Experimental Crash Analysis of a Thin-Walled Box Column Using Cylinder as Web C.P.Santhosh Ananda, R.Gobinath, K.Mathiyazhakan, S.Arulmurugan, P.Lokesh Department of Mechanical Engineering, Nandha College of Technology, Perundurai, Erode Abstract In engineering industries the use of thin walled structure is enormous. This is because of their high strength to weight ratio, low cost and excellent energy absorption capability during crashworthiness. The Specific Energy Absorption of the column is analyzed by performing a crashworthiness analysis of a model in a UTM machine. The model consists of a thin-walled column with cylinder as web with constant thickness throughout its length. During modeling Specific Energy Absorption is set as design objective, side length and thickness as design variable and maximum crushing force is set as design constrains. Based on the set of experimental results from the crushing of model, the response surface model is developed and the accuracy of the model is calculated using response surface method. After obtaining the responses from the experiment and Response Surface model the significant increase in energy absorption efficiency of column with web is investigated and validated. 1. Introduction The use of thin-walled structure is growing continuously. Such columns most appear in truss and frame structures as major energy-absorbing components and absorb a substantial amount of crash energy when the impact occurs. Therefore, such columns receive a lot of research interests and literatures have demonstrated their responses and performances during crashworthiness analyses. In designing such columns, maximizing their energy-absorption capability should always be a major objective. There are two approaches to enhance the performance of the multi-corner thin-walled columns that is either using advanced materials with high mechanical properties or designing optimized wall thickness and cross-sectional dimensions for such columns that can provide the best crash performances. However, little effort has been spent on the optimization of the cross-sectional dimensions of the thin-walled columns. Also, besides the straight columns, curved thin-walled column is another important energy-absorbing component which mainly takes bending moment during the crashes. During the optimum design, an advanced technique, the Response Surface Method (RSM) is applied to approximately formulate the columns energy-absorption capabilities. The RSM is presented by Myers and Montgomery and extensively developed by other researchers, which is to use some simple basis functions such as polynomials to approximate the crash behavior of a structure. The RSM method has been employed to optimize several other thin-walled structures with crashworthiness criterion models. 2. Literature Review Over the past decades the thin-walled structures are known to be effective energy absorbing components and enormous efforts have been taken to understand the behavior of the structure. With the advance in analytical and numerical methods the columns with more complex section become the focus of attention. 2.1 Crash worthiness model and analysis Yecheng liu (2008) presents crashworthiness design of a regular multi-corner thin walled column with different types of cross sectional profiles. The profile includes octagonal columns and curved hexagonal columns. The straight octagonal columns were optimized first which mainly takes axial crash load during crash. Following the same approach to the curved hexagonal column and is subjected to bending. During the optimization, specific energy absorption were set as design objective, the maximum crushing force is the design constrain and the side length of the cross section and wall thickness were taken as design variables. The material used for the cross section is steel. Using the material properties LS_DYNA is used to create the model. The formulation optimization problems are solved using FEA. From the analysis it is observed that increasing the side length will increase the structures energy absorption capability. Jandaghi Shahi et al (2012) analyzed the crush behavior of aluminium alloy 6061 circular tubes which are segmented and subjected to quasi-static axial loading. It was investigated analytically and 97
experimentally. The aluminium tubes were modeled by integrating available analytical models and superposition principle. The authors suggested that one successful approach towards obtaining the high energy absorption capacity and lightweight energy absorber is the use of thin-walled tailor made tubes. The authors also suggested that the crush force can be controlled by changing the thickness and length of each tube. By this method the performance of energy absorbing system gets improved. The axial crush behavior and energy absorbing characteristics of four segmented aluminium tubes with circular cross sections were investigated. The mode shapes in a TMT depends t/d ratio of the specimen and L/D ratios of different sections. Concertina mode was mostly observed in the specimen with higher t/d ratio and in the segments with higher L/D ratio. 2.2 Optimum Design Yucheng liu (2007) provided an optimum design for regular thin walled box section beams. The cross sectional beams perform ideally during the crash worthiness analysis. The author utilized the response surface method to formulate the complex design problems. During the optimum design the specific energy absorption is set as design objective and it is constrained by the maximum crushing force and the cross- sectional dimensions. Detailed finite element models are created for such thin walled box beams. Optimum design obtained with the simplified model is compared to the design achieved from the detailed models to verify the accuracy and efficiency of the simplified models. Using UTM machine the crash worthiness analysis on the design samples and generated the RS functions for the SEA and Pm. Finally Minitab is applied to solve this optimization problem and the optimal design. 3. Result and Discussion 3.1 Modeling the Samples for Conducting Experiment The models are created based on the design objective and design variables and the crashworthiness analysis are performed on the models to obtain the response. Sl. No Side length (mm) Fig.1 Element Model Table 1 Table for SEA comparison between Experimental and RSM SEA by Thickness weight (kg) Experiment (mm) (kj/kg) SEA by RSM (kj/kg) 1 30 1 0.481 19.53 18.39 2 30 2.5 1.121 17.33 18.75 3 30 3 1.311 17.76 17.47 98
4 45 1 0.733 16.16 18.02 5 45 2.5 1.749 21.23 20.03 6 45 3 2.064 19.96 19.31 7 60 1 0.985 13.35 12.65 8 60 2.5 2.378 16.5 16.29 9 60 3 2.821 15.19 16.13 Fig.2 Force Displacement curve for side length 30 mm and thickness 1mm Figure 3 Force Displacement curve for side length 30 mm and thickness 2.5mm 99
Figure 4 Force Displacement curve for side length 30 mm and thickness 3mm Figure 5 Force Displacement curve for side length 45 mm and thickness 1mm 100
Figure 6 Force Displacement curve for side length 45 mm and thickness 2.5mm Figure 7 Force Displacement curve for side length 45 mm and thickness 3mm 101
Figure 8 Force Displacement curve for side length 60 mm and thickness 1mm Figure 9 Force Displacement curve for side length 60 mm and thickness 2.5mm Using an UTM machine the models are and the results obtained are plotted in a force displacement graph for each model as shown above. From the analysis the force displacement graphs obtained for all the design variables and the energy absorbed by each model are calculated. This is possible only when the area under curve is calculated. This area is calculated using a trapezoidal method. But calculating the area under a curve is a difficult process. By using a graph pad prism6 software the area under the curve for each model is calculated. This area under the curve is the energy absorbed by the structure during crashworthiness analysis. The formula for calculating the SEA is the total energy absorbed by the structure to its mass. Using the basic mathematical formulas the volume of the thin-walled column for each model is calculated as the density of the material is defined in the material properties the weight of each column is found. By substituting the energy value and weight of each column in SEA formula the response SEA by experimental method for each model is listed in the table. 102
3.2 Calculation of SEA by RSM φ = By following the method described earlier the matrix consisting of basic functions evaluated using sampling points which is 1 30 1 900 30 1 1 30 2.5 900 75 6.25 1 30 3 900 90 9 1 45 1 2025 45 1 1 45 2.5 2025 112.5 6.25 1 45 3 2025 135 9 1 60 1 3600 60 1 1 60 2.5 3600 150 6.25 1 60 3 3600 180 9 By substituting the matrix φ in equation B= (φ T φ) -1 φ T y the regression coefficients β i values in equation (3) is obtained[5].the regression coefficients as well as the polynomial response functions are also determined based on the matrices and the experimental results. The response polynomial function of SEA in quadratic form is SEA= 2.63361+ (0.736500*side length) + (2.86991*thickness) - (0.0111407*sidelength 2 ) - (1.38556*thickness 2 ) + (0.073717*side length*thickness). Using the quadratic form the response obtained using the RSM method is listed in the table. The accuracy of the model obtained by finite element and response surface can be verified by calculating the relative error between them. Using the following formula the error is calculated RE ŷ (x) = the original response of SEA by RSM y (x) = the original response of SEA from FEA By using the formula the relative error between the original and observed response is shown in the table below Table for Relative error between Finite Element and RSM Ŷ(x) Y(x) Relative Error (RE) SS E SS T 18.39 19.53-0.059 1.299 4.343 18.75 17.33 0.082 2.016 0.013 17.47 17.76-0.016 0.084 0.099 18.02 16.16 0.115 3.460 1.654 20.03 21.23-0.057 1.440 14.319 19.31 19.96-0.033 0.423 6.320 12.65 13.35-0.052 0.490 16.777 16.29 16.5-0.013 0.044 0.895 16.13 15.19-0.004 0.884 5.090 103
Other two important properties in evaluating the model s accuracy are the sum of squares of the residuals (SS E ) and the total sum of squares (SS T ). The values obtained by using the formulas below are listed in the table SS E j ŷ j ) 2, SS T j - ȳ j ) 2 where ȳ j is the mean value of y j. Using the values of SS E and SS T the model fitness can be evaluated by F statistic coefficient of multiple determination R 2, adjusted R 2 statistic and root mean square error (RMSE). According to the RSM theory smaller the value of RMSE the better the model fit. The table below shows the value of R 2, adjusted R 2 and RMSE Model fitting RS model R 2 R 2 adj RMSE Quadratic polynomial 0.7951 0.4356 2.2516 From the table the values obtained is less and not within the range (0, 1) and also RMSE is higher compared to R 2 and R 2 adj value. This shows that fitting of the model is less. This is due to the selection of number of sampling design points and the order of polynomials. By increasing the number of sampling points and the order of polynomial will provide a best approximation on the column s response and such function can be used in the design. At the same time the SEA is higher for the models having cylinder as web compared to the box column without web. The increase in SEA is shown in table 5.4. The SEA value for box column without web is obtained from [1]. 3.3 Discussion The result shows that the fitness of the model is not in the range (0, 1). The model fitting can be improved by the selection of sampling points and order of polynomial. The results from the crashworthiness analysis and the response surface model for the box with web shows there is an increase in specific energy absorption. Table for SEA comparison of column with and without web Side length (mm) Thickness (mm) SEA of Column without web by experimental method (kj/kg) SEA of Column with web by experimental method (kj/kg) 30 1 7.09 19.53 30 2.5 11.54 17.33 30 3 12.77 17.76 45 1 5.53 16.16 45 2.5 9.09 21.23 45 3 9.70 19.96 60 1 4.57 13.35 60 2.5 10.37 16.50 60 3 13.73 15.19 104
The comparison table shows that the proposed model with cylinder as web inside a box column shows that the increase in SEA by 45% over the column without web. The increase of SEA in box column is due to the presence of web in the column. References [1]. Liu Y (2008), Crashworthiness design of multi-corner thin-walled columns. Thin-Walled Structures. Vol.46, No.12, pp 1329-1337. [2]. Liu Y (2008), Optimum design of straight thin-walled box section beams for crashworthiness analysis. Finite Elements in Analysis and Design. Vol.44, No 3, pp 139-147. [3]. Raymond H.Myers, Douglas C.Montgomery, Christine M.Anderson-Cook Response Surface Methodology,Process and Product Optimization using designed experiments. Third edition [4]. Alavi Nia A and Parsapour M (2013), An investigation on the energy absorption characteristics of multi-cell square tubes. Thin-Walled Structures. Vol.68, pp 26-34. 105