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, d
y
/d
x
, 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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