The Finite Element Method for Structural Analysis (FEM/FEA)

Explore the linear elastic spring element, its axial force-displacement relation F = k delta L, and the spring stiffness k, foundational for the finite element method.

Represent the linear elastic spring as a two-node finite element with nodes at ends. Define deflection delta as u2 minus u1 and relate forces to displacements with the stiffness matrix.

Explain the two degrees of freedom 2x2 element stiffness matrix in the element coordinate system, its symmetry, and how its inverse yields nodal displacements, with boundary conditions to prevent singularity.

Learn to assemble a two-spring system in global coordinates, relate external and internal forces with free body diagrams, and derive the global stiffness matrix via displacement compatibility.

Convert nodal forces to global coordinates and assemble a global stiffness matrix from element equations in the finite element method, solving k u = f under node constraints.

Solve a two-spring finite element exercise by applying boundary conditions, forming the stiffness matrix, and computing unknown displacements U2 and U3, reaction force F1, and the springs' internal forces.

Explore a spring-based finite element model to solve a vertical displacement problem, assemble the global stiffness matrix, apply boundary conditions, and compute nodal displacements and reactions for structural analysis.

Explore a spring-assembly analysis using finite element method in pre-analysis, defining a boundary value problem, assigning stiffness and mass, assembling the global matrix, and solving displacements and post-processing reaction forces.

Define a spring-based domain in a finite element setup by creating four reference points, connecting three springs with 1000, 2000, and 3000 N/m, and assigning 10 kg masses to P1–P3.

Define a three-spring, three-mass domain, apply displacement boundary conditions to prevent rigid-body motion, and add a gravity step with a static general analysis and displacement, reaction, and spring-stress outputs.

Verify finite element results by checking consistency with the mathematical model, equilibrium of forces, and comparison with hand calculations. Analyze displacements, reaction forces, and spring internal forces via post-processing.

Examine the dot inp file to map nodes, sets, masses, springs, and boundary conditions in a static FEM analysis. Learn to edit inputs and outputs using Notepad Plus Plus.

Explore the elastic bar element and its axial loading, strain energy, Castigliano’s first theorem, and minimum potential energy principles to derive stiffness matrices for complex elements.

Analyze the elastic bar element in local coordinates, using nodal displacements and linear shape functions to interpolate along length, apply boundary conditions, and express results in matrix form.

Explain bar element deflection as the variation in length and derive stiffness k = AE/L, linking stress and strain by Hooke’s law for nodal displacements.

Apply the finite element method to a tapered elastic bar with linearly varying cross-section under a load, solving for end displacement using one element and two elements, deriving stiffness matrix.

Explore solving a tapered bar with two finite elements, assemble the global stiffness matrix, apply boundary conditions, and compare FEM results to the analytic solution.

Compare one- and two-element finite element method solutions to the exact bar displacement, showing two elements yield closer results. Displacement converges more readily than stress, which shows discontinuities.

Explore strain energy and elastic potential energy in springs and bars, apply Castigliano's theorems to compute stiffness matrices, and relate work, stress, and volume.

Discover Castigliano's first theorem and its use of strain energy and deflection to derive the stiffness matrix for bar elements in elastic systems in equilibrium.

Apply Castigliano's first theorem to a spring assembly within the finite element method, deriving the stiffness matrix, solving for node displacements under boundary conditions, and computing reaction forces.

Apply the minimum potential energy principle to derive the total potential energy, relate displacements to boundary conditions and external loads, and obtain the stiffness matrix for stable equilibrium.

Apply the minimum potential energy theorem to derive the stiffness matrix of a spring assembly by calculating strain energy and node work, leading to a global stiffness matrix.

Apply finite element method in abaqus to a two-bar elastic assembly, computing displacements and axial forces under gravity from 250 kg mass, comparing results with hand calculations under massless bars.

Define the mathematical model and domain, apply boundary conditions, and compute bar-element cross sections and stiffness to assemble the global stiffness matrix for nodal displacements.

Set up a bar element analysis in Abacus CAE, define the work directory, create bar lines, and assign R20 and R10 truss sections with steel modulus of elasticity.

Mesh a bar domain with one element per line, apply fixed boundary conditions, create an assembly, add mass, apply gravity, and request displacements, rotations, forces, and stresses.

Solve a 3D bar element model, fix mesh and element type, and post-process results to verify displacements, reactions, and axial stresses.

Explore the abacus input file (.inp) and output (.odb) for a bar element, defining nodes, elements, node and element sets, materials, and a static gravity load step.

Learn to form nodal equilibrium equations for two-dimensional truss structures, assemble the global stiffness matrix, and apply boundary conditions to prepare for calculating element strain and stress.

Derive the six-by-six nodal equilibrium equations for a two-dimensional truss by linking local bar coordinates to global displacements and assemble the stiffness matrix from axial deflections.

Transform element stiffness matrices from local to global coordinates and enable direct assembly of the global stiffness matrix using the transformation matrix R and its transpose.

Assemble global stiffness matrix for a two-element bar assembly by transforming stiffness from local to global and using a nodal-displacement correspondence table. Exploit symmetry and main-diagonal terms to simplify computation.

Demonstrate assembling the global stiffness matrix from element stiffness matrices using the transformation matrix in the global coordinate system, with a two-element example and shared-node considerations.

Adopt a more efficient global stiffness assembly by mapping element stiffness into the global matrix using node IDs and a displacement location vector L.

Learn how to apply boundary conditions and compute reaction forces in a two-bar assembly by partitioning the stiffness matrix into constrained and active parts and solving for active displacements.

Learn to compute bar element strain and stress in the global coordinate system using interpolation (shape) functions and the transformation matrix R, with nodal displacements and a partitioning exercise.

Solve a two-element assembly by computing element stiffness, assembling the global stiffness, partitioning into constrained and active dofs, solving for active displacements, and deriving stresses and strains via transformation matrices.

Solve a comprehensive eight element bar assembly by calculating element stiffness, assembling the global stiffness matrix, and using an Excel spreadsheet to automate boundary conditions, forces, and material properties.

The lecture shows assembling the global stiffness matrix for eight bar elements using displacement and node connectivity techniques, applying boundary conditions, solving for active displacements, and post-processing stresses and reactions.

Explore solving a two-dimensional truss with an Excel spreadsheet, assembling the global stiffness matrix, applying boundary conditions, and computing displacements, reactions, stresses, and forces in an FEM workflow.

Extend the 2d truss to a 3d truss by modeling bars as three-dof elements, using a unit vector and transformation matrix to assemble the 6×6 global stiffness.

Solve a simple 3D truss by defining nodes and elements, assembling the global stiffness matrix from element stiffness using unit vectors, and solving for active displacements under external load.

Set up and analyze a 2D truss in abacus with boundary conditions, using a 200 GPa steel modulus to compute displacements, axial forces, and reaction forces, comparing results with Excel.

Perform pre-analysis by defining a boundary value problem, setting boundary conditions, and applying the finite element method with Excel for verifying truss equilibrium.

Open abaqus c and model a 2d planar truss in a new directory. Define steel as elastic isotropic, assign cross-sectional areas, mesh one element per span.

Define boundary conditions and create a load step for a 2D truss model in Abaqus, setting zero displacements at nodes, applying concentrated forces, and preparing for a static analysis.

Verify the finite element model by comparing reaction forces, displacements, and stresses against hand calculations and check equilibrium.

Analyze the dot inp file for a 2d truss, detailing nodes, elements, materials, and sections. Learn to read Abaqus input, set boundary conditions and loads, and review outputs.

Introduce the beam element and transverse bending, detailing bending moments in two planes, sign conventions for shear and moment, and core assumptions: small deflections, linear isotropic materials, and prismatic cross-sections.

Analyze a beam under transverse load by defining the neutral surface and radius of curvature, derive deflection v(x), and obtain strain from y and curvature via the bending framework.

Derive the internal bending moment in a beam cross-section via an area integral, relate stress to bending using E and I, and apply neutral surface and cross-section geometry.

Develop the beam element from elementary beam theory into a two-node first-order element by defining assumptions, degrees of freedom, and interpolation functions, and derive the stiffness and stress expressions.

This lecture derives the beam element stiffness matrix using Castigliano's first theorem, linking total strain energy to nodal displacements and rotations. It covers energy methods and the prismatic cross-section assumption.

Derives the beam element 4x4 stiffness matrix from four degrees of freedom and equations. Computes k_mn terms using E I z and the dimensionless length.

Explore the element load vector, assemble the global stiffness matrix for a two-element flexure beam, and solve for nodal displacements and reactions using the finite element method.

Learn to handle distributed loads on a beam by deriving equivalent nodal forces and moments that produce the same work, using q(x), interpolation functions, and nodal displacements.

Apply the finite element method to a simply supported beam under a uniform distributed load. Use two beam elements and compare nodal results with analytical solutions, noting small errors.

Study the flexure element with axial loading, adding three dof per node to form a 6x6 stiffness matrix. Transform to global coordinates; note buckling, stress stiffening, and small-deflection assumptions.

Practice a two-beam flexure problem using the finite element method: transform the distributed load to equivalent nodal forces and moments, compute cross-sectional area and inertia, and assemble the stiffness matrix.

Apply finite element analysis to a flexure element with axial loading: assemble the stiffness matrix, apply boundary conditions, compute active displacements and nodal forces, and evaluate axial and bending stresses.

analyze the general three dimensional beam element, including axial and bending actions about y and z axes, with displacements u, v, w, rotations theta, and stiffness matrices.

Apply torsional stiffness to a 3D beam element by incorporating torsional bending moments, polar moment J, and shear modulus G, deriving the angle of twist and a 12×12 stiffness matrix.

Analyze a steel mezzanine frame with beam and bar elements to assess von Mises stress against 345 MPa yield and deflection limits under a 3 kN/m^2 floor load.

Define the mathematical model and domain, apply boundary conditions and area-to-line loads, and outline the FEM procedure from element stiffness to solving displacements.

Model a three-dimensional steel frame in abacus by creating a working folder, datum points and wires for beam elements, and defining a W 250 17.9 profile with a steel material.

Continue boundary value problem setup in Abaqus: define beam material and i-shaped profile, assign sections, mesh beams, set orientations, add cross members, and discuss buckling and A36 material properties.

Finish setting the boundary value problem by assembling the global stiffness matrix, applying fixed nodes to prevent rigid body motion, and applying distributed and concentrated loads for static FEA.

Verify a finite element model by confirming global force equilibrium, correcting boundary conditions, and re-running analyses to ensure displacements and results align with the mathematical model.

Analyze stress and displacement using von Mises criteria vs yield strength and the displacement limit L/350; adjust the plastic curve to elastic and reinforce with higher inertia beams.

Explain constitutive equations linking stress and strain in isotropic linear elasticity, defined by modulus of elasticity and Poisson's ratio. Explore uniaxial tests and the matrix form D for multi-axial relations.

Derive the equilibrium equations for static structural analysis in the finite element method, linking normal and shear stress components to boundary conditions and body forces.

Summarizes the expressions developed in this module, including the strain-displacement relations, derivative operator matrix, and the stress-strain relations via the constitutive matrix D with modulus of elasticity and Poisson's ratio.

Master matrix mathematics essential to the finite element method, solving the stiffness matrix times displacement vector equals the force vector, and key matrix concepts like transpose, square, identity, and symmetry.

Learn simple matrix operations, including addition, subtraction, and scalar multiplication of 3x3 matrices, and basic matrix multiplication with a spring stiffness matrix example.

Learn to invert matrices using adjoint and cofactors, and apply the stiffness matrix inverse to solve for nodal displacements from a force vector in the finite element method.

Apply Gauss elimination to solve the stiffness matrix equation in the finite element method, transforming A to an upper triangular form and computing displacements without inverting the matrix.

Learn how the LU decomposition method solves linear systems in structural analysis by factoring A into L and U, connecting to Gauss elimination, and comparing direct and iterative solvers.

Explore the Gauss-Seidel iterative solver for a 3x3 linear system, starting from a zero guess and converging under a 0.001 error threshold in abacus.

Derive and apply the plane stress elasticity equations, material relations, and energy-based stiffness concepts for shell elements in structural analysis.

Explains forming the constant strain linear triangle finite element in 2d, defining nodal degrees of freedom, stiffness matrix, and shape (interpolation) functions to express displacements inside the element.

Present the constant strain triangle CST formulation for a linear triangular element in plane stress, deriving the strain-displacement matrix B and showing constant normal and shear strains across the element.

Formulate the constant strain triangle to derive total strain energy and the stiffness matrix via Castigliano's theorem and area integral.

Derive stress–strain with the D matrix for isotropic, homogeneous, linear elastic materials, build B from displacement shape functions, and compute the stiffness matrix K from total strain energy and equilibrium.

Apply distributed loads to finite element edges by converting them into equivalent nodal forces for triangular elements, using normal and tangential pressure concepts, area, and shape functions.

Compute nodal equivalent forces for a triangular element under distributed loads by deriving p_y and p_x variations, constructing shape functions, and applying edge loads to nodes 1-3.

Learn how body forces act through element volume under plane stress, and transform distributed forces into equivalent nodal forces using shape functions, density, and thickness in FEM.

Learn how body forces, notably gravity, generate equivalent nodal forces in triangular finite elements, using interpolation functions, area integration, and direct assembly into the stiffness system.

Apply the finite element method to analyze a rectangular plate with a central circular hole under plane stress, using linear triangular elements, and perform pre-analysis with convergence to under 3%.

Set up a 2D planar shell domain in abacus, define a 3500 mm by 1000 mm plate with a central 100 mm hole, apply plane-stress and symmetry.

Finish defining the boundary value problem in Abaqus by setting initial step boundary conditions, creating a load step with a uniform edge pressure, and requesting stress, strain, and displacement outputs.

Apply symmetry condition to streamline finite element analysis in Abaqus, assemble the global stiffness matrix, apply boundary conditions, and post-process results to study stress concentrations and displacements under plane stress.

Refine the mesh and run a convergence study by tracking S11 at point A and comparing with the analytical value to verify convergence.

Finish the verification step by checking equilibrium and acceptable numerical error on the converged mesh, compare results with hand calculations, and analyze reaction forces and mesh quality criteria.

Learn to assess mesh quality in Abaqus using shape factor and aspect ratio, verify meshes, and apply quality criteria from the Abaqus user guide.

Explore the plane strain equations of elasticity, the four-node rectangle finite element, and Gaussian quadrature, and relate stress and strain via the D matrix for xy-plane loading.

Formulate a four-node rectangular element for plane strain, deriving its quadratic displacement polynomials using Pascal's triangle, computing shape functions from nodal displacements and coordinates.

Explore finite element formulation for a four-node rectangular element, introducing a natural coordinate system with r and s, deriving shape functions and their use in displacement interpolation.

Define interpolation functions in natural coordinates, derive eight dof stiffness matrix for a four-node rectangle under plane strain, and apply the chain rule to obtain the B and D matrices.

Transform the volume integral to an area integral in r-s coordinates using the B and D matrices to derive the eight-by-eight stiffness matrix for a four-node rectangle under plane strain.

Use Gaussian quadrature to compute stiffness integrals in finite element method. Transform x to r to map limits to -1 to 1 using sampling points with weights for quadratic accuracy.

Apply two-dimensional gaussian quadrature to finite element area integrals, choosing full or reduced integration with r and s points, and implement in abacus.

Explore pre-analysis of a c-clamp shaped structure under plane strain, highlighting numerical singularities at 90-degree corners, four-node element issues like reduced integration and shear locking, and hand calculations.

Define a boundary value problem for a C-clamp with the finite element method in Abacus, build a plane-strain domain, mesh with CPE4R reduced integration, and apply left and right pressures.

Explore how four-node quadrilateral elements with full integration cause shear locking and hourglass in bending problems, and compare reduced integration and eight-node elements to improve displacement accuracy in FEM/FEA.

Explore hourglass and incompatible mode elements in the finite element method for structural analysis, focusing on reduced integration, center-point quadrature, and bending-related stiffness accuracy.

Shows reduced integration with four elements along the height and hourglass control, compares to fully integrated elements to address shear locking, and notes numerical singularity at 90 degrees affecting convergence.

Learn the isoparametric formulation of the four-node quadrilateral element. Map coordinates from a parent element using geometric interpolation functions, enabling accurate curved-boundary representation with a coarse mesh.

Explore how four-node quadrilateral elements express normal and shear strains from displacements via chain rule in natural coordinates, using Jacobian and its inverse to obtain x,y derivatives for stiffness calculations.

Finish the four-node quadrilateral element using the isoparametric approach, where geometry and displacement share the same interpolation in r and s, then assemble the plane-stress/plane-strain stiffness with Gaussian quadrature.

Compute the stiffness matrix for a four-node quadrilateral element under plane stress using isoparametric mapping and Gaussian quadrature, with steel properties (elastic modulus 30e6 psi, Poisson's ratio 0.3).

Explore how distortion leads to singularity in the Jacobian matrix for four-node quadrilateral elements, affecting derivatives, stiffness matrix, and Gaussian quadrature, and learn mesh quality criteria to prevent it.

Analyze a surface with plane-stress in Abaqus. Apply 50 MPa uniform pressure and a 30 kN downward force on the hole, and outline the finite element method.

Workshop 08 defines a bvp in abacus, builds a 2d plane-stress domain, meshes with quadrilaterals, applies boundary conditions and 50 MPa pressure, defines steel, runs an analysis, and prepares results.

Master post-processing of a static plane-stress FEM model, verify the solution, compute stresses such as S11, perform a convergence study, and analyze reaction forces and XY data along a path.