Machine Design for SSC JE Mechanical: Stresses, Shafts, Keys, Springs, Gears, Bearings, Failure Theories and Fatigue

Machine design applies the principles of strength of materials and mechanics to design the parts of machines. SSC JE Mechanical asks formula-based and concept-based questions on shafts, keys, springs, gears, bearings, the theories of failure and fatigue. This guide on machine design for SSC JE gives all of these in tables, with solved examples, practice questions and FAQs.

Basic Concepts

• Factor of safety = ultimate stress / working stress (for brittle materials) or yield stress / working stress (for ductile materials).
• Stress concentration occurs near holes, notches and sudden changes of section; the stress concentration factor K_t = maximum stress / nominal stress; it is more important for brittle materials and under fatigue loading.
• Preferred numbers (Renard series R5, R10, R20 and R40) are used for standard sizes; they follow a geometric progression with ratios of 1.58, 1.26, 1.12 and 1.06.
• Design stress depends on the material, the type of load (static, fatigue or impact) and the factor of safety.

Stress Formulas

Theories of Failure

Design of Shafts

• A shaft rotates and transmits power; an axle does not rotate and carries only the bending load; a spindle is a short shaft.
• For a shaft under combined bending moment M and torque T: equivalent torque T_e = √(M² + T²) (maximum shear stress theory) and equivalent bending moment M_e = ½[M + √(M² + T²)] (maximum principal stress theory).
• The design of a shaft is based on strength (stress) and on rigidity (angle of twist, deflection) and critical speed.
• The critical speed of a shaft is the speed at which resonance occurs, N_c = 60/(2π) √(g/δ).
• A hollow shaft is stronger and lighter than a solid shaft of the same weight.

Keys and Couplings

Springs

• Helical spring stiffness k = G d⁴ / (8 D³ n), where G is the modulus of rigidity, d is the wire diameter, D is the mean coil diameter and n is the number of active coils.
• Shear stress in the wire τ = K_w (8 W D / π d³), where K_w is the Wahl factor, which depends on the spring index C = D/d.
• Springs in series: 1/k = 1/k₁ + 1/k₂; in parallel: k = k₁ + k₂.
• Leaf spring: bending stress σ = 3WL/(2nbt²).
• Energy stored in a spring = ½ k x².

Gears

Belt Drives and Bearings

• For a flat belt, the ratio of tensions T₁/T₂ = e^μθ; power = (T₁ − T₂) v.
• V-belts transmit more power than flat belts because of the wedging action, T₁/T₂ = e^μθ/sin(β).
• Chain drives have no slip and give a constant velocity ratio.
• Rolling contact bearings (ball and roller) have a lower starting friction; life L₁₀ = (C/P)³ million revolutions for ball bearings (exponent 10/3 for roller bearings).
• Sliding contact bearings (journal bearings) run on a film of lubricant; the Sommerfeld number S = (μN/p)(r/c)² is a design parameter.
• Bearing materials: babbitt, bronze and white metal.

Fatigue

• Fatigue failure occurs under repeated or fluctuating stress at stress levels below the yield stress.
• Endurance limit is the maximum stress amplitude that a material can withstand for an infinite number of cycles; for steels it is about 0.5 times the ultimate tensile strength (less for other materials).
• The S–N curve shows stress against the number of cycles; for steel the curve becomes horizontal after about 10⁶ to 10⁷ cycles.
• Failure criteria for fluctuating loads: Goodman line, Soderberg line (most conservative), Gerber parabola.
• Surface finish, size, reliability and the stress concentration reduce the endurance limit; shot peening increases the fatigue life.

Fasteners and Joints

• In a riveted joint, the failure modes are tearing, shearing, crushing and tearing at the edge; efficiency = least of the strengths / strength of the solid plate.
• A threaded fastener: the stress area is smaller than the major diameter area; the lock nut and the spring washer prevent loosening.
• Welded joints: the strength of a fillet weld depends on the throat thickness (0.707 × size).
• Power screws: square threads have the highest efficiency; ACME threads are used in lead screws and a lathe; a self-locking screw has an efficiency of less than 50%.

Solved Examples

Example 1. A shaft transmits 10 kW at 300 rpm. Find the torque.
T = 60P / 2πN = 60 × 10,000 / (2π × 300) = 318.3 N m.

Example 2. Find the diameter of a solid shaft for the above torque if the allowable shear stress is 40 N/mm².
d³ = 16T/(πτ) = 16 × 318,300 / (π × 40) = 40,528 mm³, so d = 34.4 mm (say 35 mm).

Example 3. Find the stiffness of a helical spring with G = 80,000 N/mm², d = 8 mm, D = 64 mm and n = 10.
k = 80,000 × 8⁴ / (8 × 64³ × 10) = 15.6 N/mm.

Example 4. A gear has a pitch circle diameter of 120 mm and 30 teeth. Find the module.
m = 120/30 = 4 mm.

Quick Tips

• Learn the theories of failure with the materials for which they are suitable.
• Memorise the shaft formulas T = 60P/2πN and τ = 16T/πd³.
• Remember the minimum teeth 17 to avoid interference at 20°.
• Remember that Soderberg is the most conservative failure criterion.

Quick Revision Points

• The von Mises theory is the best for ductile materials.
• Module = D/z.
• The endurance limit of steel is about 0.5 of its ultimate strength.
• Rolling bearing life: L₁₀ = (C/P)³.
• Wahl factor accounts for the curvature and direct shear in springs.

Practice Questions With Answers

1. Which theory of failure is best for ductile materials?
   Answer: Distortion energy (von Mises) theory
2. Which theory is suitable for brittle materials?
   Answer: Maximum principal stress theory
3. What is the module of a gear?
   Answer: D/z
4. What is the minimum number of teeth to avoid interference for 20° pressure angle?
   Answer: 17
5. What is the stiffness of a helical spring?
   Answer: Gd⁴/(8D³n)
6. What is the life of a ball bearing formula?
   Answer: L₁₀ = (C/P)³ million revolutions
7. Which key is semi-circular?
   Answer: Woodruff key
8. What is the most conservative fatigue failure line?
   Answer: Soderberg line
9. What is the unit of the endurance limit?
   Answer: N/mm² (stress)
10. What is the polar moment of inertia of a solid shaft?
    Answer: πd⁴/32

How to Practise This Chapter

Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Mechanical Engineering notes page, read the previous topic Heat and Mass Transfer and the next topic Steam Boilers, Turbines and Power Plants.

Frequently Asked Questions

What is the factor of safety?

The ratio of the failure stress to the working stress.

What is stress concentration?

The localised increase in stress near a hole, notch or change of section.

What is the difference between a shaft and an axle?

A shaft rotates and transmits torque; an axle does not rotate and carries only bending.

What is module?

The ratio of the pitch circle diameter to the number of teeth.

What is interference in gears?

The mating of non-conjugate parts of the tooth profiles, avoided by a minimum number of teeth.

What is the endurance limit?

The stress level below which a material does not fail by fatigue.

What is the Wahl factor?

A correction factor for the stress in a helical spring due to curvature and direct shear.

What is a key?

A machine element used to connect a shaft and a hub to transmit torque.

What is the law of gearing?

The common normal at the point of contact must pass through the pitch point.

Is this topic asked in SSC JE?

Yes. Machine design is asked in the mechanical paper.

What is the critical speed of a shaft?

The speed at which the shaft resonates and whirls violently.

What is the Sommerfeld number?

A dimensionless number used in the design of journal bearings.