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.
Unit 3: Solid mechanics
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 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 Isection 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, Ix
and Iy, possesses a geometric quantity called the
product moment of area, Ixy, 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