Cantilever Beam — Concentrated Load at Any Point

1

Geometry and load

Span, load magnitude and load position (fixed on the left, free on the right)

Span ℓ

m

Point load P

kN

Load distance a

m

b = ℓ − a = 2.00 m · a ≤ ℓ is required

2

R, M_max

R = V_maxat the fixed end = P = —kN

M_maxat the fixed end (negative) = Pa = — = —kN·m

3

Define E and I for δ_max

Elastic modulus E

MPa

Moment of inertia I

cm⁴

δ_maxat the free end = Pa²(3ℓ − a)6 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) a ℓ (free)

V_x = P (x ≤ a) / 0 (x > a) = —kN

M_xx ≤ a = −P(a − x) = — = —kN·m

δ_x = Px²(3a−x)6EI = —mm