Draw qualitatively the deformed shapes of the beams. Note that if the bending moment in a portion of a beam is zero, that portion moves as a rigid body, and its displaced centerline must remain straight. Observe the support and continuity conditions.

To set-up a problem for determining the elastic line do the following:

a)  Identify beam segments over which M/EI has different expressions. Obtain the function representing M/EI in each beam segment.

b) Write the boundary conditions and continuity conditions by which the integration constants are to be determined.

2.1 Set-up the following problems.  In all cases, choose the origin of the x axis at the left end of the beam.  Assume all loads are positive as shown, EI constant, and the beam span is L.   Since EI is constant, give only M instead of M/EI in step a).  In step b), do not repeat the boundary conditions if they remain the same as in the preceding case. 

Case (a) is done for illustration.

Note: A linear function f having the values f₁ at x = 0, and f₂ at x = L has the expression

f =  f₁(1-x/L) + f₂x/L.

(a) M = M₁(1 - x/L)

     v_{x= 0} = 0,  v_x=L = 0

(b) M =

(c)

2.2. Set-up the following problems similarly to the problems of part 2.1.

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3.1. Superposition

a) The deflection function of segment AB is the same as that of a simply supported member subjected to end moments. Draw a figure showing segment AB, its end moments and its deformed shape. What are the rotations at A and B from the formulas in the book appendix?

b) The overhang to the left of A is similar to a cantilever whose end A has a known rotation. Obtain the deflection at the load point by superposition of 1) the rigid-body displacement of the overhang induced by the rotation at A, and 2) the deflection due to the deformation of the cantilever considered fixed at A. Obtain needed formulas from the book appendix.

c) Obtain the rotation at the point load by superposition.

d) Obtain the deflection and rotation at the tip of the overhang at B by superposition.

The joint at A between the vertical and horizontal members is rigid.  Are the calculations in this problem any different from those in the preceding problem for obtaining deflections and rotations?

a) The bending deflection of AB is the same as that of a cantilever fixed at B and subjected at A to a moment. Draw a figure showing AB, its end moment and its deformed shape. Neglect the axial deformation of AB.  What are the deflection and rotation at A from the formulas in the book appendix?

b) The deflection and rotation at A are known from part a).  Obtain the displacements and rotation at C by superposition of the rigid body motion of AC induced by the continuity at A, and the deformation of AC as a cantilever fixed at A. Obtain needed formulas for deformations from the book appendix.  For the rigid body motion of AC,  you may superimpose a translation downward to a small rotation about A.