MECHA ATELIERENGINEERING STUDIO
MOM-01 · Mechanics of Materials

Beam Deflection Stage

A split-stage instrument: geometry and load on the left, the bending beam on the right, and a live solver panel underneath. Everything shares one source of truth.

Euler–BernoulliReal-time

Control panel

Pinned left, roller right, point load P at midspan.

Boundary condition

δmax = PL³ / 48EI·Mmax = PL / 4

Material

locked to preset
GPa

Cross-section

I = 20.48e6 mm⁴
mm
mm

Geometry

2,000mm
2508000

Point load

5.00kN
0100

Stage

visual only
1.00×
0.254.00

The solver is physically exact; the stage amplifies the curve so microns stay visible at metre scales.

Loading stageLoading stage
Shear forceV [kN]
+2.50-2.50
Bending momentM [kN·m]
+2.50

Sagging-positive convention · V = dM/dx · dashed red cursor tracks the probe

Live results

Converged · linear-elastic
δmaxMax deflection
0.203mm

Midspan · x = L/2

MmaxMax moment
2.500kN·m

Midspan · x = L/2

QuantitySymbolValueUnitLocation
Maximum deflectionδmax0.203mmMidspan · x = L/2
Maximum bending momentMmax2.500kN·mMidspan · x = L/2
Maximum shear forceVmax2.500kNSupports · x = 0, L
Maximum bending stressσmax9.77MPaExtreme fibre · y = c
Maximum slopeθmax0.0175degSupports · x = 0, L
Second moment of areaI20,480,000mm⁴Section · bh³/12
Flexural rigidityEI4.1kN·m²Material × section (Rectangle)
Reaction R_A (pin)R2.500kNSupports
Reaction R_B (roller)R2.500kNSupports
δmax = PL³ / 48EIMmax = PL / 4σ = Mc / I200 GPa · L 2,000 mm

Mechanics of Materials: Beam Deflection & Flexure Theory

Mecha Atelier evaluates beam bending under the Euler–Bernoulli beam hypothesis, assuming linear-elastic material response and small angles of rotation:

1. Flexural Rigidity & Cross-Section Properties

The flexural rigidity is EI, where E is Young's Modulus and I is the area moment of inertia. For a rectangular cross-section of width b and depth h, I = (b · h³) / 12.

2. Boundary Condition Formulations

- Simply-Supported Beam with Central Point Load P: δ_max = (P · L³) / (48 · E · I), M_max = (P · L) / 4. - Cantilever Beam with Tip Point Load P: δ_max = (P · L³) / (3 · E · I), M_max = P · L at the fixed support. - Simply-Supported Beam with Uniform Load w: δ_max = (5 · w · L⁴) / (384 · E · I), M_max = (w · L²) / 8.

3. Maximum Bending Stress

Normal bending stress distribution is evaluated via the flexure formula: σ(y) = (M · y) / I. Peak tensile and compressive stresses occur at outer fibers: σ_max = (M_max · (h / 2)) / I.