Continuous Beam — Two Unequal Spans — Uniformly Distributed Load

1

Geometry and Load

UDL w on both spans; unequal spans: ℓ₁ ≠ ℓ₂

Left span ℓ₁

m

Right span ℓ₂

m

Load w

kN/m

2

M₁, R₁, R₂, R₃, V₁–V₄, M_x₁,max, M_x₂,max

M₁= −w(ℓ₁³+ℓ₂³)8(ℓ₁+ℓ₂) = —kN·m

R₁= M₁ℓ₁ + wℓ₁2 = —kN

R₃=V₄= M₁ℓ₂ + wℓ₂2 = —kN

R₂= w(ℓ₁+ℓ₂) − R₁ − R₃ = —kN

V₂wℓ₁−R₁= —kN V₃wℓ₂−R₃= —kN

M_x₁,maxx₁=R₁/w= R₁²2w = —kN·m

M_x₂,maxx₂=R₃/w= R₃²2w = —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_x₁,max, M_x₂,max)

—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