Geometric stiffness matrix for solid elements

[3343534645]

Hello, We have recently attempted to calculate the geometric stiffness matrix for solid elements using the methods described in the ANSYS help documentation, as well as the theories of Bathe, Cook, Borst, Němec, and Wriggers. However, the resulting geometric stiffness matrices consistently differ from those output by ANSYS APDL. We would like to confirm whether the computational principles used by ANSYS APDL for the geometric stiffness matrix differ from those documented in publicly available literature, or if there are any specific internal reference materials on this topic. Thank you for your response!

[Akshat Gupta]

Hello,

The stiffness matrices obtained using APDL may differ from the traditional stiffness matrices derived in textbooks. This discrepancy may arise because APDL calculates stiffness matrices through numerical integration methods.

For a deeper understanding of how these matrices are formulated, please go to the help document and refer to the sections shared below:

Derivation of Structural Matrices:

• Ansys Help > MAPDL > 2. Structures > 2.2 Derivation of Structural Matrices

Stress Stiffening:

• Ansys Help > MAPDL > 3. Structures with Geometric Nonlinearities > 3.4. Stress Stiffening

Thanks.

[3343534645]

Hello, thank you for your reply. I have a question: the method for calculating the geometric stiffness matrix described in Section 3.4 "Stress Stiffening" of the ANSYS help documentation is consistent with the literature by Bathe et al., which leads to the presence of "zero" elements in the calculated results. Could you clarify how the APDL method you mentioned derives the geometric stiffness matrix using numerical integration? We found that the formulas derived in Section 3.5 "Linear Buckling Analysis" of the material provided by Borst, "Non‐Linear Finite Element Analysis of Solids and Structures," are exactly the same as those provided by Bathe et al.

 

 

 

[Akshat Gupta]

Hello,

Thanks for your query.

In section 3.4, ‘Stress Stiffening’, you can find sub-section 3.4.3 ‘, Implementation’, which discusses how these matrices are calculated. Equation (3-60) onwards, will give more clarity in this regard.

Also, please mention the procedure (commands) used to determine APDL geometric stiffness.

Are you taking out element stiffness or global stiffness?

In addition, I want to check the procedure you are using to calculate the stiffness theoretically. What kind of elements/mesh are you using?  Is the mesh/element type comparable to the APDL mesh and elements being used?

Thanks.

[3343534645]

Hello, thank you for your reply. Initially, I also followed the formula (3-60) from the ANSYS help documentation, as you mentioned, to calculate the geometric stiffness matrix. This theoretical expression produces a geometric stiffness matrix with zero elements, which can be verified using Mathematica. When extracting the geometric stiffness matrix in ANSYS APDL, I first performed a static analysis, then a buckling eigenvalue analysis, and used the following command to output the .out file, in which the geometric stiffness matrix is included. I calculated the 8-node hexahedral element (cube, full integration, single element) and the 20-node hexahedral element (cube, full integration, single element), and their extracted results are as follows. Finally, I validated the results using MATLAB and found that the eigenvalues calculated from the extracted geometric stiffness matrix and the material stiffness matrix matched the results output by ANSYS APDL. This implies that the formula used by ANSYS APDL to calculate the geometric stiffness matrix for solid elements is different from what is documented in the ANSYS help manual or existing literature. That’s why I am reaching out to you for assistance. Thank you for your time and response!

 

 

[3343534645]

Dear Ansys Team,

I sincerely hope that you can publish the theoretical basis for calculating the geometric stiffness matrix for solid elements. Compared to existing literature or textbooks, the method you use in ANSYS APDL for calculating the geometric stiffness matrix seems unique and difficult to trace. It would be a great pity if all the theories on geometric stiffness matrices for solid elements published over the past few decades turned out to be incorrect! Thank you for your response.

Best regards,