Propped Cantilever Beam — Concentrated Load at Center

1

Geometry and load

Span and concentrated load at midspan (fixed on the left, pinned on the right)

Span ℓ

m

Point load P

kN

2

R_A, R_B, M_A, M_max+

R_Aat the fixed end = 11P16 = —kN

R_Bat the roller support = 5P16 = —kN

M_Afixed-end moment = 3Pℓ16 = —kN·m

M_max+at x = ℓ/2 = 5Pℓ32 = —kN·m

M = 0inflection point @ x = 3ℓ/11 = —m

3

Define E and I for δ_max

Elastic modulus E

MPa

Moment of inertia I

cm⁴

δ_maxat x ≈ 0.553ℓ = —mm

δ(x) = Px²(9ℓ − 11x)96 EI (x ≤ ℓ/2)

δ(x) = P[9ℓx²−11x³+16(x−ℓ/2)³]96 EI (x > ℓ/2)

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

M_max is taken as the larger of |M_A| and M_max+.

— σ / F_b

5

V_x, M_x, δ_x at a given point x

Distance x

m

0 (fixed) ℓ/2 ℓ (support)

V_x = R_A (x < ℓ/2) / −R_B (x > ℓ/2) = —kN

M_x = — = —kN·m

δ_x = —mm