**Strength of materials** (sometimes called *Mechanics of materials*) is the branch of engineering science concerned with the ability of structures and components to withstand loads imposed during service without experiencing fracture, collapse or unacceptable levels of deflection.

The subject has wide application particularly in structural and civil engineering, the design of machinery and process equipment and in aerospace and marine engineering.

Key concepts are the **stresses** generated in components by specific configurations of loading and the resulting deformations known as **strains**. The relationship between stress and strain is unique to each material.

The tutorials listed below cover some core elements of the subject at a level broadly equivalent to the first year of an engineering first degree. They are not a substitute for a text book or formal lecture notes but might help in understanding tricky points of detail not necessarily mentioned elsewhere.

Tutorials in the **Strength of materials** series are as follows.

Stress and strain - Basic concepts

Definitions of normal and shear stress; triaxial and biaxial stress models; Mohr's circle and definition of principal stresses; definition of strain and shear strain; Poisson's ratio; relationship between stress and strain, modulus of elasticity and shear modulus.

Beams - Terminology and sign conventions

Types of support and loading configurations; sign convention for loads and reaction forces/moments; definitions of positive and negative bending; definitions and sign conventions for bending moments and shearing forces.

Beams - bending moments and shearing forces - simply supported beam with point loads

Calculation of reaction forces, shearing forces (V) and bending moments (M) from first principles for a beam with simple supports and point loads. Demonstrates relationship between V and M. Construction of shearing force and bending moment diagrams.

Beams - bending moments and shearing forces - cantilever with uniform distributed load

Calculation of reaction force, reaction moment, shearing forces (V) and bending moments (M) from first principles for a cantilever beam with uniform distributed load (w). Demonstrates relationship between V, M and w. Construction of shearing force and bending moment diagrams. Derivation by integration of a general expression for M for this example.

Beams - bending moments and shearing forces - simply supported beam with variable distributed load

Calculation of shearing forces and bending moments for a simply supported beam with a variable distributed load w = f(x) by integration using the established relationships between V, M and w.

Bending moments and shearing forces in beams - Singularity functions

Definition, interpretation and rules for application of singularity functions to beam loading for applied moments (M), concentrated forces (F), uniformly distributed loads (w) and linear variable distributed loads (s) with a worked example.

Normal bending stresses in beams

Derivation and interpretation of the expression **σ = -My/I _{z}** for normal stress over the cross sectional area of a loaded beam of uniform cross-section; mathematical derivation of the second moment of area about a centroidal axis for a rectangular cross section.

Shear bending stresses in beams

Derivation and interpretation of the expression **τ = VQ/I _{z}b** for shear stress over the cross sectional area of a loaded beam of uniform rectangular cross-section.

Deflection of beams in bending

Derivation of the relationship **M/EI = d ^{2}y/dx^{2}** for a beam in bending; derivation of expressions for slope and deflection for a simply supported beam with uniform distributed load.

Example of determining tensile and compressive forces in the bars of a structural truss by drawing free body diagrams for each pin joint.

Derivation of the relationship **τ = Tr/J** for shear stress over the cross section of a cylindrical shaft produced by a torque load; qualitative analysis of normal stress on a cylindrical shaft arising from torque loading.

Derivation of the Euler column formula **P _{cr} = π^{2}(EI/L^{2})** for critical loading of a slender column with end pivot supports; introduction of factor

**K**to account for combinations of different end supports.

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