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Equilibrium of Non-concurrent Force Systems

  1. Translational Equilibrium: The sum of all external forces in every direction must be zero
$$\sum F_x = 0,\quad \sum F_y = 0,\quad \sum F_z = 0 \quad \text{(for 3D systems)}$$
  1. Rotational Equilibrium: The sum of all moments about any point must also be zero
$$\sum M = 0$$

In engineering mechanics, a non-concurrent force system refers to a group of forces whose lines of action do not intersect at a single point. These forces may act on different parts of a rigid body and can include forces and moments (torques). To achieve equilibrium in such systems, a body must satisfy both conditions above.


    General Resultant of Non-concurrent Force Systems:
$$R = \sqrt{(\sum F_x)^2 + (\sum F_y)^2 + (\sum F_z)^2}$$


    For Forces Lying on a 2D Plane Only
$$\tan(\theta) = \frac{\sum F_y}{\sum F_x}$$
    Directional Cosines:
\(\cos(\theta_x) = \frac{F_x}{F}\)

\(\cos(\theta_y) = \frac{F_y}{F}\)

\(\cos(\theta_z) = \frac{F_z}{F}\)
    The location of the resultant force is determined by Varignon's Theorem.
$$ R(d_x) = F_1(x_1) + F_2(x_2) + \cdots + F_n(x_n) $$ $$ R(d_y) = F_1(y_1) + F_2(y_2) + \cdots + F_n(y_n) $$
Concept Concept Concept Concept Concept Concept Concept Concept Concept

Problem 1:

a. Calculate the location of the resultant from point D.
b. Prove that the resultant force intersects point E (as indicated below)
c. Determine the distance dx and dy from point D to the resultant force and indicate the location relative to D.
d. Determine the distance from point B to the resultant force.
e. Resolve the system into a resultant-couple system.

Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 1: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 1: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 1: – Diagram

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Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 1: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 1: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 1: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 1: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 1: – Diagram
    Below is a scaled drawing of the figure (modeled using AutoCad).
Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 1: – Diagram
    We can obtain the moment caused by the resultant force about point D so that we can transfer the resultant so that it intersects point D. The moment (couple) will be applied at point D for clarity.
Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 1: – Diagram
    Finally, we have this resultant-couple system that replaces the system above.
Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 1: – Diagram

Problem 2:

For the beam shown, beam ABC is supporting beam CD.
Which of the following most nearly gives the reaction at C in kN.
A. 250
B. 150
C. 100
D. 200

Which of the following most nearly gives the reaction at D in kN.
A. 250
B. 150
C. 100
D. 200

Which of the following most nearly gives the value of "x" if the reaction at B is 380 N.
A. 1
B. 1.5
C. 2
D. 3

Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 2: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 2: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 2: – Diagram

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Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 2: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 2: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 2: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 2: – Diagram

Problem 3:

The uniform 30kg bar with end rollers is supported by the horizontal and vertical surfaces and the wire AC.

a. Determine the tension in the wire.
b. Determine the reaction at A.
c. Determine the moment at B.

Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 3: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 3: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 3: – Diagram

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Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 3: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 3: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 3: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 3: – Diagram

Problem 4:

The uniform rod weighs 420N and has its center of gravity at G as shown.

a. What is the reaction at B?
b. What is the tension in the cable?
c. What is the reaction at A?

Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 4: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 4: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 4: – Diagram

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Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 4: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 4: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 4: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 4: – Diagram

Problem 5:

A three-legged stool is subjected to the load P as shown. If P=200N, compute the following:

a. Reaction under leg A
b. Reaction under leg B
c. Reaction under leg C

Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 5: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 5: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 5: – Diagram

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Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 5: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 5: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 5: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 5: – Diagram

Problem 6:

For the figure shown, compute the reaction at the hinged support at A if the weight of the cylinder is 1000N.

Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 6: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 6: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 6: – Diagram

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Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 6: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 6: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 6: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 6: – Diagram

Problem 7:

For the figure shown, calculate the angle of tilt, θ, so that the contact force at B will be one-half that of A for the smooth cylinder.

Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 7: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 7: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 7: – Diagram

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Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 7: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 7: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 7: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 7: – Diagram

Problem 8:

Refer to the image shown:

Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 8: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 8: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 8: – Diagram

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Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 8: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 8: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 8: – Diagram Equilibrium of Non-Concurrent Force Systems | Statics of Rigid Bodies – Problem 8: – Diagram
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