Find the slope and deflection of a simply-supported beam with the conjugate beam method (Mohr's theorem). The real beam's M/EI diagram is placed as a distributed load on a fictitious conjugate beam, and its shear force is read as slope and its bending moment as deflection. Change span, stiffness or load and the results update in real time.
Parameters
Load type
Choose a central point load or a uniformly distributed load
Beam span L
m
Flexural rigidity EI
kN·m²
Product of Young's modulus E and second moment of area I
Load magnitude
kN for a point load, kN/m for a distributed load
Evaluation position x
m
Slope and deflection are reported at this position
Results
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Max deflection δ_max (mm)
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Deflection at x δ(x) (mm)
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Slope at x θ(x) (rad)
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End slope θ_end (rad)
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Conjugate end reaction =θ_end (rad)
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Total M/EI area =total Δθ (rad)
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Real beam and conjugate beam — deflection animation
The top shows the real beam and its deflected shape; the bottom shows the conjugate beam loaded with the M/EI diagram. A marker sweeps the evaluation position x while tracing the deflected shape.
The heart of the method. The slope θ of the real beam equals the shear force V of the conjugate beam, and the deflection δ of the real beam equals the bending moment M of the conjugate beam.
$$w_{conj}(x)=\frac{M(x)}{EI}$$
The distributed load carried by the conjugate beam is exactly the M/EI diagram — the real beam's bending moment diagram divided by the flexural rigidity EI.
Maximum mid-span deflection of a simply-supported beam: left for a central point load P, right for a uniformly distributed load w. L: span, EI: flexural rigidity.
What is the Conjugate Beam Method?
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The "conjugate beam method" sounds intimidating just from the name. I heard it's a way to find beam deflection — but what does "conjugate" even mean here?
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Don't let the name scare you. In plain terms, it's a trick that swaps a deflection calculation for a shear-force and bending-moment calculation. Finding a real beam's deflection directly means integrating a differential equation like EI·d²δ/dx²=−M(x) twice and plugging in boundary conditions — tedious, right? So you bring in a second, imaginary beam, the "conjugate beam", and turn the problem into one about that beam instead. It's also called Mohr's theorem.
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A second imaginary beam... So what do you load that conjugate beam with?
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You load it with the real beam's "M/EI diagram", as a distributed load. That is the real beam's bending moment diagram M(x) divided by the flexural rigidity EI. For a simply-supported beam with a central point load, the M diagram is a triangle, so the conjugate beam carries a triangular load. For a uniformly distributed load the M diagram is a parabola, so the conjugate beam carries a parabolic load. Switch the load type on the left and watch the load shape change in the canvas below.
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I see. And after loading it, what do you do — just compute shear and moment like an ordinary beam?
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That's exactly the key. You find the shear force V and bending moment M for the conjugate beam's M/EI load. Then the magic happens: the conjugate beam's shear V matches the real beam's slope θ, and the conjugate beam's moment M matches the real beam's deflection δ. So θ=V_conj and δ=M_conj. Instead of solving a differential equation, the deflection drops out of the shear and moment diagrams you practised endlessly as a student.
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That's handy! But how do you decide the supports of the conjugate beam? Are they the same as the real beam?
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That needs one extra step called "conjugating" the supports. You swap the supports so that the boundary conditions on slope and deflection are consistent with those on shear and moment. A real simple support stays a simple support, a fixed end becomes a free end, and a free end becomes a fixed end. The simply-supported beam we use here has simple supports at both ends, so after the swap it is still a simply-supported beam. That's why it's the most straightforward case to learn with.
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One last thing — there's a number called "total area of the M/EI diagram". What does that represent?
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Good question. The total area of the M/EI diagram equals the total change of slope from one end of the beam to the other. It is also the total load placed on the conjugate beam, which equals the sum of its two end reactions. For a symmetric load like ours, the total area is exactly 2·θ_end. So the relation "conjugate-beam end reaction = real beam end slope" can be checked numerically right there.
Frequently Asked Questions
The conjugate beam method (Mohr's theorem) is an analysis technique for finding the slope and deflection of a beam. You imagine a fictitious beam — the conjugate beam — of the same span as the real beam, and load it with the real beam's M/EI diagram as a distributed load. The shear force in the conjugate beam then equals the slope of the real beam, and the bending moment in the conjugate beam equals the deflection of the real beam. Instead of solving a differential equation, you obtain the deflection with familiar shear-and-moment calculations.
In the conjugate beam method the support conditions are also conjugated. The boundary conditions on slope and deflection must be consistent with the boundary conditions on shear force and bending moment of the conjugate beam. A real simple support stays a simple support, a real fixed end becomes a free end, and a real free end becomes a fixed end. For a simply-supported beam both ends remain simple supports, so the conjugate beam is again a simply-supported beam and is easy to handle.
The maximum deflection is δ_max = PL³/(48EI) for a central point load P and δ_max = 5wL⁴/(384EI) for a uniformly distributed load w. The end slope is θ_end = PL²/(16EI) for the point load and θ_end = wL³/(24EI) for the distributed load. The point load gives a triangular M/EI diagram while the distributed load gives a parabolic one, so the shape of the conjugate-beam load changes. This tool lets you switch between the two with a select box and compare them.
The total area of the M/EI diagram equals the total change of slope over the whole length of the beam. It corresponds to the total load carried by the conjugate beam and is also the sum of the two end reactions of the conjugate beam. For a symmetric loading like the cases here, the total area equals the sum of the two end slopes, i.e. 2·θ_end. This also confirms that the conjugate-beam end reaction equals the real beam's end slope.
Real-World Applications
Hand calculation and checking of beam deflection: The conjugate beam method is a standard way to estimate the deflection of bridge girders, floor beams and machine frames by hand. Without memorising a table of formulas, you can derive the deflection and slope at any point as long as you can draw the M diagram. It is also useful as a sanity check on FEM results: if the conjugate-beam estimate and the FEM value differ by an order of magnitude, suspect a mistake in the mesh or boundary conditions.
Groundwork for solving statically indeterminate structures: When continuous beams or rigid frames are solved with the three-moment equation, the slope-deflection method or the force method, the slopes and deflections of each member are needed repeatedly. The conjugate beam method serves as a tool to extract these deformations quickly from the moment diagram. It is especially the basis for writing compatibility conditions in problems with support settlement or thermal deformation.
Education and exam preparation in structural mechanics: The conjugate beam method is an excellent topic for deepening understanding, bridging the two worlds of slope-and-deflection and shear-and-moment. It appears often in qualifying exams for structural designers in building and civil engineering, and learning it together with the moment-area method (the integral version of Mohr's theorem) builds intuition for beam deformation.
Deflection assessment of shafts and spindles in machine design: The deflection of rotating bodies — machine-tool spindles, conveyor rollers and the like — directly affects machining accuracy and vibration. Even for a stepped shaft carrying several loads, drawing the M/EI diagram segment by segment and applying the conjugate beam method lets you evaluate the slope (tilt at the support bearings) and the deflection (maximum displacement) separately. This helps decide whether the allowable bearing misalignment angle is met.
Common Misconceptions and Pitfalls
The most common misconception is using the conjugate beam's supports exactly as in the real beam. For the simply-supported beam in this tool the conjugate beam happens to be simply supported too, so no problem arises — but for a cantilever the story changes completely. A real fixed end must become a free end in the conjugate beam, and a real free end must become a fixed end. Skip this conjugation and the correspondence "shear = slope, moment = deflection" breaks down, giving a completely wrong answer. Always keep the support-conversion table at hand and check it.
Next, confusing the M in the M/EI diagram with the load itself. What you place on the conjugate beam is not the load P or w acting on the real beam, but the bending moment diagram M(x) that the load produces, divided by EI. Even with the same load, the conjugate-beam load changes when EI changes. That is exactly why moving EI in this tool changes the deflection and slope — deflection is inversely proportional to EI. Mixing up the M diagram with the load diagram throws the deflection off by one or two orders of magnitude.
Finally, do not forget that the conjugate beam method is valid only within the linear, small-deflection regime. Its starting point is the basic deflection differential equation EI·δ''=−M, which assumes Hooke's law (linear elasticity) and small deflections. It cannot be applied to large deformations near buckling, to beams that have entered the plastic range, or to nonlinear materials. Also, for a stepped beam where EI changes greatly between segments, the M/EI diagram must be drawn correctly for each segment. Treat it as a convenient tool only on the playing field of an "elastic, small-deflection beam".
How to Use
Set beam length (L) in mm using LRange slider; typical spans: 2000–6000 mm for steel beams
Define flexural rigidity (EI) in N·mm² using EIRange; e.g., steel I-beam W310×39 gives EI ≈ 8.5×10⁹ N·mm²
Input distributed or point load in N using loadRange; enter distance x (mm) where you need slope θ(x) or deflection δ(x)
Simulator converts bending moment diagram to M/EI conjugate loading, calculates shear (slope) and moment (deflection) on conjugate beam
Read outputs: δ_max in mm, θ(x) in radians, conjugate end reaction equals θ_end, total M/EI area equals total Δθ
Worked Example
Simply-supported steel beam: L=3000 mm, W250×28.4 section (EI=5.2×10⁹ N·mm²), uniform load w=12 N/mm (36 kN total). M/EI curve creates parabolic loading on conjugate beam. At mid-span x=1500 mm: δ(x)=8.4 mm downward. Maximum deflection δ_max=11.6 mm at x≈1549 mm. End slope θ_end=0.00445 rad (0.255°). Total rotational change Δθ=0.00890 rad matches conjugate moment equilibrium.
Practical Notes
Conjugate beam method avoids integration; useful when M(x) is piecewise-polynomial (point loads, step changes)
For cantilever beams, set conjugate support conditions: fixed end becomes free; free end becomes fixed with reaction=slope
Verify M/EI diagram sign: sagging moment (compression top fiber) is positive; conjugate loading acts upward as distributed shear source
Deflection limit checks: L/240 (live load) or L/180 (total); 3000 mm beam allows δ_max≤12.5 mm under typical codes
High EI (solid sections) reduces deflection nonlinearly; doubling EI halves deflection for same load