Consider a normal weight concrete having a $28-$day design strength of $4000$ psi. The plant

where the concrete is produced has standard deviation of compressive strength equal to $500$ psi

for this mix design.

$($a$)$ What true tensile strength should be counted on in design for this concrete?

$($b$)$ What mean $28-$day compressive strength is likely for the concrete if it was produced so

that less than one in one hundred samples will have strength less than $500$ psi below the

design concrete strength? $($Note: Codes actually have more detailed requirements.$)$

$($c$)$ Suppose concrete cylinders prepared and stored according to the standard ASTM procedures

test out at $4000$ psi at $28-$days. What mean strength would you expect from $3-$in by $3-$in

cylinders cored from the structure at $28$ days?

$($d$)$ For the same concrete described in part $($c$),$ what strength would you expect from $6-$in by

$12-$in$.$ lab cured cylinders at age of $7$ days?

$($e$)$ A slab of concrete is cast and cured in a manner similar to a standard cylinder, so that its

in$-$situ uniaxial compressive strength is $4000$ psi $.$ The slab is loaded under biaxial principal

stressed $f_1$ and $f_2.$

I. The loading plan is to increase $f_2$ to $2000$ psi while $f_1$ is held at $0,$ and then

increase $f_l.$ When do you expect failure?

II$.$ The loading plan is to increase $f_2$ to $5000$ psi while $f_1$ is held at $0,$ and then

increase $f_l.$ When do you expect failure?

$($f$)$ A plain concrete column is subjected to hydrostatic confining stressed of $1000$ psi in all

three principal directions. A load is then applied to the column along one principal axis.

At what stress in addition to the hydrostatic stress will the column fall? $($Assume the

concrete has exactly $4000-$psi uniaxial compressive strength in the column.$)$

$($g$)$ If the concrete in part $($c$)$ is cycled for $1$ million cycles between $1000$ psi compression

and some maximum stress, what maximum stress will result in failure?

$($h$)$ What true tensile strengths are expected for the concrete described in part $($c$)$ if it is $120$

pcf lightweight concrete?

$($i$)$ A finite element analysis results in the following stresses at a point. If the concrete is that

which is described in part $($c$),$ will is fail?