BIZOMAT MAKEOS · Strength
Beam Cases: Off-Centre, Fixed and Propped
What reactions, moment, stress and deflection does this beam case give?
Reactions, maximum bending moment, bending stress and deflection for an off-centre point load, cantilever with uniform load, fixed–fixed beams and a propped cantilever.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Beam case | choice | The support and load arrangement. |
| Load (point load, or total uniform load) | N | Point load, or the total of a uniformly distributed load (w × L). |
| Span | mm | Distance between supports, or the cantilever length. |
| Load position from left support (off-centre case) | mm | Distance of the point load from the left support (off-centre case only). |
| Elastic modulus | GPa | Elastic modulus from the material data sheet or an approved materials database (steel is typically about 200 GPa). |
| Second moment of area | mm⁴ | Second moment of area about the bending axis, from a section table or the section properties calculator. |
| Distance to extreme fibre | mm | Distance from the neutral axis to the outermost fibre (half the depth for symmetrical sections). |
| Allowable bending stress | MPa | Allowable bending stress from your design code or company standard. |
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 case
- What it is
- The support and loading arrangement of the beam. It decides the formulas for reactions, maximum moment and deflection.
- Units and format
- Choose one of 5 options from the list: Simply supported · point load at distance a; Cantilever · uniform load; Fixed both ends · central point load; Fixed both ends · uniform load; Fixed one end, supported the other · uniform load.
- Example
- The form opens with “Simply supported · point load at distance a” selected as a worked example. Choose the option that matches your case.
- Get it right
- Real supports are rarely perfectly fixed; a fixed-end case gives lower moments and deflections than a partly fixed beam.
What this means: Load (point load, or total uniform load)
- What it is
- The load on the beam: a single point load, or for the uniform-load cases the total of the distributed load over the whole span.
- 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 5000 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 all rise in proportion to the load.
- Get it right
- For a uniform load enter the total, which is the load per millimetre times the span, not the load per unit length.
What this means: Span
- What it is
- The span of the beam between its supports, or the projecting length of a cantilever from its fixed 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 the same total load.
- Get it right
- Use the distance between the support centres or bearing points.
What this means: Load position from left support (off-centre cas…
- What it is
- The distance from the left support to the point load. Used only for the simply supported off-centre case.
- 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 600 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- It must lie between the two supports; it is ignored for the other cases.
What this means: Elastic modulus
- What it is
- The elastic modulus (Young’s modulus) of the beam material, from the material data sheet. It sets how far the beam deflects.
- 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 as the modulus rises; stress is unaffected.
- Get it right
- Enter GPa, not MPa: 1 GPa is 1000 MPa.
What this means: Second moment of area
- What it is
- The second moment of area of the beam section about the axis it bends about, from a section table or section properties calculation.
- 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 and stress both fall in proportion as it rises.
- Get it right
- Use the value about the bending axis; for an I-beam on edge that is the strong axis.
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 highest.
- 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 rises in proportion to it.
- Get it right
- For a symmetrical section it is half the depth; for an unsymmetrical one use the larger distance.
What this means: Allowable bending stress
- What it is
- The highest bending stress the beam is allowed to reach, from your design code or company standard. The calculated stress is compared with 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
- This is a design limit you supply, not the material’s yield strength unless your code says so.
Method and formulas
- Off-centre: R₁ = Pb ÷ L, M = Pab ÷ L; δmax = Pb(L² − b²)^1.5 ÷ (9√3 EIL) for the shorter segment b.
- Cantilever UDL: M = WL ÷ 2, δ = WL³ ÷ 8EI. Fixed–fixed point: M = PL ÷ 8, δ = PL³ ÷ 192EI. Fixed–fixed UDL: M = WL ÷ 12, δ = WL³ ÷ 384EI.
- Propped UDL: R = 5W/8 and 3W/8, M = WL ÷ 8 at the fixed end, δmax at 0.4215L from the support.
- Stress σ = M c ÷ I.
- Example: 5000 N at 600 mm on a 2000 mm span gives M = 2,100 N·m, σ = 13.125 MPa and δ = 0.41766 mm.
Limitations
- Linear elastic, small-deflection beams of constant section; self-weight is not added automatically.
- Lateral-torsional buckling, shear deflection and local bearing are not checked.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check frames, rails and support beams
- Compare support conditions
- Size a section for stress and stiffness
Who it is for
- Design engineers
- Structural and mechanical engineers
- Engineering students
What you get
- Reactions
- Maximum bending moment
- Bending stress and its ratio to the allowable
- Maximum deflection and span-to-deflection ratio
Interpreting the result
- Typical deflection limits are span/250 to span/1000 depending on use; take yours from the design brief.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 257–277. Topic reference; no source tables or recommended settings are reproduced.
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Open the Beam Cases: Off-Centre, Fixed and Propped calculator