BIZOMAT MAKEOS · Mechanics
Resultant of Non-Concurrent and Parallel Forces
What single force and line of action replaces these forces acting at different points?
Resultant magnitude, direction, moment about the origin and line of action of up to four coplanar forces acting at different points, including parallel forces.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Force 1 | ||
| Magnitude | N | Magnitude of force 1 (0 if not used). |
| Direction from +X | ° | Direction of force 1 anticlockwise from +X (−90° is straight down). |
| X of a point on its line | mm | X coordinate of any point on force 1’s line of action. |
| Y of that point | mm | Y coordinate of that point. |
| Force 2 | ||
| Magnitude | N | Magnitude of force 2 (0 if not used). |
| Direction from +X | ° | Direction of force 2 anticlockwise from +X (−90° is straight down). |
| X of a point on its line | mm | X coordinate of any point on force 2’s line of action. |
| Y of that point | mm | Y coordinate of that point. |
| Force 3 | ||
| Magnitude | N | Magnitude of force 3 (0 if not used). |
| Direction from +X | ° | Direction of force 3 anticlockwise from +X (−90° is straight down). |
| X of a point on its line | mm | X coordinate of any point on force 3’s line of action. |
| Y of that point | mm | Y coordinate of that point. |
| Force 4 | ||
| Magnitude | N | Magnitude of force 4 (0 if not used). |
| Direction from +X | ° | Direction of force 4 anticlockwise from +X (−90° is straight down). |
| X of a point on its line | mm | X coordinate of any point on force 4’s line of action. |
| Y of that point | mm | Y coordinate of that point. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
Force 1
What this means: Magnitude
- What it is
- The size of force 1, one of up to four forces acting on the same flat body. Its direction is given separately, so the magnitude is always positive.
- Units and format
- Enter a number in N: newtons. The form accepts 0 N or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1000 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Set 0 to leave the force out. Reverse a force by changing its direction by 180°, not by its sign.
What this means: Direction from +X
- What it is
- The direction force 1 points, as an angle measured anticlockwise from the +X axis; straight down is −90° or 270°.
- Units and format
- Enter a number in °: degrees of angle. The form accepts -360° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -90° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Give the direction the force acts in, not the direction of the member it acts along.
What this means: X of a point on its line
- What it is
- The X coordinate of any point on force 1’s line of action, measured from the origin you choose for the whole system.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Any point on the line gives the same moment; the application point is usually easiest.
What this means: Y of that point
- What it is
- The Y coordinate of the same point on force 1’s line of action, measured from the same origin.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the same point as for the X coordinate.
Force 2
What this means: Magnitude
- What it is
- The size of force 2, acting on the same flat body as force 1. Its direction is given separately, so the magnitude is always positive.
- Units and format
- Enter a number in N: newtons. The form accepts 0 N or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1500 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Set 0 to leave the force out.
What this means: Direction from +X
- What it is
- The direction force 2 points, as an angle measured anticlockwise from the +X axis; straight down is −90° or 270°.
- Units and format
- Enter a number in °: degrees of angle. The form accepts -360° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -90° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Give the direction the force acts in, measured from the same +X axis as the others.
What this means: X of a point on its line
- What it is
- The X coordinate of any point on force 2’s line of action, measured from the common origin.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 400 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the same origin for every force.
What this means: Y of that point
- What it is
- The Y coordinate of the same point on force 2’s line of action, measured from the common origin.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the same point as for the X coordinate.
Force 3
What this means: Magnitude
- What it is
- The size of force 3, acting on the same flat body as the others. Its direction is given separately, so the magnitude is always positive.
- Units and format
- Enter a number in N: newtons. The form accepts 0 N or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 500 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Set 0 to leave the force out.
What this means: Direction from +X
- What it is
- The direction force 3 points, as an angle measured anticlockwise from the +X axis; straight down is −90° or 270°.
- Units and format
- Enter a number in °: degrees of angle. The form accepts -360° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -90° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Give the direction the force acts in, measured from the same +X axis as the others.
What this means: X of a point on its line
- What it is
- The X coordinate of any point on force 3’s line of action, measured from the common origin.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1000 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the same origin for every force.
What this means: Y of that point
- What it is
- The Y coordinate of the same point on force 3’s line of action, measured from the common origin.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the same point as for the X coordinate.
Force 4
What this means: Magnitude
- What it is
- The size of force 4, the last of the forces acting on the same flat body. Its direction is given separately, so the magnitude is always positive.
- Units and format
- Enter a number in N: newtons. The form accepts 0 N or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Leave it at 0 when only three forces act.
What this means: Direction from +X
- What it is
- The direction force 4 points, as an angle measured anticlockwise from the +X axis; straight down is −90° or 270°.
- Units and format
- Enter a number in °: degrees of angle. The form accepts -360° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -90° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Ignored while the magnitude of force 4 is 0.
What this means: X of a point on its line
- What it is
- The X coordinate of any point on force 4’s line of action, measured from the common origin.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Ignored while the magnitude of force 4 is 0.
What this means: Y of that point
- What it is
- The Y coordinate of the same point on force 4’s line of action, measured from the common origin.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Ignored while the magnitude of force 4 is 0.
Method and formulas
- Rx = Σ F cos θ; Ry = Σ F sin θ.
- Moment about the origin M = Σ (x Fy − y Fx).
- Line of action crosses the X axis at x₀ = M ÷ Ry (or the Y axis at −M ÷ Rx); if R = 0 the system reduces to a couple M.
- Example: the default downward loads total 3,000 N acting at x = 366.67 mm.
Limitations
- Coplanar forces only; 3D systems need a wrench reduction.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find where the resultant of several loads acts on a beam or frame
- Find the centre of loads on a foundation
- Check equilibrium of a free body
Who it is for
- Design engineers
- Structural engineers
- Engineering students
What you get
- Resultant components and magnitude
- Moment about the origin
- Line of action of the resultant
- Couple when the resultant is zero
Interpreting the result
- For parallel loads, x₀ is the load centre: put a support or lifting point there to avoid tipping.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 153–163. Topic reference; no source tables or recommended settings are reproduced.
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Open the Resultant of Non-Concurrent and Parallel Forces calculator