For the cantilever beam of span 3 m (shown below), a concentrated load of 20 kN applied at the free end causes a vertical displacement of 2 mm at a section located at a distance of 1 m from the fixed end. If a concentrated vertically downward load of 10 kN is applied at the section located at a distance of 1 m from the fixed end (with no other load on the beam), the maximum vertical displacement in the same beam (in mm) is _________.

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GATE CE 2014 Official Paper: Shift 1

__Concept__

__Maxwell-Betti Theorem:__

Maxwell-Betti's law of real work is a basic theorem in structural analysis.

Using this theorem, it can be established that the flexibility coefficients in compatibility equations, formulated to solve indeterminate structures by the flexibility method, form a symmetric matrix and this will reduce the number of deflection computations.

The Maxwell-Betti law also has applications in the construction of influence lines diagrams for statically indeterminate structures.

The Maxwell-Betti law applies to any stable elastic structure (a beam, truss, or frame, for example) on unyielding supports and at a constant temperature, states.

Let’s suppose two systems of forces

According to Maxwell-Betti theorem,

∴ Pi × y1 = Pj × y2

Where, y = Deflection, and P is the force

**Calculation:**

P_{1} × Δ_{12} = P_{2} × Δ_{21}

10 × 2 = 20 × Δ21

⇒ Δ21 = 1 mm