engineering
1
Properties in Shear
Shear stress plays important role in failure of ductile materials as they resist
to normal stress by undergoing large plastic deformations, but actually fail
by rupturing under shear stress.
Figure 7a Figure 7cFigure 7b
Direct or transverse shear tests are employed to obtain approximate
measure of “shear strength” for specific applications while torsion test is
usually employed to evaluate shear behaviour and properties of materials.
Shear stresses are categorized into two main types:
1. Direct (transverse) shear in rivets (Fig. 7a) and beams (Fig. 7c).
2. Pure (torsional) shear in shafts subjected to pure torsion (Fig. 7b).
2
Direct Shear Tests
In direct shear tests, the specimen is clamped in a fixture and shearing force
is applied through a shear tool so that the maximum load is determined.
–
–
–
Tests are done in three ways:
Single shear test (Fig. 8a)
Double shear test (Fig. 8b)
Punch shear test (Fig. 8c)
Figure 8
(a) (b) (c)
2
2
single shear :
double shear : 2
punch shear :
F r
F r
F dh
These tests do not provide reliable information on properties of materials in
shear (i.e. yield, stiffness, resilience). They only provide the approximate
ultimate strength of material in shear. Even so, it is not always reliable due to
certain factors (e.g. hardness, sharpness and correct setting of shearing
tools as well as bending stresses and friction between the parts).
3
Torsion Test
Torsion tests are carried out by applying
a twisting moment (torque T) to one end
of a specimen while measuring angular
deformation (angle ) at the other end.
Figure 9Cylindrical specimens having
square/hexagon shaped ends
are usually used.
A torsion test machine (Fig. 9) consists of units for loading (A) and indicating (B).
Unit A is stationary (mounted on bed) while unit B is movable (to permit adjustment
of varying specimen lengths and automatic compensation of changes in length).
4
Torsion Test
Torsion test is useful in determining the material properties such as shear
modulus of elasticity, torsional yield strength and shear modulus of rupture.
Such tests can also be carried out on full-sized engineering components to
determine their behaviour under service conditions.
Torsion test offers certain advantages over tensile test:
During torsion test, no necking occurs. Therefore, the torque increases up
to the moment of failure.
Plastic deformation is almost uniform over entire length of specimen,
which enables the determination of deformations and stresses reliably for
highly ductile materials (especially pure metals).
Brittle or low ductility materials, that are often difficult to test in tension,
can undergo quite measurable deformation in torsion test, which enables
the determination of their mechanical properties.
In addition, torsion tests can easily be conducted at high strain rates.
5
Torsion Test
During torsion test, torque (T) and
angle of twist ( ) are measured.
Based on this data, “torque-twist
diagram” is constructed.
Fig. 10 gives torque-twist diagrams
for ductile (normalised mild steel)
and brittle (cast iron) material.
Using torque-twist diagram, shear
stresses and corresponding shear
strains are calculated (by equations
given in the next slide).
Figure 10
Normalised mild steel
(D = 6.35 mm, L = 152.4 mm)
Cast iron
(D = 6.35 mm, L = 76.2 mm)
6
Shear Stress & Shear Strain
: shear stress (kg/mm2)
: shear strain (mm/mm)
T : applied torque (kg*mm)
R : radius of the bar (mm)
J : polar moment of inertia (mm4)
: angle of twist (radians)
L : length of the bar (mm)
T R J
4 2 (for solid bar)J R
tan S L R L
Take a cylindrical bar (Fig. 11a) that is fixed at one end and subjected to a torsional
moment (torque, T) at the other end, and hence will be twisted through an angle ( ).
Such torsional deformation (S) induces shear stress ( ) resulting in shear strain ( )
along the circumference (Fig. 11b).
Figure 11
T R
O A
B
B
C
C
L
T
S
(a) (b)
7
Solid vs. Tubular (Hollow) Specimens
Torsion test is not used in material specifications to the same extent as tension test
since no uniform shear stress can be generated. Magnitude of shear stress ( ) varies
from zero at symmetry axis to the maximum at surface (Fig. 12a). When the surface
reaches to elastic limit, the interior will still be in elastic range. Thus, start of yielding
at the surface cannot be detected until large amount of plastic deformation occurs.
In order to overcome this problem,
thin-walled tubular specimens
are used so that shear stress is
assumed to be uniform along cross
section (Fig. 12b). However, there
is a danger of buckling when using
tubular (hollow) specimens if ratios
of length-to-diameter and diameter-
to-thickness are not kept within
elastic limits.
Figure 12
(b) Tubular (thin-walled)
m a x
T
m in
max min
m a x
T
(a) Solid
m in
max >>
min = 0
8
Stiffness in Torsion
“Stiffness in torsion” is measured
by “shear modulus” or “modulus of
rigidity” denoted by G:
It can also be obtained from Young’s
Modulus (E) provided that Poisson’s
Ratio ( ) of that material is known:
Poisson’s ratio and modulus of rigidity
for various materials are given in table
( is about 0.3 for most materials).
G T L J
2 1G E
9
Elastic Shear Strength
“Elastic shear strength” is measured by the maximum stress in specimen
corresponding to torque during the transition from elastic to plastic range.
In torsion test, the first onset of yielding is usually
not apparent for most materials, and hence “offset
yield point” is employed (point P in Fig. 13). Offset
angle of twist is usually taken as 4*10-5 rad/mm of
gauge length. Thus, “elastic shear strength (Ssy)”
is defined by:
Tubular specimens are also used to precisely determine torsional elastic
limit or yield strength. Using a specimen having a thin-walled circular cross-
section, shear stress along the wall is uniform for all practical purposes as
such specimens do not benefit from “strengthening effect” of inner fibers.
Figure 13T
sy T
P
offset O
//
Tsy : torque at proportional limit
(i.e. at offset angle of twist)sy sy S T R J
10
Elastic Shear Strength
Strengthening effect of inner fibers is illustrated in Fig. 14 for Al alloy 6061-T4.
In theory, thinner is the wall thickness, more reliable is the measurement of
elastic shear strength as all fibers are at about the same stress.
However, if a thin-walled specimen is subjected to torsion, it would first fail by
buckling before the shear strength of material is reached. Thus, a tubular
specimen with ratio of length-to-diameter of at least 10 and ratio of diameter-to-
wall thickness of about 10 is recommended for torsion test without buckling.
Figure 14For a tubular specimen, the elastic
shear strength is defined by:
2 2
sy
sy
T S
R t
2
2
o i
o i
t d d
R d d
11
Resilience, Ultimate Strength, Toughness
The area under torque-twist
diagram represents the total
work in stressing specimen to
the proportional limit. Thereby,
“modulus of resilience” (Us)
is defined as:
Tsy : torque at proportional limit
sy : corresponding angle of twist
Ssy : elastic shear strength
A : cross-sectional area
L : gauge length
G : modulus of rigidity
2
2 4
sy sy sy
s
T S U
A L G
Plastic shear strength ( u) is the approximate
definition of “modulus of rupture” or “ultimate
shear strength” (Ssu):
2
solid specimen :
tubular specimen : 2
su u
su u
S T R J
S T R t
For accurate definition of ultimate shear strength,
tubular specimens with short reduced sections
are recommended (i.e. L/d = 0.5 and d/t = 10).
Definition for “toughness” based on torque-twist
diagram is also approximate. Toughness index
number in torsion (To) can be defined as:
f : fracture angle of twist u f
o
T T
A L
12
Failure Types in Torsion
The torsional fracture is quite distinct from either
tension or compression fracture. There is almost
no localized reduction or area (i.e. no necking).
Possible fractures are as follows:
Ductile materials (Fig. 15a) fracture at 90° to
the torsional axis in maximum shear plane.
Brittle materials (Fig. 15b) fracture at 45° to
the torsional axis in maximum tension plane.
Buckling (Fig. 15c) occurs in hollow specimen
if L/d and d/t ratios are not kept withing limits.
Figure 15
(a)
(b)
(c)
Torsional failure of a ductile material
Torsional failure of a brittle material
13
Failure Examples in Torsion