Using FEA to determine Kt (Stress conc Factor)
Using FEA Results To Determine Stress Concentration Factors:
Stress Concentration factors (Kt’s) for numerous simple geometries have been determined by researchers (analytical equations), which have been compiled into tables. eg: Roark's Handbook.
Determining stress concentration factors (Kt) for complex geometries can be difficult and expensive but Kt can be obtained easily using FEA results.
The general Kt equation is:
`Kt = σ_(max) / σ_(nom)`
FEA results can be used to easily determine the maximum stress (σmax). Determining the nominal stress (σ_nom) can be more difficult. Knowledge of the stress gradient across the lateral critical section can provide a means of determining the nominal stress (σnom).
Kt calulation Methodology:
- If the stress concentration were not present, the uniform stress gradient is the nominal stress (σ_nom).
- But in reality, due to stress concentration, the stresses redistribute along the critical section.
- Using the plot of the σ_max across the critical section shown above obtained from the FEA analysis, calculate area under the curve (A).
- The area under the two stress gradient curves (stress concentration and uniform stress field) must be equal (equivalent energy).
- σ_nom can be calculated as `σ_(nom)=("Area under the curve (A)")/("Length of integration (x)"`
- Kt obtained now can be compared with that from the handbook to estimate the error%.
ANSYS Kt Evaluation:
The plate is subject to a force of 10kN tension.
Plate dimensions: Length=300mm, Height=120mm, Thickness=30mm.
Circular Hole at the center: Diameter=60mm.
Here a quarter model with symmetry (Y and Z) will be setup as shown below:
The normal stress along the path described before is obtained as shown below:
The Path is created using Construction Geometry. This Path is then selected as the scoping method for normal stress evaluation.
A graph is plotted for the normal stress against the path shown above. To estimate the area under the curve a polynomial is fit and is integrated to obtain the area.
`∫_0^30 y(dx) = -0.0007 x^4/4 + 0.0385 x^3/3 - 0.9119 x^2/2 + 12.244x=161.715`
Area under the curve = 161.715
`σ_(nom)= A/x=161.715/30=5.39MPa`
`K_t=σ_(max)/σ_(nom)=12.53/5.39=2.3`
Value of Kt obtained from FEA is 2.3;
Analytical Evaluation:
Kt can be obtained from Roark's handbook:
`K_t=3.00-3.13*(60/120)+3.66*(60/120)^2-1.53*(60/120)^3= 2.16`
Percentage error wrt analytical soln= `(2.3-2.16)/2.16=0.064 or 6.4%`
σ_nom= 10000N/1800mm^2 = 5.55MPa
σ_max= σ_nom * Kt= 11.98MPa
Inference:
The FEA is a more reasonable method of determining Kt as it more accurate and less expensive. The results in FEA can take the actual geometry into evaluation, unlike the handbook results which make assumptions (Length is considered to be infinite).
Another feature that cannot be considered analytically is the stress variation across thickness:
References: