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Stress and Deformation Analysis

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					                                                   LEARNING OUTCOME
                                                Use the principle of stress and deformation analysis in
                                                stress analyses problem.
          CHAPTER 2
     STRESS AND
DEFORMATION ANALYSIS



                                            1                                                         2




             CONTENT                                           CONTENT
Representing stress on a stress element.         Torsion in members having noncircular cross
Mohr’s circle for plane stress
Mohr’                                            sections.
Direct stresses: tension and compression.        Vertical shearing stress.
Deformation under direct axial loading.          Stress due to bending.
Direct shear stress.                             Flexural center for beams.
Torsional shear stress.                          Combined normal stresses: superposition principle.
Torsional deformation.                           Stress concentrations factors.
                                                 Types of loading and stress ratio


                                            3                                                         4




                                                                                                          1
REPRESENTING STRESS ON A
    STRESS ELEMENT                                          General case of combined stress
Positive shear stresses tend to rotate the element in a
clockwise direction                                       Consider a small stress
                                                          element which combined
Negative shear stresses tend to rotate the element in a
                                                          stresses act.
counterclockwise direction
                                                          Stresses can be normal
                                                          stresses and/or shear
                                                          stresses.
                                                          Consider only two
                                                          dimensional stresses, i.e. in
                                                                        direction.
                                                          the x and y direction.

                                                      5                                            6




                        Principal stresses
Combination of applied stresses                           Max shear stress
(normal and shear) that produces
maximum normal stress.                                                ⎛ σ x −σ y   ⎞
                                                                                     2

Max/min principal stress                                    τ max = ⎜
                                                                    ⎜              ⎟ + τ xy 2
                                                                                   ⎟
                                                                      ⎝ 2          ⎠
                                     2
          σ x +σ y     ⎛ σ x −σ y   ⎞
   σ1 =              + ⎜
                       ⎜ 2          ⎟ + τ xy 2
                                    ⎟                     Angle for maximum shear stress element
             2         ⎝            ⎠
                                                            φσ = arctan[− (σ x − σ y ) 2τ xy ]
                                                                  1
                                     2
          σ x +σ y     ⎛σ x −σ y ⎞                                2
   σ2 =                ⎜ 2 ⎟ + τ xy
                     − ⎜         ⎟
                                    2

             2         ⎝         ⎠                        Average normal stress
Angle of principal stress element                           σ avg = (σ x + σ y ) 2
                                 x-
  Measured from the positive x-axis
  (shown in figure). Positive sign
  calls for a clockwise rotation of
  element and vise versa.
      φσ = arctan[2τ xy (σ x − σ y )]
             1
             2
                                                      7                                            8




                                                                                                       2
                                                                     MOHR’S CIRCLE FOR PLANE
                                                                              STRESS
Summary                                                                   Mohr’
                                                                     The Mohr’s circle enables the following computation
                                                                     with relative ease and better accuracy:
                                                                          Max/min principal stresses and their
                                                                          directions.
                                                                          Max shear stress and the orientation of the
                                                                          planes.
                                                                          Value of normal stresses that act on the planes
                                                                          where the max shear stresses act.
                                                                          Values of the normal and shear stresses that
                                                                          act on an element with any orientation.


                                                                 9                                                       10




               Mohr’
Drawing the Mohr’s circle
Perform stress analysis and determine the magnitudes and
directions of the normal and shear stresses acting at the point of
interest, using the convention:
     Tensile stresses are positive; compressive negative.
     Shear stresses that tend to rotate element clockwise are
     positive; otherwise negative.
                                         (σ
Set up the normal and shear stress (σ, τ) plane and locate the
point on it.
Connect the points. The line that cuts the σ-axis is the center of
the circle, O, and radius of the circle, R:
                                                                       Locating the points:              Mohr’
                                                                                                     The Mohr’s circle
                               σ x −σ y
                          R=              + τ xy
                                  2

                                                                11                                                       12




                                                                                                                              3
                                                                         Case when the both principle
                                                                          stresses have the same sign




                  using
                MDESIGN
                                                                 13                                     14




The true maximum shear stress on the element will not
be found if the two principal stresses are of the same
sign.
sign.
Hoop stress, σ x = pD 2t          Where;
                                    p = internal pressure in cylinder
                                    D = diameter of the cylinder
Longtitudinal stress, σ y = pD 4t   t = thickness of the cylinder wall
    3-
Use 3-D stress element
                                                                                                 g N
                                                                                              in
                                                                                            us SIG
                                                                                                E
                                                                                              D
                                                                                            M

                                                                 15                                     16




                                                                                                             4
      Mohr’s circle special stress
             conditions                                  Pure torsional shear
Pure uniaxial tension




Pure uniaxial compression                                Uniaxial tension combined with torsional shear




                                                17                                                                 18




DIRECT STRESSES: TENSION
   AND COMPRESSION
Stress: Internal resistance offered by a unit area              normal stress, σ =
                                                                                      force F
                                                                                      area
                                                                                           =
                                                                                             A
                                                                                                     [N   m2   ]
of material to an external load
Perpendicular to element                             Conditions:
Compressive stresses: Crushing action. Negative      •   load-
                                                         load-carry member must be straight
by convention                                        •   line of action of the load must pass through the
                                                         centroid of cross section of the member
Tensile: Pulling action. Positive by convention.     •   member must be of uniform cross section
                                                     •   material must be homogeneous and isotropic
                                                     •   member must be short in the case of compression
                                                         members
                                                19                                                                 20




                                                                                                                        5
    DEFORMATION UNDER
                                                                 DIRECT SHEAR STRESS
    DIRECT AXIAL LOADING
                       FL σL
                                                           Occurs when the applied force tends to cut through the
                  δ=     =        [m]                      member as scissors. Ex: tendency for a key to be sheared
                       EA E                                off at the section between the shaft and the hub of a
                                                           machine element when transmitting torque(see next
Where:                                                     slide).
 δ = total deformation of the member carrying the axial    Apply force is assumed to be uniformly distributed
                                                           across the cross section.
 F = direct axial load
 L = original load length of the member                               τ=
                                                                           shearing force F
                                                                            area in shear
                                                                                          =
                                                                                            As
                                                                                                          [N   m2   ]
 E = modulus of elasticity of the material
 A = cross-sectional area of the member
     cross-
 σ = direct/normal stress
                                                      21                                                                22




                                                             TORSIONAL SHEAR STRESS
                                                           A torque will twist a member, causing a shear stress in
                                                           the member

                                                                         τ max =
                                                                                   Tc T
                                                                                     =
                                                                                    J Zp
                                                                                                [N   m2   ]

                                                           A general shear stress formula:
                                                                                           Tr
                                                                                     τ=
                                                                                            J


                                                      23                                                                24




                                                                                                                             6
Where:                                                       The distribution of stress is not uniform across the cross
                                                             section
 T  = torque
 c  = radius of shaft to its outside surface
 J                               (Appendix)
    = polar moment of inertia (Appendix)
 r  = radial distance from the center of the shaft to the
      point of interest
 Zp = polar section modulus (Appendix)
                               (Appendix)



                                                        25                                                                26




  TORSIONAL DEFORMATION                                           TORSION IN MEMBERS
                                                                  HAVING NONCIRCULAR
                    θ=
                         TL
                               [deg]                                CROSS-SECTIONS
                         GJ
                                                                    CIRCULAR                     NONCIRCULAR
Where:                                                              SECTION                        SECTION
 T = torque                                                      τ max =
                                                                           Tc T
                                                                             =
                                                                            J Zp
                                                                                              τ max =
                                                                                                        T
                                                                                                        Q
                                                                                                            [N   m2   ]
 L = length of the shaft over which the angle of twist is
     being computed                                              θ=
                                                                      TL                      θ=
                                                                                                   TL
                                                                                                            [rad ]
                                                                      GJ                           GK
 G = modulus of elasticity of the shaft material in shear
 J = polar moment of inertia (Appendix)
                               (Appendix)

                                                        27                                                                28




                                                                                                                               7
                                                               VERTICAL SHEAR STRESS
                                                             Beam carrying transverse loads experience shearing
 TORSION IN                                                         (V
                                                             forces (V) which cause shearing stress:
  MEMBERS
   HAVING                                                            τ=
                                                                          VQ
                                                                           It
                                                                                [N     ]
                                                                                     m 2 ; Q = Ap y   [m ]
                                                                                                         3


NONCIRCULAR
   CROSS-
   CROSS-
  SECTIONS



                                                        29                                                         30




 Where:
  V = shearing force                                         In the analysis of beams, it is usual to compute the
                                                             variation in shearing force across the entire length of
  Q = first moment                                           the beam and to draw the shearing force diagram.
  I = moment of inertia                                      Vertical shear stress = Horizontal shear stress, because
  t = thickness of the section                               any element of material subjected to a shear stress on
  Ap = area of the section above the place where the         one face must have a shear stress of the same
        shearing force is to be computed                     magnitude on the adjacent face for the element to be in
                                                             equilibrium.
  y = distance from the neutral axis of the section to the
        centroid of the area Ap


                                                        31                                                         32




                                                                                                                        8
  STRESS DUE TO BENDING
A beam is a member that carries load transverse to its axis.       Where:
Such loads produce bending moments in the beam, which
result in the development of bending stress.                        M = magnitude of bending moment at the section
Bending stress are normal stresses, that is, either tensile or      c = distance from the neutral axis to the outermost
compressive.
                                                                        fiber of the beam cross section
The maximum bending stress in a beam cross section will
occur in the part farthers from the neutral axis of the section.    I = moment of inertia
At that point, the flexure formula gives the stress:

          flexure formula; σ =
                                  Mc
                                   I
                                          [N   m2   ]

                                                              33                                                          34




The flexure formula was developed subject to the following
conditions:
- Beam must pure bending. No shearing stress and axial loads.
- Beam must not twist or be subjected to torsional load.
                                 Hooke’
- Material of beam must obey Hooke’s law
- Modulus of elasticity of the material must be the same in
  both tension and compression.
- Beam is initially straight and has constant cross section.
- No part of the beam shape fails because of buckling or
  wrinkling.


                                                              35                                                          36




                                                                                                                               9
                                                                      FLEXURAL CENTER FOR
For design, it is convenient to define the term section                     BEAMS
modulus, S:                                                       To ensure symmetrical bending i.e. no tendency to twist
                                                                  under loading, action of load pass through the line of
                        S=I c     [m3 ]
                                                                  symmetry:
The flexure formula then becomes:
                      σ =M S      [ N m2 ]


Then, in design, it is usual to define a design stress, σd ,and
with the bending moment known, then:
                       S = M σd     [m3 ]

                                                           37                                                           38




                                                                      COMBINED NORMAL
                                                                    STRESSES: SUPERPOSITION
                                                                           PRINCIPLE
 If there is no vertical axis symmetry:
                                                                                                    load-
                                                                  When the same cross section of a load-carrying member
                                                                  is subjected to both a direct tensile and compressive
                                                                  stress and a stress due to bending, the resulting normal
                                                                  stress can be computed by the method of superposition:

                                                                                 σ =±
                                                                                        Mc F
                                                                                         I
                                                                                           ±
                                                                                             A
                                                                                                 [N   m2   ]


                                                           39                                                           40




                                                                                                                             10
  COMBINED
    NORMAL                                                                   EXAMPLE 3.8
   STRESSES:
SUPERPOSITION
                                                                               Shigley’s text book
  PRINCIPLE




                                                          41                                                                 42




   STRESS CONCENTRATIONS
           FACTORS
                                                               Example:
  Any geometric discontinuities will cause the actual
  maximum stress to be higher than the calculated value        Diagram shows a round bar subjected to an axial force, F.
  Stress concentration factor, Kt= factor by which the         Compute maximum stress.
  actual maximum stress exceeds the nominal stress             Given Kt = 1.60 (Appendix), F = 9800N
  (σnom,τnom) predicted by calculations. That is:

         σ max = K tσ nom ; τ max = K tτ nom   [ N m2 ]

                                                                          σnom = F/A = ( 9800N)/ [π( 10mm)2/4] = 124.8 MPa
                                                                          σmax = Kt σnom = (1.60)(124.8) = 199.6 MPa

                                                          43                                                                 44




                                                                                                                                  11
                                                                      TYPES OF LOADING AND
                                                                          STRESS RATIO
From figure below, the highest stress occurs in the fillet

                                                                             Mean stress,   σ m = (σ max + σ min ) 2
                                                                        Alternating stress, σ a = (σ max − σ min ) 2

                                                                              Stress ratio, R = σ min σ max

                                                                              Stress ratio, A = σ a σ m




                                                             45                                                             46




Static                                                            Repeated and Reversed
   Load is applied slowly without shock and is held at constant                         load-
                                                                     Reversed: when a load-carrying component is subjected to
   value.                                                            certain level of tensile load followed by a same level of
   R = 1.0 because σmax = σmin                                       compressive load.
                                                                     Repeated: when loading is repeated many thousand times.
                                                                     Also known as fatigue loading.




                                                             47                                                             48




                                                                                                                                 12
                                                                            Fluctuating load
                                                                                                        non-
                                                                               Alternating loading with non-zero mean.
                                                                               Example:




                                                   R.R Moore Fatigue test
                                                   device




                                                                     49                                                                 50




Fluctuating load
                                                                            Shock or Impact loading
                                                     one-
   A special case of fluctuating stress is repeated, one-direction
   stress.                                                                     Loads applied suddenly and rapidly.
                                                                               E.g. Hammer blow, Rock crushing.

                                                                            Random Loading
                                                                               Varying loads that are not regular in their amplitude.




                                                                     51                                                                 52




                                                                                                                                             13
                 CONCLUSION
         NORMAL STRESS



DIRECT STRESS:
                  STRESS DUE
  TENSION &

                                                                THANK YOU
                  TO BENDING
COMPRESSION                            SHEAR STRESS



         SUPERPOSITION           DIRECT          VERTICAL
           PRINCIPLE           SHEAR STRESS   SHEARING STRESS


                                        TORSIONAL
                                       SHEAR STRESS


                                                           53               54




                                                                                 14