A simply supported post$-$tensioned concrete beam spanning over $20m$ length has a rectangular

cross$-$section $300mm$ wide and $1000mm$ deep and is subjected to a uniformly distributed load of

$16k\frac{N}{m}$ including self$-$weight. The beam is post$-$tensioned with two high tensile strength steel

cables $1$ and $2,$ tensioned successively in the same sequence, to prestressing force of $2000kN$ in

each cable. Cable $1$ that is post$-$tensioned first has a parabolic cable profile with eccentricity of

$100mm$ below the centroidal axis at the midspan and $100mm$ above the centroidal axis at the

endspans, while cable $2$ that is tensioned subsequently has a parabolic profile with an eccentricity

of $300mm$ below the centroidal axis at midspan and $100mm$ below the centroidal axis at endspans.

Given that the ultimate tensile strength of the post$-$tensioning steel ${}_{su}=900$MPa, area of cross$-$

section of each post$-$tensioning tendon is $4000m{m}^2$ and the grade of concrete is M$35$:

$($a$)$ Calculate the prestress loss in Cable $1$ and Cable $2$ due to elastic deformation of concrete

$($elastic axial shortening of concrete$).$ Assume the modulus of elasticity as per IS: $456-2000.$

Use Simpson's rule for numerical integration for $5$ points $($midspan$,$ quarter and end spans$).$

$($b$)$ Calculate the prestress losses in cables $1$ and $2$ due to shrinkage of concrete in accordance

with IS $1343-2012$ assuming dry humidity conditions $($relative humidity

$($c$)$ Calculate the prestress losses in the cables $1$ and $2$ due to creep of concrete using the Creep

Coefficient method as per IS: $1343-2012$ assuming dry humidity conditions. Assume the

grade of concrete as M$35$ and modulus of elasticity as per IS: $456-2000.$ Use Simpson's rule

for numerical integration with five points $($midspan$,$ quarter$-$spans and end spans$).$

$($d$)$ Calculate the prestress loss in Cables $1$ and $2$ due to relaxation of steel as per IS: $1343-2012.$

$($e$)$ Considering that some types of prestress losses $($loss due to friction and anchorage slip$)$ can

be compensated during post$-$tensioning operation by over$-$stressing the cables, compute the

total prestress loss in Cables $1$ and $2($loss due to elastic deformation of concrete, shrinkage of

concrete, creep of concrete and relaxation of steel$)$ as a percentage of the original prestress.

$($f$)$ Calculate the effective prestressing force in each of the post$-$tensioning tendons i$.$e$.$ Cable $1$

and Cable $2$ after the total prestress loss computed in part $($e$)$ above.

$($g$)$ Based on IS: $1343-2012,$ calculate the over$-$tensioning force that needs to be applied in each

cable in order to compensate for losses due to friction and anchorage slip during the post$-$

tensioning operation using a jacking device at one end Assume that the Coefficient of friction

between duct and cable $=0\mathrm{.}20,$ Friction coefficient for wave effect $=\frac{0\mathrm{.}0018}{m}$ and

Anchorage slip at jacking end $=11mm.$

$($h$)$ Calculate effective prestressing force in Cables $1$ and $2$ after prestress losses due to all the

factors mentioned above assuming that no over$-$tensioning force has been applied to

compensate or overcome the prestress losses due to friction and anchorage slip.

Useful Formulae: Simpson's $1/3$ Rule for numerical integration.

${\displaystyle \underset{a}{\overset{b}{}}}f(x)*dx$~~$\frac{(b-a)}{6}[f(a)+4f(\frac{a+b}{2})+f(b)]$

Derive and use Simpson's five$-$point rule for numerical integration of the prestress in concrete

${}_{cp}(x)$ at the level of post$-$tensioning steel at any section $x$ in questions $($a$)$ and $($c$)$ above.