Simple Beam — Load Increasing Uniformly to One End

1

Geometry and Load

Define the span and the total triangular load

Span ℓ

m

Total load W

kN

w_peak = 2W/ℓ = 20 kN/m peak load intensity at the right end

2

R₁, R₂, M_max

R₁left support (lightly loaded) = W3 = — = —kN

R₂right support (heavily loaded) = 2W3 = — = —kN

M_maxx* = ℓ/√3 ≈ 0.5774ℓ = 2Wℓ9√3 ≈ — = —kN·m

3

Define E and I for δ_max

Elastic modulus E

MPa

Moment of inertia I

cm⁴

δ_maxx ≈ 0.5193ℓ = 0.01304 · Wℓ³EI = — = —mm

For the triangular load, δ_max occurs at x* = ℓ·√(1 − √(8/15)) ≈ 0.5193ℓ (slightly right of midspan).

Deflected Shape Diagram

4

Compare σ_max against the allowable stress

Section modulus S

cm³

Allowable bending F_b

MPa

σ_max = M_maxS = — = —MPa

— σ / F_b

5

V_x, M_x, δ_x values at a given x

Distance x

m

0 ℓ/2 ℓ

V_x = W3 − Wx²ℓ² = — = —kN

M_x = Wx3ℓ²(ℓ² − x²) = — = —kN·m

δ_x = Wx180 EIℓ²·(3x⁴ − 10ℓ²x² + 7ℓ⁴) = —mm