Hello, I need help finding the deflection of this beam at point B. W.pdf

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amittripathi2002

Hello, I need help finding the deflection of this beam at point B. Where Force F is located. Using castiglianos theorem. https://drive.google.com/file/d/0B4sP15AuGJ6lVXVhdDhHT0ZRZkxlRXppM1pkVWJ1V3lhR kdN/view?usp=sharing Solution Since no values are mentioned for the variables like F and moment therefore I am explaining in general terms so you can apply this in any question of castigliano\'s theorem. For applying castigliano\'s theorem we have to calculate the strain energy of the beam. For that purpose we have to solve the reaction of beam i.e. RA and RC in term of M1 and F by applying equilibrium conditions. Once we have calculated the value of reactions then we are able to write the equation of strain energy. Ubeam = UAB + UBC So first we write for UAB For this we have to write bending moment equation taking a section before point B Mx-x = RA X - M1 ( assumption is taking anticlockwise moment as negative) We know that strain energy is U given by the above expression. Now puttting the value of Mx-x in the strain energy expression will give the strain energy in the term of F and M1 and the limit will be 0 - 3L/4 Now we have to write the strain energy of part 2 therefore we will take a section before point C as y-y So My-y = RA((3L/4) +y) - Fy - M1 , for writing the equation for strain energy we have to replace Mx-x by My-y and the limit will be 0- L/4 so we will get two expression for strain energy we will add up them and get the final or complete strain energy of the beam. Now we are in a position to apply the castigliano\'s theorem to find deflection at point B YB (Deflection at poinnt B) = d/dF(U) where d/dF is the differential and U is the complete strain energy..

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