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Solid mechanics Learning summary

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Solid mechanicsLearning summary

By the end of this chapter you should have learnt about:

- Combined loading
- Yield criteria
- Deflection of beams
- Elastic-plastic deformations
- Elastic instability
- Shear stresses in beams
- Thick cylinders
- Asymmetrical bending
- Strain energy

An Introduction to Mechanical Engineering: Part Two

Solid mechanicsLearning summary

- Fatigue
- Fracture mechanics
- Thermal stresses.

An Introduction to Mechanical Engineering: Part Two

3.2 Combined loading – key points

By the end of this section you should have learnt:

- the basic use of Mohr’s circle for analysing the general state of plane stress
- how the effect of combined loads on a component can be analysed by considering each load asinitially having an independent effect
- how to use the principle of superposition to determine the combined effect of these loads.

An Introduction to Mechanical Engineering: Part Two

3.3 Yield criteria – key points

By the end of this section you should have learnt:

- the difference between ductile and brittle failure, illustrated by the behaviour of bars subjected touniaxial tension and torsion
- the meaning of yield stress and proof stress, in uniaxial tension, for a material
- the Tresca (maximum shear stress) yield criterion and the 2D and 3D diagrammatic representationsof it
- the von Mises (maximum shear strain energy) yield criterion and the 2D and 3D diagrammatic representations of it.

An Introduction to Mechanical Engineering: Part Two

3.4 Deflection of beams – key points

By the end of this section you should have learnt:

- how to derive the differential equation of the elastic line (i.e. deflection curve) of a beam
- how to solve this equation by successive integration to yield the slope, dy/dx, and the deflection, y, of abeam at any position along its span
- how to use Macaulay’s method, also called the method of singularities, to solve for beam deflections
- where there are discontinuities in the bending moment distribution arising from discontinuousloading

An Introduction to Mechanical Engineering: Part Two

3.4 Deflection of beams – key points

- how to use different singularity functions in the bending moment expression for different loadingconditions including point loads, uniformly distributed loads and point bending moments
- how to use Macaulay’s method for statically indeterminate beam problems.

An Introduction to Mechanical Engineering: Part Two

3.5 Elastic-plastic deformations – key points

By the end of this section you should have learnt:

- the shapes of uniaxial stress-strain curves and the elastic–perfectly plastic approximation touniaxial stress-strain curves
- the kinematic and isotropic material behaviour models used to represent cyclic loading behaviour
- the elastic-plastic bending of beams and the need to use equilibrium, compatibility and behaviour to solve these types of problems

An Introduction to Mechanical Engineering: Part Two

3.5 Elastic-plastic deformations – key points

- the elastic–plastic torsion of shafts and the need to use equilibrium, compatibility and behaviour to solve these types of problems
- how to determine residual deformations and residual stresses.

An Introduction to Mechanical Engineering: Part Two

3.6 Elastic instability – key points

By the end of this section you should have learnt:

- Macaulay’s method for determining beam deflection in situations with axial loading
- the meanings of and the differences between stable, unstable and neutral equilibria
- how to determine the buckling loads for ideal struts
- the effects of eccentric loading, initial curvature and transverse loading on the buckling loads
- how to include the interaction of yield behaviour with buckling and how to represent this interactiongraphically.

An Introduction to Mechanical Engineering: Part Two

3.7 Sheer stresses in beams – key points

By the end of this section you should have learnt:

- that in addition to longitudinal bending stresses, beams also carry transverse shear stresses arisingfrom the vertical shear loads acting within the beam
- how to derive a general formula, in both integral and discrete form, for evaluating the distributionof shear stresses through a cross section
- how to determine the distribution of the shear stresses through the thickness in a rectangular,circular and I-section beam

An Introduction to Mechanical Engineering: Part Two

3.7 Sheer stresses in beams – key points

- that we can identify the shape of required pumps by calculating the specific speed without knowingthe size of the pump.

An Introduction to Mechanical Engineering: Part Two

3.8 Thick cylinders – key points

By the end of this sections you should have learnt:

- the essential differences between the stress analysis of thin and thick cylinders, leading to anunderstanding of statically determinate and statically indeterminate situations
- how to derive the equilibrium equations for an element of material in a solid body (e.g. a thickcylinder)
- the derivation of Lame’s equations
- how to determine stresses caused by shrink-fitting one cylinder onto another

An Introduction to Mechanical Engineering: Part Two

3.8 Thick cylinders – key points

- how to include ‘inertia’ effects into the thick cylinder equations in order to calculate the stresses in arotating disc.

An Introduction to Mechanical Engineering: Part Two

3.9 Asymmetrical bending – key points

By the end of this section you should have learnt:

- that an asymmetric cross section, in addition to its second moments of area about the x- and y- axes, Ix and Iy, possesses a geometric quantity called the product moment of area, Ixy, with respect tothese axes
- how to calculate the second moments of area and the product moment of area about aconvenient set of axes

An Introduction to Mechanical Engineering: Part Two

3.9 Asymmetrical bending – key points

- that an asymmetric section will have a set of axes at some orientation for which the product moment ofarea is zero and that these axes are called the principal axes
- that the second moments of area about the principal axes are called the principal second moments ofarea
- how to determine the second moments of area and the product moment of area about anyoriented set of axes, including the principal axes, using a Mohr’s circle construction

An Introduction to Mechanical Engineering: Part Two

3.9 Asymmetrical bending – key points

- that it is convenient to analyse the bending of a beam with an asymmetric section by resolving bendingmoments onto the principal axes of the section
- how to follow a basic procedure for analysing the bending of a beam with an asymmetric cross section.

An Introduction to Mechanical Engineering: Part Two

3.10 Strain energy – key points

By the end of this section you should have learnt:

- the basic concept of strain energy stored in an elastic body under loading
- how to calculate strain energy in a body/structure arising from various types of loading, includingtension/compression, bending and torsion
- Castigliano’s theorem for linear elastic bodies, which enables the deflection or rotation of a body ata point to be calculated from strain energy expression.

An Introduction to Mechanical Engineering: Part Two

3.11 Fatigue – key points

By the end of this section you should have learnt:

- the various stages leading to fatigue failure
- the basis of the total life and of the damage-tolerant approaches to estimating the number ofcycles to failure
- how to include the effects of mean and alternating stress on cycles to failure using the Gerber,modified Goodman and Soderberg methods
- how to include the effect of a stress concentration on fatigue life
- the S–N design procedure for fatigue life.

An Introduction to Mechanical Engineering: Part Two

3.12 Fracture mechanics – key points

By the end of this section you should have learnt:

- the meaning of linear elastic fracture mechanics (LEFM)
- what the three crack tip loading modes are
- the energy and stress intensity factor (Westergaard crack tip stress field) approaches to LEFM
- the meaning of small-scale yielding and fracture toughness
- the Paris equation for fatigue crack growth and the effects of the mean and alternatingcomponents of the stress intensity factor.

An Introduction to Mechanical Engineering: Part Two

3.13 Thermal stresses – key points

By the end of this section you should be able to:

- understand the cause of thermal strains and how ‘thermal stresses’ are caused by thermal strains
- include thermal strains in the generalized Hooke’s Law equations
- include the temperature distribution within a solid component (e.g. a beam, a disc or a tube) in thesolution procedure for the stress distribution
- understand that stress/strain equations include thermal strain terms but the equilibrium and compatibility equations are the same whether the component is subjected to thermal loading ornot.

An Introduction to Mechanical Engineering: Part Two

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