LECTURE 40 ILLUSTRATIVE PROBLEMS Using Castigliano's Theorem : 1. The cantilever beam CD supports a uniformly distributed Load w. and a concentrated load P as shown in figure below. Suppose L = 3m; w = 6KN/m ; P = 6KN and E. I = 5 MN m^{2} determine the deflection at D The deflection 'Y_{0} ‘at the point D Where load ‘P' is applied is obtained from the relation Since P is acting vertical and directed downward d ; represents a vertical deflection and is positions downward. The bending moment M at a distance x from D And its derivative with respect to ‘P' is Substituting for M and ¶ M/ ¶ P into equation (1) 2. Areas a_{1} = 500 mm^{2} a_{2} = 1000 mm^{2} For the truss as shown in the figure above, Determine the vertical deflection at the joint C. Solution: Since no vertical load is applied at Joint C. we may introduce dummy load Q. as shown below Using castigliano's theorem and denoting by the force F_{i} in a given member i caused by the combined loading of P and Q. we have Free body diagram : The free body diagram is as shown below Force in Members: Considering in sequence, the equilibrium of joints E, C, B and D, we may determine the force in each member caused by load Q. Joint E: F_{CE} = F_{DE} = 0 Joint C: F_{AC} = 0; F_{CD} = -Q Joint B: F_{AB} = 0; F_{BD} = -3/4Q The total force in each member under the combined action of Q and P is
P = 60 KN Sub-(2) in (1) Deflection of C. Since the load Q is not the part of loading therefore putting Q = 0 3. For the beam and loading shown, determine the deflection at point D. Take E = 200Gpa, I = 28.9x10^{6} mm^{4} Solution: Castigliano's Theorem : Since the given loading does not include a vertical load at point D, we introduce the dummy load Q as shown below. Using Castigliano's Theorem and noting that E.I is constant, we write. The integration is performed seperatly for portion AD and DB Reactions Using F.B.D of the entire beam Portion AD of Beam : From Using the F.B.D.we find Portion DB of Beam : From Using the F.B.D shown below we find the bending moment at a distance V from end B is Deflection at point D: Recalling eq (1) . (2) and (3) we have 4. For the uniform loaded beam with following supports. Determine the reactions at the supports Solution: |