Simple Beam — Concentrated Load at Center

1

Geometry and Load

Define the span and the concentrated load at midspan

Span ℓ

m

Concentrated load P

kN

2

R₁, R₂, M_max

R₁left support (symmetric) = P2 = — = —kN

R₂right support (symmetric) = P2 = — = —kN

M_maxx* = ℓ/2 (under the load) = Pℓ4 = — = —kN·m

3

Define E and I for δ_max

Elastic modulus E

MPa

Moment of inertia I

cm⁴

δ_maxx = ℓ/2 (under the load) = Pℓ³48 EI = — = —mm

Under a concentrated load at midspan, δ_max occurs exactly at midspan (x = ℓ/2). By symmetry, the moment and deflection peaks coincide.

Deflected Shape Diagram

4

Compare σ_max against the allowable stress

Section modulus S

cm³

Allowable bending F_b

MPa

σ_max = M_maxS = — = —MPa

— σ / F_b

5

V_x, M_x, δ_x values at a given x

Distance x

m

0 ℓ/2 ℓ

V_xx < ℓ/2 (left half) = P2 = — = —kN

M_xx < ℓ/2 (left half) = Px2 = — = —kN·m

δ_xx < ℓ/2 (left half) = Px48 EI·(3ℓ² − 4x²) = —mm