What is the finite element method and why is it used in structural analysis?

The finite element method (FEM) is a numerical technique that divides a complex structure into smaller, simpler sub-regions called elements, connected at nodes. For each element, shape functions interpolate displacements between nodes, leading to element stiffness matrices. Assembling all elements yields the global stiffness equation [K]{d}={f}, solved for nodal displacements � from which strains (e=[B]{d}) and stresses (s=[D]e) are recovered. FEM is used because analytical solutions to the governing partial differential equations (equilibrium, compatibility, constitutive) only exist for simple geometries. FEM handles arbitrary shapes, mixed boundary conditions, nonlinear materials, and dynamic loadings � making it the engineering industry standard (ANSYS, Abaqus, NASTRAN).

How does bending stress vary through a beam's cross-section?

In Euler-Bernoulli beam theory, plane sections remain plane during bending, so longitudinal strain e_x = -y/?, where y is distance from the neutral axis and ? is the radius of curvature. The bending stress follows s_x = E�e_x = -E�y/? = M�y/I, where M is the bending moment and I is the second moment of area (I = bh�/12 for a rectangular section of breadth b, depth h). Stress is therefore linear through the depth: zero at the neutral axis (y=0), maximum tensile stress +M�(h/2)/I at the bottom fibre, and maximum compressive stress -M�(h/2)/I at the top fibre (for positive sagging moment). The colour map in this simulator shows this variation: blue (compression), neutral grey at mid-depth, red (tension at peak bending location).

What does the stress intensity factor K describe in fracture mechanics?

The stress intensity factor K_I (Mode I, opening) characterises the magnitude of the singular stress field at a crack tip. From linear elastic fracture mechanics (LEFM), the near-tip stress field is s_ij = K_I/v(2pr) � f_ij(?), where r is distance from the crack tip and f_ij(?) is an angular function derived by Williams (1957). For an edge crack of half-length a in a plate stressed by s at infinity, K_I = sv(pa). Fracture initiates when K_I reaches the material's fracture toughness K_Ic (e.g. steel � 50 MPavm, glass � 0.7 MPavm). Irwin's plastic zone correction gives a small zone of plasticity ahead of the crack tip of radius r_p � (K_I/s_Y)�/(6p), visible as the orange dashed circle in the crack propagation preset.

Home▸Physics & Mechanics▸FEM Structural Mechanics Simulator – Beams, Trusses, Pressure Vessels & Vibration

🧪 FEM Structural Mechanics Simulator – Beams, Trusses, Pressure Vessels & Vibration

Interactive finite element method simulator. Six presets: Cantilever Beam, Simply Supported Beam, Bridge Truss, Pressure Vessel, Crack Propagation, and Vibration Modes. Visualise stress distributions, deflection, bending moments, and natural frequencies. Load, elastic modulus, and thickness sliders.

Physics & Mechanics2DModerate60 FPS

⚙ Under the hood

This simulation demonstrates the principles of Finite Element Method (FEM) analysis for structural mechanics. It allows users to explore how forces and loads affect the behavior of beams, trusses, pressure vessels, and their vibrational characteristics.

FEM AnalysisStructural Mechanics

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install