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.

FEM Structural Mechanics Simulator � Beams, Trusses, Pressure Vessels & Vibration

Structural Mechanics Physics

Euler-Bernoulli Beam Theory

The Euler-Bernoulli beam model assumes that plane cross-sections remain plane and perpendicular to the neutral axis after bending (no shear deformation). The governing differential equation relates the bending moment M to curvature:

Governing equation: EI � d�y/dx� = M(x) (beam bending DE) Cantilever beam (point load P at free end): Bending moment: M(x) = P(L - x) Deflection: y(x) = Px�(3L - x) / (6EI) Tip deflection: d_tip = PL� / (3EI) Max bending stress:s_max = PL�(h/2) / I [at fixed end, top/bottom fibre] Simply-supported beam (UDL w N/m): Bending moment: M(x) = wx(L - x)/2 Deflection: y(x) = wx(L� - 2Lx� + x�) / (24EI) Max deflection: d_max = 5wL4 / (384EI) [at mid-span] Bending stress: s(x,y) = M(x)�y / I [y from NA] Second moment of area (rectangular section, breadth b, depth h): I = bh�/12 ? elastic section modulus Z = I/(h/2) = bh�/6

Truss Analysis (Direct Stiffness Method)

Member stiffness matrix (local, 1-D bar element): [k]_local = (EA/L) � [ 1 -1 ; -1 1 ] Transformation to global coords (angle a from horizontal): c = cosa, s = sina [k]_global = (EA/L) � [ c� cs -c� -cs ] [ cs s� -cs -s� ] [-c� -cs c� cs ] [-cs -s� cs s� ] Global system: [K]{d} = {f} ? Solve for displacements {d} ? Member force: F = (EA/L)�(d_j - d_i)�cos(a) + ... Pratt truss (6 panels): diagonals in tension, verticals under shear Bottom chord: tension T = M / h_truss Top chord: compression C = -M / h_truss (M = bending moment at panel)

Pressure Vessel � Thin-Walled Theory

Thin-walled assumption: t/R < 0.1 (t = wall thickness, R = inner radius) Hoop (circumferential) stress: s_? = p�R / t [controls burst] Axial (longitudinal) stress: s_z = p�R / (2t) [closed ends] Radial stress: s_r � 0 [thin wall approx] Von Mises equivalent stress: s_VM = v(s_?� - s_?�s_z + s_z�) = (p�R/t)�v(3)/2 � 0.866�s_? Hoop strain: e_? = (s_? - ?�s_z) / E = pR(2 - ?) / (2Et) Radial expansion: dR = e_?�R = pR�(2 - ?) / (2Et) Design requirement: s_VM < s_Y / SF (SF = safety factor, typically 2�4)

Linear Elastic Fracture Mechanics (LEFM)

Mode-I stress intensity factor (edge crack, infinite plate): K_I = s_8 � v(p�a) [Pavm or MPavm] General form: K_I = F�s�v(p�a) (F = geometry correction factor) Near-tip stress field (Williams expansion): s_xx = K_I/v(2pr) � cos(?/2)�[1 - sin(?/2)�sin(3?/2)] s_yy = K_I/v(2pr) � cos(?/2)�[1 + sin(?/2)�sin(3?/2)] t_xy = K_I/v(2pr) � sin(?/2)�cos(?/2)�cos(3?/2) Fracture criterion: K_I = K_Ic (material fracture toughness) Steel: K_Ic � 50 MPavm Aluminium: K_Ic � 20�35 MPavm Glass: K_Ic � 0.7 MPavm Irwin plastic zone: r_p = (1/6p)�(K_I/s_Y)� [plane stress] Paris fatigue law: da/dN = C�(?K)^m (C, m material constants)

Vibration Modes � Euler-Bernoulli Beam

Equation of motion: EI�?4w/?x4 + ?A�?�w/?t� = 0 Natural frequencies (cantilever, clamped-free BC): ωₙ = (�?L)� � v(EI / ?AL4) [rad/s] fₙ = ωₙ / (2π) [Hz] Characteristic values �?L: Mode 1: �1L = 1.8751 f1 ? L?�v(EI/?A) Mode 2: �2L = 4.6941 Mode 3: �3L = 7.8548 Mode 4: �4L = 10.9955 (higher modes approach (2n-1)p/2) Mode shapes (cantilever): f?(x) = cosh(�?x) - cos(�?x) - s?�[sinh(�?x) - sin(�?x)] s? = (cosh(�?L) + cos(�?L)) / (sinh(�?L) + sin(�?L)) Simply-supported beam (pinned-pinned): �?L = np ? f?(x) = sin(npx/L) f? = (np)�/(2pL�) � v(EI/?A) = n��f1

Preset Guide

Preset	Key Physics	What to observe
🏗️ Cantilever	d_tip=PL�/3EI; s_max=PL(h/2)/I	Increase load ? deflection grows cubically; reduce h ? very large deflection (I?h�)
📐 Simply Supported	d_max=5wL4/384EI; uniform load	Stress peaks at mid-span; reduce E ? large mid-span deflection
🌉 Bridge Truss	F=M/h_truss; direct stiffness	Tension members (red), compression (blue); increase load ? all forces scale linearly
⚙️ Pressure Vessel	s_?=pR/t; s_z=pR/2t	Reduce thickness ? hoop stress grows inversely; von Mises controls yielding
🔬 Crack Prop.	K_I=svpa; r_p?K_I�	Increase load ? K grows; plastic zone (orange) scales with K�; 1/vr singularity visible
📊 Vibration	fₙ∝(�?L)�v(EI/?A)	Increase E or reduce h: frequencies shift; four animated mode shapes shown

Explore Structural Theory

Go deeper into solid mechanics, beam theory, and the finite element method with our full-length articles.

Structural Mechanics ? Finite Element Method ?

Curriculum Links

Level	Topic	Covered
GCSE Physics / DT	Forces, structures, materials	Beam bending concepts, stress/strain, load-bearing structures
A-Level Physics	Materials, Young's modulus	s = F/A, e = ?L/L, E = s/e; elastic/plastic deformation
AP Physics / Mechanics	Statics, elasticity, oscillations	Truss analysis, equilibrium, simple harmonic motion
IB Physics / Design Tech	Materials and structures	Stress distribution, safety factors, structural failure modes
University Engineering	Solid mechanics, FEM, dynamics	Euler-Bernoulli theory, direct stiffness, LEFM, modal analysis

FEM Structural Mechanics Simulator #87