Structural mechanics and finite elements

Pull on a solid and two questions appear: how far does it move, and what forces does it transmit? We begin with a bar you can solve on paper, then build a two-dimensional elastic body from equilibrium, geometry and a material law. After solving it with Eqiora, we open the reusable components that contain the same equations.

The central investigation is a square held along its left edge and pulled by a uniform force throughout its interior. Its exact displacement is quadratic in position. That simple answer lets us distinguish a good displacement prediction from a good stress prediction without trusting a picture alone.

Follow the investigation

1. Loads, supports and elastic energy — draw a free body, derive a bar’s extension and calculate stored energy.
2. Displacement, strain and stress — pass from scalar stretching to tensors, traction and two-dimensional laws.
3. Virtual work and finite elements — turn equilibrium into a weak equation and understand what Q1 approximates.
4. A square with mixed boundaries — derive the answer, read the .eqi equations, run the example, and find the same laws in components.
5. Reading displacement, reactions and error — explain mesh refinement, boundary stress and physical applicability.

You need basic differentiation, integration and vector algebra. Each chapter introduces its own mechanics notation. If you already know equilibrium and small-strain elasticity, start at the square investigation.

Connections that help

Mathematical modeling explains why a material law and boundary conditions belong to the model, while a mesh belongs to the numerical realization. In heat transfer, temperature gradients drive heat flux; here displacement gradients create strain and stress. The common integration-by-parts idea is developed further in numerical simulation.

The stress tensor also appears in fluid mechanics. Its traction meaning stays the same. The solid’s elastic law depends on displacement gradients; a Newtonian fluid’s viscous law depends on velocity gradients.

Use Get started for the shared Python environment. The executable chapter uses the source revision installed there and a Cartesian Q1 mesh; the opening bar calculation is a hand derivation. The mechanics in this path is static, isotropic and small-strain.