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4 Equilibrium of Bodies (3D)
In many ways 3D can be viewed as an extension/generalization of 2D.
Noteworthy, however, that some formulas don’t work anymore (recall
the elimination of couple from the force-couple system).
In 3D 2 vector EE’s (force & moment) reduce to a system of 6 scalar
EE’s:
Forces:
∑ 𝐹𝑥 = 0, ∑ 𝐹𝑦 = 0, ∑ 𝐹𝑧 = 0
Moments:
∑ 𝑀𝑥 = 0, ∑ 𝑀𝑦 = 0, ∑ 𝑀𝑧 = 0
Note: three moments about any ONE arbitrary point
Cathegories of 3D Equilibrium
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Types of Supports and Reactions
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Equilibrium Equations
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Similarly to 2D, EE’s are necessary and sufficient for equilibrium of the
body.
Basic approach for solving equilibrium problems in 3D is the same as in
2D:
1.
Construct FBD with all external forces shown/included
2.
Apply EE’s
To properly draw a correct FBDs one has to completely understand ways
forces can be applied. Very important.
Similarly to 2D there are simplified, special cases. One has to learn
which equations are satisfied by default and which should be used for
solving. See the different cathegories of equilibrium in the table.
Solving Equilibrium Equations
Maximum number of independent EE’s in 3D is 6. Hence, the maximum
number of unknowns is also 6. In special cases this number is lower.
If number of unknowns exceeds number of EE’s, then the problem is
statically indeterminate.
Use of formal and straightforward computation is more frequently used
in 3D, compared to 2D. Nonetheless, at times solutions can be
significantly simplified by taking a moment equilibrium equation about
a point with most unknown forces applied.
Similarly, moment can be taken about an axis, collinear with most
unknown forces.
Considering computational complexity for obtaining moment of a force
in 3D, generally it can be advantageous to select the point with largest
total number of forces passing through it (known and unknown).
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Note: three moment axes should not necessarily be perpendicular (see
table for cathegories of equilibrium). Any 3 non-parallel and noncoplanar axes will do.
Example 1
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Example 2
Example 3
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Example 4
Example 5