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Observe that in this example, the secondary moment at B _ 150% of the primary moment due to prestress. The secondary moment is thus secondary in nature, but not in magnitude.

Example 2A.

Given. The two-span prestressed concrete beam shown in Fig. 7.31A(a) has a parabolic tendon in each span with zero eccentricity at the A and C ends, and at the center support B. Eccentricity of the tendon at the center of each span _ 1.7 ft. The prestress force P _ 263.24 kips.

Required. Secondary reactions and moments.

Solution. The approach here is similar to that typically used in commercially available computer programs. However, in the computer programs, statically indeterminate structures such as the example problem, are typically analyzed using a stiffness matrix approach. Here we take the easy street: We use beam formulas to analyze the two-span continuous beam. It should be noted that the analysis could be performed using other classical methods such as the moment distribution method or slop-deflection method.

First we determine the equivalent load due to prestress P _ 263.24 kips acting at eccentricity e _ 1.7 ft at the center of the two spans. The equivalent load consists of: 1) an upward uniformly distributed load Wp due to drape in the tendon; 2) a horizontal compression P equal to 263.24 kips at the ends; 3) downward loads at A, B, and C to equilibrate the upward load Wp; and 4) additional reactions at A, B, and C due to the restraining effect of support at B. The last set of loads need not be considered for this example, because the loads are implicitly included in the formulas for the statically indeterminate beam.

Of the equivalent loads shown in Fig. 7.31A(b), only the uniformly distributed load Wp corresponding to P acting at eccentricity e induces bending action in the beam. Wp is

Figure 7.31A. Concept of secondary moments—example 2A: (a) two-span continuous prestressed beam; (b) equivalent loads due to prestress, consisting of upward UDL, horizontal compression due to prestress Wp, and downward loads at A, B, and C; (c) shear force diagram, statically indeterminate beam; (d) moment diagram, statically indeterminate beam; (e) primary shear force diagram; (f) primary moment diagram; (g) secondary shear forces; (h) secondary moments.

Figure 7.31A. Concept of secondary moments—example 2A: (a) two-span continuous prestressed beam; (b) equivalent loads due to prestress, consisting of upward UDL, horizontal compression due to prestress Wp, and downward loads at A, B, and C; (c) shear force diagram, statically indeterminate beam; (d) moment diagram, statically indeterminate beam; (e) primary shear force diagram; (f) primary moment diagram; (g) secondary shear forces; (h) secondary moments.

determined by the relation WpL2

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