Various questions related to flexibility factor
What does it mean to have flexibility factor = 1?
Answer:
it means that the component does not have any flexibility taken into consideration. For example, a straight pipe has a flexibility factor equal to 1.
How are flexibility factors incorporated into AutoPIPE's calculations?
Let us first understand AutoPIPE's methodology of handling stiffness for a straight pipe and a bend. Then discuss how flexibility factors influences these components.
AutoPIPE’s 12x12 stiffness matrix is used to solve Hooke’s law. That is, the relationship between deflection, x, and reaction, F:
F = k x
The matrix, or k in our example, is 12x12 for the six degrees of freedom (DOF) and the two points of a pipe element.
You can think of the matrix as being split into 4 quadrants, each a 6x6:
Where
- kii is the relationship of deflection of node I to reaction of node I
- kij is the relationship of deflection of node J to reaction of node I
- kji is the relationship of deflection of node I to reaction of node J
- kjj is the relationship of deflection of node J to reaction of node J
The full matrix looks like this for tangent (straight) pipes, where the red boxes represent the four quadrants explained above:
AutoPIPE uses a different stiffness matrix for curved pipe which is much more complicated in nature.
Calculation of DOF Flexibility on Node J
Let the pipe wall shape factor
The stiffness relationship between loads in the bend axial direction to translation in the axial direction consists of three components – axial, shear, bending. Here, you can see the flexibility factor, k, in the bending component.
The stiffness relationship between loads in the X direction to deflections in the Y direction consists of the same three components evaluated in different ways. Again, you see the flexibility factor, k, in the bending component.
Rotation to moment stiffness relationship is different altogether. For example, the axial rotation to axial moment, with torsional term applied is
This forms a 6x6 matrix relationship of a nodes load to its deflections. This matrix is then reflected and inverted
Calculation of DOF Flexibility on Node I
The 6x6 stiffness block for relating loads from node I to deflection from node I, a matrix multiplication of and translation matrix H is performed
Calculation of DOF Flexibility on Node I-J
Using the H translation matrix from the previous section, \mathcal{K}_{i,j} can be calculated as
Complete Matrix
The complete matrix can be visualized as four quadrants: I-I, J-J, I-J, J-I:
Conclusion:
A straight pipe has a flexibility factor of 1 and therefore does not affect the stiffness matrix. However a bend component considers a flexibility factor, k, under bending.
Bend & Miter Piping Components - Modeling Approaches
Confirm and modify the Bend and Tee flexibility factors
Bentley AutoPIPE