Simple Beam — Concentrated Load at Any Point

1

Geometry and Load

Span ℓ, load P, distance a of the load from the left support (b = ℓ − a)

Span ℓ

m

Concentrated load P

kN

Distance a

m

b = ℓ − a = — · R₁ is the pinned support (x=0), R₂ is the roller support (x=ℓ). At a = ℓ/2 this reduces to the Figure 7 values.

2

R₁, R₂, V_max, M_max

R₁ = V₁= Pbℓ = —kN

R₂ = V₂= Paℓ = —kN

V_maxthe larger of the two= max(R₁, R₂) = —kN

M_maxunder the load, x=a= Pabℓ = —kN·m

M_xx < a= Pbxℓ M_xx > a= Pa(ℓ−x)ℓ

3

Define E and I for Δ_max and the deflection under the load

Elastic modulus E

MPa

Moment of inertia I

cm⁴

Δ_maxx = √(a(a+2b)/3)= Pab(a+2b)√(3a(a+2b))27EIℓ = —mm

Δ_aunder the load= Pa²b²3EIℓ = —mm

Deflected Shape Diagram

4

σ_max = M_max / S

Section modulus S

cm³

Allowable bending F_b

MPa

σ_max=—MPa

—σ / F_b

5

V_x, M_x, δ_x at a given location

Distance x

m

0aℓ

V_x=—kN

M_x=—kN·m

δ_x=—mm