.e., by replacing
true or real work and strain energy by external and internal work.
19) Write the formula for finding strain energy per unit volume due to a tensile stress (f)
U=f^2
2E
Where,
P= tensile stress
E=Young’s Modulus
20) Write the formula for finding deflection of a beam of length (L) simply supported at one end caries a
point load (W) at its centre.
δ = WL^3
48EI