🧱Structural Analysis Unit 10 – Displacement Method for Structures

Key Concepts and Definitions

• Displacement method analyzes structures by determining unknown displacements at nodes and using them to calculate internal forces and reactions
• Nodes are points where structural elements connect and unknown displacements (translations and rotations) are determined
• Degrees of freedom (DOF) refer to the number of independent displacements needed to define the deformed shape of a structure
  • Each node typically has 3 DOF in 2D (two translations and one rotation) and 6 DOF in 3D (three translations and three rotations)
• Stiffness matrix $[K]$ represents the force-displacement relationship of a structure and is assembled from individual element stiffness matrices
• Load vector $\{F\}$ contains the applied nodal forces and moments acting on the structure
• Displacement vector $\{D\}$ contains the unknown nodal displacements and rotations to be determined
• Boundary conditions specify the known displacements (usually zero) at support nodes and are used to modify the stiffness matrix and load vector

Fundamental Principles of Displacement Method

• Based on the principle of minimum potential energy, which states that a structure will deform to a configuration that minimizes its total potential energy
• Assumes small displacements and linear elastic behavior of materials, allowing the use of linear force-displacement relationships
• Compatibility of displacements ensures that the deformed shape of the structure remains continuous and no gaps or overlaps occur between elements
• Equilibrium of forces is satisfied at each node, meaning that the sum of internal forces and external loads at each node must be zero
• Superposition principle allows the total response of a structure to be determined by summing the individual responses caused by each load case
• Stiffness matrix is symmetric due to Maxwell's reciprocal theorem, which states that the displacement at point A due to a unit force at point B is equal to the displacement at point B due to a unit force at point A

Matrix Formulation in Displacement Analysis

• The fundamental equation of the displacement method is $[K]\{D\} = \{F\}$, where $[K]$ is the stiffness matrix, $\{D\}$ is the displacement vector, and $\{F\}$ is the load vector
• Element stiffness matrices are derived using force-displacement relationships and the principle of virtual work
  • For example, the element stiffness matrix for a 2D beam element is a 4x4 matrix that relates the end forces and moments to the end displacements and rotations
• Global stiffness matrix is assembled by summing the contributions of individual element stiffness matrices based on the connectivity of nodes and elements
• Boundary conditions are applied by modifying the stiffness matrix and load vector
  • Rows and columns corresponding to fixed DOFs are removed or constrained
• The unknown displacements are obtained by solving the system of linear equations $[K]\{D\} = \{F\}$ using methods such as Gaussian elimination or LU decomposition

Step-by-Step Procedure for Displacement Method

1. Identify the structure, material properties, and loading conditions
2. Determine the number of nodes and elements, and assign DOFs to each node
3. Derive the element stiffness matrices using force-displacement relationships and the principle of virtual work
4. Assemble the global stiffness matrix by summing the contributions of individual element stiffness matrices
5. Apply boundary conditions by modifying the stiffness matrix and load vector
6. Solve the system of linear equations $[K]\{D\} = \{F\}$ to obtain the unknown nodal displacements
7. Calculate the element forces and reactions using the obtained displacements and the element stiffness matrices
8. Interpret the results and check for accuracy and reasonableness

Applications to Different Structural Types

• Trusses: Displacement method is well-suited for analyzing pin-jointed truss structures, where elements carry only axial forces and have two DOFs per node (translations)
• Beams and frames: The method can analyze structures with beam and frame elements that carry axial forces, shear forces, and bending moments, and have three DOFs per node (two translations and one rotation) in 2D
• Plane stress and plane strain problems: Displacement method is used to analyze 2D continuum structures, such as plates and membranes, by discretizing them into finite elements with two DOFs per node (translations)
• Three-dimensional structures: The method can be extended to analyze 3D structures, such as space trusses and frames, by considering six DOFs per node (three translations and three rotations)
• Composite structures: Displacement method can handle structures made of multiple materials by using appropriate element stiffness matrices and ensuring compatibility at the interfaces

Advantages and Limitations

• Advantages:
  • Systematic and straightforward procedure that can be easily implemented in computer programs
  • Applicable to a wide range of structural types and loading conditions
  • Provides a complete solution, including displacements, internal forces, and reactions
  • Allows for easy incorporation of boundary conditions and structural modifications
• Limitations:
  • Assumes small displacements and linear elastic behavior, which may not be valid for structures with large deformations or nonlinear materials
  • Requires the structure to be discretized into elements, which may introduce approximation errors
  • Computational effort increases rapidly with the number of DOFs, especially for large and complex structures
  • May not provide accurate results for structures with stress concentrations or singularities, as these require a very fine mesh or special elements

Numerical Examples and Problem-Solving

• Simple truss: Analyze a statically determinate truss with pin-jointed elements and calculate the nodal displacements and element forces
• Beam with distributed load: Determine the deflection and bending moment distribution of a simply supported beam subjected to a uniformly distributed load
• Frame with multiple load cases: Analyze a portal frame with fixed supports subjected to various load combinations, including nodal loads and distributed loads
• Plane stress problem: Calculate the stress and strain distribution in a thin plate with a hole subjected to uniaxial tension using quadrilateral elements
• Convergence study: Investigate the effect of mesh refinement on the accuracy of displacement and stress results for a cantilever beam with a concentrated end load

Comparison with Other Structural Analysis Methods

• Force method (flexibility method):
  • Solves for unknown forces and determines displacements indirectly
  • Better suited for statically indeterminate structures with few redundant forces
  • Requires more problem-dependent formulation and less systematic than the displacement method
• Finite difference method:
  • Approximates the governing differential equations using finite difference equations
  • Easier to formulate but less accurate than the displacement method, especially for complex geometries and boundary conditions
• Finite element method (FEM):
  • Generalizes the displacement method to handle more complex structural and continuum problems
  • Uses various element types (e.g., beams, plates, shells, solids) and interpolation functions to approximate the displacement field
  • Provides more accurate results and can handle nonlinear and dynamic problems, but requires more computational resources and expertise