Summarize for me the formula of calculating the forces

For a body in static equilibrium, the sum of all external forces and moments must be zero.

$$\sum F_x = 0$$ $$\sum F_y = 0$$ $$\sum M = 0$$

Where:

• $\sum F_x$ is the sum of all forces in the horizontal direction.
• $\sum F_y$ is the sum of all forces in the vertical direction.
• $\sum M$ is the sum of all moments (torques) about any arbitrary point.

This method applies the equilibrium conditions ($\sum F_x = 0$ and $\sum F_y = 0$) to each joint of the truss.

• Resolve Forces: Each force acting on a joint (member forces, external loads, reactions) is resolved into its horizontal and vertical components. If a force $F$ acts at an angle $\theta$ with the horizontal: $$F_x = F \cos(\theta)$$ $$F_y = F \sin(\theta)$$
• Apply Equilibrium: For each joint, set the sum of horizontal components to zero and the sum of vertical components to zero. $$\sum F_{x, \text{joint}} = 0$$ $$\sum F_{y, \text{joint}} = 0$$
• Sign Convention: Consistently define positive directions (e.g., right and up are positive). Assume unknown member forces are in tension (pulling away from the joint); a negative result indicates compression.

This method involves cutting the truss into two sections and applying the three equilibrium conditions ($\sum F_x = 0$, $\sum F_y = 0$, $\sum M = 0$) to one of the sections. This is useful for finding forces in specific members without analyzing every joint.

These formulas allow you to set up a system of equations that can be solved to find unknown forces (reactions and internal member forces).