1. ME 201 Engineering Mechanics: Statics
Chapter 5 – Part D
Equilibrium in Three Dimensions
5.5 Free-Body Diagrams
5.6 Equations of Equilibrium
5.7 Constraints and Statical Determinancy

2. Constraints for a Rigid Body
Statically Determinate – reactive forces can be determined from equations of equilibrium
Statically Indeterminate – more unknown forces than equations of equilibrium. Requires use of deformation conditions to solve (Strength of Materials).
Redundant Constraints – more supports than necessary to hold body in equilibrium
Improper Constraints - # of unknowns may be okay, but improperly constrained – may cause instability

3. Equilibrium in 3-Dimensions
Support Reactions
A force is developed by a support that restricts the translation of the attached member
A couple is developed when rotation of the attached member is prevented
∑F = 0
∑M = 0
∑Fx = 0
∑Fy = 0
∑Fz = 0
∑Mx = 0
∑My = 0
∑Mz = 0

4. Common Support Reactions (See also Text pages 238-239)2D 3D
Simple or Roller Support
1 unknown: 1F, 0M
Smooth Surface Support
Roller Support
Pin or Hinge Support
2 unknowns: 2F, 0M
Ball & Socket Support
3 unknowns: 3F, 0M
Fixed Support
3 unknowns: 2F, 1M
6 unknowns: 3F, 3M

5. Example Problem Given: A-ball joint BC, BD-cables F=1 kN Find:
A, TBC, TBD z x y 6m A C B D 3m F
(0,0,6)
(-3,6,0)
(6,0,0)
How many unknown?
5 unknowns: Ax, Ay, Az, TBC, TBD

6. Example Problem Solution
Given:
A-ball joint
BC, BD-cables
F=1 kN
Find:
A, TBC, TBD
Solution:
1-FBD
2-Force (vectors) z x y 6m Ax C B D 3m
1 kN
(0,0,6)
(-3,6,0)
(6,0,0) Ay Az
TBC
TBD z x y 6m A C B D 3m F
(0,0,6)
(-3,6,0)
(6,0,0)

7. Example Problem Solution
1-FBD
2-Force (vectors)
3-Eqn of Equil z x y 6m Ax C B D 3m
1 kN
(0,0,6)
(-3,6,0)
(6,0,0) Ay Az
TBC
TBD
5 unknowns
3 from ∑F , 2 from ∑M

8. Example Problem Solution
1-FBD
2-Force (vectors)
3-Eqn of Equil z x y 6m Ax C B D 3m
1 kN
(0,0,6)
(-3,6,0)
(6,0,0) Ay Az
TBC
TBD
5 unknowns
3 from ∑F , 2 from ∑M

9. Example Problem Solution
1-FBD
2-Force (vectors)
3-Eqn of Equil z x y 6m Ax C B D 3m
1 kN
(0,0,6)
(-3,6,0)
(6,0,0) Ay Az
TBC
TBD
Substituting TBD and TBC into ∑F equations: