Properties of Metals, Stress - Strain and Elastic Constants
ELONGATION OF BARS
1. A bar of uniform cross-sectional area
2. Non-uniform bar
3. Tapering bar of circular cross-section whose diameter changes from ‘d1’ at one end to ‘d2’ at the other end.
where P is the load applied & E is the Young’s modulus of elasticity. L is the length of the specimen.
4. Tapering bar of rectangular cross-section having uniform thickness ‘t’ but width of the bar varies from ‘a’ to ‘b’.
5. Bars joined together parallel to each other A structural member composed of two or more elements of different materials to form a parallel arrangement and subjected to axial loading is termed as a compound bar. Such a section is also known as composite section
Where A1 and E1 are properties of one bar and A2 and E2 are properties of other bar.
6. Elongation due to self weight
(a) For bar of uniform section (Prismatic bar): -
Where l = Unit - weight of the material = W/AL
L = Length of the specimen OR
W = Total weight of specimen
A = Cross-sectional area.
(b) For bar of tapering section (Conical bar) :
OR Deflection of prismatic bar
7. Bar of Uniform Strength :
Where A1 is the area at upper section where the bar is fixed with immovable support. A2 is area at lower section which is freely hanging.
Free expansion in length is given by
ΔL = LaT Where,
L = Length of specimen
α = Coefficient of thermal change
T = Change in temperature
TEMPERATURE STRESSES IN COMPOSITE SECTIONS
(a) When ends of a compound bar are free to expand :
If the temperature is increased then the metal with greater value of a will be in compression and the other metal will be in tension. On the other hand if the temperature is decreased then the nature of stresses will be change i.e., metal with greater value of a will be in tension and the other metal will be in compression.
Example:- In copper and steel composite section, since coeff. of expansion of copper (αc) is greater than coefficient of expansion of steel (αs) hence free expansion of copper is more than that of steel.
Also for composite section Actual expansion of steel = Actual expansion of copper, this gives following condition
Where Es and Ec are modulus of elasticity of steel and copper respectively.
Note : When ends of a compound bar are restrained, then on increasing the temperature metal with greater value of a will be in compression and the other metal will be in tension and vice versa.
(b) When two bars are connected end to end between two fixed supports
Total compressive force = σsAs = σcAc ...(i)
Total extension = 0 Therefore
1. Poisson’s ratio (µ)
The value of m lies between 3 and 4 for most of the metals.
µ = 0.05 to 0.1 for glass = 0.1 to 0.2 for concrete
= 0.25 to 0.42 for metals
= 0.5 for pure rubber (for perfectly plastic body)
2. Volumetric strain ( Îv )
or Îv = Îx +Îy +Îz
Volumetric strain ‘ Îv’ is given by
Îv = 2Îd +Î l = (2× Diametral strain) + Longitudinal strain
Îv = Îx +Îy +Îz
Under hydrogtatic loading
3. Shear Modulus
Modulus of rigidity.
Shear modulus is also known as modulus of rigidity.
RELATIONSHIP BETWEEN ELASTIC CONSTANTS
Young’s modulus, E = Linear Stress / Linear Strain
Modulus of rigidity, G = Shear stress/ Shear strain
Bulk modulus, K = Direct Stress/ volumetric Strain
Poisson’s ratio, μ = 1/m = Lateral strain/ Linear Strain
The strain energy is equal to the work done by the load provided no energy is added or subtracted in the form of heat. It is denoted by U. Thus,
stress strain volume = (1/2)
For a prismatic bar, Δ = PL/ AE
It is denoted by ‘u’. Thus,
When the stress (σ) is equal to proof stress (σf) at the elastic limit, the corresponding resilience is known as proof resilience. Thus,