RESPONSE, DAMPING, AND TRANSMISSIBILITY CHARACTERISTICS OF TOP-LOADED CORRUGATED CONTAINERS

U.S.D.A. FOREST SERVICE RESEARCH PA PER FPL 160 1971

U. S. Department of Agriculture Forest Service Forest Products Laboratory Madison, Wis. ABSTRACT

During transportation, stacked corrugated contain­ ers receive dynamic loading forces from which, added to the weight of the stacked load, may cause failure of the lower containers in the stack This study, in cooperation with the Fibre Box Associa­ tion, experimentally verified that top-loaded con­ tainers are frequency-sensitive systems with resonant ranging from 8.4 to 18.2 cycles per second. Transmissibility amplification ratios as high as 6.7 were found at resonance, with a representative value of system damping of 0.115. These resonant responses occur within the range of frequencies likely to be present in common carrier transportation ve­ hicles. Thus consideration must be given to these dy­ namic effects when selecting or designing corrugated containers. FREQUENCY RESPONSE, DAMPING, AND TRANSMlSSIBILITY CHARACTERlSTICS OF TOP-LOADED CORRUGATED CONTAINERS

By W.O.GODSHALL, Engineer

Forest Products Laboratory1 Forest Service U.S. Department of Agriculture

INTRODUCTION and shipment environment. Storage stacking loads are static “dead” loads which do not vary as a Corrugated fiberboard containers are used al­ function of time, and failure of containers in stor­ most universally for the packaging and shipment age is related primarily to the creep characteris­ of a wide variety of products. Corrugated con­ tics of the container material. Relationships have tainers are economical, light in weight, and use a been developed2, 3 which can be used to predict minimum amount of material to perform their the static load-carrying ability of containers, function because of their efficient structural expressed as a ratio of the top-to-bottom com­ design. However, they do not possess the same pressive strength as determined in a testing inherent compressive strength of heavier and machine. more costly containers. This is an important cri­ During transportation the stacking loads may terion of performance because it represents a not be as great, because stack height is limited limiting factor in the utilization of corrugated by the vehicle, but the Containers experience dy­ fiberboard containers. namic loading forces from the up-and-downvibra­ A shipping container necessarily experiences tion of the vehicle on its suspension system in top-to-bottom compressive loading. In shipment addition to the existing stacking load. and storage, containers must be stacked on top of A previous study,4 conducted at the Forest each other, often to considerable heights, to effi­ Products Laboratory in cooperation with the Fibre ciently utilize available space. Stacking thus im­ Box Association, investigated the effects of trans­ poses large compressive loads on the lower con­ portation vibration on the stacking loads which can tainers. In many instances the packaged item may be safely supported by corrugated containers. be fragile or unable to support a load without Vibration transmissibility theory was used to suffering damage; therefore, the container itself analyze a vertical stack of loaded containers as must be capable of supporting most or all of the a simplified spring-mass system with vibration load. Thus containers must be designed and excitation applied to the base of the stack. This selected to withstand the rigors of the storage analysis showed that such a system was sensitive

1Maintained at Madison, Wis., in cooperation with the University of Wisconsin. 2 KelIicutt, K. Q., and Landt, E. F. Safe Stacking Life of Corrugated Boxes. Fibre Containers 36(9): 28-29. 1951. 3Moody, R. C., and Skidmore, K. E. How Dead Load, Downward Creep Influence Corrugated Box Design. Package Engineering 11(8): 75-81. 1966. 4 Godshall, W. D. Effects of Vertical Dynamic Loading on Corrugated Fiberboard Containers. U.S.D.A. Forest Serv. Res. Pap. FPL 94. 1968. Forest Products Lab., Madison, Wis. to certain frequencies. The dynamic loads im­ includes a substantial portion of the range ex­ posed on the bottom container could be several pected in transportation, but not the range of pre­ times greater than the static stacking load when dicted resonant frequencies, from 10 to 17 Hz. the frequency of the applied vibration was at or This machine also restricted the experimental near the resonant frequency of the system. Cal­ program to discrete fixed frequency and ampli­ culations based on static repeated loading data tude levels. Nevertheless, the experimental data obtained from a universal testing machine pre­ conclusively supported the behavior predicted by dicted that the resonant frequencies of the load the theoretical analysis, within the range of opera­ and container would range Between 10 and 17 Hz tion of the vibration machine. (Hertz) (cycles per sec.), depending on load and Since the response Characteristics of top- acceleration levels, loaded containers depend on the frequency of the A survey of environmental data showed that the vibration input, the desirability and usefulness of range of the most probable experienced more completely determining those characteris­ in rail or truck shipment was from 3 to 20 Hz at tics of the load-container system over the entire acceleration levels between 0.1 and 0.8 G. Thus range of frequency of possible interest is appar­ the range of probable vibration inputs included ent. Therefore, a second of the study was the range of resonant responses of the load- initiated, again in cooperation with the Fibre Box container systems; it was highly probable that Association, to determine those dynamic char­ the containers would receive large dynamic acteristics of top-loaded corrugated containers loading forces. This would cause an effective which are of the greatest value and interest. reduction in the actual stacked load a corrugated Those characteristics include the actual resonant container could safely sustain. frequency (f ), the maximum transmissibility (T ) r r An experimental test program verified the (which occurs at resonance), and the effective conclusions reached in the preceding analysis. damping coefficient (c/c ) of the system. Know­ The experimental procedure used was a highly c simplified simulation of the transportation en­ ledge of these dynamic characteristics will greatly vironment. Single empty containers were top- aid in the design and selection of containers to loaded with weights to various loads ranging be­ withstand expected transportation environments. tween 10 and 70 percent of maximum compressive Additional objectives of this study were to further strength of the box, as determined by previous confirm the applicability of the analysis used. and teats in a universal testing machine. Containers to determine how accurately resonant frequencies could be calculated from machine compression were then subjected to a sinusoidally varying 4 vibration along the vertical axis at discrete fre­ test data using the previously developed method quencies ranging from 4 to 10 Hz at peak accel­ of analysis. eration levels ranging from 0.2 to 0.8 G. Analysis of Stacked Containers This experimental test program showed that as a Mechanical System the load-container systems were indeed frequency sensitive and that the load a container could sus­ For vibration analysis, any mechanical system tain was drastically reduced as the exciting fre­ can be represented by one or more spring-maps quency of vibration approached the calculated systems, as shown in figure 2. Actually a stack resonant frequency of the system A summary of of loaded containers is a complicated system, the test results is shown in figure 1. The load, but for the purposes of our analysis we have rep­ frequency, and vibration level conditions below resented it as a single linear spring-mass sys­ and left of a frequency curve represent the safe tem without damping, with the bottom container loading conditions at which the containers expe­ acting as an elastic spring, and the upper con­ rienced at least 21,600 cycles of loading without tainers acting as a unified mass. The mass exerts failure. The areas above and to the right of a a compressive force on the spring equal to its frequency curve represent the loading conditions weight, which is the force due to the earth’s grav­ which produced failure, itational acceleration, as stated by Newton’s First The capabilities of the mechanically driven Law: Force = Mass x Acceleration. This is a vibrator limited the experimental data to fre­ static loading, such as would be applied to a con­ quencies of 10 Hz or less. This frequency range tainer in storage.

2 M 139 075 Figure 1.--Summary plot of safe loading limits as determined in first phase of study. The area below and to the left of a frequency line defines the combination of loading conditions that will not cause failure of the container at that specified frequency. The area above and to the right of that frequency line defines the conditions under which the container will fail.

M 133 480

Figure 2.--Simple spring-mass system with sinusoidal acceleration applied to the base; F = force transmitted through the spring.

3 If some additional force were applied and then farces, which added to the static laadweight, may suddenly removed, the mass would oscillate ver­ cause compressive overloading and failure of the tically about its static resting position at some container. At frequencies above 1.4 times the particular frequency. This frequency of oscilla­ natural frequency, the vibration input is attenu­ tion is called the natural frequency(f ) of the sys­ ated and the dynamic loading forces become insig­ n nificant. The amount of amplification or tem, and is determined by the magnitude of the attenuation is inversely related to the damping in mass (M) and the stiffness (K) of the spring, as the system. expressed by the relationship: Thus the dynamic characteristics of greatest interest are the resonant frequency, maximum transmissibility, and damping coefficient.

A different action occurs when a sinusoidally MATERIAL varying vibration is applied to the base of the system, in the vertical axis, similar to the actual transportation environment. The mass will oscil­ The corrugated fiberboard containers used for late, not at the natural frequency, but at the fre­ this study were from the same production run as quency of the forcing vibration. However, the the containers evaluated in the first phase of this of vibration of the mass will vary study.4 They were 18-9/16 inches long by greatly, dependent on the relationship between 12-3/8 inches wide by 14 inches high, and were the forcing frequency and the natural frequency. regular slotted containers (RSC) of 200-pound test The ratio of the amplitude of vibration of the single-wall, C-flute. A glued manufacturer's joint mass (A ) to the amplitude of the forcing vibra­ m was used, with a crushed tab inside the container. tion (A ) is called the transmissibility ratio (T) The containers were manufactured according to f normal commercial practice except that a parti­ and is expressed by the relationship cularly high level of quality control was main­ tained to obtain uniformity throughout the production run of 2,100 containers.

SAMPLING PROCEDURE The transmissibility is a function of the ratio of the forcing frequency to the natural frequency and of the damping in the system, as shown in The sampling plan used for this study was the figure 3. This relationship holds for acceleration, same blocked sampling procedure adopted ear­ force and displacement ratios. The resonant fre­ lier? This plan assured that data and conclusions quency (f ) of the mass-spring system is the fre­ r obtained could be directly compared with those quency at which maximum transmissibility (T ) of the previous study. Of the three replicate r specimens used for each test condition, one each occurs, and is for all practical purposes the same was selected from the beginning, center, and end as the natural frequency. sections of the production run. It may be seen from figure 3 that the accelera­ tion of the mass and the resulting forces trans­ mitted through the spring (lower container) vary CONDITIONING greatly, dependent on the frequency of the applied forcing vibration. At frequencies below approxi­ mately one-half the resonant frequency, very little Prior to testing, all containers were precondi­ amplification occurs, and the dynamic forces on tioned and conditioned in accordance with ASTM the lower container are small. Method D 641-49. The hocked-down containers At forcing frequencies between about 0.5 and 1.4 were stored in a preconditioning chamber at 80° F. times the natural frequency, large amplifications and 30 percent relative humidity. They were set of the vibratory input occur producing dynamic up and closed by stapling, top and bottom while

FPL 160 4 Figure 3.--PictoriaI representation showing how the response of a single degree of freedom system varies as a function of the ratio of the forcing frequency to the natural frequency of the system.

5 in this condition. Then they were conditioned at Repeated Compressive Loading Tests 73° F. and 50 percent relative humidity for a mini­ mum of 48 hours prior to testing. This condition The repeated loading tests were conducted to was maintained for the tests in the universal obtain data that could he used to predict the res­ testing machine. onant frequencies of load-container systems at The vibration tests were conducted in a room various levels of loading and at several vibration conditioned to 68° F. and 65 percent relative humi­ . Single containers were loaded in top­ dity. The containers were sealed in a bag of poly­ to-bottom compression in a universal testing ethylene film while at 73° F. and 50 percent machine (fig. 4) but were not loaded directly to relative humidity and were transported and tested failure, as in a conventional test. Instead, the con­ while sealed in the bag. Since the vibration test tainer was first loaded to some selected percen­ durations ranged from only a few seconds to a tage of the previously determined average maxi­ maximum time of 1-1/2 hours, the moisture con­ mum compressive strength. Second, the container tent of the containers did not change significantly was then cycled five times between the maximum during the test period. and minimum loads that would result from this selected load weight being accelerated to a given RESEARCH METHODS vibration amplitude. After the cyclic loading, the AND EQUIPMENT container was loaded to failure. A load-deforma­ tion graph was recorded for the complete loading Two separate evaluation procedures were used procedure. for this study--repeated cyclic loading in a univer­ sal testing machine and sweep frequency tests.

Figure 4.--Repeated loading top-to-bottom compression tests in a universal testing machine. M 139 051

FPL 160 6 Sweep Frequency Tests piston The experimental arrangement is shown in figure 5. Because of the limits of the vibration testing The vibration system employs a closed electro­ equipment then available, the frequency range hydraulic servoloop. A hydraulic power supply investigated in the first phase4 did not fully cover provides the basic source of power for the double- the range of probable transportation vibration, acting piston. This hydraulic power is controlled nor did it cover the range of predicted resonant by an electromagnetic servovalve which is driven frequencies for the load-container systems. by a servoamplifier. Programmed command sig­ Therefore, to accomplish the objectives of this nals are compared with a feedback signal from a study, a vibration exciter was obtained which had displacement transducer attached to the piston; sufficient frequency range and the capability to the difference of these two signals is applied to maintain constant acceleration levels while being the servovalve, which regulates the flow of hy­ swept continuously across the entire range of fre­ draulic fluid in the direction to reduce the differ­ quencies of interest. This was accomplished by ence signal to zero. While the system controls converting an existing electrohydraulic structural in the displacement mode, control signals are testing systeminto a vibration exciter. A hydraulic programmed in the acceleration mode and are piston actuator was mounted vertically on a large then converted to equivalent displacement com­ inertial mass, and a table large enough to hold mands by double integration. This allows acceler­ the test containers was mounted on the end of the ation levels to be held at selected constant values

Figure 5.--Experimental arrangement for sweep frequency transmissibility tests of con­ tainers on electrohydrauIic vibration machine. M 135 305

7 over any range of frequency sweep desired. A tainer occurred or until an arbitrary time limit provided sine-wave command of 1-1/2 hours was reached. Because in many in­ signals which could be manually swept across the stances the resonant frequency of the load- desired range of frequencies, container system shifted with time, the input fre­ An electric hoist handled the loading weights quency was readjusted to maintain resonance. and restrained the load upon. failure of the con­ These sweep frequency tests were conducted at tainer. A steel plate weighing approximately 23 different combinations of dead load and accel­ 500 pounds was fastened to the vibration table to eration input levels to determine the response provide a more suitable operating load for the over the entire range of probable loading hydraulic system and to improve the waveformof conditions. the vibration applied to the test containers. ANALYSIS OF RESULTS Instrumentation Repeated Compressive Loading Tests The vibration input applied to the container and The resonant frequencies of the top-loaded con­ the dynamic response of the load were measured tainers were computed from data obtained in the by a separate instrumentation system, indepen­ repeated compressive loading tests performed in dent of the vibrator control system Two identical a universal testing machine. The analysis was strain gage accelerometers with a maximum range made by considering the loaded container to be a of 15 G and a flat frequency response from d.c. to simple single-degree-of-freedom system, and 250 Hz were used. One accelerometer was mounted computing the resonant frequency from the on the vibrating base, and the other was mounted relationship on the load. The output signals from the acceler­ ometers were amplified by wide-band d.c. ampli­ fiers and recorded by an optical oscillograph, using fluid-damped galvanometers with a flat fre­ quency response from 0 to 600 Hz. The input and response signals were recorded simultaneously Where f = natural frequency of vibration of the n as a function of time. Frequency was determined system visually from the calibrated function generator K = spring factor of the container dial and recorded on the oscillograph chart by the M = mass (W/g) of the load operator. Transmissibility was determined as the ratio of the magnitude of the response signal to the The spring factor was determined from the load- magnitude of the input vibration signal. deformation graph obtained from the repeated loading tests by drawing tangents to the loading curves at the dead load value. A more detailed Dynamic Test Procedure explanation of this technique is given in FPL 94.4 A summary plot of the resonant frequencies com­ The container, enclosed in a vapor barrier, was puted is shown in figure 6. It can be seen that they placed on the vibration table and top-loaded with range from 10.4 Hz to 17.8 Hz and are functions weights to some predetermined percentage of the of both the input acceleration level and of the sup­ top-to-bottom compressive strength of the con­ ported dead load. The resonant frequency de­ tainer. The vibration exciter controls were set to creases as the amplitude of the input vibration produce a specified peak acceleration level, and increases, and tends to increase as the dead load vibration was begun at the low-frequency limit of percentage increases. These results differ from 4 Hz. The frequency was then slowly swept upward those which might be expected of a linear system, through resonance until a frequency was reached and result because a corrugated container has at which substantial attenuation occurred. The nonlinear load-deformation characteristics. The frequency was then swept downward until reso­ container acts as a soft spring with small loads nance was reached and vibration was continued and stiffens as the load increases. The calculated at the resonant frequency until failure of the con­ spring factor varies from 850 pounds per inch at

FPL 160 a Figure 6.--Summary of resonant frequencies of Figure 7.--Summary of resonant frequencies of loaded containers, computed from repeated loaded containers as determined experimental­ compressive Ioading tests, for various ly by sweep frequency tests for various vibration input levels. M 139 076 vibration input levels. M 139 074 a load of 10 percent of maximum compressive at some of the scheduled test conditions, became strength to 14.030 pounds per inch at a load of the severity of the test condition caused failure TO percent of maximum compressive strength. of the container before resonance was reached, as the input frequency was slowly swept upward toward the resonant frequency. These failure con­ Resonant Frequencies ditions were at the same levels as previously de­ termined in the first phase of this study 4 with The resonant frequencies of the load-container fixed vibration frequencies. systems as determined by the sweep frequency Nonlinear systems are known to exhibit frequen­ teats are shown in figure 7. Resonant frequencies cy instability, where the resonant amplitude will could be determined at only 16 of the 23 test con­ shift up or down, depending on whether resonance ditions. The other seven test conditions were is approached from a higher or from a lower fre­ severe enough that the container failed before the quency. This phenomenon was noted only a few resonant frequency was reached. times in these tests, and the change in amplitude A comparison of the predicted and experimen­ was not large. However, at a number of the differ­ tally determined resonant frequencies can be ob­ ent test conditions, a low-level resonant response tained from table 1, which summarizes results. was noted at approximately one-half the resonant Reasonably good agreement is obtained, with the frequency. Within thelimits of the frequency range predicted values running slightly above the initial employed in these tests, no multi-order harmonic experimental values. However, for test conditions responses were noted, that did not severely load the container, a gradual increase of the resonant frequency with time was noted. This appeared to be due to working and Transmissibility stiffening of the corners of the boxes, causing the spring factor or stiffness of the box to increase. The maximum transmissibility found at each For most of these instances, the predicted reso­ test condition is given in table 1. Values range nant frequency fell between the initial and final from 1.65 to 6.70, but do not necessarily repre­ values of the actual resonant frequency. Thus the sent the transmissibility at resonance because, simplified analysis, using spring factors easily at many test conditions, the container failed before obtained from the repeated loading tests, is capa­ resonance was reached. Because of the high trans­ ble of accurately predicting the resonant frequen­ missibilities, peak dynamic loads sufficient to cies of top-loaded corrugated containers. cause the containers to fail were reached even The resonant frequency could not be determined though the static load weight was relatively small.

9 Table 1.--Summary of frequency response, damping, and transmissibility characteristics1

This may be seen from the calculated maximum 70 percent of the machine compressive strength dynamic loading forces given in table 1. These of the container, and did not fail (with one excep­ values were calculated using the relationship tion) when the maximum dynamic lead was less than 70 percent. The exception occurred when the input acceleration level was 1.0 G, which allowed the container to leave the vibrator bed and impact against it. Previous studies2,3 on where F = maximum dynamic loading force static stacking strength of corrugated containers m have shown that containers can sustain about W = static load weight 70 percent of their machine compressive strength G = input acceleration level G’s i for periods of a day or two. T = maximum transmissibility ratio m and F and W are expressed as a percentage of Effective System Damping m the machine compressive strength of the The transmissibility of a vibrating spring-mass container. system is a function of the damping characteris­ The containers failed at all test conditions tics of the system An undamped spring-mass where the maximum dynamic load exceeded system theoretically would have a transmissibi-

FPL 160 10 lity of infinity at resonance. The higher the damp­ vide a useful estimate of the damping ing in a system, the lower the resonant response characteristics of corrugated fiberboard will be. Since the actual transmissibilities have containers. been determined experimentally, approximate Damping ratios for each of the 23 load-container values of equivalent system damping can be de­ systems evaluated are given in table 1. Values termined, using the relationships range from 0.07 to 0.30. Highest values were ob­ tained from containers which were dynamically overloaded and failed in a relatively low number of cycles. The transmissibilities recorded for these containers were not obtained at resonance, because failure occurred before resonance was where = damping factor reached; thus the highest range of damping ratios = fraction of critical damping, or cannot be considered typical of corrugated con­ damping ratio tainers. The average of the damping ratios for all T = transmissibility ratio, or amplifica­ of the containers which did not fail during the tion factor. scheduled test period is 0.115 and this value can These relationships are strictly valid only for be considered more representative. This estimate linear single-degree-of-freedom systems with of the damping ratio of corrugated containers is linear viscous damping, but nevertheless are suf­ useful for prediction of the response of load- ficiently applicable to nonlinear systems to pr­ container systems at other service conditions.

11 CONCLUSIONS

This study has experimentally proven that top- loaded containers may be subjected to large dy­ namic loading forces, in addition to the static stacking loads, when they are subjected to vibra­ tion, such as in transportation. The high dynamic loads occur when the input vibration is at or near the resonant frequency of the load-container sys­ tem. The resonant frequencies for the containers evaluated ranged from 8.4 to 18.2 Hz at various loadings. Transmissibilities (amplification fac­ tors) as high as 6.3 were found at resonance. It was confirmed that simple linear vibra­ tion theory can be utilized with data from com­ pression tests, which can be easily performed in universal testing machines, to satisfactorily pre­ dict resonant frequencies of load-container systems. Because the resonant frequencies of the load- container systems fall within the range of probable transportation vibration frequencies, stacking load weights must be reduced in shipment, or stronger containers must be used, to avoid con­ tainer failure .from the large dynamic loading forces which may occur.

FPL 160 32 5.-17-7-71

GPO 1971 750- 016