Page 76 - Handbook of Civil Engineering Calculations, Second Edition

STATICS, STRESS AND STRAIN, AND FLEXURAL ANALYSIS 1.59

FIGURE 40

Calculation Procedure:

1. Apply a unit horizontal load
Apply the unit horizontal load at A, directed to the right.
2. Evaluate the bending moments in each member
Let M and m denote the bending moment at a given section caused by the load P and by
the unit load, respectively. Evaluate these moments in each member, considering a mo-
ment positive if it induces tension in the outer fibers of the frame. Thus:
Member AB: Let x denote the vertical distance from A to a given section. Then M 0;
m x.
Member BC: Let x denote the horizontal distance from B to a given section. Then M
Px; m a.
Member CD: Let x denote the vertical distance from C to a given section. Then M
Pb; m a x.
3. Evaluate the required deflection
b c
Calling the required deflection , we apply [Mm/(EI)] dx; EI 0 Paxdx 0
2
c
2
2
b
Pb(a x) dx Pax /2] 0 Pb(ax x /2)] 0 Pab /2 Pabc Pbc /2;
2
2
[Pb/(2EI)](ab 2ac c ).
If this value is positive, A is displaced in the direction of the unit load, i.e., to the right.
Draw the elastic curve in hyperbolic fashion (Fig. 40b). The above three steps constitute
the unit-load method of solving this problem.
4. Construct the bending-moment diagram
Draw the diagram as shown in Fig. 40c.
5. Compute the rotation and horizontal displacement by the
moment-area method
Determine the rotation and horizontal displacement of C. (Consider only absolute values.)
2
Since there is no rotation at D, EI C Pbc; EI 1 Pbc /2.
6. Compute the rotation of one point relative to another and the
total rotation
2
2
Thus EI BC Pb /2; EI B Pbc Pb /2 Pb(c b/2). The horizontal displacement of
B relative to C is infinitesimal.