Two$-$span post$-$tensioned concrete beam

The figure shows a post$-$tensioned concrete beam with an idealized parabolic tendon, consisting

of two parabolas with a "kink" in the middle.

The eccentricity of the tendon in relation to the centroid of the cross section is defined by

$e_1=0\mathrm{.}4m$ and $e_2=0\mathrm{.}4m.$

The prestressing force is assumed to be constant with a value $F=3000kN.$

The figure also shows the moment diagram along the beam due to prestressing.

$F=$

a$)$ Calculate and show the diagram for the restraining moment $($secondary moment$)$ along the

beam.

b$)$ Calculate the reactions at the supports due to the prestressing.

c$)$ Determine the eccentricities $e_1$ and $e_2$ to achieve a concordant prestressing, which gives the

same moment diagram as shown in the figure.

d$)$ Define the geometry of a concordant prestressing system for the $2-$span continuous beam

that is loaded by its self$-$weight $q_1$ and a uniformly distributed load applied only on one span

of value $q_2.$ Leave the results in terms of the variables $q_1,q_2$ and the length of each span $L.$