**Strength of Materials**

In engineering point of view the terms Strength of materials may be defined as one of the major parts in engineering and science.Generally this engineering methodology deals with nature of behavior of various materials of failure under various action and forces.

Definition of

StressWhenever any external forces applies on a body or mass , on that same time equal and opposite forces try to react and resist at various section of the body.This internal forces per unit area at any section of the body is defined as

Stress.

Basics of STRENGTH OF MATERIAL

MECHANICAL PROPERTY OF MATERIAL

StrengthThe capacity of material to withstand load is called Strength.

Ability to sustain under loading condition without undue distortion, rupture or any collapse.

Maximum stress that any material can withstand is called Ultimate Strength.

ElasticityOn material, external load is applied, it undergoes deformation and on removal of load, it returns back to it’s original shape. This property of material is called Elasticity.

PlasticityIf material does not regain its original shape, on removal of the external load.

Ability of material to deform without breaking is called Plasticity.

DuctilityIf material can undergo a considerable deformation without rupture, it is called Ductility.

Ductile material is most suitable material for tension member.

Tension test performed on ductile material.

Stainless steel is most ductile material & Grey cast iron is least ductile material.

Higher the percentage of elongation, more ductile is material.

As temperature increase, ductility increase.

Metals having elongation more than 15% are ductile.

BrittlenessIf material can not undergo any deformation and it fails by rupture.

Stronger in compression and weak in tension.

Materials having less than 5% elongation are considered as brittle.

Compression test performed on brittle material.

MalleabilityProperty by which material can be converted in to thin sheets by hammering.

Material can easily rolled & forged without cracking.

Ex.:- Gold, Silver, Copper, Aluminium, Tin, Lead, Steel etc.

Gold is most malleable material while C.I is the least.

ToughnessResistance to impact or shock loading.

Capacity of material to absorb energy before rupture.

Work required to cause rupture.

Ability to withstand large stress and strains without fracture.

Ex.:- Mild steel, W.I., Manganese etc.

Steel used for cutting tools = High carbon steel.

HardnessResistance to abrasion, wear or scratches.

C.I. is hardest material.

Magnesium alloy is softest material.

StiffnessResistance to deformation or strain.

Force required to produce unit deformation.

Stiffness measured by modulus of elasticity.

Higher E, more stiffness.

Steel is stiffer material.

CreepIncrease in strain under sustained load.

Inelastic deformation due to sustained load.

Creep phenomena used for steel cables, nuclear reactor field etc.

Slow extension of material with time under load.

Fatigue StrengthMaximum stress which material can withstand under repeated stress cycles without fracture is called Fatigue Strength.

Endurance Limit :-Stress at which a material factures under large number of reversals of stress is called endurance limit.

Homogeneous material :-Material of the member is of the same kind through its length.

Isotropic material :-It possesses the same elastic properties in all the direction.

STRESSOn a body when external force is applied, it undergoes some deformation and internal resisting forces are set up. This resistance to force per unit area is called Stress.

Stress = Force/ Area = P/A = ς

S.I. Unit = N/mm

^{2}(MPa) 1 Pa = 1 N/m^{2 }

STRAINRatio of change in length to the original length.

Strain = Change in length /Original length = δl/l = ε

No unit.

Direct or Normal Stress :-The stresses which act normal to plane on which the forces act are called Normal or Direct Stress.

TENSION STRESSBody subjected to two equal & opposite pulls, the stress produced is Tensile stress.

Stress is tensile, corresponding strain is Tensile Strain.

COMPRESSIVE STRESSBody subjected to two equal & opposite pushs, the stress produced is Compressive Stress.

Stress is compressive, corresponding strain is Compressive Strain.

Elastic Limit :-Maximum value of force up to and within which to deformation entirely disappears on removal of force is elastic limit.

Hooke’s Law :-“Within elastic limit, Stress is proportional to Strain.”

MODULUS OF ELASTICITYAlso known as Young’s Modulus.

Ratio of Direct stress to Direct strain.

M.O.E. = Stress/Strain = ς/ε

S.I. Unit = N/mm

^{2}For Mild Steel E = 2 x 105 N/mm

^{2}Modulus of elasticity is Highest for Steel.

Modulus of elasticity is least for Wood.

For Perfectly rigid body, M.O.E. = Infinity.

Reciprocal of M.O.E is called Coefficient of Compressibility (StrainStress).

Relation between δl & E is equal to δl = Pl/AE

Deformation of body due to self weight for uniform prismatic bar = δl = wl

^{2}/2AEDeformation of body due to self weight for uniform conical bar = δl = wl

^{2}/6AEStresses in bars of uniformly tapering circular section = δl = 4Pl/πEd

_{1}d_{2}

Modular ratio:- Modular ratio of two material is the ratio of theirs modulus of elasticises.The direction of force = Linear Direction.

The direction perpendicular to linear direction = Lateral Direction.

LINEAR STRAIN (ε)Linear strain = Change in lengthoriginal length = δl/l

Strain has No unit.

Also called Longitudinal strain.

Strain rosetters are used to measure linear strain.

LATERAL STRAIN (ε’)Lateral strain = Change in diameter/original diameter = δd/d

Lateral strain is always less than linear strain.

POISSON’S RATIO (μ)Poisson’s ratio = Lateral strain/Linear strain = 1/m

For steel μ = 0.23 to 0.27.

For cast iron μ = 0.25 to 0.33.

For concrete μ = 0.08 to 0.18.

This ratio is not greater than 0.5 for any material.

Poisson’s ratio is Highest for Rubber.

If modulus of elasticity = 2 x modulus of rigidity, then μ=0.

VOLUMETRIC STRAIN (εv)Volumetric strain = Change in volume/Original volume = δv/V = εv

Equation to find change in volume = ε (1 – 2μ) = ε (1 – 2/m)

BULK MODULUS (k)When body is subjected to three mutually perpendicular stresses of equal intensity, the ratio of direct stress to volumetric strain is Bulk modulus.

k = Direct stress/Volumetric strain=ς/εv

Unit = N/mm2

MODULUS OF RIGIDITY (G or N or C)Also known as Shear Modulus.

C = Shear stress/Shear strain = τ/∅

Unit = N/mm2

Bar fixed at ends is cooled, it will develop Tensile Stress.

Bar fixed at ends is heated, it will develop Compressive Stress.

RELATION BETWEEN K & E & GBetween E & K

k = mE/3 (m−2)

Between E & G

G = mE/2 (m+1)

Between G, K & E

E = 9GK/G+3K

SHEAR STRESS (τ)When body is subjected to two equal and opposite force acting tangentially, body tends to shear off across the section, then the stress induced is called Shear stress.

Shear stress = Shear force/Shearing area = FAs

Unit = N/mm2

THERMAL STRESS (ς)With increase in temperature material expands, With decrease in temperature material contracts. If this free expansion or contraction is provided, stress will be generated in the material.

Allow expand or contract freely, no stress will be produced in material.

If temperature increased & expansion prevented, compressive stress will be provided.

If temperature decreased & contraction prevented, tensile stress will be provided.

Thermal stress for support do not yield = ς = αtE.

Thermal stress for supports are yielding = ς = αt− (δ/l) E.

End of body yield, magnitude of thermal stress decrease.

Thermal stress in bar of tapering circular section

Maximum stress = ς = αtE (d1/d2) Where, d1 = Larger dia

Minimum stress = ς = αtE (d2/d1), d2 = Smaller dia.

THERMAL STRAIN (ε)Thermal strain = Change in length due to change in temperature/Original length = αt

Thermal strain for support do not yield = ε = αt.

Thermal strain for supports are yielding = ε = αt – δ/l

.

STRAIN ENERGYWhen body is strained within elastic limit, energy stored in it is called strain energy.

Work done on body = u = (ς

^{2}/2E) x VUnit = N.m.

RESILIENCETotal strain energy stored in body within elastic limit.

Resilience = u = (ς

^{2}/2E) x V

PROOF RESILIENCE (up)Maximum strain energy that can be stored in body at elastic limit.

Proof resilience = up = (ςE

^{2}/2E) x VςE = Stress at elastic limit

MODULUS OF RESILIENCE (um)Maximum strain energy that can be stored in body per unit volume at elastic limit is called M.O.R.

M.O.R = (ςE )

^{2}/2EUnit = N.mm/mm3

Instantaneous Stress :-When body is subjected to sudden load or impact load, the stress produced.

Gradual Load :-Load increasing gradually from zero to P.ς = P/A

Sudden Load :-Body is subjected to total load P at a time without small increment.ς = 2P/A

Impact Load :-Load fall on a body from some height.

STRESS-STRAIN CURVEArea under curve is indicate energy required to cause failure.

Points on curve

A = Proportional Limit

B = Elastic Limit

C = Upper Yield Point (Yield stress)

D = Lower Yield Point

E = Ultimate Stress (Tensile stress)

F = Breaking Point (nominal)

G = Breaking Point (actual)

Different stage in curve

OA = Elastic stage

CD = Yield stage

DE = Strain Hardening

EF = Necking (waisting)

At proportional limit, stress is proportional to strain.

At elastic limit, the deformation entirely disappear on removal of the force.

Beyond elastic limit, strain increase more rapidly than stress.

Factor of safety = Ultimate stress

Working stressF.O.S is always greater than 1.

Yield stress = Yield load/Original c.s area

Ultimate stress = Ultimate stress/Original c.s area

Breaking stress = Breaking stress/Original c.s area

Actual stress = Breaking stress /Final cross sectional area

For mild steel, percentage elongation = 23 to 25%.

For mild steel, percentage reduction in area = 40 to 65%.

GAUGE LENGTH (L0)Distance between point A & B on stress-strain curve is called Gauge Length.

L0 = 5.65 √A

_{0}

BENDING MOMENTUnit = N.m or kN.m.

B.M is zero at free end of cantilever & at hinged support.

POINT OF CONTRAFLEXUREThe point in B.M diagram at which B.M change sign from +ve to –ve or –ve to +ve.

At point of contra flexure, B.M is zero.

Rate of change of S.F with respect to distance = Intensity of loading.

Rate of change of B.M with respect to distance = S.F at that section.

The point at which S.F changes sign or zero, B.M will maximum.

Slope = θ = Unit =Radian or Degree. ( 1 degree = π/180 )

Deflection = y = Unit = mm or cm.

Flexural Rigidity = Modulus of elasticity x Moment of inertia = EI

SLOPE-DEFLECTION EQUATION

BeamSlopeDeflectionCantilever beam with a point load at the free end Wl ^{2}/2EIWl ^{3}/3EICantilever beam with a uniformly distributed load wl ^{3}/6EIwl ^{4}/8EISimply supported beam with central point load Wl ^{2}/16EIWl ^{3}/48EISimply supported beam with uniformly distributed load wl ^{3}/24EI(5/384)(wl ^{3}/EI )Fixed beam with central point load Wl ^{3}/192 EIFixed beam with uniformly distributed load wl ^{4}/384 EICantilever beam with moment at the free end Ml/EI Ml ^{2}/2EICantilever beam with partially u.d.l. (7/384)(wl ^{4}/EI )Cantilever beam with gradually varying load (wl ^{4}/30EI)Simply supported beam with eccentric load (Wa ^{2}b^{2}/3EIl^{3})Fixed beam with eccentric load

BENDING EQUATION ( Flexure Equation)M/I= f/y= E/R

MAXIMUM BENDING MOMENT

BeamMaximum Bending MomentSimply supported beam with central point load Wl/4 Simply supported beam with eccentric load Wab/l Simply supported beam with uniformly distributed load wl ^{2}/8Cantilever beam with a point load at the free end WL Cantilever beam with a uniformly distributed load wl ^{2}/2

EQUATION OF SHEAR STRESSτ = FAȳ/Ib

STRUT

Member subjected to axial compressive force.

Used in roof truss and bridge trusses.

C/s area is small.

COLUMNStrut is vertical, it is known as column.

Used in concrete and steel buildings.

C/s area is large.

RADIUS OF GYRATIONRadius of gyration = k =√ I/A

SLENDERNESS RATIO (λ)Slenderness ratio = λ

= Effective length of column/Minimum radius of gyration = l

_{e}/k_{min }For column, λ is more then load carrying capacity is less.

For column, λ is less then load carrying capacity is more.

LONG COLUMNL

_{e}/d ≥ 12.L

_{e}/k_{min}≥ 50.For mild steel, λ ≥ 80 is called Long Column.

SHORT COLUMNl

_{e}/d < 12.l

_{e}/kmin < 50.For mild steel, λ< 80 is called Short Column.

Crushing Load :-The load at which short column fails by crushing is called Crushing Load.

Crippling Load :-The load at which long column starts buckling is called Critical Load or Buckling Load.Buckling Load is always less than Crushing Load.

COLUMN END CONDITIONS & EFFECTIVE LENGTH

Column ends conditionEffective LengthBoth ends Hinged le = l Both ends Fixed le = l/2 One end Fixed & Other end Hinged le = l/√2 One end Fixed & Other end Free le = 2l Rankine constant = 1/7500 for mild steel column.

PRINCIPAL PLANEPlane on which only direct stress is acting is called Principal Plane.

Shear stress on principal plane is zero.

Major principal plane

Carry maximum direct stress.

Minor principal plane

Carry minimum direct stress.\

ANGLE OF OBLIQUITY (∅)Angle made by resultant stress with normal stress.

tan ∅ = ςt/ςn

∅ = tan−1 (𝜎𝑡/𝜎𝑛)

Axial Load :-Load acting on the longitudinal axis of column.(e = 0)

ECCENTRIC LOADLoad acting away from the longitudinal axis of column.

Eccentric load produces both direct stress(ςo) and bending stress(ςb).

Maximum stress = Direct stress + Bending stress

Maximum stress is always Compressive.

Minimum stress = Direct stress – Bending stress

Minimum stress is Compressive if ςo > ςb.

Minimum stress is Tensile if ςo < ςb.

Eccentricity (e)Horizontal distance between longitudinal axis of column and line of action of load.

Limit of Eccentricity (ςo = ςb)

Maximum distance of load from the centre of column, up to which there is no tensile stress in column is called Limit of eccentricity.

For No Tension condition in column, e ≤ ZA

For Rectangular Section, e ≤ b/6 & e ≤ d/6

For Circular section, e ≤ d/8

PRESSURE AT BASE OF DAMWeight of dam

w = (a + b) ρ (H/2)

ρ = Density of water = 10 kN/m3 = 1000 kg/m3

Total Water Pressure

P = wh

^{2}/2

Eccentricitye = (d – b/2 )

Maximum pressure

ς

_{max}= (W/b) (1+ 6e/b)Minimum pressure

ς

_{min }= (W/b) (1− 6e/b )For No sliding, Resisting force > Sliding force (μW > P)

For No crushing, Maximum pressure < Crushing pressure (ςmax < ςc)

THIN CYLINDRICAL SHELLIf t ≤ d10, then it’s Thin Cylindrical Shell.

If t > d10, then it’s Thick Cylindrical Shell.

Hoop or Circumferential Tensile Stress

Design of thin cylindrical shell is based on Hoop stress.

ςc = pd/2t

Longitudinal Stressςl = pd/4t

Circumferential Strainεc = (pd/2tE) (1 – 1/2m)

Longitudinal Strainεl = (pd/2tE) (1/2 −1/m)

Volumetric StrainδvV = (pd/4tE)( 5− 4/m )

Maximum Shear Stressτ

_{max}= pd/8t

THIN SPHERICAL SHELL

Hoop Stressςc = pd/4t

Circumferential Strainεc = (pd/4tE) (1− 1/m)

Volumetric StrainΔv/V = (pd/2tE)( 1− 1/m)

THICK CYLINDRICAL SHELLTangential Stress is Maximum at inner surface & Minimum at outer surface.

Radial Stress is Maximum at inner surface & Zero at outer surface.

Torsion :-When turning force is applied on pulley mounted on the shaft, deformation is produced in the shaft. This deformation is called Torsion.

TORQUE OR TURNING MOMENT OR TWISTING MOMENT (T)Due to turning force, moment is produced in the shaft, This moment is called Torque.

Unit = N.m or kN.m.

For Circular Shaft, T = (π/16) τ D

^{3 }

Shear Stress in Shaft :-Shaft is subjected to torque, stress resisting torque is produced in material of shaft. This stress is called Shear stress in shaft.

POLAR MOMENT OF INERTIA (J)M.I of plane area respect to axis perpendicular to plane of lamina.

Unit = mm

^{4 }For Circular Shaft, J = (π/32) D

^{4}

POLAR SECTION MODULUS (Zp)Zp = J/R

Unit = mm

^{3}For Circular Shaft, Zp = (π/16) D

^{3}

TORSIONAL RIGIDITYTorsional Rigidity = Modulus of Rigidity x Polar Moment of Inertia = C x T.

Unit = N.mm2

EQUATION OF MOTIONT/J=Cθ/l=τ/R

Tθ = Torsional Rigidity

Bending Stress = ς = 32 M/πD

^{3}Shear Stress = τ = 16 T/πD

^{3}

POWER IN H.PP = 2πNT/4500 h.p

POWER IN WATT

P = 2πNT/60 watt

Elongation of conical bar/ Elongation of prismatic bar = 1/3

In composite bar, Stress produced in part with higher coefficient of thermal expansion is Tensile.

In composite bar, Stress produced in part with lower coefficient of thermal expansion is Compressive.

There are four elastic constant E, k, G & μ.

E & μ are independent constant.

Castellated Beam used for Light Construction.

In triangular section, Maximum shear stress occurs at Mid of the height.

For Rectangular section, τ

_{max}= 1.5 τ_{avg }For Circular section, τ

_{max}= 1.33( 4/3 ) τ_{avg }

EULER’S FORMULAPE = π

^{2}EI /le 2

BeamEuler’s EquationBoth ends Hinged π ^{2}EI/l^{2 }Both ends Fixed 4π ^{2}EI/l^{2 }One end Fixed & Other end Free π ^{2}EI/4l^{2}One end Hinged & Other end Fixed 2π ^{2}EI/l^{2}In engineering material, Modulus of rigidity is always less than half the young’s modulus.

The purpose of flitched beam is to improve moment of resistance over the section & equalise the strength in tension and compression.

SPRINGSpring absorbs shock and vibration is called Leaf spring.

Spring StiffnessLoad required to produce unit deflection in spring is called Spring Stiffness.

Spring in InstrumentUseIn spring balance To measure force In watches Absorb shock and vibration In car Store strain energy In break & clutches To apply forces Radius of Mohr circle of stress represent Shear stress.

The extremities of any diameter on Mohr’s circle represent normal stresses on planes at 45°.

Reaction in Propped Cantilever Beam = 4.

The Self Weight of Beam will be taken as Uniformly Distributed Load.

Example of Shearing Failure = Making hole in paper using punch.

Shear centre of section is which the load must applied to produce zero twisting moment on the section.

SHAPE OF S.F. & B.M. DIAGRAM

BeamS.FB.MCantilever beam with point load on free end Rectangle Right angle triangle Cantilever beam with u.d.l Right angle triangle Parabolic Cantilever beam with gradually varying load Right angle triangle Parabolic Simply supported beam with central point load Two same rectangle Triangle Simply supported beam with u.d.l Two right angle triangle Parabolic Simply supported beam with eccentric load Two rectangle Triangle Strain energy due to the Shear stress = τ

^{2}/2C x VStrain energy due to the Torsion = τ

^{2}/4C x V

RIVETSRivets made of ductile type material.

Tearing of plate at an edge can be avoided by keeping the margin atleast = 1.5 d.

Tearing of the plate across the row of rivets, required resistance or pull P

_{t}= p.d.t.ς_{t}.Shearing of rivets, pull required Ps = n.(π/4) d

^{2}τ for single shear &Ps = (π/2) d

^{2}τ for double shear.Crushing of rivets, pull required Pc = n.d.t.ςc

Strength of rivet joint is equal to least of Pc, Ps & Pt .

Efficiency = least of Pc Ps Pt /ptd

_{t}

Chain rivetRivets in the various rows are opposite to each other is called chain rivet.

Zigzag rivetRivets in adjacent row are staggered in such a way that every rivet is in middle of two rivets of opposite row then joint is called zigzag rivet.

Diamond rivetNumber of rivets decreases from inner most row to outer most row.

PitchDistance from the centre of one rivet to centre of the next rivet measured parallel to the seam.

Back pitchPerpendicular distance between the centre lines of the successive rows.

Diagonal pitchDistance between the centres of the rivets in adjacent rows.

MarginDistance between the centre of rivet hole to the nearest edge of the plate.

For Riveting, size of hole drilled is greater than shank diameter of rivet.

The object of caulking in riveted joint is to make joint leak proof.

Transverse fillet welded joints are designed for Tensile strength.

Parallel fillet welded joints are designed for Shear or Bending strength.

The problem of thick cylinder is complex and it solved by Lame’s theory.

As per Rankine’s theory the horizontal thrust offered by the retaining wall

P = wh

^{2}/2((1−sin∅)/(1+sin∅ ))

FOR RECTANGULAR BEAMIf the width is double, then the strength is 2 times greater.

If the depth is double, then strength is 4 times greater.

If the length is double, then strength is ½ times greater.

In case of beam, Compressive stress is on Bottom layer.

In case of Beam, Tensile stress is on Top layer.

Unwin formula, d = 6√t