MET 415 Lecture 04 Notes, SP08

First Principles:

Newton’s Laws of motion give:             S F = m a_G                     S M_G = I_G a

What is “G” ?                What is the parallel axis theorem ?

For a static analysis:             S F = 0                      S M_O = 0

Rigid Body Motion: Movement without strain

Stress and Strain

• Normal stress (s), shear stress (t)
• SX or s_X
• SY or s_Y
• SZ or s_Z
• SXY or t_XY
• SYZ or t_YZ
• SXZ or t_XZ
• Principal stress quantities
• S1 or s₁
• S2 or s₂
• S3 or s₃
• SEQV or s_EQ        (equivalent stress, vonMises)
• SINT or 2*t_max      (stress intensity
• Plane stress (biaxial stress): No stress normal to the plane of loading (2D)
• Plane strain: No strain normal to the plane of loading (2D)

Fundamental Stress States

(from Strength of Materials, MCH T 213 or MET 320)

1. Direct stress (uniform bar under uniaxial load):  s = F/A

2. Beam in Flexure: Maximum bending stress at outer fiber, s = Mc/I

3. Beam in Shear: Maximum shear stress at neutral axis, t = VQ/Ib

4. Cylinder in Torsion: Maximum shear stress on O.D., t = Tr/J

5. Hollow cylinder, internal pressure: 

(With thin walls) Maximum hoop stress, s_H = p r_o/t  
 If ends are capped: Maximum axial stress, s_Z = p r_i/2t  
(In both cases, true only far from the end caps)

Thin-walled cylinder: t < (0.05) r
Thick-walled vessel formulae are more complicated.

6. Determine stress caused by contact of parts: formulae for spheres and cylinders

7. Determine stress caused by constrained thermal expansion, e_th = a DT

8. Properly apply stress concentration factors: K_t

COMMON ASSUMPTIONS:

• 2D or 3D ?  Solid, Shell, or Line elements ?
• Symmetry ?
• Isotropic and homogeneous material
• Linear elastic behavior i.e.,
  • material obeys Hooke’s Law [ s = E e ] and is independent of load rate
  • small deformation effects
    (Small angles approx. cos q = 1, sin q = q , tan q = 0)
  • no change in contact between bodies; no sliding
  • final load distribution is known and defined initially
• How well do you know the material properties ?
• Static loading: Not varying with time. No impact, no vibration
• Initially straight / Constant cross-sections
• Nominal dimensions
• What about alignment of loads ?
• How is the structure really constrained ?
• Is gravity load (weight) important ?
• Ignore rounds/chamfers on outside corner ?
• Ignore cosmetic features ?
• Ignore small features in regions of low stress ?
• Ignore environmental factors: temperature, humidity, moisture, corrosion/oxidation, etc.