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Shear and Bending Moment Diagrams: Concentrated Moments on a Beam

Shear and Bending Moment Diagrams: Concentrated Moments on a Beam
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2019-06-11 03:34
VectorFunctionCalculusIntegralGraphing
① This example demonstrates how to construct shear force and bending moment diagrams for a beam subjected to concentrated moments, clearly showing the effect of applied moments on internal forces. ② Core mathematical concepts include static equilibrium, the differential relationships between load, shear, and moment, piecewise functions, and handling discontinuities at the point of moment application, involving integration and differentiation. ③ Construction highlights: using GeoGebra commands such as Point, Vector, Segment, and Polygon to build the beam geometry, If, Max, Min, and member to manage piecewise conditions, Textfield for parameter input, and the Moment object to compute the bending moment contribution from concentrated moments, finally generating the shear (V) and moment (M) diagrams. ④ Suitable for university courses in engineering mechanics, structural analysis, or calculus when teaching shear-moment diagrams, and also useful for high school physics extensions or classroom demonstrations.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. If(x(A) ≟ 0, -0.5, 0.5) → wall_h
  3. Point(xAxis) → D
  4. Point(xAxis) → C
  5. Point(xAxis) → B
  6. Point(xAxis) → E
  7. Vector(A, C) → r_{P_C}
  8. Vector(A, B) → r_B
  9. If(x(D) < x(E), D, E) → start
  10. If(start ≟ D, E, D) → end
  11. Point(x = ((-AxisStepX[])) / 4) → V
  12. Max(x({A, C, B, D, E})) → right
  13. Min(x({A, C, B, D, E})) → left
  14. member((Min[x({A, C, E, D, B})], 0), (Max[x({A, C, E, D, B})], 0), (0.2 * AxisStepY[])) → beam, beam1, beam2, beam3, beam4
  15. Segment((x(A), v'(x(A))), (x(A), v'(x(A) + 0.001))) → v_A
  16. Segment((x(B), v'(x(B))), (x(B), v'(x(B) + 0.001))) → v_B
  17. Point(x = ((-AxisStepX[])) / 4) → M
  18. Segment((x(C), v'(x(C))), (x(C), v'(x(C) + 0.001))) → v_C
  19. Segment(V', (x(V'), v'(right))) → v_r
  20. Textfield(Scale_V) → InputBox1
  21. Textfield(Scale_M) → InputBox2
  22. Polygon((right, y(Corner[2])), (right, y(Corner[3])), Corner[3], Corner[2]) → mask, mask_l, mask_t, mask_r, mask_b
  23. Segment((0, y(V)), (right, y(V))) → Vaxis
  24. Segment((0, y(M)), (right, y(M))) → Maxis
  25. Segment((x(D), m'(x(D))), (x(D), m'(x(D) + 0.001))) → m_D
  26. Moment(D, M_D) → MD_1, MD_2, MD_3
  27. Moment(A, 1) → MA_1, MA_2, MA_3
  28. Segment((x(A), m'(x(A) - 0.0001)), (x(A), m'(x(A) + 0.001))) → m_A
  29. Segment((x(A), m'(x(A))), M') → m_r
  30. Segment((0, m'(0.0001)), (0, y(M))) → m_L
  31. Max(Join[{{0}, y(Sequence[Max[v, points(k), points(k + 1)], k, 1, Length[points] - 1])}]) → V_{max}
  32. Min(RemoveUndefined[Join[{{0}, y(Sequence[Min[v, points(k), points(k + 1)], k, 1, Length[points] - 1])}]]) → V_{min}
  33. Max(Join[{{0}, y(Sequence[Max[m, points(k), points(k + 1)], k, 1, Length[points] - 1])}]) → M_{max}
  34. Min(Join[{{0}, y(Sequence[Min[m, points(k), points(k + 1)], k, 1, Length[points] - 1])}]) → M_{min}
  35. If(F_A ≥ 0, Vector[A - j, A], Vector[A, A - j]) → a
  36. If(F_B ≥ 0, Vector[B, B + (1.5 * j)], Vector[B + (1.5 * j), B]) → b
  37. Polygon(A + {(0, 2), (wall_h, 2), (wall_h, -2), (0, -2)}) → wall
  38. If(P_C ≥ 0, Vector[C, C + (1.5 * j)], Vector[C + (1.5 * j), C]) → p_C

AI Drawing Prompt Reference

Draw a simply supported beam with concentrated moments applied at points D and E. Start by setting up the coordinate axes, then place points A, C, B, D, and E on the x-axis. Define the beam's geometry using vectors and create its segments with points and vectors. Use Max and Min functions to determine the beam's boundaries, and define its shape with Piecewise functions. Next, draw the shear force and bending moment diagrams, set the scale with input boxes. Finally, add walls and arrows to mark key points, showing the internal forces under the concentrated moments.
The prompt above was compiled from the original drawing commands and is for reference only.