BIZOMAT MAKEOS · Strength
Overhanging Beam
What reactions, moments and deflections does this beam with overhanging ends have?
Reactions, maximum positive and negative bending moments, stress and deflections of a beam on two supports with one or two overhangs, under end point loads or a uniform load.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Arrangement | choice | Overhang arrangement and load type. |
| Span between supports | mm | Distance between the two supports. |
| Overhang length | mm | Length of each overhang beyond a support. |
| Point load (each), or uniform load per metre | N or N/m | Point load at each overhang tip (N), or uniform load intensity w (N per metre run) for the uniform cases. |
| Elastic modulus | GPa | Elastic modulus from the material data sheet (steel is typically about 200 GPa). |
| Second moment of area | mm⁴ | Second moment of area of the section about the bending axis. |
| Distance to extreme fibre | mm | Neutral axis to extreme fibre. |
| Allowable bending stress | MPa | Allowable bending stress from your design code. |
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: Arrangement
- What it is
- How the beam overhangs its two supports and how it is loaded: one or two overhangs, with a point load at each overhang tip or a uniform load along the whole beam.
- Units and format
- Choose one of 4 options from the list: One overhang · point load at its tip; Equal overhangs · point load at each tip; Equal overhangs · uniform load over whole length; One overhang · uniform load over whole length.
- Example
- The form opens with “Equal overhangs · uniform load over whole length” selected as a worked example. Choose the option that matches your case.
- Get it right
- The load input changes meaning with this choice: newtons for point-load cases, newtons per metre for uniform-load cases.
What this means: Span between supports
- What it is
- The distance between the two supports, measured centre to centre of the bearings, not the overall length of the beam.
- 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 3000 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Do not include the overhangs; they are entered separately.
What this means: Overhang length
- What it is
- How far the beam sticks out beyond a support to its free tip. With two overhangs, both are taken as this same length.
- 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 800 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The support moment grows with the overhang (Pa for tip loads, wa²/2 for uniform loads).
- Get it right
- Measure from the support centre line to the free end.
What this means: Point load (each), or uniform load per metre
- What it is
- For the point-load cases, the force at each overhang tip. For the uniform-load cases, the load per metre of beam length, spread over supports and overhangs alike.
- Units and format
- Enter a number in N or N/m: N or N/m. The form accepts 0.000001 N or N/m or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 4000 N or N/m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Reactions, moments, stress and deflection are all directly proportional to the load.
- Get it right
- Uniform load is per metre, not per millimetre and not the total; include self-weight here if it matters.
What this means: Elastic modulus
- What it is
- Young’s modulus of the beam material, its stiffness in tension and bending. Take it from the material data sheet or standard.
- 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
- Deflections fall in proportion as E rises; moments and stress do not change.
- Get it right
- Enter GPa, not MPa: a value of 200 000 MPa is 200 GPa.
What this means: Second moment of area
- What it is
- The second moment of area of the beam section about the axis it bends around, from a section table, CAD or the section properties tools.
- Units and format
- Enter a number in mm⁴: millimetres to the fourth power (second moment of area). The form accepts 0.000001 mm⁴ or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 8000000 mm⁴ 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 as I rises; stress also falls for the same extreme-fibre distance.
- Get it right
- Use the value for the bending axis actually loaded; I about the weak axis is much smaller.
What this means: Distance to extreme fibre
- What it is
- The distance from the neutral axis of the section to its outermost fibre, where bending stress is greatest; half the depth for a symmetric section.
- 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
- Bending stress is directly proportional to this distance.
- Get it right
- For an unsymmetric section use the larger distance to find the highest stress.
What this means: Allowable bending stress
- What it is
- The highest bending stress your design code or company standard allows for this beam, material and load type; the calculated stress is compared against it.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 100000 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
- It is a limit you supply, normally yield or ultimate strength divided by a code factor; do not enter raw yield strength unless that is the rule.
Method and formulas
- One overhang, tip load P: R₂ = P(L + a)/L, R₁ = −Pa/L; M = −Pa at the support; tip δ = Pa²(L + a) ÷ 3EI.
- Two overhangs, tip loads P: R = P each; M = −Pa between supports; tip δ = Pa²(3L + 2a) ÷ 6EI; mid-span rises Pa L² ÷ 8EI.
- Equal overhangs, uniform w: R = w(L + 2a)/2; M_support = −wa²/2; M_centre = wL²/8 − wa²/2; δ_centre = wL²(5L² − 24a²) ÷ 384EI; tip δ = wa(3a³ + 6a²L − L³) ÷ 24EI.
- One overhang, uniform w: R₁ = w(L² − a²)/2L; M_support = −wa²/2; M_span max = R₁²/2w; tip δ = wa(4a²L − L³ + 3a³) ÷ 24EI.
- Example: the default beam has a support moment of -1.28 kN·m and a mid-span moment of 3.22 kN·m.
Limitations
- Linear elastic, constant section; self-weight is part of the uniform load only if included.
- Deflection signs: positive downward; negative values are upward.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check platform, crane-runway and machine-bed beams with overhangs
- Choose support positions to balance moments
- Find tip deflection of an overhang
Who it is for
- Design engineers
- Structural and mechanical engineers
- Engineering students
What you get
- Reactions
- Maximum negative (support) and positive (span) moments
- Stress
- Tip and mid-span deflections
Interpreting the result
- Overhangs of about 0.35 × span with uniform load roughly balance support and span moments.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 257–262. Topic reference; no source tables or recommended settings are reproduced.
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