Question $7$: Consider a rectangular sandwich plate with composite face sheets that is simply supported along all four plate edges. The top and bottom face sheets are identical and manufactured using a GFRP material system with the elastic properties quoted in the Table $1$ and failure properties given in Table $2.$ The sandwich core material is a cross linked PVC material with shear moduli Gc$,$xz$=$Gc$,$yz$=38$ MPa, and Young$$s modulus Ec$=110$ MPa $($isotropic core$).$ The sandwich plate dimensions are: Length a $($x$-$direction$)$; Width b $($y$-$direction$),$ Core thickness tc$.$ The sandwich plate is subjected to uniform in$-$plane compressive loading in the x$-$direction. Use ESAComp and the CLPT analyses presented in the lecture notes to answer the following: a$.$ Consider the sandwich plate discussed above and assume that that GFRP laminate face sheets composed of $4$ plies with a stacking sequence of $[0,90]$S$,$ and a ply thickness of $1$mm$,$ a$=$b$=750$ mm$,$ and tc$=25$ mm: Calculate the critical load for global buckling $($expressed as the critical value of the compressive normal stress resultant$)$ and determine the corresponding buckling mode $($number of half waves in the x and y directions$).$ b$.$ For the sandwich plate problem defined in question a above: Discuss the result obtained in the context of the answer to question a above, and in the context of what the estimated ultimate load corresponding to $$material$$ failure $($compression failure$)$ would be$.$ c$.$ Consider the sandwich plate configuration defined in question a$,$ but now change with a$=250$ mm and a$=2250$ mm$,$ whilst b$=750$ mm $($i$.$e$.$ two cases a$/$b$=1/3$ and a$/$b$=3)$: Calculate the critical loads $($expressed as critical value of compressive normal stress resultant$)$ and determine the corresponding buckling modes $($number of half waves in the x and y directions$).$ Discuss the results obtained. Hint: You can use the CLPT$-$code you have derived in Part $1$ directly to calculate the bending stiffness coefficients in $[$D$].$ You need to expand your CLPT$-$code to calculate the transverse shear stiffness coefficients for the sandwich plate$/$laminate to accommodate for the effect of finite shear stiffness on the buckling response

Table $1$ Lamina engineering elastic constants Property Symbol $($Units$)$ Value Young$$s modulus in the $1-$direction E$1($GPa$)54$ Young$$s modulus in the $2-$direction E$2($GPa$)18$ Major Poisson$$s ratio $120\mathrm{.}28$ Shear Modulus G$12($GPa$)6$ Table $2$ Failure stresses $($strengths$)$ and strains $$ data sheet values: Property Value Xt $($MPa$)1040$ Yt $($MPa$)35$ Xc $($MPa$)570$ Yc $($MPa$)114$ S $($MPa$)72$ $1$t $0\mathrm{.}021$ $2$t $0\mathrm{.}002$ $1$c $0\mathrm{.}011$ $2$c $0\mathrm{.}0064$ $60\mathrm{.}038$ X and Y are the stresses to failure $($strengths$)$ in the directions parallel and transverse to the fibres, and S is the shear strength. The subscripts t and c represent tension and compression. Notation according to lecture notes.