Propped Cantilever — Concentrated Load at Any Point

1

Geometry and Load

Span, load magnitude and load location (fixed at left, pinned at right)

Span ℓ

m

Concentrated load P

kN

Load distance a

m

b = ℓ − a = 3.00 m · a ≤ ℓ is required

2

R_A, R_B, M_A, M_max+

R_Bat the free (roller) support = Pa²(3ℓ−a)2ℓ³ = —kN

R_Aat the fixed end = P − R_B = —kN

M_Afixed-end moment = Pab(2ℓ−a)2ℓ² = —kN·m

M_max+at x = a = R_A·a − M_A = —kN·m

M = 0inflection point @ x = M_A / R_A = —m

3

Define E and I for δ_max

Elastic modulus E

MPa

Moment of inertia I

cm⁴

δ_maxnumerical = —mm

δ(x) = x²(3M_A − R_Ax)6EI (x ≤ a)

δ(x) = 3M_Ax²−R_Ax³+P(x−a)³6EI (x > a)

Deflected Shape Diagram

4

Compare σ_max with the allowable stress

Section modulus S

cm³

Allow. bending F_b

MPa

σ_max = M_maxS = — = —MPa

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

— σ / F_b

5

V_x, M_x, δ_x at a specific point x

Distance x

m

0 (fixed) a ℓ (support)

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

M_x = — = —kN·m

δ_x = —mm