🍏Simple Beam

1. Simply supported beam

Example properties:

  • line load (q): 5 kN/m

  • span (l): 6 m

  • modulus of elasticity (E): 2.1e10+7 N/mm²

  • moment of inertia (I): 6.75e10+8 mm⁴

1.1 Bending moment

Mmax=18ql2M_{max} = \frac{1}{8}ql^2
Mmax=18⋅6⋅52=22,5  kNmM_{max} = \frac{1}{8}\cdot 6 \cdot 5^2 = \mathbf{22,5}\; kNm

1.2 Shear force

Vmax=12qlV_{max} = \frac{1}{2}ql
Vmax=12⋅5⋅6=15 kNV_{max} = \frac{1}{2}\cdot 5\cdot6= \mathbf{15}\: kN

1.3 Displacement

δmax=5 ql4384 EI\delta _{max}=\frac{5\: ql^4}{384\: EI}
δmax=5⋅5⋅64384⋅2.07e+11⋅6,75e−4=0.349  mm\delta _{max}=\frac{5\cdot 5\cdot 6^4}{384 \cdot 2.07e+11 \cdot 6,75e-4} = 0.349\; mm

1.4 Benchmark

simply supportedStandard Beam EquationAlpaca4dDifference

Shear force

15,0 kN

14,7 kN

2.04%

Bending moment

18,75 kNm

18,74 kNm

0.05%

Displacement

0.349 mm

0.353 mm

1.13%

1.5 Download Grasshopper benchmark file

2. Cantilever beam

2.1 Bending moment

2.2 Shear force

2.3 Displacement

2.4 Benchmark

simply supportedStandard Beam EquationAlpaca4dDifference

Shear force

- kN

- kN

-%

Bending moment

- kNm

- kNm

-%

Displacement

- mm

- mm

-%

2.5 Download Grasshopper benchmark file