Chapter – 2 : JSSC JE Mechanical Notes

ENGINEERING MECHANICS & STRENGTH OF MATERIALS

Equilibrium of Forces

• A body is in equilibrium when the resultant force and resultant moment are zero.
• For coplanar concurrent forces:

  \( \sum F_x = 0, \quad \sum F_y = 0 \)

• For general coplanar forces:

  \( \sum F_x = 0, \quad \sum F_y = 0, \quad \sum M = 0 \)

• Resultant of two forces \( P \) and \( Q \) with included angle \( \theta \):

\( R = \sqrt{P^2 + Q^2 + 2PQ\cos\theta} \)

• Direction of resultant:

\( \tan\alpha = \frac{Q\sin\theta}{P + Q\cos\theta} \)

• Moment of force: \( M = F \times \) perpendicular distance.
• Couple: Two equal, opposite and parallel forces separated by a distance.
• Moment of a couple = \( F \times \) arm.
• Free-body diagram (FBD) shows all external forces and reactions acting on the body.

Laws of Motion

• Newton’s First Law: A body remains at rest or in uniform motion unless acted upon by an external unbalanced force.
• Newton’s Second Law:

\( F = ma = \frac{dp}{dt} \)

• Newton’s Third Law: Every action has an equal and opposite reaction.
• Linear momentum: \( p = mv \)
• Impulse: \( J = F\Delta t = \) change in momentum.
• Work: \( W = F \times \) displacement in the direction of force.
• Kinetic energy: \( KE = \frac{1}{2}mv^2 \)
• Potential energy: \( PE = mgh \)
• Power: \( P = \) Work/time

Friction

• Friction opposes relative motion or tendency of relative motion between contacting surfaces.
• Types: static friction, limiting friction, kinetic/sliding friction, rolling friction.
• Static friction adjusts itself up to a maximum value.
• Limiting friction:

\( F_{lim} = \mu N \)

• Coefficient of friction:

\( \mu = \frac{F_{lim}}{N} \)

• Angle of friction \( \phi \): \( \tan\phi = \mu \)
• Angle of repose: \( \tan\theta = \mu \)
• Therefore, angle of repose = angle of friction.
• For impending motion: friction = limiting friction.
• For sliding motion: friction ≈ \( \mu_k N \).
• Generally, \( \mu_{static} > \mu_{kinetic} \).

Stress and Strain

• Stress: Internal resisting force per unit area.

\( \sigma = \frac{P}{A} \)

• Unit: Pa = N/m²; commonly MPa or N/mm².
• Types of normal stress: tensile and compressive.
• Shear stress:

\( \tau = \frac{V}{A} \)

• Strain: Deformation per unit original dimension.
• Longitudinal strain:

\( \varepsilon = \frac{\Delta L}{L} \)

• Strain is dimensionless.
• Shear strain: \( \gamma = \) angular deformation.
• Volumetric strain:

\( \varepsilon_v = \frac{\Delta V}{V} \)

Stress-Strain Curve

• Proportional limit: Stress is proportional to strain.
• Elastic limit: On removal of load, material returns to original shape.
• Yield point: Large deformation occurs with little/no increase in stress.
• Ultimate tensile stress: Maximum engineering stress.
• Fracture point: Material breaks.
• Area under stress-strain curve up to elastic limit → modulus of resilience.
• Total area under curve up to fracture → modulus of toughness.

Elastic Limit & Elastic Constants

• Young’s modulus:

\( E = \frac{\sigma}{\varepsilon} \)

• Modulus of rigidity / shear modulus:

\( G = \frac{\tau}{\gamma} \)

• Bulk modulus:

\( K = \frac{\text{volumetric stress}}{\text{volumetric strain}} \)

• Poisson’s ratio:

\( \nu = -\frac{\text{lateral strain}}{\text{longitudinal strain}} \)

• For isotropic elastic materials:

\( E = 2G(1+\nu) \)

\( E = 3K(1-2\nu) \)

\( E = \frac{9KG}{3K+G} \)

• For a stable isotropic material generally: \( -1 < \nu < 0.5 \).

Shear Force & Bending Moment

• Shear force (SF): Algebraic sum of vertical forces on one side of a section.
• Bending moment (BM): Algebraic sum of moments of forces about the section.
• Important relation:

\( \frac{dM}{dx} = V \)

• Therefore, slope of BMD = shear force.
• \( \frac{dV}{dx} = -w \) for a distributed load \( w \) under the common sign convention.
• Therefore, slope of SFD = −load intensity.
• Point load → sudden jump in SFD.
• Point moment → sudden jump in BMD.
• UDL → linear SFD and parabolic BMD.
• UVL → parabolic SFD and cubic BMD.
• Maximum/minimum BM generally occurs where \( SF = 0 \).
• At a simple support with no applied couple, BM is generally zero.

Standard Beam Results

Bending Stress

• Flexure equation:

\( \frac{M}{I} = \frac{\sigma}{y} = \frac{E}{R} \)

• Section modulus:

\( Z = \frac{I}{y_{max}} \)

• Maximum bending stress:

\( \sigma_{max} = \frac{M}{Z} \)

• For a rectangular section:

\( I = \frac{bd^3}{12} \)

• For a solid circular section:

\( I = \frac{\pi d^4}{64} \)

Stress in Composite Bars

• A composite bar consists of two or more materials acting together.
• Example: steel + brass, steel + concrete.
• For members rigidly connected and subjected to axial loading, deformation compatibility must be satisfied.
• For equal length and bonded materials:

\( \varepsilon_1 = \varepsilon_2 \)

• Since \( \sigma = E\varepsilon \):

\( \frac{\sigma_1}{E_1} = \frac{\sigma_2}{E_2} \)

• Hence:

\( \frac{\sigma_1}{\sigma_2} = \frac{E_1}{E_2} \)

• Total load:

\( P = \sigma_1 A_1 + \sigma_2 A_2 \)

• Material with higher \( E \) carries higher stress when both materials have the same strain.

Torsion of Circular Shafts

• Torsion = twisting of a shaft due to applied torque.
• Basic torsion equation:

\( \frac{T}{J} = \frac{\tau}{R} = \frac{G\theta}{L} \)

• \( T \) = torque
• \( J \) = polar moment of inertia
• \( \tau \) = shear stress
• \( R \) = outer radius
• \( G \) = modulus of rigidity
• \( \theta \) = angle of twist
• \( L \) = shaft length

Polar Moment of Inertia

• Solid circular shaft:

\( J = \frac{\pi d^4}{32} \)

• Hollow circular shaft:

\( J = \frac{\pi (D^4 – d^4)}{32} \)

• Maximum shear stress in solid shaft:

\( \tau_{max} = \frac{16T}{\pi d^3} \)

• Power transmitted:

\( P = \frac{2\pi N T}{60} \)

• In kW:

\( P = \frac{2\pi N T}{60 \times 1000} \)

Buckling of Columns

• A column is primarily subjected to compressive axial load.
• Buckling = sudden lateral deflection of a slender column under compressive load.
• Short column → mainly crushing.
• Long column → mainly buckling.
• Critical load = load at which buckling begins.
• Effective length:

Slenderness Ratio

Slenderness ratio = \( \frac{L_e}{k} \)

• \( k \) = radius of gyration.
• Radius of gyration:

\( k = \sqrt{\frac{I}{A}} \)

• Higher slenderness ratio → greater tendency to buckle.

Euler’s Column Theory

• Euler theory applies mainly to long, slender columns.
• Critical buckling load:

\( P_{cr} = \frac{\pi^2 EI}{L_e^2} \)

• For both ends hinged:

\( P_{cr} = \frac{\pi^2 EI}{L^2} \)

• Euler load is proportional to \( E \) and \( I \).
• Euler load is inversely proportional to \( L_e^2 \).
• Maximum buckling resistance occurs about the axis having the largest moment of inertia.

Rankine’s Theory

• Rankine theory considers both crushing and buckling.
• Rankine formula:

\( P = \frac{\sigma_c A}{1 + a(L_e/k)^2} \)

• \( \sigma_c \) = crushing stress
• \( A \) = cross-sectional area
• \( a \) = Rankine constant depending on material/end condition/convention
• \( L_e/k \) = slenderness ratio
• For short columns, Rankine result approaches crushing load.
• For long columns, Rankine behaviour approaches Euler buckling behaviour.

Euler vs Rankine

Thin-Walled Pressure Vessels

• Used for vessels containing fluid/gas under internal pressure.
• Thin cylinder condition is commonly taken approximately as:

\( t < d/20 \)

• \( t \) = wall thickness; \( d \) = internal diameter.
• For thin vessels, radial stress is small compared with membrane stresses and is usually neglected.

Thin Cylindrical Pressure Vessel

• Hoop/Circumferential stress:

\( \sigma_h = \frac{pd}{2t} \)

• Longitudinal stress:

\( \sigma_l = \frac{pd}{4t} \)

• Therefore:

\( \sigma_h = 2\sigma_l \)

• Hence, hoop stress is twice the longitudinal stress.

Thin Spherical Pressure Vessel

• For a thin spherical shell:

\( \sigma = \frac{pd}{4t} \)

• Spherical vessels develop equal membrane stress in all tangential directions.
• For the same \( p \), \( d \) and \( t \), spherical vessel has lower membrane stress than a cylindrical vessel.

Change in Dimensions of Thin Cylinder

• Using plane stress relations, circumferential strain:

\( \varepsilon_h = \frac{\sigma_h – \nu\sigma_l}{E} \)

• Longitudinal strain:

\( \varepsilon_l = \frac{\sigma_l – \nu\sigma_h}{E} \)

• Change in diameter:

\( \Delta d = \varepsilon_h d \)

• Change in length:

\( \Delta L = \varepsilon_l L \)

High-Yield Formula Sheet

• Equilibrium: \( \sum F_x = 0 \), \( \sum F_y = 0 \), \( \sum M = 0 \)
• Newton: \( F = ma \)
• Momentum: \( p = mv \)
• Limiting friction: \( F = \mu N \)
• Stress: \( \sigma = P/A \)
• Strain: \( \varepsilon = \Delta L/L \)
• Young’s modulus: \( E = \sigma/\varepsilon \)
• Shear modulus: \( G = \tau/\gamma \)
• Poisson: \( \nu = – \)lateral strain/longitudinal strain
• Elastic relation: \( E = 2G(1+\nu) \)
• Bending: \( M/I = \sigma/y = E/R \)
• Section modulus: \( Z = I/y_{max} \)
• Torsion: \( T/J = \tau/R = G\theta/L \)
• Shaft power: \( P = 2\pi N T/60 \)
• Euler: \( P_{cr} = \pi^2 EI/L_e^2 \)
• Rankine: \( P = \sigma_c A/[1+a(L_e/k)^2] \)
• Thin cylinder hoop: \( \sigma_h = pd/(2t) \)
• Thin cylinder longitudinal: \( \sigma_l = pd/(4t) \)
• Thin sphere: \( \sigma = pd/(4t) \)

One-Minute Revision

• Equilibrium → \( \sum F = 0 \) and \( \sum M = 0 \)
• Newton 2nd law → \( F = ma \)
• Friction → \( \mu N \) at limiting condition
• Stress → Force/Area
• Strain → Deformation/Original dimension
• \( E \) → Normal stress/Normal strain
• SFD slope → −load intensity
• BMD slope → Shear force
• Maximum BM → generally where \( SF = 0 \)
• Composite bar → compatibility + equilibrium
• Torsion → \( T/J = \tau/R = G\theta/L \)
• Long column → buckling
• Euler → long slender columns
• Rankine → crushing + buckling
• Thin cylinder → hoop stress = 2 × longitudinal stress
• Thin sphere → \( \sigma = pd/(4t) \)