1.6 Torsional stress and Deformations.......................
1.6 Torsional stress and Deformations
Torsion: Consider a shaft rigidly clamped at one end and twisted at the other end by a
torqueT =F.dapplied in a plane perpendicular to the axis of the bar (Figure1.16)^3.
Such a shaft is said to be in torsion. Therefore, torsion refers to the twisting of a structural
member when it is loaded by couples that produce rotation about its longitudinal axis
[ 3 ]. Torsion is a variation of pure shear, wherein a structural member is twisted in the
F dFT= F.dshould be at least four times this diameter; 60 mm is common. Gauge length
is used in ductility computations, as discussed in Section 6.6; the standard value is
50 mm (2.0 in.). The specimen is mounted by its ends into the holding grips of the
testing apparatus (Figure 6.3). The tensile testing machine is designed to elongate
the specimen at a constant rate and to continuously and simultaneously measure the
instantaneous applied load (with a load cell) and the resulting elongations (using an
extensometer). A stress–strain test typically takes several minutes to perform and is
destructive; that is, the test specimen is permanently deformed and usually fractured.
[The (a) chapter-opening photograph for this chapter is of a modern tensile-testing
apparatus.]
The output of such a tensile test is recorded (usually on a computer) as load
or force versus elongation. These load–deformation characteristics are dependent
on the specimen size. For example, it will require twice the load to produce the same
elongation if the cross-sectional area of the specimen is doubled. To minimize these1214 in. 26.2 Concepts of Stress and Strain• 153TTFFFFFFFA 0A 0A 0(a) (b)(c) (d)!"l l 0 l 0 lFigure 6.2
A standard tensile
specimen with
circular cross
section.Figure 6.1
(a) Schematic
illustration of how a
tensile load produces
an elongation and
positive linear strain.
Dashed lines
represent the shape
before deformation;
solid lines, after
deformation.
(b) Schematic
illustration of how a
compressive load
produces contraction
and a negative linear
strain. (c) Schematic
representation of
shear strain , where
!tan.
(d) Schematic
representation of
torsional
deformation (i.e.,
angle of twist )
produced by an
applied torque T.fuggGauge length2"Reduced section
2 "
"Diameter
"(^14)
(^34)
(^38) Radius
0.505" Diameter
JWCL187_ch06_150-196.qxd 11/5/09 9:36 AM Page 153
d
A A^0 > A
d 0 > d , Δd = (d 0 - d)
l 0 < l , Δl = (l - l 0 )
longitudinal strain = Δl/l 0
d 0 lateral strain = -^ Δd/d^0
z
x
Poisson s ratio = ( l/ld/d 00 )
=dl^00 dl
T
Figure 1.16: A shaft subjected to torsion.
manner of Figure1.16. Torsion is utilised in a large number of engineering devices, of
which some examples are shown in Figure1.17. Shear modulus in the context of torsion
is calledRigidity Modulus.
Twisting Couple: The two equal and opposite forces (with magnitudeF) acting at two
di erent points at a distancedon the shaft as shown in Figure1.16form the twisting
coupleT(also see Figures1.18(a) and (b)). E ects of a torsional load applied to a bar
are (i) To impart an angular displacement of one end cross – section with respect to the
other endand(ii) To setup shear stresses on any cross section of the bar perpendicular
to its axis.
Torsional stress: The shear stress produced inside a material as a reaction to an externally
applied torque to the end of a shaft about its axis is known as torsional stress.
Torsional deformation and shear strain: If a line a-b is marked on the surface of the
unloaded bar, then after the twisting moment or torqueT has been applied, this line
moves to a-bÕas shown in Figure1.18. The angle“, measured in radians, between ab and
abÕis defined as the shear strain at the surface of the bar or shaft^4 The same definition
will hold at any interior point of the bar.
Angle of Twist: If a shaft of lengthLis subjected to a constant twisting momentTalong
its length, then the angle◊through which one end of the bar will twist relative to the
other is known is the angle of twist as shown in Figure1.18. (In Figure1.2(d), angle of
twist was denoted by„.)
(^3) The shaft is a machine element which is used to transmit power in machines. Acoupleis a pair of
forces, equal in magnitude, oppositely directed, displaced by perpendicular distance and do not share a
line of action. Torque(or moment of the force) is the product of the magnitude of the force and the
perpendicular distance of the line of action of force from the axis of rotation.
(^4) Strictly speaking, the shear strain (“) is the tangent of the angle between ab and abÕ.Butwhenthe
angle is small (and expressed in radians), the tangent is nearly equal to the angle itself.