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Mechanical Engineering

What is required to relate the stress distribution to the resultant bending moment?

Mechanics of Materials (MindTap Course List)

9th Edition

ISBN:9781337093347

Author:Barry J. Goodno, James M. Gere

Publisher:Barry J. Goodno, James M. Gere

Chapter7: Analysis Of Stress And Strain

Section: Chapter Questions

Problem 7.4.8P: - A specimen used in a coupon test has norm al stress ??. = 15 MPa (see figure). Using Mohr’s...

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Concept explainers

Combined Loading

Any functioning component, such as machine parts, structural members, pressure vessels, and so on, are all under the influence of more than one force. When anybody is under the influence of more than one force such as shear forces, bending moments, axial forces, torsion, and so on, then such body is said to be under combined loading. A very practical example of combined loading is the crankshaft of an internal combustion engine (IC engine). The crankshaft of the IC engine is under high rpm, which is caused due to the torsional forces. Due to the presence of piston, counterbalances, and internal imbalances, the crankshaft experiences lateral forces and due to the rise in temperature, the crankshaft undergoes thermal expansion, which pulls the crankshaft along the axial direction and lateral direction. The presence of unbalanced loads along the periphery of the crankshaft also induces a bending moment.

Question

What is required to relate the stress distribution to the resultant bending moment?

Expert Solution

Step 1

The restoring force per unit area which is equal in magnitude of the applied force but in opposite direction is known as stress.

The distribution of stress in a curved flexural member is expressed as $σ = ε E = \frac{E (r - ρ) d x}{ρ x}$

now the sum of the normal force acting is equal to zero that is

$\int_{}^{} σ d A = E \frac{d x}{x} \int_{}^{} \frac{(r - ρ) d A}{ρ} = 0 E \frac{d x}{x} [\int_{}^{} \frac{(r) d A}{ρ} - \int_{}^{} d A] = 0 \int_{}^{} \frac{(r) d A}{ρ} - A = 0 r = \frac{A}{\int_{}^{} \frac{d A}{ρ}}$

this equation indicates that the neutral axis and centroidal axis doesn't coincide

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