Stress and Strain (3.8-3.12, 3.14)

1
Stress and Strain(3.8-3.12, 3.14)

• MAE 316 Strength of Mechanical Components
• NC State University Department of Mechanical
  Aerospace Engineering

2
Introduction

• MAE 316 is a continuation of MAE 314 (solid
  mechanics)
• Review topics
• Beam theory
• Columns
• Pressure vessels
• Principle stresses
• New topics
• Contact Stress
• Press and shrink fits
• Fracture mechanics
• Fatigue

3
Normal Stress (3.9)

• Normal Stress (axial loading)
• Sign Convention
• s gt 0 Tensile (member is in tension)
• s lt 0 Compressive (member is in compression)

4
Shear Stress (3.9)

• Shear stress (transverse loading)
• Single shear
• Double shear

average shear stress
5
Stress and Strain Review (3.3)

• Bearing stress
• Bearing stress is a normal stress

Single shear case
6
Strain (3.8)

• Normal strain (axial loading)
• Hookes Law

Where E Modulus of Elasticity (Youngs modulus)
7
Stress and Strain Review (3.12)

• Torsion (circular shaft)
• Shear strain
• Shear stress
• Angle of twist

T
Where G Shear modulus (Modulus of rigidity)and
J Polar moment of inertia of shaft cross-section
8
Thin-Walled Pressure Vessels (3.14)

• Thin-walled pressure vessels
• Cylindrical
• Spherical

Circumferential hoop stress
Longitudinal stress
9
Beams in Bending (3.10)

• Beams in pure bending
• Strain
• Stress

Where ? Poissons Ratio and ? radius of
curvature
Where I 2nd moment of inertia of the
cross-section
10
Beam Shear and Bending (3.10-3.11)

• Beams (non-uniform bending)
• Shear and bending moment
• Shear stress
• Design of beams for bending

Where Q 1st moment of the cross-section
11
Example
Draw the shear and bending-moment diagrams for
the beam and loading shown.
12
Example Problem

• An extruded aluminum beam has the cross section
  shown. Knowing that
• the vertical shear in the beam is 150 kN,
  determine the shearing stress
• a (a) point a, and (b) point b.

13
Combined Stress in Beams

• In MAE 314, we calculated stress and strain for
  each type of load separately (axial, centric,
  transverse, etc.).
• When more than one type of load acts on a beam,
  the combined stress can be found by the
  superposition of several stress states.

14
Combined Stress in Beams

• Determine the normal and shearing stress at point
  K. The radius of the bar is 20 mm.

15
2D and 3D Stress(3.6-3.7)

• MAE 316 Strength of Mechanical Components
• NC State University Department of Mechanical
  Aerospace Engineering

16
Plane (2D) Stress (3.6)

• Consider a state of plane stress sz txz tyz
  0.

f
f
f
f
f
f
f
f
Slice cube at an angle f to the x-axis (new
coordinates x, y)
f
17
Plane (2D) Stress (3.6)

• Sum forces in x direction and y direction and
  use trig identities to formulate equations for
  transformed stress.

f
f
f
f
f
18
Plane (2D) Stress (3.6)

• Plotting a Mohrs Circle, we can also develop
  equations for principle stress, maximum shearing
  stress, and the orientations at which they occur.

19
Plane (2D) Strain

• Mathematically, the transformation of strain is
  the same as stress transformation with the
  following substitutions.

20
3D Stress (3.7)

• Now, there are three possible principal stresses.
• Also, recall the stress tensor can be expressed
  in matrix form.

21
3D Stress (3.7)

• We can solve for the principle stresses (s1, s2,
  s3) using a stress cubic equation.
• Where i 1,2,3 and the three constant I1, I2,
  and I3 are expressed as follows.

22
3D Stress
Let nx, ny and nz be the direction cosines of the
normal vector to surface ABC with respect to x,
y, and z directions respectively.
23
3D Stress (3.7)

• How do we find the maximum shearing stress?
• The most visual method is to observe a 3D Mohr's
  Circle.
• Rank principle stresses largest to smallest s1 gt
  s2 gt s3

24
3D Stress (5.5)

• A plane that makes equal angles with the
  principal planes is called an octahedral plane.

25
3D Stress (3.7)

• For the stress state shown below, find the
  principle stresses and maximum shear stress.

Stress tensor
26
3D Stress (3.7)

• Draw Mohrs Circle for the stress state shown
  below.

Stress tensor
27
Curved Beams(3.18)

• MAE 316 Strength of Mechanical Components
• NC State University Department of Mechanical
  Aerospace Engineering

28
Curved Beams (3.18)

• Thus far, we have only analyzed stress in
  straight beams.
• However, there many situations where curved beams
  are used.

Hooks
Chain links
Curved structural beams
29
Curved Beams (3.18)

• Assumptions
• Pure bending (no shear and axial forces present
  will add these later)
• Bending occurs in a single plane
• The cross-section has at least one axis of
  symmetry
• What does this mean?
• s -My/I no longer applies
• Neutral axis and axis of symmetry (centroid) are
  no longer the same
• Stress distribution is not linear

30
Curved Beams (3.18)
Flexure formula for tangential stress
Where M bending moment about centroidal axis
(positive M puts inner surface in tension)y
distance from neutral axis to point of interestA
cross-section areae distance from centroidal
axis to neutral axis rn radius of neutral axis
31
Curved Beams (3.18)

• If there is also an axial force present, the
  flexure formula can be written as follows.
• Table 3-4 in the textbook shows rn formulas for
  several common cross-section shapes.

32
Example
Plot the distribution of stresses across section
A-A of the crane hook shown below. The cross
section is rectangular, with b0.75 in and h 4
in, and the load is F 5000 lbf.
33
Curved Beams (3.18)

• Calculate the tangential stress at A and B on the
  curved hook shown below if the load P 90 kN.