The stiffness method, also known as the displacement method or equilibrium method, is a structural analysis technique that determines the unknown displacements at the joints of a structure to solve for internal forces and stresses.
Understanding the Stiffness Method
The stiffness method fundamentally differs from the flexibility method (also known as the force method or compatibility method). While the flexibility method focuses on solving for redundant forces, the stiffness method directly tackles displacement variables. Here's a breakdown of key aspects:
- Focus on Displacements: The primary unknowns are the displacements (translations and rotations) at the nodes or joints of the structure.
- Equilibrium Equations: The method utilizes the equilibrium conditions at each node to relate the applied loads to the nodal displacements.
- Stiffness Matrix: A key element is the creation and assembly of the structure stiffness matrix. This matrix relates the nodal forces to the nodal displacements for the entire structure:
{F} = [K] {u}, where{F}is the force vector,[K]is the stiffness matrix, and{u}is the displacement vector. - Solving for Displacements: The equation
{F} = [K] {u}is solved for the unknown displacements{u}. - Calculating Internal Forces: Once the displacements are known, the internal forces (axial forces, shear forces, bending moments) in each element can be calculated using the element's stiffness properties and the nodal displacements of that element.
Key Steps in the Stiffness Method
- Discretization: Divide the structure into a finite number of elements (e.g., beams, columns, trusses) connected at nodes.
- Element Stiffness Matrices: Determine the stiffness matrix for each individual element. This matrix relates the forces and displacements at the element's nodes.
- Global Stiffness Matrix: Assemble the individual element stiffness matrices into a global stiffness matrix for the entire structure, considering the connectivity of the elements. This step involves transforming element stiffness matrices to the global coordinate system and summing them appropriately.
- Apply Boundary Conditions: Apply known support conditions (e.g., fixed supports, pinned supports). This involves modifying the global stiffness matrix and the force vector to account for the constraints.
- Solve for Displacements: Solve the system of equations
{F} = [K] {u}for the unknown nodal displacements{u}. - Calculate Element Forces: Using the calculated nodal displacements and the element stiffness matrices, determine the internal forces (axial force, shear force, bending moment) in each element.
Advantages of the Stiffness Method
- Suited for Computer Implementation: The stiffness method is highly systematic and can be easily implemented in computer programs. This is because the assembly of the global stiffness matrix is a well-defined process.
- Applicable to Complex Structures: It can handle complex structures with multiple degrees of freedom, including those with complex geometry and loading conditions.
- Direct Solution for Displacements: It directly provides the displacements, which are often of primary interest in structural design.
Example Application
Consider a simple two-span continuous beam. The stiffness method would involve:
- Dividing the beam into two elements.
- Developing the element stiffness matrix for each beam element.
- Assembling the global stiffness matrix for the entire beam.
- Applying the support conditions (e.g., pinned or fixed supports).
- Solving for the unknown displacements at the intermediate support.
- Calculating the bending moments and shear forces in each span.
In summary, the stiffness method is a powerful and widely used technique for analyzing structures by directly solving for nodal displacements based on equilibrium and the structure's stiffness properties. It lends itself well to computational analysis and is applicable to a wide range of structural problems.
