Question

For the beam below, determine the deflection at point C. Use E 200 GPa and I = 13.4 x 10 mm". 20 kN/m в С W150 X 24 30 KN -1.6 m-

Answer 1

The solution is done using Castigliano's theorem. We shall place an imaginary load p where we want to find the deflection.  

Potential Energy U = $\int (M^2/2EI)dx$

After integration of the potential energy deflection,   (delta) = $\frac{\partial }{\partial p} (U)$

After differentiating with imaginary load p. substitute p =0 in it.  

There is a little bit of integration we need to do (cumber some)

la P is imaginary load Patro = 20 (1/408) +30 + P 1-6 18+P Ralu) + P(1.6) +20 (1-6) + 28 (2-4) (3-4) - RB (204) = 0. 1.68 +48 + 576 = R(24). 16P + 105-6 = (24) he --0.666 P + 4h > RA= [184p] - [ resp. +45] RA. = 34. + 0.333P Considering a section. Ivo ha - 20(1) - V- V . 34 +0 p -10 (). At x = 1.6 Atx=0 V 2 +0.333P ZMO Rull) -2012) (5) Vel) - M Mx = 3hto map] x – 10 x²

Potential Energy U r 2010N/m MA Now consider section from B from varies from sto os IVO RB - 20(x) - Vx=0 x = 0.6668 +44201 IM - Race) +20 (x) (1) +Mx=0 Mx = RB (x) 20x2. Mx = (0.666 P + 24 dx - 10x U= || (3424 0.333 PX - 10x (Gone 223x - 10x² + 1 [0.666px +4hx - 10x leave '2E1' we can include it at last. (14x) + (0.333px)² + (102)² + 2 (34x) (0.333px) + 2, (0.33384) (102) -2 (10x) (34x) (64x) + (1027+ Co-bromu) + 2 (0.66683)(4x) +2 (448) (-03) -2 Cox') (0.66668)

u= u56x² + o.lll p2 it loox + 22-644P x² + (-666 P x²) - 68023 1936x² + look" - 0.444 p 3x2 + 58-602 Px² - 88013 - - 13.32px - Customs - - 156 C6om 10 or 2 ss maio 2013 10) + 14.699, 1.0), o me" com a se car (ab 20% 136) U= 1578.32 + 0.1515 P² + 209.71 +30 916 P - 10.997P – 1112.112 + 330.410 + 6.553 + 0.015711 + 10.002P-90 112 ~ 136319 BET U= [920.769 + 0.22726 8² +28.6604 P. 2ET Sed - / (0.22726) (3P) + 2866040) + de 0.45452P 28.6604 & 2 E1 28.6604 ac 2 EI

Es. 2008 109 No. = 200 x 10 I = 13.4 x 100 mm KN/m². -13-48106 (163 m) = 13.4 x 10 tx10-12 m = 13-4X156 mb. EI = 200 x 13.4 X 100-6 = 2680 KN-m² dc= 28.6604 (2680) = 8.347 x 103 m. (metres). = 5.347 mm. (Half centimetre deflection).

Kindly go through every step for better understanding.

All the best.!

I hope you shall understand.

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