🍏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

$$M_{max} = \frac{1}{8}ql^2$$

$$M_{max} = \frac{1}{8}\cdot 6 \cdot 5^2 = \mathbf{22,5}\; kNm$$

1.2 Shear force

$$V_{max} = \frac{1}{2}ql$$

$$V_{max} = \frac{1}{2}\cdot 5\cdot6= \mathbf{15}\: kN$$

1.3 Displacement

$$\delta _{max}=\frac{5\: ql^4}{384\: EI}$$

$$\delta _{max}=\frac{5\cdot 5\cdot 6^4}{384 \cdot 2.07e+11 \cdot 6,75e-4} = 0.349\; mm$$

1.4 Benchmark

simply supported	Standard Beam Equation	Alpaca4d	Difference
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 supported	Standard Beam Equation	Alpaca4d	Difference
Shear force	- kN	- kN	-%
Bending moment	- kNm	- kNm	-%
Displacement	- mm	- mm	-%

2.5 Download Grasshopper benchmark file