Time$-$Effect in Reinforced Concrete Column

a$)$ Show the equilibrium equations for total concrete and steel stresses incorporating time$-$effects from creep and shrinkage and define each term.

b$)$ Calculate initial and the ultimate stress in the concrete and steel. The reinforced concrete column with a cross section of $12$ by $24$ in$.$ and reinforcement ratio of $0\mathrm{.}02($Grade $60$ steel $($i$.$e$.$ yield strength is $60,000).$ Lab testing of plain concrete specimens correspond to $1,500$ in compression. The concrete has a $28$ day strength of $5,000$ and $28$ day conerete elastic modulus $Ec=4\mathrm{.}0x{10}^{{?}^{{?}^{?}}}6$ psi and steel modulus, Es $-29{10}^{{?}^{{?}^{?}}}6$ psi. assume ultimate creep coefficient $($C$($u$)=2\mathrm{.}35$ and ultimate shrinkage strain of $).$

c$)$ Calculate the stress in the steel and the concrete at load$-$durations: $30$ days; $365$ days; and $10,950$ days $(30$ years$)$ based on information above. Then, plot stress results including on a log$-$time scale. Assume the following equation is valid for time$-$dependent development of $$ coefficient:

$C(t)=((t{)}^{{?}^{{?}^{?}}}\frac{0\mathrm{.}6}{10+(t{)}^{{?}^{{?}^{?}}}0\mathrm{.}6}]$

where, $t$ is the load$-$duration time in days since start of loading, and $Cu$ is ultimate creep coefficient $(2\mathrm{.}35).$

Assume shrinkage strain development over time $rsh(t)=(t)[35+t]400{10}^n-6$ in$/$in$/,$ where $t$ is time in days similar to the time since start of loading.

d$)$ Demonstrate through calculations that the total concrete and steel strains are the same after $30$ years of loading.

$2.$ Time Effect in Prestressed Concrete

a$)$ Determine immediately after prestressing release the initial strains in the concrete and steel. Assume that the steel tensile force prior to pre$-$stressing $Ts(0)$ is the maximum allowable based on $70\%$ of yield stress.

Concrete design strength $fe=5,000.$

Concrete design elastic modulus $Ee=4{10}^{{?}^{{?}^{?}}}6$

Steel $s=$ AsiAc $=0\mathrm{.}005$; steel yicld stress $=270,000$ psi.

Beam cross section has dimensions of $12$ inches $($width$)$ by $24$ inches $($height$)$

b$)$ In this case pre$-$stressing along the neutral axis will result in a uniform stress block within the cross section. Determine the maximum bending moment stress $($i$.$e$.$ maximum tensile stress at bottom of the beam and maximum compression at the top$)$ such that the net initial cross section tensile stress is zero at the bottom. Plot the stress distribution for each component.

c$)$ Determine the effect of creep and shrinkage on the steel and concrete strains at ultimate, and the effect on the net concrete stress at bottom and top of the cross section. Plot the new net stress distribution from top to bottom and comment on results. Assume the ultimate creep coefficient of the concrete $Cu$ is $2\mathrm{.}35,$ and the ultimate concrete shrinkage is $700{10}^{{?}^{{?}^{?}}}-6$ in$/$in$.$

d$)$ Estimate percent loss of pre$-$stress at end of service life.

e$)$ Explain using equations why ultimate strain in the concrete cannot be determined from theory of elasticity, while the ultimate concrete stress is obtained from theory of elasticity.