Dynamic Buckling of Columns Due to Slamming Loads D. Karagiozova*, N

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

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

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

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.

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 ]

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. This is consistent with the observations made in the present theoretical study and in the experimental results reported in Reference [1]. When the deformed column (having initial imperfections w<, = 0.25 + 1.34 (mm)) is dropped from the next level, H = 90 mm, buckling occurs at a lower axial strain (Figure 2(d)). The plastic deformations in bending are far beyond the elastic limit having values 8-10 times the elastic limit strain. Plastic deformations develop also in springs 1 of the cells at both sides of the central one. The deformed shape of the column has a maximum transverse displacement w ~ 10 mm which is shown in Figure 4(a). The numerical calculations reveal that the forces due to a flat-bottom slamming with drop heights from 40 mm to 90 mm cause only elastic deformations and do not produce any plastic deformations in uniform compression. It transpires that the duration of the slamming force is long enough to cause sufficient bending, or buckling, of the column which produces plastic strains. The maximum resultant strain, e|^ at the middle of the column versus the peak axial compressive strain near the impacted end | eJ , is shown in Figure 4(b), where good agreement is obtained with the experimental results, reported in [1]. The maximum value of |e, corresponds to the maximum drop height, which the column can withstand without developing a significant change of the initial shape.

4 Conclusions

The numerical simulation of the dynamic elastic-plastic buckling phenomena, which are caused by a slamming load, reveals that the representation of the pressure-time history by a modified semi-sine wave is in good agreement with

-0.00 1 - H = 60mm -0.002 h H = 70mm

J 40 60 t (msec) 40 60 t (msec)

(a) (b)

H = 85mm H = 90mm -0.015 -

20 40 60 WAAAAAAAAAA 22 t (msec) ^R t (msec) 0.015

Figur e 3: Resultant strains at the inside, e\, and the outside, ^2, in the middle of the column for different drop heights

model predictions; residual strains, e\* and C2R, measured experimentally [ 1 ]

400

/(mm) H = 85mm \ H = 90mm

200

-10 0 10 20 w (mm)

(a)

max experiment 0.009 model

0.006

0.003

0.000 0.0004 0.0006 0.0008 0.0010 led (b)

Figure 4: Buckling of a column having a critical buckling stress near to the yield stress, Oy. (a)r final buckling shapes for two successive drop

heights; (b) maximum resultant strain, I c Lax, in the middle of the column versus peak axial strain near the impacted end, I £c I

the experimentally obtained force acting on the column. It is shown that if the elastic response of the column is known from experimental tests, then a relationship between the peak value of the slamming force and the drop height can be obtained which is valid also for slamming forces which produce plastic strains. It is evident from the results obtained that the response of a column is a quasi-static one and that it buckles in the fundamental mode largely because the slamming force acts over a relatively long time (i.e., milliseconds). Plastic deformations develop only in the middle of the column, so that an even simpler model consisting of two rigid links and discussed in [3] would be adequate to predict the response of the column to a hydrodynamic slamming load within the range of the considered parameters. It should be remarked that a second bucking mode can be initiated elastically for a slender column [1] but with a further rearrangement with the passage of time into the first mode [1,5]. In this case, the model in Figure 1 can give an adequate prediction for the response of the column. However, it is not necessary for a large number of degrees of freedom to be involved in the analysis as the response time of the structure is much larger than the time of the elastic wave propagation along the column (~ 200) and the wave effects are not important.

Acknowledgments

The first author is grateful to the Royal Society for their partial support of this study through an exagreement exchange award. The authors wish to acknowledge the assistance of Mr. H. Parker with the figures and Mrs. M. White for her secretarial assistance.

References

1. Zhang, Q., Li, S. & Zheng, J. J. Dynamic response, buckling and collapsing of elastic-plastic straight columns under axial solid-fluid slamming compression -1. Experiments, International Journal of Solids

and Structures, 1992, 29, 381-397.

2. Jones, N. Structural Impact, Cambridge University Press, Cambridge,

U.K., 1989.

3. Karagiozova, D. & Jones, N. Some observations on the dynamic elastic-plastic buckling of a structural model, International Journal of Impact Engineering, 1995, 16, 4, 621-635.

4. Karagiozova, D. & Jones, N. A note on the inertia and strain-rate effects in the Tarn and Calladine model, International Journal of Impact Engineering, 1995, 16, 4, 637-649.

5. Karagiozova, D. & Jones, N. Multi-degrees of freedom model for dynamic buckling of an elastic-plastic structure, International Journal of Solids and Structures (In Press).

6. Chuang, Sh.-L. Experiments on flat-bottom slamming, Journal of Ship Research, 1966, 10, 10-17.