Q$2\mathrm{.}2$

$25$ MARKS

The flexural design of a beam was completed for ultimate limit states. The beam now needs to be checked for deflections. The beam was first subject to loading $($self$-$weight of $3\mathrm{.}38k\frac{N}{m},$ live load of $4\mathrm{.}0k\frac{N}{m}($no live load sustained$))$ when concrete was $28-$days old.

Three months after this point, non$-$structural finishes likely to be damaged by large deflections were installed. Loading at this point is now: self$-$weight of $3\mathrm{.}38k\frac{N}{m},$ superimposed dead load of $5\mathrm{.}0k\frac{N}{m},$ and live load of $6\mathrm{.}0k\frac{N}{m}(20\%$ of this live load is sustained$).$ An additional $30$ kN point load $($live load, $50\%$ sustained$)$ is also applied at midspan of the beam at this time.

Many properties relating to the beam were previously determined:

Gross moment of inertia: $I=3\mathrm{.}13{10}^{9}m{m}^4$ Distance from uncracked centroid to tension face: Concrete compressive strength:

Concrete elastic modulus: $E_c=27,700$MPa Tensict steel reinforcement ratio: $=0\mathrm{.}0157$

Cracked moment of inertia: $I=1\mathrm{.}42{10}^{9}m{m}^4$

$y_t=250mm$

$f_c^{\text{'}}=40$MPa

Steel elastic modulus: $,E_s=200,000$MPa

Compression steel reinforcement ratio: $,=0\mathrm{.}0078.$

$1$ of $10$

$($a$)$ Determine the maximum instantaneous deflection $($sustained and transient$)($do not account for long term deflections at this point$)$ that occurs immediately prior to the installation of the non$-$structural components.

$($b$)$ Determine the maximum instantaneous deflection $($sustained and transient$)($do not account for long term deflections at this point$)$ that occurs after the installation of the non$-$structural components.

$($c$)$ Does the deflection that occurs after the installation of the non$-$structural finishes violate relevant requirements from Table $9\mathrm{.}3$ of A$23\mathrm{.}3$:$24$ when the beam is over $5$ years old?