ABSTRACT
The prestress can be replaced by end moments as shown in Fig. 6.2. When the simply supported beam is prestressed it hogs up but the reactions are not affected. On the other hand, in the case of the continuous beam, the central reaction prevents the beam from deflecting at the centre. The resulting moment distribution due to
the end moments of Pe can be determined from matrix analysis. Assuming uniform flexural rigidity EI and span length L, the stiffness matrix in terms of the rotations at the supports θ1, θ2 and θ3 is given by
Using the Gaussian elimination method, reduce the stiffness matrix to the upper triangular form:
Solving, θ1 = −0.25 (L/EI), θ2 = 0, θ3 = 0.25 (L/EI) The final bending moments are M12 = −Pe, M21 = −0.5Pe, M23 = 0.5Pe, M32 = Pe The reactions are V1 = 1.5 Pe/L, V2 = −3 PE/L, V3 = 1.5Pe/L. Fig. 6.3 shows the three bending moment diagrams. They are
• Due to end moments on the continuous beam with zero deflection at support. This is the final bending moment due to prestress and is the one used in design.
