Introduction of structural mechanics

Strength Of Material

1. Introduction Scope of subject
Properties of materials
St. Venant's Principle
Principle of Superposition
Simple stress (σ)
Simple strain (e)
Lateral strain
Poisson's Ratio
Hook's law
Young's Modulus or Modulus of elasticity (E)
Modulus of Rigidity or Shear Modulus(C, G, N)
Bulk Modulus(K)
Volumetric Strain(e_v)
Concept of uni-axial loading
Stress strain behavior for ductile material
Stress strain behavior for brittle material
Working Stress
Factor of Safety
Ultimate Stress

2. Shear force and bending moment.

Beam Type of beams Type of loads Type of supports Shear Force (S.F.) Bending Moment (B.M.) Relation between SF and rate of loading Relation between SF and BM Shear Force and Bending Moment Diagram Point of contraflexure Variation of SF and BM for different loading Horizontal Thrust Diagram

Scope of subject:

Various structures in society which may be any building or any machine or any object which carry load are subjected with various kinds of loading such as its self weight, wind load, earth quake, etc. due to this loading some effects are observed in structure. In structural mechanics we study and analysis the properties of various materials and various structure under such kinds of loading.

Properties of materials:

1. Strength - It is property of material which sustain the load.
2. Elasticity - It is property of material by virtue of which it can regains its original size and shape after deformation on removing load causing deformation.

3. Plasticity - It is property of material by virtue of which it can not regains its original size and shape after deformation on removing load causing deformation.

4. Ductility - It is property of material by which it can under goes considerable deformation under tension before its failure.

5. Brittleness - It is property of material by which it can not under goes considerable deformation under tension before its failure.
6. Malleability - It is property of material by which it can under goes permanent deformation under compression with out its failure.
7. Toughness (impact strength) - It is property of material by virtue of which it can absorb suddenly applied load with out its failure.
8. Hardness - It is property of material by virtue of which it can resist wear, abrasion and scratching.
9. Stiffness - It is property of material by virtue of which it can resist elastic deformation.
OR
Mathematically it is load required to produce unit deformation.
10. Creep - It is property of material by virtue of which it undergoes continuous deformation under steady load with time.
It is also called as 'Time yield' or ' Plastic flow'.

St. Venant's Principle:

It state that;
" The difference between effect of two different statically equivalent load become very small at sufficiently large distance from load."

Principle of Superposition:

It state that;
" When a number of forces are acting on a body, then resulting strain will be the algebraic sum of strains caused by individual forces."

Simple stress (σ):

It is simply defined as the load applied per unit area. It is denoted by σ (sigma).

∴ σ = ^P / _A

Where,
P - Load,
A - Area.
It's unit is ^N / _mm2.

Explanation: Consider any material say one bar of having length (l) and having cross sectional area (A) is subjected to load (P). Due to this loading length of bar will increase, this increase in length is (δl). This increasing length is resisted by internal resisting force (R). After continuous increase in load (P); increase in length (δl) is continued till internal resisting force(R) equal to external applied load (P). After increasing load (P) than internal resisting force (R) bar will break. And the situation before failure of bar internal resisting force(R) equal to external applied force(P); and this internal resisting force per unit area is called as stress (σ).

Simple strain (e):

It is simply defined as change in length per unit original length. It is denoted by (e). It also called as "Linear Strain".

∴ e = ^δl / _l

Where,
δl - change in length,
l - original length.
It is unit less because change in length and original length is measured in same unit and same unit become cancelled.

Lateral strain:

Strain in perpendicular direction of applied load is called as "Lateral Strain or Secondary Strain".

Mathematically,
Lateral Strain = ^{Change in length in lateral dimension} / _{Original lateral dimension}

For rectangular bar: Lateral Strain = ^δb / _b or ^δt / _t
Where,
             δb - change in width or breadth and b - original width or breadth,
             δt - change in thickness and t - original thickness.
For circular bar:Lateral Strain = ^δd / _d
Where,
             δd - change in diameter and d - original diameter,

Poisson's Ratio:

When material is loaded within elastic limit the ratio of lateral strain to the linear strain is constant and this called as "Poisson's Ratio". And it denoted by μ (greek letter mu) or ¹ / _m.
The value of Poison's ratio varies from 0.1 to 0.5. The extreme values for concrete and rubber are 0.1 and 0.5 respectively.

Mathematically,
Poisson's Ratio = ^{Lateral strain} / _{Linear strain}
μ or ¹ / _m = ^{Lateral strain} / _e

For rectangular bar: As shown in fig. bar having length(l), width(b) and thickness(t) subjected to pulling force(P) Due to this loading length of bar increases while decrease in width and thickness take place. Therefore linear strain is positive and lateral strain is negative. In this case,
            Linear Strain = e = ^+δl / _l
            Lateral Strain = ^-δb / _b or ^-δt / _t
∴ form Poisson's ratio, μ or ¹ / _m = ^{Lateral strain} / _e
             Lateral strain = -μ X e

Hook's law:

It state that;
"When material is loaded within elastic limit then stress is directly proportional to the strain."

Mathematically, stress ∝ strain.
σ ∝ e
σ = constant X e
^σ/_e = constant

Young's Modulus or Modulus of elasticity:

It state that the ratio of stress to the strain.
form Hook's law,

Mathematically, stress ∝ strain.
σ ∝ e
σ = constant X e
^σ/_e = constant
this constant is called as Young's Modulus or Modulus of elasticity

It is denoted by (E).

Mathematically, E = ^σ/_e.

∴ It's S.I. unit is ^N / _mm2.

Modulus of Rigidity or Shear Modulus(C, G, N):

When material is loaded within elastic limit the ratio of shear stress to the shear strain is constant and this called as "Modulus of Rigidity". And it denoted by C (capital letter C) or G (capital letter G) or N (capital letter N). It is also known as "Shear Modulus"

Mathematically, Modulus of Rigidity or Shear Modulus = ^{Shear Stress} / _{Shear Strain}.
C or G or N = ^q / _Φ.

S.I. unit:^N / _m2 or Pascal.

Bulk Modulus(K):

When material is subjected to three mutually perpendicular like stresses for same time then ratio of direct stress to the volumetric strain is called as "Bulk Modulus". And it denoted by K (capital letter K).

Mathematically, Bulk Modulus = ^{Direct Stress} / _{Volumetric Strain}.
K = ^σ / _(δV/V).

Where,
δV - change in volume,
V - original volume.
S.I. unit: Due to volumetric strain has no unit it's unit is ^N / _m2 or Pascal.

Volumetric Strain(e_v):

When body is subjected to external forces, there will be change in volume. The ratio of change in volume to the original volume called as "Volumetric Strain". And it denoted by e_v.

∴ e_v = ^{Change in Volume} / _{Original Volume}
∴ e_v = ^δV / _V

Where,
δV - change in volume,
V - Original volume.
Volumetric strain is algebraic sum of all axial or linear strains.

∴ e_v = e_x + e_y + e_z

Where,
             e_x - Strain in X-direction,
             e_y - Strain in y-direction,
             e_z - Strain in z-direction.

Concept of uni-axial loading:

To finding volumetric strain we have to consider three axis of body i.e. X, Y, Z. If we apply load in any one direction then such kind of loading is called as uni-axial loading. Here consider we apply load in X-direction.

Volumetric strain for rectangular bar: Consider a rectangular bar of having dimensions length(L), breadth(b) and thickness(t) is subjected to tensile load(P) in X-direction as shown in fig.

Stress in X-direction,
σ_x = ^P / _A
σ_y = σ_z = 0 _________________________(Due to no stress(load) in Y and Z directions.)
Now, strain in X-direction,
e_x = ^σ_x / _E __________________________(Young's modulus E = ^σ/_e)
strain in Y and Z-directions are lateral strains,
e_y = e_z = -μ X Linear strain _____________(Poisson's Ratio = ^{Lateral strain} / _{Linear strain})
     = -μ X e_x
     = -μ X ^σ_x / _E
∴ Volumetric strain of bar,
e_v = ^δV / _V
     = e_x + e_y + e_z
     = ^σ_x / _E - μ X ^σ_x / _E - μ X ^σ_x / _E
     = ^σ_x / _E (1 - μ - μ)
     = ^σ_x / _E (1 - 2μ)
     = e(1 - 2μ) __________________________(^σ_x / _E = e_x = linear strain i.e. e)

∴ Volumetric strain of bar = e_v = ^δV / _V = e(1 - 2μ)

Stress strain behavior for ductile material:

Consider any ductile material, like mild steel bar of uniform cross sections is subjected with uniformly increasing tensile load up to breaking point. It behave differently at different conditions of increasing load. It behavior can be easily understood by graphical representation. For this purpose we plot graph stress vs strain. Which is also called as stress strain curve.

If we plot graph as mentioned above it seem like as following.

In above graph we can seen that stress strain behavior line start form origin after than we obtaints various points which are described as following-
A. Limit of proportionality - Up to this point material obeys Hook's law i.e. stress is directly proportional to the strain. From origin to upto this point graph obtained is straight line. From this point straight line loose its proportional law(i.e. stress is directly proportional to the strain) hence this point is called as "limit of proportionality(A)."
B. Elastic limit - From point A graph changes form straight line and it become non-liner A to B. Up to this point material possese it's elasticity property hence this point called as "elastic limit(B)." If we remove load applied on material before getting this point material gains its original size and shape.
C. Upper yield point and D. Lower yield point- The state between point "C" and "D" is called as yield state. In this state increase in stain is take place while no increase in stress is occurred. The stating of this state is called as "Upper yield point(C)", while ending of this state is called as "Lower yield point(D)."
E. Ultimate load point - After the yield state stress strain curve increases up to point "E". At this point load(stress) caring capacity of material is maximum hence this point called as "Ultimate load point(E)."
F. Breaking point - After continue loading of material we reach point "F". At this point material fails to carry the load hence this point called as "Failure point(F) or Breaking point(F)."

Stress strain behavior for brittle material:

Consider any brittle material, like cast iron, aluminium, concrete, etc. is subjected with uniformly increasing tensile load up to breaking point. It behave differently at different conditions of increasing load. It behavior can be easily understood by graphical representation.

If we plot graph as mentioned above it seem like as following.

In above graph we can see that behavior of brittle materials under load subjected to tension. In brittle materials do not show the yield point. In such cases where yield point is not clearly defined it is taken as point of some definite amount of permanent strain generally 0.2%. It is obtained by offset method as shown in above fig. The stress corresponding to that point is called as "Yield Stress or Proof Stress". For brittle materials graph obtained is continuous curve form origin.

Working Stress:

It is the ratio of actual load applied and the original cross sectional area of the specimen.
Mathematically, Working Stress = ^{Actual axial load applied}/_{Original cross sectional area}
Working stress is maximum stress which can be applied during it's service period. In practice we take working stress with in elastic limit by using factor of safety.

Factor of Safety:

The ratio of ultimate stress and the working stress for the material is called "Factor of Safety"..
Mathematically, Factor of Safety = ^{Ultimate stress}/_{Working Stress}

Ultimate Stress:

The ratio of maximum load taken by material safely and the original cross sectional area of material is called "Ultimate Stress"..
Mathematically, Ultimate Stress = ^{Maximum load}/_{Original cross sectional area}