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Distributed Loads Analysis

Distributed Loads Analysis
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2019-06-11 11:48
VectorCoordinate GeometryCalculusOptimizationGeometric Construction
This example demonstrates the mechanical effects and moment distribution of distributed loads on a beam, involving integral calculus, derivatives, vector operations, and geometric constructions. Key GeoGebra commands and objects used include Point, Vector, If, Sort, Max, Min, member, Segment, Textfield, Polygon, Sequence for constructing load distributions, beam geometry, and force analysis. Suitable for high school students and engineering professionals studying structural mechanics and calculus applications.

GeoGebra Commands & Parameters

  1. Point(xAxis) → C
  2. Point(xAxis) → B
  3. Point(xAxis) → A
  4. Vector(A, Midpoint[B, C]) → r_{w_1}
  5. Point(xAxis) → E
  6. Point(xAxis) → F
  7. Vector(A, F) → r_F
  8. Point(xAxis) → D
  9. Vector(A, Midpoint[E, D]) → r_{w2}
  10. If(abs(w_1) > abs(w_2), {1.4, 1}, If[abs(w_1) < abs(w_2), {1, 1.4}, {1, 1}]) → height
  11. If(x(D) < x(E), D, E) → start
  12. If(start ≟ D, E, D) → end
  13. Vector(A, C) → r_{P_1}
  14. Sort(Unique[x({A, B, C, D, E, F})]) → points
  15. Vector(A, B) → r_B
  16. Max(x({A, C, B, D, E, F})) → right
  17. Point(x = ((-AxisStepX[])) / 4) → V
  18. Min(x({A, C, B, D, E})) → left
  19. If(w_2 < 0, Polygon[E + (0.2 * j), E + (height(2) * j), D + (height(2) * j), D + (0.2 * j)], Mirror[Polygon[E + (0.2 * j), E + (height(2) * j), D + (height(2) * j), D + (0.2 * j)], xAxis]) → load_2
  20. member(A, (right, 0), (0.2 * AxisStepY[])) → beam, beam_t, beam_b, beam_r, beam_L
  21. Segment((x(A), v'(x(A))), (x(A), v'(x(A) + 0.001))) → v_A
  22. Segment((x(B), v'(x(B))), (x(B), v'(x(B) + 0.001))) → v_B
  23. Point(x = ((-AxisStepX[])) / 4) → M
  24. Segment((x(C), v'(x(C))), (x(C), v'(x(C) + 0.001))) → v_C
  25. Segment(V', (x(V'), v'(right))) → v_r
  26. Max(Join[{{0}, y(Sequence[Max[m, points(k), points(k + 1)], k, 1, Length[points] - 1])}]) → M_{max}
  27. Min(Join[{{0}, y(Sequence[Min[v, points(k), points(k + 1)], k, 1, Length[points] - 1])}]) → V_{min}
  28. Textfield(Scale_V) → InputBox1
  29. Textfield(Scale_M) → InputBox2
  30. Polygon((right, y(Corner[2])), (right, y(Corner[3])), Corner[3], Corner[2]) → mask, mask_l, mask_t, mask_r, mask_b
  31. Segment((0, y(V)), (right, y(V))) → Vaxis
  32. Segment((0, y(M)), (right, y(M))) → Maxis
  33. Max(Join[{{0}, y(Sequence[Max[v, points(k), points(k + 1)], k, 1, Length[points] - 1])}]) → V_{max}
  34. Min(Join[{{0}, y(Sequence[Min[m, points(k), points(k + 1)], k, 1, Length[points] - 1])}]) → M_{min}
  35. If(w_1 < 0, Polygon[B + (0.2 * j), B + (height(1) * j), C + (height(1) * j), C + (0.2 * j)], Mirror[Polygon[B + (0.2 * j), B + (height(1) * j), C + (height(1) * j), C + (0.2 * j)], xAxis]) → load_1
  36. If(F_A ≥ 0, Vector[A - (1.5 * j), A], Vector[A, A - (1.5 * j)]) → a
  37. If(F_F ≥ 0, Vector[F - (1.5 * j), F], Vector[F, F - (1.5 * j)]) → f
  38. Sequence(Vector[If[w_1 < 0, Vector[B + (height(1) * j) + (n * delta), B + (0.2 * j) + (n * delta)], Vector[B - (height(1) * j) + (n * delta), B - (0.2 * j) + (n * delta)]]], n, 0, Length[Segment[B, C]] / abs(delta)) → distLoad_1
  39. Sequence(Vector[If[w_2 < 0, Vector[D + (height(2) * j) + (n * delta), D + (0.2 * j) + (n * delta)], Vector[D - (height(2) * j) + (n * delta), D - (0.2 * j) + (n * delta)]]], n, 0, Length[Segment[D, E]] / abs(delta)) → distLoad_2
  40. Segment((x(F), v'(x(F))), (x(F), v'(x(F) + 0.001))) → v_F

AI Drawing Prompt Reference

Draw a cantilever beam with two distributed loads applied at both ends. Use vectors to represent the direction and magnitude of the loads. Construct the beam's geometry using points and vectors. Calculate and display the maximum and minimum bending moments and shear forces on the beam, as well as the boundary conditions. Draw the shear force and bending moment diagrams for the beam.
The prompt above was compiled from the original drawing commands and is for reference only.