Wrench: For 3D systems, any system of forces and couples can be reduced to a resultant force $\vec{F}_R$ and a resultant couple moment $\vec{M}_R$ acting at a specific point. If $\vec{F}_R$ and $\vec{M}_R$ are parallel, it's a wrench.

5. Equilibrium of a Rigid Body

5.1. Conditions for Equilibrium

$\sum \vec{F} = 0 \implies \sum F_x = 0, \sum F_y = 0, \sum F_z = 0$
$\sum \vec{M}_O = 0 \implies \sum M_x = 0, \sum M_y = 0, \sum M_z = 0$ (about any point O)

5.2. Supports and Reactions

Support Type	2D Reactions	3D Reactions
Cable/Rope	1 force (tension)	1 force (tension)
Smooth Surface	1 normal force	1 normal force
Roller	1 normal force	1 normal force
Pin (hinge)	2 forces ($F_x, F_y$)	3 forces ($F_x, F_y, F_z$)
Fixed (cantilever)	2 forces ($F_x, F_y$), 1 moment ($M_z$)	3 forces ($F_x, F_y, F_z$), 3 moments ($M_x, M_y, M_z$)

5.3. Two- and Three-Force Members

Two-Force Member: A member subjected to forces only at two points. The forces must be equal, opposite, and collinear along the line connecting the two points.
Three-Force Member: A member subjected to forces only at three points. The forces must be concurrent or parallel.

6. Trusses, Frames, and Machines

6.1. Trusses

Assumptions: Members are connected by pins, loads applied at joints, members are two-force members.
Method of Joints: Apply particle equilibrium ($\sum F_x=0, \sum F_y=0$) at each joint.
Method of Sections: Cut through members to isolate a section, then apply rigid body equilibrium ($\sum F_x=0, \sum F_y=0, \sum M_O=0$).
Zero-Force Members:
1. If only two non-collinear members connect at a joint and no external load or reaction acts at that joint, both members are zero-force members.
2. If three members connect at a joint, two of which are collinear, and no external load or reaction acts at that joint, the third member is a zero-force member.

6.2. Frames and Machines

Components are generally multi-force members.
Disassemble the frame/machine into its component parts.
Draw FBDs for each component and for the entire structure.
Apply rigid body equilibrium equations to each FBD. Remember Newton's Third Law for internal forces.

7. Internal Forces

Procedure:
1. Determine external reactions.
2. Pass an imaginary section through the point of interest.
3. Draw FBD of either segment.
4. Apply equilibrium equations to find internal normal force (N), shear force (V), and bending moment (M).
Sign Convention (for positive internal forces):
- Normal Force (N): Tension is positive (pulling away from section).
- Shear Force (V): Causes clockwise rotation of the element (down on right face, up on left face).
- Bending Moment (M): Causes compression in the top fibers and tension in the bottom fibers (smiles).
Shear and Moment Diagrams:
- $\fr