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# Solid mechanics Learning summary - PowerPoint PPT Presentation

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.

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Presentation Transcript
Solid mechanicsLearning summary

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

• 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

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

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

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

• 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

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

• 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

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

• how to include the interaction of yield behaviour with buckling and how to represent this interactiongraphically.

An Introduction to Mechanical Engineering: Part Two

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

• 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

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

• 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

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

• 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

• 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

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

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

By the end of this section you should have learnt:

• the meaning of linear elastic fracture mechanics (LEFM)

• 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

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