Answer Happy

A two-stepped beam of flexural rigidity 3EI, over the first half of its length and EI, over the other half is fixed at the end x=0 and pinned at the end x=L and is supported on a linear elastic spring of spring constant K at the midpoint. A transverse load P₁ (N) also acts at the mid-point. Use the multi-part Rayleigh-Ritz approach to analyze this problem. A two-part Rayleigh-Ritz approach makes sense here: P₁ [a₁ + R₂ { } } + x₁ [ ] ² for 0≤x≤ L/2 FL. u₁(x) = X X α₂ + B₂₁ +127 for L/2≤x≤ L L/2 L/2 K_(N/m) where a₁,₁,₁ ₂,272 are constants. Let u, be the mid-point deflection. Given: L=4m; E=2.1x10¹¹ Nm³²; I₂ =4x10-ºm¹; K= 10000N; P₁ = 20,000N. (3.i) By imposing the appropriate geometric boundary conditions at x=0 and x=L as well as continuity of deflection and slope at x=L/2, obtain α₁,₁,₁; α₂,B₂,72 in terms of the mid-point deflection u₁. [7 points] (3.ii) Crank the PMPE machinery to obtain the RR parameter u₁. [6 points] (3.iii) Determine where the beam deflection is largest and obtain the largest deflection. [2 points] X