Stress Transformation 9.1-9.3 Plane Stress Stress

1
Stress Transformation

• 9.1-9.3
• Plane Stress
• Stress Transformation in Plane Stress
• Principal Stresses Maximum Shear Stress

2
Introduction

• We have learned
• Axially
• In Torsion
• In bending
• These stresses act on cross sections of the
  members.
• Larger stresses can occur on inclined sections.

3
Introduction

• We will look at stress elements to analyze the
  state of stress produce by a single type of load
  or by a combination of loads.
• From the stress element, we will derive the
  Transformation Equations
• Give the stresses acting on the sides of an
  element oriented in a different direction.

4
Introduction

• Stress elements only one intrinsic state of
  stress exists at a point in a stressed body,
  regardless of the orientation of the element for
  that state of stress.
• Two elements with different orientations at the
  same point in a body, the stress acting on the
  faces of the two elements are different, but
  represent the same state of stress
• The stress at the point under consideration.

5
Introduction

• Remember, stresses are not vectors.
• Are represented like a vector with magnitude and
  direction
• Do not combine with vector algebra
• Stresses are much more complex quantities than
  vectors
• Are called Tensors (like strain and I)

6
Plane Stress

• Plane Stress The state of stress when we
  analyzed bars in tension and compression, shafts
  in torsion, and beams in bending.
• Consider a 3 dimensional stress element
• Material is in plane stress in the xy plane
• Only the x and y faces of the element are
  subjected to stresses
• All stresses act parallel to the x and y axis

7
Plane Stress

• Normal stress
• subscript identifies the face on which the stress
  acts
• Sign Convention
• Tension positive
• compression negative

8
Plane Stress

• Shear Stress -
• Two subscripts
• First denotes the face on which the stress acts
• Second gives the direction on that face
• Sign convention
• Positive when acts on a positive face of an
  element in the positive direction of an axis ()
  or (--)
• Negative when acts on a positive face of an
  element in the negative direction of an axis (-)
  or (-)

9
Plane Stress

• A 2-dimensional view can depict the relevant
  stress information, fig. 9.1c
• Special cases
• Uniaxial Stress
• Pure shear
• Biaxial stress

10
Stresses on Inclined Planes

• First we know ?x, ?y, and ?xy,
• Consider a new stress element
• Located at the same point in the material as the
  original element, but is rotated about the z axis
• x and y axis rotated through an angle ?

11
Stresses on Inclined Planes

• The normal and shear stresses acting on they new
  element are
• Using the same subscript designations and sign
  conventions described.
• Remembering equilibrium, we know that

12
Stresses on Inclined Planes

• The stresses in the xy plane can be expressed
  in terms of the stresses on the xy element by
  using equilibrium.
• Consider a wedge shaped element
• Inclined face same as the x face of inclined
  element.

13
Stresses on Inclined Planes

• Construct a FBD showing all the forces acting on
  the faces
• The sectioned face is ?A.
• Then the normal and shear forces can be
  represented on the FBD.
• Summing forces in the x and y directions and
  remembering trig identities, we get

14
Stresses on Inclined Planes

• These are called the transformation equations for
  plane stress.
• They transfer the stress component form one set
  of axes to another.
• The state of stress remains the same.
• Based only on equilibrium, do not depend on
  material properties or geometry
• There are Strain Transformation equations that
  are based solely on the geometry of deformation.

15
Stresses on Inclined Planes

• Special case simplifications
• Uniaxial stress- ?y Txy 0
• Pure Shear - ?x ?y 0
• Biaxial stress - Txy 0
• Transformation equations are simplified
  accordingly.

16
Principal Maximum Shear Stresses

• Since a structural member can fail due to
  excessive normal or shear stress, we need to know
  what the maximum normal and stresses are at a
  point.
• We will determine the maximum and minimum stress
  planes for which maximum and minimum normal and
  shear stresses act.

17
Principal Maximum Shear Stresses

• Principal stresses maximum and minimum normal
  stresses.
• Occurs on planes where
• Applying to eq 9.1 we get
• ?pthe orientation of the principal planes
• The planes on which the principal stresses act.

18
Principal Maximum Shear Stresses

• Two values of the angle 2?p are obtained from the
  equation.
• One value 0-180, other 180-360
• Therefore ?p has two values 0-90 90-180
• Values are called Principal Angles.
• For one angle ?x is maximum, the other ?x is
  minimum.
• Therefore Principal stresses occur on mutually
  perpendicular planes.

19
Principal Maximum Shear Stresses

• We could find the principal stress by
  substituting this angle into the transformation
  equation and solving
• Or we could derive general formulas for the
  principal stresses.

20
Principal Stresses

• Consider the right triangle
• Using the trig from the triangle and substituting
  into the transformation equation for normal
  stress, we get
• Formula for principal stresses.

21
Shear Stresses on the Principal Planes

• If we set the shear stress ?xy equal to zero in
  the transformation equation and solve for 2?, we
  get equation 9-4.
• The angles to the planes of zero shear stress are
  the same as the angles to the principal planes
• ThereforeThe shear stresses are zero on the
  principal planes

22
The Third Principal Stress

• We looked only at the xy plane rotating about the
  z-axis.
• Equations derived are in-plane principal stresses
• BUT, stress element is 3D and has 3 principal
  stresses.
• By Eigenvalue analysis it can be shown that ?z0
  when oriented on the principal plane.

23
Maximum In-Plane Shear Stress

• Consider the maximum shear stress and the plane
  on which they act.
• The shear stresses are given by the
  transformation equations.
• Taking the derivative of ?xy with respect to ?
  and setting it equal to zero we can derive
  equation 9-7

24
Maximum Shear Stress

• The maximum negative shear stress ?min has the
  same magnitude but opposite sign.
• The planes of maximum shear stress occur at 45 to
  the principal planes

25
Maximum Shear Stress

• If we use equation 9-5, subtract ?2 from ?1, and
  compare with equation 9-7, we see that
• Maximum shear stress is equal to ½ the difference
  of the principal shear stress.

26
Average Normal Stress

• The planes of maximum shear stress also contain
  normal stresses.
• Normal stresses acting on the planes of maximum
  positive shear stress can be determined by
  substituting the expressions for the angle ?s
  into the equations for ?x.
• Result is Equation 9-8.

27
Important Points

• The principal stresses are the max and min normal
  stress at a point
• When the state of stress is represented by the
  principal stresses, no shear stress acts on the
  element
• The state of stress at the point can also be
  represented in terms of max in-plane shear
  stress. In this case an average normal stress
  also acts on the element
• The element in max in-plane shear stress is
  oriented 45 from the element in principal
  stresses.

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