\chapter{Verification} \section{Introduction} \SLAP\ was verified by comparison of the analytical results of this code with accelerometer data from the half-scale Defense High Level Waste (DHLW) cask test program [3]. The half-scale DHLW cask geometry and instrumentation locations are shown in Figure B.1. The cask was dropped from a height of 30 ft. $\pm$ 1 in. at an initial angle of 10$^\circ \pm$ 1$^\circ$. such that the narrower end (opposite the closure) of the cask contacted the unyielding target first. Comparisons of experimental and analytical accelerations and velocities (integrated accelerations) were made at the accelerometer locations A7, A1, and A3. A7 is located on the cask bottom end (initial impact end). A1 is located slightly toward the closure from the center of gravity. A3 is located on the closure (secondary impact) end. \begin{figure} \vspace{3.5 in} \caption{Half-Scale DHLW Cask Model Instrumentation Drawing} \end{figure} \section{\SLAP\ Input Development} A typical input to \SLAP\ is shown in Table B.1. The total mass was determined from the measured weight of the half-scale cask. The moment of inertia was calculated from the geometry, weight, and location of the individual cask components. The moment of inertia calculation was not experimentally verified. The location of the cask center of gravity (length from nose to CG and tail to CG) was also calculated and verified experimentally. Agreement between the calculation and measurement of the center of gravity location was within .09 in. \begin{table} \begin{center} \caption{Input Parameters to \SLAP\ for the 30 Foot Drop Test of the DHLW Half-Scale Model at 10$^\circ$} \makeqnum \begin{tabular}{||l|c||} \hline Geometry: & \\ \quad Nose to CG &$??25.68$\\ \quad Tail to CG &$??23.5?$\\ \quad Mass &$??15.2?$\\ \quad Moment of Inertia &$8900.??$\\ Nose Spring: & \\ \quad Loading &$400000.$ at 1.\\ &$973000.$ at 3.\\ \quad Unloading &$8000000.$\\ Tail Spring: &\\ \quad Loading &$388000.$ at 1.\\ &$888000.$ at 2.8\\ \quad Unloading &$8000000.$\\ \hline Initial Conditions: & \\ \quad Initial Velocity &$-527.45$\\ \quad Initial Angle &$??10.??$\\ \hline \end{tabular} \end{center} \end{table} The springs were defined using the results of finite element analysis and testing of a honeycomb structure similar to (but much simpler to analyse) that used for side impact limiters on the DHLW cask. A ring of aluminum honeycomb with an inside diameter of 20 in., an outside diameter of 31.75 in. and an axial length of 10.125 in. was applied to the axial center of a 5500 lb. cylinder. The honeycomb cell structure was aligned radially, as in the DHLW impact limiters. A sketch of the test structure is shown in Figure B.2. \begin{figure} \vspace{3.5 in} \caption{Honeycomb Crush Test Structure} \end{figure} The cylinder was dropped from 15 and 22 ft. The cylinder drops were analysed using DYNA3D [4]. The analysis results for acceleration, final honeycomb crush (permanent impact limiter displacement), and footprint were in good agreement with the experimental results. These analysis results were then used to define the spring behavior of the DHLW impact limiters required by \SLAP . The results for the 22-foot test cylinder drop were used because the amount of crush matched that expected in the DHLW impact limiters due to a 30-foot drop. The acceleration of the center of gravity of the cylinder predicted by DYNA3D for the 22-foot drop is shown plotted against the center of gravity displacement in Figure B.3. \begin{figure} \vspace{3.5 in} \caption{Acceleration versus Displacement for Honeycomb Crush Test Structure from DYNA3D for a 22-Foot Drop} \end{figure} This curve was smoothed, using the Butterworth low pass filter implemented in GRAFAID [5], to eliminate the contribution of the high frequency deformation modes to the acceleration. In order to apply a Butterworth filter, the curve must be single valued. A series of initial and trailing zeros will also facilitate the filtering operation. Thus the analytical curve of Figure B.3 was modified as shown in Figure B.4 by eliminating the unloading portion of the curve and adding the leading and trailing zeros. \begin{figure} \vspace{3.5 in} \caption{Acceleration versus Displacement for Honeycomb Crush Test Structure from DYNA3D for a 22-Foot Drop - Filtered} \end{figure} After filtering, the acceleration was converted into load by multiplying by the mass, and then scaled to the length used in the DHLW impact limiter. This result was approximated with bi-linear spring definition shown in Fig B.5 and used in the \SLAP\ input described in Table B.1. \begin{figure} \vspace{3.5 in} \caption{Approximation of Load versus Displacement Behavior of Honeycomb Impact Limiters on the DHLW Half-Scale Model} \end{figure} The unloading modulus was estimated directly from Figure B.3. The effects of friction were ignored in the {\em spring} definition. \section{Comparison of \SLAP\ with Experiment} The \SLAP\ program writes results at the locations of the nose spring, the center of gravity, and the tail spring. For the half-scale DHLW cask test, these locations do not coincide with the accelerometer locations. However, because the cask is analysed as a rigid body, the analytical data may be determined at any arbitrary location by linear interpolation. Thus, the \SLAP\ results were interpolated to be consistent with the accelerometer locations shown in Figure B.1. Variation in drop height, within the experimental uncertainty, has a negligible effect on the cask behavior and thus was ignored. The initial angle, however, has an appreciable effect on the duration of the slapdown event along with a minor effect on the magnitudes of the accelerations and velocities. Therefore, slapdown analyses were run with initial angles between 9$^\circ$ and 14$^\circ$. The values of acceleration and velocity from \SLAP\ for the 13$^\circ$ initial angle are compared to the experimental data (at the three accelerometer locations A7, A1, and A3) in Figures B.6 - B.11. The 13$^\circ$ initial angle was chosen for display here because it provided the best match for the experimental data, perhaps indicating some rotation during the 30-foot free drop of the test. \begin{figure} \vspace{3.5 in} \caption{Comparison of Analytical and Experimental Vertical Accelerations at Location A7} \end{figure} \begin{figure} \vspace{3.5 in} \caption{Comparison of Analytical and Experimental Vertical Accelerations at Location A1} \end{figure} \begin{figure} \vspace{3.5 in} \caption{Comparison of Analytical and Experimental Vertical Accelerations at Location A3} \end{figure} \begin{figure} \vspace{3.5 in} \caption{Comparison of Analytical and Experimental Vertical Velocities at Location A7} \end{figure} \begin{figure} \vspace{3.5 in} \caption{Comparison of Analytical and Experimental Vertical Velocities at Location A1} \end{figure} \begin{figure} \vspace{3.5 in} \caption{Comparison of Analytical and Experimental Vertical Velocities at Location A3} \end{figure} Because the actual cask is not perfectly rigid, the accelerometers record some high frequency response. The experimental data in the Figures B.6 - B.11 has been filtered at 500 Hz to allow for easier comparison. 500 Hz was the lowest frequency which did not significantly alter the rise and fall times and the total pulse width of the data. Unfortunately, there is still a significant high frequency component in the accelerometer response remaining. Because of this remaining high frequency response, even after filtering, it is difficult to make accurate quantitative comparisons of acceleration values. However, when the accelerometer data are integrated to give velocities, the comparison is remarkably good. When the coarseness of the spring definition, the neglect of friction, and the experimental uncertainties are considered, these results indicate that slapdown events can be analysed with sufficient accuracy for a great number of purposes including determination of worst initial angles for testing and the effects of variations in impact limiter and cask parameters.