BIZOMAT MAKEOS · Strength
Beam Sizing for Strength and Deflection
What section modulus and second moment of area does this beam need for its loads, span and deflection limit?
Required section properties for a beam carrying a uniform load and a point load together: maximum bending moment by superposition (sampled along a simple span, or at the critical sections of a cantilever or fixed-ended beam), the section modulus for the allowable stress, the second moment of area for a span-over deflection limit, and the utilisation of a chosen section.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Supports | choice | Support arrangement. |
| Span | mm | Span (or cantilever length). |
| Uniform load | N/mm | Uniformly distributed load including self-weight. |
| Point load | N | Concentrated load. |
| Point load position from the left support (or fixed end) | mm | Position of the point load (ignored for fixed ends, where it is taken at mid-span). |
| Allowable bending stress | MPa | Allowable bending stress for the material and code. |
| Elastic modulus | GPa | Elastic modulus of the beam material. |
| Deflection limit: span ÷ | Deflection limit as span divided by this number (360 for floors, 180–250 for many machine frames). | |
| Chosen section modulus (0 = none) | mm³ | Section modulus of a section you want to check. |
| Chosen second moment of area (0 = none) | mm⁴ | Second moment of area of that 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: Supports
- What it is
- How the beam is supported: resting on two supports, built into one end only, or built in at both ends.
- Units and format
- Choose one of 3 options from the list: Simply supported; Cantilever; Fixed both ends (point load at mid-span).
- Example
- The form opens with “Simply supported” selected as a worked example. Choose the option that matches your case.
- Get it right
- With both ends fixed, the tool places the point load at mid-span whatever position is entered.
What this means: Span
- What it is
- Distance between the supports, or the overhanging length from the fixed end for a cantilever.
- 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 3000 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Moment rises with span and deflection with its third or fourth power.
- Get it right
- Use the clear span between support centres.
What this means: Uniform load
- What it is
- Load spread evenly along the beam per millimetre of length, including the beam’s own weight.
- Units and format
- Enter a number in N/mm: newtons per millimetre. The form accepts 0 N/mm to 1000000 N/mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5 N/mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Moment and deflection rise in proportion.
- Get it right
- Convert from kN/m: 1 kN/m equals 1 N/mm.
What this means: Point load
- What it is
- A single concentrated load acting on the beam at one point.
- 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 10000 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Moment and deflection rise in proportion.
- Get it right
- Enter 0 if there is only the uniform load.
What this means: Point load position from the left support (or f…
- What it is
- Distance from the left support, or from the fixed end of a cantilever, to where the point load acts.
- 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 1500 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 fixed end for a cantilever; ignored for fixed-both-ends beams.
What this means: Allowable bending stress
- What it is
- Allowable bending stress for the beam material under your design code or practice, used to find the section modulus required.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 2000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 165 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Required section modulus falls in proportion as the allowable rises.
- Get it right
- Take it from the design code for your material and loading.
What this means: Elastic modulus
- What it is
- Elastic (Young’s) modulus of the beam material, used to find the second moment of area needed for the deflection limit.
- 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 206 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Required second moment of area falls in proportion as E rises.
- Get it right
- Enter gigapascals; 1 GPa = 1,000 MPa.
What this means: Deflection limit: span ÷
- What it is
- Divisor that sets the largest deflection you accept: allowed deflection is span divided by this number.
- Units and format
- Enter a plain number with no unit. The form accepts 50 to 5000. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 360 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Required second moment of area rises in proportion.
- Get it right
- A larger number means a stiffer requirement; take it from your code or machine design practice.
What this means: Chosen section modulus (0 = none)
- What it is
- Elastic section modulus of a specific beam section you want to check against the requirement.
- Units and format
- Enter a number in mm³: mm³. The form accepts 0 mm³ or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm³ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter 0 to skip the check; the check runs only when both this and the second moment of area are above 0.
What this means: Chosen second moment of area (0 = none)
- What it is
- Second moment of area of the same section you want to check, about the bending axis.
- Units and format
- Enter a number in mm⁴: millimetres to the fourth power (second moment of area). The form accepts 0 mm⁴ or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm⁴ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter 0 to skip; it must belong to the same section and axis as the section modulus.
Method and formulas
- Simple span: M(x) = w x (L − x) ÷ 2 + point-load moment, maximum found along the span; deflection = 5wL⁴ ÷ 384EI + P b (L² − b²)^1.5 ÷ (9√3 L E I) (sum of the two maxima, conservative).
- Cantilever: M = wL² ÷ 2 + P a; tip deflection = wL⁴ ÷ 8EI + P a² (3L − a) ÷ 6EI.
- Fixed ends: M = wL² ÷ 12 + PL ÷ 8; deflection = wL⁴ ÷ 384EI + PL³ ÷ 192EI.
- Required Z = M ÷ allowable; required I from the deflection limit.
- Example: the default beam needs Z ≥ 79,545 mm³ and I ≥ 6,348,600 mm⁴; both checks governs the choice.
Limitations
- Elastic, laterally restrained beam; slender beams also need a lateral-torsional buckling check.
- Shear and local bearing at supports and loads are not checked.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Choose a steel or aluminium section for a frame member
- Check whether strength or stiffness governs
- Compare an available section against the loads
Who it is for
- Design engineers
- Mechanical engineers
- Engineering managers
What you get
- Maximum moment and its position
- Required section modulus and second moment of area
- Utilisation of the chosen section
Interpreting the result
- If stiffness governs, a deeper section is far more effective than a stronger material.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 258–275. Topic reference; no source tables or recommended settings are reproduced.
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