A sandwich composite beam is to be constructed using woven$-$roving fiberglass $($GFRP$)$ skins and a PVC core. The simply$-$supported beam has a span of $800-mm,100-mm$ width, $25-mm$ thick core, and two identical $2\mathrm{.}5-$mm FRP skins. The beam is subjected to a uniformly distributed load of $4\frac{N}{m}m.$ Determine the following:

a$)$ The maximum normal stress in the FRP skins

b$)$ The maximum shear stress in the PVC core

c$)$ Select an appropriate Divinycell $H$ core from the ones listed in the table below for a factor of safety of $3$ against shear failure in the core

d$)$ The beam deflection at the mid$-$span

The properties of the FRP skins are: $E_{11}=E_{22}=14,000$MPa,$v_{12}=0\mathrm{.}25,G_{12}=2400$MPa.

MECHANICAL PROPERTIES DIVINYCELL'H

$\backslash$table$[[$Property$,$Test Procedure,Unit,,H$45,$H$6$O$,$H$8$O$,$H$100,$H$130,$H$160,$H$200,$H$250],[\backslash$table$[[$Compressive$],[$Strength$]],$ASTMD $1621,$MPa,Nominal,$0\mathrm{.}6,0\mathrm{.}9,1\mathrm{.}4,2\mathrm{.}0,3\mathrm{.}0,3\mathrm{.}4,5\mathrm{.}4,7\mathrm{.}2],[$Minimum$,0\mathrm{.}5,0\mathrm{.}7,1\mathrm{.}15,1\mathrm{.}65,2\mathrm{.}4,2\mathrm{.}8,4\mathrm{.}5,6\mathrm{.}1],[$Compressive Modulus',ASTM D$1621-$B$-73,$MPa,Nominal,$50,70,90,135,170,200,310,400],[$Minimum$,45,60,80,115,145,175,265,350],[$Tensile Strength,ASTMD$1623,$MPa,Nominal,$1\mathrm{.}4,1\mathrm{.}8,2\mathrm{.}5,3\mathrm{.}5,4\mathrm{.}8,5\mathrm{.}4,7\mathrm{.}1,9\mathrm{.}2],[$Minimum$,1\mathrm{.}1,1\mathrm{.}5,2\mathrm{.}2,2\mathrm{.}5,3\mathrm{.}5,4\mathrm{.}0,6\mathrm{.}3,8\mathrm{.}0],[$Tensile Modulus $^(1),$ASTMD $1623,$MPa,Nominal,$55,75,95,130,175,205,250,320],[$Minimum$,45,57,85,105,135,160,210,260],[$Shear Strength,ASTMC $273,$MPa,Nominal,$0\mathrm{.}56,0\mathrm{.}76,1\mathrm{.}15,1\mathrm{.}6,2\mathrm{.}2,2\mathrm{.}6,3\mathrm{.}5,4\mathrm{.}5],[$Minimum$,0\mathrm{.}46,0\mathrm{.}63,0\mathrm{.}95,1\mathrm{.}4,1\mathrm{.}9,2\mathrm{.}2,3\mathrm{.}2,3\mathrm{.}9],[$Shear Modulus,ASTMC $273,$MPa,Nominal,$15,20,27,35,50,60,73,97],[$Minimum$,12,16,23,28,40,50,65,81],[$Shear Strain,ASTM C $273,\backslash$bar $(\%),$Nominal,$12,20,30,40,40,40,45,45],[$Density$,150845,$kg$/$m$^(\text{'}),$Nominal,$48,60,80,100,130,160,200,250]]$