1. What Is the Stiffness Matrix?

The stiffness matrix (k) is a fundamental concept in finite element analysis that relates nodal displacements to applied forces. It represents the resistance of a structure to deformation under load.

2. How Does the Calculator Work?

The calculator uses the stiffness matrix equation:

Where:

• \( b \) — Strain-displacement matrix
• \( d \) — Constitutive matrix (material properties)
• \( dv \) — Volume element
• \( k \) — Resulting stiffness matrix

Explanation: The equation integrates the product of the strain-displacement matrix (transposed), constitutive matrix, and strain-displacement matrix over the element volume.

3. Importance of Stiffness Matrix

Details: The stiffness matrix is crucial for solving structural mechanics problems in finite element analysis, determining displacements, stresses, and natural frequencies of structures.

4. Using the Calculator

Tips: Enter the strain-displacement matrix (b), constitutive matrix (d), and element volume. The matrices should be entered in proper matrix format (rows separated by newlines, columns by spaces).

5. Frequently Asked Questions (FAQ)

Q1: What is the physical meaning of the stiffness matrix?
A: Each entry k_ij represents the force at degree of freedom i due to a unit displacement at degree of freedom j, with all other displacements held fixed.

Q2: How does element type affect the stiffness matrix?
A: Different element types (beam, shell, solid) have different strain-displacement matrices and thus different stiffness matrices.

Q3: What are the units of the stiffness matrix?
A: Typically N/m (force/length) for structural problems, though units depend on the specific problem.

Q4: How is the global stiffness matrix assembled?
A: Element stiffness matrices are combined using the direct stiffness method, accounting for connectivity between elements.

Q5: What affects the accuracy of the stiffness matrix?
A: Element formulation, integration scheme, mesh density, and material property accuracy all influence results.