The Finite Element Method (FEM/FEA)

Explore the linear elastic spring element with axial loading, its force–displacement relation f = k delta l, the two-node 2×2 symmetric stiffness matrix, and how constraints prevent rigid body motion.

Assemble a two-spring system in global coordinates, apply displacement compatibility and free body diagrams to build the global stiffness matrix for a static analysis.

Practice building the stiffness matrix for a two-spring assembly with a fixed constraint, and solve for active displacements u2 and u3, then determine reaction and spring forces.

Perform a practical Ansys based finite element analysis of a four-node spring assembly. Assemble the global stiffness matrix, apply constraints, solve for u2, u3, u4, and compute reaction forces.

Apply the finite element method in Ansys to a three-spring, three-weight assembly, perform the pre-analysis by defining the domain and boundary conditions, and obtain nodal displacements and reactions.

Open Ansys Workbench, set up a static structural analysis, define a steel material with density 7850 kg/m3, model three 100 mm cubes in SpaceClaim, and save the workshop project.

Set up a spring-based finite element model by assigning flexible steel parts, configuring boundary conditions, creating body–body and body-ground springs with specific stiffness, and generating the mesh.

apply gravity as a body force, apply frictionless boundary conditions to prevent rigid motion, solve, and post-process directional deformation and spring forces with results shown in millimeters.

Verify the results against the mathematical model and hand calculations, confirming a consistent reaction force of 230.95 N and exact node displacements in this static analysis.

Learn to navigate Ansys help to locate information on the Combine 14 spring element and its theory. Discover the element library, stiffness matrices, and longitudinal stiffness concepts for basic analysis.

This lecture introduces the elastic bar element in finite element analysis, deriving its 2x2 axial stiffness matrix from shape functions and nodal displacements U1 and U2 under axial loading.

In this exercise, the finite element method models a tapered bar under end load, comparing one and two element assemblies to obtain end displacement and convergence of stresses.

Understand how external work becomes elastic energy in springs and bars, and apply vector notation to relate stress, strain, and energy through F equals K delta in three-dimensional space.

Explains Castigliano's first theorem by deriving force from the change in strain energy with respect to deflection, and applies it to a spring assembly to find nodal displacements and reactions.

Apply the minimum potential energy principle to determine stable, neutral, and unstable equilibria using total potential energy (strain energy minus conservative work) and derive the stiffness matrix from displacement derivatives.

Learn to model a two-bar assembly in Ansys, perform problem specification and pre-analysis, and compute displacements, forces, and reactions using the finite element method.

Set up the bar element analysis in Ansys Workbench, create a steel material with low density and assign isotropic elasticity, including Young's modulus and Poisson's ratio.

Use SpaceClaim to create two lines in the xy plane, assign circular profiles of 20 and 10 millimeters, and configure them as link or truss elements with shared topology.

Build a two-bar finite element model in mechanical, apply boundary conditions and loads, select truss elements with flexible behavior, and mesh one element per bar before gravity and post-processing.

Post-process bar element results in the finite element method by extracting node displacements, directional y displacement, and reaction forces; verify against hand calculations and obtain axial forces from Bing results.

Verify the FEM model by checking equilibrium (sum of vertical forces equals zero) and comparing node displacements with hand calculations, noting minor differences due to non-massless bars.

Learn how nodal equilibrium equations apply to 2d truss structures, linking local bar forces to global coordinates, enforcing node compatibility, and forming node equations in x and y.

Explain how global nodal displacements relate to local bar displacements in a two-dimensional truss, with two degrees of freedom per node, using theta and small elastic rotations under linear analysis.

Compute bar deflection from U2 minus U1 using the element stiffness K, noting perpendicular components do not generate force. Form the six-by-six symmetric global stiffness matrix from global displacements.

Transform the element stiffness matrix from local to global coordinates using R in a two-dimensional bar element, illustrating the global stiffness matrix, and the global displacements and forces.

Directly assemble the global stiffness matrix for a two-element truss by mapping element matrices to global displacements and using a table to align local and global degrees of freedom.

Practice transforming element stiffness matrices for two-bar elements into the global frame and assembling the global stiffness matrix via direct assembly, noting symmetry and reduced efficiency for computer implementation.

Assemble the global stiffness matrix by combining element stiffness matrices using the nodal displacement mapping and the element connectivity table in a 2d truss setting.

Apply general boundary conditions to a stiffness matrix, partitioning it into constraint and active parts to compute active displacements and reaction forces in a two-bar assembly.

apply global displacements from the finite element solution to compute element strain and stress in a truss, using local-to-global coordinate transforms, shape-function interpolation, and the transformation matrix.

This exercise analyzes a two-bar truss to compute displacements, reactions, and stresses using global–local coordinates and a 45-degree element.

Solve a complex 2D truss with eight bar elements, compute element stiffness, transform to global coordinates, assemble the global stiffness matrix, and obtain displacements and reaction forces.

Continue the comprehensive FEM example by assembling the global stiffness matrix and applying boundary conditions. Derive active displacements, reaction forces, and element strains and stresses via Hooke's law.

Solve a 2D truss using an Excel-based FEM workflow, from element stiffness to assembly, active displacements, reactions, and stresses and strains, with equilibrium verification and workshop comparing results to Ansys.

Explain three dimensional trusses with bar elements in a global x, y, z frame, six degrees of freedom, and the transformation matrix with cosines to form the stiffness matrix.

Assemble the global stiffness matrix for a four-node, three-element three-dimensional truss, then solve for the active displacements at node four and compute reaction forces.

Solve a 2D truss with Ansys and worksheet to obtain displacements, reaction forces, and strain and stress, and perform pre-analysis by formulating a boundary-value problem and finite element method steps.

Solve a 2D truss in Excel, compute element areas and lengths, assemble the global stiffness, and obtain active displacements, reaction forces, and post-processing results for Ansys modeling.

Solve a 2D truss structure in ANSYS Workbench by creating geometry in SpaceClaim, defining circle cross-sections and beam profiles, using structural steel, and sharing coincident topology.

Define material properties and create a one-element-per-bar mesh for a 3D truss in FEM/FEA, using structural steel and defined cross-sections.

Define boundary conditions by applying node forces with x and y components, constrain displacements to prevent rigid motion, and review linear analysis settings, including direct solver and large deflection.

Post-process finite element results to extract displacements and axial forces, compare node values, and compute axial stress via cross-sectional area using probe and solution tools.

Verify the finite element solution by checking consistency with the equilibrium equations and hand calculations, extract reaction forces at nodes, and assess numerical error, bar elements yield exact solutions.

Explore the elementary beam theory in the element context by examining a beam element under transverse and distributed loads, detailing bending, shear, and bending moment sign conventions about symmetry axes.

Explore beam bending concepts by defining the neutral surface and radius of curvature, deriving arc lengths and strain from deflection, and obtaining stress via Hooke's law.

Compute bending moments along a beam by integrating over cross-section and relate axial stress to bending via sigma = M y / I, using moment of inertia and neutral surface.

Explore the beam element in the finite element method using cubic interpolation, boundary conditions at two nodes, and shape functions N1–N4 to derive displacement and bending stress.

Apply Castigliano's first theorem to derive beam element stiffness matrix from displacement and strain energy, using modulus E, moment of inertia I, and interpolation functions for nodal degrees of freedom.

Apply Castigliano's first theorem to derive the beam-element stiffness matrix terms, relating nodal displacements and rotations to forces and moments, and assemble a 4x4 symmetric stiffness matrix using interpolation functions.

Transform dimensionless length to simplify beam element stiffness matrix calculations, derive derivatives, and assemble the 4x4 symmetric matrix, noting its singularity and how constraints remove it.

Explore the element load vector and sign conventions in finite element formulation, including beam theory. Practice a mid-span deflection problem with two beam elements and their stiffness matrices.

Assemble the global stiffness matrix from element to global coordinates, relate local to global displacements, and analyze singularity while solving with the stiffness matrix inverse in constrained systems.

Constrain nodal degrees of freedom to prevent rigid body motion and solve for active displacements and reaction forces in FEM beam analysis.

Replace a uniformly or varying distributed load on a beam with equivalent nodal forces and moments by equating the distributed-load work, through Q(x) times V(x) and interpolation, to nodal work.

Compute equivalent nodal loads for a distributed beam load by equating the work of Q with nodal forces and moments using interpolation functions.

Explore work equivalence for distributed loads in a two-element simply supported beam using finite element methods, assembling global stiffness, applying boundary conditions, and comparing nodal displacements with exact strength-of-materials solutions.

Explains the flexure element with axial loading by combining axial and bending stiffness matrices for three dof per node, notes buckling under compression and negligible stress stiffening under small deflections.

Transform the flexure element stiffness from local to global coordinates using the transformation matrix, derive local-to-global displacements, and assemble the global stiffness matrix for beam-axial elements.

Practice assembling the global stiffness matrices for a two-beam axial element under a distributed load, transforming local to global coordinates, and computing displacements U4–U6 and axial plus bending stresses.

study a three-dimensional beam element with 12 degrees of freedom per element and a 12×12 stiffness matrix, in global and local coordinates, under bending, torsion, and axial loading.

Assemble axial, bending, and torsional stiffness matrices into a 12 by 12 beam element, then transform to global coordinates and apply equivalent torsional stiffness for non-circular sections.

Learn to set up a beam element in fea software: define geometry, cross-section, materials, mesh, boundary conditions, loads, and post-process axial, bending, and shear results.

Analyze a steel frame under a 3 kN/m^2 floor load to verify stresses stay below yield and displacements stay under L/350, using ASTM A572 GR50 beams and diagonal bar bracings.

Define the mathematical model as a boundary value problem in the domain, convert area loads to per-length forces, and apply the finite element method to obtain nodal displacements and reactions.

Create isotropic materials in Ansys workbench for a steel frame, set Young's modulus and Poisson's ratio, assign tensile yield strength to ASTM A572 GR 50 and ASTM A36, then save.

Define the analysis geometry by building the steel frame in SpaceClaim, creating eye shaped and circular cross sections, positioning and copying beams, orienting components, and manually sharing topology.

Generate a 3d finite element mesh for beam and bar elements in a steel frame using edge selection, one division per edge, and linear elements, then archive project for sharing.

Define boundary conditions by fixing nodes and applying line pressure forces to beam elements; assemble the global stiffness matrix, apply constraints, and solve for active displacements to obtain reaction forces.

Post-process finite element results to extract node displacements, axial forces, moments, and stresses, while balancing direct and iterative solvers for accuracy and computational cost in static analyses.

Extract reaction forces at fixed nodes and analyze axial forces, noting 20 kilonewtons, then assess how diagonal bracings affect deformation and bending moments along the y and z axes.

Analyze post-processing results for a steel structure by extracting shear force, bending moment, and von Mises stress, compare to yield strength, and evaluate display options.

Verify finite element solutions by checking consistency with the mathematical model, refining the mesh for numerical convergence using work equivalence for distributed loads, and ensuring deflection stays under L/350.

Discover how u, v, w depend on position and how strain components arise from spatial derivatives, with epsilon_x = ∂u/∂x, epsilon_y = ∂v/∂y, epsilon_z = ∂w/∂z, from infinitesimal length changes.

Derive small-displacement strain-displacement relations, express alpha and beta as dv/dx and du/dy, form the shear strain gamma_xy, and relate to normal and shear stresses via E and G.

Define constitutive equations and Hooke's law for homogeneous, isotropic, linear elastic materials using modulus of elasticity and Poisson's ratio; cover elastic region and yield strength for FEM.

Apply equilibrium equations in three dimensions by summing forces to zero under static conditions, and relate surface to interior normal and shear stresses via Taylor series across dx, dy, dz.

Review equilibrium in a fixed beam by analyzing shear stress and bending moments. Note how normal stresses vary across the cross-section and about the neutral surface, and tau_xy equals tau_yx.

Explore how gravity and other accelerations create a body force in a solid, computed as density times acceleration and represented as force per unit volume, leading to equilibrium equations.

Integrate equilibrium equations, strain-displacement relations, and stress-strain laws using modulus of elasticity and Poisson's ratio to form a solvable 15-equation, 15-variable system for two- and three-dimensional structural elements.

Master matrix notation and linear equation solutions in finite element contexts, and study transposes, diagonal and identity matrices, plus the symmetric stiffness matrix linking displacement and force.

Master matrix operations including addition, subtraction, scalar multiplication, and matrix multiplication via row-by-column products, plus determinants and minors of square matrices.

Explore determinants, minors, and cofactors to compute inverses and solve linear systems, using the adjoint and determinant to invert a 3×3 matrix and discuss singularity.

Learn Gauss elimination to solve Ax = f by transforming A into an upper triangular form using simple algebraic operations, with back substitution yielding x and avoiding matrix inversion.

Select direct or iterative solvers in Ansys, using the sparse direct (Lu) decomposition to transform A into L and U and solve Ax = F.

Explore iterative solvers in finite element analysis, highlighting the preconditioned conjugate gradient method and Gauss-Seidel, convergence to an approximate solution, and the Pxg solver's iterative approach.

Learn the equations of elasticity for plane stress by applying key assumptions: constant, small thickness; loading in the xy plane; and a linear, isotropic, homogeneous material.

Develop the equilibrium equations for plane stress, extracting the in-plane stresses sigma_x, sigma_y, and tau_xy, noting Poisson's ratio induced z-strain. Apply plane-stress stress-strain relations with the D matrix.

Formulate total strain energy and total potential energy under plane stress, then derive the stiffness matrix from nodal displacements, using the D material matrix in a matrix-based energy expression.

Develop the finite element formulation for the constant strain triangle. Explore first order triangles with vertices and second order with mid-side nodes, using interpolation functions N and nodal displacements.

Derive displacement in x and y using linear interpolation for a constant strain triangle, obtain constant strains from derivatives, and note limitations for varying strain, via the strain-displacement (B) matrix.

Learn finite element formulation for constant strain triangles, derive the six-by-six stiffness matrix from strain energy and work, and apply the minimum potential energy principle.

Compute the element stiffness matrix for a triangular CST element by multiplying the D matrix, the B strain-displacement matrix, and derivatives of interpolation functions with respect to x and y.

Convert distributed edge loads to equivalent nodal forces for a triangular element by resolving p_n and p_t into x and y components and computing work.

The lecture demonstrates deriving nodal equivalent forces for a triangular finite element under distributed loads using node coordinates and thickness.

Compute equivalent nodal body forces from gravity by integrating differential forces over a differential area, using density and thickness, and assemble into nodal forces via interpolation functions and displacements.

Compute nodal forces in a plane-stress 3-node triangle under body forces. Use gravity, density, and thickness to derive F1Y, F2Y, F3Y and connect to the element stiffness equation.

Demonstrate a simple beam bending with the constant strain triangle in Ansys, fixed at the left and loaded with 50 kN downward, plane-stress analysis yields 125 MPa compressive stress.

model a beam in Ansys workbench with 3-node triangular elements under plane-stress, using SpaceClaim 2d geometry and steel properties, Young's modulus and Poisson's ratio, then generate a triangular mesh.

Explore beam analysis with the finite element method, building a 24 dof stiffness matrix, applying fixed supports and a distributed load, solving with a direct solver, and evaluating stress.

Refine the mesh by adjusting edge sizing and enabling face meshing to obtain a convergent 2d finite element solution, comparing stress results with the analytical value and exploring second-order elements.

Model a plane-stress rectangular plate with a central hole via finite element method, apply a right-edge distributed load, and perform a convergence study against the exact solution.

Define material properties and geometry for a 2D plane-stress analysis in Ansys, using isotropic elastic steel with E and Poisson's ratio, for a plate with a central circular hole.

Configure a 2d plane stress model by importing geometry, selecting plane stress, and defining isotropic linear elastic material with Young's modulus and Poisson's ratio, plus thickness and small-displacement settings.

Learn to generate linear triangular meshes, compare linear vs second-order elements, refine locally with edge sizing, and assemble the global stiffness matrix for boundary conditions.

Apply boundary conditions in static structural, placing a 100,000 newton force on right edge and fixing left edge; solve with stiffness matrix to obtain active displacements, reactions, and stresses.

Perform post-processing to compute reaction forces at the fixed left edge, then use probe tools to evaluate normal stress in global coordinate system and stress at points A and B.

Learn how to perform verification in finite element analysis by checking force balance, conducting convergence studies, refining meshes, and comparing results with hand calculations in Ansys.

Verify finite element convergence by comparing results to exact 300 MPa stress at A and B, noting hole-induced concentration and nominal stress, and assess mesh quality with Ansys metrics.

Explore the equations of elasticity for plane strain, formulate the four-node rectangular element, and learn Gaussian quadrature for finite element method analysis of isotropic, homogeneous, linear elastic materials.

Explain plane strain stress–strain relations using the elastic property matrix D, relate stresses to strains in x, y, z, and discuss energy and matrix notation in finite element method.

Formulate a four-node rectangular element with a quadratic interpolation in x and y, solving four coefficients from nodal displacements via boundary conditions and Pascal's triangle.

The lecture reformulates a four-node rectangular element in a natural R–S coordinate system to simplify interpolation, deriving x–y relations, center coordinates, and isoparametric displacement using N1–N4 and U1–U4.

Derive the stiffness matrix for a four-node rectangular element by formulating total strain energy in plane strain, using the elastic property matrix D, the B matrix, and nodal displacements.

Derive chain-rule based transformations from x,y to r,s to form the B and stiffness matrices for a four-node rectangle, and convert the volume integral to an area integral with thickness.

Derive the eight-by-eight stiffness matrix for a four-node rectangle using interpolation via Pascal's triangle, the B and D matrices, and plane stress or plane strain assumptions.

Learn how to perform numerical integration with Gaussian quadrature, applying one- and two-dimensional schemes for the four-node rectangle to compute stiffness matrix integrals using Gauss points and weights.

Apply Gaussian quadrature twice to compute double integrals over a four-node rectangle in finite element method, enabling efficient stiffness calculations and highlighting full vs reduced integration and shear locking.

Work through problem specification and pre-analysis for a clamped C-clamp under pressure using plane strain, estimate stresses from direct load and bending, and review the finite element method workflow.

Create a C-clamp shaped geometry in a 2D plane using SpaceClaim and Workbench, sketch in XY, apply dimensions, repair geometry, and save the project.

define structural steel with a 200 gpa modulus and a 0.3 poisson's ratio, then perform a plane stress analysis using imported geometry and a linear 4-node quadrilateral mesh.

Finish defining the plane-strain FEM model by applying boundary conditions, fixing the upper edge, and applying a 200 N nodal force and a 1.33 MPa pressure, then solve.

Verify the mathematical model by checking force balance and numerical errors. Refine the mesh and plot normal stress along a mid-section path to compare with hand calculations.

Refine the mesh and monitor von Mises stress in a region of interest to assess convergence and stress variation through targeted, coordinate-based probing.

Continuing the verification step, we assess numerical error and mesh refinement, compare averaged versus actual element stresses, and confirm convergence with hand calculations and equilibrium checks.

Refine the mesh to analyze von Mises stress near notches and reveal singularities that prevent convergence, highlighting rounding corners to reduce stress concentration.

Apply the isoparametric formulation to evaluate the stiffness matrix of a four-node quadrilateral element. Map coordinates from a parent rectangular element using geometric interpolation functions.

Explore the isoparametric formulation for a four-node quadrilateral element, deriving line mappings with node coordinates using interpolation functions, and show that the same functions compute both geometry and displacements.

Visualize the isoparametric four-node quadrilateral mapping in Excel, plotting parent and quadrilateral elements and deriving interpolation functions to relate R, S coordinates to X, Y for stiffness matrix work.

Derive the quadrilateral element stiffness matrix in isoparametric formulation using B, D, the Jacobian, and the chain rule under plane stress or plane strain.

Compute displacement derivatives via the inverse Jacobian and gaussian quadrature for a four-node isoparametric element, linking nodal displacements to strain through B and G matrices.

Explore the isoparametric formulation for a four-node quadrilateral element by evaluating its stiffness matrix under plane stress or plane strain, using a parent element, jacobian mapping, and Gaussian quadrature.

Analyze the singularity of the Jacobian matrix, the determinant, and the inverse’s role in finite element calculations. See how distorted elements cause near-zero determinants, leading to stiffness matrix errors.

Use the isoparametric formulation to derive stiffness matrices by mapping a simple parent element to complex second-order quadrilateral and triangular elements, via interpolation functions, the jacobian, and curved boundaries.

Extract the mid-surface and analyze with 2D plane-stress elements using isoparametric formulation in Ansys, then compare results to 3D solid elements.