STRESS CONCENTRATION
INTRODUCTION
• Development of the basic stress equations
for tension, compression, bending, and
torsion
– Assumption: no geometric irregularities occurred
in the member under consideration.
• But it is quite difficult to design a machine
without permitting some changes in the cross
sections of the members.
In developing a machine it is impossible to avoid changes in cross-section,
holes, notches, shoulders etc.
Stress Concentration and Causes
• Mathematical analysis and experimental measurement show that in a loaded
structural member, near changes in the section, distributions of stress occur in
which the peak stress reaches much larger magnitudes than does the average
stress over the section.
• This increase in peak stress near holes, grooves, notches, sharp corners,
cracks, and other changes in section is called stress concentration.
• Any discontinuity in a machine part alters the stress distribution in the
neighborhood of the discontinuity so that the elementary stress
equations no longer describe the state of stress in the part at these
locations.
• Stress concentrations can arise from some irregularity not inherent in
the member, such as tool marks, holes, notches, grooves, or threads.
• The section variation that causes the stress concentration is referred to as a
stress raiser.
• ……….. Then how to quantify this phenomenon ?.....
INTRODUCTION
• Example
– Rotating shafts must have
shoulders designed on them
so that the bearings can be
properly seated and so that
they will take thrust loads.
– Shafts must have key slots
machined into them for
securing pulleys and gears.
INTRODUCTION
• Example
– A bolt has a head on one end and screw threads
on the other end, both of which account for
abrupt changes in the cross section.
– Other parts require holes, oil grooves, and
notches of various kinds.
Let’s look back the theory of uniform stress
• The assumption of a uniform distribution of stress is frequently
made in design.
• The result is then often called pure tension, pure compression, or
pure shear.
• The stress is said to be uniformly distributed with
=
Let’s look back the theory of uniform stress
This assumption of uniform stress distribution requires
that :
 The bar be straight and of a homogeneous
material
 The line of action of the force contains the
centroid of the section
 The section be taken remote from the ends and
from any discontinuity or abrupt change in cross
section
Let’s look back the theory of uniform stress
• Direct shear is usually assumed to be uniform across the cross
section, and is given by
=
• The assumption of uniform stress is not accurate, particularly in
the vicinity where the force is applied, but the assumption
generally gives acceptable results.
Normal Stress for Beams in Bending:
Assumptions
The equations for the normal bending
stresses in straight beams are based
on the following assumptions
• The beam is subjected to pure bending.
Means
– Shear force is zero
– No torsion
– Axial loads are present
For most engineering applications it is assumed
that these loads affect the bending stresses
minimally.
• The material is isotropic and
homogeneous.
• The material obeys Hook’s law.
Normal Stress for Beams in Bending:
Assumptions
• The beam is initially straight with a
cross section that is constant
throughout the beam length.
• The beam has an axis of symmetry
in the plane of bending.
• The proportions of the beam are
such that it would fail by bending
rather than by crushing, wrinkling, or
sidewise buckling.
• Plane cross sections of the beam
remain plane during bending.
Normal Stress for Beams in Bending:
Analysis
•
•
•
A portion of a straight beam acted upon by a positive
bending moment M shown by the curved arrow showing
the physical action of the moment together with a
straight arrow indicating the moment vector.
Elements of the beam coincident with the neutral plane
have zero stress.
The bending stress varies linearly with the distance
from the neutral axis, y, and is given by
where I is the second-area moment about the z
axis.
Normal Stress for Beams in Bending:
Analysis
• Designating σmax as the maximum
magnitude of the bending stress, and c as
the maximum magnitude of y often
written as
where Z = I/c is called the section modulus
Shear Stress for Beams
•
For a beam segment of constant cross section
subjected to a shear force V and a bending
moment M at x, the net force in the x direction will
be directed to the left with a value of
•
For equilibrium
Shear Stress for Beams
•
Define the first moment of the area A′ with respect
to the neutral axis
•
The transverse shear stress, which is always
accompanied with bending stress.
Torsion
• Any moment vector that is collinear with an axis of a mechanical
element is called a torque vector, because the moment causes the
element to be twisted about that axis.
• The angle of twist, in radians, for a solid round bar
where T = torque , l = length, G = modulus of rigidity
J = polar second moment of area
• Shear stresses develop throughout the cross section are given by
with
Torsion
• The assumptions used in the analysis are
 The bar is acted upon by a pure torque, and the
sections under consideration are remote from
the point of application of the load and from a
change in diameter.
 Adjacent cross sections originally plane and
parallel remain plane and parallel after twisting,
and any radial line remains straight.
 The material obeys Hooke’s law.
What is stress concentration???
Any discontinuity in a machine part alters the stress distribution
in the neighborhood of the discontinuity so that the
elementary stress equations no longer describe the state of
stress in the part at these locations. Such discontinuities are
called stress raisers, and the regions in which they occur are
called areas of stress concentration.
How to Estimate SCF
• Difficult to determine SCF, theoretically, due to complex geometric
shapes.
• SCFs are found using EXPERIMENTAL TECHNIQUES
Photo-elasticity, electrical strain-gauge methods
• Finite Element Methods use very fine mesh at the SC region and
transition of mesh from rest of the part to SC region must be gradual
How to Estimate SCF
• Photoelasticity is an experimental method to determine the
stress distribution in a material.
• The method is mostly used in cases where mathematical
methods become quite cumbersome.
• Unlike the analytical methods of stress determination,
photoelasticity gives a fairly accurate picture of stress
distribution, even around abrupt discontinuities in a material.
• The method is an important tool for determining critical stress
points in a material, and is used for determining stress
concentration in irregular geometries. (Source: wikip)
Photo elasticity technique
• Suppose a small circular hole subjected to tensile stress
• In photo elasticity technique, an identical model of the
plate is made of epoxy resins.
• The model is placed in a circular polariscope and loaded at
the edges.
• It is observed that there is a sudden rise in the magnitude
of stresses in the vicinity of the hole.
• The localized stress in the neighbourhood of the hole are
far greater than the stress obtained by the elementary
equations
Difficulty to Estimate SCF
• Most stress-concentration factors are found by using
experimental techniques.
• Though the finite-element method has been used, the fact
that the elements are indeed finite prevents finding the true
maximum stress.
• Grid and strain-gauge methods both suffer from the same
drawback as the finite-element method.
Distribution of elastic stress
• The distribution of elastic stress across a section of a
member may be
– Uniform as in a bar in tension
– Linear as a beam in bending
– Even rapid and curvaceous as in a sharply curved beam
• Stress concentrations can arise from some irregularity not
inherent in the member, such as tool marks, holes, notches,
grooves, or threads.
• The nominal stress is said to exist if the member is free of
the stress raiser.
• This definition is not always honored.
Check the definition on the stress-concentration chart or table
you are using.
Theoretical stress-concentration factor
• A theoretical, or geometric, stress-concentration factor or
is used
to relate the actual maximum stress at the discontinuity to the nominal
stress. The factors are defined by the equations
=
=
=used for normal stresses
=used for shear stresses
• The nominal stress or is more difficult to define.
• Generally, it is the stress calculated by using the elementary stress
equations and the net area, or net cross section. But sometimes the
gross cross section is used instead, and so it is always wise to double
check your source of or
before calculating the maximum stress.
Example: Thin plate loaded in tension
• Thin plate in tension or simple compression with a transverse
central hole.
Net tensile force
=
t =thickness of the plate
Nominal stress is
=
=
Charts of Theoretical Stress-Concentration
Factors
Refer Shigley’s book
Recommendation to apply stressconcentration factors
In static loading, stress-concentration factors are applied as
follows.
• In ductile (ε ≥ 0.05 ) materials
– Stress-concentration factor is not usually applied to predict the
critical stress, because plastic strain in the region of the stress is
localized and has a strengthening effect.
• In brittle materials (ε < 0.05
– Geometric stress concentration factor is applied to the nominal
stress before comparing it with strength. Gray cast iron has so many
inherent stress raisers that the stress raisers introduced by the
designer have only a modest (but additive) effect.
Design suggestion stress-concentration
factors
• Stress concentration is a highly localized effect. In some
instances it may be due to a surface scratch.
• If the material is ductile and the load is static, the design
load may cause yielding in the critical location in the notch.
• This yielding can involve strain strengthening of the material
and an increase in yield strength at the small critical notch
location.
• Since the loads are static and the material is ductile, that
part can carry the loads satisfactorily with no general
yielding. In these cases the designer sets the geometric
(theoretical) stress concentration factor to unity.
Stress-concentration factors in static
loading of a ductile material
• In static loading stress concentration is not so serious as in brittle
materials because in ductile materials local deformation or yielding takes
place which reduces the concentration.
• This is the reason designers do not apply in static loading of a ductile
material loaded elastically, instead setting = 1.
• When using this rule for ductile materials with static loads, be careful to
assure yourself that the material is not susceptible to brittle fracture in
the environment of use.
• The usual definition of geometric (theoretical) stress centration factor for
normal stress and shear stress is
=
=
• Since your attention is on the stress-concentration factor, and the
definition of
or
stress is appropriate for the section carrying
the load
Stress-concentration factors in static
loading of a brittle material
• Brittle materials do not exhibit a plastic range. A brittle material
“feels” the stress concentration factor or
, which is applied
by using
=
=
• An exception to this rule is a brittle material that inherently
contains micro discontinuity stress concentration, worse than the
macro discontinuity that the designer has in mind.
• Example: Sand molding introduces sand particles, air, and water
vapor bubbles. The grain structure of cast iron contains graphite
flakes (with little strength), which are literally cracks introduced
during the solidification process. When a tensile test on a cast iron
is performed, the strength reported in the literature includes this
stress concentration. In such cases or
need not be applied.
How to mitigate SC/ Design to Minimize
SC
• Force should be transmitted from point to point as smoothly
as possible. The lines connecting the force transmission
path are sometimes called the force (or stress) flow.
• Sharp transitions in the direction of the force flow should be
removed by smoothing contours and rounding notch roots.
• When stress raisers are necessitated by functional
requirements, the raisers should be placed in regions of low
nominal stress if possible.
• When notches are necessary, removal of material near the
notch can alleviate(make less severe) stress concentration
effects.
• A type of stress concentration called an interface notch is
commonly produced when parts are joined by welding.
Smoothen the force flow
Smoothen the force flow
Sharp transitions in the direction of the force flow should be
removed by smoothing contours and rounding notch roots
Guiding the lines of stress by means of notches that are not functionally essential
is a useful method of reducing the detrimental effects of notches that cannot be
avoided. These are termed relief notches.
When notches are necessary, removal of material near the
notch can alleviate stress concentration effects
Grooves near a hole can reduce the stress concentration around the hole
A type of stress concentration called an interface notch is
commonly produced when parts are joined by welding
THINK ??
Which one has the lowest SC in each
group?
Example-1
The 2-mm-thick bar shown in the figure is loaded axially with a constant
force of 10 KN. The bar material has been heat treated and quenched to
raise its strength, but as a consequence, it has lost most of its ductility. It is
desired to drill a hole through the center of the40 mm face of the plate to
allow a cable to pass through it. A 4 mm hole is sufficient for the cable to
fit, but an 8mm drill is readily available. Will a crack be more likely to
initiate at the larger hole, the smaller hole, or at the fillet?
Example-1
Solution:
Since the material is brittle , the effect of stress concentration near the discontinuities
must be considered. Dealing with the hole first, for a 4mm hole, the nominal stress is
=
=
=
= 139 MPa
The theoretical stress concentration factor taken from table with d/w=4/40=0.1 is
2.7
The maximum stress is
=
=
=2.7(139) = 380 MPa
Similarly for an 8mm hole
=
=
=
= 156 MPa
The theoretical stress concentration factor taken from table with d/w=8/40=0.2 is =
2.5
The maximum stress is
=
=2.5(156) = 390MPa
though the stress concentration is higher with 4 mm hole, in this case the increased
nominal stress with 8 mm hole has more effect on the maximum stress.
Example-1
Solution:
For the fillet,
= =
= 147 MPa
The theoretical stress concentration factor taken from table with
D/d=40/34=1.18 and r/D=1/34=0.026. Then = 2.5
The maximum stress is
=
=2.5(147) = 368MPa
The crack will most likely occur with the 8 mm hole, next likely would be 4
mm hole, and least likely at the fillet.
Example-2
Find the maximum stress induced in the following cases taking stress
concentration into account:
a)A rectangular plate 60 mm × 10 mm with a hole 12 diameter as shown in
Figure (a) and subjected to a tensile load of 12 kN.
b) A stepped shaft as shown in Figure(b) and carrying a tensile load of 12
kN.
Example-2
Solution:
Example-2
Solution:
Thank you