BIZOMAT MAKEOS · Strength
Beam Cases: Fixed, Guided, Propped and Partial Loads
What moments and deflection does this less common beam case give?
Reactions, maximum bending moment, stress and maximum deflection for a fixed–fixed beam with an off-centre load, beams fixed at one end and guided at the other, a propped cantilever with a point load anywhere, two symmetrical loads, and a uniform load on part of a span.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Beam case | choice | Support and load arrangement. |
| Load (each point load, or total uniform load) | N | Each point load, or the total of a uniform load. |
| Span | mm | Span between supports. |
| Distance a | mm | Load position as described for the case. |
| Loaded length c (partial load) | mm | Length of a part-span uniform load. |
| 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: Beam case
- What it is
- The support and loading arrangement: both ends fixed, fixed and guided, fixed and propped, or simply supported with two loads or a part-length uniform load. It sets where Distance a is measured from.
- Units and format
- Choose one of 6 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; Simply supported · uniform load of length c starting at a.
- Example
- The form opens with “Fixed both ends · point load at distance a” selected as a worked example. Choose the option that matches your case.
- Get it right
- Read the option text: a is measured from the fixed end for the propped case and from each support for the two-load case.
What this means: Load (each point load, or total uniform load)
- What it is
- The size of each point load, or for the uniform-load cases the total load spread along the loaded length (not the load per metre).
- 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
- Moments, stress and deflection are all directly proportional to the load.
- Get it right
- For the two-load case this is each of the two loads, not their sum.
What this means: Span
- What it is
- The span between the two supports or fixed ends, measured between their centre lines or faces of fixity.
- 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.
- Get it right
- Measure between the support or fixing centre lines, not the overall length of the member.
What this means: Distance a
- What it is
- Where the load sits, as defined by the chosen case: from the left end for a fixed–fixed beam, from the fixed end for a propped beam, from each support for two loads, or to the start of a part-length load.
- 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 two fixed–guided cases. For two symmetrical loads it must be no more than half the span.
What this means: Loaded length c (partial load)
- What it is
- The length of beam covered by a part-span uniform load, starting at Distance a. Only the simply supported part-load case uses it.
- 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 800 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- The start position plus this length must stay within the span.
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
- Deflection falls 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.
- 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 from the governing code, not a property of the material alone.
Method and formulas
- Fixed–fixed, load at a: M_A = −Pab²/L², M_B = −Pa²b/L², M_load = 2Pa²b²/L³; δ_load = Pa³b³ ÷ 3EIL³.
- Fixed–guided, end load: M = PL/2 at both ends, δ = PL³/12EI. Uniform: M = WL/3 fixed, WL/6 guided, δ = WL³/24EI.
- Propped, load at a from fixed end: R_B = Pa²(3L − a)/2L³; deflection from cantilever superposition.
- Two symmetrical loads: M = Pa, δ_centre = Pa(3L² − 4a²)/24EI. Part-span load: moment from statics, deflection by numerical integration of M/EI.
- Example: 5000 N at 600 mm on a fixed–fixed 2000 mm beam gives an end moment of 1.47 kN·m.
Limitations
- Linear elastic, constant section, small deflections.
- Fixed ends are assumed perfectly rigid; real connections rotate and reduce end moments.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check beams with built-in or guided ends
- Check a simply supported beam with two loads or a part-length load
- Fill gaps in standard beam tables
Who it is for
- Design engineers
- Structural and mechanical engineers
- Engineering students
What you get
- Reactions and maximum bending moment
- Bending stress
- Maximum deflection
Interpreting the result
- If the real end fixity is uncertain, check both the fixed and the simply supported cases.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 257–270. Topic reference; no source tables or recommended settings are reproduced.
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Open the Beam Cases: Fixed, Guided, Propped and Partial Loads calculator