Multi-Station Beam Reaction Test Bed
Rigid structural bed supporting simply supported beam across a 1000 mm span.
Civil Engineering / Maxwell-Betti Reciprocity
FREE TO EXPLORELABORATORY 14 · CIVIL ENGINEERING
Verification of Maxwell–Betti Reciprocal Deflection Theorem on Elastic Beams
Verify the Maxwell–Betti reciprocal deflection theorem on an elastic simply supported beam by comparing deflection at station A due to load at station B (δ_AB) with deflection at station B due to load at station A (δ_BA). Explore interactive structural mechanics, materials simulation, and civil engineering telemetry.
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Your virtual civil engineering laboratory.
CROSS-STATION DEFLECTION Δ_AB—
CIVIL INFRASTRUCTURE TELEMETRY
Ready to sample with Applied Load = 15.0 kN.
Adjust Applied Load, then execute civil engineering measurement.
Civil engineering telemetry is evaluated rigorously using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations. Continuous data structures support high-fidelity structural mechanics and geotechnical continuum modeling.
Civil engineering assumptions and boundary condition limits
The civil engineering physical models apply within the specified calibrated operational window (5.0 to 25.0 kN). Beam span L = 1000 mm. Station A located at x_A = 300 mm; Station B located at x_B = 600 mm. Closed-form flexibility coefficient f_AB = x_A (L - x_B) (2 L x_B - x_A^2 - x_B^2) / (6 E I L) = 0.3150 mm/kN.
Laboratory apparatus & testing equipment.
Twin Precision Digital LVDT Transducers
Independent displacement transducers mounted at station A (300 mm) and station B (600 mm).
Portable Hydraulic Jack Loading Assembly
Applies concentrated vertical loads up to 30 kN with precision electronic load cell.
Structural Steel Test Beam (Rectangular Section)
Uniform, homogeneous prismatic steel beam with known flexural rigidity EI.
Magnetic Articulated Transducer Stands
Clamps LVDTs rigidly to isolated datum beam to prevent frame deflection errors.
Precision Metric Vernier Layout Rule
Marks exact load and displacement stations to within 0.1 mm precision.
FOLLOW THE METHOD
A reason behind every step.
Verify the Maxwell–Betti reciprocal deflection theorem on an elastic simply supported beam by comparing deflection at station A due to load at station B (δ_AB) with deflection at station B due to load at station A (δ_BA). Standardized parameter settings: 5.0, 10.0, 15.0, 20.0, 25.0 kN.
01
Beam Station Layout & LVDT Mounting
Set beam clear span L = 1000.0 mm; mark Station A at x_A = 300.0 mm and Station B at x_B = 600.0 mm; position digital LVDTs at stations A and B; zero both gauges.
02
Case 1 Loading: Load at Station B, Measure at Station A
Position loading jack at Station B (x_B = 600 mm); apply loads P_B incrementally at 5.0, 10.0, 15.0, 20.0, and 25.0 kN.
03
Deflection Logging for Case 1
At each load step, record deflection at Station A (δ_AB) and direct deflection at Station B (δ_BB); release load completely.
04
Transducer Verification & Re-Zeroing
Verify both transducers return to zero (residual deflection < 0.005 mm) confirming purely elastic response.
05
Case 2 Loading: Load at Station A, Measure at Station B
Move loading jack to Station A (x_A = 300 mm); apply identical loads P_A at 5.0, 10.0, 15.0, 20.0, and 25.0 kN.
06
Deflection Logging & Reciprocal Comparison
Record deflection at Station B (δ_BA) and direct deflection at Station A (δ_AA); compare δ_AB with δ_BA at every load step to verify reciprocity.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into civil engineering insight.
Points are your recorded observations. The solid regression line validates Maxwell-Betti Reciprocity mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
δ_AB = 0.3150 · P_B
| # | Reciprocal Test Load P (kN) | Cross-Station Deflection δ_AB (mm) | Applied load P_B at station B (kN) | Measured deflection δ_AB at station A (mm) | Remove |
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No readings yet. Execute a civil engineering trial and record it.
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CHECK YOUR UNDERSTANDING
Explain what the civil engineering metrics mean.
The Maxwell–Betti reciprocal theorem proves that the structural flexibility matrix [F] is always:
If a 10 kN load applied at station B produces a 3.15 mm deflection at station A, a 10 kN load at station A will produce at station B a deflection of:
Maxwell-Betti reciprocity strictly requires which of the following conditions?
The Müller-Breslau principle for constructing influence lines is derived directly from:
THEORY AND STRUCTURAL CONTEXT
Civil engineering background & physical principles.
The Maxwell–Betti reciprocal work theorem forms one of the foundational cornerstones of linear structural analysis, energy methods, and modern matrix structural mechanics. Formulated by James Clerk Maxwell in 1864 and generalized by Enrico Betti in 1872, the theorem states: For any linear elastic structure subjected to two independent equilibrium load systems, the work done by the first system acting through the displacements produced by the second system is identically equal to the work done by the second system acting through the displacements produced by the first system. When applied to two discrete load points A and B on a structural beam, the theorem simplifies to the classical reciprocal deflection law: the deflection at station A caused by a unit load at station B (flexibility coefficient f_AB) is identically equal to the deflection at station B caused by a unit load at station A (f_BA), meaning the global structural flexibility matrix is perfectly symmetric: [F] = [F]^T. This mathematical symmetry drastically reduces computational effort in structural indeterminate frame analysis and validates Müller-Breslau influence line principles. In this laboratory experiment, twin digital displacement transducers monitor reciprocal deflections across asymmetric beam stations to verify Maxwell-Betti reciprocity across loads from 5 to 25 kN.
δ_AB = 0.3150 · P_B
Beam span L = 1000 mm. Station A located at x_A = 300 mm; Station B located at x_B = 600 mm. Closed-form flexibility coefficient f_AB = x_A (L - x_B) (2 L x_B - x_A^2 - x_B^2) / (6 E I L) = 0.3150 mm/kN.