12.2 Stiffness matrix method for trusses

The stiffness matrix method for trusses is a powerful tool in structural analysis. It uses matrices to represent how trusses resist deformation, combining individual element behaviors into a global system. This approach allows engineers to analyze complex structures efficiently.

By creating and assembling stiffness matrices, we can solve for displacements and forces in trusses. This method connects to the broader topic of matrix analysis, providing a foundation for understanding more complex structural systems.

Stiffness Matrices

Global and Element Stiffness Matrices

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Top images from around the web for Global and Element Stiffness Matrices

• 

  Frontiers | Using Influence Matrices as a Design and Analysis Tool for Adaptive Truss and Beam ... View original

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• 

  Frontiers | Vibration Suppression Through Variable Stiffness and Damping Structural Joints View original

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• 

  New Formula for Geometric Stiffness Matrix Calculation View original

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  Frontiers | Using Influence Matrices as a Design and Analysis Tool for Adaptive Truss and Beam ... View original

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  Frontiers | Vibration Suppression Through Variable Stiffness and Damping Structural Joints View original

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• Global stiffness matrix represents the entire structure's resistance to deformation
• Contains information about all elements and their connections
• Element stiffness matrix describes the behavior of individual structural members
• Relates forces to displacements for a single element
• Size of element stiffness matrix depends on the number of degrees of freedom per element
• For truss elements, typically 4x4 matrix (2 DOFs at each end)
• Element stiffness matrix incorporates material properties (Young's modulus) and geometric characteristics (cross-sectional area, length)

Assembly of Global Stiffness Matrix

• Process of combining individual element stiffness matrices into the global stiffness matrix
• Involves mapping local element DOFs to global structure DOFs
• Uses element connectivity information to determine which elements contribute to each global DOF
• Follows the principle of superposition, adding contributions from all elements
• Results in a symmetric, banded matrix for the entire structure
• Size of global stiffness matrix equals the total number of DOFs in the structure
• Sparsity of the matrix increases with the number of elements and nodes

Coordinate Systems and Transformations

Degrees of Freedom and Node Numbering

• Degree of freedom (DOF) represents possible independent motions at a node
• For 2D trusses, each node typically has 2 DOFs (horizontal and vertical displacement)
• 3D trusses have 3 DOFs per node (displacements in x, y, and z directions)
• Node numbering assigns a unique identifier to each node in the structure
• Influences the arrangement of the global stiffness matrix
• Efficient numbering can reduce the bandwidth of the stiffness matrix, improving computational efficiency

Local and Global Coordinate Systems

• Local coordinate system aligns with individual element axes
• Typically, local x-axis runs along the length of the element
• Global coordinate system defines the overall structure orientation
• Usually fixed and consistent for the entire analysis
• Transformation between local and global systems necessary for assembly and analysis
• Transformation matrix converts quantities between local and global coordinates
• Depends on the angle between local and global axes
• For 2D trusses, transformation matrix involves sine and cosine of the element angle

Boundary Conditions and Results

Application of Boundary Conditions

• Boundary conditions define constraints on the structure's movement
• Essential for creating a solvable system of equations
• Types include fixed supports, roller supports, and pin connections
• Implemented by modifying the global stiffness matrix and force vector
• Fixed DOFs removed from the system of equations
• Can involve setting diagonal terms to large values (penalty method) or eliminating rows and columns

Analysis Results and Interpretation

• Nodal displacements obtained by solving the system of equations
• Represent the deformed shape of the structure under applied loads
• Member forces calculated using element stiffness matrices and nodal displacements
• Axial forces in truss members determined from local coordinate displacements
• Stress in members computed by dividing axial force by cross-sectional area
• Results used to check against design criteria (deflection limits, material strength)
• Post-processing often involves visualizing deformed shape and stress distribution