\chapter{Effect of Aspect Ratio} \section{Introduction} An important issue for radioactive materials transportation casks is the aspect ratio at which shallow angle slapdown impact events result in a higher velocity secondary impact than primary impact. Two primary parameters have been identified, slenderness and location of the center of gravity. \section{Slenderness} The slenderness issue was studied using solid and hollow cylinders (simple models which capture the essential geometry of most radioactive materials transportation casks). For the solid cylinder model, the initial geometry had a radius of 30 and a length of 120. The length was reduced in steps to 50 while the radius was increased to keep the overall model volume constant. The hollow cylinder model was treated similarly. The initial hollow cylinder model had a length of 120 with an outer radius of 30 and an inner radius of 21.21. These radii gave a cross-sectional area of the hollow cylinder equal to half the area enclosed by the outer radius. As for the solid cylinder, the length was reduced to 50 while the overall model volume and the relationship between cross-sectional areas were held constant. The details of the model geometries are shown in Tables 4.1 and 4.3. The slapdown analysis was performed for an initial angle of 15$^\circ$. Two linear springs were investigated, one of which was elastic and the other plastic. The elastic spring had a linear spring constant for both loading and unloading of 600,000. Thus no energy was absorbed in the spring. The plastic spring had a loading constant of 600,000 (same as the elastic spring) but had an unloading constant of 600,000,000,000 (6 orders of magnitude higher than the loading constant). The effect of this high unloading constant was to prevent energy being returned to the structure on spring unloading thus giving an almost perfectly plastic spring. The initial angle of 15$^\circ$ was sufficient to ensure that for all cases, the nose was rebounding from the target prior to the tail impact. No frictional effects were considered here. \begin{table} \begin{center} \caption{Effect of Aspect Ratio on Slapdown for Solid Cylinder Model with a Linear-Elastic Spring} \makeqnum \begin{tabular}{||r|r|r|r|r|r|r||} \hline &\multicolumn{1}{c|}{Outer} &\multicolumn{1}{c|}{Moment of} &\multicolumn{1}{c|}{Radius of} &\multicolumn{1}{c|}{Aspect} &\multicolumn{1}{c|}{Tail} &\multicolumn{1}{c||}{Tail}\\ \multicolumn{1}{||c|}{Length} &\multicolumn{1}{c|}{Radius} &\multicolumn{1}{c|}{Inertia} &\multicolumn{1}{c|}{Gyration} &\multicolumn{1}{c|}{(L/r)} &\multicolumn{1}{c|}{Velocity} &\multicolumn{1}{c||}{Displ}\\ \hline $120$ &$30.00$ &$??114,000$ &$?37.75$ &$3.18$ &$?-979$ &$6.028$\\ $?80$ &$36.75$ &$???69,700$ &$?29.51$ &$2.71$ &$?-831$ &$5.689$\\ $?60$ &$42.33$ &$???60,000$ &$?27.39$ &$2.19$ &$?-613$ &$4.745$\\ $?55$ &$44.31$ &$???59,400$ &$?27.25$ &$2.02$ &$?-526$ &$4.243$\\ $?50$ &$46.48$ &$???61,200$ &$?27.26$ &$1.83$ &$?-409$ &$3.468$\\ \hline \end{tabular} \end{center} Analysis performed at the following initial conditions: \makeqnum \begin{tabular}{lll} Initial vertical velocity &= &$-527.5$\\ Initial angle &= &$???15.0^\circ$\\ Total mass &= &$???80.0$\\ Spring - elastic Load &= &$??600.0\times10^3$\\ Spring - elastic Unload &= &$??600.0\times10^3$\\ \end{tabular} \end{table} \begin{table} \begin{center} \caption{Effect of Aspect Ratio on Slapdown for Solid Cylinder Model with a Plastic (Energy-Absorbing) Spring} \makeqnum \begin{tabular}{||r|r|r|r|r|r|r||} \hline &\multicolumn{1}{c|}{Outer} &\multicolumn{1}{c|}{Moment of} &\multicolumn{1}{c|}{Radius of} &\multicolumn{1}{c|}{Aspect} &\multicolumn{1}{c|}{Tail} &\multicolumn{1}{c||}{Tail}\\ \multicolumn{1}{||c|}{Length} &\multicolumn{1}{c|}{Radius} &\multicolumn{1}{c|}{Inertia} &\multicolumn{1}{c|}{Gyration} &\multicolumn{1}{c|}{(L/r)} &\multicolumn{1}{c|}{Velocity} &\multicolumn{1}{c||}{Displ}\\ \hline $120$ &$30.00$ &$??114,000$ &$?37.75$ &$3.18$ &$-754$ &$5.031$\\ $?80$ &$36.75$ &$???69,700$ &$?29.51$ &$2.71$ &$-678$ &$4.804$\\ $?60$ &$42.33$ &$???60,000$ &$?27.39$ &$2.19$ &$-567$ &$4.426$\\ $?55$ &$44.31$ &$???59,400$ &$?27.25$ &$2.02$ &$-523$ &$4.261$\\ $?50$ &$46.48$ &$???61,200$ &$?27.26$ &$1.83$ &$-465$ &$3.990$\\ \hline \end{tabular} \end{center} Analysis performed at the following initial conditions: \begin{tabular}{lll} Initial vertical velocity &= &$-527.5$\\ Initial angle &= &$???15.0^\circ$\\ Total mass &= &$???80.0$\\ Spring - plastic Load &= &$??600.0\times10^3$\\ Spring - plastic Unload &= &$??600.0\times10^9$\\ \end{tabular} \end{table} \begin{figure} \vspace{3.5 in} \caption{Effect of Aspect Ratio (L/r) on Slapdown Severity} \end{figure} The results of the slapdown analysis for the solid cylinders with elastic and plastic springs are presented in Tables 4.1 and 4.2 respectively. For the hollow cylinders, the results are shown in Tables 4.3 and 4.4. The ratio of length to radius of gyration (L/r) was selected to describe the slenderness of an object subjected to shallow angle slapdown. The vertical velocity of the tail at the secondary impact and the maximum tail spring displacement were chosen to represent the severity of the secondary impact event. The maximum spring displacement is directly related to the energy required to stop the tail of the object while the vertical velocity at impact provides a clear representation of the slapdown event independent of the tail spring characteristics. Tail vertical velocity at impact, non-dimensionalized by the initial velocity, is plotted against aspect ratio for both the solid and hollow cylinders in Figure 4.1. In Figure 4.1, non-dimensional tail velocities less than one, indicate that slapdown did not occur (secondary impact was less severe than primary impact). Slapdown did not occur when the aspect ratio was less than 2 for both model geometries and both spring types. The plastic spring brings the tail velocity at secondary impact closer to the initial velocity for all aspect ratios. Thus, with the plastic spring, secondary impact velocities are lower for aspect ratios greater than 2 and higher (but still less than the initial velocity) for aspect ratios less than 2. \begin{table} \begin{center} \caption{Effect of Aspect Ratio on Slapdown for Hollow Cylinder Model with a Linear-Elastic Spring} \makeqnum \begin{tabular}{||r|r|r|r|r|r|r|r||} \hline &\multicolumn{1}{c|}{Outer} &\multicolumn{1}{c|}{Inner} &\multicolumn{1}{c|}{Moment of} &\multicolumn{1}{c|}{Radius of} &\multicolumn{1}{c|}{Aspect} &\multicolumn{1}{c|}{Tail} &\multicolumn{1}{c||}{Tail}\\ \multicolumn{1}{||c|}{Length} &\multicolumn{1}{c|}{Radius} &\multicolumn{1}{c|}{Radius} &\multicolumn{1}{c|}{Inertia} &\multicolumn{1}{c|}{Gyration} &\multicolumn{1}{c|}{(L/r)} &\multicolumn{1}{c|}{Velocity} &\multicolumn{1}{c||}{Displ}\\ \hline $120$ &$30.00$ &$21.21$ &$??123,000$ &$?39.21$ &$3.06$ &$?-944$ &$5.974$\\ $?80$ &$36.75$ &$25.99$ &$???83,200$ &$?32.25$ &$2.48$ &$?-741$ &$5.360$\\ $?62$ &$41.74$ &$29.52$ &$???77,900$ &$?31.20$ &$1.99$ &$?-508$ &$4.128$\\ $?55$ &$44.31$ &$31.34$ &$???79,100$ &$?31.44$ &$1.75$ &$?-373$ &$3.202$\\ $?50$ &$46.48$ &$32.87$ &$???81,500$ &$?31.91$ &$1.57$ &$?-261$ &$2.315$\\ \hline \end{tabular} \end{center} Analysis performed at the following initial conditions: \begin{tabular}{lll} Initial vertical velocity &= &$-527.5$\\ Initial angle &= &$???15.0^\circ$\\ Total mass &= &$???80.0$\\ Spring - elastic Load &= &$??600.0\times10^3$\\ Spring - elastic Unload &= &$??600.0\times10^3$\\ \end{tabular} \end{table} \begin{table} \begin{center} \caption{Effect of Aspect Ratio on Slapdown for Hollow Cylinder Model with a Plastic (Energy-Absorbing) Spring} \makeqnum \begin{tabular}{||r|r|r|r|r|r|r|r||} \hline &\multicolumn{1}{c|}{Outer} &\multicolumn{1}{c|}{Inner} &\multicolumn{1}{c|}{Moment of} &\multicolumn{1}{c|}{Radius of} &\multicolumn{1}{c|}{Aspect} &\multicolumn{1}{c|}{Tail} &\multicolumn{1}{c||}{Tail}\\ \multicolumn{1}{||c|}{Length} &\multicolumn{1}{c|}{Radius} &\multicolumn{1}{c|}{Radius} &\multicolumn{1}{c|}{Inertia} &\multicolumn{1}{c|}{Gyration} &\multicolumn{1}{c|}{(L/r)} &\multicolumn{1}{c|}{Velocity} &\multicolumn{1}{c||}{Displ}\\ \hline $120$ &$30.00$ &$21.21$ &$??123,000$ &$?39.21$ &$3.06$ &$-736$ &$4.962$\\ $?80$ &$36.75$ &$25.99$ &$???83,200$ &$?32.25$ &$2.48$ &$-633$ &$4.659$\\ $?62$ &$41.74$ &$29.52$ &$???77,900$ &$?31.20$ &$1.99$ &$-515$ &$4.224$\\ $?55$ &$44.31$ &$31.34$ &$???79,100$ &$?31.44$ &$1.75$ &$-448$ &$3.896$\\ $?50$ &$46.48$ &$32.87$ &$???81,500$ &$?31.91$ &$1.57$ &$-391$ &$3.562$\\ \hline \end{tabular} \end{center} Analysis performed at the following initial conditions: \makeqnum \begin{tabular}{lll} Initial vertical velocity &= &$-527.5$\\ Initial angle &= &$???15.0^\circ$\\ Total mass &= &$???80.0$\\ Spring - plastic Load &= &$??600.0\times10^3$\\ Spring - plastic Unload &= &$??600.0\times10^9$\\ \end{tabular} \end{table} A secondary impact less severe than the primary impact has been shown, in Chapter 2, to be a general relationship for aspect ratios less than two. In Chapter 2, the aspect ratio was defined in terms of $\beta_{n} = l_{n}/r$ and $\beta_{t} = l_{t}/r$ where: \begin{tabbing} $r \;$ \= is the radius of gyration,\\ $l_{n}$ \> is the distance from the center of mass to the nose, and\\ $l_{t}$ \> is the distance from the center of mass to the tail.\\ \end{tabbing} Here $l_{n}$ = $l_{t}$ = ${{L}\over{2}}$, so that \begin{equation} \beta_{n} = \beta_{t} = {{L}\over{2r}}. \end{equation} Thus \begin{equation} \beta_{n}\cdot\beta_{t} = {{L^{2}}\over{4r^{2}}}. \end{equation} As shown in Equation 2.3.19, the tail impact velocity is less than the nose impact velocity when \begin{equation} \beta_{n} \cdot \beta_{t}\leq 1 \end{equation} or when \begin{equation} {{L}\over{r}} \leq 2. \end{equation} \section{Center of Gravity Location} In most transportation systems, due to the impact limiters, closure systems, gamma shielding, and contents, the center of gravity rarely coincides with the midpoint between loading springs. Thus, variation in the location of the center of gravity are of concern for transportation system design. Two of the solid cylinder models, those with length 120 and length 55, were chosen to investigate this phenomenon. These two models provide a wide variation in aspect ratio (3.18 and 2.02, respectively). The center of gravity was shifted independently of all other parameters including the moment of inertia. A physical interpretation of this shift is difficult to provide. It can be regarded as resulting from the appropriate distribution of a variable density material within the confines of the solid cylinder model geometry. Alternately, it can be regarded as a structure with the given mass and moment of inertia placed at varying locations along a rigid massless bar. The elastic and plastic springs described in Section 4.2 were also used here. \begin{figure} \vspace{3.5 in} \caption{Effect of the Location of the center of gravity on Slapdown Severity} \end{figure} The results of shifting the center of gravity axially on the two solid cylinder models are shown in Figure 4.2 and in Tables 4.5 and 4.6. As can be seen in these tables, both the aspect ratio and the nose spring characteristics, influence the effects shifting the center of gravity. For the long cylinder, the tail velocity is maximized when the center of gravity is at the 30\% location (closer to the nose than the tail) for both the linear and the nonlinear springs. For the short cylinder, maximum tail velocity occurs when the center of gravity coincides with the cylinder center (50\% location). As expected, for an arbitrary center of gravity location, maximum energy absorption at the tail does not coincide with maximum tail velocity. The maximum tail energy occurs at a different center of gravity location for each case investigated. For the long cylinder with an elastic nose spring, maximum tail energy occurs at a center of gravity location of 60\% (slightly toward the tail). With a plastic nose spring, maximum tail energy occurs at the 90\% location. For the short cylinder, the maximum tail energy center of gravity location is 70\% for an elastic nose spring and 80\% for a plastic nose spring. \begin{table} \begin{center} \caption{Effect of Shift of center of gravity on Slapdown for Solid Cylinder Model Length=120 with Elastic and Plastic Spring} \makeqnum \begin{tabular}{||c|r|r|c|r|r||} \hline \multicolumn{3}{||c|}{Elastic Spring} &\multicolumn{3}{c||}{Plastic Spring}\\ \hline \multicolumn{1}{||c|}{C. G.} & & &\multicolumn{1}{c|}{C. G.} & &\\ \multicolumn{1}{||c|}{Location} &\multicolumn{1}{c|}{Tail} &\multicolumn{1}{c|}{Tail} &\multicolumn{1}{c|}{Location} &\multicolumn{1}{c|}{Tail} &\multicolumn{1}{c||}{Tail}\\ \multicolumn{1}{||c|}{\% Length} &\multicolumn{1}{c|}{Velocity} &\multicolumn{1}{c|}{Displ} &\multicolumn{1}{c|}{\% Length} &\multicolumn{1}{c|}{Velocity} &\multicolumn{1}{c||}{Displ}\\ $10$ &$?-406$ &$1.557$ &$10$ &$-470$ &$1.791$\\ $20$ &$?-967$ &$4.085$ &$20$ &$-743$ &$3.165$\\ $30$ &$-1132$ &$5.357$ &$30$ &$-827$ &$4.080$\\ $40$ &$-1093$ &$5.865$ &$40$ &$-809$ &$4.654$\\ $50$ &$?-979$ &$6.028$ &$50$ &$-754$ &$5.031$\\ $60$ &$?-847$ &$6.065$ &$60$ &$-689$ &$5.263$\\ $70$ &$?-721$ &$6.049$ &$70$ &$-626$ &$5.466$\\ $80$ &$?-608$ &$5.945$ &$80$ &$-570$ &$5.607$\\ $90$ &$?-501$ &$5.613$ &$90$ &$-520$ &$5.722$\\ \hline \end{tabular} \end{center} Analysis performed at the following initial conditions: \makeqnum \begin{tabular}{llllll} Initial vertical velocity &= &$-527.5$ &Initial vertical velocity &= &$-527.5$\\ Initial angle &= &$???15.0^\circ$ &Initial angle &= &$???15.0^\circ$\\ Total mass &= &$???80.0$ &Total mass &= &$???80.0$\\ Spring - elastic Load &= &$??600.0\times10^3$ &Spring - plastic Load &= &$??600.0\times10^3$\\ Spring - elastic Unload &= &$??600.0\times10^9$ &Spring - plastic Unload &= &$??600.0\times10^9$\\ \end{tabular} \end{table} \begin{table} \begin{center} \caption{Effect of Shift of center of gravity on Slapdown for Solid Cylinder Model Length=55 with Elastic and Plastic Spring} \makeqnum \begin{tabular}{||c|r|r|c|r|r||} \hline \multicolumn{3}{||c|}{Elastic Spring} &\multicolumn{3}{c||}{Plastic Spring}\\ \hline \multicolumn{1}{||c|}{C. G.} & & &\multicolumn{1}{c|}{C. G.} & &\\ \multicolumn{1}{||c|}{Location} &\multicolumn{1}{c|}{Tail} &\multicolumn{1}{c|}{Tail} &\multicolumn{1}{c|}{Location} &\multicolumn{1}{c|}{Tail} &\multicolumn{1}{c||}{Tail}\\ \multicolumn{1}{||c|}{\% Length} &\multicolumn{1}{c|}{Velocity} &\multicolumn{1}{c|}{Displ} &\multicolumn{1}{c|}{\% Length} &\multicolumn{1}{c|}{Velocity} &\multicolumn{1}{c||}{Displ}\\ $10$ &No Hit &$0.???$ &$10$ &$-198$ &$1.110$\\ $20$ &$-193$ &$1.146$ &$20$ &$-358$ &$2.185$\\ $30$ &$-401$ &$2.658$ &$30$ &$-460$ &$3.075$\\ $40$ &$-500$ &$3.655$ &$40$ &$-510$ &$3.755$\\ $50$ &$-526$ &$4.243$ &$50$ &$-523$ &$4.261$\\ $60$ &$-508$ &$4.522$ &$60$ &$-516$ &$4.636$\\ $70$ &$-467$ &$4.562$ &$70$ &$-496$ &$4.902$\\ $80$ &$-416$ &$4.398$ &$80$ &$-471$ &$5.044$\\ $90$ &$-361$ &$4.048$ &$90$ &$-444$ &$5.031$\\ \hline \end{tabular} \end{center} Analysis performed at the following initial conditions: \makeqnum \begin{tabular}{llllll} Initial vertical velocity &= &$-527.5$ &Initial vertical velocity &= &$-527.5$\\ Initial angle &= &$???15.0^\circ$ &Initial angle &= &$???15.0^\circ$\\ Total mass &= &$???80.0$ &Total mass &= &$???80.0$\\ Spring - elastic Load &= &$??600.0\times10^3$ &Spring - plastic Load &= &$??600.0\times10^3$\\ Spring - elastic Unload &= &$??600.0\times10^9$ &Spring - plastic Unload &= &$??600.0\times10^9$\\ \end{tabular} \end{table} \section{Conclusions} The ratio of length to radius of gyration is the most appropriate parameter to describe the body geometry for shallow angle slapdown. The conclusion reached in Chapter 2 for linear behavior, that for a length to radius of gyration ratio less than two, no slapdown (increase in tail velocity at impact) can occur, has been numerically confirmed for nonlinear spring behavior as well. It is difficult to draw general conclusions concerning location of the center of gravity. For off-center location of the center of gravity, a simple relationship between tail velocity at impact and required energy absorption no longer exists. The maximum tail velocity occurs at a different center of gravity location than does the maximum tail spring energy absorption. Therefore, severity of the secondary (tail) impact is difficult to define. Thus, off-center location of the center of gravity can have an important effect on slapdown but each case must be investigated individually.