Permanent set of a bottom shell plate due to slamming loading

UNIVERSITY

INSTITUTE OF EN

BERKELEY

PERMANENT SET OF A BOTTOM SHELL

PLATE DUE TO SLAMMING LOADING

by

Tamotsu Nagai

Contract MA 2620; Task Order No. 1

Department of Commerce, Maritime Administration

Series No. 186, Issue No. 2 August 6, 1962

CALIFORNIA

ERING RESEARCH

LIFORNIA

by

Tamotsu Nagai

Contract MA 2620; Task Order No. 1

Department of Commerce, Maritime Administration

Series 186 Issue 2

Institute of Engineering Research University of California

Berkeley, California August 6, 1962

Abstract

The theoretical analysis of a bottom shell plate designed to attain a small permanent set is carried out based on the unproven assumptions that plastic actions will develop on the first application of the maximum slamming load, and that deformations of the resultant dished panel will take place elastically. Purpose of the analysis is to determine the magnitude of the small permanent set, the deflection-time relations and the energy dissipated in plastic flow. Supplementary data are also given on the rough estimation of large permanent set. The theoretical methods are applied to a sample problem illustrating how plate thickness is

Table of Contents Page Abstract, , , . e e e o o e e e s i. Table of Contents ii Nomenclature iii Listoflllustrations. . . .

IINTRODUCTION.

o o o e o o o o e s s s C

i

II ORIGIN OF THE PROBLEM 3

III THEORETICAL ANALYSIS 6

3-l. Fundamental Knowledge and Basic Assumptions. . 6

3-2.

Theoretical Results and Their Numerical

Representation 11

IV AN APPLICATION OF THEORETICAL RESULTS 15

V CONCLUSIONS 19

VI ACKNOWLEDGMENTS ₂₁

BIBLIOGRAPHY ₂₂

APPENDICES. ₂₃

APPEIXio

o o o s e e o e o e s s e s e s s 24

A Unit Strip Due to the Impact of Massive

Water Having Initial Velocity ₂₄

APPENDIX2. . . 35

Yield Criteria. . . 35 TABLES AND GRAPHS. . . 23

(2:

14:

¡'4':

(4e:

t:

t,:

(J:

7f:

Nomenc la turc

span length of bottom shell plate in the I-direction length of bottom shell plate in the' -direction

slamming load which is a function of time peak value of F(t)

thickness of bottom shell plate

mass of unit strip per unit length of span total mass of bottom shell plate,

('4 =

mass of striking water per unit area

total mass of striking water,

t4"= o8m'

yield moment per unit length

time

time to reach the limit, '

time elapsed uñtil when becomes zero time ratio,

energy dissipated in permanent set velocity

velocity of striking water before impact initial velocity after impact

deflection of bottom shell plate

final permanent set of bottom shell plate

?4:

motion of the center of bottom shell plate

2jY:

permanent set at the center of bottom shell plate function of

,

dynamic yield stress

stress in the li-direction

List of Illustrations

Deformation Process Assumed in Case of Slamming Loading.

_.8

Effect of Rate of Strain on Dynamic Yield in Steel. . . . 9 Velocity Distribution Determined from the Given Shape

of Slamming Load ₁₅

Position of Hinge as a Function of Time

₄₄

620.

Final Shape of Bottom Plate for Various Weight Ratios.

_45-57

21-35. Deflection at Center as a Function of Time

_58-72

Energy Dissipated in Permanent Set as a Function of

Velocity. . . , . . . 73

Energy Dissipated as a Function of Permanent Set at

the Center0 , * , , , .

. ...

₇₄

Sketch of Double Bottom Structure. . . , . . . 24

Graphical Determination of Plate Thickness ₇₅

40. Yield Curve s. , . . . .35

Table _Page

L Final Permanent Set as a Function of Position 37

2. _{Motion of the Center of Plate as a Function of Time. .}

. 39

3. Energy Dissipated in Permanent Set as a Function of

Ve i oc i

ty

. . . .43

4. _{Energy Dissipated in Permanent Set as a Function of}

Central Permanent Set

₄₃

5. Example of Numerical Computation. _.16

6. Hinge Position as a Function of Time, 0 0 * O .

.43

Figure

I, Introduction

In this report are presented graphs which will allow the determination of slamming loads that, acting upon the bottom shell plate, will produce a given permanent set0 The basic concept used in the preparation of these graphs has not been proven yet, and is that in an initially flat panel designed to attain small permanent set plastic actions will develop on

the first

application

of

the maximum slamming load, and

sub-sequent deformations

of

the dished panel will take place elastically0

It is well known that theoretical treatment makes it necessary to construct a simplified mathematical model not only of the structure but also of the plastic behavior of the

materiaL Analysis will be carried out on a unit strip df

bottom shell plate0

In order to get an estimate of small permanent set, the elastic responses and plastic vibrations are neglected. As-suming the dynamic yield condition is kept unchanged during motion only the bending action is taken into consideration by applying the traveling hinge concept.

The above simplifications have in part been proven by theory and experiment, but in part are based on unproven assumptions0 Therefore there no doubt remains some research to be undertaken in the following areas:

a) Experimental confirmation of the applicability

of

the new theoretical concepts to

structurai elements such as beams

and plates subjected to slamming0

b) The effect of strain rate on dynamic yield of medium,

high tensile or high yield steel, and how to estimate the average strain rate throughout the bottom shell plate.

In the absence of such information, the determination of plate thickness under given slamming loading and magnitude of permanent set is nonetheless obtained with reasonable approxi-mation. Supplementary data are also given on the rough

estimation of large permanent set with sufficient margin of

safety.

The deflection-time relations and the energy dissipated in plastic flow are also obtained as part of the theoretical development for both cases of small and large deformations.

Overestimated values for permanent set are given as a result of neglecting the stretching in the plate caused by the limited movement of the plate supports during dishing.

II. Origin of the Problem

The bow section of a ship pitching severely in heavy seas is subjected to rapidly varying buoyancy forces due to immersion into oncoming wave crests0

Superimposed on these loads during the immersion process are others resulting from impact of portions of the ship with the water surface. This pressure impact is called slamming loading, and leads to extremely heavy external loads restricted to the bottom plating0 Therefore they can result in large

plastic deformation of the bottom plates0

For completeness it should be mentioned that the response of the bottom plates starts simultaneously with the deceler-ation of the plates and the large accompanying water mass that occurs upon slamming, and the response continues until the dish-shape is finally formed, During this motion the bottom plates are assumed to move together with the large water mass.

This phenomenon can be reasonably explained: slamming loading produces an "instantaneoust' velocity change of the bottom plate between stiffeners, whereas the stiffeners

them-selves have smaller initial velocity because of their larger mass per unit of bottom surface area. This velocity differ-ence, in addition to differences in stiffness, will lead to

the relative deformation of the plate between stiffeners, causing the 'tdished" or"hungry horse" look.

In design problems involving repetitive loading such as slamming it might be possible and even desirable to allow yielding to occur with the first loading, as it is realized

that the s1aining loads previously discussed [1] are scarcely different from underwaer explosion loads [2] The only

difference is that in the former case the loading lasts

longer and the bottom plates are completely enveloped by the water mass. Designing bottom plates so that the stresses in an initially fiat and stress-free panel do not exceed the yield point will lead to much heavier plating [1] than if a small permanent set is tolerated0 In an initially flat panel designed to accept a small permanent seta plastic actions will develop on the first application of the maximun.!iaImnin load.

On subsequent loadings a favorable system of residual stresses and a reduction in stress gradients due to the permanent set will probably enable the deformations to take place

elasti-cally0

In other words it might be even possible to maintain that design of bottom plates for a given permanent set is really elastic esign and that the main justification for the method is that an initially flat plate will become so much weaker

elastically than one

with small deformation in the direction of loading0

In order to zake an assnption regarding acceptable levels of permanent set it is first necessary to determine the permanent set resulting in a plate subjected to given slamming loading. In this paper we are primarily concerned with the problem of estimating the

permanent

set of the

bot-tom plate0 After solving this problem we can estïmate what to a first approxLation would constitute a reasonably

acceptable permanent se with adequate margin of safety. The results of theoretical analysis presented herein would lead to a simple procedure in which an acceptable permanent set is assumed and elastic design criteria only are used, if these phenomena could be proven by experiments. This design procedure does however rest on the intuitively correct assumption that if a plate requires a certain amount of permanent set to withstand a given lateral load then it will acquire that permanent set on the first application of load if it is not already present. Such an argument may appear to have gone full circle from elastic to plastic de-sign and then back again to elastic dede-sign.

III. Theoretical Analysis

3-1 Fundamental Knowledge and Basic Assumptions

Although equations governing the plastic behavior of the bottom plate, including the interaction with the water, can be written down from the theories of hydrodynamics and plasticity, they cannot be solved at the present time due to

the high degree of non-linearity involved.

Therefore, the exact approach is discarded in favor of an approximate, considerably simpler analysis which is quite readily amenable to numerical computations. It is well known

that the theoretical treaient makes it necessary to con-struct a mathematical model of the con-structure, simplifying not only the structure but also the plastic behavior of the material.

The damaged bottom plate is generally of rectangular

tU

shape and subjected to a heavy slamming load that is assumed

constant along the plate length, i.e. the direction of the floors (see Fig. 1), Following the results of experimental studies of direct shock-wave impact [2], [6], the deformation process of a section of bottom plate at a considerable dis-tance from the girders is illustrated by Fig. 2. It is seen that the central portion of the panel remains plane while moving in the direction of the slamming load, and that this plane section shrinks as the outer edges grow to a

cylindrical shape.

We can, therefore, restrict ourselves to the investigation of the bending of an elemental strip cut out from the bottom plate by two planes perpendicular to the direction of the

t

Go

o

o

-J INNER BOTTOM PL T E BOTTOM SHELL PL4TE LONGL GIRDER UNIT STRIP cr:

o

o

_-J

u-SECTION ®-©

FIG. I

Skcc

_c,.

Dou5l

o.i

c

Q0

(2

/

Before loading

tf_t

plastic hinge

At the onset of yielding /

7plastic

Intermediate stage with moving hinges toward the

center

Final deformed shape

Fig. 2. Deformation Process Assumed in Case of Slamming Loading

floors and unit distance (say i in.) apart, and with assumed clamped 4ges at the floors (Fig. 1). Then the deflection is that of a bent beam, as in Fig. 2.

Before solving the problem, some information regarding the dynamic characteristics of plate material is reviewed. That information is the result of past theoretical and ex-perimental investigations [3] _{[4], [5], [6], [7],}

Materials which exhibit a yield point, such as mild steel, show a great dependence of the stress-strain curve on the

3

ç)2

fo (Q3

yield strength increases almost linearly with the log of the average strain rate0 In

Fig0

3 results obtained by G. I. Taylor are shown [4] On the other hand, steels support very

high loads elastically for a

short time before yielding. This

so-called tdelay time" before yielding is a function of the Average rate of strain per overstress and the temperature.

second

Therefore,, if the full load is Fig. 3. Effect of Rate of

Strain on Dynamic applied to a mild-steel specimen Yield in Steel

in less than the delay time, the upper yield strength will always be equal to the applied load, and the lower yield strength will be a function of the applied stress and the temperature, but will be independent of the rate of loading0 The delay time is found to be the same for tensile as for compressive loads0 Therefore, the shear stress and not the normal stress must be the controlling variable for yield strength.

The

quantitative predictions of delay times by dislocation theory are in good agreement with experimental observations.

The ratio between these yield stresses may vary by a factor of two to three for such materials as annealed steels, but heat-treated, high'strength steels show only a small

strain-rate effect0 For those types of materials the asstunp tion of strain-rate independence will give only small errors in quantitative calculations and thus appear, to a first approximation, to verify the unchanged yield stress.

The quantities predicted by the hexagonal yield condition (See Appendix 2) lie between the two values predicted by the inscribed and circumscribed square. That this is necessarily the case for the static loads follows from the theorems of limit analysis0 Comparable upper and lower bound theorems may also be available for problems in dynamic plasticity0 This may well be true for the energy absorption,, although as yet no such theorem has been proved. However it is not likely to be generally the case in predicting a purely local phenomenon

such as the maximum displacement.

That the work-hardening effect on the permanent set is small follows from the comparison between experiment and theory [8], therefore workhardening does not appear to be very

important.

Based on the information reported above, several reason able assumptions are made to simplify the problem.

The first assumption is that the impact of the water mass with the bottom plates is of sufficiently large duration to be considered uniformly distributed over a panel considered to be made up of rigidplastic unit strips (see Fig. 1).

Further assumptions are given below The material is isotropic in yielding.

The dynamic elastic responses and the plastic vibrations are neglected.

Both ends of the unit strip are clamped (See Fig. 2). Dynamic yield stress due to variable strain-rate is constant throughout motïon therefore constant yield

it1a0

2

J

4xa64-moment is used with the average dynamic yield stress.

(5) Traveling hinge concept is applied with use of linear

velocity distribution along the length of strip, by the following reason:

In the case of bottom plate the deflection at the onset of

yielding on both sides is small compared with the plate thick-ness, therefore the plastic region will quickly s?read out from both sides on the floor of the plate before reaching the steady state with a small permanent sete The case of small deflections of a cantilever due to the concentrated load at the tip has already been investigated [9], where the bending energy is predominant and where the traveling hinge concept proved extremely helpful. Under this circumstance the travel-ing htravel-inge concept is here applied for calculation of permanent

set on the bottom plate. Incidentally, the deflection-time relations and energy of deformation are also obtained.

3-2 Theoretical Results and Their Numerical Representation

The complete theoretical analysis of a unit strip, due to the impact of a mass of water having initial velocity, is given in Appendix I.

[A] The final permanent set of the bottom plate as a function of position (see Fig. 1) is

222

I

t'

M

f4Z

(/±NÌ)

I

I+

3+,__2A

M±M21+3(/

(-)Q±

J

2[ifI+)J[I

_oI/+

Mi

2t(I± M')J

e

(i+)/O±)J

(,)[2±±(/±

M2

4[i+(i±

)]

where is the velocity of striking water before impact,

frl

the total mass of bottom plate, ("1' the total mass of

striking water, and

('1

_{the yield moment per unit length which}

equals (See Appendix 2).

The final permanent set

24

as a function of

position is

shown in Table i and Figs. 6 to 20, corresponding to

differ-M'

ent ratios of and

[B] The motion of the center

24

as a function of time becomes

M'

2

a0

M

frl"3

o

(2xo.64

M')

(1)

X

r4i±(/+

₎

v(T_

I Q

j

2[f±(/+

J

(4

H'

3f:

_t1Lt

a0

t1,

7.

73f

12x0. 4-

- t'10

_(i

)2

X[{'_

7(X

L /

4+

)

H

/+,

3(3t)

The motion of the center of plate as a function of time is shown in Table 2 and Figs. 21 to 35, corresponding to

a

different ratios of 0 and

[C] The energy dissipated in permanent set is given by

/1

f

f1

LI

_2xo.42

2. (2) (4) (3)

/

or X

wo'

57±4o7(i)

Nl

/+

4(3+,)Z

,

where is the permanent set at the center of bottom plate. Eqs. (4) and (5) are shown in Tables 3, 4 and Figs.

A/I'

36 and 37, choosing ' ' as a parameter.

F

IV. An Application of Theoretical Results

One example is here given as an application of theoreti-cal results obtained above:

The problem is to find the central permanent set

when a bottom plate of dimensions

_{X -4,}

X _{(28"x 183" x}

0.913") and with a dynamic yield point of = 60 x lO3 psi

is subjected to a triangular slamming load with the maximum

FM = 60 psi of 0.02 sec duration. This is the same example as that shown in report [1].

In that report it was found that the maximum elastic response happened at

t

= 0.012 sec. We may choose the

initial point

t

= O roughly at

t

= 0.01 sec. Then velocity

=

V

directly after getting plastic hinges at both

t-0

sides of the floor is obtained from Newton's second law:

//

0.0/ 0.02

t

(in sec.)

(6)

in which m' is the mass of striking water per unit area. After integration:

'riz.'

o.00

5- F

lb. sec

or '7fl!\7= 0.3 (With in. in this

case)

(7) O

oo/ t (in sec.)

Fige 4 Velocity Dis- Assuming we have also ?7Z'

tribution Determined from

the Given Shape of Slamming from Eq. (7). Load.

Using

141284x0,913 \ lb sec2 for mild steel,

frl

= 386 )x28x183 in

and = lb0 in. from Eq. (38)

of Appendix 2 and assuming the following velocities

la

la

in

30 sec , 45 sec , 90 sec

we can obtain permanent set at the center of bottom plate, corresponding to each velocity given above, by the use of curves in Fig. 10 for as shown in following table,

Table 5 Example of Numerical Computation

i 2 3 4 7 30 45 90 1Ç7 /

fbs2

lo

(2/3)lo_2 (113).1o_1 (from Eq.(7))

tv1',

?7Z' 1 1 0 'w;, 45 28 11 ?2J01 (from Fig. 10) 2005 31.7 69 o _sec (from E.,. (14)) '

245

0.0224 3.65 0.0333 6.8

(%)

in 0.0621

X-

₌

yZf\,j

/

x=

o

and inserting the previous values for

frl

and ("1

as a

function of into

X

, the plate thickness now reduces to = 7,93x10

Using the curves, we first obtain the linear relation be-tween and as shown in Fig. 38, and then by getting cross point with the above -equation for each corresponding velocity 7,1

of the fifth line in Table 5, we have 48 which gives

= 0.583",

if the velocity before impact, 74 , is asstmied equal to

in ft in

69 _{c (} 5,75

c), or

to 90 ¡c. This thickness

corresponds to

0.638 compared with

_{the original thickness} = 0.913".

If we next choose the more severe case such as 180 psi under the same velocity:

V=

90 sec or

74=

62.4 c

( 5.2 , we have 34 from Fig. 38,

6 2

7X

The

second problem is to

determine the plate thickness required in each case to withstand the saine slamming load without exceeding a permanent set of l57 of plate thickness,

that is, = 0.15.

by applying the same graphical procedure as the above shown in Fig. 38; which gives

= 0.825".

And this magnitude corresponds to 0.516 compared with the original thickness 1.6" from Fig. 43 of the previous report [1]. Hence from these results we can see the great reduction of weight of the bottom plate.

V. Conclusions

As there has been no way to evaluate small permanent

sets caused by slamming loading, in this paper the theoretical analysis is carried out under some reasonable assumptions by which the following results are obtained as a first approxi-mation.

The final magnitude of permanent set as a function of position on the bottom plate is obtained from Figs. 6-20, by which the plate thickness is determined as a function

of a given acceptable permanent set.

The deflection-time relations and the energy dissipated in plastic flow are obtained from Figs. 21-37, by which an instantaneous deflection at the center and the energy dissipated are both determined.

Supplementary data are also given that allow a rough

estimation of large permanent set with adequate margin of safety, and in terms of which the amount of slamming load is reversely estimated. Therefore the energy dissipated can also be obtained.

To confirm the estimates, the theory should be supple-mented by experiments on the effect of strain rate on

dynamic yield in medium and high tensile and in high yield steels. Moreover the experiments would serve to estimate the average strain rate throughout the bottom

shell plate.

If a new design criterion of bottom shell plate with a

application of the heavy slamming load would be accepted, a great reduction of weight of the bottom plate could be expected as was shown in the example of Chapter IV.

VI. Acknowledgments

Financial aid from the Maritime Administration Office which has supported this work is gratefully acknowledged.

The author would like to express his thankfulness to Professors H. A. Schade and J. V. Wehausen, moreover to his department colleagues Messrs. O. J. Sibul, W. M. Maclean and R. Glasfeld who gave him helpful advice and discussions.

He is indebted also to Mr. K. Kojima, graduate student of the Department of Naval Architecture, for the numerical computation, to Mr. William Kot for the preparation of draw-ings, to Mmes. Joan Sherwin, Marci Thomas for the typing of

[3]

Bibliography

[1] Nagai, T., "Elastic Response of a Stiffened Plate Under Slamming Loading", Inst. of Engineering Research,

University of California, Series No, 186, Issue No. 1, April 1962.

[2] Keil, A. H., "Problems of Plasticity in Naval Structures, Explosive and Impact Loading", Plasticity, Proc. of the

Second Symposium on Naval Structural Mechanics, Pergamon Press, New York, N.Y., 1960.

Hodge, P. G., Jr., "Approximate Yield Conditions in Dynamic Plasticity", Third Midwestern Conf, on Solid Mechanics, University of Michigan, Ann Arbor, Michigan,

1957.

[4] Simmons, J, A., F. Hauser and J. E. Dorn, "Mathematical Theories of Plastic Deformation under Impulsive Loading", University of California Publications in Engineering, Vol. 5, No. 7, 1962.

Manjoine, M. J., "Influence of Rate of Strain and Temper-ature on Yield Stresses of Mild Steel", Journ. Appi. Mechs., Vol. 11, 1944

Lisanby, J A,, J. E. Rasmussen and H. M. Schauer, "Corn-parison of Dynamic Yield Effects of Steels", Underwater Explosions Research Division, Norfolk Naval Shipyard, Report No. 10, 1957.

Nagai, T., "Large Plastic Deformations of Corrugated Bulkhead for All Clamped Edges under Transverse Impact-Third Report", Journ. Naval Arch, of Japan (in Japanese), No. 108, 1960.

Fredrick, D., "A Simplified Analysis of Circular Membrane Subjected to an Impulsive Loading Producing Large Plastic Deformations", Fourth Midwestern Conf. on Solid Mechanics, University of Texas, Austin, Texas, 1959.

Parkes, E. W., "The Permanent Deformation of a Cantilever Struck Transversely at Its Tip", Proc. Roy. Soc. of

London, A 228, 1955.

Naghdi, P. M., "Stress-Strain Relations in Plasticity and Thermoplasticity",Proc. of the Second Symposium on Naval Structural Mechanics, Pergainon Press, New York, N.Y., 1960.

Appendices

Appendix i

Appendix 2

SIDE

N

SIDE

Appendix 1

Response of a unit strip due to the impact with a water mass having initial velocity

We shall consider a unit strip (say 1 inch width beam) which is suddenly loaded by the massive water

over all its

span (See

Fig0

1) To be precise, a massive water of uniform mass i-n..' per unit span is suddenly attached to the total beam span moving with its upward velocity ?

Since the beam is assumed to be rigid plastic, it must undergo sufficient plastic deformation to absorb the kinetic energy of the massive water0 This deformation can take place

only by means of a plastic

hinge0

At the moment the massive water hits the beam this hinge must be at both sjdes (See Fig. 38). Indeed,, if the hinge were initially in the beam

interior, a finite

portion

of the beam

would begin to move

instantaneously0 This, in turn, would generate a large inertia force which would require a moment greater than the

yield moment at the hinge0

However, it is obvious that the hinge cannot stay at the sides0

Therefore., it must move along the

beam, toward the center of the

beam (See Fig0 25, Fig0 38)

Let

(CENTER

us choose a coordinate system t,

zJ for the beam, as indicated in Fig, 38. in which the origin

of zT

is on the center and that

of x, at the left side0 This coordinate system is differ-ent from that given in

Fig0

1, And let be the distance of the hinge from the

side0

We shall assume throughout that the deformations are small so that geometry changes may be neglected.

Now, the moment: at the hinge is the yield moment ¡1 , and the moment everywhere else is not greater than M0 (See

Appendix 2) 0 This implies that the moment is a maximum at the hinge, hence, the shear force is zero there, ₍

f

= O

in Fig. 38), _{Therefore, we may analyze the portion of the} beam between the hinge and the side as a rigid body, subject only to a yield moment î-id and the inertia forces (See Fig. 38). Assuming as the reaction at the side direct-ly after impact by the massive water ' per unit length of the beam span, because of reasonably large _{mass ratio be} tween water and beam, we can approximately construct the equations of vertical motion and angular motion about the

hinge: 's

+ (±')

= û

Z

dtz

J

(7n I

_-

-

o

Z '

In which is the mass of beam per unit length of the span.

At any instant, the section between the hinge and the side rotates about the point . Therefore, assuming that W is still reasonably small, the velocity at any point X.' is

where a dot indicates differentiation with respect to

L

After inserting Eq. (10) for 2J into Eq. (8) by differenti-ating Eq. (10) with respect to time, we have the space integral of Eq. (8) resulting

o

The parenthetical term in Eq. (11) is an exact differential;

hence, we can

integrate Eq. (11) with respect to time,

obtain-ing

Oz±('it

=

(12)

where C,

is a constant

to be determined.

At

L

= O, the hinge is still at the side, and the velocity of the massive water directly after getting the hinge at the

side, V, is determined by , using the principle of

mo-mentum conservation

at _o (13)

and

(14)

m')

Using Eq. (13) to

evaluate

_C, of Eq. (12), we finally write Eq. (12) in the form

For , the hinge is at the center of the beam. It

is no longer necessary for the hinge moment to be a relative

th

V

- /(/)

a0 (15)

Eq. (15) will remain valid until the motion stops or until the hinge reaches the center of the beam. According to Eq. (15)

a,.

zj

is never zero; hence, the limit of validity is '

Returning now to the solution of Eq. (9), we similarly have the space integral of Eq. (9) after inserting Eq. (10) for

into Eq. (9) by differentiating Eq. (10) with respect to time and combining Eq. (9) with Eq. (8) to obtain an exact

differ-ential:

i2u1

7'

resulting

/2/1

'-'2 ' (16)

where is a constant to be determined.

Using Eq. (13) to evaluate C' , we finally obtain

L=

(17)

Inserting

_{1= 4-}

into Eq. (17), time to reach the limit becomes

maximum at the hinge, since moments for are not defined.

Now we consider the equation of angular motion. Setting

in

Eq0 (9),

this latter becomes

2t7

//_

2r

-

O

"(I

_LLÍ

_{__}

Integrating Eq. (19) with respect to time so that 12 is

con-tinuous at = , , we obtain

s

(20)

4±,

Eq. (20) will remain valid until the motion stops or until t' when J will vanish.

From Eq. (20) t' becomes

?7? V2O

21)

L

In other words, until t becomes

L'

the motion is continuous and the beam moves to the final deformed shape. Therefore we have the time ratio from Eqs. (17) and (21):

,,

t

2(--)()

t' -

i (22)

-

i"i

Table 6 and Fig. 5 show the hinge position

'Ç as a

function of time for various mass (weight) ratio or

The velocities at any point in the beam may be found by substituting Eq. (15) and Eq. (20) into Eq. (10)

/-2

Í3t

7n

mJ,2J

t1tt'

(19) (24)

o

(23)

2J=

rdÌ.

Evaluating ?J from Eq0 (23) and from Eq. (17), we obtain 0

a

(

92»

(

7fl)2()

fl1Z

J-12 tÇ (i -t

_-)

in a0 -r-

₊

_{Z 2 Iab+ (:t}

₊

_{)] [a04- (}

-t-0z0-1-

(--)*)]

+

f

(26) for

;t

t,

(27)

Since the motion at a particular point on the beam does not start until the hinge reaches that point, i.e., until

Eqs. (26), (27) should be supplemented by

for

(28)

For t, , Eq0 (24) is to be integrated so that the displace-.

a.»

ment is continuous at Thus,

Although Eq. (23) can he integrated directly by solving

Eq0 (17) for as a function of time, it is more convenient to note that

zu

p.77

(Y()4Vz

_¿±

_(»)V2a2

+J2m2(()1

_J

/2f«/*)2

2

r

a0

+ (,+

) t

z[*

+)

a0 L2a0±

(i

+

22±

_m

'+

(')

_»1J

}

t,

'

(29) z

Inserting A'= for into the above equation reduces the final deformed shape 2J as a function of position Z

2Z

2m

I

J

I - -

ao

(

L

??? 2 3 7ì

(i)

(J)ô

Th2I

2 J 2

(i)

(i-i-- 'Th2

I T -f

(30)

x(

-f

z--)(»--)

I

ci

)[i

±('±)J

)}

2X

'2

J -;

3Vma2

_4H0[(H)J

4+

where

1=Jì

,

From Eqs. (26) and (29) the motion of the center ¿J as

M') c,

(2)

f

-

I

Q

3(3+))

a function of time becomes after setting

'.

M

ajo / 1,_.Z/

7;i--

/2 X C.

_&

₁₁₀ (

I

(

¡'-J r

ri

1J

-

('±)J

+(i±).

2Z

In order to show the final deformed shape more conveniently

in curves, we change the notation ¿ and the direction

of j

as shown in Fig L Thus, in the case of large value such as the bottom plate,

&/

approximately becomes 2 2LL /

2ít1

/CL0 i

íM'

I-4

o, 42

b ()

))

₍₄₊

,)(: +

I-1

xL3

'+

±

J-1

r

2 3*

(3+J +

(i±)t

[i+

_)J

1 L

2(

_M

M')'

(+

)í

(i+

,)]

*

j)j

4[i+(i,]2

I

t

(i)

I

±

/2 X

O.L

(

M

74û±7)

4(3+

t(41)

J

the limiting case such that M

- gx42

ìio

(T)

M

fi

+ i

11'i _b'

M

Under the assumptions with which we started if the mass of uniformly distributed water becomes very large compared with. the mass of the bottom plate, the required time for which the moving hinge reaches the center will close to the value of

M'

_{tends to infinity, as shown}

in Fig. 5.

From Eq. (1) we can obtain the final deformed shape 2J as a function of position as shown in Figs. 6 to 20.

Also from Eqs. (2) and (3) the motion of the center as a function of time is given as shown in Figs. 21 to 35, By those figures

the effects of different massive water t] on the behavior of the bottom plate are clearly shown.

For smaller values of M' or , part of the deformation

will take place while the hinge is still on the bottom plate, as shown in Fig. 5.

Energy dissipated in permanent set of a bottom plate (See Fig. 38) is next evaluated by

3(3+)

U ==-2-,J

o

Inserting the following values:

U o?T) iV210₂

Using the permanent

rÍ1

=

L

J0

2Z

(ì

j{/ ki)J

[]}

set at the center which is given by

3

t - í64z

(L

ít-1 ' ti') p7

f2

57+4

()

1>íi

tfd'A _/-1'} (31) (32) (33) (4) (34)

obtained from Eqs. (13), (23) and (24) into Eq (32), we finally obtain

±

/2

J

C2

(

)

Thus, QO

u =

-û Ö

from Eq. (3), we obtain the relation between energy and permanent set

_2J0,

as follows:

U

3 /o

₍₄

j J(i)

In ()

M'

.7*4

*7(!,,)l

(5)

where (from Eq. (38)),

Using Eq. (4) or (5), the energy relation is now shown by _J or 2J,, in Fig. 36 or 37, for different striking masses of water.

Those curves show that the amount of energy dissipated in permanent set increases parabolically with the velocity of striking water, but on the contrary, it increases linearly with the permanent set at the center of bottom plate.

Fig. 40 Yield Curves

Appendix 2

,6-Yield Criteria

Let us set stresses and

4/ 5j in the directions of 2

andt (See Fig. 1). As the

:

a-case of static loading, all conceivable yield curves in impact loading satisfying isotropy, convexity, absence By von of Bauschinger effect and Mises

Criterion independence from the effect of hydrostatic stress, will be assumed to lie between the two hexagonal curves ABCDEF and A'B'C'D'E'F', as shown in Fig.

40 [3] [10]. It should be emphasized that all curves with convex lying between the two hexagonal curves, are admissible. The elliptic yield curve of von Mises (as shown in Fig. 40) is

such a curve, as is the yield curve of Tresca which is the inner hexagonal curve in Fig. 40, and the outer hexagon in Fig. 40 is not a well-known yield condition but is shown here only to serve as a bound on possible admissible isotropic yield curves.

It may be noted, in particular, if the point _A

is

moved to

_A"

corresponding to agreement of the yield curves in pure shear, a- = - , the Tresca hexagon would serve as the outer bound. In any event, taking now the Mises yield

condition as a reference, the maximum deviation of any admissible yield condition becomes about l5.477, as ex plained below:

- c5 ₂ = (35)

where c5 is defined as a dynamic yield stress determined by the average strain rate throughout the bottom shell plate.

And from the assumption of no change of plastic strain increment in the floor direction, we have

(38) z

(36)

From Eqs. (35) and (36) we have

-

/.1547

(37)

Finally, using of Eq. (37) yields as the yield moment per unit length of the beam span _M0

M=

₄

/!57R2

a02x

20

frV'

a0-15 10 7 5 3 1 0.000 20.98 15.10 9.41 6.05 3.77 1.83 0.24 0.250 16.74 12.04 7.52 4.84 3.02 1.43 0.20 10 0.500 11.89 8.57 5.36 3.45 2.17 1.06 0.14 0.750 6.47 4.62 2.88 1.87 1.18 0.58 0.08 1.000 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000 31.47 22.65 14.12 9.08 5.66 2.75 0.36 0.250 25.11 18.06 11.28 7.26 4.53

215

0.30 15 0.500 17.84 12.86 8.04 5.18 3.26 1.59 0.21 0.750 9.71 6.93 4.32 2.81 1.77 0.87 0.12 1.000 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000 41.96 30.20 18.82 12.10 7.54 3.66 0.48 0.250 33.48 24.08 15.04 9.68 6.04 2.86 0.40 20 0.500 23.78 17.14 10.72 6.90 4.34 2.12 0.28 0.750 12.94 9,24 5.76 3.74 2.36 1.16 0.16 1.000 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000 52.45 37.75 23.53 15.13 9.43 4.58 0.60 0.250 41.85 30.10 18.80 12.10 7.55 3.58 0.50 25 0.500 29.73 21.43 13.40 8.63 5.43 2.65 0.35 0.750 16.18 11.55 7.20 4.68 2.95 1.45 0.20 1.000 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000 62.94 45.30 28.23 18.15 11.31 5.49 0.72 0.250 50.22 36.12 22.56 14.52 9.06 4.29 0.60 30 0.500 35.67 25.71 16.08 10.35 6.51 3.18 0.42 0.750 19.41 13.86 8.64 5.61 3.54 1.74 0.24 1.000 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000 83.92 60.40 37.64 24.20 15.08 7.32 0.96 0.250 66.96 48.16 30.08 19.36 12.08 5.72 0.80 40 0.500 47.56 34.28 21.44 13.80 8.68 4.24 0.56 0.750 25.88 18.48 11.52 7.48 4.72 2.32 0.32 1.000 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000 104.90 75.50 47.05 30.25 18.85 9.15 1.20 0.250 83.70 60.20 37.60 24.20 15.10 7.15 1.00 50 0.500 59.45 42.85 26.80 17.25 10.85 5.30 0.70 0.750 32.35 23.10 14.40 9.35 5.90 2.90 0.40 1.000 0.00 0.00 0.00 0.00 0.00 0.00 0.00

50 60 70 80

Table i (continued)

2x

a0

0.000

0.250

90

0.500

0.750

10000

00000

0.250

100

0.500

0.750

1.000

0,000

0.250

110

0.500

0.750

1.000

00000

0.250

120

0.500

0.750

1.000

/v1/M

0.000

0.250

0.500

0.750

1.000

0.000

0,250

0.500

0.750

1,000

0.000

0.250

0.500

0.750

1.000

0,000

0.250

0.500

0.750

1.000

60

50

336

279 269 222 191 159 103 91 0 0

403

335

322

266

229 190 123 109 0 0

470

390

376

310

268

221

144 127 0 0 537

446

429

355

306

253 164 145 0 0

604

502

483

399

344

285 185 164 o o

671

558

536

444

382

317

205

182 O O 739 613 591 487

421

348

226

199 o o

806

669 644

532

459

380

246

218 O O

40

30 20 10 5 220 162 105 47 19

1.2

175 129 84 38 15

1.0

124 92 59 27 11

0.7

67 48 32 14 6

0.4

0 0 0 0 0 0 265 195 126 57 23

1.4

210 155 101

46

19

1.1

150 111 72 33 14

0.8

81 59 39 17 7

0.4

0 0 0 0 0 0 309 227 147 66 27

1.7

245 181 117 53 22

1.4

175 128 83 37 17

1,0

94 68 45 19 8

0.6

0 0 0 0 0 0

353

260 168

76

31

1.9

280 207 134 61 25

1.6

200 147 95 43 19

1.1

108 78 52 23 9

0.6

0 0 0 0 0 0 397 292 189 85 35

2.1

315 233 151 68 28

1.7

225 165 107 48 22

1.3

121 88 58 25 10

0.6

o o o o o o

441

325

210

95 39

2.4

350 259 168

76

32

2.0

249 184 119 54 24

1.5

134 98 65 28 12

0.8

O O O O O O

486

357

231

104

43

2.6

386

284

184 83 35

2.1

275 202 131 59 27

1.6

149 107 71

30

13

07

o o o o o o 530 390 252 114 47

4.3

421

311 201 101 38

3.8

300

221 143 65 30

3.2

162 117

78

34 14

2.3

O O O O O O

Table 2

Motion of the Center of Plate as a Function of Time

a0

20 15 10 7 5 3 1

0.00

0.000

0.000

0.000

o,000

o.,000

0.000

0.000

2

0.25

0.495

0.347

0.232

0.158

0.105

0,053

0.011

7

0.50

1.634

1.192

0.779

0.511

0.344

0.177

0.032

0.75

3.082

2.300

1.462

0.973

0.646

0.328

0.056

1.00

4.814

3.527

2.235

1.473

0.972

0.492

0.080

10

0.2

7.738

5.612

3.503

2.259

1.380

0.658

0.093

- 0.4

13.477

9.768

6.087

3.918

2.439

1.157

0.158

t

0.6

17.576

12.736

7.933

5.103

3.196

1.514

0.203

0.8

20.035

14.516

9.041

5.814

3.651

1.726

0.231

1.0

20.860

15.110

9.410

6.051

3.802

1.800

0.240

0.00

r

0.000

000

0.0

000

¿.000

00

2

0.25

0.743

0.521

0.348

0.237

0.158

0.080

0.017

0.50

2.451

1.788

1.169

0.767

0.516

0.266

0.048

-°

_0.75

_4.623

_3.450

_2.193

_1.460

_0.969

_0.492

_0.080

1.00

7.221

5.291

3.353

2.210

1.458

0.738

0.120

15

0.2

11.607

8.418

5.255

3.389

2.070

0.987

0,140

- 0.4

20.216

14.652

9.131

5.877

3.659

1.736

0.237

t

0.6

26.364

19.104

11.900

7.655

4.794

2.271

0.305

0.8

30.053

21.774

13.562

8.721

5.477

2.589

0.347

l.0

31.290

22.665

14.115

9.077

5.703

2.700

0.360

0.00

0.000

0.000

0000

0.000

0.000

0.000

0.000

.2. 0.25

0.990

0.694

0.464

0.316

0.210

0.106

0.022

0.50

i

3.268

2.384

1.558

1.022

0.688

0.354

0.064

°

_0.75

_6.164

_4.600

_2.924

_1.946

_1.292

_0.656

_0.112

1.00

9.628

7.054

4.470

2.946

1.944

0.984

0.160

20

0.2

15.476

11.224

7.006

4.518

2.760

1.316

0.186

- 0.4

26.954

19.536

12.174

7.836

4.878

2.314

0.316

t

0,6

35.152

25.472

15.866

10.206

6.392

3.028

0.406

0.8

40.070

29.032

18.082

11.628

7.302

3.452

0.462

1.0

41.720

30.220

18.820

12.102 7.6043.600

0.480

0000

0.000

0.000

0.000

0.000

0..OÓO

0.000

0.000

2Ç 0025

1.238

0.868

0.580

0.395

0.263

0.133

0,028

0.50

4.085

2.980

1.948

1.278

0.860

0.443

0.080

0.75

7.705

5.750

3.655

2.433

1.615

0.820

0.140

1.00

12.035

8.818

5.588

3.683

2.430

1.230

0.200

25

-0.2

19.345

14.030

8.758

5.648

3.450

1.645

0.233

- 0.4

33.693

24.420

15.218

90795

6.098

2.893

0.395

0.6

43.940

31.840

19.833

12.758

7.990

3.785

0.508

0.8

50.088

36.290

22.603

14.535

9.128

4.315

0.577

1,0

52.150

37.775

23.525

15.128

9.505

4.500

0.600

Table 2 (continued)

Z

a

°

J'1

20 15 10 7 5 3 1

0.00

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.25

1.485

1.041

0.696

0.474

0.315

0.159

0.033

4.902

3.576

2.337

1.533

1.032

0.531

0,096

0.75

9.246

6.900

4.386

2.919

1.938

0.984

0.168

1.00

14.442

10.581

6.705

4.419

2.916

1.476

0.240

0.2

23.214

16.836

10.509

6.777

4.140

1.974

0.279

0.4

40.431

29.304

18.261

11.754

7.317

3.471

0.474

0.6

52.728

38.208

23.799

15.309

9.588

4.542

0.609

0.8

60.105

43.548

27.123

17.442 10.953

5.178

0.693

1.00

62.580

45.330

28.230

18.153 11.406

5.400

0.720

0.00

0.000

0.000

0,000

0.000

0.000

0.000

0.000

0.25

1.980

1.388

0.928

0.632

0.420

0.212

0.044

0.50

6.536

4.768

3.116

2.044

1.376

0.708

0.128

0.75

12.328

9.200

5.848

3.892

2.584

1.312

0.224

1.00

19.256

14.108

8,940

5.892

3.888

1.968

0.320

0.2

30.952

22.448

14.012

9.036

5.520

2.632

0.372

0.4

53.908

39.072

24.348

15.672

9.756

4.628

0.632

0.6

70.304

50.944

31.732

20.412 12.784

6.056

0.812

0.8

80.140

58.064

36.164

23.256 14.604

6.904

0.924

1.0

83.440

60.440

37.640

24.204 15.208

7.200

0.960

0.00

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.25

2.475

1.735

1.160

0.790

0.525

0.265

0.055

0.50

8.170

5.960

3.895

2.555

1.720

0.885

0.160

0.75

15.410

11.500

7.310

4.865

3.230

1.640

0.280

1.00

24.070

17.635

11.175

7.365

4.860

2.460

0.400

0.2

38.690

28.060

17.515

11.295

6.900

3.290

0.465

0.4

67.385

48.840

30.435

19.590 12.195

5.785

0.790

0.6

87.880

63.680

39.665

25.515 15.980

7.570

1.015

0.8

100.175

72.580

45.205

29.070 18.255

8.630

1.155

1.0

104.300

75.550

47.050

30.255 19.010

9.000

1.200

-2

° 30

-40

-t,

50

50 60

70

80

90

Table 2 (continued)

J 60 40 30 20 15 10 7 5 3

0.2

124.5

81.5

60.2

38.690

28.060

17.515

11.295

6.900

3.290

0,4

217.1

142.1

104.8

67.385

48.840

30.435

19.590

12195

5.785

0.6

283.2

1855

136.7

87.880

63.680

39.665

25.515

15.980

7.570

0.8

322.9

211.5

155.9

100.175

72.580

45.205

29.070

18.255

8.630

1.00 337.9

220.1

162.3

104.300

75.550

47.050

30.255

19.010

9.000

0,2

149.3

97.9

72.2

46.4

33,7

21.0

13.6

8,3

4.0

0.4

260.5

170.5

125.8

80.9

58.6

36.5

23.5

14.6

7.0

0.6

339.8

222.5

164.0

105.5

76.4

47.6

30.6

19.2

9.1

0.8

387.5

253.7

187.0

120.2

87.1

54.2

34.9

21.9

10.4

1.00 405.5

264.1

194.7

125.2

90.7

56.5

36.3

22.8

10.8

0.2

174.2

114.2

84.2

54.2

39.3

24.5

15.8

9.7

4.6

0,4

303.9

198.9

146.7

94.4

68.4

42.6

27.4

17.1

8.1

0.6

396.5

259,6

191.4

123.1

89.2

55.5

35.7

22.4

10.6

0.8

452.1

296.0

218.2

140.3

101.6

63.3

40.7

25.6

12.1

1.00 473.1

308.1

227.2

146.0

105.8

65.9

42.4

26.6

12.6

0.2

199.1

130.5

96.2

61.9

44.9

28.0

18.1

11,0

5.3

0.4

347.3

227.4

167.7

107.8

78.2

48.7

31.4

19.5

9.3

0.6

453.1

296.7

218.7

140.6

101.9

33.4

40.8

25.6

12.1

0.8

516.6

338.3

249.4

160.3

116.2

72.3

46.5

29.2

13.8

1.00 540.7

352.2

259.6

166.9

120.9

75.3

48.4

30.4

14.4

0.2

224.0

146.8

108.3

69,7

50.5

31.5

20.3

12.4

509

0,4

390.7

255.8

188.6

121.3

87.9

54.8

35.3

22.0

10.4

0.6

509.8

333.8

246.1

158.2

114.7

71.4

45,9

28.8

13.6

0,8

581.2

380.6

280.5

180.4

130.7

81.4

52.3

32.9

15.6

1.00 608.3

396.2

292.1

187.7

136.0

84.7

54.5

34.2

16.2

Table 2 (continued)

5 3

2,%

60 40 30 20 15 10 7

0,2

248.9

16301

120.3

77.4

56.1

35.0

22.6

0.4

434.1

284.2

209.6

13408

97.7

60.9

39.2

0.6

566.4

37009

273.4

175.8

127,4

79.3

51.0

0.8

645 .8

422.9

311.7

200.4

145.2

90.4

58.1

1.00 675 09

440.2

324.5

208.6

151.1

94.1

60.5

0.2

273.8

179.4

13203

85.1

61.7

38.5

2409

0.4

47705

31206

230.6

148.3

107.5

67.0

4301

0.6

62300

408.0

300.7

19304

140.1

87.2

56.1

0.8

71004

465.2

34209

22004

159.7

9904

6309

1.00 743.5

484.2

357.0

229.5

166.2

103.5

66.6

0.2

298.7

19507

144.4

9209

67.3

42.0

27.1

0.4

520.9

341.0

251.5

16108

117.2

73.1

47.0

0.6

679.7

445.1

328.1

211,0

152.9

95.2

61.2

0.8

775.0

507.5

37400

240.5

174.2

108.5

69.7

1.00 811.1

528.2

389,4

250.3

181.3

112.9

72.6

13.8

606

24.4

11.6

32.0

1501

3605

17.3

38.0

18.0

15.2

7.3

26.8

12.8

35.2

10.6

40.2

19,0

41.8

19.8

16.6

7.9

29.3

13.9

38.4

18.1

43.8

20.8

45.6

21.6

2

Table 3

Energy Dissipated in Permanent Set as

a Function of Velocity 20 15 10 7 5 3 Table 4

Energy Dissipated in Permanent Set as a Function of

Central Permanent Set

1.0 0.75 0.50 0.25 0.00 O O O O O O O O 20 443 429 404 374 339 275 122 40 1772 1717 1614 1495 1356 1099 488 60 3986 3862 3632 3364 3052 2473 1099 80 7087 6867 6457 5980 5425 4397 1953 100 11073 10729 10089 9344 8477 6870 3052 Table 6

Hinge Position as a Function of Time

0 0.1 0.5 .1 3 5 7 10 15 20 0.3333 0.2821 0.2000 0.1667 0.1333 0.1250 0.1212 0.1183 0.1159 0.1148 0.2500 0.2018 0.1327 0.1076 0.0836 0.0778 0.0752 0.0732 0.0716 0.0708 0.1667 0.1222 0.0714 0.0556 0.0417 0.0385 0.0370 0.0359 0.0351 0.0347 0.0833 0.0482 0.0227 0.0166 0.0119 0.0109 0,0104 0.0101 0.0098 0.0097 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0,0000 0.0000 0.0000 0.0000 20 15 10 7 0 .000 00000 0.000 0.000 0.000 0.500 0.393 0.415 0.458 0.513 10000 0.785 0,829 0.915 1.026 1.500 1.178 1.244 1.373 1.539 2.000 1.570 1.658 1.830 2.052 s 3 1 0.000 0.000 0.000 0.587 0.757 1.591 1.173 1.514 3.181 1.760 2.271 4.772 2.346 3.028 6.362

I-o

0.75

2e

Q0

o-5

_OE25 0-I

t-.

02

2 I +

M I 01

3[i +('+.')

i]

ao

FIG. 5

Position of Hinge as a Function of Time

W.f h

vM

bo M0

20

15

Io

5

o

FIG.. 6

Final Shape of Botton Plate for Various Weight

Rat io s

o

0.25

OE50 OE75 1.00

W1r h

vM

b0 M0

30

20

I0

o

o

0.25

0.50

075

LOO

a0

FIG. 7

Final Shape of Bottom Plate for Various Weight

Ratios

Wf h

vM

b0 M0

40

30

20

OE25 OE75

FIG.8

Final Shape of Bottom Plate for Various Weig1t

Ratios

OE25

0.50

075

LOO

a0

FIG. 9 Final Shape of Bottom Plate for Various Weight

60

50

40

Wf h

vM

b0 M0 3 2

o

FIG. IO

Final Shape of Bottom Plate for Various Weight

Ratios

ciO30

Ç7

"s

7

0

O

0.25

OE50

0J5

LOO 2

---a0

0.25

OE50

015

LOO

a0

FIG. II _{Final Shape of Bottom Plate for Various Weight} Ratios

Wf h voRM b0 M0 loo

80

6 4 o

FIG. 12 Final Shape of Bottom Plate for Various Weight

Ratios

50

h

=

/

102x4

wf

h 2

v.M

b0 M0

o

00

FIG. 13 Final Shape of Bottom Plate for Various Weight

Ratios IO x4 3 Wf h v

M 2

b0 M0 O O

260

O

0.25

025

0.50

OE50 2

----a0

0.75

0.75

l.00

LOO

FIG.. 14 Final Shape of Bottom Plate for Various Weight

Ratios

=50

OQ

4 Wf 3 h

vM

b'0 M0 2 2

lOx

Wf h

vM

b0 M0 O

0

0.25

0.50

)

X

u0

FIG 15 Final Shape of Bottom Plate for Various Weicht Ratios

075

1.00

=7oj

6o_

so

uuuI

0.25

050

0.75

I-00

Go

FIG. 16 Final Shape of Bottom Plate for Various Weight

2

l0x7

6 5 4 Wf h

vgM

b0M0

2 o o OE25

0.50

2

--Go 0.75

I-00

FIG. 17

Final Shape of Bottom Plate for Various Weight

Ratio s

hi

-IO2x 7 6 5

4

h y02 M b0

M0

2

o

h°00

o

0.25

0.50

075

1.00 2 _Go

FIG. 18 Final Shape of Bottom Plate for Various Weight Ratios

2

10x8

7

6

5

4

Wf h

vcM

b0 M0 2

o

o

025

₀₅₅

_0.75

1.00

2 L

a0

FIG.. 19

_{Final Shape of Bottom Plate for Various}

102x 9 o '-I L

-

=120

_64

vo2M

b0M04

:.

::

o

0.25

OE50 2 Go

0.75

FIG. 20 Final Shape of Bottom Plate for Various Weight Ratios

h y02 M b0 M0 20 15 Io 5 o

FIG. 21 Deflection at Center as a Function of Time

Go'0

-I,

-\O

-

---'T

f15

o

02

0.4 0.6 0.8 LO t

30

25 20 wo h

vgM

b0 M0 15

Io

5

-h

-I

0.2

04

0.6 0.8 I-0

t

wo h

vM

b0 M0

40

30

20

I0

FIG..23

_{Deflection at Center as a Function of Time}

h

-20

'I

-)

\O II -i:

0.2

04

0.6

08

I-0

t

50

40

wo h

vM 30

b0 M0

20

Io

o

FIG. 24

_{Deflection at Center as a Function of Time}

-=25

-

_h

-I

H

o

0.2

0.4

0.6

0.8 ¡.0

t

60

50

40

wo h

vM

b0 M0 30 20

Io

o

t

FIG. 25 Deflection at Center as a Function of Time

i

-I

\O

--r I n'

wo h

vM

b0 M0 80

60

40

20 o t

FIG.. 26 _{Deflection at Center as a Function of Time}

Go

ArAs

I

wo h

vM

b0M0

8(

6c

40

20 o o

_0.2

_OA

-

_0.6

t

0.8

FIG. 27

_{Deflection at Center as a Function of Time}

10x14

8 WO h

vM 6

b0 M0 4 o

04

0.6

t

FIG. 29

Deflection at Center as a Function of Time

i

--=6O

u

40x10

I

30

20

o

_

______rAP!---4

__4'111

0

s:--I__o

__H

o

0.2

0.8

i-0

10x14 Q) o u u) 8 wo h

vM

6 b0M0 4 2

o

o

02

04

0.6

t

08

o

I-0

FIG. 30

Deflection at Center as a Function of Time

Io

50x10

wo h

vM

b0M0 20

o

rI

1,

o

Ça

/

/

/

/

/

,1

/

lOx 14

12

Io

0.6

t

0.8

FIG. 31

Deflection at Center as a Function of Time

50xI0

(n > Q Q)

vM

b0 M0

20

IO o I-0

o

h

/ '\

/

/

/

/

r -I.

'Ar,,

Ì4

wo 8 h

vM

b0M0 4 2

o

o

02

0.4

10x14 u, Q) > 12 L C-) o u, L

eio

Q) D L) V) 8 wo h

vM 6

b0M0 4 2

0.2

0.4

-

0.6

t

08

FIG. 32

Deflection at Center as a Function of Time

0x10

50

w

o

I-0

30

wo h

vM

b0M0

20

(1 LIQ

-=u

_h e

/

/

\O

-I-_

8 wo h

vM

b0M0 ₆

4

o

70x10

ch a) > L Q

60

50

40

_wo h

vM

b0M0

30

20

l0

o

o=IOO

o

02

0.4

0.6

0.8 I-0

t

10x14 8 WO h

vM

b0 M06 2

0.2

0.4

06

08

t

FIG.. 34 Deflection at Center as a Function of Time o o 0x10

o

20

Io O U) Q) > L o V Q) -c U) o -o L Q) o o (J) h

vM

b0M0

=ii

/

/

/

/

f

/

/

/

/

/

/

r

r

/

/

/

/

/

/

/

z

z

/7

V

lì

//1

/

r

15_

-10x14 (n a) > 12 t-Q -o o (0)

t-I0

a) o (J (f) 8 wo h

vM

6 b0M0 4 2 o 0x10

o

50--h 2 vo 40 b0M0 30 20 lo o

FIG. 35 Deflection at Center as a Function of Time

(n > t-Q -C u, o -o t.-o '4-a) o Q (J)

Go_120

/

/

/

o

ri

o

0.2

04

0.6

0.8

I-0 -t

o

o

I0 9 8 7 6 u

M

(in2 se2) ₅ 4 3 2 o =20 15 I0 7 5 3 o

20

40

60

v0(in sec

FIG. 36 Energy Dissipated in Permanent Set as a Function

of Velocity

Ua0 cr b0h3 2 3

M'3

M4 5

0.25

0.5

0.75

I-0 wo_i h

FIG. 37

Energy Dissipated as a Function of Permanent Set at the Center

W.f h

vM

b0 M0 180 150 _loo 50 o

4:'

\5 2.0 5 (69) o 20

40

0.0

60

80

h