Triaxial State of Stress at any Critical Point in a Loaded

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Triaxial State of Stress at any Critical Point in
a Loaded Body

• Cartesian stress components are found first in
  selected x-y-z coordinate axes (Fig. 4.1)
• Three mutually perpendicular principal planes are
  found at unique orientations
• NO Shear stresses on these planes
• Principal Normal Stresses ?1, ?3, ?2, one of
  which is maximum normal stress at the point
• Three mutually perpendicular Principal Shearing
  Planes (Planes of max. shear)
• Principal Shearing Stresses ?1, ?2, ?3, one of
  which is the maximum shear stress at the point
• Normal stresses are NOT zero, NOT principal
  stresses and depend on the type of loading

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3-D Stress Transformations Equations

• Relate Known Cartesian Stress Components at Any
  Point with Unknown Stress Components on Any Other
  Plane through the SAME Point
• From equilibrium conditions of infinitesimal
  pyramid
• Important for the Unique Orientations of the
  Principal Normal and Shearing Planes
• Stress Cubic Equation- three real roots are the
  Principal Normal Stresses, ?1, ?2, ?3.
• Principal Shearing Stresses can be calculated
  from the Principal Normal Stresses as follows

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Mohrs Circle Analogy for Stress Graphical
Transformation of 2-D Stress State

• Principal stress solution of stress cube eq. for
  stresses in the x-y plane (Fig. 4.12)
• Analogy with the equation of a circle plotted in
  the ?-? plane leads to Mohrs circle for biaxial
  stress
• Sign convention for plotting the Mohrs circle
  (normal stress is positive for tension, shear is
  positive for clockwise (CW) couple)
• Two additional Mohrs circle for triaxial stress
  states
• Find orientation of principal axes from the
  Mohrs circle

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Strain Cubic Equation and Principal Strains

• STRAIN a measure of loading severity, defining
  the intensity and direction of deformation at a
  point, w.r.t. specified planes through that
  point.
• Strain state at a point is completely defined by
• Three normal and three shearing strain components
  in the selected x-y-z coordinate system, OR
• Three PRINCIPAL strains and their directions from
  Strain Cubic Equation (similar to Stress Cubic
  Equation, where ?s are replaced by ?s, and
  shear stresses, ?s, are replaced by one-half of
  ?s.)

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Summary of Example Problems

• Example 4.8 Principal Stresses in Beam
• Hollow cylindrical member subjected to transverse
  forces (four-point bending), axial force and
  torque
• Sketch state of stress at critical point (bottom
  edge)
• Use stress cubic equation to find principal
  stresses from calculated Cartesian stresses at
  critical point
• Find principal shear stresses from principal
  normal ones
• Example 4.9 Mohrs Circle for Stress
• Semi-graphical analysis of biaxial stress state
  at critical point of previous example ?cube is
  replaced by 2-D sketch in x-y plane
• Principal normal and shear stresses are found
  graphically from the basic and the two additional
  Mohr circles, respectively

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Summary of Textbook Problems Problem 4.31,
Principal Stresses

• Identify critical points for each of the three
  types of loading applied on the bar
• Locations where stresses are amplified by
  superposition of effects from different loads
  top end of vertical diameter and left end of
  horizontal one
• Sketch infinitesimal cube elements for the states
  of stress at critical points
• Calculate stresses at critical points, in the
  given system of Cartesian coordinates
• Use stress cubic equation to find the principal
  stresses at each of the two critical points
• Top edge ?126,047 psi, ?20, ?3-7683 psi
• Left edge ?131,124 psi, ?20, ?3-31,124 psi
• Calculate the maximum shearing stress at each
  point
• Top edge ?max 16,865 psi, while at the left
  edge ?max 31,124 psi

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