KEVOS® MAKEOS · Strength
Beam of Uniform Strength: Safe Load, Size and Outline
For a cantilever or simply supported beam shaped to uniform strength (tapered depth or width), what load can it carry at the safe stress, what depth or width does it need for my load, and what is its depth or width at a point along the span?
Beams of uniform strength throughout their length by the handbook’s tables: P = S b h² ÷ 6l for a cantilever with an end load (parabolic depth or triangular width), P = S b h² ÷ 3l for a cantilever with a uniformly distributed load (triangular depth or parabolic width), P = S b h² l ÷ 6ac for a simply supported beam with a load at any point (P = 2 S b h² ÷ 3l at the centre) and P = 4 S b h² ÷ 3l for a uniformly distributed load, solved for the load, the depth or the width, with the section at a chosen point from the outline the table gives.
Open the calculatorRuns in your browser. No account needed.
How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Beam and outline | choice | Support, load and outline as in the handbook’s Tables 3a and 3b. |
| Calculate | choice | Whether to find the safe load or the depth or width for your load. |
| Safe stress S | MPa | Safe bending stress of the material for your design. |
| Width b at the largest section | mm | Width of the beam at its largest section. |
| Depth h at the largest section | mm | Depth of the beam at its largest section. |
| Length l (cantilever) or span (supported) | mm | Length of a cantilever, or span between supports. |
| Load point from the left support a (load at any point) | mm | Distance of a concentrated load from the left support. |
| Load P (total, for a distributed load) | kN | Load the beam must carry; the whole load for a distributed load. |
| Point for the section (from the loaded end, or the left support) | mm | Point along the beam at which to give the section. |
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.
What this means: Beam and outline
- What it is
- Supports, load and tapered outline of the beam from the handbook tables.
- Units and format
- Choose one of 9 options from the list: Cantilever, end load: uniform width, depth parabolic; Cantilever, end load: uniform depth, width triangular; Cantilever, end load: approximately uniform strength, not to a point; Cantilever, distributed load: uniform width, depth triangular; Cantilever, distributed load: uniform depth, width two parabolas; Supported both ends, load at any point: uniform depth, width triangles; Supported both ends, load at any point: uniform width, depth parabolas; Supported both ends, distributed load: uniform depth, width parabolas; Supported both ends, distributed load: uniform width, depth half ellipse.
- Example
- The form opens with “Cantilever, end load: uniform width, depth parabolic” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Sets the formula constant and the outline at a point.
- Get it right
- Pick the row whose sketch matches the support, the load and which dimension tapers.
What this means: Calculate
- What it is
- Whether to find the safe load, the depth needed or the width needed.
- Units and format
- Choose one of 3 options from the list: Safe load for this section; Depth h needed for the load; Width b needed for the load.
- Example
- The form opens with “Safe load for this section” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Chooses which of P, h and b is worked out.
- Get it right
- Size a new beam with depth or width; check an existing one with the safe load.
What this means: Safe stress S
- What it is
- Bending stress you allow in the beam material for the duty.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 5000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 100 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The safe load rises in proportion.
- Get it right
- Take it from your design code or the yield strength over your factor of safety.
What this means: Width b at the largest section
- What it is
- Breadth of the beam at its largest section, across the load.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The safe load rises in proportion.
- Get it right
- For a beam tapered in width this is the width at the root or the load point.
What this means: Depth h at the largest section
- What it is
- Height of the beam at its largest section, in the direction of the load.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 100 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The safe load rises with its square.
- Get it right
- For a beam tapered in depth this is the depth at the root or the load point.
What this means: Length l (cantilever) or span (supported)
- What it is
- Length of a cantilever, or distance between the supports.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 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.
- Effect on the result
- A longer beam carries less load.
- Get it right
- Measure from the fixed end to the load, or between the support centres.
What this means: Load point from the left support a (load at any…
- What it is
- Distance from the left support to a concentrated load.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 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.
- Effect on the result
- A load near the centre gives the lowest safe load.
- Get it right
- Only used for a load at any point; the other distance is the span minus this.
What this means: Load P (total, for a distributed load)
- What it is
- Load the beam must carry; for a distributed load, the whole load.
- Units and format
- Enter a number in kN: kilonewtons (1 kN = 1000 N). The form accepts 0.000001 kN to 1000000 kN. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5 kN as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Used to size the beam and compared with the safe load.
- Get it right
- Include the weight of anything the beam supports and any shock allowance.
What this means: Point for the section (from the loaded end, or…
- What it is
- Position along the beam at which you want its depth or width.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 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.
- Effect on the result
- Gives the section from the tapered outline at that point.
- Get it right
- Measure from the loaded free end of a cantilever, or from the left support.
Method and formulas
- Cantilever, end load: P = S b h² ÷ 6l. Cantilever, uniformly distributed load: P = S b h² ÷ 3l.
- Supported at both ends, load at any point: P = S b h² l ÷ 6ac, with a and c the distances from the load to the supports (2 S b h² ÷ 3l at the centre). Uniformly distributed: P = 4 S b h² ÷ 3l.
- Depth needed h = √(K P ÷ S b) and width needed b = K P ÷ S h², with P = S b h² ÷ K.
- Section at a point: the depth follows the parabola, triangle or ellipse of the outline, and the width the triangle or parabola, so that S stays the same along the beam.
- Example: the beam carries 8.3333 kN at a safe stress of 100 MPa.
Limitations
- The outlines taper to nothing at the ends where the moment is zero; leave enough section there to carry the shear and any fixing.
- The formulas give the bending strength only; check deflection, which is larger than for a uniform beam of the same largest section.
- S is your safe stress for the material and duty.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Size a tapered lever, arm or cantilever for its load
- Find the safe load of a beam shaped for uniform strength
- Lay out the outline of a tapered beam
Who it is for
- Machine designers
- Structural designers
- Design engineers
What you get
- Safe load, or the depth or width needed
- Bending stress under your load
- Depth or width at the chosen point from the outline
- Whether the load is within the safe load
Interpreting the result
- A beam of uniform strength uses less material than a uniform beam because every section is stressed to S.
- Leaf and plate springs are practical forms of the triangular width outline.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 271–272. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These KEVOS® Library readings explain the method this tool uses, with worked examples.
Useful alongside · KEVOS® original book
Torsion bars and flat springs
What it covers: How to size a torsion bar for stress and twist and balance a heavy panel with it, and how to check flat cantilever springs, with their limits at the ends and in large deflection.
KEVOS® tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.
Open the Beam of Uniform Strength: Safe Load, Size and Outline calculator