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    Old Dominion University: Torsion

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    Muhammad Fahim
    The document summarizes key concepts about torsion and shear stresses from a chapter on engineering mechanics. It discusses how a circular bar deforms under pure torsion, with cross sections remaining circular and planes. Shear strains increase linearly from the center to the outer surface. For linearly elastic materials, the torsion formula gives shear stress proportional to the distance from the center. The chapter also covers torsion of thin-walled tubes, non-uniform torsion, stresses and strains in pure shear, and other topics.

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    © All Rights Reserved
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    Old Dominion University: Torsion

    Uploaded by

    Muhammad Fahim
    55 views54 pages
    The document summarizes key concepts about torsion and shear stresses from a chapter on engineering mechanics. It discusses how a circular bar deforms under pure torsion, with cross sections remaining circular and planes. Shear strains increase linearly from the center to the outer surface. For linearly elastic materials, the torsion formula gives shear stress proportional to the distance from the center. The chapter also covers torsion of thin-walled tubes, non-uniform torsion, stresses and strains in pure shear, and other topics.

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    Chap 03

    Original Title

    Chap 03-Torsion

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    The document summarizes key concepts about torsion and shear stresses from a chapter on engineering mechanics. It discusses how a circular bar deforms under pure torsion, with cross sections remaining circular and planes. Shear strains increase linearly from the center to the outer surface. For linearly elastic materials, the torsion formula gives shear stress proportional to the distance from the center. The chapter also covers torsion of thin-walled tubes, non-uniform torsion, stresses and strains in pure shear, and other topics.

    Copyright:

    © All Rights Reserved
    Download as PPT, PDF, TXT or read online from Scribd
    Download as ppt, pdf, or txt
    55 views54 pages

    Old Dominion University: Torsion

    Uploaded by

    Muhammad Fahim
    The document summarizes key concepts about torsion and shear stresses from a chapter on engineering mechanics. It discusses how a circular bar deforms under pure torsion, with cross sections remaining circular and planes. Shear strains increase linearly from the center to the outer surface. For linearly elastic materials, the torsion formula gives shear stress proportional to the distance from the center. The chapter also covers torsion of thin-walled tubes, non-uniform torsion, stresses and strains in pure shear, and other topics.

    Copyright:

    © All Rights Reserved
    Download as PPT, PDF, TXT or read online from Scribd
    Download as ppt, pdf, or txt
    You are on page 1of 54

    Old Dominion University

    Chapter 03
    Torsion

    Engr. Muhammad Fahim


    mfahi001@odu.edu
    hmf_engr@yahoo.com
    1
    3.2 TORSIONAL DEFORMATIONS OF A CIRCULAR BAR
    • Since every cross section of the bar is
    identical, and since every cross section is
    subjected to the same internal torque T,
    we say that the bar is in pure torsion.
    • From considerations of symmetry, it can be
    proved that cross sections of the bar do
    not change in shape as they rotate about
    the longitudinal axis. (No Warping)
    • In other words, all cross sections remain
    plane and circular and all radii remain FIG. 3-3 Deformations of a
    straight. circular bar in pure torsion
    • Furthermore, if the angle of rotation
    between one end of the bar and the other
    is small, neither the length of the bar nor
    its radius will change.

    Chap 04 ` Engr. M. Fahim 2


    3.2 TORSIONAL DEFORMATIONS OF A CIRCULAR BAR

    • The angle of twist changes along the axis of the bar, and at intermediate cross sections
    it will have a value φ(x) that is between zero at the left-hand end and φ at the right-
    hand end.
    • If every cross section of the bar has the same radius and is subjected to the same
    torque (pure torsion), the angle φ(x) will vary linearly between the ends.

    Chap 04 ` Engr. M. Fahim 3


    3.2 TORSIONAL DEFORMATIONS OF A CIRCULAR BAR
    Shear Strains at the Outer Surface

    • The angles at the corners of the element (Fig. 3-4b) are no longer equal to 90°.
    • The element is therefore in a state of pure shear. (Shear strains but no axial strains.)

    Chap 04 ` Engr. M. Fahim 4


    3.2 TORSIONAL DEFORMATIONS OF A CIRCULAR BAR
    Shear Strains at the Outer Surface

    Chap 04 ` Engr. M. Fahim 5


    3.2 TORSIONAL DEFORMATIONS OF A CIRCULAR BAR
    Shear Strains at the Outer Surface
    • In the special case of pure torsion, the rate of twist is equal to the total angle of
    twist φ divided by the length L, that is,
    θ =φ/L.
    • Therefore, for pure torsion only, we obtain:

    • This equation can be obtained directly from the geometry of Fig. 3-3a.

    Chap 04 ` Engr. M. Fahim 6


    3.2 TORSIONAL DEFORMATIONS OF A CIRCULAR BAR
    Shear Strains Within the Bar

    Chap 04 ` Engr. M. Fahim 7


    3.2 TORSIONAL DEFORMATIONS OF A CIRCULAR BAR
    Circular Tubes

    Chap 04 ` Engr. M. Fahim 8


    3.2 TORSIONAL DEFORMATIONS OF A CIRCULAR BAR

    All of the preceding equations for the strains in a circular bar are based upon geometric
    concepts and do not involve the material properties.
    Therefore, the equations are valid for any material, whether it behaves elastically
    or inelastically, linearly or nonlinearly. However, the equations are limited to bars having
    small angles of twist and small strains.

    Chap 04 ` Engr. M. Fahim 9


    3.3 CIRCULAR BARS OF LINEAR ELASTIC MATERIALS

    The torque T tends to rotate the right-hand end


    of the bar counterclockwise when viewed from
    the right. Therefore the shear stresses τ acting
    on a stress element located on the surface of the
    bar will have the directions shown in the figure.

    Chap 04 ` Engr. M. Fahim 10


    3.3 CIRCULAR BARS OF LINEAR ELASTIC MATERIALS

    The shear stresses acting on a cross-sectional plane are accompanied by shear


    stresses of the same magnitude acting on longitudinal planes (Fig. 3-7). This
    conclusion follows from the fact that equal shear stresses always exist on mutually
    perpendicular planes, as explained in Section 1.6. If the material of the bar is weaker
    in shear on longitudinal planes than on cross-sectional planes, as is typical of wood
    when the grain runs parallel to the axis of the bar, the first cracks due to torsion
    will appear on the surface in the longitudinal direction.

    Chap 04 ` Engr. M. Fahim 11


    3.3 CIRCULAR BARS OF LINEAR ELASTIC MATERIALS

    Chap 04 ` Engr. M. Fahim 12


    3.3 CIRCULAR BARS OF LINEAR ELASTIC MATERIALS
    The Torsion Formula

    3.7b

    3.9

    Chap 04 ` Engr. M. Fahim 13


    3.3 CIRCULAR BARS OF LINEAR ELASTIC MATERIALS
    The Torsion Formula
    3.7b

    Equation (3-13) is a generalized torsion formula, and we see once again that the
    shear stresses vary linearly with the radial distance from the center of the bar.

    Equation (3-12) shows that the shear stress is inversely proportional to the cube
    of the diameter. Thus, if the diameter is doubled, the stress is reduced by a factor
    of eight.

    Chap 04 ` Engr. M. Fahim 14


    3.3 CIRCULAR BARS OF LINEAR ELASTIC MATERIALS
    Angle of Twist
    3.7a

    Chap 04 ` Engr. M. Fahim 15


    3.3 CIRCULAR BARS OF LINEAR ELASTIC MATERIALS
    Circular Tubes

    The limits of integration are from ρ = r1 to ρ = r2.

    Chap 04 ` Engr. M. Fahim 16


    3.3 CIRCULAR BARS OF LINEAR ELASTIC MATERIALS
    Circular Tubes
    The torsion formula (Eq. 3-11) may be used for a circular tube of
    linearly elastic material provided IP is evaluated according to Eq. (3-16). The
    same comment applies to the general equation for shear stress (Eq. 3-13), the
    equations for rate of twist and angle of twist (Eqs. 3-14 and 3-15), and the
    equations for stiffness and flexibility (Eqs. a and b).

    Chap 04 ` Engr. M. Fahim 17


    3.3 CIRCULAR BARS OF LINEAR ELASTIC MATERIALS

    Chap 04 ` Engr. M. Fahim 18


    3.3 CIRCULAR BARS OF LINEAR ELASTIC MATERIALS
    Examples 3.1
    Examples 3.2
    Examples 3.3

    Chap 04 ` Engr. M. Fahim 19


    3.4 NON UNIFORM TORSION
    Case 1: Prismatic Segments with Constant Torque in each Segment

    An internal torque is positive when its vector points away from the
    cut section and negative when its vector points toward the section.
    Chap 04 ` Engr. M. Fahim 20
    3.4 NON UNIFORM TORSION
    Case 1: Prismatic Segments with Constant Torque in each Segment

    Chap 04 ` Engr. M. Fahim 21


    3.4 NON UNIFORM TORSION
    Case 2: Continuously varying cross sections and constant Torque

    Chap 04 ` Engr. M. Fahim 22


    3.4 NON UNIFORM TORSION
    Case 3: Continuously varying cross sections and
    continuously varying Torque

    Chap 04 ` Engr. M. Fahim 23


    3.4 NON UNIFORM TORSION
    Limitations

    Chap 04 ` Engr. M. Fahim 24


    3.4 NON UNIFORM TORSION

    Examples 3.4
    Examples 3.5

    Chap 04 ` Engr. M. Fahim 25


    3.5 STRESSES AND STRAINS IN PURE SHEAR

    Chap 04 ` Engr. M. Fahim 26


    3.5 STRESSES AND STRAINS IN PURE SHEAR
    Stresses on Inclined Planes

    FIG. 3-23 Stress elements oriented at θ = 0 and θ = 45°for pure shear

    If a stress element is oriented at an angle other than 45°, both normal and shear
    stresses will act on the inclined faces.

    Chap 04 ` Engr. M. Fahim 27


    3.5 STRESSES AND STRAINS IN PURE SHEAR
    Strains in Pure Shear

    FIG. 3-25 Strains in pure shear: (a) shear distortion of an element oriented at θ = 0, and
    (b) distortion of an element oriented at θ = 45°

    Chap 04 ` Engr. M. Fahim 28


    3.5 STRESSES AND STRAINS IN PURE SHEAR
    Strains in Pure Shear

    The strain in the perpendicular direction is a negative strain of the same amount. In other
    words, pure shear produces elongation in the 45°direction and shortening in the 135°
    direction.

    Chap 04 ` Engr. M. Fahim 29


    3.5 STRESSES AND STRAINS IN PURE SHEAR
    Examples 3.6

    Chap 04 ` Engr. M. Fahim 30


    3.6 RELATIONSHIP BETWEEN MODULI OF
    ELASTICITY E AND G

    Chap 04 ` Engr. M. Fahim 31


    3.7 TRANSMISSION OF POWER BY CIRCULAR SHAFTS

    Chap 04 ` Engr. M. Fahim 32


    3.8 STATICALLY INDETERMINATE TORSIONAL MEMBERS

    Chap 04 ` Engr. M. Fahim 33


    3.8 STATICALLY INDETERMINATE TORSIONAL MEMBERS
    Examples 3.9

    Chap 04 ` Engr. M. Fahim 34


    3.9 STRAIN ENERGY IN TORSION AND PURE SHEAR

    Chap 04 ` Engr. M. Fahim 35


    3.9 STRAIN ENERGY IN TORSION AND PURE SHEAR
    Strain-Energy Density in Pure Shear

    Chap 04 ` Engr. M. Fahim 36


    3.9 STRAIN ENERGY IN TORSION AND PURE SHEAR
    Examples 3.10

    Examples 3.11

    Examples 3.12

    Chap 04 ` Engr. M. Fahim 37


    3.10 THIN WALLED TUBES

    Chap 04 ` Engr. M. Fahim 38


    3.10 THIN WALLED TUBES
    Shear Stresses and Shear Flow

    Chap 04 ` Engr. M. Fahim 39


    3.10 THIN WALLED TUBES
    Shear Stresses and Shear Flow

    Chap 04 ` Engr. M. Fahim 40


    3.10 THIN WALLED TUBES
    Torsion Formula for Thin-Walled Tubes

    Am: Area enclosed by median line.

    Chap 04 ` Engr. M. Fahim 41


    3.10 THIN WALLED TUBES
    Torsion Formula for Thin-Walled Tubes: Circular Tube

    Polar Moment of inertia

    Chap 04 ` Engr. M. Fahim 42


    3.10 THIN WALLED TUBES
    Torsion Formula for Thin-Walled Tubes: Rectangular Tube

    Chap 04 ` Engr. M. Fahim 43


    3.10 THIN WALLED TUBES
    Strain Energy and Torsion Constant
    Strain Energy Density: Volume of Element = tdsdx

    Chap 04 ` Engr. M. Fahim 44


    3.10 THIN WALLED TUBES
    Strain Energy and Torsion Constant

    Torsion Constant:

    It has the same form as the equation for strain energy in a circular bar (3-51a), J has replaced the polar
    moment of inertia IP. Note that the torsion constant has units of length to the fourth power.

    Chap 04 ` Engr. M. Fahim 45


    3.10 THIN WALLED TUBES
    Strain Energy and Torsion Constant: Circular Tube

    Chap 04 ` Engr. M. Fahim 46


    3.10 THIN WALLED TUBES
    Strain Energy and Torsion Constant: Rectangular Tube

    Chap 04 ` Engr. M. Fahim 47


    3.10 THIN WALLED TUBES
    Angle of Twist

    Chap 04 ` Engr. M. Fahim 48


    3.10 THIN WALLED TUBES
    Limitations

    Chap 04 ` Engr. M. Fahim 49


    3.10 THIN WALLED TUBES
    Example 3.13

    Chap 04 ` Engr. M. Fahim 50


    3.10 THIN WALLED TUBES
    Example 3.14

    Chap 04 ` Engr. M. Fahim 51


    3.11 STRESS CONCENTRATION IN TORSION

    Chap 04 ` Engr. M. Fahim 52


    3.11 STRESS CONCENTRATION IN TORSION

    Chap 04 ` Engr. M. Fahim 53


    PROBLEMS

    Chap 04 ` Engr. M. Fahim 54

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