Verify on a simply supported beam that delta_AB times P_A equals delta_BA times P_B. The symmetry of the influence function C(a,b) is exactly what makes the FEM stiffness matrix symmetric.
Parameters
Beam length L
m
Position of point a
m
Position of point b
m
Load P_A
kN
Load P_B is fixed at 50 kN and bending stiffness EI at 1e4 kN m squared. Points a and b are clamped to (L minus 0.1) m.
Results
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delta_BA at point a (mm)
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delta_AB at point b (mm)
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delta_BA times P_A (left, kN m)
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delta_AB times P_B (right, kN m)
Deflection curves for systems A and B
Top: system A with P_A at point a, blue deflection curve, yellow marker at point b shows delta_AB. Bottom: system B with P_B at point b, red curve, yellow marker at point a shows delta_BA.
Theory & Key Formulas
For a simply supported beam of length $L$ with constant bending stiffness $EI$ and a concentrated load $P$ at $x = a$, the influence function $C(a,b)$ giving the deflection at $x = b$ is:
Maxwell reciprocity: $C(a,b) = C(b,a)$. Extending to two general load systems gives Betti's identity:
$$\delta_{AB}\,P_A = \delta_{BA}\,P_B$$
Here $\delta_{BA} = C(a,b)\,P_B$ is the deflection at $a$ caused by $P_B$ acting alone, and $\delta_{AB} = C(b,a)\,P_A$ is the deflection at $b$ caused by $P_A$ acting alone.
About this Maxwell-Betti simulator
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I've heard of the "reciprocal theorem" before, but what exactly is being "reciprocated" here?
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Roughly speaking, when you apply two independent load systems to a linear elastic body, the work each system does through the displacements caused by the other is the same. The identity is $\delta_{AB}\,P_A = \delta_{BA}\,P_B$. Slide P_A, a, or b around above. The bottom two cards (left and right side) should always match.
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They really do agree exactly! That can't be a coincidence, right?
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No coincidence. The influence function itself satisfies $C(a,b) = C(b,a)$ — Maxwell's reciprocity. If you look at the closed form, the term 2La b − a^2 − b^2 is symmetric in a and b. FEM users will recognize this as the same fact that makes the stiffness matrix symmetric (K = K^T) — that symmetry is just the discrete version of this theorem.
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So does the theorem hold for any structure at all?
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Linear elasticity is the strict prerequisite. Once you have plasticity, contact, or large deformation, the response depends on load history and reciprocity breaks down. For example, if you yield a steel member with one load and then add another, the two sides no longer match. In practice the theorem is used inside the small-strain, linear-material range — building influence lines or sanity-checking FEM results.
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What does "sanity-checking" look like in practice?
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A favorite is saving experimental effort. Suppose you want the displacement at 100 points on a turbine blade for a load at one fixed point. Reciprocity lets you fix a single sensor and tap the 100 points instead — you swap the role of load and probe. The same principle underlies adjoint methods in shape optimization and sensitivity analysis.
Frequently asked questions
delta_AB is the displacement at point b (the application point of system B) caused by system A acting alone, and delta_BA is the displacement at point a (the application point of system A) caused by system B acting alone. A useful mnemonic: the first subscript is the observation point, the second is the load source. This simulator computes both analytically from the simply supported beam influence function and checks delta_AB times P_A equals delta_BA times P_B.
In linear FEM, nodal forces and displacements satisfy F = K u, and the Betti identity is equivalent to K_{ij} = K_{ji}, i.e. K equals K transpose. If the assembled stiffness has a non-symmetric component, it usually indicates the presence of non-conservative loads (follower forces) or non-symmetric damping. Symmetry is also why production FEM codes can use optimized symmetric Cholesky-type solvers.
The influence line for the deflection at point b is, by Maxwell reciprocity C(a,b) = C(b,a), the same as the deflection curve produced by placing a unit load at point b — exactly Mueller-Breslau's principle. Moving a in this simulator and reading delta_BA is therefore equivalent to reading the influence-line ordinate of the b-point deflection.
Analytically, delta_AB times P_A equals delta_BA times P_B exactly. The simulator uses IEEE 754 double precision, so the relative difference between the two sides is typically below 1e-10, and to the displayed precision (kN m to three decimals) the two cards always read identical values. That is concrete evidence that reciprocity is not a textbook idealization but holds to round-off in any consistent numerical implementation.
Real-world applications
Influence lines and surfaces: Bridges, crane runways, and floor slabs all carry moving loads. Maxwell's reciprocity lets engineers obtain the influence line for any response quantity by placing a single unit load at the response point and reading the resulting deformation — Mueller-Breslau's principle. This shortcut has been the backbone of bridge design for over a century.
Saving experimental effort: In modal testing of large structures, swapping the driver and receiver leaves the transfer function unchanged. A test engineer can fix a shaker at one location and roam accelerometers, or fix an accelerometer and roving-hammer the structure — the choice is dictated by access and cost, not physics.
FEM verification: Sanity-checking a linear analysis is as simple as confirming K_{ij} = K_{ji}. A non-symmetric assembled stiffness should immediately raise flags about follower loads, asymmetric coupling, or buggy boundary conditions.
Inverse problems and adjoint sensitivity: Sensor placement, force identification, and topology optimization all exploit reciprocity. The adjoint method, central to large-scale gradient-based optimization, is essentially Betti's theorem in disguise: it swaps the load case and the response sensor to obtain gradients in a single extra solve.
Common misconceptions and caveats
The most common misconception is that reciprocity is a universal law of mechanics. It is not — it requires linearity, elasticity, small strain, and conservative loading. Plasticity, damage, contact, friction, buckling, and large deformation all violate it in general. This simulator deliberately keeps EI constant and stays in the small-deflection regime so that the linearity assumption holds. When reciprocity appears to be violated in practice, the first suspect should always be hidden nonlinearity.
Another misconception is that the perfect agreement between the two sides is somehow coincidental. As you vary L, a, b, and P_A, the cards on the left and right of Betti's identity remain identical because the influence function is symmetric in (a,b) by construction. Plug in a = 1.5, b = 3.5, L = 5, EI = 1e4: you get C(a,b) = 1.5375e-4 m/kN, which gives delta_BA times P_A = delta_AB times P_B = 0.2306 kN m — by direct algebra, not luck.
Finally, students often confuse Maxwell with Betti. Maxwell's theorem (1864) states the symmetry of the influence function for unit loads: C(a,b) = C(b,a). Betti's theorem (1872) generalizes this to two arbitrary load systems. They describe the same underlying property of linear elastic systems and are usually treated together as the Maxwell-Betti reciprocal theorem. The key takeaway is: in linear elasticity, swapping the observation point and the load source preserves the response, and the FEM identity K = K transpose is just one numerical face of that same fact.