 # Unit 3: Solid mechanics An Introduction to Mechanical Engineering: Part Two Solid mechanics Learning summary By the end of this chapter you should have.

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Unit 3: Solid mechanics An Introduction to Mechanical Engineering: Part Two Solid mechanics Learning 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

Unit 3: Solid mechanics An Introduction to Mechanical Engineering: Part Two Solid mechanics Learning summary Fatigue Fracture mechanics Thermal stresses.

Unit 3: Solid mechanics 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 as initially having an independent effect how to use the principle of superposition to determine the combined effect of these loads.

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 to uniaxial 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 representations of it the von Mises (maximum shear strain energy) yield criterion and the 2D and 3D diagrammatic representations of it. Unit 3: Solid mechanics 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 a beam 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 discontinuous loading Unit 3: Solid mechanics 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 loading conditions including point loads, uniformly distributed loads and point bending moments how to use Macaulay’s method for statically indeterminate beam problems. Unit 3: Solid mechanics 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 to uniaxial 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 Unit 3: Solid mechanics 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. Unit 3: Solid mechanics 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 interaction graphically. Unit 3: Solid mechanics 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 arising from the vertical shear loads acting within the beam how to derive a general formula, in both integral and discrete form, for evaluating the distribution of 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 Unit 3: Solid mechanics 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 knowing the size of the pump. Unit 3: Solid mechanics 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 an understanding 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 thick cylinder) the derivation of Lame’s equations how to determine stresses caused by shrink-fitting one cylinder onto another Unit 3: Solid mechanics 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 a rotating disc. Unit 3: Solid mechanics 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, I x and I y, possesses a geometric quantity called the product moment of area, I xy, with respect to these axes how to calculate the second moments of area and the product moment of area about a convenient set of axes Unit 3: Solid mechanics 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 of area 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 of area how to determine the second moments of area and the product moment of area about any oriented set of axes, including the principal axes, using a Mohr’s circle construction Unit 3: Solid mechanics 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 bending moments 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. Unit 3: Solid mechanics 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, including tension/compression, bending and torsion Castigliano’s theorem for linear elastic bodies, which enables the deflection or rotation of a body at a point to be calculated from strain energy expression. Unit 3: Solid mechanics 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 of cycles 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. Unit 3: Solid mechanics 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 alternating components of the stress intensity factor. Unit 3: Solid mechanics 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 the solution 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 or not. Unit 3: Solid mechanics An Introduction to Mechanical Engineering: Part Two

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