Cantilever Beam — Concentrated Load at Free End

1

Geometry and load

Define the span and the concentrated load at the free end (fixed on the left, free on the right)

Span ℓ

m

Point load P

kN

2

R, M_max

R = V_maxat the fixed end = P = —kN

M_maxat the fixed end (negative: tension on top fibre) = Pℓ = — = —kN·m

3

Define E and I for δ_max

Elastic modulus E

MPa

Moment of inertia I

cm⁴

δ_maxat the free end = Pℓ³3 EI = — = —mm

Deflected Shape Diagram

4

Compare σ_max against the allowable stress

Section modulus S

cm³

Allow. bending F_b

MPa

σ_max = M_maxS = — = —MPa

— σ / F_b

5

V_x, M_x, δ_x values at a given point x

Distance x

m

0 (fixed) ℓ/2 ℓ (free)

V_x = P = —kN

M_xnegative (tension on top fibre) = −P(ℓ − x) = — = —kN·m

δ_x = Px²6 EI·(3ℓ − x) = —mm