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Strength of Materials

Beam deflection and stress, stress analysis, buckling safety and section properties.

Beam Deflection & Bending Stress

4 load cases, point load or uniformly distributed load

Load case

Force F [N]

Length L [mm]

Young's modulus E [N/mm²] (steel 210,000)

I [cm⁴] (IPE 200: 1,943)

W [cm³] (IPE 200: 194)

σ_allow [N/mm²] (S235: 156)

Max. deflection f–

Max. bending moment M_b–

Bending stress σ_b–

Utilization σ_b/σ_allow–

Deflection ratio L/f–

Guideline for mechanical engineering: f ≤ L/500 to L/1000. For a distributed load, enter q in N/mm (total load = q·L).

Theory

Deflection and moment per load case:

Simply supported + point load: f = F·L³48·E·I , M = F·L4

Simply supported + distributed load: f = 5·q·L⁴384·E·I , M = q·L²8

Cantilever + point load: f = F·L³3·E·I , M = F·L

Cantilever + distributed load: f = q·L⁴8·E·I , M = q·L²2

The bending stress follows from the section modulus:

σ_b = M_bW ≤ σ_allow

σ_allow = R_e/S, e.g. S235: 235/1.5 ≈ 156 N/mm².

Stresses & Equivalent Stress

tension/compression, bending, torsion → von Mises

Axial force F [N]

Cross-sectional area A [mm²]

Bending moment M_b [Nm]

W [cm³]

Torsional moment M_t [Nm]

W_p [cm³]

Allowable stress σ_allow [N/mm²]

Tensile/compressive stress σ_z–

Bending stress σ_b–

Torsional stress τ_t–

Equivalent stress σ_v (von Mises)–

Utilization–

Theory

Basic stress types:

σ_z = FA · σ_b = M_bW · τ_t = M_tW_p

Combined loads are merged into one equivalent stress using the distortion energy hypothesis (von Mises):

σ_v = (σ_z+σ_b)² + 3·τ_t²

von Mises applies to ductile materials (steel)
for brittle materials: maximum normal stress hypothesis
Check: σ_v ≤ σ_allow = R_e/S

Euler Buckling

critical buckling load of slender columns

End conditions (Euler case)

Column length L [mm]

Compressive force F [N]

Young's modulus E [N/mm²]

I_min [cm⁴]

Cross-sectional area A [mm²] (for slenderness ratio)

Effective length L_k–

Slenderness ratio λ–

Critical buckling load F_k–

Buckling safety factor S = F_k/F–

Required in mechanical engineering: S ≥ 3–5. Euler applies only in the elastic range (λ ≳ 105 for S235). Below that, use Tetmajer.

Theory

A slender column deflects sideways at the critical load:

F_k = π² · E · IL_k²

The effective length L_k depends on the end conditions (Euler cases 1–4). The slenderness ratio shows whether Euler applies:

λ = L_ki with i = I / A

λ ≥ approx. 105 (S235): elastic buckling → Euler
λ < 105: inelastic range → Tetmajer lines
λ < 60: plain compressive stress check

Section Properties I and W

rectangle, circle, round tube, rectangular tube

Cross-section shape

Width b [mm]

Height h [mm]

Area A–

Second moment of area I–

Section modulus W–

Bending about the horizontal axis (height h or diameter governs).

Theory

The second moment of area I describes the bending stiffness, the section modulus W = I/z_max the load capacity of the cross-section:

Rectangle: I = b·h³12 , W = b·h²6

Circle: I = π·d⁴64 , W = π·d³32

Tube: I = π·(D⁴−d⁴)64 , W = 2·ID

Material far from the neutral axis is most effective (h³!)
which is why I-beams and tubes are so efficient

Rough estimate for preliminary design. Does not replace a code-compliant design calculation (e.g. Eurocode 3).