The shear force and bending moment diagrams are shown in Figs. 7.31A, parts (c) and (d).

Since the formulas account for the beam continuity, the resulting shear force and bending moments shown in Figs. 7.31A(c) and (d) include the effect of secondary moments. The resulting moment due to prestress, then, is the algebraic sum of the primary and secondary moments. Once the resulting moments are determined, the secondary moments can be calculated by the relation

Mbal = the resulting moment, also referred to as the total moment in the redundant beam due to equivalent loads Mp = the primary moment that would exist if the beam were a statically determinate beam (Mp is given by the eccentricity of the prestress multiplied by the prestress.)

Msec = the secondary moment due to redundant secondary reactions

With the known primary moment acting on the continuous beam, the secondary moment caused by induced reactions can be computed from the relation

A similar equation is used to calculate the shear forces.

The resulting secondary shear forces and bending moments are shown in Figs. 7.31A(g) and (h), while the primary shear forces and bending moments are shown in Figs. 7.31A(e) and (f).

Example 2B. Compatibility Method. To firm up our concept of secondary reactions and moments, perhaps it is instructive to redo the previous example using a compatibility approach. In this method the beam is rendered statically determinate by removing the redundant reaction at B. The net vertical deflection (which happens to be upward in our case) is calculated at B due to Wp _ 1.225 kip-ft acting upward and a vertical downward load _ 1.225 x 54 _ 66.15 kips acting downward at B. Observe that the reaction at B, along with those at A and C, equilibrates the vertical load of 1.225 kip-ft action on the tendon in its precise profile but does not necessarily guarantee compatibility at B.

Given. A two-span continuous beam analyzed previously, shown again for convenience in Fig. 7.31B.

Required. Secondary moments and shear forces using a compatibility approach.

Solution. The equivalent loads required to balance the effect of prestressed, draped tendons are shown in Fig. 7.31B(b). As before, Wp _ 1.225 kip-ft. However, the reactions at A, B, and C do not include those due to secondary effects. The reactions are in equilibrium with load Wp, and do not necessarily assure continuity of the beam at support

264.24 kip,

264.24 kip

33.07 kip

33.07 kip

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