4.3 Strength of Materials

4.3 Strength of Materials

1. Centre of Gravity and Centroid

1. Centre of Gravity (CG):

   • Point where total weight appears to act.

   • Depends on weight distribution and gravity field.

   • For homogeneous bodies in uniform g-field: coincides with centroid.

2. Centroid:

   • Geometric center of area/volume.

   • Depends only on shape, not material.

   • Also called "center of area" or "center of volume".

3. Calculation Methods:

   • Simple shapes: use symmetry.

   • Composite shapes: use weighted averages.

   • For area: $\bar{x} = \frac{\sum A_i x_i}{\sum A_i}, \quad \bar{y} = \frac{\sum A_i y_i}{\sum A_i}$

4. Applications:

   • Determines where resultant forces act.

   • Used in beam bending, stability analysis.

2. Moment of Inertia

1. Area Moment of Inertia ($I$):

   • Measures resistance to bending.

   • $I = \int y^2 dA$ (about x-axis).

   • Units: m⁴ or mm⁴.

2. Radius of Gyration ($k$):

   • $k = \sqrt{\frac{I}{A}}$

   • Characteristic length for buckling.

3. Parallel Axis Theorem:

   • Transfer between parallel axes.

   • $I = I_c + A d^2$

   • $I_c$ = centroidal moment, $d$ = distance between axes.

4. Perpendicular Axis Theorem:

   • For thin plates: $I_z = I_x + I_y$

   • Only valid for planar laminas.

5. Polar Moment of Inertia ($J$):

   • Resistance to torsion.

   • $J = \int r^2 dA = I_x + I_y$

3. Shear Force and Bending Moment

1. Types of Beams:

   • Simply supported, Cantilever, Fixed, Continuous, Overhanging.

2. Load Types:

   • Point load, Distributed load (UDL, UVL), Moment.

3. Shear Force (V):

   • Algebraic sum of vertical forces to left/right of section.

   • Positive: upward on left face.

4. Bending Moment (M):

   • Algebraic sum of moments about section.

   • Positive: causes sagging (compression on top).

5. Relationships:

   • $\frac{dV}{dx} = -w$ (load intensity).

   • $\frac{dM}{dx} = V$.

   • Area under shear diagram = change in moment.

6. Shear and Moment Diagrams:

   • Essential for determining maximum values.

   • Identify points of zero shear for max moment.

4. Beam Deflection

1. Deflection ($v$ or $y$):

   • Vertical displacement of beam neutral axis.

   • Important for serviceability limits.

2. Slope ($\theta$):

   • First derivative of deflection: $\theta = \frac{dv}{dx}$.

3. Differential Equation:

   • $EI \frac{d^2v}{dx^2} = M(x)$

   • $EI \frac{d^4v}{dx^4} = w(x)$

4. Boundary Conditions:

   • Simply supported: $v=0$ at supports.

   • Fixed: $v=0$ and $\theta=0$.

   • Free end: no displacement/rotation constraints.

5. Methods:

   • Double integration (basic).

   • Macaulay's method (discontinuities).

   • Moment-area method.

   • Superposition principle.

6. Maximum Deflection:

   • Occurs where slope = 0.

   • Limited by design codes (e.g., L/360).

5. Truss Analysis

1. Truss Definition:

   • Structure of straight members connected at joints.

   • Loads applied only at joints.

   • Members carry only axial forces (tension/compression).

2. Assumptions:

   • Members are weightless (or weight lumped at joints).

   • Joints are frictionless pins.

   • Loads act only at joints.

3. Analysis Methods:

   • Method of Joints:

     • Equilibrium at each joint ($\sum F_x=0, \sum F_y=0$).

     • Suitable for finding all member forces.

   • Method of Sections:

     • Cut through truss, apply equilibrium to section.

     • Efficient for specific members.

4. Zero-Force Members:

   • Identified by joint geometry and loading.

   • Can carry no load under current configuration.

5. Stability and Determinacy:

   • $m + r = 2j$ (statically determinate).

   • $m + r < 2j$ (unstable).

   • $m + r > 2j$ (statically indeterminate).

6. Torsion of Shafts

1. Pure Torsion:

   • Twisting moment about longitudinal axis.

   • Produces shear stress, no bending.

2. Torsion Formula:

   • $\frac{T}{J} = \frac{\tau}{r} = \frac{G\theta}{L}$

   • $T$ = applied torque, $J$ = polar moment.

   • $\tau$ = shear stress at radius $r$.

   • $\theta$ = angle of twist (radians).

3. Shear Stress Distribution:

   • Linear from center: $\tau = \frac{Tr}{J}$.

   • Maximum at outer surface: $\tau_{max} = \frac{TR}{J}$.

4. Angle of Twist:

   • $\theta = \frac{TL}{GJ}$.

   • Proportional to length and torque.

   • Inversely proportional to $GJ$ (torsional rigidity).

5. Power Transmission:

   • $P = T\omega = \frac{2\pi NT}{60}$ (SI).

   • $P$ in watts, $T$ in N·m, $N$ in RPM.

6. Hollow vs Solid Shafts:

   • More efficient material use for same torque.

   • Same weight: hollow has larger diameter, less stress.

7. Composite Shafts:

   • Series/parallel combinations.

   • Compatibility: same angle of twist in series.

   • Equilibrium: sum of torques in parallel.