Beam Overhanging One Support — Concentrated Load Between Supports

1

Geometry and Load

Main span ℓ, load position a (from R₁), overhang c (unloaded), concentrated load P

Main span ℓ

m

Load position a

m

Overhang c

m

Concentrated load P

kN

b = ℓ − a = — · The overhang is unloaded, so V = M = 0 there; it deflects upward as a straight line due to the rotation at R₂ (x₁ measured from the free end).

2

Calculation details

R₁, R₂, V_max, M_max

R₁ = V₁= Pbℓ= —kN

R₂ = V₂= Paℓ= —kN

V_max= max(R₁, R₂)= —kN

M_maxunder the load= Pabℓ= —kN·m

M_xx < a= Pbxℓ M_xa < x < ℓ= Pa(ℓ−x)ℓ

3

Deflection calculation

Downward in the span, upward at the overhang tip — define E and I

Elastic modulus E

MPa

Moment of inertia I

cm⁴

Δ_maxx = √(a(a+2b)/3), downward= Pab(a+2b)√(3a(a+2b))27EIℓ= —mm

Δ_aunder the load= Pa²b²3EIℓ= —mm

Δ_x₁=coverhang tip, upward= Pabc(ℓ+a)6EIℓ= —mm

Deflected Shape Diagram

4

Stress check

σ_max = M_max / S

Section modulus S

cm³

Allowable bending F_b

MPa

σ_max=—MPa

—σ / F_b

5

Point analysis

V_x, M_x, δ_x — x measured from R₁

Distance x

m

0 (R₁)ℓ (R₂)ℓ+c

V_x=—kN

M_x=—kN·m

δ_x+ downward=—mm