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Dynamic response of elastic-plastic beams with axial constraints
Inr. J. Impact &/"(I Vol. 15. No. I. pp. 3-16. 1994 Printed in Great Britain
DYNAMIC
073&743X/94 $6.00+0.00 I$ 1993 Pergamon Press Lrd
RESPONSE OF ELASTIC-PLASTIC WITH AXIAL CONSTRAINTS ~LO
Department
of Theoretical
(Received
Mechanics,
20 February
Tartu
BEAMS
LEPIK University, Estonia
Vanemuise
1992; itI reuised/onn
I March
str. 46250,
EE24CKl Tartu.
1993)
Summary-Dynamic response of axially restrained beams under transverse impulsive loading is discussed. The material of the beam is elastic-plastic with linear strain-hardening. The ends of the beam are clamped against end rotations and axial displacements, The equations of motion are integrated by the method of finite differences. Computations for diRerent material and beam parameters are carried out. These results enable us to investigate the character of the beam’s response. After reaching the peak deflection the beam responds with nonlinear vibrations. In order to analyse this phase of motion an approximate solution is proposed. According to this the motion subsequent to the peak deflection is regarded as elastic. More details of the beam‘s single-degree-of-freedom model are discussed. In spite of its artificiality, this model describes anomalous behavior of the beam, which was found by Symonds er al. for a Shanley-type beam.
NOTATION
B, h, L P f x. i I‘, \\’ TM 6, e ” 0, E rl 1. U, K w, A K I Y YI. Y2. Y,
width, height and length of the beam density nondimensional time nondimensional coordinates nondimensional displacements nondimensional axial force and bending moment nondimensional stress and strain nondimensional initial velocity yield stress Young’s modulus nondimensional viscosity coefficient strain hardening coefficient nondimensional internal and kinetic energy nondimensional plastic energy (WU2 wJ,lwLY nondimensional mid-span deflection f, -f solutions of the cubic equation (18)
Subscripts * m e, P Superscripts (.) (3
marker of a dimensional quantity peak value of deflections elastic or plastic component
I. INTRODUCTION
In the present paper, response of elastic-plastic fully-clamped beams under dynamic loading is discussed. The beam is constrained so that neither rotations nor axial displacements at its ends can occur. The loads must be so large that plastic deformations take place. For a rigid-plastic material this problem was solved by Symonds and Mentel [I] in 1958. After that it has been discussed in many papers, from which we shall discuss [24]. In all these papers the axial displacement of the beam was neglected, but as Jones had
shown [S] the response is very sensitive to the degree of end fixity. The interest in axially constrained beams has remained until recently: to confirm this statement the papers [6-73 may be cited. The response of rigid-plastic beams with stepwise varying thickness has been discussed in [8-91. It is not possible to deal with all the papers about the problem in question, therefore we would like to turn our attention to the monograph of Jones [lo] and to several review papers (e.g. [ 11,123). In spite of the great number of papers on the response of rigid-plastic pinned beams, all these solutions have been approximate: to obtain an exact solution for the rigid-plastic material is mathematically quite complicated and to our mind it is lacking at present, For some problems it is essential to take into account elastic deformations. Here also several approximate methods of solution have been proposed. As an example we shall mention the method which has been evaluated by Symonds [13-141. Here the course of the solution is divided into four stages: (i) the elastic phase of motion, (ii) small deformations for a rigid-plastic body, (iii) membrane phase for rigid-plastic material, (iv) elastic recovery. In this case numerical examples are in good agreement with the experimental data. In some papers comparisons have been made between the results obtained for rigid-plastic and elastic-plastic materials, as in the paper of Symonds and Frye [ 15). Numerical methods of solution for impulsively loaded structures have been discussed by Mikkola et al. [16]. An interesting fact was discovered by Symonds and his co-workers [ 171 in 1985. They demonstrated that for short pulse loading the permanent deflection of the pin-ended beam may be in the opposite direction of the load. They have called this phenomenon the “counter-intuitive behavior” of the beam. For examining this effect Symonds et al. have applied a Shanley type model, which consists of two rigid-bars connected by a small elastic-plastic cell [ 171. This beam model has a single degree of freedom and if damping is omitted it is possible to obtain a closed-form solution. It turns out that the counter-intuitive behavior occurs over a wide range of load parameters. Such anomalous behavior of inelastic structures was also stated by Galiev and Nechitailo [ 181. Poddar et al. [19] have suggested that in the Symonds’ problem a chaotic motion may be possible. For confirmation of this assertion they have put together the phase and Poincare’ diagrams; they affirm also that the boundaries in the space of the initial conditions are fractal. In their reply to the paper [19], Symonds et al. [20] have shown that the results of [19] are mechanically incorrect, since the damping coefficient was changed during the course of the solution (at the initial stage damping was absent). Using energy diagrams Symonds and his coworkers have shown convincingly that for the model with a single degree of freedom the motion is fully determined and there cannot be any chaos in the case of a short pulse loading. The problem of chaos remained open for models with two or more degrees of freedom. The two-degree-of-freedom beam is examined in the papers of Lee and Symonds [21] and Lee et al. [22]. In these papers, making use of the energy plots, Poincare’ diagrams and Lyapunov exponents, the possibility of chaotic vibrations has been shown. In this paper a method of solution for the response of elastic-plastic beams is developed. Numerical results for some material and beam parameters are obtained. By discussing these results special attention is paid to the beam’s response after peak deffection. For this case a simple approximate model is presented. This model enables us to find the minimum and maximum values of the deflection and to investigate the possibility of the counter-intuitive behavior of the beam.
2. BASIC
EQUATIONS
We shall consider an elastic-plastic beam with a rectangular cross-section; B, h and L are the width, thickness and length of the beam, respectively. The beam is subjected to impulsive transverse loading (the initial velocity is prescribed). The ends of the beam are fixed so as to prevent all the displacements and rotations.
Dynamic
response
The equations of the beam’s motion
ol elastic-plastic
beams
are
dT* ~ =pBh-
ax*
d2U*
at*2 (1)
Here x* is axial coordinate, u* is axial displacement, w* is deflection, p the density, t* the time, T* and M* denote axial force and bending moment, respectively. It is convenient to introduce the following dimensionless quantities
x=--,
X* c=
L v=-,
Lv* hc
J
as -,
U* t =
M=-
a,Bh’
W* WC-
u=--,
L
P T*
T=-
Ct*,
L
h’
4M* u,Bh”
(2)
In these formulae (T, is the yield stress and v* the initial velocity. The Eqns (1) take now the form T’=ii,
M”+4(Tw’)‘=48.
(3)
Henceforth primes and dots will denote differentiation with respect to x and t. The dimensionless membrane force T and bending moment M are obtained from the formulae 0.5
0.5
adz,
T=
az dz,
M=4 s
s -0.5
-0.5
where a = a*,fa, and z = z*/h. We shall assume that the hypotheses of Kirchhoff
(4)
hold. It follows that
-z*- d2w* dx*’
or in dimensionless
quantities
(2) e = u’ + A($w’~ - zw”),
(5)
where A = (h/L)2. One possible method for solving Eqns (3)-(5) is given in the Appendix. Some numerical results obtained by this technique are discussed in the next Section. 3. DISCUSSION
OF
THE
NUMERICAL
RESULTS
To begin with we shall neglect the strain-hardening and viscosity effects; so we take I, = 1, p= 0. The calculations show that the beam behavior depends substantially upon the following parameter
Some typical cases are shown in Figs 1-3, where for v= 10 the mid-span deflection f
u.
6
LEPIK
$G---+z (b)TOI.:,
2
1
t
1 r\
Cd)
n
-
o++Iy
t
1 FIG.
I. (a) Mid-point deflection force and bending moment
01 ’
2
t
versus time for L’= 10, A=O.16 x lo-‘. x=0.8; (b) and (c) axial for the mid-span x=0.5; (d) plastic energy of the mid-span.
1
3
2
FIG. 2. (a) Mid-point deflection versus time for u= 10, A = IO-“, bending moment for the mid-span x=0.5; (d) plastic
4
t
x=0.05; (b) and (c) axial force and energy of the mid-span.
versus time t is plotted. In the first case, where ~=0.8 (Fig. l), the deflection grows up to the peak value, but after that it practically remains constant. In the second case (Fig. 2) we have ~=0.05; now after reaching the peak value the beam vibrates, but the deflection remains positive. In the third case (Fig. 3), where ~=0.02 the deflection attains negative values (counter-intuitive behavior of the beam). In Figs l-3 the plots for the axial force T and bending moment M at the mid-span x = 0.5 are shown. In the same diagrams the plastic energy W, versus time t is also given. The increment of the plastic energy is
ss + h/2
dW;=B
-h/2
or in nondimensional
L/Z
CT*de; dz* dx*
0
form +0.5
dWP=
+0.5
dx s 0
LTde, dz
(d Wi = o,BhL d W,).
(7)
s -0.5
It is of interest to see how the behavior of the beam depends upon the initial velocity U. Results of calculations, which were carried out for h/L=O.Ol and ~=0.02 are presented
Dynamic
(a)
response
of elastic-plastic
beams
f 64-
20. -2. -4.
(b)
(c)
M(0.5.I I.
fd)
riii-;T
--7 1
2
3
4
t
1
2
3
4
t
FIG. 3. (a) Mid-point deflection versus time for L‘= IO. A= 10m4. ~=0.02; (b) and (c) axial force and bending moment for the mid-span I =0.5: (d) plastic energy of the mid-span.
fi 864-
2. 0. -2.
-4. -6.
FOG. 4. Mid-span deflections 2.3-maximal and minimal
versus initial velocity for A= lo-‘. deflections after the peak deflection: rigid-plastic material from [3].
~=0.02; l-peak &approximate
deflection; solution for
in Fig. 4, where curve 1 presents the peak value of mid-span deflection, curves 2-3 the maximal and minimal values of vibrations subsequent to the peak value. To this diagram the following comments can be made. In the case of small initial velocities the beam material remains elastic and the beam carries out elastic vibrations around the equilibrium position f=O; the curves 1 and 2 coincide. By increasing u the motion is elastic-plastic, but the curve 2 remains in the positive side of the diagram. After that comes a region of u, where the curve 2 has negative values (counter-intuitive behavior of the beam). If we shall increase u still more the beam response changes vary abruptly and the deflections are positive all the time. It should be mentioned that the region of counter-intuitive behavior exists only for small values of the parameter K. An analogical problem (beam subjected to short pulse of force at its mid-point) was discussed by Borino et al. [23]. They plotted the mid-point deflections as a function of magnitude of pulse load and noticed many “slots”, formed by vertical line segments (see Fig. 2 in [23]). Such “slots” were not established for the solution, presented in this paper. This seeming contradiction may be caused by the two following reasons (i) In the paper [23] envelopes
of final vibrations
are presented,
but the curve 2 of
8
ti.
FIG. 5. Response
FIG. 6. Approximate
history
LEPlK
lor beams of aluminum 7075-T6 for impulse for three values of viscosity coefficient q.
per unit area
solution (24) Tor/,,=4.5, ~=0.02; l-case u, #O,f, #O, j2=O; .(I $0. j” #O; 3--“real solution” from Fig. 3.
100 1b.s in-’
24ase
u, =O.
Fig. 4 in this paper presents the first maximum of vibrations after the peak deflection was reached. (ii) The end constraints and loading conditions in both cases are different. Dependence of the deflection curvef(t) upon the material parameters I and ‘7 is shown in Figs l-3 and 5. In Figs 1-3 the calculations were carried out also for J=O.95 (strain-hardening material). It follows from these diagrams that in some cases (Figs 2-3) the effect of strain-hardening is not essential, but in the case of Fig. 1 it considerably decreases the deflections. Figure 5 corresponds to the diagram from Fig. 2; the material is aluminum alloy 7075-T6; the impulse is lOOlb.in -2. It follows from this figure that the value of the minimal deflection is very sensitive to the coefficient of viscosity ‘I. Dependence of the beam response upon material parameters evidently requires complementary calculations. 4. APPROXIMATE
METHOD
OF
SOLUTION
The presented method for integration of Eqns (3)-(5) is somewhat complicated and demands much computational time. Therefore it would be useful to develop an approximate method. We shall assume that the peak deflection f,, is prescribed (it can be calculated, e.g. using the concept of a rigid-plastic material) and we shall concentrate on the subsequent motion. We shall propose a solution which uses the following assumptions. (i) The initial velocity is so large that forf = f, the beam is practically in the membrane state. (ii) The motion after the peak valuejm is wholly elastic. According to Figs l-3 the first assumption holds good. More problematic is the second assumption: in some cases (Figs l-2) it is fulfilled quite well, but in the case from Fig. 3 the plastic-energy W, increases from the first peak value to the first minimum. In this case we shall artificially eliminate plastic flow after the peak deflection, This allows us to obtain simple results for which all the characteristic features of the real problem are preserved. We would like to turn our attention to Fig. 6 from which the accordance between the approximate and fiducial solutions seems to be quite satisfactory. The plastic flow after
Dynamic
response
of elastic-plastic
beams
9
the first peak has been neglected in the recent paper by Symonds et al. [24], who termed this solution “the elastic recovery case”. Let us note the values of (T and e at the peak moment by urn and e,. Since the following motion is elastic, we have
u-u, In the membrane
= :(e-em). s
state we have s -0.5 0.5
-0.5 0.5
T, =
Making
M,=4
o,,,dz= 1;
use of Eqns (5) and calculating
T= 1 +E
u,z dz = 0.
the integrals (4) we obtain
1
u'-u:,++-wl) 0s
1E M = - - -A(w”2 fJs
w”),
Here symbols u;, w’,,,, WI denote the values of the corresponding quantities moment. In the following we shall assume that uk=O. The equations of motion (3) are integrated by the method of Galerkin: 0.5
0.5
[M"+4(T~')'-4ti]6wdx=O.
(T’-ii)&dx=O, s 0
Integrating
at the peak
s 0
these equations
by parts we get 0.5
(T&'+ii&)
dx=O
s 0 0.5
(M”6~-4T’6~‘-4K@?w)dx=O.
(9)
s0 We shall seek displacements
u and w in the form
u=u,x(l-2x) w= 16x71 -x)‘[fl
+f2(x-0.5)2],
for which all boundary and symmetry conditions are satisfied. The coefficients fz are calculated from Eqns (9). Calculations have been carried out for the two following cases. (i) Ifs, =O, we get from (9) the system of differential
(10) u,,fi
and
equations
(11)
Is.
IO
LEPIK
(ii) If u1 =O, we obtain 4% + 72 fi = - 48f, - 6k.f2 - 168~(f, - f,)
ii;
+;
= -4f2-22~[4(fr-f,)+3fJ 1155 - 8 @(3f:
where A= 17.459, B=6.418
- .m2
+ 3CfI.G + of:1
x 10m2, C= 1.689 x 10e3, D=6.333
(12)
x 10e4.
These differential equations can be integrated numerically making use of the initial conditions a(r,,,)= li(t,)=0,.f,(t,)=.fm,~~(t,)=0,f2(t,)=.~2(t,)=0. Some calculations, which were carried out forf,,=4.5 are presented in Fig. 6 for ~=0.02. The curve 1 is calculated according to Eqns (11) and curve 2-from Eqns ( 12). The curve 3 in Fig. 6 corresponds to the “actual solution” from Fig. 3. Now we shall consider in detail the single-degree-of-freedom model (SDoF), for which u1 =f2 =O, andf, ~0. For better interpretation of the results, we shall introduce the linear viscosity; so we get from (11) the equation
where c is the viscosity coefficient. By multiplying both sides of (13) with fr and integrating ;.i:=
-U-c
we obtain
’ j;dt. s fin
(14)
Here CJ=6(f:-f~)+21~(f~-~~)~+
1536 143 a-:
- .fzJ21
(15)
is the nondimensional potential energy (the constant in the energy expression has been chosen so that U = 0 for f, = f,). Equation ( 13) was integrated numerically for K = 0.01, c = 0.05. The permanent deflections versus peak deflectionf, are shown in Fig. 7. Dotted lines present maximal and minimal values offin the case of an undamped model. This picture is very similar to that which was found by Symonds et al. for the Shanley type beam [24]: here we also have narrow slots, inside which the counter-intuitive behavior takes place. In the case of the proposed model two questions are of interest: (i) for what peak values f, is the counter-intuitive behavior of the beam possible; (ii) to find minimum and maximum values forfr subsequent to the peak deflection. Let us try to obtain answers to these problems. It is convenient to introduce the quantity y =f, - fr. Equation (15) now takes the form U = -y[12f,-(6+21~+4a~~f~)y+4arcf,+aicy~].
(16)
In the case of an undamped beam, it follows from (14) that forf=O (extremal points off) we have U =O. According to Eqn (16) this is possible only for positive values of y;
Dynamic
FIG. 7. SDoF
beam,
permanent
FIG. 8. Energy
response
of elastic-plastic
deflectionsj‘versus
diagrams
beams
peak deflectionsj,
for the SDoF
for x=0.01
and c=O.O5.
beam for ~=0.02.
consequently f,> II; and during the whole motion the mid-span deflections fi cannot exceed the peak valuef,. Typical diagrams of the “energy approach” [23] are presented in Fig:8. (Calculations were carried out for k-=0.02.) Here two curves are plotted in each of the four cases: one is the potential energy (elastic strain energy) given by Eqn (16); the other (zig-zag lines) is the total energy, obtained by numerical integration of Eqn (14). The motion will terminate at a minimum point of II. Counter-intuitive motion is possible in case b, where the motion may be terminated at the right-hand minimum where y >f, andf, ~0. Let us denote the interval of the values off, for which negative permament deflections are possible by (f;,fz). It follows from Fig. 8 that for small values off,, the curve U = U(y) has only one minimum. By increasingf, a point of inflection appears and after that two minima are possible. In the inflection point we have U”(y)=0 and it follows from Eqn (16) that
12
u.
This inflection
LEPIK
point appears first if the expression under the squareroot is zero; i.e. (17)
Counter-intuitive behavior of the beam is possible up to the critical situation shown in Fig. 8(c). Hence the upper limit/i can be calculated from the conditions U(y)= L”(y) =O. This means that the equation U(y)=0 must have a double root and (18)
u = ak-yb - Y 112(Y - J’J,
where y, and y, are roots of the equation By comparing (16) and (18) one finds y3=4fm+
-2Y1
3y:-8f,fy, y:-2f;y:= By eliminating
U(y)=O.
from (19) the quantitiesj,
3(2 + 7~) - ~
+4f,i2=
aK
- df; UK and y3 we get the biquadratic
143 YY - ___ 3584~~
equation (20)
After solving this equation we find from (19X that 256~~; - 143’
I: = 512~~;
y3=4fZ
-2Y,.
(21)
Equation (20) has real solutions only if its discriminant is non-negative, this requirement gives ~2&=0.03571; so we see that negative permanent deflections can be realized only for small values of ti. For calculating the minimal values offi for an undamped beam we shall make use of Eqn (16) and find for U(y)=0 the largest root y=y,. The minimal value of deflections is thenfmi, =f, - y,. These values versus peak deflectionsf, are shown in Fig. 9. If K < 0.0357 these curves have abrupt changes at f,=fz (negative values of fmin turn positive). The motion proceeds between fmin and f,,.
FIG.
9. Minimum values for the mid-span deflections versus peak values.
Dynamic
FIG.
10. Peak
valuesj,
response
of elastic-plastic
and extreme
values
beams
ofjversus
13
parameter
K
The critical situation from Fig. 8(c) deserves special attention. For a prescribed K we can find from Eqns (20) and (23) the quantities f,‘, fmi”=f~ -y,, f,,,=fz -y,. These values are shown in Fig. 10. If we take a valuef,, which is a bit less fromfz, and introduce small damping, the vibrations-depending on the magnitude of the viscosity coefficientoccur in one of the intervals (f rn,“~f max) or (f,,,,fi). In the first case we have counter-intuitive behavior of the beam. 5. CONCLUSIONS
A method for calculating the dynamic response of elastic-plastic beams with edge constraints has been proposed. In the discussion of numerical results special attention is turned to the nonlinear vibrations which proceed after the peak value of deflections. Counter-intuitive behavior of the beam, where the mid-point deflections remain negative, is also examined. Numerical results show that motion of the beam after the peak deflection is very sensitive to material and beam parameters. The main purpose of this paper is to propose an approximate solution for the beam response. More details of the SDoF case are examined. In spite of the conditionality of this solution, it yields simple closed-form solutions for finding the extreme values of deflections and intervals for parameters, in which counter-intuitive behavior of the beam may be expected. These results are in accordance with the solutions obtained by Symonds and his collaborators for a Shanley-type beam. The case of two and more degrees of freedom needs a complementary investigation. As shown by Symonds et al. [21-231 here chaotic motion of the beam could take place. Acknowledgements-The about the problems
author has had many useful discussions with dealt with in this paper. Their valuable suggestions
Professors are highly
P. S. Symonds appreciated.
and N. Jones
REFERENCES 1. P. S. SYMONDS and T. J. MENTAL, Impulsive loading of plastic beams with axial constraints. J. Mech. Phys. Solids 6, 186202 (1958). 2. N. JONES, A theoretical study of the dynamic plastic behavior of beams and plates with finite deflections. IHI. J. Solids Strucr. 7, 1007-1029 (1971). 3. P. S. SYMONDS and N. JONES, Impulsive loading of fully clampped beams with finite plastic deformations and strain-rate sensitivity. Int. J. Mech. Sci. 14, 49-69 (1972). 4. N. JONES, Plastic failure of ductile beams loaded dynamically. J. Engng Industry Trans. ASME 98, Series B, No. 1, 131-136 (1976). 5. N. JONES, Influence of in-plane displacements at the boundaries of rigid-plastic beams and plates. Inc. J. Mech. Sci. 15, 547-561 (1973). 6. R. VAZIRI, M. D. OLSON and D. L. ANDERSON, Dynamic response of axially constrained plastic beams to blast loads. Inl. J. Solids Strucf. 23, 153-174 (1987). 7. R. B. SCHUBAK, D. A. ANDERSON and M. D. OLSON, Simplified dynamic analysis of rigid-plastic beams. Int.
J. Impact
Engng
8, 27-42
(1989).
8. u. LEPIK, Optimal design of dynamically loaded rigid-plastic beams effects. Transact. Turru University 721, 25-38 (in Russian, 1981).
taking
into
account
the membrane
u. LEPIK
14
9. T. LEPIKLJLT, Calculation oldynamically loaded rigid-plastic beams in the case of fixed ends. Transact. firfu Unirrr.sif>, 853, 25-37 (in Russian, 1989). 10. N. JONES. Sfruc~ural Impacf. Cambridge Press. U.K. (1989). 1 I. P. S. SYMONDS. Twenty years of developments in inelastic dynamics. Proc. Srh Et1qiwrritlg Mechanical Specidiry Conj: ASCE. New York. pp. l-23 (1984). 12. N. JONES, Recent studies on the dynamic plastic behavior of structures. Appl. Mech. Rec. 42(4),95-l 15 (1989). 13. P. S. SYMONDS. Finite elastic and plastic deformations of pulse loaded structures by an extended mode technique. Inf. J. Mech. Sri. 22. 597-605 (1980). 14. P. S. SYMONDS, Elastic, finite deflection and strain rate effects in a mode approximation technique for plastic deformation of pulse loaded structures. J. Mech. Enyny Sci. 22(4). 189-197 (1980). 15. P. S. SYMONDS and CH. W. G. FRYE. On the relation between rigid-plastic and elastic-plastic predictions on response to pulse loading. Inf. J. lmpocf Euyuy 7(l), 139-149 (1988). 16. M. I. MIKKOLA, M. TUOMALA and H. SINISALO. Comparisons ol numerical integration methods in the analysis of impulsively loaded elastic-plastic and viscoplastic structures. Contpufers Sfrurf. 14,469478 (198 I). 17. P. S. SYMONDS and T. X. Yu. Counter-intuitive behavior in a problem of elastic-plastic beam dynamics. J. Appl. Mech. 52, 517-522 (1985). 18. SH. U. GALIEV and N. V. NECHITAILO, Unexpected behavior of plates under impulsive and hydrodynamic loading. Prohlemy prochnosti 12, 63-72 (in Russian. 1986). 19. B. PODDAR, F. C. MOON and S. MUKHERJEE, Chaotic motion of an elastic-plastic beam. ASME J. Appl. Mech. 55, 185-189 (1988). 20. P. S. SYMONDS. G. BORINO and U. PEREGO, Chaotic motion of an elastic-plastic beam. ASME J. Appl. Mech. 55, 745-746 ( 1988). 21. J.-Y. LEE and P. S. SYMONDS, Extended energy approach to chaotic elastic-plastic response to impulsive loading. Inf. J. Mech. Sci. 34(2), 139-157 (1992). 22. J.-Y. LEE, P. S. SYMONDS and G. BORINO, Chaotic responses of a fixed ended elastic-plastic beam model to short pulse loading. I-11 (to be published in J. Appl. Mech.). 23. G. BORINO, I-J. PEREGO and P. S. SYMONDS, An energy approach to anomalous damped elastic-plastic response response to short pulse loading. ASME J. Appl. Mech. 56, 430438 (1989). 24. P. S. SYMONDS, F. GENNA and A. CIULLINI. Special cases in study of anomalous dynamic elastic-plastic response of beams by a simple model. Inr. J. So/ids Struct. 27(3), 299-314 (1991).
APPENDIX Here an iterative numerical scheme for integrating the equations of motion (3)-(5) is presented. We shall assume that the material or the beam is elastic-plastic with linear strain hardening (Fig. Al). Here E is the Young’s modulus; E” = E( 1 - 2.). where i. is a constant, characterizing the strain hardening (A= 0 for elastic and %= 1 for perfectly plastic material). The segment BD corresponds to elastic unloading, the segment CD to secondary plastic loading. We shall also include the viscosity in the form
u* = Q(e)
+ qP,
(AlI
where the [unction Q(e) will be determined according to Fig. Al. For elastic deformations Eqn (Al) describes the Voight’s body, ror ideally plastic materials we have Bingham’s body. Further on we shall use the dimensionless quantities o=a*/o, and q=tl*/u.. It is convenient to write Eqn (Al) in a dilTerential form. Let us assume that we know the quantities u and e for the two instants f and f +At and are aware ol the segment of Fig. Al on which we find ourselves at the instant f. If we are on the segments OA or BC, the following deformation is elastic and we have:
u(f+Af)=u(f)+
6”
0
9
-
0E
FIG. Al.
Stress-strain
642)
E[e(f+Af)-e(f)]+qP(f). US
diagram
E”
e
for strain-hardening
material.
Dynamic lfwearron
response
of elastic-plastic
beams
15
thesegmentsABorDCandcT(r~Ae(r)zOtheloadingprocess~sactiveforr~tr.r a(r+Ar)=o(r)
+ L(l u<
+Ar)andwe
have
-i.)[e(r+Ar)-e(t)]+~P(t).
(A31
However iT o(r).Ae(r)
quantities
for the instant
r +Ar
Step 1. We extrapolate
+2u(s.t)-U(.YJ-AI) +2rr(s,t)-w(s.r + ti(.x,t -AI) + tifx,r -AI).
we shall find in the following
the values
Tand
M for I +A/
T+ = T(s,r - )A/)M,
-Ar) (A41 way
according
37-(s,r -AI)+
IO parabolic
interpolation:
3T(.v.r)
=M(s.r-ZAI)-~IM(.\..I-AI)+~M(.~...I).
(AS)
2. Making use of the cubic splines. we calculate the derivatives T’, (Tw’)‘, M” and find the accelerations ii(s,r +Ar), i+(.x,r +Ar) from Eqns (3). Step 3. Here we shall calculate the quantities u. w, ri. rc from Eqns (A4). Step 4. With the aid of cubic splines we find 11’. w and w” for the instant f +Ar. Step 5. We shall calculate the dimensionless deformation e from Eqn (5). Step 6. For each point .@O, O.OS], ~-0.5. OS] we shall clearly indicate whether the material of the beam deforms plastically or if there is elastic loading or unloading. In the first case we calculate the dimensionless stress 0 from (A3). in the second case. from Eqn (AZ). Step 7. Evaluating the integrals (4) numerically we find the quantities T(.Y.r+Ar), M(.u.f +Ar). Step 8. We have lo check the inequalities Step
IT+-T(x,r+Ar))<.a,
w3
~I~+-M(.YJ+A~))
for SE[O, 0.51. where E>O is a prescribed small quantity. If the inequalities (A6) are not fulfilled, we shall take r= T(r,r +A[). M + =,44(.x-.1 +Ar) and repeat the steps 228 until they are satisfied. Only after that shall we go on IO the following step. Step
9. We shall increase
the time by AI and return
to step I.
The reliability of the obtained results can be checked by the energy balance criterion Li* and K* denote the internal and kinetic energies, the energy balance criterion is
(Mikkola
et al. [16]).
If
AW*=AU*+AK*=O, where A”*=~nL{~*A[~+;(~)‘]-M*A($)}d.~
AK*
= J-1 ,,,[A(gy
+ A(gy]dr*.
Here AW* is the error which we have made in calculating In dimensionless quantities (2) the energy criterion obtains only one half of the beam): AW=
0.5 s[( T
0
Au’ + f AA\v”
the energy during the time step At. the form (because of the symmetry we shall consider
- AMAw”
+ Ati
+ hAti
1 dx.
(A71
The energy error AW must be sufficiently small; if it is not so, the step size At must be reduced. We have calculated the integrals (4) numerically by taking eight Gaussian points. As to the spanwise step size AX, then the mid-span of the beam was divided into n equal parts; for n the values IO and 20 have been taken.
16
u.
LEPIK
FIG. A2. Mid-span deflections as functions of initial impulse for beams of aluminum I -peak values; 2-minimal values; 3--estimated final deflection: 0 and A-test points for maximum transient and permanent deflections.
707ST6; from [ 133
The case n = 10 gave quite good results for the peak deflection, but the minimum values of the curve deflection versus time were somewhat overestimated. Therefore some of the calculations have been carried out for n = 20. The size of the time step Ar was controlled in two ways. First the number of iterations for the steps 2-8 must be small; secondly the energy error AW in (A7) must be smaller than 5 x lo-? For most problems these requirements were satisfied for the time step Ar = 0.5 x 10m3 (if a = IO). For n = 20 the time step must be considerably smaller. which to a large extent increases the computation time. For checking the efficiency of the proposed algorithm and for comparing the numerical results with experimental data, calculations were carried out for beams of aluminium alloy 707ST6; the beam height was 0.075 in, length 5.1 in. These dimensions correspond to the experimental data of Lindholm and Bessey, which have been taken from Fig. 5 of the paper [l3]. Numerical results achieved according to our method have been presented for various impulses in Fig. A2 (for parameter i. the value 0.99 has been taken). Curve I gives maximal deflections (peak deflections). curve 2 values of the first minimum. Test points for maximum transient and permanent deflections are marked with circles and triangles, respectively. The permanent deflections have been estimated as the average of the peak displacement and the subsequent first minimum displacement. Of course this is a rather crude approximation, but somewhat encouraging is the good agreement between the curve 3 and test points. Analogical comparisons have also been made for beams of mild steel. Here the deflections are somewhat higher than for the test data (this is caused by a circumstance that we have not taken into account, the strain-rate sensitivity of the material).
DYNAMIC
073&743X/94 $6.00+0.00 I$ 1993 Pergamon Press Lrd
RESPONSE OF ELASTIC-PLASTIC WITH AXIAL CONSTRAINTS ~LO
Department
of Theoretical
(Received
Mechanics,
20 February
Tartu
BEAMS
LEPIK University, Estonia
Vanemuise
1992; itI reuised/onn
I March
str. 46250,
EE24CKl Tartu.
1993)
Summary-Dynamic response of axially restrained beams under transverse impulsive loading is discussed. The material of the beam is elastic-plastic with linear strain-hardening. The ends of the beam are clamped against end rotations and axial displacements, The equations of motion are integrated by the method of finite differences. Computations for diRerent material and beam parameters are carried out. These results enable us to investigate the character of the beam’s response. After reaching the peak deflection the beam responds with nonlinear vibrations. In order to analyse this phase of motion an approximate solution is proposed. According to this the motion subsequent to the peak deflection is regarded as elastic. More details of the beam‘s single-degree-of-freedom model are discussed. In spite of its artificiality, this model describes anomalous behavior of the beam, which was found by Symonds er al. for a Shanley-type beam.
NOTATION
B, h, L P f x. i I‘, \\’ TM 6, e ” 0, E rl 1. U, K w, A K I Y YI. Y2. Y,
width, height and length of the beam density nondimensional time nondimensional coordinates nondimensional displacements nondimensional axial force and bending moment nondimensional stress and strain nondimensional initial velocity yield stress Young’s modulus nondimensional viscosity coefficient strain hardening coefficient nondimensional internal and kinetic energy nondimensional plastic energy (WU2 wJ,lwLY nondimensional mid-span deflection f, -f solutions of the cubic equation (18)
Subscripts * m e, P Superscripts (.) (3
marker of a dimensional quantity peak value of deflections elastic or plastic component
I. INTRODUCTION
In the present paper, response of elastic-plastic fully-clamped beams under dynamic loading is discussed. The beam is constrained so that neither rotations nor axial displacements at its ends can occur. The loads must be so large that plastic deformations take place. For a rigid-plastic material this problem was solved by Symonds and Mentel [I] in 1958. After that it has been discussed in many papers, from which we shall discuss [24]. In all these papers the axial displacement of the beam was neglected, but as Jones had
shown [S] the response is very sensitive to the degree of end fixity. The interest in axially constrained beams has remained until recently: to confirm this statement the papers [6-73 may be cited. The response of rigid-plastic beams with stepwise varying thickness has been discussed in [8-91. It is not possible to deal with all the papers about the problem in question, therefore we would like to turn our attention to the monograph of Jones [lo] and to several review papers (e.g. [ 11,123). In spite of the great number of papers on the response of rigid-plastic pinned beams, all these solutions have been approximate: to obtain an exact solution for the rigid-plastic material is mathematically quite complicated and to our mind it is lacking at present, For some problems it is essential to take into account elastic deformations. Here also several approximate methods of solution have been proposed. As an example we shall mention the method which has been evaluated by Symonds [13-141. Here the course of the solution is divided into four stages: (i) the elastic phase of motion, (ii) small deformations for a rigid-plastic body, (iii) membrane phase for rigid-plastic material, (iv) elastic recovery. In this case numerical examples are in good agreement with the experimental data. In some papers comparisons have been made between the results obtained for rigid-plastic and elastic-plastic materials, as in the paper of Symonds and Frye [ 15). Numerical methods of solution for impulsively loaded structures have been discussed by Mikkola et al. [16]. An interesting fact was discovered by Symonds and his co-workers [ 171 in 1985. They demonstrated that for short pulse loading the permanent deflection of the pin-ended beam may be in the opposite direction of the load. They have called this phenomenon the “counter-intuitive behavior” of the beam. For examining this effect Symonds et al. have applied a Shanley type model, which consists of two rigid-bars connected by a small elastic-plastic cell [ 171. This beam model has a single degree of freedom and if damping is omitted it is possible to obtain a closed-form solution. It turns out that the counter-intuitive behavior occurs over a wide range of load parameters. Such anomalous behavior of inelastic structures was also stated by Galiev and Nechitailo [ 181. Poddar et al. [19] have suggested that in the Symonds’ problem a chaotic motion may be possible. For confirmation of this assertion they have put together the phase and Poincare’ diagrams; they affirm also that the boundaries in the space of the initial conditions are fractal. In their reply to the paper [19], Symonds et al. [20] have shown that the results of [19] are mechanically incorrect, since the damping coefficient was changed during the course of the solution (at the initial stage damping was absent). Using energy diagrams Symonds and his coworkers have shown convincingly that for the model with a single degree of freedom the motion is fully determined and there cannot be any chaos in the case of a short pulse loading. The problem of chaos remained open for models with two or more degrees of freedom. The two-degree-of-freedom beam is examined in the papers of Lee and Symonds [21] and Lee et al. [22]. In these papers, making use of the energy plots, Poincare’ diagrams and Lyapunov exponents, the possibility of chaotic vibrations has been shown. In this paper a method of solution for the response of elastic-plastic beams is developed. Numerical results for some material and beam parameters are obtained. By discussing these results special attention is paid to the beam’s response after peak deffection. For this case a simple approximate model is presented. This model enables us to find the minimum and maximum values of the deflection and to investigate the possibility of the counter-intuitive behavior of the beam.
2. BASIC
EQUATIONS
We shall consider an elastic-plastic beam with a rectangular cross-section; B, h and L are the width, thickness and length of the beam, respectively. The beam is subjected to impulsive transverse loading (the initial velocity is prescribed). The ends of the beam are fixed so as to prevent all the displacements and rotations.
Dynamic
response
The equations of the beam’s motion
ol elastic-plastic
beams
are
dT* ~ =pBh-
ax*
d2U*
at*2 (1)
Here x* is axial coordinate, u* is axial displacement, w* is deflection, p the density, t* the time, T* and M* denote axial force and bending moment, respectively. It is convenient to introduce the following dimensionless quantities
x=--,
X* c=
L v=-,
Lv* hc
J
as -,
U* t =
M=-
a,Bh’
W* WC-
u=--,
L
P T*
T=-
Ct*,
L
h’
4M* u,Bh”
(2)
In these formulae (T, is the yield stress and v* the initial velocity. The Eqns (1) take now the form T’=ii,
M”+4(Tw’)‘=48.
(3)
Henceforth primes and dots will denote differentiation with respect to x and t. The dimensionless membrane force T and bending moment M are obtained from the formulae 0.5
0.5
adz,
T=
az dz,
M=4 s
s -0.5
-0.5
where a = a*,fa, and z = z*/h. We shall assume that the hypotheses of Kirchhoff
(4)
hold. It follows that
-z*- d2w* dx*’
or in dimensionless
quantities
(2) e = u’ + A($w’~ - zw”),
(5)
where A = (h/L)2. One possible method for solving Eqns (3)-(5) is given in the Appendix. Some numerical results obtained by this technique are discussed in the next Section. 3. DISCUSSION
OF
THE
NUMERICAL
RESULTS
To begin with we shall neglect the strain-hardening and viscosity effects; so we take I, = 1, p= 0. The calculations show that the beam behavior depends substantially upon the following parameter
Some typical cases are shown in Figs 1-3, where for v= 10 the mid-span deflection f
u.
6
LEPIK
$G---+z (b)TOI.:,
2
1
t
1 r\
Cd)
n
-
o++Iy
t
1 FIG.
I. (a) Mid-point deflection force and bending moment
01 ’
2
t
versus time for L’= 10, A=O.16 x lo-‘. x=0.8; (b) and (c) axial for the mid-span x=0.5; (d) plastic energy of the mid-span.
1
3
2
FIG. 2. (a) Mid-point deflection versus time for u= 10, A = IO-“, bending moment for the mid-span x=0.5; (d) plastic
4
t
x=0.05; (b) and (c) axial force and energy of the mid-span.
versus time t is plotted. In the first case, where ~=0.8 (Fig. l), the deflection grows up to the peak value, but after that it practically remains constant. In the second case (Fig. 2) we have ~=0.05; now after reaching the peak value the beam vibrates, but the deflection remains positive. In the third case (Fig. 3), where ~=0.02 the deflection attains negative values (counter-intuitive behavior of the beam). In Figs l-3 the plots for the axial force T and bending moment M at the mid-span x = 0.5 are shown. In the same diagrams the plastic energy W, versus time t is also given. The increment of the plastic energy is
ss + h/2
dW;=B
-h/2
or in nondimensional
L/Z
CT*de; dz* dx*
0
form +0.5
dWP=
+0.5
dx s 0
LTde, dz
(d Wi = o,BhL d W,).
(7)
s -0.5
It is of interest to see how the behavior of the beam depends upon the initial velocity U. Results of calculations, which were carried out for h/L=O.Ol and ~=0.02 are presented
Dynamic
(a)
response
of elastic-plastic
beams
f 64-
20. -2. -4.
(b)
(c)
M(0.5.I I.
fd)
riii-;T
--7 1
2
3
4
t
1
2
3
4
t
FIG. 3. (a) Mid-point deflection versus time for L‘= IO. A= 10m4. ~=0.02; (b) and (c) axial force and bending moment for the mid-span I =0.5: (d) plastic energy of the mid-span.
fi 864-
2. 0. -2.
-4. -6.
FOG. 4. Mid-span deflections 2.3-maximal and minimal
versus initial velocity for A= lo-‘. deflections after the peak deflection: rigid-plastic material from [3].
~=0.02; l-peak &approximate
deflection; solution for
in Fig. 4, where curve 1 presents the peak value of mid-span deflection, curves 2-3 the maximal and minimal values of vibrations subsequent to the peak value. To this diagram the following comments can be made. In the case of small initial velocities the beam material remains elastic and the beam carries out elastic vibrations around the equilibrium position f=O; the curves 1 and 2 coincide. By increasing u the motion is elastic-plastic, but the curve 2 remains in the positive side of the diagram. After that comes a region of u, where the curve 2 has negative values (counter-intuitive behavior of the beam). If we shall increase u still more the beam response changes vary abruptly and the deflections are positive all the time. It should be mentioned that the region of counter-intuitive behavior exists only for small values of the parameter K. An analogical problem (beam subjected to short pulse of force at its mid-point) was discussed by Borino et al. [23]. They plotted the mid-point deflections as a function of magnitude of pulse load and noticed many “slots”, formed by vertical line segments (see Fig. 2 in [23]). Such “slots” were not established for the solution, presented in this paper. This seeming contradiction may be caused by the two following reasons (i) In the paper [23] envelopes
of final vibrations
are presented,
but the curve 2 of
8
ti.
FIG. 5. Response
FIG. 6. Approximate
history
LEPlK
lor beams of aluminum 7075-T6 for impulse for three values of viscosity coefficient q.
per unit area
solution (24) Tor/,,=4.5, ~=0.02; l-case u, #O,f, #O, j2=O; .(I $0. j” #O; 3--“real solution” from Fig. 3.
100 1b.s in-’
24ase
u, =O.
Fig. 4 in this paper presents the first maximum of vibrations after the peak deflection was reached. (ii) The end constraints and loading conditions in both cases are different. Dependence of the deflection curvef(t) upon the material parameters I and ‘7 is shown in Figs l-3 and 5. In Figs 1-3 the calculations were carried out also for J=O.95 (strain-hardening material). It follows from these diagrams that in some cases (Figs 2-3) the effect of strain-hardening is not essential, but in the case of Fig. 1 it considerably decreases the deflections. Figure 5 corresponds to the diagram from Fig. 2; the material is aluminum alloy 7075-T6; the impulse is lOOlb.in -2. It follows from this figure that the value of the minimal deflection is very sensitive to the coefficient of viscosity ‘I. Dependence of the beam response upon material parameters evidently requires complementary calculations. 4. APPROXIMATE
METHOD
OF
SOLUTION
The presented method for integration of Eqns (3)-(5) is somewhat complicated and demands much computational time. Therefore it would be useful to develop an approximate method. We shall assume that the peak deflection f,, is prescribed (it can be calculated, e.g. using the concept of a rigid-plastic material) and we shall concentrate on the subsequent motion. We shall propose a solution which uses the following assumptions. (i) The initial velocity is so large that forf = f, the beam is practically in the membrane state. (ii) The motion after the peak valuejm is wholly elastic. According to Figs l-3 the first assumption holds good. More problematic is the second assumption: in some cases (Figs l-2) it is fulfilled quite well, but in the case from Fig. 3 the plastic-energy W, increases from the first peak value to the first minimum. In this case we shall artificially eliminate plastic flow after the peak deflection, This allows us to obtain simple results for which all the characteristic features of the real problem are preserved. We would like to turn our attention to Fig. 6 from which the accordance between the approximate and fiducial solutions seems to be quite satisfactory. The plastic flow after
Dynamic
response
of elastic-plastic
beams
9
the first peak has been neglected in the recent paper by Symonds et al. [24], who termed this solution “the elastic recovery case”. Let us note the values of (T and e at the peak moment by urn and e,. Since the following motion is elastic, we have
u-u, In the membrane
= :(e-em). s
state we have s -0.5 0.5
-0.5 0.5
T, =
Making
M,=4
o,,,dz= 1;
use of Eqns (5) and calculating
T= 1 +E
u,z dz = 0.
the integrals (4) we obtain
1
u'-u:,++-wl) 0s
1E M = - - -A(w”2 fJs
w”),
Here symbols u;, w’,,,, WI denote the values of the corresponding quantities moment. In the following we shall assume that uk=O. The equations of motion (3) are integrated by the method of Galerkin: 0.5
0.5
[M"+4(T~')'-4ti]6wdx=O.
(T’-ii)&dx=O, s 0
Integrating
at the peak
s 0
these equations
by parts we get 0.5
(T&'+ii&)
dx=O
s 0 0.5
(M”6~-4T’6~‘-4K@?w)dx=O.
(9)
s0 We shall seek displacements
u and w in the form
u=u,x(l-2x) w= 16x71 -x)‘[fl
+f2(x-0.5)2],
for which all boundary and symmetry conditions are satisfied. The coefficients fz are calculated from Eqns (9). Calculations have been carried out for the two following cases. (i) Ifs, =O, we get from (9) the system of differential
(10) u,,fi
and
equations
(11)
Is.
IO
LEPIK
(ii) If u1 =O, we obtain 4% + 72 fi = - 48f, - 6k.f2 - 168~(f, - f,)
ii;
+;
= -4f2-22~[4(fr-f,)+3fJ 1155 - 8 @(3f:
where A= 17.459, B=6.418
- .m2
+ 3CfI.G + of:1
x 10m2, C= 1.689 x 10e3, D=6.333
(12)
x 10e4.
These differential equations can be integrated numerically making use of the initial conditions a(r,,,)= li(t,)=0,.f,(t,)=.fm,~~(t,)=0,f2(t,)=.~2(t,)=0. Some calculations, which were carried out forf,,=4.5 are presented in Fig. 6 for ~=0.02. The curve 1 is calculated according to Eqns (11) and curve 2-from Eqns ( 12). The curve 3 in Fig. 6 corresponds to the “actual solution” from Fig. 3. Now we shall consider in detail the single-degree-of-freedom model (SDoF), for which u1 =f2 =O, andf, ~0. For better interpretation of the results, we shall introduce the linear viscosity; so we get from (11) the equation
where c is the viscosity coefficient. By multiplying both sides of (13) with fr and integrating ;.i:=
-U-c
we obtain
’ j;dt. s fin
(14)
Here CJ=6(f:-f~)+21~(f~-~~)~+
1536 143 a-:
- .fzJ21
(15)
is the nondimensional potential energy (the constant in the energy expression has been chosen so that U = 0 for f, = f,). Equation ( 13) was integrated numerically for K = 0.01, c = 0.05. The permanent deflections versus peak deflectionf, are shown in Fig. 7. Dotted lines present maximal and minimal values offin the case of an undamped model. This picture is very similar to that which was found by Symonds et al. for the Shanley type beam [24]: here we also have narrow slots, inside which the counter-intuitive behavior takes place. In the case of the proposed model two questions are of interest: (i) for what peak values f, is the counter-intuitive behavior of the beam possible; (ii) to find minimum and maximum values forfr subsequent to the peak deflection. Let us try to obtain answers to these problems. It is convenient to introduce the quantity y =f, - fr. Equation (15) now takes the form U = -y[12f,-(6+21~+4a~~f~)y+4arcf,+aicy~].
(16)
In the case of an undamped beam, it follows from (14) that forf=O (extremal points off) we have U =O. According to Eqn (16) this is possible only for positive values of y;
Dynamic
FIG. 7. SDoF
beam,
permanent
FIG. 8. Energy
response
of elastic-plastic
deflectionsj‘versus
diagrams
beams
peak deflectionsj,
for the SDoF
for x=0.01
and c=O.O5.
beam for ~=0.02.
consequently f,> II; and during the whole motion the mid-span deflections fi cannot exceed the peak valuef,. Typical diagrams of the “energy approach” [23] are presented in Fig:8. (Calculations were carried out for k-=0.02.) Here two curves are plotted in each of the four cases: one is the potential energy (elastic strain energy) given by Eqn (16); the other (zig-zag lines) is the total energy, obtained by numerical integration of Eqn (14). The motion will terminate at a minimum point of II. Counter-intuitive motion is possible in case b, where the motion may be terminated at the right-hand minimum where y >f, andf, ~0. Let us denote the interval of the values off, for which negative permament deflections are possible by (f;,fz). It follows from Fig. 8 that for small values off,, the curve U = U(y) has only one minimum. By increasingf, a point of inflection appears and after that two minima are possible. In the inflection point we have U”(y)=0 and it follows from Eqn (16) that
12
u.
This inflection
LEPIK
point appears first if the expression under the squareroot is zero; i.e. (17)
Counter-intuitive behavior of the beam is possible up to the critical situation shown in Fig. 8(c). Hence the upper limit/i can be calculated from the conditions U(y)= L”(y) =O. This means that the equation U(y)=0 must have a double root and (18)
u = ak-yb - Y 112(Y - J’J,
where y, and y, are roots of the equation By comparing (16) and (18) one finds y3=4fm+
-2Y1
3y:-8f,fy, y:-2f;y:= By eliminating
U(y)=O.
from (19) the quantitiesj,
3(2 + 7~) - ~
+4f,i2=
aK
- df; UK and y3 we get the biquadratic
143 YY - ___ 3584~~
equation (20)
After solving this equation we find from (19X that 256~~; - 143’
I: = 512~~;
y3=4fZ
-2Y,.
(21)
Equation (20) has real solutions only if its discriminant is non-negative, this requirement gives ~2&=0.03571; so we see that negative permanent deflections can be realized only for small values of ti. For calculating the minimal values offi for an undamped beam we shall make use of Eqn (16) and find for U(y)=0 the largest root y=y,. The minimal value of deflections is thenfmi, =f, - y,. These values versus peak deflectionsf, are shown in Fig. 9. If K < 0.0357 these curves have abrupt changes at f,=fz (negative values of fmin turn positive). The motion proceeds between fmin and f,,.
FIG.
9. Minimum values for the mid-span deflections versus peak values.
Dynamic
FIG.
10. Peak
valuesj,
response
of elastic-plastic
and extreme
values
beams
ofjversus
13
parameter
K
The critical situation from Fig. 8(c) deserves special attention. For a prescribed K we can find from Eqns (20) and (23) the quantities f,‘, fmi”=f~ -y,, f,,,=fz -y,. These values are shown in Fig. 10. If we take a valuef,, which is a bit less fromfz, and introduce small damping, the vibrations-depending on the magnitude of the viscosity coefficientoccur in one of the intervals (f rn,“~f max) or (f,,,,fi). In the first case we have counter-intuitive behavior of the beam. 5. CONCLUSIONS
A method for calculating the dynamic response of elastic-plastic beams with edge constraints has been proposed. In the discussion of numerical results special attention is turned to the nonlinear vibrations which proceed after the peak value of deflections. Counter-intuitive behavior of the beam, where the mid-point deflections remain negative, is also examined. Numerical results show that motion of the beam after the peak deflection is very sensitive to material and beam parameters. The main purpose of this paper is to propose an approximate solution for the beam response. More details of the SDoF case are examined. In spite of the conditionality of this solution, it yields simple closed-form solutions for finding the extreme values of deflections and intervals for parameters, in which counter-intuitive behavior of the beam may be expected. These results are in accordance with the solutions obtained by Symonds and his collaborators for a Shanley-type beam. The case of two and more degrees of freedom needs a complementary investigation. As shown by Symonds et al. [21-231 here chaotic motion of the beam could take place. Acknowledgements-The about the problems
author has had many useful discussions with dealt with in this paper. Their valuable suggestions
Professors are highly
P. S. Symonds appreciated.
and N. Jones
REFERENCES 1. P. S. SYMONDS and T. J. MENTAL, Impulsive loading of plastic beams with axial constraints. J. Mech. Phys. Solids 6, 186202 (1958). 2. N. JONES, A theoretical study of the dynamic plastic behavior of beams and plates with finite deflections. IHI. J. Solids Strucr. 7, 1007-1029 (1971). 3. P. S. SYMONDS and N. JONES, Impulsive loading of fully clampped beams with finite plastic deformations and strain-rate sensitivity. Int. J. Mech. Sci. 14, 49-69 (1972). 4. N. JONES, Plastic failure of ductile beams loaded dynamically. J. Engng Industry Trans. ASME 98, Series B, No. 1, 131-136 (1976). 5. N. JONES, Influence of in-plane displacements at the boundaries of rigid-plastic beams and plates. Inc. J. Mech. Sci. 15, 547-561 (1973). 6. R. VAZIRI, M. D. OLSON and D. L. ANDERSON, Dynamic response of axially constrained plastic beams to blast loads. Inl. J. Solids Strucf. 23, 153-174 (1987). 7. R. B. SCHUBAK, D. A. ANDERSON and M. D. OLSON, Simplified dynamic analysis of rigid-plastic beams. Int.
J. Impact
Engng
8, 27-42
(1989).
8. u. LEPIK, Optimal design of dynamically loaded rigid-plastic beams effects. Transact. Turru University 721, 25-38 (in Russian, 1981).
taking
into
account
the membrane
u. LEPIK
14
9. T. LEPIKLJLT, Calculation oldynamically loaded rigid-plastic beams in the case of fixed ends. Transact. firfu Unirrr.sif>, 853, 25-37 (in Russian, 1989). 10. N. JONES. Sfruc~ural Impacf. Cambridge Press. U.K. (1989). 1 I. P. S. SYMONDS. Twenty years of developments in inelastic dynamics. Proc. Srh Et1qiwrritlg Mechanical Specidiry Conj: ASCE. New York. pp. l-23 (1984). 12. N. JONES, Recent studies on the dynamic plastic behavior of structures. Appl. Mech. Rec. 42(4),95-l 15 (1989). 13. P. S. SYMONDS. Finite elastic and plastic deformations of pulse loaded structures by an extended mode technique. Inf. J. Mech. Sri. 22. 597-605 (1980). 14. P. S. SYMONDS, Elastic, finite deflection and strain rate effects in a mode approximation technique for plastic deformation of pulse loaded structures. J. Mech. Enyny Sci. 22(4). 189-197 (1980). 15. P. S. SYMONDS and CH. W. G. FRYE. On the relation between rigid-plastic and elastic-plastic predictions on response to pulse loading. Inf. J. lmpocf Euyuy 7(l), 139-149 (1988). 16. M. I. MIKKOLA, M. TUOMALA and H. SINISALO. Comparisons ol numerical integration methods in the analysis of impulsively loaded elastic-plastic and viscoplastic structures. Contpufers Sfrurf. 14,469478 (198 I). 17. P. S. SYMONDS and T. X. Yu. Counter-intuitive behavior in a problem of elastic-plastic beam dynamics. J. Appl. Mech. 52, 517-522 (1985). 18. SH. U. GALIEV and N. V. NECHITAILO, Unexpected behavior of plates under impulsive and hydrodynamic loading. Prohlemy prochnosti 12, 63-72 (in Russian. 1986). 19. B. PODDAR, F. C. MOON and S. MUKHERJEE, Chaotic motion of an elastic-plastic beam. ASME J. Appl. Mech. 55, 185-189 (1988). 20. P. S. SYMONDS. G. BORINO and U. PEREGO, Chaotic motion of an elastic-plastic beam. ASME J. Appl. Mech. 55, 745-746 ( 1988). 21. J.-Y. LEE and P. S. SYMONDS, Extended energy approach to chaotic elastic-plastic response to impulsive loading. Inf. J. Mech. Sci. 34(2), 139-157 (1992). 22. J.-Y. LEE, P. S. SYMONDS and G. BORINO, Chaotic responses of a fixed ended elastic-plastic beam model to short pulse loading. I-11 (to be published in J. Appl. Mech.). 23. G. BORINO, I-J. PEREGO and P. S. SYMONDS, An energy approach to anomalous damped elastic-plastic response response to short pulse loading. ASME J. Appl. Mech. 56, 430438 (1989). 24. P. S. SYMONDS, F. GENNA and A. CIULLINI. Special cases in study of anomalous dynamic elastic-plastic response of beams by a simple model. Inr. J. So/ids Struct. 27(3), 299-314 (1991).
APPENDIX Here an iterative numerical scheme for integrating the equations of motion (3)-(5) is presented. We shall assume that the material or the beam is elastic-plastic with linear strain hardening (Fig. Al). Here E is the Young’s modulus; E” = E( 1 - 2.). where i. is a constant, characterizing the strain hardening (A= 0 for elastic and %= 1 for perfectly plastic material). The segment BD corresponds to elastic unloading, the segment CD to secondary plastic loading. We shall also include the viscosity in the form
u* = Q(e)
+ qP,
(AlI
where the [unction Q(e) will be determined according to Fig. Al. For elastic deformations Eqn (Al) describes the Voight’s body, ror ideally plastic materials we have Bingham’s body. Further on we shall use the dimensionless quantities o=a*/o, and q=tl*/u.. It is convenient to write Eqn (Al) in a dilTerential form. Let us assume that we know the quantities u and e for the two instants f and f +At and are aware ol the segment of Fig. Al on which we find ourselves at the instant f. If we are on the segments OA or BC, the following deformation is elastic and we have:
u(f+Af)=u(f)+
6”
0
9
-
0E
FIG. Al.
Stress-strain
642)
E[e(f+Af)-e(f)]+qP(f). US
diagram
E”
e
for strain-hardening
material.
Dynamic lfwearron
response
of elastic-plastic
beams
15
thesegmentsABorDCandcT(r~Ae(r)zOtheloadingprocess~sactiveforr~tr.r a(r+Ar)=o(r)
+ L(l u<
+Ar)andwe
have
-i.)[e(r+Ar)-e(t)]+~P(t).
(A31
However iT o(r).Ae(r)
quantities
for the instant
r +Ar
Step 1. We extrapolate
+2u(s.t)-U(.YJ-AI) +2rr(s,t)-w(s.r + ti(.x,t -AI) + tifx,r -AI).
we shall find in the following
the values
Tand
M for I +A/
T+ = T(s,r - )A/)M,
-Ar) (A41 way
according
37-(s,r -AI)+
IO parabolic
interpolation:
3T(.v.r)
=M(s.r-ZAI)-~IM(.\..I-AI)+~M(.~...I).
(AS)
2. Making use of the cubic splines. we calculate the derivatives T’, (Tw’)‘, M” and find the accelerations ii(s,r +Ar), i+(.x,r +Ar) from Eqns (3). Step 3. Here we shall calculate the quantities u. w, ri. rc from Eqns (A4). Step 4. With the aid of cubic splines we find 11’. w and w” for the instant f +Ar. Step 5. We shall calculate the dimensionless deformation e from Eqn (5). Step 6. For each point .@O, O.OS], ~-0.5. OS] we shall clearly indicate whether the material of the beam deforms plastically or if there is elastic loading or unloading. In the first case we calculate the dimensionless stress 0 from (A3). in the second case. from Eqn (AZ). Step 7. Evaluating the integrals (4) numerically we find the quantities T(.Y.r+Ar), M(.u.f +Ar). Step 8. We have lo check the inequalities Step
IT+-T(x,r+Ar))<.a,
w3
~I~+-M(.YJ+A~))
for SE[O, 0.51. where E>O is a prescribed small quantity. If the inequalities (A6) are not fulfilled, we shall take r= T(r,r +A[). M + =,44(.x-.1 +Ar) and repeat the steps 228 until they are satisfied. Only after that shall we go on IO the following step. Step
9. We shall increase
the time by AI and return
to step I.
The reliability of the obtained results can be checked by the energy balance criterion Li* and K* denote the internal and kinetic energies, the energy balance criterion is
(Mikkola
et al. [16]).
If
AW*=AU*+AK*=O, where A”*=~nL{~*A[~+;(~)‘]-M*A($)}d.~
AK*
= J-1 ,,,[A(gy
+ A(gy]dr*.
Here AW* is the error which we have made in calculating In dimensionless quantities (2) the energy criterion obtains only one half of the beam): AW=
0.5 s[( T
0
Au’ + f AA\v”
the energy during the time step At. the form (because of the symmetry we shall consider
- AMAw”
+ Ati
+ hAti
1 dx.
(A71
The energy error AW must be sufficiently small; if it is not so, the step size At must be reduced. We have calculated the integrals (4) numerically by taking eight Gaussian points. As to the spanwise step size AX, then the mid-span of the beam was divided into n equal parts; for n the values IO and 20 have been taken.
16
u.
LEPIK
FIG. A2. Mid-span deflections as functions of initial impulse for beams of aluminum I -peak values; 2-minimal values; 3--estimated final deflection: 0 and A-test points for maximum transient and permanent deflections.
707ST6; from [ 133
The case n = 10 gave quite good results for the peak deflection, but the minimum values of the curve deflection versus time were somewhat overestimated. Therefore some of the calculations have been carried out for n = 20. The size of the time step Ar was controlled in two ways. First the number of iterations for the steps 2-8 must be small; secondly the energy error AW in (A7) must be smaller than 5 x lo-? For most problems these requirements were satisfied for the time step Ar = 0.5 x 10m3 (if a = IO). For n = 20 the time step must be considerably smaller. which to a large extent increases the computation time. For checking the efficiency of the proposed algorithm and for comparing the numerical results with experimental data, calculations were carried out for beams of aluminium alloy 707ST6; the beam height was 0.075 in, length 5.1 in. These dimensions correspond to the experimental data of Lindholm and Bessey, which have been taken from Fig. 5 of the paper [l3]. Numerical results achieved according to our method have been presented for various impulses in Fig. A2 (for parameter i. the value 0.99 has been taken). Curve I gives maximal deflections (peak deflections). curve 2 values of the first minimum. Test points for maximum transient and permanent deflections are marked with circles and triangles, respectively. The permanent deflections have been estimated as the average of the peak displacement and the subsequent first minimum displacement. Of course this is a rather crude approximation, but somewhat encouraging is the good agreement between the curve 3 and test points. Analogical comparisons have also been made for beams of mild steel. Here the deflections are somewhat higher than for the test data (this is caused by a circumstance that we have not taken into account, the strain-rate sensitivity of the material).