BIZOMAT MAKEOS · Strength
Continuous Beam over Two Spans
What reactions, moments and deflections does this beam continuous over three supports have?
Reactions, the support moment (three-moment equation), maximum span moments, stress and deflections of a beam continuous over two equal or unequal spans with uniform or point loads.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Loading | choice | Uniform loads or one point load per span. |
| Span 1 | mm | Length of the first span. |
| Span 2 | mm | Length of the second span. |
| Load on span 1 (N/m, or N for a point load) | N/m or N | Uniform load on span 1 in N per metre, or the point load in N. |
| Load on span 2 (N/m, or N for a point load) | N/m or N | Uniform load on span 2, or its point load. |
| Point load position in span 1 (from end support) | mm | Point load distance from the end support in span 1. |
| Point load position in span 2 (from centre support) | mm | Point load distance from the centre support in span 2. |
| 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: Loading
- What it is
- Whether each of the two spans carries a uniform load along its length or a single point load at a stated position.
- Units and format
- Choose one of 2 options from the list: Uniform load on each span; One point load in each span.
- Example
- The form opens with “Uniform load on each span” selected as a worked example. Choose the option that matches your case.
- Get it right
- The load inputs are N per metre for uniform loads but newtons for point loads.
What this means: Span 1
- What it is
- The length of the first span, from the outer end support A to the centre support B, measured between support centre lines.
- 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
- The two spans may differ; enter each one separately.
What this means: Span 2
- What it is
- The length of the second span, from the centre support B to the other end support C, measured between support centre lines.
- 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
- The two spans may differ; enter each one separately.
What this means: Load on span 1 (N/m, or N for a point load)
- What it is
- The load carried by the first span: the uniform load per metre of span, or the single point load force when the point-load case is chosen.
- Units and format
- Enter a number in N/m or N: N/m or N. The form accepts 0 N/m or N or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2000 N/m or N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Set one span’s load to zero to check pattern loading, which can give larger span moments.
What this means: Load on span 2 (N/m, or N for a point load)
- What it is
- The load carried by the second span: the uniform load per metre of span, or the single point load force when the point-load case is chosen.
- Units and format
- Enter a number in N/m or N: N/m or N. The form accepts 0 N/m or N or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2000 N/m or N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Set one span’s load to zero to check pattern loading, which can give larger span moments.
What this means: Point load position in span 1 (from end support)
- What it is
- Where the point load sits in span 1, measured from the outer end support A towards the centre support.
- 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 1500 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Measured from the end support, not the centre support. Used only in the point-load case; it must not exceed span 1.
What this means: Point load position in span 2 (from centre supp…
- What it is
- Where the point load sits in span 2, measured from the centre support B towards the far end support.
- 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 1500 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Measured from the centre support, unlike span 1. Used only in the point-load case; it must not exceed span 2.
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, reactions 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 largest 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 a strength divided by a code factor; take it from the governing code.
Method and formulas
- Three-moment equation with pinned ends: M_B = −(T₁ + T₂) ÷ 2(L₁ + L₂), with T = wL³/4 for a uniform load and P a(L² − a²)/L for a point load (a measured from the outer support).
- Reactions by statics: R_A = (simple-span reaction) + M_B ÷ L₁, similarly R_C; R_B = total load − R_A − R_C.
- Deflection of each span by superposition of the simple-span load and the end moment; maxima found by sampling.
- Example: two equal 3000 mm spans at 2000 N/m give a centre reaction of 7,500 N and a support moment of -2.25 kN·m.
Limitations
- Supports at the same level and rigid; constant section.
- Pattern loading (one span loaded) can govern; check it by setting one load to zero.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check purlins, rails and shafts over three supports
- Find the centre support load
- Compare continuous and simply supported spans
Who it is for
- Design engineers
- Structural and mechanical engineers
- Engineering students
What you get
- End and centre reactions
- Centre support moment
- Maximum span moments and stress
- Maximum deflection in each span
Interpreting the result
- The centre support carries about 5/4 of a span’s uniform load for equal spans.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 264–267. Topic reference; no source tables or recommended settings are reproduced.
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