Transcript Document

Mechanics of Materials – MAE 243 (Section 002)
Spring 2008
Dr. Konstantinos A. Sierros
Problem 2.7-2
A bar of circular cross section having two different diameters d and 2d is shown
in the figure. The length of each segment of the bar is L/2 and the modulus of
elasticity of the material is E.
(a) Obtain a formula for the strain energy U of the bar due to the load P.
(b) Calculate the strain energy if the load P = 27 kN, the length L = 600 mm, the
diameter d = 40 mm, and the material is brass with E = 105 GPa.
Problem 2.7-8
The statically indeterminate structure shown in the figure consists of a
horizontal rigid bar AB supported by five equally spaced springs. Springs 1, 2,
and 3 have stiffnesses 3k, 1.5k, and k, respectively. When unstressed, the
lower ends of all five springs lie along a horizontal line. Bar AB, which has
weight W, causes the springs to elongate by an amount δ.
(a) Obtain a formula for the total strain energy U of the springs in terms of the
downward displacement of the bar.
(b) Obtain a formula for the displacement by equating the strain energy of the
springs to the work done by the weight W.
(c) Determine the forces F1, F2, and F3 in the springs.
(d) Evaluate the strain energy U, the displacement δ, and the forces in the
springs if
W = 600 N and k = 7.5 N/mm.
3.1: Introduction to torsion
• Torsion is the twisting of a straight bar when it is loaded by moments (or
torques) that tend to produce rotation about the longitudinal axis of the bar
• It is a little bit more complicated behaviour than axial tensile/compressive
loading
FIG. 3-1
Torsion of a
screwdriver
due to a torque
T applied to
the handle
Copyright 2005 by Nelson, a division of Thomson Canada Limited
3.1: Introduction to torsion
• Torsional loading of a bar
• Moment of a couple may be represented by a vector (fig 3-2b) using the
right-hand rule
• Moment of a couple may be represented by a curved arrow (fig 3-2C)
Circular bar
subjected to torsion
by torques T1 and T2
FIG. 3-2
Copyright 2005 by Nelson, a division of Thomson Canada Limited
3.2: Torsional deformations of a circular bar
• Consider a prismatic bar twisted by torques T acting at the ends
• The bar is under pure torsion since every cross-section is subjected to the
same torque
• Φ is the angle of twist
• Longitudinal line pq become a helical curve pq’
Deformations of a
circular bar in pure torsion
FIG 3-3
Copyright 2005 by Nelson, a division of Thomson Canada Limited
3.2: Shear strains at the outer surface
• Consider an element which is under pure shear
Angle of twist per unit length
Shear strain of outer surface
General cases of
torsion
Special case: Pure torsion
where L is length of prismatic bar
Deformation of an element of length
dx cut from a bar in torsion
FIG. 3-4
Copyright 2005 by Nelson, a division of Thomson Canada Limited
3.2: Shear strains within the bar
Deformation of an element of length
dx cut from a bar in torsion
FIG. 3-4
Copyright 2005 by Nelson, a division of Thomson Canada Limited
• Previous equations for shear strains apply not only to solid circular bars but
also to circular tubes
FIG. 3-5
Shear strains in
a circular tube
Copyright 2005 by Nelson, a division of Thomson Canada Limited
3.3: Circular bars of linearly elastic materials
• Torque T tends to rotate the right-hand
end of the bar counterclockwise
• Therefore the shear stresses τ acting on
the surface stress element will have the
directions shown in fig 3-6b
Max shear stress at
the outer surface
G is shear modulus of elasticity
FIG. 3-6
Shear stresses in a circular bar in torsion
Copyright 2005 by Nelson, a division of Thomson Canada Limited
Shear stress at an
interior point of
radius ρ
3.3: The torsion formula
• Determine a relationship between the shear stresses and the torque T
The torsion formula
Determination of the resultant of
the shear stresses acting on a cross section
FIG. 3-9
Copyright 2005 by Nelson, a division of Thomson Canada Limited
Polar moment of inertia
Substituting r = d/2 and Ip = πd4/32 into the torsion formula we get;
3.3: Angle of twist
• The angle of twist of a bar of linearly elastic material can be related to the
applied torque T
…and for a
bar in pure
torsion…
Torsional rigidity
…and circular tubes
FIG. 3-10
Circular tube in torsion
Copyright 2005 by Nelson, a division of Thomson Canada Limited
3.3:Limitations
• The equations shown in this section are valid for bars of circular cross-section
(either solid or hollow) that behave in a linearly elastic manner
• These equations can not be used for bars of other shapes such as rectangular
bars and non-circular bars
Coulomb
Young
***Torsion theory originated from the work of C.A. de Coulomb and
further developments were due to Thomas Young
Homework 2
due Wednesday 20 February 2008
is this
…plus 1st
midterm Friday
22nd February…