Continuous Beam — Two Unequal Spans — Midspan Point Load on Each Span

1

Geometry and Loads

P₁ at the center of the left span, P₂ at the center of the right span (a=ℓ₁/2, b=ℓ₂/2)

Left span ℓ₁

m

Right span ℓ₂

m

Left load P₁

kN

Right load P₂

kN

Derivation: Three-moment equation (Clapeyron) — P₁ and P₂ at the midpoint of each span (a = ℓ₁/2, b = ℓ₂/2)

2

M₁, R₁–R₃, V₁–V₄, M_m₁, M_m₂

M₁= −316P₁ℓ₁²+P₂ℓ₂²ℓ₁+ℓ₂ = —kN·m

R₁= M₁ℓ₁ + P₁2 = —kN

R₃=V₄= M₁ℓ₂ + P₂2 = —kN

R₂= P₁ + P₂ − R₁ − R₃ = —kN

V₂=P₁−R₁= —kN V₃=P₂−R₃= —kN

M_m₁x=ℓ₁/2= R₁·a = —kN·m

M_m₂x=ℓ₂/2= R₃·b = —kN·m

3

σ_max = M_cr / S

Section modulus S

cm³

Allow. bending F_b

MPa

σ_max=—MPa

M_cr = max(|M₁|, M_m₁, M_m₂)

—σ / F_b

4

V_x and M_x at a given location

Distance x

m

0ℓ₁ℓ₁+ℓ₂

V_x=—kN

M_x=—kN·m