STRENGTH OF MATERIALS INTRODUCTION CLASS NOTES FOR MECHANICAL ENGINEERING - ME Subjects

STRENGTH OF MATERIALS

INTRODUCTION CLASS NOTES FOR

MECHANICAL ENGINEERING

Strength of materials tells which

materials are strong in

tension/compression/

bending /torsion/buckling. Hence

depending upon the application, one can

select the material for that application.

Application includes chair, table, rod,

shaft or any item one can think of. The

selection is possible after performing

tests under various kinds of

forces/moments/torques etc. Tests help

to know which material is stronger under

which type of load.

Examples strength of materials

wires/ropes/belts/chains should be of materials strong in tension
Short columns/chair legs/table legs/walls should be of materials strong in compression
Beams should be of materials strong in tension as well as in compression
Shafts should be of materials strong in tension, compression, shear and should have high endurance strength
Medium and long columns should be of materials strong in tension and compression

Sources of force in strength of materials

Symbols used for loads are W or P or F.

Self weight of the component/body under consideration
Weight being supported by the component /body being considered
Wind Load
Seismic load –due to earth quake
Thermal load
Pressure load
Inertia force
Centrifugal force
Frictional force

STRESS IN STRENGTH OF MATERIALS

It is the internal resistance of a material against the external force applied. Since it is an internal resistance, it is invisible.

Mathematically stress = force/area =N/mm ².

TYPES OF STRESSES IN STRENGTH OF MATERIALS: Three

(i) Tensile stress

(ii) Compression stress

(iii) Shear stress

Symbols of simple stresses in strength of materials

For tensile stress = σ (sigma) = tensile force/Area perpendicular to the force = + in nature

compression stress symbol = σ (sigma)= compression force/Area perpendicular to the force = – in nature

Symbol for shear stress = τ (tou) = shear force/area parallel to force = (+/- ) in nature

Shear stress is always in a pair. One is applied and the other is induced. Induced is perpendicular to the applied. Thus, it is accompanied by a complementary shear stress. Complementary shear stress is always at right angle to the applied shear shear stress. This complementary shear stress is also called the induced shear stress.

Normal stress: Stress due to a weight is a normal stress.

Tensile stress is a normal stress. Compression stress is a normal stress due to weight.

Bending stresses are normal stresses. Buckling stress is a normal stress.

UNITS OF STRESS IN STRENGTH OF MATERIALS

Each and every stress unit is N/mm².

1 MN/m² = 1 N/mm² = 1 M Pa

Unit Stress or Engineering stress

Engineering stress= Force/original area =nominal stress = Unit stress

In this, stress is calculated with the original area.

STRESSES IN THE STRESS-STRAIN CURVE

Fig. STRESS STRAIN-CURVE FOR MILD STEEL

Proportional limit stress, σ_p,,

Elastic limit stress, σ_e,

Upper yield point stress, σ_uyp,

Lower yield point stress, σ_lyp,

Ultimate stress, σ_ult,

Breaking stress OR failure stress, σ_f

Yield Point Stress

A Yield point stress is that stress below which strain increases with decrease of stress. There are upper and lower yield stresses. It is found only in a ductile material. It is much lower than the ultimate stress for a ductile material. There is no yield point stress for a brittle material.

Ultimate stress

An Ultimate stress is the maximum stress which can be induced in a material. Ultimate and elastic limit stress are equal for a brittle material. Ultimate stress is much greater than the elastic limit stress and yield for a ductile material.

Stress due to suddenly applied load

It is two times the stress if the same load was applied gradually. P_eq.g = 2 P_sud,

σ_sudden = 2σ_gal

σ_galis the stress due to gradually applied load

Stress due to impact load

It is many times the stress if the same load was applied gradually. It can be 50, 100 or any times the nominal stress.

P_{eq. g} = Pi[1+ (1+2hAE/P_iL)^0.5], σ_i = Pi[1+ (1+2hAE/P_iL)^0.5] / A

Stresses in a composite section

σ₁= P_total E₁/ (E₁A₁ + E₂A₂)

σ₂= P_total E₂/ (E₁A₁ + E₂A₂)

THERMAL STRESS

Thermal stress (σ_th) is there only when contraction due to fall of temperature is restricted——It is a tensile stress.=EαΔt

Thermal stress (σ_th) is there only when expansion due to rise of temperature is restricted——- It is a compressive stress.= EαΔt

Thermal strain = ε_th= σ_th/ E= αΔt

Where is the coefficient of thermal expansion

Thermal stress in a single material σ_th= Eα Δt, ε_th = α Δt

THERMAL STRESSES IN A COMPOSITE MATERIAL

COMMON THERMAL LOAD = P_th= [(α1 –α2 ) Δt] /(1/A₁E₁ +1/A₂E₂)

σ_th1 = P_th/A₁= [(α1 –α2 ) Δt] /A₁(1/A₁E₁ +1/A₂E₂)

σ_th2 = P_th/A₂= [(α1 –α2 ) Δt] /A₂(1/A₁E₁ +1/A₂E₂)

Definition of strain

Strain is change/original

Strain is change in length/original length

TYPES OF STRAINS

Tensile strain,
Compressive strain
Shear strain
Linear strain
Lateral strain
Volumetric strain

1-dimensional Strain–Strain is change in width/original width, Strain is change in thickness/original thickness.

2-dimensional Strain Strain is change in area/original area

3-dimensional Strain- Strain is change in volume/original volume.

SYMBOLS

tensile strain =Є (epsilon) = δL/L = + in nature

compressive strain = Є (epsilon) = δL/L

shear strain =γ (gamma), it is an angular strain.

Volumetric strain = change in volume/ original volume= δV/ V. It is used in pressure vessels.

Engineering strain OR Unit Strain

Engineering strain= Change in dimension/original dimension=Nominal strain=Unit strain

It is called an unit strain If strain is calculated with the original length

UNITS

There is no units for each and every strain since strain is a ratio of length vs length or volume vs volume = δL/L or δV/ V

Modulus

Young’s Modulus

Ratio of tensile stress to tensile strain OR Ratio of compressive stress to compressive strain. Its symbol is E.

Young’s modulus = E = tensile stress/ tensile strain

Young’s modulus = E = compressive stress/ compressive strain

Modulus of rigidity OR Shear modulus

Shear modulus is a ratio of shear stress to shear strain. Its symbol is G. G = τ / γ Its units are GN/m²_. Its value for steel is 80 GN/m². But when used in solving problems it is used as 80 x 1000 N/mm²_.

Shear modulus = G = shear stress/ shear strain

Bulk modulus

Bulk modulus is the ratio of volumetric stress to volumetric strain. Its symbol is ‘K’. Its units are GN/m². It is due to pressure exerted by a fluid on a solid causing change in volume of a solid. Fluid pressure causes change in volume and thus volumetric strain in a solid. It is given as

Bulk modulus = K =Volumetric stress/ volumetric strain= pressure/volumetric strain=

K = p / δV/V

Units of each modulus is GN/m². and Modulus in N/mm² = (GN/m² ) x 1000

POISSON’S RATIO

It is ratio of lateral strain to linear strain.

Lateral strain is in a direction to perpendicular direction of force. Thus it will be less than linear strain.

Linear strain is in the direction of the force. Thus it will be more than lateral strain.

Lateral strain is of opposite type than the linear strain.

Lateral strain is compressive when linear strain is tensile.

Therefore, there is a negative sign with the Poisson’s Ratio.

Its symbol is μ (mu). Its value is less than 1.

Elastic Constants

E,K,G and μ are called elastic constants since these are applicable only within elastic limit.

Relation Between Elastic Constants

Firstly E = 2G (1 + μ)

Secondly E = 3K (1 – 2μ)

Thirdly E = 9KG/(3K+G)

St. Venant’s principle

Strain is not uniform near the area where forces are applied. Yet, strain becomes uniform at some perpendicular distance from the area on which force is applied. Say 5 % of the length from either end of a beam in simple tension. Say the length of the beam under simple tension is 2 m. Then strain is not uniform for 100 mm on either side of the rod in simple tension. Similarly it applies to simple compression and simple strain cases also.

Factor of Safety

Factor of safety accounts for uncertainties in the actual situation and the assumed situation. Because of many uncertainties, factor of safety is FACTOR OF IGNORANCE.

(i) First, uncertainty about the homogeneous nature of the material

(ii) Second, Uncertainty about the isotropic nature of the material

(iii)Third, Uncertainty about the continuous nature of the material

(iv) Fourth, uncertainty about the method of manufacture of the material component

(v) Fifth, uncertainty about the exact type of loading

(vi)Sixth, Uncertainty about the load application-truly axial or eccentric

The value of Factor of safety will be

For Ductile materials FOS = σ_yp/σ_all

For Brittle Materials FOS = σ_ult/σ_all

σ_yp= yield point stress for ductile materials

Its value is very near to proportional limit stress

σ_all = allowable stress = 1/3 to 2/3 of elastic limit stress

σ_ult = ultimate stress

Normal factor of safety

(i) Gradually applied load 2 to 4

(ii) Suddenly applied load 4 to 8

(iii) Impact load 8 to 16

Stress concentration factor k

k is strain energy= Stress at a discontinuity / Nominal stress

It is always >1

MECHANICAL PROPERTIES

Ductility=Either % elongation OR % reduction in area of cross section at fracture

Toughness = Energy required to fracture a specimen under an impact load= Area of load extension curve up to the point of fracture.

Hardness is the resistance to indentation.

Resilience is strain energy per unit volume. It is represented by ‘u’.

u = σ² /2E

u=σ_t² /2E in tension

u= σ_c² /2E in compression

u= τ² /2G in shear

Proof resilience=u_e = σ_e² /2E

_{σe is the elastic limit stress.}

https://www.mesubjects.net/wp-admin/post.php?post=2310&action=edit Q. Ans. Simple stresses

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