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Large deflection, rotation, and plastic strain in cantilevered beams
Int. I. Engng Sci. Vol. 28, No. 3, pp. 231-239, 1990
Printedin GreatBritain. All fightsreserved
0020-7225[90 $3.00+ 0.00 Copyright© 1990PergamonPressplc
LARGE DEFLECTION, ROTATION, AND PLASTIC STRAIN IN CANTILEVERED BEAMS J A M E S F. BELL Johns Hopkins University, Baltimore, MD 21218, U.S.A. Al~lraet--Relatively slender cantilever beams, having a ratio of length to height in excess of 50, are subjected to a series of loads at the free end, resulting in end deflections in which the final displacement is greater than half the unreformed length of the bar. As each load is added to the free end, the curve of deflection, angles of rotation, plastic and elastic strain histories, and rigid body displacement of generic material points are accurately recorded. From these data, moments in the undeformed and deformed reference configurations are compared. Since the role of rigid body rotation is found to be negligible, having been cancelled by the rigid body displacement, one may introduce moments in the unreformed reference configuration that correlate with experiment in the context of a general incremental theory of finite strain plasticity. The theory satisfies the analytical requirement of material objectivity when, as is found by experiment, the role of the rigid body rotation of principal axes is negligible.
1. I N T R O D U C T I O N A discriminating laboratory factor in assessing a p r i o r i conjecture in current theories of finite strain plasticity, is the precise specification of the role of rigid-body rotation of principal axes between undeformed and deformed reference configurations. As long as experimentists fail to provide proper bounds, the role of the spin tensor is restricted only by the imagination of the theorist, albeit modified by his or her preference or lack of preference for analytical simplicity. In current papers [1, 2], by describing the kinematics of grossly deformed thin-walled tubes, doubts have been cast upon the efficacy of analytical freedom in these matters. That kinematical analysis is based upon the precise measurement, during loading to very large strain, of inside diameters, outside diameters, specimen length, angles of twist, and helicies formed from initially straight generators. In every instance, to four decimal places, those measurements were contrasted with similar measurements made prior to deformation. The tubes were loaded in various combinations of twisting and extension, along both proportional and non-proportional loading paths. Measurements were made just prior to the formation of gross local non-homogeneity in the form of shear bands, necking, or buckling. The results were both unanticipated and striking. For 85 thin-walled tubes of annealed copper, annealed aluminum, and annealed mild steel subjected to arbitrary stress paths that included twisting from 9 ° to 347 ° and simultaneous extension by amounts in individual tests ranging from 0 to 25%, the average measured angle of rigid body rotation is 3.91 °. In 300 tests, the largest measured angle of principal axis rotation observable for homogeneous deformation just before failure, is a rigid body rotation of principal axes of only 7.5 °. For the rigid body rotation of principal axes, R = I to a close approximation for deformation in solids that remain homogeneous and isotropic to large finite strain. Introducing R = I leads to an incremental theory of finite strain plasticity in homogeneous, isotropic solids that satisfies the principle of material objectivity [2]. Inescapably one must conclude that whatever is the choice of a general continuum theory for finite strain plasticity, for the largest homogeneous deformation obtainable for thin-walled tubes the role of the rotation tensor R is negligible. A natural sequence to such a finding is to examine situations for which large rigid body rotation of principal axes are known to occur. One such situation is that of a grossly deflected cantilever beam. The homogeneous strains of these beams do not exceed 3%, but the rigid body rotation of their principal axes reaches 40 ° . A decade ago, G r e e n s p o n [3] showed that experiments for the dynamically (blast) loaded simply supported beam, and the present writer's theory of finite strain plasticity, are in close 231
232
J . F . BELL
correlation. In a simply supported beam, when the deflection at the center is equal to the height of the beam, the difference between finite plastic strain response in the undeformed and deformed reference configurations is too small to be considered. To obtain large angles for the rigid body rotation of principal axes, the displacements must exceed the specimen height by many times. In the tests considered below, the displacement at the end of a cantilever beam, W(Lo) = 6e, has a maximum 27 times larger than the height h of the beam. The displacement 6p at the free end, the point at which the load is applied, exceeds half the length of the undeformed beam. At that very large end displacement the maximum measured rigid body rotation of ~ = 40° is achieved. As in the study of thin-walled tubes, the modus operandi is a precise measurement of dimensions during loading to large deformation.
2. THE DATA The first cantilever beam test described is a 16.45 in. active length, rectangular bar of fully annealed, hot-rolled aluminum, the maximum end displacement perpendicular to the bar being 8.6 in. The bar was annealed at 593°C for 2 h and cooled in the furnace, thereby achieving what I have defined as a "genesis reference configuration" ([4], Section 4). The bar is clamped at one end and loaded vertically at the other. The width of the beam is b = 0.750 in.; the height is h = 0.3125 in. The area moment of inertia is I = (bh3/12) = 0.00191 in. 4, and the distance from the center of the cross-section of the beam to the outermost fiber is 0.15625 in. The manner of loading permitted for each applied load the accurate, detailed tracing on an adjacent screen of the shape of the deformed beam.t In addition, after the load had been removed, curves were drawn for the post-deformation permanent set. The curves in the deformed states under load are shown in Fig. 1, for test 1832. To determine the plastic strain, E=, for each applied load P, post-yield electric resistance gages were located at a distance of 1 in. from the clamped end. Let Lo be the active length of the undeformed beam; Le the distance parallel to Lo from the fixed end to the deflected end in the deformed configuration for each load P; X, the distance from the fixed end to a generic material point in the undeformed reference configuration; and x, the distance from the fixed end to the same material point in the deformed reference configuration. (X - x) is the X component of the rigid body translation. Xin 0 t
2
4
i
J
6 _h . . . .
8 I
I0
12
I
14
16
I
I
~ - .
18
Lo
-
i
o"
x
"I
|
-2
\ -8
Lp--
"~
-tO ~in. Fig. 1. Displacement curves for the cantilever beam of test 1832 for each of the 11 applied loads P at the free end. t I am indebted to the ingenuity and persistence of Faith Paquet Moeckel for the high degree of precision obtained for the curves reproduced in Fig. 1.
Large deflection, rotation, and plastic strain in cantilevered beams
233
Let ¢ be the angle of rigid body rotation of principal axes m e a s u r e d at x; M, the applied moment at X in the undeformed reference configuration; and M', the corresponding applied moment at x in the deformed reference configuration. For M in the undeformed reference configuration we have (1)
M = P(L~, - X )
For a horizontal axis through the centroid of any cross-section of the beam, the moment in the deformed reference configuration becomes the product of P cos ~ times the distance ( L e - x ) s e c O or (2)
m' = P(Lp - x).
The rigid body rotation of principal axes ~ appears only in the integration along the displacement curves of the beam while determining the location of the material point x. The percentage difference between M and M' is determined from the ratio, (M' - M)/M'
(3)
= (X - x)/(Lp - x).
Figure 1 is not schematic. The original drawing is a careful tracing of the displacement curves of the beam. A detailed study of these displacement curves W ( X ) were made, like that shown for the curve of maximum deflection in Fig. 2. The section shown in Fig. 2 is for the plastic zones in the first 9 in. from the fixed end. Beyond 6 in. from the fixed end, the magnitude of the plastic strains decreases until it becomes infinitesimal at the end of the zone. It is only in this region of small plastic strain, far from the fixed end of the beam, that non-negligible differences between M and M ' are recorded. In the important high stress, large strain region, within 4 in. from the fixed end in this example, the difference between the moments M and M' becomes a fraction of a percent. These differences in moments are detailed in Tables 1 and 2. As we found for the gross twisting of thin-walled tubes, for the large deflection of a cantilever beam the role of the rigid body rotation of principal axes is negligible. For the beam, however, while the plastic strains are relatively small, the angles are large. The decrease one might anticipate from the rigid body rotation is offset by the rigid body translation that acts in opposition. Tabulated in Table 1 are moments calculated from equation (1) in the undeformed reference configuration compared with those from equation (2) in the deformed reference configuration at integral values of X in the plastic zone. They are determined along four of the displacement trajectories of Fig. 1. The double lines ( = ) in the table indicate that the moment is in the linear 0
I
2
3
4
5
I
I
I
I
I
,
6
7
8
9
f
I
I
I
0
-I X
-2 in. -3
-4
T e s t 1832 rigid body rotation o x position in deformed configuration + X position in undeformed configuration o.--+(X-x), X component of rigid body displacement
~
,d~=40 I
\ \\ \-,
Fig. 2. A detail from Fig. 1 of the displacement curve for the maximum load in the plastic region between X - - 0 and X = 9in. Included are the location of material points X undeformed, the corresponding value x in the deformed reference configuration, and the angle of rotation 0 at x.
234
J. F. BELL
Table 1 dip = 1,7 in. P = 2.31 lb
die = 4.9 in.
dip = 8.6 in. P = 6.95 lb
6 e = 6.65 in. P = 5.45 lb
P = 4.32 Ib
X in.
M
M'
M
M'
M
M'
M
M'
~b at X
(in.)
(in.-lb) 37.7 35.3 33.0 30.7
(in.-lb) 37.7 35,3 33.0 30.7 ~
(in.-lb) 67.2 62.9 58.5 54.2 49.9 45.6 41.3 36.9 32.6 ~
(in.-lb) 67.2 62.9 58.5 54.4 50.3 46.2 42.1 38.0 34.3 32.0
(in.-lb) 80.7 75.2 69.8 64.3 58.9 53.4 48.0 42.5 37.1 31.6 ~
(in.-Ib) 80.7 75.2 69.8 64.4 59.4 54.5 49.3 44.1 39.2 34.9 33.8
(in.-lb) 94.5 87.6 80.6 73.7 66.7 59.8 52.8 45.9 38.9 32.0
(in.-lb) 94.5 87.6 81.3 75.1 68.8 63.2 57.0 51.6 45.9 38.9 34.8 30.6
(deg.) 0° 7.0 ° 18.5 ° 23.0 ~ 28.5 ° 33.0 ° 34.5 ° 36.0 ° 39.0 ° 39.5 ° 40.0 ° 40.0 °
0 1 2 3 4 5 6 7 8 9 10 11
Indicates linear domain.
elastic region and hence the strain is infinitesimal. These data and those of Tables 2 and 3 below are independent of any theory of finite strain plasticity. The data are from direct measurements in a study of the kinematics of the deformed beam. In the final column are listed the measured angles of rigid body rotation of principal axes ~p at X for the deflection curve produced by the maximum load. This is in contrast to the angles for the same material points at x in the deformed reference configuration shown in Fig. 2. The angles for all other deflection curves are smaller. We note that although the difference in the moments may be small, the corresponding angles of rigid body rotation of axes are large. To further detail the small difference in moments in the two reference configurations, I have tabulated in Table 2 the difference in moments from zero deflection to the maximum for two material points initially at X = 4 in. and X = 6 in. At X = 4 in. from the fixed end, the largest difference between M and M' is 2% for the maximum free end deflection of 6e = 8.6 in. In contrast, the angle of rigid body rotation of principal axes at that same point x is the large angle of cp = 28°! Further away from the fixed end at X = 6 in. where the plastic strains already are becoming small, the difference between the moments M and M' is found to be 7.4%. For that same material point, x = 6, the angle of rigid body rotation of axes has increased to the very large value of ~ = 33 °. Tables 1 and 2 document in detail from the measured curves that, like the gross twisting of thin-walled tubes, the role of the rigid body rotation of principal axes in the gross plastic deflection of beams also is negligible. Similar to the continuum mechanics for thin-walled tubes, one may describe, with very small error, the response from applied moments and calculated stresses and strains by referring only to the undeformed reference configuration. Unlike the situation for thin-walled tubes for which a similar conclusion is reached because the angles of rotation are minuscule, for test 1832 the angles of rigid body rotation are large. Table 2 X = 6.0 in.
X = 4.0 in.
die (in.) 1.7 2.9 3.9 4.9 5.5 6.65 7.7 8.6
P
M
M'
(lb)
(in.-lb)
(in.-lb)
2.31 3.28 3.80 4.32 4.85 5.45 6.03 6.95
39.7 45.2 49.9 55.0 58.9 61.5 66.7
39.7 45.2 50.5 55.4 59.4 62.4 68.1
M
M'
% diff.
(in.-lb)
(in.-lb)
% diff.
2.0%
33.1 37.6 41.2 45.4 48.0 49.5 52.8
33.3 38.0 42.1 46.6 50.1 52.2 57.0
0% 1.0% 2.1% 2.6% 4.2% 5.0% 7.4%
Large deflection, rotation, and plastic strain in cantilevered beams
235
Table 3. Test 1832 P(lb)
be(in.)
LeOn.)
M(in.-Ib)(X = 1.000)
M2(in.-Ib)2
E x x ( X = 1.000)
0.54 1.19 1.53 1.94 2.31 3.28 3.80 4.32 4.85 5.45 6.03 6.95
--
0.1 0.3 0.8 1.7 2.9 3.9 4.9 5.5 6.65 7.7 8.6
16.45 16.45 16.45 16.40 16.30 16.10 15.90 15.55 15.35 14.80 14.20 13.60
8.5 18.3 23.6 29.9 35.6 49.6 56.6 63.3 69.6 75.3 79.6 87.6
73.0 335.2 558.5 893.7 1266.2 2458.7 3204.0 4004.5 4843.1 5665.4 6332.6 7673.3
0.001~ 0.00013 0.00023 0.00028 0.00046 0.0034 0.0052 0.0061 0.0081 0.0116 0.0121 0.0140
3. THE D E T E R M I N A T I O N OF STRESS AND STRAIN In the general incremental theory of finite strain plasticity, with proportional loading chosen to be along a single principal axis, as in the present situation for uniaxial tension and uniaxial compression, constitutive statements reduce to equation (4) [5]: (4)
o=, = (3/2)3/4fl(E= - E=,b) 1/2,
where tl= and E = are the uniaxial stress and strain components of a and E, defined in the original undeformed reference configuration. The constant E=b designates the intercept on the strain axis at zero stress. For fully annealed hot-rolled aluminum at room temperature, the measured material constant is fl = 4.13 x 104 psi, or fl = 29.0 kg/mm 2. For the general theory to include the plastic deformation of beams, equation (4) must apply, with opposite signs, to both the tension and the compression plastic domains of the bent beam. Since the response functions are parabolic, it is convenient to compare the experimental data with the straight lines in a t r 2 vs Exx plot. For fully annealed aluminum, equation (4), with fl = 4.13 x 104psi, the constant becomes (3/2)3c2fl2 = 31.5 x 108. Hence o~x = 31.5 x 10S(E= - e=b)(psi) 2.
(5)
At X = 1.000 in. for test 1832 of Fig. 1, Table 3 lists the applied loads P, the length Le from the fixed end to the point of load application for the deformed beam, the vertical displacement 6p of the point of application of the load P, the moment M and M E at X = 1.000 where the strain gage is located, and finally, the measured strain E= at X -- 1.000 for each load P. For the data in Table 3, Fig. 3 shows a plot of the measured M E v s Exx at X -- 1.000 in. That M E vs gxx is found to be parabolic is compatible with expectation from the continuum theory referred to above, provided the stress is proportional to the moment. When well-documented prior experiment provides constitutive equations that, like 19th
8000
o
a~6ooo .~_
~ 2
4OOO
/ ~
¢4
X= i.O00 in
2000 ~ 0
i
M=P(Lp-X) I
018
0'.8
,10
,.12
,.;
E x x */.
Fig. 3. A plot of M 2 vs Exx from the data of Table 2. The straight line in the plastic domain indicates that the response is parabolic.
236
J . F . BELL
century linear elasticity or 20th century non-linear rubber elasticity, correlate in detail with acceptable continuum theory, they can delimit and characterize the study of a new problem. On the other hand, when constitutive equations are solely a matter of a priori conjecture delineating what may be or what might be, one only can resort to speculation. From a prior correlation of experiment and continuum theory that includes experimentally based constitutive statements, we find that the plastically deformed cantilever beam follows from general principles when o~ = Mc/l. Prompted by the data in Fig. 3 and by the fact that the role of the rigid body rotation is negligible, we introduce O~x(y)/Oxx(C) = y/c. For the moment M, we have M =
ff
oxx(y)yb dy = [oxx(c)/c c
,f[
y2 dA
=
o,,x(c)I/c
(6)
c
or
Oxx(C) = [Mc]/I = [Pc](Le - X ) / I
(7)
where c = h/2 = 0.156 in., is the maximum value of y from the center of the beam cross-section to the outermost fiber, and I = [bh3]/12 = 0.00191 in. 4 is the area moment of inertia. For test 1832, c / I = 81.9 in. -3. For test 1832 of Fig. 1, Fig. 4(b) shows a ~ x vs Exx plot of the measured data compared with equation (5). The slope is precisely in accord with the general theory. The coefficient of 31.5 x 108 (psi) z of equation (5) is the slope of the solid line in the figure! The maximum stress trx= on the cross-section at y = c is of the form of equation (7). Equation (7) implies that the stress varies linearly with the distance y from the center of the cross-section. To check that this is indeed the situation, in a second test, 1831, the height of the beam was reduced from 0.3125 to 0.2521 in. Thus c in equation (7) is reduced from 0.156 to 0.126 in. The width of this beam is approximately the same: b = 0.745 in. The area moment of inertia becomes I = 0.00099 in. 4 instead of I = 0.00191 in. 4. The length of the beam is slightly shorter: L0 = 15.156 in. For this test, in addition to the change in the maximum value of y from 0.156 to 0.126 in., the range of applied loads is from 0 to 4.4 lb instead of from 0 to 6.95 lb, in order to provide a similar large end deflection 6p for the thinner beam. To further check on the analysis, the post-yield strain gage was located at X = 1.533 in. instead of at X = 1.000 in. The parameter c / l of equation (7) increases from 81.9 in.-3 for test 1832 of Fig. 1, to 126.8 in.-3 for the thinner beam of test 1831. As may be seen in comparing the O2xx vs E x x plot of Fig. 4(b) for test 1832 with a similar plot in Fig. 4(a) for the beam of test 1831 measured at a smaller value of y and at a different (0)
60
~..~J S I o pe :
40
~ / ~
~/o Test
oJ
c.O
b
o 0
40
(Bell)
c=126.8 in-3 I c=O. 126 in I = 0 . 0 0 0 9 9 ,n
'~ 20 o x
18.3 I
--Theory
¢
, 0.20
. 0.60
. 0.80 Exx %
0.40
(b)
.
. 1.00
L20
°
Slope
.
"i'e st
1852
--Theory c 81,9 in -5 ~-=
20
(Bell)
c=0.156 m I =0.0019f
o / I
0.20
I
0.40
I
0.60
I
0.80
I
k
1.00
k20
k
140
,n k
i
t.60
1.80
Exx % Fig. 4. Plots of o~x vs Exx for tests 1831(a) and 1832(b). The solid lines are the predicted parabolicity from equation (9); the slope is in accord with the general theory, equations (4)-(6).
Large deflection, rotation, and plastic strain in cantilevered beams
(CI) +c!+y
i
+Exx
+5000 +10000 psi
+ .01
/
C
-Exx
+ct+Y +o"~x
, t_----~ +.o2
I
-Y
237
-o-~
J--C
-y~
P +c
+yt
(c)
Tension Plostic Lineor
~
Elostic
X (inches}
Compression Plostic --C
-y
Fig. 5. (a) The parabolic distribution of strain over an initially planar surface. (b) The linear distribution of stress over the same cross-section. (c) The separation surfaces between the parabolic plastic and the linear elastic domains.
location X, equation (7) is admissible. The general continuum theory of finite strain plasticity is extended to include the large deflection, rotation, and plastic strain of beams. As will be shown in Fig. 6, the m e a s u r e d elastic limit is cry = 2500 psi. By setting oxx = ey, equation (7) may be used to determine the separation surface between parabolic plastic and linear elastic domains, as shown in Fig. 5(c). The calculated linear stress distribution with y at X = 0, is given in Fig. 5(b). For the plastically deforming beam in parabolic plasticity, instead of plane sections remaining plane as in the linear theory, initially plane sections immediately become parabolic and remain parabolic. This is shown at X = 0 in Fig. 5(a) using equation (8). From equation (4), Exx = 02=/[(3/2)3/4/5]2, or from equation (7) E=, = {[P/II2[(3/2)3/4[31-Z[Lp - Xl2}y 2.
(8)
From the calculation of stress distribution of Fig. 5(c) we see that the zones of parabolic plastic deformation are bounded at loads below the elastic limit Cry = 2500 psi. For all y, at any cross-section of the beam the response to the initial loading is linear elastic. At X --- 1.000 in. this corresponds to the first four initial loadings for test 1832 in Table 3. From the measured moments and strains we may determine and plot o~, vs E~x for the initial loading of the beam. Such plots are shown in Fig. 6(a) for the initial region of test 1831, and in Fig. 6(b) for test 1832. In each instance, the above analysis and measurements reveal that not only is the measured slope linear as predicted, but also both slopes provide the known bar modulus for aluminum: E = 10 x 106psi, (7030kg/mm2). From tests 1831 and 1832, the initial loading provides a direct measure of the elastic limit, namely, o r = 2500 psi. To extend the analysis of the beam into the linear elastic region, to X > 9 in., a post-yield strain gage for test 1831 has been located at X = 13~ in. from the fixed end, i.e. in the totally linear elastic region toward the free end of the beam. If the analysis of experiment presented above is indeed a proper correlation between theory and experiment, then the application of equation (7) to the linear elastic domain beyond X = 9 in. should provide a or= vs E= plot with a linear slope of E = 10 x 106 psi (7030 kg/mm2), the bar modulus for aluminum. That the data listed in Table 3 does in fact meet the test is illustrated in the a= vs E~x plot of Fig. 7. The slope determined from equation (7) for the measured data at X---13~ in. for test 1831 is precisely E = 10 x 106 psi.
238
J . F . BELL
(a)
(b)
3000
3000 o
O"y
2500
2500 i
•-
2000
•-
g
2000
x
b~15oo
b x 1500
I000
I000
Test 1851 --Theory X= I 17/32 in c =0,126 in ! = 0 . 0 0 0 9 9 in 4
500
I
0
I
2 y X= I.O00in c=0.156 in I=0.00191 in 4
/
500
/ !
I
0.0002 0.0004 0.0006 Exx
0
0.0002
0 . 0 0 0 4 0.0006 Exx
Fig. 6. The initial portions of tests 1831 and 1832 in which the low strain data from Table 2 in equation (11) provide the linear elastic slopes in oxx vs Exx plots. The slopes have the standard modulus for aluminum: 10 x 106psi. The measured elastic limit is o v = 2500 psi.
For the engineer seeking a failure criteria for the cantilever beam, the calculated maximum stress at the fixed end is oxx(0) - 7842 psi for test 1832 and axe(0) = 8696 psi for test 1831. For this fully annealed solid in tension, the ultimate stress is 12,300 psi.
4. S U M M A R Y A N D C O N C L U S I O N S This laboratory study of very large deflection in a cantilever beam was undertaken as part of an effort to contribute to an understanding of the role of the rigid body rotation of principal axes in finite strain plasticity. 1400
1200
I000
"
800
x
b
E=IOXIO
6 psi~
"~
6o0 0
Test 1831
400
--Theory X= 13 1/16 in c =0,126 m ] = 0 . 0 0 0 9 9 in 4
200
I
0
I
L
0.0002 0.0004- 0.0006
1
0.0008 Exx
I
0.0010
J
0.0012
I
0.0014-
I
0.0016
Fig. 7. At 13~ from the fixed end, a plot of oxx vs Exx determined from measured moments and strain in equation (11). Note that the response is linear elastic as anticipated in this domain, with a slope of 10 x 106 psi, the standard value for aluminum.
Large deflection, rotation, and plastic strain in cantilevered beams
239
As in the kinematical study of the large twisting of thin-walled tubes [1, 2], the kinematical study of the gross deflection of a cantilever beam in the presence of relatively small plastic strain, once again relegates the role of the rigid body rotation of principal axes to a negligible parameter. The unanticipated by-product is of considerable current and historical interest: it is the finding of a complete description of the detailed plastic strain response of grossly deflected beams that is compatible with general finite strain continuum theory. Included are the details of the spatial distribution of stress and strain over all cross-sections; the delineation of the surface of separation that bounds parabolic plastic and linear elastic domains; and the provision of maximum stress and strain for a known applied load. For a linear response function with a stress that linearly increases with the distance from the center of the beam, the classical theory provides that plane sections remain plane. For a parabolic response function in which the stress also linearly increases with the distance from the center of the beam, examination of the stretches reveals that initially plane sections become parabolic, and these parabolic sections remain parabolic. The constitutive equations for structural metal alloys of steel, aluminum, and copper are known [4]. They lead to the same type of parabolicity in simple tension and in compression. One anticipates that the above analysis will include the gross deflection of beams of these hardened metals. Acknowledgement--I wish to thank Dean L. Pendleton for her carefully made drawings for the figures. Note--To avoid introducing error, the nearly 200 numbers measured in this study have been presented in their original form. The conversion to SI units is as follows: pounds force (Ib)x4.45=newtons (N); pounds force per square inch x 3.89 = kilopascals (kPa); inches (in.) x 25.4 = millimeters (mm).
REFERENCES [1] J. F. BELL, Int. J. Solids & Structures 25, No. 3,267-278 (1989). [2] J. F. BELL, Int. J. Plasticity In press. [3] J. E. GREENSPON, Energy approaches to structural vulnerability with application of the new Bell stress-strain laws, U.S. Army, Ballistics Research Laboratory Contract Report, No. DAADO-75-C-0731.
(1975).
[4] J. F. BELL, Int. J. Plasticity, 1, No. 1, 3-27 (1985). [5] J. F. BELL, Int. J. Plasticity, 4, No. 2, 127-148 (1988).
(Received 27 April 1989; received for publication 30 August 1989)
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0020-7225[90 $3.00+ 0.00 Copyright© 1990PergamonPressplc
LARGE DEFLECTION, ROTATION, AND PLASTIC STRAIN IN CANTILEVERED BEAMS J A M E S F. BELL Johns Hopkins University, Baltimore, MD 21218, U.S.A. Al~lraet--Relatively slender cantilever beams, having a ratio of length to height in excess of 50, are subjected to a series of loads at the free end, resulting in end deflections in which the final displacement is greater than half the unreformed length of the bar. As each load is added to the free end, the curve of deflection, angles of rotation, plastic and elastic strain histories, and rigid body displacement of generic material points are accurately recorded. From these data, moments in the undeformed and deformed reference configurations are compared. Since the role of rigid body rotation is found to be negligible, having been cancelled by the rigid body displacement, one may introduce moments in the unreformed reference configuration that correlate with experiment in the context of a general incremental theory of finite strain plasticity. The theory satisfies the analytical requirement of material objectivity when, as is found by experiment, the role of the rigid body rotation of principal axes is negligible.
1. I N T R O D U C T I O N A discriminating laboratory factor in assessing a p r i o r i conjecture in current theories of finite strain plasticity, is the precise specification of the role of rigid-body rotation of principal axes between undeformed and deformed reference configurations. As long as experimentists fail to provide proper bounds, the role of the spin tensor is restricted only by the imagination of the theorist, albeit modified by his or her preference or lack of preference for analytical simplicity. In current papers [1, 2], by describing the kinematics of grossly deformed thin-walled tubes, doubts have been cast upon the efficacy of analytical freedom in these matters. That kinematical analysis is based upon the precise measurement, during loading to very large strain, of inside diameters, outside diameters, specimen length, angles of twist, and helicies formed from initially straight generators. In every instance, to four decimal places, those measurements were contrasted with similar measurements made prior to deformation. The tubes were loaded in various combinations of twisting and extension, along both proportional and non-proportional loading paths. Measurements were made just prior to the formation of gross local non-homogeneity in the form of shear bands, necking, or buckling. The results were both unanticipated and striking. For 85 thin-walled tubes of annealed copper, annealed aluminum, and annealed mild steel subjected to arbitrary stress paths that included twisting from 9 ° to 347 ° and simultaneous extension by amounts in individual tests ranging from 0 to 25%, the average measured angle of rigid body rotation is 3.91 °. In 300 tests, the largest measured angle of principal axis rotation observable for homogeneous deformation just before failure, is a rigid body rotation of principal axes of only 7.5 °. For the rigid body rotation of principal axes, R = I to a close approximation for deformation in solids that remain homogeneous and isotropic to large finite strain. Introducing R = I leads to an incremental theory of finite strain plasticity in homogeneous, isotropic solids that satisfies the principle of material objectivity [2]. Inescapably one must conclude that whatever is the choice of a general continuum theory for finite strain plasticity, for the largest homogeneous deformation obtainable for thin-walled tubes the role of the rotation tensor R is negligible. A natural sequence to such a finding is to examine situations for which large rigid body rotation of principal axes are known to occur. One such situation is that of a grossly deflected cantilever beam. The homogeneous strains of these beams do not exceed 3%, but the rigid body rotation of their principal axes reaches 40 ° . A decade ago, G r e e n s p o n [3] showed that experiments for the dynamically (blast) loaded simply supported beam, and the present writer's theory of finite strain plasticity, are in close 231
232
J . F . BELL
correlation. In a simply supported beam, when the deflection at the center is equal to the height of the beam, the difference between finite plastic strain response in the undeformed and deformed reference configurations is too small to be considered. To obtain large angles for the rigid body rotation of principal axes, the displacements must exceed the specimen height by many times. In the tests considered below, the displacement at the end of a cantilever beam, W(Lo) = 6e, has a maximum 27 times larger than the height h of the beam. The displacement 6p at the free end, the point at which the load is applied, exceeds half the length of the undeformed beam. At that very large end displacement the maximum measured rigid body rotation of ~ = 40° is achieved. As in the study of thin-walled tubes, the modus operandi is a precise measurement of dimensions during loading to large deformation.
2. THE DATA The first cantilever beam test described is a 16.45 in. active length, rectangular bar of fully annealed, hot-rolled aluminum, the maximum end displacement perpendicular to the bar being 8.6 in. The bar was annealed at 593°C for 2 h and cooled in the furnace, thereby achieving what I have defined as a "genesis reference configuration" ([4], Section 4). The bar is clamped at one end and loaded vertically at the other. The width of the beam is b = 0.750 in.; the height is h = 0.3125 in. The area moment of inertia is I = (bh3/12) = 0.00191 in. 4, and the distance from the center of the cross-section of the beam to the outermost fiber is 0.15625 in. The manner of loading permitted for each applied load the accurate, detailed tracing on an adjacent screen of the shape of the deformed beam.t In addition, after the load had been removed, curves were drawn for the post-deformation permanent set. The curves in the deformed states under load are shown in Fig. 1, for test 1832. To determine the plastic strain, E=, for each applied load P, post-yield electric resistance gages were located at a distance of 1 in. from the clamped end. Let Lo be the active length of the undeformed beam; Le the distance parallel to Lo from the fixed end to the deflected end in the deformed configuration for each load P; X, the distance from the fixed end to a generic material point in the undeformed reference configuration; and x, the distance from the fixed end to the same material point in the deformed reference configuration. (X - x) is the X component of the rigid body translation. Xin 0 t
2
4
i
J
6 _h . . . .
8 I
I0
12
I
14
16
I
I
~ - .
18
Lo
-
i
o"
x
"I
|
-2
\ -8
Lp--
"~
-tO ~in. Fig. 1. Displacement curves for the cantilever beam of test 1832 for each of the 11 applied loads P at the free end. t I am indebted to the ingenuity and persistence of Faith Paquet Moeckel for the high degree of precision obtained for the curves reproduced in Fig. 1.
Large deflection, rotation, and plastic strain in cantilevered beams
233
Let ¢ be the angle of rigid body rotation of principal axes m e a s u r e d at x; M, the applied moment at X in the undeformed reference configuration; and M', the corresponding applied moment at x in the deformed reference configuration. For M in the undeformed reference configuration we have (1)
M = P(L~, - X )
For a horizontal axis through the centroid of any cross-section of the beam, the moment in the deformed reference configuration becomes the product of P cos ~ times the distance ( L e - x ) s e c O or (2)
m' = P(Lp - x).
The rigid body rotation of principal axes ~ appears only in the integration along the displacement curves of the beam while determining the location of the material point x. The percentage difference between M and M' is determined from the ratio, (M' - M)/M'
(3)
= (X - x)/(Lp - x).
Figure 1 is not schematic. The original drawing is a careful tracing of the displacement curves of the beam. A detailed study of these displacement curves W ( X ) were made, like that shown for the curve of maximum deflection in Fig. 2. The section shown in Fig. 2 is for the plastic zones in the first 9 in. from the fixed end. Beyond 6 in. from the fixed end, the magnitude of the plastic strains decreases until it becomes infinitesimal at the end of the zone. It is only in this region of small plastic strain, far from the fixed end of the beam, that non-negligible differences between M and M ' are recorded. In the important high stress, large strain region, within 4 in. from the fixed end in this example, the difference between the moments M and M' becomes a fraction of a percent. These differences in moments are detailed in Tables 1 and 2. As we found for the gross twisting of thin-walled tubes, for the large deflection of a cantilever beam the role of the rigid body rotation of principal axes is negligible. For the beam, however, while the plastic strains are relatively small, the angles are large. The decrease one might anticipate from the rigid body rotation is offset by the rigid body translation that acts in opposition. Tabulated in Table 1 are moments calculated from equation (1) in the undeformed reference configuration compared with those from equation (2) in the deformed reference configuration at integral values of X in the plastic zone. They are determined along four of the displacement trajectories of Fig. 1. The double lines ( = ) in the table indicate that the moment is in the linear 0
I
2
3
4
5
I
I
I
I
I
,
6
7
8
9
f
I
I
I
0
-I X
-2 in. -3
-4
T e s t 1832 rigid body rotation o x position in deformed configuration + X position in undeformed configuration o.--+(X-x), X component of rigid body displacement
~
,d~=40 I
\ \\ \-,
Fig. 2. A detail from Fig. 1 of the displacement curve for the maximum load in the plastic region between X - - 0 and X = 9in. Included are the location of material points X undeformed, the corresponding value x in the deformed reference configuration, and the angle of rotation 0 at x.
234
J. F. BELL
Table 1 dip = 1,7 in. P = 2.31 lb
die = 4.9 in.
dip = 8.6 in. P = 6.95 lb
6 e = 6.65 in. P = 5.45 lb
P = 4.32 Ib
X in.
M
M'
M
M'
M
M'
M
M'
~b at X
(in.)
(in.-lb) 37.7 35.3 33.0 30.7
(in.-lb) 37.7 35,3 33.0 30.7 ~
(in.-lb) 67.2 62.9 58.5 54.2 49.9 45.6 41.3 36.9 32.6 ~
(in.-lb) 67.2 62.9 58.5 54.4 50.3 46.2 42.1 38.0 34.3 32.0
(in.-lb) 80.7 75.2 69.8 64.3 58.9 53.4 48.0 42.5 37.1 31.6 ~
(in.-Ib) 80.7 75.2 69.8 64.4 59.4 54.5 49.3 44.1 39.2 34.9 33.8
(in.-lb) 94.5 87.6 80.6 73.7 66.7 59.8 52.8 45.9 38.9 32.0
(in.-lb) 94.5 87.6 81.3 75.1 68.8 63.2 57.0 51.6 45.9 38.9 34.8 30.6
(deg.) 0° 7.0 ° 18.5 ° 23.0 ~ 28.5 ° 33.0 ° 34.5 ° 36.0 ° 39.0 ° 39.5 ° 40.0 ° 40.0 °
0 1 2 3 4 5 6 7 8 9 10 11
Indicates linear domain.
elastic region and hence the strain is infinitesimal. These data and those of Tables 2 and 3 below are independent of any theory of finite strain plasticity. The data are from direct measurements in a study of the kinematics of the deformed beam. In the final column are listed the measured angles of rigid body rotation of principal axes ~p at X for the deflection curve produced by the maximum load. This is in contrast to the angles for the same material points at x in the deformed reference configuration shown in Fig. 2. The angles for all other deflection curves are smaller. We note that although the difference in the moments may be small, the corresponding angles of rigid body rotation of axes are large. To further detail the small difference in moments in the two reference configurations, I have tabulated in Table 2 the difference in moments from zero deflection to the maximum for two material points initially at X = 4 in. and X = 6 in. At X = 4 in. from the fixed end, the largest difference between M and M' is 2% for the maximum free end deflection of 6e = 8.6 in. In contrast, the angle of rigid body rotation of principal axes at that same point x is the large angle of cp = 28°! Further away from the fixed end at X = 6 in. where the plastic strains already are becoming small, the difference between the moments M and M' is found to be 7.4%. For that same material point, x = 6, the angle of rigid body rotation of axes has increased to the very large value of ~ = 33 °. Tables 1 and 2 document in detail from the measured curves that, like the gross twisting of thin-walled tubes, the role of the rigid body rotation of principal axes in the gross plastic deflection of beams also is negligible. Similar to the continuum mechanics for thin-walled tubes, one may describe, with very small error, the response from applied moments and calculated stresses and strains by referring only to the undeformed reference configuration. Unlike the situation for thin-walled tubes for which a similar conclusion is reached because the angles of rotation are minuscule, for test 1832 the angles of rigid body rotation are large. Table 2 X = 6.0 in.
X = 4.0 in.
die (in.) 1.7 2.9 3.9 4.9 5.5 6.65 7.7 8.6
P
M
M'
(lb)
(in.-lb)
(in.-lb)
2.31 3.28 3.80 4.32 4.85 5.45 6.03 6.95
39.7 45.2 49.9 55.0 58.9 61.5 66.7
39.7 45.2 50.5 55.4 59.4 62.4 68.1
M
M'
% diff.
(in.-lb)
(in.-lb)
% diff.
2.0%
33.1 37.6 41.2 45.4 48.0 49.5 52.8
33.3 38.0 42.1 46.6 50.1 52.2 57.0
0% 1.0% 2.1% 2.6% 4.2% 5.0% 7.4%
Large deflection, rotation, and plastic strain in cantilevered beams
235
Table 3. Test 1832 P(lb)
be(in.)
LeOn.)
M(in.-Ib)(X = 1.000)
M2(in.-Ib)2
E x x ( X = 1.000)
0.54 1.19 1.53 1.94 2.31 3.28 3.80 4.32 4.85 5.45 6.03 6.95
--
0.1 0.3 0.8 1.7 2.9 3.9 4.9 5.5 6.65 7.7 8.6
16.45 16.45 16.45 16.40 16.30 16.10 15.90 15.55 15.35 14.80 14.20 13.60
8.5 18.3 23.6 29.9 35.6 49.6 56.6 63.3 69.6 75.3 79.6 87.6
73.0 335.2 558.5 893.7 1266.2 2458.7 3204.0 4004.5 4843.1 5665.4 6332.6 7673.3
0.001~ 0.00013 0.00023 0.00028 0.00046 0.0034 0.0052 0.0061 0.0081 0.0116 0.0121 0.0140
3. THE D E T E R M I N A T I O N OF STRESS AND STRAIN In the general incremental theory of finite strain plasticity, with proportional loading chosen to be along a single principal axis, as in the present situation for uniaxial tension and uniaxial compression, constitutive statements reduce to equation (4) [5]: (4)
o=, = (3/2)3/4fl(E= - E=,b) 1/2,
where tl= and E = are the uniaxial stress and strain components of a and E, defined in the original undeformed reference configuration. The constant E=b designates the intercept on the strain axis at zero stress. For fully annealed hot-rolled aluminum at room temperature, the measured material constant is fl = 4.13 x 104 psi, or fl = 29.0 kg/mm 2. For the general theory to include the plastic deformation of beams, equation (4) must apply, with opposite signs, to both the tension and the compression plastic domains of the bent beam. Since the response functions are parabolic, it is convenient to compare the experimental data with the straight lines in a t r 2 vs Exx plot. For fully annealed aluminum, equation (4), with fl = 4.13 x 104psi, the constant becomes (3/2)3c2fl2 = 31.5 x 108. Hence o~x = 31.5 x 10S(E= - e=b)(psi) 2.
(5)
At X = 1.000 in. for test 1832 of Fig. 1, Table 3 lists the applied loads P, the length Le from the fixed end to the point of load application for the deformed beam, the vertical displacement 6p of the point of application of the load P, the moment M and M E at X = 1.000 where the strain gage is located, and finally, the measured strain E= at X -- 1.000 for each load P. For the data in Table 3, Fig. 3 shows a plot of the measured M E v s Exx at X -- 1.000 in. That M E vs gxx is found to be parabolic is compatible with expectation from the continuum theory referred to above, provided the stress is proportional to the moment. When well-documented prior experiment provides constitutive equations that, like 19th
8000
o
a~6ooo .~_
~ 2
4OOO
/ ~
¢4
X= i.O00 in
2000 ~ 0
i
M=P(Lp-X) I
018
0'.8
,10
,.12
,.;
E x x */.
Fig. 3. A plot of M 2 vs Exx from the data of Table 2. The straight line in the plastic domain indicates that the response is parabolic.
236
J . F . BELL
century linear elasticity or 20th century non-linear rubber elasticity, correlate in detail with acceptable continuum theory, they can delimit and characterize the study of a new problem. On the other hand, when constitutive equations are solely a matter of a priori conjecture delineating what may be or what might be, one only can resort to speculation. From a prior correlation of experiment and continuum theory that includes experimentally based constitutive statements, we find that the plastically deformed cantilever beam follows from general principles when o~ = Mc/l. Prompted by the data in Fig. 3 and by the fact that the role of the rigid body rotation is negligible, we introduce O~x(y)/Oxx(C) = y/c. For the moment M, we have M =
ff
oxx(y)yb dy = [oxx(c)/c c
,f[
y2 dA
=
o,,x(c)I/c
(6)
c
or
Oxx(C) = [Mc]/I = [Pc](Le - X ) / I
(7)
where c = h/2 = 0.156 in., is the maximum value of y from the center of the beam cross-section to the outermost fiber, and I = [bh3]/12 = 0.00191 in. 4 is the area moment of inertia. For test 1832, c / I = 81.9 in. -3. For test 1832 of Fig. 1, Fig. 4(b) shows a ~ x vs Exx plot of the measured data compared with equation (5). The slope is precisely in accord with the general theory. The coefficient of 31.5 x 108 (psi) z of equation (5) is the slope of the solid line in the figure! The maximum stress trx= on the cross-section at y = c is of the form of equation (7). Equation (7) implies that the stress varies linearly with the distance y from the center of the cross-section. To check that this is indeed the situation, in a second test, 1831, the height of the beam was reduced from 0.3125 to 0.2521 in. Thus c in equation (7) is reduced from 0.156 to 0.126 in. The width of this beam is approximately the same: b = 0.745 in. The area moment of inertia becomes I = 0.00099 in. 4 instead of I = 0.00191 in. 4. The length of the beam is slightly shorter: L0 = 15.156 in. For this test, in addition to the change in the maximum value of y from 0.156 to 0.126 in., the range of applied loads is from 0 to 4.4 lb instead of from 0 to 6.95 lb, in order to provide a similar large end deflection 6p for the thinner beam. To further check on the analysis, the post-yield strain gage was located at X = 1.533 in. instead of at X = 1.000 in. The parameter c / l of equation (7) increases from 81.9 in.-3 for test 1832 of Fig. 1, to 126.8 in.-3 for the thinner beam of test 1831. As may be seen in comparing the O2xx vs E x x plot of Fig. 4(b) for test 1832 with a similar plot in Fig. 4(a) for the beam of test 1831 measured at a smaller value of y and at a different (0)
60
~..~J S I o pe :
40
~ / ~
~/o Test
oJ
c.O
b
o 0
40
(Bell)
c=126.8 in-3 I c=O. 126 in I = 0 . 0 0 0 9 9 ,n
'~ 20 o x
18.3 I
--Theory
¢
, 0.20
. 0.60
. 0.80 Exx %
0.40
(b)
.
. 1.00
L20
°
Slope
.
"i'e st
1852
--Theory c 81,9 in -5 ~-=
20
(Bell)
c=0.156 m I =0.0019f
o / I
0.20
I
0.40
I
0.60
I
0.80
I
k
1.00
k20
k
140
,n k
i
t.60
1.80
Exx % Fig. 4. Plots of o~x vs Exx for tests 1831(a) and 1832(b). The solid lines are the predicted parabolicity from equation (9); the slope is in accord with the general theory, equations (4)-(6).
Large deflection, rotation, and plastic strain in cantilevered beams
(CI) +c!+y
i
+Exx
+5000 +10000 psi
+ .01
/
C
-Exx
+ct+Y +o"~x
, t_----~ +.o2
I
-Y
237
-o-~
J--C
-y~
P +c
+yt
(c)
Tension Plostic Lineor
~
Elostic
X (inches}
Compression Plostic --C
-y
Fig. 5. (a) The parabolic distribution of strain over an initially planar surface. (b) The linear distribution of stress over the same cross-section. (c) The separation surfaces between the parabolic plastic and the linear elastic domains.
location X, equation (7) is admissible. The general continuum theory of finite strain plasticity is extended to include the large deflection, rotation, and plastic strain of beams. As will be shown in Fig. 6, the m e a s u r e d elastic limit is cry = 2500 psi. By setting oxx = ey, equation (7) may be used to determine the separation surface between parabolic plastic and linear elastic domains, as shown in Fig. 5(c). The calculated linear stress distribution with y at X = 0, is given in Fig. 5(b). For the plastically deforming beam in parabolic plasticity, instead of plane sections remaining plane as in the linear theory, initially plane sections immediately become parabolic and remain parabolic. This is shown at X = 0 in Fig. 5(a) using equation (8). From equation (4), Exx = 02=/[(3/2)3/4/5]2, or from equation (7) E=, = {[P/II2[(3/2)3/4[31-Z[Lp - Xl2}y 2.
(8)
From the calculation of stress distribution of Fig. 5(c) we see that the zones of parabolic plastic deformation are bounded at loads below the elastic limit Cry = 2500 psi. For all y, at any cross-section of the beam the response to the initial loading is linear elastic. At X --- 1.000 in. this corresponds to the first four initial loadings for test 1832 in Table 3. From the measured moments and strains we may determine and plot o~, vs E~x for the initial loading of the beam. Such plots are shown in Fig. 6(a) for the initial region of test 1831, and in Fig. 6(b) for test 1832. In each instance, the above analysis and measurements reveal that not only is the measured slope linear as predicted, but also both slopes provide the known bar modulus for aluminum: E = 10 x 106psi, (7030kg/mm2). From tests 1831 and 1832, the initial loading provides a direct measure of the elastic limit, namely, o r = 2500 psi. To extend the analysis of the beam into the linear elastic region, to X > 9 in., a post-yield strain gage for test 1831 has been located at X = 13~ in. from the fixed end, i.e. in the totally linear elastic region toward the free end of the beam. If the analysis of experiment presented above is indeed a proper correlation between theory and experiment, then the application of equation (7) to the linear elastic domain beyond X = 9 in. should provide a or= vs E= plot with a linear slope of E = 10 x 106 psi (7030 kg/mm2), the bar modulus for aluminum. That the data listed in Table 3 does in fact meet the test is illustrated in the a= vs E~x plot of Fig. 7. The slope determined from equation (7) for the measured data at X---13~ in. for test 1831 is precisely E = 10 x 106 psi.
238
J . F . BELL
(a)
(b)
3000
3000 o
O"y
2500
2500 i
•-
2000
•-
g
2000
x
b~15oo
b x 1500
I000
I000
Test 1851 --Theory X= I 17/32 in c =0,126 in ! = 0 . 0 0 0 9 9 in 4
500
I
0
I
2 y X= I.O00in c=0.156 in I=0.00191 in 4
/
500
/ !
I
0.0002 0.0004 0.0006 Exx
0
0.0002
0 . 0 0 0 4 0.0006 Exx
Fig. 6. The initial portions of tests 1831 and 1832 in which the low strain data from Table 2 in equation (11) provide the linear elastic slopes in oxx vs Exx plots. The slopes have the standard modulus for aluminum: 10 x 106psi. The measured elastic limit is o v = 2500 psi.
For the engineer seeking a failure criteria for the cantilever beam, the calculated maximum stress at the fixed end is oxx(0) - 7842 psi for test 1832 and axe(0) = 8696 psi for test 1831. For this fully annealed solid in tension, the ultimate stress is 12,300 psi.
4. S U M M A R Y A N D C O N C L U S I O N S This laboratory study of very large deflection in a cantilever beam was undertaken as part of an effort to contribute to an understanding of the role of the rigid body rotation of principal axes in finite strain plasticity. 1400
1200
I000
"
800
x
b
E=IOXIO
6 psi~
"~
6o0 0
Test 1831
400
--Theory X= 13 1/16 in c =0,126 m ] = 0 . 0 0 0 9 9 in 4
200
I
0
I
L
0.0002 0.0004- 0.0006
1
0.0008 Exx
I
0.0010
J
0.0012
I
0.0014-
I
0.0016
Fig. 7. At 13~ from the fixed end, a plot of oxx vs Exx determined from measured moments and strain in equation (11). Note that the response is linear elastic as anticipated in this domain, with a slope of 10 x 106 psi, the standard value for aluminum.
Large deflection, rotation, and plastic strain in cantilevered beams
239
As in the kinematical study of the large twisting of thin-walled tubes [1, 2], the kinematical study of the gross deflection of a cantilever beam in the presence of relatively small plastic strain, once again relegates the role of the rigid body rotation of principal axes to a negligible parameter. The unanticipated by-product is of considerable current and historical interest: it is the finding of a complete description of the detailed plastic strain response of grossly deflected beams that is compatible with general finite strain continuum theory. Included are the details of the spatial distribution of stress and strain over all cross-sections; the delineation of the surface of separation that bounds parabolic plastic and linear elastic domains; and the provision of maximum stress and strain for a known applied load. For a linear response function with a stress that linearly increases with the distance from the center of the beam, the classical theory provides that plane sections remain plane. For a parabolic response function in which the stress also linearly increases with the distance from the center of the beam, examination of the stretches reveals that initially plane sections become parabolic, and these parabolic sections remain parabolic. The constitutive equations for structural metal alloys of steel, aluminum, and copper are known [4]. They lead to the same type of parabolicity in simple tension and in compression. One anticipates that the above analysis will include the gross deflection of beams of these hardened metals. Acknowledgement--I wish to thank Dean L. Pendleton for her carefully made drawings for the figures. Note--To avoid introducing error, the nearly 200 numbers measured in this study have been presented in their original form. The conversion to SI units is as follows: pounds force (Ib)x4.45=newtons (N); pounds force per square inch x 3.89 = kilopascals (kPa); inches (in.) x 25.4 = millimeters (mm).
REFERENCES [1] J. F. BELL, Int. J. Solids & Structures 25, No. 3,267-278 (1989). [2] J. F. BELL, Int. J. Plasticity In press. [3] J. E. GREENSPON, Energy approaches to structural vulnerability with application of the new Bell stress-strain laws, U.S. Army, Ballistics Research Laboratory Contract Report, No. DAADO-75-C-0731.
(1975).
[4] J. F. BELL, Int. J. Plasticity, 1, No. 1, 3-27 (1985). [5] J. F. BELL, Int. J. Plasticity, 4, No. 2, 127-148 (1988).
(Received 27 April 1989; received for publication 30 August 1989)