Fixed-Fixed Beam — Concentrated Load at Any Point

1

Geometry and load

ℓ = a + b; load P at distance a from the left fixed end

Length ℓ

m

Concentrated load P

kN

Left distance a

m

b = ℓ − a = 4.00 m

2

R₁, R₂, M₁, M₂, Mₐ

R₁ = V₁ = Pb²(3a+b)ℓ³ = —kN

R₂ = V₂ = Pa²(a+3b)ℓ³ = —kN

M₁left end = Pab²ℓ² = —kN·m

M₂right end = Pa²bℓ² = —kN·m

M_aunder the load = 2Pa²b²ℓ³ = —kN·m

3

Δₐ = Pa³b³/3EIℓ³; Δmax = 2Pa³b²/[3EI(3a+b)²] (a>b)

Elastic modulus E

MPa

Moment of inertia I

cm⁴

Δ_aunder the load = Pa³b³3EIℓ³ = —mm

Δ_maxif a>b, x=2aℓ/(3a+b) = 2Pa³b²3EI(3a+b)² = —mm

Deflected Shape Diagram

4

σ_max = M_krit / S

Section modulus S

cm³

Allow. bending F_b

MPa

σ_max = —MPa

M_krit = max(M₁, M₂, Mₐ)

—σ / F_b

5

V_x, M_x, Δ_x at the given x

Distance x

m

0aℓ

V_x=—kN

M_x=—kN·m

Δ_x=—mm