Stress Concentration

WHEN TO USE THIS

When you see a place where stress could be concentrated (like a hole or fillet or something like that) make sure you use this stuff (find nominal and then apply a correction factor for the specific thing that causes the stress concentration)

Locations

Includes:

• Shoulders on rotating shafts
• Key slots in shafts
• Bolts with heads, screw threads
• Irregularities

Distributions are altered at discontinuities

• Discontinuities raise stress
• The affected area of the distribution is the stress concentration

Geometry

Stress concentration is geometry dependant

$k_t={\frac{\sigma \max }{{\sigma }_0}}_{}$
$k_{ts}={\frac{\tau \max }{{\tau }_0}}_{}$ Where the nominal stress is ${\sigma }_0$
${\sigma }_0=\frac{Mc}{I_{net}}$
For bending in the plane of a plate
*This situation is a beam with a hole in it
$\frac{6Mw}{\left(w^3-d^3\right)t}$ On a K_t d/w graph
You can find shear stress by finding (K_t) for a given d/w and taking their product.

General strategy for finding max stress / loads

1. Determine ratios from geometry and find stress concentration factor ($k_t$) from the tables/graphs
2. Find allowable average normal stress using the material allowable stress and $k_t$ from the ratios and graphs
3. Apply the definition of normal stress to find allowable loads

Geometry Ratios

The ratio you take is given on the graph you examine

For Static Loading

For ductile materials

• Stress concentration is usually not applied to predict critical stress
• Stress concentration is localized and has a strengthening effect

For brittle materials

• k_t is applied to nominal stress before comparing with strength See Failure Theories for more details on these failures

For Dynamic Loading

Always consider stress concentration

For Normal Stress
$K_f=1+q\left(K_t-1\right)$
For Shear Stress
$K_{fs}=1+q_s\left(K_{ts}-1\right)$
Where q is the Notch Sensitivity factor

Solving for stress concentration factors

• Use geometric ratios to solve for $k_{ts}$ from graphs
• Using $S_{ut}$ given or solved for you can solve for q in Notch Sensitivity by first solving for the Neuber constant and then use the $K_f$ and $K_{fs}$ formulas above to get the factors
• You can find the factor of safety for fatigue based on infinite life using $n_f=\frac{S_e}{k_f{\sigma }_{ar}}$ where sometimes ${\sigma }_{ar}={\sigma }_o$