BIZOMAT MAKEOS · Mechanics
Three-Point Support Loads (Reaction at Each Support, Uplift, Capacity)
For a machine, plate or frame resting on three supports, how much load does each support carry from the loads on it, and does any support lift off?
Load share of a rigid body on three supports from up to three vertical loads: the three equilibrium equations (total load, and moments about the X and Y axes) solved by inverting the 3 × 3 matrix of support positions (as the 3 × 3 matrix tool), giving each support reaction, then the check that every reaction is upward (the combined load lies inside the support triangle) and within its capacity.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Support 1 | ||
| X | mm | X of support 1. |
| Y | mm | Y of support 1. |
| Support 2 | ||
| X | mm | X of support 2. |
| Y | mm | Y of support 2. |
| Support 3 | ||
| X | mm | X of support 3. |
| Y | mm | Y of support 3. |
| Load 1 | ||
| Load (0 = unused) | N | Load 1 (weight, downward). |
| X | mm | X of load 1. |
| Y | mm | Y of load 1. |
| Load 2 | ||
| Load (0 = unused) | N | Load 2. |
| X | mm | X of load 2. |
| Y | mm | Y of load 2. |
| Load 3 | ||
| Load (0 = unused) | N | Load 3. |
| X | mm | X of load 3. |
| Y | mm | Y of load 3. |
| Capacity | ||
| Capacity of each support | N | Load each support (foot, mount, jack) 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.
Support 1
What this means: X
- What it is
- The X coordinate of support 1 (a foot, mount or jack) in plan, measured from an origin you choose.
- 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 one origin and one pair of axes for all supports and loads.
What this means: Y
- What it is
- The Y coordinate of support 1 in plan, at right angles to X, 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
- Keep the sign: positions on the other side of the origin are negative.
Support 2
What this means: X
- What it is
- The X coordinate of support 2 in plan, measured from the same origin as support 1.
- 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
- The three supports must not lie on one straight line.
What this means: Y
- What it is
- The Y coordinate of support 2 in plan, 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
- Keep the sign: positions on the other side of the origin are negative.
Support 3
What this means: X
- What it is
- The X coordinate of support 3 in plan, 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 500 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- The three supports must not lie on one straight line.
What this means: Y
- What it is
- The Y coordinate of support 3 in plan, 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 800 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Keep the sign: positions on the other side of the origin are negative.
Load 1
What this means: Load (0 = unused)
- What it is
- The first vertical load on the body, such as its own weight acting at its centre of gravity, or a weight of equipment on it. 0 leaves 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 5000 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- All three reactions change with it; their total equals the sum of the loads.
- Get it right
- Enter it in newtons: multiply a mass in kg by 9.81.
What this means: X
- What it is
- The X coordinate in plan of the point where load 1 acts, such as the body’s centre of gravity.
- 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 500 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 and axes as the supports.
What this means: Y
- What it is
- The Y coordinate in plan of the point where load 1 acts.
- 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 300 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 and axes as the supports.
Load 2
What this means: Load (0 = unused)
- What it is
- A second vertical load on the body, such as equipment or contents at its own centre of gravity. 0 leaves 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 2000 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter it in newtons: multiply a mass in kg by 9.81.
What this means: X
- What it is
- The X coordinate in plan of the point where load 2 acts.
- 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 300 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 and axes as the supports.
What this means: Y
- What it is
- The Y coordinate in plan of the point where load 2 acts.
- 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 200 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 and axes as the supports.
Load 3
What this means: Load (0 = unused)
- What it is
- A third vertical load on the body. 0 leaves 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 it in newtons: multiply a mass in kg by 9.81.
What this means: X
- What it is
- The X coordinate in plan of the point where load 3 acts.
- 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 load 3 is 0.
What this means: Y
- What it is
- The Y coordinate in plan of the point where load 3 acts.
- 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 load 3 is 0.
Capacity
What this means: Capacity of each support
- What it is
- The largest load any one support can carry, compared with the largest of the three reactions. All supports are taken to have the same capacity.
- 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 4000 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 load of the foot, mount or jack from its maker; if they differ, enter the weakest.
Method and formulas
- Equilibrium: R₁ + R₂ + R₃ = ΣW; Σ R x = Σ W x; Σ R y = Σ W y.
- Matrix of [1 1 1; x₁ x₂ x₃; y₁ y₂ y₃] inverted (as the 3 × 3 matrix tool); reactions = inverse × [ΣW, ΣWx, ΣWy].
- A negative reaction means the load lies outside the support triangle and that support lifts.
- Example: supports carry 2,712.5, 1,912.5 and 2,375 N.
Limitations
- Rigid body and supports at one level; three supports always share load statically, four or more do not.
- Vertical loads only.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the load on each foot of a three-legged machine or table
- Place a heavy item on a three-point base
- Check a kinematic mount or levelling feet
Who it is for
- Design engineers
- Riggers
- Machine installers
What you get
- Total load and where it acts
- Reaction at each support
- Uplift and capacity checks
Interpreting the result
- With three supports the share depends only on where the loads are, not on stiffness, which is why precision instruments stand on three feet.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 122–125, 153–163. Topic reference; no source tables or recommended settings are reproduced.
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Open the Three-Point Support Loads (Reaction at Each Support, Uplift, Capacity) calculator