The Direct Stiffness Method for Truss Analysis with Python

Apply the direct stiffness method and finite element methods by building element-level models. Assemble them with matrices to form a structural model and relate stress to strain using Young's modulus.

Explore two-dimensional stress and strain, extending concepts from one-axis stress to normal and shear stresses on a 2D element, using Poisson's ratio, Young's modulus, and the shear modulus.

Relate strain to displacement in a 2D element under axial load, showing epsilon_xx equals du/dx and shear strains from ux and uy. Introduce matrix notation and cover elongation and translation.

Expand from 2D to 3D elements by deriving full 3D stress-strain relations and matrix notation with the D and C matrices, enabling direct stiffness analysis for truss systems.

Plane stress reduces 3d behavior to a 2d model when one dimension is small, with zero stress on z faces and in‑plane stresses and shear governing the response.

Learn how plane strain applies to long structures where the dominant strains occur in the x-y plane, allowing a reduced stress-strain matrix and ignoring z-direction stresses.

Characterize material behavior using the C matrix across 3D, plane stress, and plane strain models, linking stress and strain for the direct stiffness method and finite element analysis.

Learn to derive the direct stiffness method and finite element equations, build element stiffness matrices, and transform them to a global framework to analyze beams, 2D, and 3D structures.

Explore finite element equations and the direct stiffness method to derive element stiffness matrices for truss elements, assemble the structure stiffness matrix, and solve for nodal displacements.

Derive two finite element equations—the stiffness matrix and a force vector—using the minimum potential energy principle by expressing energy as strain energy minus work and differentiating w.r.t. nodal displacements.

Introduce the displacement interpolation matrix H and the strain displacement matrix B to express nodal and internal displacements, and derive the finite element equations from the potential energy.

Apply the finite element method to derive the bar element stiffness matrix, introducing the displacement interpolation matrix and the strain-displacement matrix B, with EA over L.

Review the finite element equation to build the stiffness matrix for a bar element, detailing the displacement interpolation matrix and its derivative—the strain-displacement matrix—and the integration steps.

Transform local reference frames into the global frame for the direct stiffness method, deriving a simple transformation matrix to convert coordinates and forces for assembling a global stiffness matrix.

Explore the direct stiffness method through a simple truss example, clarifying foundational theory, and learn to streamline implementation by coding solutions and downloading PowerPoint slides for large matrices.

Calculate the global stiffness matrix for each element, transform to coordinate system, assemble the primary stiffness matrix, apply boundary conditions, solve for nodal displacements, and compute reactions and member forces.

Label nodes and members, assign global degrees of freedom, and compute element stiffness matrices by measuring each member's orientation angle anticlockwise from the global positive x axis.

Break down each element's global stiffness into four quadrants, build a 3x3 template, and expand to a 6x6 primary stiffness matrix by inserting quadrants at node-based locations.

Reduce the primary stiffness matrix to the structure stiffness matrix by imposing boundary conditions on six degrees of freedom, zeroing displacements at supports, and forming a reduced system.

Compute nodal displacements for node two by applying boundary conditions to stiffness matrix, yielding equations for two unknowns; invert it and multiply by force vector (horizontal zero, vertical minus 150).

Solve for unknown reactions using the primary stiffness matrix and known displacements, yielding forces at nodes and mapping them back onto the structure for equilibrium checks.

Use the direct stiffness method to calculate truss member forces from stiffness times displacement, transforming between local and global frames and solving with a global stiffness matrix.

Translate the direct stiffness method for truss analysis into an algorithmic Python solution. Refactor code into reusable functions to build a generalized solver for larger structures without commercial software.

begin by setting up a Python notebook, importing dependencies, and defining material properties to compute element stiffness matrices for a two-element truss, then assemble to a global stiffness matrix.

Build primary stiffness matrix for a truss by manually assembling element quadrants into a template, then reduce it to structure stiffness matrix with boundary conditions on four degrees of freedom.

The lecture demonstrates calculating nodal displacements, reaction forces, and member forces in a truss using the direct stiffness method, including matrix extraction and transformation.

Visualize the truss output by plotting the deflected shape in Python, setting up a figure and axes with equal aspect, and applying a displacement scale factor.

Refactor the code by encapsulating repeated tasks into functions to compute member stiffness matrices, assemble stiffness matrix from restrained degrees of freedom, and compute member forces from nodal displacements.

Apply the direct stiffness method to an eight-bar truss, first conceptually with slides, then implement the same solution in Python.

Label the six nodes to establish 12 degrees of freedom. Compute the eight element stiffness matrices and measure each angle from the global x axis.

Assemble the primary stiffness matrix by placing each element's four quadrants into a six-by-six template (expanding to twelve-by-twelve) for nodes A–H, with zeros between blocks, using the dupere notebook to sum.

Apply the direct stiffness method in Python to reduce the full force-displacement system to the structure stiffness matrix, imposing restrained displacements and deriving an eight-by-eight matrix.

Invert the stiffness matrix and multiply by the force vector to obtain nodal displacements, applying the direct stiffness method for truss analysis.

Multiply the primary stiffness matrix by the unknown displacements to obtain the reaction force vector, then identify the vertical reactions at nodes six, one, and four.

Apply the direct stiffness method in python to find element forces, transforming global nodal displacements to local frames and computing axial forces for each truss member A, B, and C.

Generalize the direct stiffness method for truss analysis in Python by building a reusable function to add element contributions to the stiffness matrix, preparing a fully generalized data-driven solver.

Define the 12 degree-of-freedom system and map node indices, then implement a function to assemble the 12 by 12 primary stiffness matrix by adding each element.

Identify restrained degrees of freedom and convert them to zero-based indices, then delete rows and columns from the primary stiffness matrix to obtain an 8 by 8 structure stiffness matrix.

Apply the direct stiffness method to a truss in Python: solve displacements, reconstruct the global displacement, and compute nodal reactions, updating Young's modulus and cross-sectional area.

Apply the direct stiffness method to calculate member forces in a truss using a function, feeding E, A, length, angle, and global displacements to obtain axial forces after local-frame transformation.

Visualize the truss output by plotting the structure and its deformed shape with matplotlib, including nodal positions and displacements, using a scale factor of 50 and annotations.

Generalize the direct stiffness method in Python to solve any structure by a notebook that inputs geometry and loading and outputs a complete solution.

Generalize truss analysis notebook by defining input data for nodes, node coordinates, members, supports, and loads. Import material properties (Young's modulus and cross-sectional area) and a displacement scale for analysis.

Define a Python function to calculate a truss member's length and orientation from node coordinates, using dx, dy, quadrants, and angle calculations for the direct stiffness method.

Assemble the primary stiffness matrix from each member's local stiffness into the global system, then reduce to the structure stiffness matrix by removing restrained degrees of freedom.

Compute displacements using the inverse stiffness matrix and reduced force vector, assemble a displacement vector with restrained degrees of freedom, then derive reactions and member axial forces via transformation matrix.

Automate plotting of truss results in Python by showing undeformed and deformed shapes, computing node coordinates and displacements from degrees of freedom, and rendering with axes and nodes.

Automate the text summary of the structural output by extracting reactions, member forces, and nodal displacements from calculations, and format and print them in kilonewtons and meters.

Parse and input example truss structures into the solver notebook, loading definitions correctly so the code can compute results using the direct stiffness method.

Apply the direct stiffness method to a statically determinate truss with unsymmetrical loading using python. Define nodes and restraints, input material data, and plot the structure to identify zero-force members.

demonstrates implementing the direct stiffness method for a large truss in python, including node numbering, coordinates, degrees of freedom, loads, and interpreting deflections and member forces.

Work through the final test question on applying the direct stiffness method to a statically indeterminate truss in Python. Note node coordinates, degrees of freedom, and the zero member.

Combine your core structural analysis knowledge with Python, and enhance your course tools by adding interactive widgets, animating structural deflection, and extending to treaty structures or space frames.