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BIZOMAT MAKEOS · Mechanics

Force Group Support Check (Resultant, Line of Action, Support Reactions, Uplift)

For several forces acting on a frame, base or beam resting on two supports, what is their resultant, where does it act, and what does each support carry?

Check of a rigid member or base on a pinned support A and a roller support B: the resultant of up to four forces in any direction, its moment about the origin and where its line of action crosses the support line (as the force system tool), then the reactions at A (both directions) and B (perpendicular to the support line), any uplift at either support, and the reactions against the support capacities.

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How this tool works

Inputs, units and where to find them

InputUnitTypical source
Force 1
MagnitudeNSize of force 1 (0 if unused).
Direction from +X°Direction of force 1, anticlockwise from +X (−90 is straight down).
X of a point on its linemmX of any point on the line of force 1.
Y of that pointmmY of that point.
Force 2
MagnitudeNSize of force 2.
Direction from +X°Direction of force 2.
X of a point on its linemmX of a point on force 2.
Y of that pointmmY of that point.
Force 3
MagnitudeNSize of force 3.
Direction from +X°Direction of force 3.
X of a point on its linemmX of a point on force 3.
Y of that pointmmY of that point.
Force 4
MagnitudeNSize of force 4.
Direction from +X°Direction of force 4.
X of a point on its linemmX of a point on force 4.
Y of that pointmmY of that point.
Supports (on the X axis)
Pinned support A at XmmPosition of the pinned support (takes both X and Y loads).
Roller support B at XmmPosition of the roller support (takes Y load only).
Capacity of support ANLoad support A can carry.
Capacity of support BNLoad support B can carry.

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
Illustration: MagnitudeA block with an arrow pushing on it; the arrow is the force Magnitude.F₁Example: 1000 NMagnitude
What it is
The size of the first force acting on the member, without a sign; its direction is set by the angle. Set it to 0 to leave the force out.
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
Enter a positive size and use the angle for direction: a downward load is −90° or 270°, not a negative magnitude.
What this means: Direction from +X
Illustration: Direction from +XTwo lines meeting at a point with an arc between them, marking the angle Direction from +X.θ₁Example: -90°Direction from +X
What it is
The direction in which the first force acts, measured anticlockwise from the positive X axis.
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
Straight down is −90° (or 270°), straight up is 90°, along +X is 0°.
What this means: X of a point on its line
Illustration: X of a point on its lineA bar with a dimension line and arrows from one end to the other, marking the length X of a point on its line.x₁Example: 0 mmX of a point on its line
What it is
The X coordinate of any point on the line of action of the first force, measured from the origin of the X and Y axes.
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 result, so pick the easiest one, such as where the force is applied.
What this means: Y of that point
Illustration: Y of that pointA block with a vertical dimension line from its base to its top, marking the height Y of that point.y₁Example: 0 mmY of that point
What it is
The Y coordinate of the same point on the line of action of the first force, measured from the X axis on which the supports sit.
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
Leave it 0 for a vertical force applied on the support line; it matters for horizontal or sloping forces.

Force 2

What this means: Magnitude
Illustration: MagnitudeA block with an arrow pushing on it; the arrow is the force Magnitude.F₂Example: 1500 NMagnitude
What it is
The size of the second force acting on the member, without a sign; its direction is set by its angle. Set it to 0 to leave it out.
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
Enter a positive size and use the angle for direction.
What this means: Direction from +X
Illustration: Direction from +XTwo lines meeting at a point with an arc between them, marking the angle Direction from +X.θ₂Example: -90°Direction from +X
What it is
The direction in which the second force acts, measured anticlockwise from the positive X axis.
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
Straight down is −90° (or 270°), straight up is 90°.
What this means: X of a point on its line
Illustration: X of a point on its lineA bar with a dimension line and arrows from one end to the other, marking the length X of a point on its line.x₂Example: 400 mmX of a point on its line
What it is
The X coordinate of any point on the line of action of the second force, measured from the 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 and both supports.
What this means: Y of that point
Illustration: Y of that pointA block with a vertical dimension line from its base to its top, marking the height Y of that point.y₂Example: 0 mmY of that point
What it is
The Y coordinate of the same point on the line of action of the second force, measured from the X axis.
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
It only changes the result if the force has a horizontal component.

Force 3

What this means: Magnitude
Illustration: MagnitudeA block with an arrow pushing on it; the arrow is the force Magnitude.F₃Example: 500 NMagnitude
What it is
The size of the third force acting on the member, without a sign; its direction is set by its angle. Set it to 0 to leave it out.
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
Enter a positive size and use the angle for direction.
What this means: Direction from +X
Illustration: Direction from +XTwo lines meeting at a point with an arc between them, marking the angle Direction from +X.θ₃Example: -90°Direction from +X
What it is
The direction in which the third force acts, measured anticlockwise from the positive X axis.
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
Straight down is −90° (or 270°), straight up is 90°.
What this means: X of a point on its line
Illustration: X of a point on its lineA bar with a dimension line and arrows from one end to the other, marking the length X of a point on its line.x₃Example: 1000 mmX of a point on its line
What it is
The X coordinate of any point on the line of action of the third force, measured from the 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 and both supports.
What this means: Y of that point
Illustration: Y of that pointA block with a vertical dimension line from its base to its top, marking the height Y of that point.y₃Example: 0 mmY of that point
What it is
The Y coordinate of the same point on the line of action of the third force, measured from the X axis.
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
It only changes the result if the force has a horizontal component.

Force 4

What this means: Magnitude
Illustration: MagnitudeA block with an arrow pushing on it; the arrow is the force Magnitude.F₄Example: 0 NMagnitude
What it is
The size of the fourth force acting on the member, without a sign; its direction is set by its angle. Set it to 0 to leave it out.
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
Enter a positive size and use the angle for direction.
What this means: Direction from +X
Illustration: Direction from +XTwo lines meeting at a point with an arc between them, marking the angle Direction from +X.θ₄Example: -90°Direction from +X
What it is
The direction in which the fourth force acts, measured anticlockwise from the positive X axis.
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
Straight down is −90° (or 270°), straight up is 90°.
What this means: X of a point on its line
Illustration: X of a point on its lineA bar with a dimension line and arrows from one end to the other, marking the length X of a point on its line.x₄Example: 0 mmX of a point on its line
What it is
The X coordinate of any point on the line of action of the fourth force, measured from the 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 origin for every force and both supports.
What this means: Y of that point
Illustration: Y of that pointA block with a vertical dimension line from its base to its top, marking the height Y of that point.y₄Example: 0 mmY of that point
What it is
The Y coordinate of the same point on the line of action of the fourth force, measured from the X axis.
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
It only changes the result if the force has a horizontal component.

Supports (on the X axis)

What this means: Pinned support A at X
Illustration: Pinned support A at XA bar with a dimension line and arrows from one end to the other, marking the length Pinned support A at X.x_AExample: 0 mmPinned support A at X
What it is
The X position of the pinned support A on the X axis. A pin resists both horizontal and vertical force, so it takes all the horizontal load.
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
Measure from the same origin as the force points; the two supports must not be at the same X.
What this means: Roller support B at X
Illustration: Roller support B at XA bar with a dimension line and arrows from one end to the other, marking the length Roller support B at X.x_BExample: 1000 mmRoller support B at X
What it is
The X position of the roller support B on the X axis. A roller resists force only at right angles to the support line.
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
Measure from the same origin as the force points; the two supports must not be at the same X.
What this means: Capacity of support A
Illustration: Capacity of support AA block with an arrow pushing on it; the arrow is the force Capacity of support A.cap_AExample: 2500 NCapacity of support A
What it is
The largest force support A can carry, compared with the combined (resultant) reaction at the pin.
Units and format
Enter a number in N: newtons. The form accepts 0.000001 N or more. Required: the tool does not calculate until every input is valid.
Example
The form opens with 2500 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Get it right
Use the rated or allowable load of the pin, bracket or bearing, from its maker or your own design check.
What this means: Capacity of support B
Illustration: Capacity of support BA block with an arrow pushing on it; the arrow is the force Capacity of support B.cap_BExample: 2000 NCapacity of support B
What it is
The largest force support B can carry at right angles to the support line, compared with the size of its reaction.
Units and format
Enter a number in N: newtons. The form accepts 0.000001 N or more. Required: the tool does not calculate until every input is valid.
Example
The form opens with 2000 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Get it right
The tool flags uplift separately; this capacity check uses the size of the reaction either way.

Method and formulas

  1. R_x = Σ F cos θ; R_y = Σ F sin θ; moment about the origin M = Σ (x F_y − y F_x); line of action crosses the X axis at M ÷ R_y (as the force system tool).
  2. Roller B: R_By = −(M − x_A R_y) ÷ (x_B − x_A); pin A: R_Ay = −R_y − R_By, R_Ax = −R_x.
  3. A negative (downward) reaction means uplift: that support must be held down.
  4. Example: resultant 3,000 N acting at X 366.67 mm; A carries 1,900 N and B 1,100 N.

Limitations

  • Rigid member, forces in one plane, supports on the X axis.
  • Static loads; dynamic and impact loads need factors.
  • Example values are illustrations. Replace them with values for your actual task.

Typical uses

  • Find the support loads under a machine with several forces
  • Check a base for uplift at one support
  • Check bracket or beam supports for inclined loads

Who it is for

  • Design engineers
  • Structural detailers
  • Engineering students

What you get

  • Resultant, moment and line of action
  • Reactions at A and B
  • Uplift and capacity checks

Interpreting the result

  • When the line of action of the resultant falls outside the supports, the far support lifts off unless it is held down.

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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