CES Mechanics I Lecture 9

Topics:

• General (3D) support reactions: 5.2

  The range of common supports in 3D systems increases significantly compared to 2D systems, with number of support reactions ranging from 1 (cable, frictionless surface) to 6 (cantilever support). See supplementary material below.

• General (3D) equilibrium equations: 5.2
  1. General system:

     \sum \mathbf{F} = \mathbf{0}

     \sum \mathbf{M}_O = \mathbf{0},

     (6 equations).

  2. Concurrent force system (all forces intersecting at a point):

     \sum \mathbf{F} = \mathbf{0},

     (3 equations).

  3. Parallel force system (align z axis with the forces):

     \sum F_z = 0,

     \sum M_x = 0,

     \sum M_y = 0,

     (3 equations).

  4. All forces intersecting a line (align, e.g., y axis with the line):

     \sum \mathbf{F} = \mathbf{0},

     \sum M_x = 0,

     \sum M_z = 0,

     (5 equations, see example that follows).

• Example: 3D equilibrium (see PDF link below)

  

  The horizontal boom OC, which is supported by a ball-and-socket joint and two cables, carries the vertical force P = 8000 \mathrm{N}. Calculate T_{AD} and T_{CE}, the tensions in the cables, and the components of the force exerted on the boom by the joint at O (the weight of the boom is negligible).

  The cable forces \mathbf{T}_{AD} and \mathbf{T}_{CE} can be found from:

  \sum \mathbf{M}_O = \mathbf{0} \Rightarrow \mathbf{r}_{OA} \times \mathbf{T}_{AD} + \mathbf{r}_{OC} \times \mathbf{T}_{CE} + \mathbf{r}_{OB} \times \mathbf{P} = \mathbf{0}

  with \mathbf{r}_{OA} = 3 \mathbf{j} \; \mathrm{m}, \mathbf{r}_{OC} = 6 \mathbf{j} \; \mathrm{m}, \mathbf{r}_{OB} = 5 \mathbf{j} \; \mathrm{m}

  \mathbf{T}_{AD} = T_{AD} \left( \frac{2.5 \mathbf{i} - 3 \mathbf{j} + 3 \mathbf{k}}{\sqrt{2.5^2 + 3^2 + 3^2}} \right)

  \mathbf{T}_{CE} = T_{CE} \left( \frac{-3.5 \mathbf{i} - 6 \mathbf{j} + 3 \mathbf{k}}{\sqrt{3.5^2 + 6^2 + 3^2}} \right)

  \mathbf{P} = - 8000 \mathbf{k} \; \mathrm{N}

  This gives rise to 3 equations, 2 of which are non-trivial and give T_{AD} = 12770 \; \mathrm{N} and T_{CE} = 7010 \; \mathrm{N}.

Key techniques:

• Identifying reactions at 3D supports.
• Identifying and solving general (3D) equilibrium equation sets.

Additional material (accessible from RWTH domains only):

• Free-body diagram support table (3D)

  One may wonder why the hinges/bearings (f) and (g) do not provide reaction couples along y and z axes. Although it makes sense that they would, some bearings are not designed to do this (ball bearings), while others are designed for such loads (journal bearings). The table shows the first kind.

• Boom equilibrium example (3D)