Solid mechanics Learning summary

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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, 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