Fixed-Fixed Beam Moment Calculator
A fixed-fixed (encastre) beam is rotationally restrained at both ends, so it develops fixed-end moments the way a simply supported beam cannot.
Enter Values
Before you rely on this: First-pass guide only. Verify safety-critical or regulated work against the relevant standards, your project requirements and a qualified professional.
How to use this calculator
- Enter the clear span L in metres, then a uniform load w (kN/m) and/or a central point load P (kN). Leave a load field blank if it doesn't apply.
- Read the end (support) moment and the mid-span moment. For a fixed-fixed beam the hogging moment over the supports is usually the design-critical value.
- Optionally enter Young's modulus E (GPa, steel ≈ 200) and second moment of area I (cm⁴) together to also get the mid-span deflection and the span-to-deflection ratio.
How it works
A fixed-fixed (encastre) beam is rotationally restrained at both ends, so it develops fixed-end moments the way a simply supported beam cannot. For a uniform load the standard fixed-end moments are M_end = wL²/12 (hogging, over each support) and M_mid = wL²/24 (sagging, at the centre). A central point load gives M_end = M_mid = PL/8. Multiple loads are superposed. Note the end moment for a UDL (wL²/12) is a third smaller than the simply supported mid moment (wL²/8), which is why fixing both ends stiffens and strengthens a span.
When both E and I are supplied the tool adds the mid-span deflection, which is much smaller than the equivalent simply supported case: wL⁴/384EI for a UDL and PL³/192EI for a central point load (versus 5wL⁴/384EI and PL³/48EI simply supported). The formulas assume a prismatic, linear-elastic, small-deflection beam with genuinely rigid rotational fixity at both ends and no support settlement or axial restraint effects.
Worked example
6 m encastre beam under a 5 kN/m uniform load. A 6 m beam is built in at both ends and carries a uniform load of w = 5 kN/m. End (support) moment = wL²/12 = 5 × 6² / 12 = 15 kN·m (hogging). Mid-span moment = wL²/24 = 5 × 6² / 24 = 7.5 kN·m (sagging) — exactly half the end moment. Adding E = 200 GPa and I = 5000 cm⁴ gives a mid-span deflection of wL⁴/384EI = 5000 × 6⁴ / (384 × 1×10⁷) = 1.688 mm, i.e. about L/3556 — well inside typical serviceability limits.
Common mistakes
- Assuming the mid-span moment governs. For a fixed-fixed beam under UDL the hogging end moment wL²/12 is twice the mid-span sagging moment wL²/24 — the supports, not the centre, are usually critical.
- Using simply supported formulas (wL²/8, 5wL⁴/384EI) for a fixed-fixed beam. Fixing both ends roughly halves the peak moment and cuts deflection to about a fifth — the two cases are not interchangeable.
- Trusting the deflection when real connections aren't fully rigid. True encastre fixity is an idealisation; partial rotational stiffness raises mid-span moment and deflection above these values, so treat the result as a lower-bound estimate.
Frequently asked questions
What's the difference between this and the Simply Supported Beam calculator?
A simply supported beam sits on a pin and a roller and is free to rotate at both ends, so it carries no support moment and its peak moment is wL²/8. A fixed-fixed beam is built in (rotationally restrained) at both ends, so it develops hogging end moments of wL²/12 plus a smaller sagging mid-span moment of wL²/24, and it deflects far less. Use this tool when both ends are genuinely fixed.
How is this different from a cantilever?
A cantilever is fixed at ONE end and free at the other, with the maximum moment at the built-in support. A fixed-fixed beam is fixed at BOTH ends and spans between two supports, so it has hogging moments at each end and a sagging moment in the middle. Different restraint, different formulas.
Why is the end moment hogging and the mid moment sagging?
The fixed ends resist the beam's tendency to rotate, pulling the top of the beam into tension over the supports (hogging). Between the supports the load sags the beam, putting the bottom fibre in tension (sagging). The tool labels each moment so you can size top and bottom reinforcement or check flange stresses correctly.
Do I need E and I?
No — the moments only need span and loads. Enter Young's modulus E (GPa) and second moment of area I (cm⁴) together only if you also want the mid-span deflection. Supplying just one of the two returns an error so you don't get a misleading half-computed result.
Can I trust this for final design?
No. It's a first-pass estimate assuming perfect rigid fixity, a prismatic linear-elastic beam and no settlement. Real supports are rarely fully fixed, and design must follow the relevant standard (AS 4100/AS 3600, Eurocode, AISC) and be checked by a competent structural engineer.
Related tools
- Simply Supported Beam Calculator
- Cantilever Beam Calculator
- Beam Deflection Calculator
- Overhanging Beam Calculator
- Fatigue Life Estimator
- Mohr's Circle Stress Calculator
Explore more in Structural, Materials, Mechanical & Workshop.
Tip: Enter any known values to calculate the remaining results.
All calculations run in your browser. Your inputs are never saved or transmitted.