Transactions on the Built Environment vol 22, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509 Dynamic buckling of columns due to slamming loads D. Karagiozova*, N. Jones* ^Institute of Mechanics, Bulgarian Academy of Sciences, Sofia, Bulgaria ^Impact Research Centre, Department of Mechanical Engineering, The University of Liverpool, Liverpool, UK Abstract The dynamic elastic-plastic buckling of a column is examined when subjected to an axial hydrodynamic slamming load such as develops during the landing of a sea plane, or the slamming behaviour of a ship or marine vehicle. The theoretical predictions of a simplified method of analysis are compared with some experimental results published recently by Zhang et al. Good agreement is obtained with the experimental data and some insight is offered on the results including the observation that a quasi-static method of analysis might be adequate for design purposes. 1 Introduction An experimental study on the dynamic buckling of columns due to hydrodynamic slamming compression has been reported recently in Reference [1]. An increased understanding of this particular topic is important for structural designs and calculations related to ship and marine vehicle slamming, the landing of flying boats and other engineering areas. The dynamic elastic-plastic buckling behaviour of structures is complex and is not properly understood, as noted in Reference [2], for example. In order to examine various features of the dynamic buckling phenomena, an idealised model, which retains most of the characteristics of an actual structural system, was studied in References [3,4]. This model gave considerable insight into several features of the dynamic elastic-plastic buckling behaviour of structures and predicted good agreement with the available experimental results for low velocity impacts. In view of the success of the idealised model, the authors have extended the work in References [3,4] in order to examine the dynamic response of Transactions on the Built Environment vol 22, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509 312 Structures Under Shock And Impact systems having higher degrees of freedom for high velocity impacts [5]. Again, good agreement has been obtained between the theoretical predictions and the corresponding experimental results. The theoretical studies reported in References [3-5] and elsewhere have focused on the dynamic elastic-plastic buckling response due to dynamic pressure pulses or dropped weight loadings. Thus, the experimental data reported in Reference [1] on the hydrodynamic slamming behaviour of a column provides an opportunity to examine the accuracy of the idealised model response for another type of loading and boundary condition. It is the objective of this article to compare the predictions of the model in Figure 1 with the experimental results reported in Reference [1] and, in so doing, clarify some aspects of the experimental observations. 2 Loading conditions and specimen characteristics The experimental data reported in Reference [1] are used to determine the hydrodynamic slamming force acting on the column in Figure 1 as well as to estimate the ability of the model to predict the dynamic response of such structural elements. The column in Reference [1] is loaded by heavy mass G attached to the upper end and with a splash plate at the lower end, thus providing clamped boundary conditions at both ends. The unit is dropped onto a water surface from a series of increasing heights. It is assumed here that upon initial contact with the water, the column and the attached mass have an initial velocity v<, = (2gH)\ where H is the drop height and that at t = 0 the slamming force, P(t), commences to subject the column to a compression load. It is observed experimentally [1], that the axial compression force is essentially a semi-sine wave within the elastic range of deformation, so that the following expression for the slamming force is assumed [6] , (1) where 2T is the wave period, t is time and p^ is determined by /'map x =t**/o ' (2) The empirical coefficient k in equation (2) is obtained using the experimental results reported in [1] when considering the wholly elastic response due to a slamming load of two specimens having equal cross-sections but different total lengths of £ = 600 mm and £ = 400 mm. It is observed in [1] that the shorter column responds with a shorter pulse and a larger magnitude, while the response of the longer column is characterised by a longer pulse with a smaller magnitude. Using this observation, it is assumed that the pulses acting on columns having different slenderness ratios should be equal for equal drop heights, so that Transactions on the Built Environment vol 22, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509 Structures Under Shock And Impact 313 W, JSI w. w. P(t) Initial Deformed (a) cr Loading Unloading L_ Reloading (b) Figure l:(a) Discrete model: w,-axial displacements, w^-lateral displacements, vv. - initial imperfections; (b) Material characteristics Transactions on the Built Environment vol 22, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509 314 Structures Under Shock And Impact = f In the present study, an elastic-plastic column having f = 400 mm and a rectangular cross-section with dimensions h = 7.7 mm and b = 14.6 mm is considered. The maximum initial imperfection measured in the narrow direction of the cross-section is w^ = 0.25 mm (specimen S31 from [1]). The material characteristics are assumed to be elastic-plastic with linear strain hardening as shown in Figure l(b) and with an elastic modulus E = 206 GPa, yield stress o^ = 263 MPa, elastic wave speed c = 5190 m/sec and a ratio between the hardening modulus and the elastic modulus of E^/E = 0.006. For these particular parameters, the peak value of the slamming force is P^ = 12700v* (TV) , (4) when T = 15 msec [1] and v<, is expressed in m/sec. The idealised model in Figure l(a) consists of N rigid weightless links each of length L connected by springs simulating the material properties. The breadth of the model is Lj = h/2, where h is the thickness of the column. The model has stress-free initial imperfections Wj (i = 0, ..., N) and a total mass, mj, which is distributed as discrete masses m = m/2N at each end of a rigid link. It is assumed that all of the springs have identical characteristics and simulate an elastic-plastic material with linear strain hardening and the Bauschinger effect (Figure l(b)). Strain rate effects are not taken into account. The equations of motion are developed assuming moderate changes in the geometry of the model, so that [(Wj + w,) - (Wj.j + WJ_I)]/L = sin0j « fy, where w^ + Wj are the total displacements at location i (i — 0, ..., N). 3 Response of the model and discussion The idealised model in Figure 1 was used to predict the exhaustion of the load carrying capacity of an elastic-plastic column when it is subjected to a slamming load. It is assumed that the model consists of N = 12 rigid links as higher dynamic buckling models are unlikely for the column in Reference [1]. The boundary conditions are assumed clamped at both ends with respect to the lateral displacements w^ but displacements in the axial direction are allowed at both ends. The numerical calculations reveal that the model in Figure 1 responds initially with an axial compression followed by an overall bending in the fundamental mode. The axial strains are distributed uniformly along the column during the compression phase until they reach their maximum values. A comparison between the model predictions for the axial strain history near the impacted end (at 33.4 mm from the lower end), and the experimentally measured axial strains at 50 mm from the lower end, is shown in Figure 2. Transactions on the Built Environment vol 22, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509 H = 60mm -000087 H = 70mm 40 60 t (msec) (b) Cc ~ H = 85mm ^ -0.00087 H = 90mm 7 A\ -0.00087 #- 0 0 0 \J^ 40~" 60 0 PV> ...- \, 0 ^^"~*"40 60 X t (msec) t (msec) /«\ ^L (d) 1 Figure 2 Axial strain histories near the impacted end for different drop heights model predictions, experimental results [ 1 ] Transactions on the Built Environment vol 22, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509 316 Structures Under Shock And Impact The sequence of the drop heights follows the experiments [1], namely H = 40 mm, 50 mm, 60 mm, 70 mm, 85 mm and 90 mm. For drop heights less than 70 mm, buckling does not occur and only elastic vibrations are observed as shown in Figure 3(a). Strains beyond the elastic limit occur at the middle of the column for H = 70 mm and they are present in the spring which models the inside of the cross-section (spring 1) (Figure 3(b)). However, these strains do not lead to a change of the initial shape of the column, so that, the next drop from H = 85 mm is performed again on an undeformed column. In this case, the slamming load causes significant plastic bending deformations at the middle of the column (springs 1 and 2) and produces the deformed shape of the column shown in Figure 4(a). The associated maximum transverse displacement at the middle is w = 1.34 mm. Buckling of the model having the particular parameters considered occurs plastically according to the buckling criterion developed in [5] when v^ = 1.27 m/s, which corresponds to a drop height H ~ 78 mm. However, this does not lead to a collapse of the column, but only to a stable deformed shape. It was shown in [5], that for a column with a particular attached mass, G, there is a range of impact velocities beyond the velocity at which buckling is initiated when the elastic-plastic column does not collapse, but responds in a new stable deformed position.