BIZOMAT MAKEOS · Strength
Special Section Beam Check (Section Properties, Beam Case, Stress, Deflection Limit)
For a beam made from a hexagon, ellipse, half-round, triangle or holed round bar, what are its section properties, and are the bending stress and deflection within limits for my supports and loads?
Check of a beam with an uncommon solid section: area, second moment of area and extreme fibre distance of an ellipse, hollow ellipse, half circle, hexagon, isosceles triangle, or round bar with a hexagonal or square hole (as the more section shapes tool), then the reactions, largest bending moment, bending stress and deflection for a fixed, guided, propped or simply supported case (as the more beam cases tool), against your allowable stress and span-to-deflection limit, with the mass per metre.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Section | choice | Shape of the cross-section, bent about its horizontal axis. |
| Dimension A | mm | First dimension of the section (see the shape list). |
| Dimension B | mm | Second dimension. |
| Dimension C | mm | Third dimension (hole or inner axis). |
| Dimension D | mm | Fourth dimension (inner axis of the hollow ellipse). |
| Beam | ||
| Beam case | choice | How the beam is supported and loaded. |
| Load (each point load, or total uniform load) | N | Each point load, or the total uniform load. |
| Span | mm | Span between supports. |
| Distance a | mm | Position of the load (see the beam case). |
| Modulus of elasticity | GPa | Modulus of elasticity of the material. |
| Allowable bending stress | MPa | Allowable bending stress. |
| Deflection limit (span ÷) | Largest deflection allowed as a fraction of the span (250 means span ÷ 250). | |
| Density | kg/m³ | Density for the mass per metre. |
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: Section
- What it is
- The cross-section of the beam. The list shows what dimensions A to D mean for each shape; bending is about the horizontal axis.
- Units and format
- Choose one of 7 options from the list: Ellipse (A, B = full axes; B vertical); Hollow ellipse (A, B outer; C, D inner axes); Half circle, flat side down (A = diameter); Hexagon, flats top and bottom (A = across flats); Isosceles triangle, base down (A = base, B = height); Round with hexagonal hole (A = dia, C = hole across flats); Round with square hole (A = dia, C = hole side).
- Example
- The form opens with “Hexagon, flats top and bottom (A = across flats)” selected as a worked example. Choose the option that matches your case.
- Get it right
- Check that the section is oriented as the option says (for example hexagon flats top and bottom, triangle base down).
What this means: Dimension A
- What it is
- The first section dimension: full horizontal axis of an ellipse, outer horizontal axis of a hollow ellipse, diameter of a half circle or round bar, hexagon size across flats, or triangle base.
- 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 60 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- All ellipse values are full axes, not semi-axes.
What this means: Dimension B
- What it is
- The second section dimension: full vertical axis of an ellipse, outer vertical axis of a hollow ellipse, or height of the triangle. Other shapes do not use it.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Ignored for the half circle, hexagon and round bar with a hole.
What this means: Dimension C
- What it is
- The third section dimension: inner horizontal axis of the hollow ellipse, the across-flats size of a hexagonal hole, or the side of a square hole.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Used only for the hollow ellipse and the round bars with a hole; the hole must fit inside the outer shape.
What this means: Dimension D
- What it is
- The fourth section dimension: the inner vertical axis of the hollow ellipse. No other shape uses it.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Used only for the hollow ellipse; it must be smaller than B.
Beam
What this means: Beam case
- What it is
- How the beam is supported at its ends and how it is loaded. Each case has its own reaction, bending moment and deflection formulas.
- Units and format
- Choose one of 5 options from the list: Fixed both ends · point load at distance a; Fixed one end, guided other · load at guided end; Fixed one end, guided other · uniform load; Fixed one end, supported other · point load at a from fixed end; Simply supported · two equal loads at a from each support.
- Example
- The form opens with “Simply supported · two equal loads at a from each support” selected as a worked example. Choose the option that matches your case.
- Get it right
- Choose fixed only where the end really cannot rotate; a bolted end plate is often closer to simply supported.
What this means: Load (each point load, or total uniform load)
- What it is
- The applied load in newtons: each point load for the point load cases (both of the two equal loads), or the total of the uniform load for the uniform load case.
- 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.
- Effect on the result
- Moment, stress and deflection rise in proportion to it.
- Get it right
- For the uniform load case enter the whole load along the span, not the load per metre.
What this means: Span
- What it is
- The length of the beam between its supports, or from the fixed end to the guided end.
- 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 2000 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Deflection rises with the cube of the span for a given load.
- Get it right
- Measure between the support centres or faces that actually restrain the beam.
What this means: Distance a
- What it is
- The position of the point load: from the left support for the fixed-both-ends case, from the fixed end for the propped case, or from each support for the two equal loads.
- 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 600 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Not used by the guided cases. For two equal loads it must not exceed half the span.
What this means: Modulus of elasticity
- What it is
- The modulus of elasticity of the beam material, its stiffness in tension and compression, used for the deflection.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 1000 GPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Deflection falls in proportion to it; stress does not change.
- Get it right
- Enter it in GPa, not MPa: divide a value in MPa by 1000. Take it from material data.
What this means: Allowable bending stress
- What it is
- The highest bending stress you accept in the beam, from the material’s strength with your safety factor, or from a design standard.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 10000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 150 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Take the material strength from the supplier’s data; the tool applies no safety factor of its own.
What this means: Deflection limit (span ÷)
- What it is
- The deflection limit written as the span divided by a number: the tool allows a deflection of span ÷ this number.
- Units and format
- Enter a plain number with no unit. The form accepts 10 to 100000. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 250 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Raising the number lowers the deflection allowed.
- Get it right
- A larger number is a stricter limit. Take it from your design standard or the function of the beam.
What this means: Density
- What it is
- The density of the beam material, used only to report the mass per metre of the section.
- Units and format
- Enter a number in kg/m³: kilograms per cubic metre. The form accepts 0.000001 kg/m³ to 25000 kg/m³. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 7850 kg/m³ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- It does not include the beam’s own weight in the load; add that to the load yourself if it matters.
Method and formulas
- Section area and I_x of the chosen shape; extreme fibre half the height, or the larger of the centroid distances for the half circle and triangle (as the more section shapes tool).
- Reactions, maximum moment and deflection for the beam case (as the more beam cases tool).
- Stress = M c ÷ I; deflection limit = span ÷ limit.
- Example: I = 779,420 mm⁴; 92.376 MPa and 6.7742 mm deflection against 8 mm allowed.
Limitations
- Bending about the horizontal axis only; unsymmetrical sections can also twist under load.
- Elastic, with rigid fixed ends where shown.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check a hex bar or half-round used as a beam
- Check a guide rail or slide of odd section
- Compare sections for stiffness
Who it is for
- Design engineers
- Machine designers
- Toolmakers
What you get
- Section area, I and extreme fibre
- Reactions, moment, stress and deflection
- Against the stress and deflection limits
Interpreting the result
- For the same area, sections with more material far from the axis (hollow shapes) are much stiffer.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 234–246, 257–270. Topic reference; no source tables or recommended settings are reproduced.
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