Cantilever Slope and Deflection Calculator: Area-Moment Method with Stepped Section

Slope and deflection of a cantilever from the area of the M/EI diagram, for point, uniform and triangular loads and a section that changes part way along.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Enter the length of the cantilever and measure every load from the free end. Use one line per load: P 0 90 is a 90 kN point load at the free end, U 0 4 10 is 10 kN/m over the first 4 m, and U 2 6 0 25 is a triangular load that rises from 0 at 2 m to 25 kN/m at 6 m.

If the section changes, enter how far from the free end the change is and how many times stiffer the fixed-end part is. A section that goes from I to 2I is a multiple of 2. Leave the length at 0 for one constant section.

The calculator draws the bending moment diagram, divides it by EI, and finds the slope at the free end as the area of that diagram and the deflection as the moment of that area about the free end. It also gives the slope and deflection at the change of section and at any other point you name.

Formulas used

  • \theta_s = \int_s^{L}\frac{M}{EI}\,dx
  • \delta_s = \int_s^{L}\frac{M}{EI}\,(x - s)\,dx
  • EI(\mathrm{kN\,m^2}) = E(\mathrm{GPa})\times I(10^6\,\mathrm{mm^4})

Worked example

With the default numbers in the calculator above:

  1. The loads read: point load 90.00 kN at 0.00 m
  2. Stiffness of each part (EI): EI 20000.00 kN m2 (free-end side), 40000.00 kN m2 (fixed-end side)
  3. Bending moment at the fixed end: 810.00 kN m
  4. Shear force at the fixed end: 90.00 kN
  5. Slope at the free end (area of the M/EI diagram): 0.13163 rad
  6. Slope at the free end in degrees: 7.5416 degrees
  7. Deflection at the free end (moment of the M/EI area about the free end): 708.75 mm
  8. Area of the M/EI diagram, in terms of the outer-part EI: 2632.50 kN m2 (divide by EI)
  9. First moment of that area about the free end, in terms of the outer-part EI: 14175.00 kN m3 (divide by EI)
  10. Slope at the change of section: 0.050625 rad
  11. Deflection at the change of section: 81.00 mm
  12. Slope at the chosen point: 0.050625 rad

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Length of the cantilever (free end to fixed end) 6 m; Length of the outer part measured from the free end (0 if the section does not change) 0 m; Elastic modulus 20 GPa; Second moment of area of the outer part (free-end side) 1800 10⁶ mm⁴; Second moment of area of the inner part (fixed-end side) as a multiple of the outer part 1 × I; Loads, one per line, with distances measured from the FREE end U 0 4 10; Also report the slope and deflection at this distance from the free end 2 m.

  1. The loads read: uniform load 10.00 kN/m from 0.00 m to 4.00 m
  2. Stiffness of each part (EI): EI 36000.00 kN m2 (one constant section)
  3. Bending moment at the fixed end: 160.00 kN m
  4. Shear force at the fixed end: 40.00 kN
  5. Slope at the free end (area of the M/EI diagram): 0.0096296 rad
  6. Slope at the free end in degrees: 0.55174 degrees
  7. Deflection at the free end (moment of the M/EI area about the free end): 42.96 mm
  8. Area of the M/EI diagram, in terms of the outer-part EI: 346.67 kN m2 (divide by EI)
  9. First moment of that area about the free end, in terms of the outer-part EI: 1546.67 kN m3 (divide by EI)
  10. Slope at the chosen point: 0.0092593 rad
  11. Deflection at the chosen point: 23.89 mm
  12. Free-end deflection as a fraction of the length: 139.66

Common questions

What is the area-moment method?

It uses two theorems. The change in slope between two points on a beam equals the area of the M/EI diagram between them, and the deflection of one point from the tangent at another equals the moment of that area about the first point. For a cantilever the fixed end has zero slope, so the free-end slope and deflection come straight from the diagram.

How do I handle a beam whose I changes?

Divide the moment diagram by the EI of each part. Where I doubles, the M/EI value drops to half just inside the stiffer part. The calculator does this for you and adds the areas of the parts with their own distances.

Does this work for a triangular load?

Yes. Enter the load as U start end load-at-start load-at-end, for example U 2 6 0 25. A trapezoid is the same with a non-zero value at the start.

Why measure from the free end?

It keeps the moment simple: the moment at a distance x is just the load between the free end and x times its lever arm. The fixed end is then the end with zero slope and zero deflection.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Hibbeler, Mechanics of Materials (moment-area theorems); Gere and Goodno, Mechanics of Materials; any structural theory text.

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