I used sympy's beam module to evaluate beam loadings for a workshop gantry crane design. I have a S4x7.7 Aluminum I-Beam¹ which has the following properties:

• Dimensions
  • Flange: 2.66 in
  • Height: 4 in
  • Web: 0.19 in
  • Thickness: 0.19 in
  • Shape Type: American Standard
  • Ix = 6.04 in^4
• Material
  • 6061-T6 Aluminum (AL)
    • Weight: 2.7 lb/ft
    • Modulus of Elasticty (E): 9.9 ksi
  • Steel (S)
    • Weight: 7.7 lb/ft
    • Modulus of Elasticity (E) : 29,000 ksi

• Span (L): 10 ft
• End Constraints: Fixed-Fixed (Best Case) or Simple-Simple (Worst Case)
• Point Load (F): 2120 lb (AL) or 2180 lb (S) @ L/2²

From my initial research, it appears this beam will handle a point load of 1,000 lb, however I want to be able to lift as much as 2,000 lb. beam-calc was inspired by the AISC's "Steel Tools" Website > BEAM ANALYSIS & DESIGN > Tool "BMREINF13.xls" to evaluate the effects of increasing Ix for various I-Beam reinforcement configurations. I did not use its analysis capabilities, merely it's ability to calculate various moments of inertia for different beam reinforcement techniques suggested. https://clearcalcs.com/freetools/free-moment-of-inertia-calculator/us also seems like a good option for calculations of composite shape properties, I could also try S4x7.7 + C-Channel section.

main.py currently has three Ix values for:

1. Member Only	2. Member + Plate Bottom (or Top)	3. Member + Plate Top and Bottom

The bar stock is 0.25 x 3 inches for cases 2. and 3.

The resultant moment of inertia list in the code is Ix = [6.04, 8.6, 12.83] #in4

If I had to do this all over again, I definitely wouldn't use sympy. It was more trouble manipulating plot_loading_results() to work as I envisioned than simply using Beam Design Formulas - Figure 24. or 25. - directly.

I updated this in the Second Release tool to

1. include a calculation with simple-simple end constraints, which has much larger deflection values
2. evaluate a steel S4x7.7 I-Beam for comparison

δ_allowable = L/450 per https://www.spanco.com/blog/understanding-overhead-crane-deflection-and-criteria/ for aluminum gantry cranes.

Aluminum	Steel

Analytical Expression for Maximum Deflection:

⎛    3 │ F │⎞
⎜   L ⋅│───│⎟
⎜L     │E⋅I│⎟
⎜─, ────────⎟
⎝2    192   ⎠

Table 1. Material: Aluminum, Constraint: Fixed-Fixed

Table 2. Material: Steel, Constraint: Fixed-Fixed

Aluminum	Steel

Analytical Expression for Maximum Deflection:

⎛    3 │ F │⎞
⎜   L ⋅│───│⎟
⎜L     │E⋅I│⎟
⎜─, ────────⎟
⎝2    48    ⎠

Table 3. Material: Aluminum, Constraint: Simple-Simple

Table 4. Material: Steel, Constraint: Simple-Simple

Clearly, Option 3) Member + Plate Top and Bottom has the least deflection as the analytical expressions show.

If Fixed-Fixed is assumed, an Aluminum reinforced I-Beam would work, but for Simple-Simple, we're up the creek without a paddle. Steel will be needed if the Simple-Simple end constraint assumption is used. I need to consult a structural engineer on the differences between these two support types; https://web.mit.edu/4.441/1_lectures/1_lecture13/1_lecture13.html looks like a decent reference. I would think that a gantry crane with a sufficiently supported post and solid, bolted, connection to the overhead beam would like a fixed support. However, from what I read, it seems like fixed is only appropriate when you're securing to something monolithic, like a large concrete structure and not a post which could deflect at the joint.

Overall, it was a good introductory project to learn some Python, PyCharm, and details of the sympy and matplotlib modules.

• [1]: S4x7.7 is the designation for the equivalent a steel I-beam. If one exists for aluminum it would be S4x2.7 following the specification HxLB/FT. See https://www.aisc.org/publications/historic-shape-references/ and https://www.aisc.org/globalassets/aisc/publications/historic-shape-references/hot-rolled-carbon-steel-structural-shapes-1948.pdf for a great selection of tables!
• [2]: Load (F) should account for the weight of the beam as well, I think this is typically referred to as the "dead" load. Estimated beam weight + trolley + chain hoist/fall weights are added to F