Solid mechanics Learning summary

1. 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

2. Solid mechanicsLearning summary • Fatigue • Fracture mechanics • Thermal stresses. An Introduction to Mechanical Engineering: Part Two

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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