Training Engineering Math Statics — Forces, Moments, and Beams
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Statics — Forces, Moments, and Beams

24 min Engineering Math

Statics is the study of rigid bodies in equilibrium — structures and machines that are not accelerating. The two conditions for static equilibrium require that both the net force and the net moment (torque) acting on a body equal zero. By drawing free-body diagrams and summing forces and moments, you can determine unknown support reactions, internal forces in truss members, and bending moments in beams. These skills are essential for civil, mechanical, and aerospace engineers who must ensure that bridges, buildings, and aircraft frames can safely support their loads.

Statics

For a structure to be in equilibrium, all forces and moments must balance — a system of linear equations.

Equilibrium Conditions

$$\Sigma F_x = 0, \quad \Sigma F_y = 0, \quad \Sigma M = 0$$

Sum of forces in every direction is zero, and sum of moments about any point is zero.

Moment (Torque)

$$M = F \times d$$

Force times perpendicular distance from the pivot.

Example 1

A 6 m beam is supported at both ends. A 600 N load sits 2 m from the left end. Find the support reactions.

  1. Let $R_L$ and $R_R$ be the support forces.
  2. $\Sigma F_y = 0$:
  3. $R_L + R_R = 600$.
  4. $\Sigma M_L = 0$: $600(2) - R_R(6) = 0$ → $R_R = 200$ N.
  5. $R_L = 600 - 200 = 400$ N.
Example 2

A seesaw: 40 kg child sits 2 m from pivot. Where must a 30 kg child sit to balance?

  1. $40(2) = 30(d)$ → $d = 80/30 = 2.67$ m from pivot.
Example 3

Two forces act on a door: 20 N at 0.8 m from the hinge and 10 N at 0.4 m, both perpendicular. Net moment?

  1. If same direction:
  2. $20(0.8) + 10(0.4) = 16 + 4 = 20$ N·m.

Practice Problems

1. A 4 m beam with 800 N load at center. Find each support reaction.
2. Moment of 50 N force applied 3 m from pivot?
3. A beam: 500 N at 1 m from left, 300 N at 3 m from left, beam is 5 m. Find $R_L$.
4. Seesaw: 60 kg at 1.5 m. Where does 45 kg sit to balance?
5. What does $\Sigma M = 0$ mean physically?
6. If a beam has two equal loads placed symmetrically, what are the reactions?
Show Answer Key

1. $R_L = R_R = 400$ N (symmetric loading)

2. $M = 150$ N·m

3. $\Sigma M_R = 0$: $R_L(5) = 500(4) + 300(2) = 2600$ → $R_L = 520$ N

4. $d = 60(1.5)/45 = 2$ m

5. No rotational tendency — the structure is in rotational equilibrium.

6. Each reaction is half the total load.

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