Addis Ababa Science and Technology University
College of Engineering
Department of Chemical Engineering
Chemical Engineering Apparatus Design (ChEg4110)
1. Overview of basic strength of materials terms
by
Adamu E. (PhD)
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Chapter One
1. Overview of basic strength of materials terms
1.1. Introduction
In materials science, the strength of a material is its
ability to withstand an applied load without failure.
The difference between Statics and strength of materials
is:
Statics : It studies about forces on a rigid body.
Strength of materials: It deals with forces and
deformations that result from their acting on a material.
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1.2. Types of loadings and stresses
A load applied to a mechanical member will induce internal
forces within the member called stresses.
The stresses acting on the material cause deformation.
Load induces stress, and stress causes deformation.
Once the state of stress and strain within the member is known,
the strength (load carrying capacity) of that member, its
deformations (stiffness qualities), and its stability (ability to
maintain its original configuration) can be determined.
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Axial stress can be either compressive or tensile.
Tensile stress: It is the stress state caused by an applied load
that tends to elongate the material along the axis of the applied
load, in other words the stress caused by pulling the material
(Fig.1).
Fig. 1 Tensile stress
Compressive stress : It is the stress state caused by an applied
load that acts to reduce the length of the material (compression
member) along the axis of the applied load (Fig. 2).
Fig. 2 Compressive stress
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Uniaxial stress is expressed by:
Where F is the force [N] acting on an area A
[m2].
The area can be the undeformed area or the deformed area,
depending on whether engineering stress or true stress is of
interest.
The deformation of materials due to tensile and compressive
stresses is shown in Fig. 3.
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1. Overview of basic strength …
(a) (b)
Fig.3 (a) tensile,(b) compressive stresses
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Shear stress: It is the stress state caused by the combined energy of a pair of
opposing forces acting along parallel lines of action through the material, in
other words the stress caused by faces of the material sliding relative to one
another (Fig.4a).
An example is cutting paper with scissors or stresses due to torsional loading.
The shear stress is the punch force divided by the sheared surface (Fig.4b).
(a) (b)
Fig.4 shear stress due to (a) opposing forces long parallel lines, (b) torsional loading
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Fig.5 Stress due to (a) tensile, (b) compressive, and (c) shear
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Combined stresses…
The normal stresses, σx and σy, could be due to a direct tensile
force or to bending (Fig.6).
If the normal stresses were compressive (negative), the vectors
would be pointing in the opposite sense, into the stress
element.
Fig.6 Normal stresses
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Combined stresses…
The shear stress could be due to direct shear, torsional shear, or
vertical shear stress.
The double-subscript notation helps to orient the direction of shear
stresses.
For example, τxy indicates the shear stress acting on the element face
that is perpendicular to the x-axis and parallel to the y-axis.
A positive shear stress is one that tends to rotate the stress element
clockwise.
In the first figure, τxy is positive and τyx is negative.
Their magnitudes must be equal to maintain the element in
equilibrium.
With the stress element defined, the objectives of the remaining
analysis are to determine the maximum normal stress, and the planes
on which these stresses occur.
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Maximum normal stresses: The combination of the applied
normal and shear stresses that produces the maximum normal
stress is called the maximum principal stress, σ1.
Minimum normal stresses: The combination of the applied
normal and shear stresses that produces the minimum normal
stress is called the minimum principal stress, σ2.
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The angle of inclination of the planes on which the principal
stresses act, called principal planes, can be found from (Fig.7):
Fig.7 Normal stresses
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Maximum shear stress: On a different orientation of the
stress element, the maximum shear stress will occur (Fig.8).
Fig.8 Normal stresses
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Types of loads
Transverse loading - Forces applied perpendicular to the longitudinal axis
of a member (Fig.9).
It causes the member to bend and deflect from its original position, with
internal tensile and compressive strains accompanying the change in
curvature of the member.
It also induces shear forces that cause shear deformation of the material and
increase the transverse deflection of the member.
Fig.9 Transverse loading
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The angle of inclination of the element on which the maximum
occurs is computed as follows:
The angle between the principal stress element and the
maximum shear stress element is always 45o.
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Axial loading: The applied forces are collinear with the longitudinal axis
of the member (Fig.10).
The forces cause the member to either stretch or shorten.
Fig. 10 Axial loading
Torsional loading: Twisting action caused by a pair of externally applied
equal and oppositely directed forces (couples) acting on parallel planes or
by a single external couple applied to a member that has one end fixed
against rotation (Fig.11).
Fig. 11 Torsional loading
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Combined loading: It can be the combination of
transverse loading, and torsional or axial loading and
torsional loading (Fig.12).
Fig. 12 Combined loading
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Class Activity 1
1. Explain the difference between statics and
strength of materials
2. Explain how does loads cause materials
deformation
3. List and explain types of loads in a given
material
4. List and explain types of stresses in materials
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1.3. Strength terms, stress strain diagram and combined
stresses
Yield strength: It is the lowest stress that produces a permanent
deformation in a material.
In some materials, like aluminium alloys, the point of yielding is difficult to
identify, thus it is usually defined as the stress required to cause 0.2%
plastic strain. This is called a 0.2% proof stress (Fig. 13).
Fig. 13 Proof stress
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1.3. Strength terms, stress strain …
Compressive strength: It is a limit state of compressive stress
that leads to failure in a material in the manner of ductile failure
or brittle failure (rupture as the result of crack propagation, or
sliding along a weak plane), Fig. 14.
Fig. 14 Stress- strain diagram
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1.3. Strength terms, stress strain …
Tensile strength or ultimate tensile strength is a limit state
of tensile stress that leads to tensile failure, Fig. 15a .
In the manner of ductile failure (yield as the first stage of
that failure, some hardening in the second stage and
breakage after a possible "neck" formation), Fig. 15b.
Brittle failure (sudden breaking in two or more pieces at a
low stress state), Fig. 15c.
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Ductile Failure
(a)
(b)
Fig. 15 (a) Stress- strain diagram,
(b) ductile failure,
and (c) brittle failure
(c)
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The ultimate tensile strength: If compressive stress less
than yield strength stress, compressive failure will not
occur, Fig. 16a .
For ductile materials: If compressive stress greater than
yield strength stress, ductile failure will occur, Fig. 16b.
For brittle materials: If compressive stress greater than
yield strength stress, brittle failure will occur, Fig. 16c.
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Fig. 16 (a) ultimate tensile strength,
(b) ductile failure,
and (c) brittle failure
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Fatigue strength: It is a measure of the strength of a material or a
component under cyclic loading, and is usually more difficult to
assess than the static strength measures.
Impact strength: It is the capability of material to withstand a
suddenly applied load and is expressed in terms of energy.
Elasticity: It is the ability of a material to return to its previous
shape after stress is released.
The slope of stress-strain diagram line is the Young's modulus.
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Plasticity or plastic deformation is the opposite of elastic
deformation and is defined as unrecoverable strain.
Deformation of the material: It is the change in geometry
created when stress is applied (as a result of applied forces,
gravitational fields, accelerations, thermal expansion, etc.).
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Strain or reduced deformation is a mathematical term that expresses
the trend of the deformation change among the material field.
Strain: It is the deformation per unit length. In the case of uniaxial
loading the displacements of a specimen (for example a bar element)
lead to a calculation of strain expressed as the quotient of the
displacement and the original length of the specimen.
Deflection( ): It is a term to describe the magnitude to which a
structural element is displaced when subject to an applied load.
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Class Activity 2
1. Explain the terms: Young’s modulus, strain,
tensile strength, yield strength, ductile failure,
brittle failure, elastic zone, plastic zone,
deformation of materials, deflection
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1.4. Failure theories
In the case of multidimensional stress at a point
we have a more complicated situation present.
Since it is impractical to test every material and
every combination of stresses σ₁, σ₂, and σ₃, a
failure theory is needed for making predictions on
the basis of a material’s performance on the
tensile test of how strong it will be under any
other conditions of static loading.
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Ductile failure
Evolution of ductile failure: necking, void nucleation, void
growth, crack propagation and fracture.
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The general failure theories are:
Maximum shear stress theory
Maximum strain energy theory
Maximum distortion energy theory.
Maximum normal stress theory
Maximum shear stress theory: This theory
postulates that failure will occur if the magnitude of
the maximum shear stress in the part exceeds the
shear strength of the material determined from
uniaxial testing.
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Maximum strain energy theory: This theory postulates that
failure will occur when the strain energy per unit volume due to
the applied stresses in a part equals the strain energy per unit
volume at the yield point in uniaxial testing.
The total strain energy per unit volume for multi dimensional
stress can be given by:
E modulus of elasticity
v Poisson ratio And for uniaxial test
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For uniaxial test,
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Maximum distortion energy theory: This theory is also known
as shear energy theory or Von Mises-Hencky theory.
This theory postulates that failure will occur when the distortion
energy per unit volume due to the applied stresses in a part
equals the distortion energy per unit volume at the yield point in
uniaxial testing.
The total elastic energy due to strain can be divided into two
parts: one part causes change in volume, and the other part
causes change in shape.
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Distortion energy is the amount of energy that is needed to
change the shape.
The total distortion energy for multi dimensional stress can be
given by:
Poisson ratio: It is the ratio of lateral strain to longitudinal
strain.
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Ud is distortion energy.
v is Poisson ratio.
E is Modulus of elasticity.
For uniaxial test,
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Maximum normal stress theory: This theory postulates that
failure will occur if the maximum normal stress in the part
exceeds the ultimate tensile stress of the material as
determined from uniaxial testing.
This theory deals with brittle materials only.
The maximum tensile stress should be less than or equal to
ultimate tensile stress divided by factor of safety.
The magnitude of the maximum compressive stress should be
less than ultimate compressive stress divided by factor of
safety.
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