Mock Course Plane Stress Transformation (ref MCHT 213

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Mock Course Plane Stress Transformation (ref
MCHT 213, Hibbeler text)

• 1 hr 15 min format
• Present Theory/Overview with Power Point, 15 min.
• Problem solving on board (example problems), 45
  min
• Industry Examples, 5 min
• Answer HW questions

Chapter 9 Plane Stress Transformations
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Stress Analysis - Review

• Determine critical point
• Solve for internal forces at that point (or
  reduce to cantilever)
• Solve for stresses at that point
• Add like stresses, i.e.
• stotal s1 s2 s3 ...
• ttotal t1 t2 t3
• Summarize stresses at that point on a stress
  element.
• May be necessary to use Stress Transformation or
  Mohrs circle to get max stresses!

Chapter 8
Chapter 9
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Recall from Chapter 8, already did steps 1 5
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Recall from Chapter 8, already did steps 1 5
Drill Bit
Drill Bit Isolator (
Thrust Load 8,000 to 10,000 lb
Bending Load 125 lbs) Kd
Drill Rod
Torsion Load 300 lb-ft
Chuck Isolator
HOT SPOT!
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Now What????
Solve for stress at a point using standard SoM
Equations.
Summarize these stresses on an initial stress
element or aligned stress element.
Compare max stresses to material allowables to
determine Is it safe, will it fail???
step 6
Must find MAXIMUM stresses at that point, may be
different then the applied stresses can occur at
some other orientation plane or angle.
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Step 6. Stresses on other planes? Really nothing
new, recall Chapter 1 Average normal stress and
shear stress (see example 1-10)
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Now, instead of stresses on planes transform
stress at a point
9.1 Stress Elements
a) Stress state for plane stress can be
summarized on a 2D element.
a) In general, can have 6 independent stresses (3
normal and 3 shear) acting at a point.
a) Many practical engineering problems involve
only three independent stresses called plane
stress.
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9.2 Plane Stress Transformation In this course
force equilibrium in 2D) Graduate course force
equilubruim in 3D but math solved with matrices
and tensors!!/ solve eigenvalue problem)
Derivation of the plane-stress transformation
equations
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A. Given Plane Stress State
B. What are new stresses at element rotation of q
??
Note, positive stress directions shown.
Note, positive angle (ccw) shown.
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Easy, cut element, sum forces in the x and y
directions
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(9.1)
(9.2)
(9.3)
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9.3 Principal Stresses and Maximum In-Plane Shear
Stresses

• What are the maximum stresses at a point?
• Will they be different than whats shown on the
  initial stress element?

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Recall equation for normal stress at new angle q
(9.1)
Want max and min stresses so what do we do?
Solving for q
(9.4)
This is the principal plane
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Next, plug 9.4 into 9.1 and get
(9.5)
Max principal
Min principal
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So, to summarize
Principal Stress/Principal plane Note-shear
stresses are zero on principal plane!
(9.5)
(9.4)
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Next, do the same thing for shear stress
(9.2)
Solve for angle, then plug into eqn 9.2 to get
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(9.6)
(9.7)
(9.8)
1. On planes of max shear, normal stress is not
zero but savg as shown in 9.8. 2. The planes for
max shear stress can be determined by orienting
an element 45deg from the position of an element
of max principal stress!
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Summary
Max principal stress
Max shear stress
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Important Points

• The principal stresses represent the maximum and
  minimum normal stress at the point. These
  stresses are shown on the principal stress
  element.
• When the state of stress is represented by the
  principal stresses, no shear stress will act on
  the element.
• The state of stress at the point can also be
  represented in terms of the maximum in-plane
  shear stress. In this case an average normal
  stress will also act on the element. This is
  called the maximum in-plane shear stress element.
• The maximum in-plane shear stress element is
  oriented 45 degrees from the principal stress
  element.

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210/418/470 Capstone Design Projects!!