Civil Engineering

Propped cantilever has 4 reactions (M, R_A at fixed end, R_B at prop) but only 3 equilibrium equations. DSI = 4 - 3 = 1.

Stiffness matrix [K] is defined by equation [K]{δ} = {F}, relating nodal displacements {δ} to nodal forces {F}.

Beam deflection formula δ = f(W,L,E,I) depends on all four parameters: load, span, elastic modulus, and second moment of inertia.

For a cantilever beam, the slope at the free end depends on the bending moment distribution and the beam's flexural rigidity.

The relationship between bending moment and slope is given by the standard beam theory equation, where the bending moment M(x) at distance x from the fixed end equals the second derivative of deflection.

For a cantilever beam with fixed end at x = 0 and free end at x = L, the bending moment at distance x from the fixed end is:

Integrating the differential equation once gives the slope:

At the fixed end (x = 0), the slope is zero: \(C = -\frac{wL^3}{6EI}\)

At the free end (x = L), the slope is:

The magnitude of slope at the free end is:

**The answer is (A) $\frac{wL^3}{6E

Superposition principle applies for linear elastic systems with small deformations where effects are additive and independent of loading sequence.

For a simply supported beam with central point load, maximum bending moment occurs at the center due to symmetry. BM_max = WL/4 = 40×8/4 = 80 kNm

For UDL on simply supported beam, BMD is parabolic (second-degree curve) with maximum at center.

Castigliano's first theorem states that ∂U/∂P = δ, where U is strain energy and δ is deflection in the direction of force P.

For a cantilever beam with point load at free end, maximum deflection = WL⁴/(3EI). This is a standard formula in structural analysis.

In the method of sections, we can cut maximum 3 members in a single section to ensure we have only 3 unknowns, which can be solved using 3 equilibrium equations (ΣFx=0, ΣFy=0, ΣM=0).