Acta Mechanica 108, 4 9 - 6 2 (1995)
ACTA MECHANICA 9 Springer-Verlag 1995
Rotation effects in the global/local analysis of cantilever beam contact problems M. Zhou and W. P. Schonberg, Huntsville, Alabama (Received January 25, 1993; revised October 12, 1993)
Summary.A modification of a previously developed method for solving the static contact problem of a rigid, cylindrical punch indenting an isotropic cantilever beam is presented. The boundary conditions at the contact site are modified by including beam upper-surface rotation effects. Local contact stress distributions, as well as global beam displacements, are computed for various ratios of contact width to beam thickness and for various positions of the indenter. The results are compared to those of a previous study in which the rotation effects were not considered, to Hertz theory of contact stress, and to beam theory solutions. It is noted that the rotation effects became prominent as the contact width approached the thickness of the beam.
1 Introduction Consider the finite layer of length L and thickness h shown in Fig. 1. The layer is fixed at one end, free on the other end and loaded by a frictionless cylindrical indenter centered at a distance L0 from the fixed end. Such a configuration can serve as a first-order model for gear interaction and turbine blade impact problems. Traditional methods used to analyze such a problem typically employ a beam-theory solution to obtain an overall load-displacement relationship and a Hertzian contact solution to calculate local indenter stresses. However, such methods are only applicable in a fairly limited class of problems [1]-[3]. A problem such as this was solved previously for isotropic and transversily isotropic materials [4], [5]. The method of solution used was the superposition of an infinite elastic layer solution with a beam theory solution to match the boundary conditions of a finite layer. In this way, local contact stresses as well as global stresses are represented in one total solution. However, a major limination of the method developed in [4], [5] was that it did not include the rotation of the beam under the indenter in the boundary conditions. Rather, the method developed presumed that beam rotation effects were negligible due to the assumption of small angle rotation. While this may be valid for relatively small contact lengths, the effects of beam rotation may be significant for problems where the contact length is large relative to the thickness of the beam. Since it is impossible to know a priori whether or not these effects can be neglected, the analytical model given in [4], [5] must be modified to include them. In this paper, the problem of the static indentation of an isotropic cantilever beam is re-solved such that the rotation of the beam under the indenter is included in the mixed boundary condition at the contact site. Using the same technique as in [4] a system of two Fredhom integral equations of the second kind is obtained and solved. The numerical solution of these two integral equations gives us the physical quantities of interest: the stress distribution under the indenter and the displacement of the beam. In the solution process, the ratio of semi-contact width to
50
M. Zhou and W. R Schonberg
beam thickness and the ratio of indenter arm length to beam thickness are treated as known parameters. By varing these and other parameters as in [4], an extensive study of the response of an isotropic cantilever beam subjected to various contact loading conditions is performed. Following the theoretical solution of the isotropic cantilever beam indentation problem, numerical results are obtained for a wide variety of geometric and indentation conditions. The results obtained are compared to those derived using the original analytical model of cantilever beam indentation. It is shown that for large area contact and for indentations near the fixed end of the beam, the effects of the rotation of the beam under the indenter are significant. The results are also compared to beam theory solutions and to Hertz theory contact solutions for validation.
2 Problem formulation 2.1 Introductory comments The mixed boundary value problem to be solved is that of an elastic layer of thickness h and length L that is indented by a cylindrical punch on its upper surface (Fig. 1). The conditions at the ends of the layer are those of a cantilever beam: clamped on the left and free on the right. The solution of the problem is achieved by a suitable superposition and matching of an elasticity solution for an infinite layer and a beam theory solution as in [4].
P
X
I IY
Fig. 1. Problem configuration: static indentation of a cantilever beam
0 ,9
IAB
------
C
IY
Fig. 2. Boundary conditions at the contact site for a cantilever beam
Rotation effects in the global/local analysis of cantilever beam contact problems
5i
2.2 Infinite layer solution The boundary conditions for the mixed boundary value problem can be written as follows (see Fig. 2): o%,(x, h) = axy(x, h) = axy(x, 0) = 0, %y(x, 0 ) = 0 ,
]x[ < oo
(1-3)
(4)
c
uy(x, O) = d + Oox - x2/2R,
[xI < c
(5)
where 00 is the beam rotation under the indenter given by ~uy Oo = ~-x (0, O)
(6)
and where 2c is the total contact length produced by the indentation; hence, the quantity c is referred to as the 'semi-contact length'. Equations (4) and (5) are the mixed boundary conditions for beam upper surface. It is noted that Eq. (5) is different from the corresponding equation in [4] due to the inclusion of beam upper-surface rotation effects via the 0o term. The geometry for the derivation of Eq. (5) is shown in Fig. 2. In the original study, line OA was coincident with the y-axis; in this study, it is not. From Fig. 2, and assuming small deflection theory, we see that
AA' = A + R - [ R 2 --(ROo)211/2_~ A + R ( l O o 2)
(7)
but also
AA' = BB' + R - [R 2 - (x - RO0)2]1/2 ~ BB' + ~ R
;
- O0
(8)
Therefore, it follows that X2
uy(x, O) = BB' = A + Oox - - -
2R
(9)
Following the procedure presented in Sneddon [6], a suitable elasticity solution that represents a general loading on the upper surface of an isotropic elastic infinite layer in plane strain, and with no loading on its lower surface, is given as follows:
f Es(~)c o s ~x + EA({)sin i x ~,, =
/32 _ sh2/3
{[/3 + sh/3ch/3 + {ysh2/3] shiy
0
_ [shZfi _/32 + iY(/3 + shflch/3)] ch{y} d~ f ~x,
=
-
o
Es(~) sin i x + EA(I) cos Ix {[l32 -- iY(fi + shflchfl)] shay + iysh2flch{y} d~ /32 _ sh2/3
(lo)
(11)
52
M. Zhou and W. E Schonberg
ax~ =
f
Es(i) cos i x + EA(I) sin i x f12 _ sh2fl {[fl + shfl -- iysh2fl] shay
0
- - [f12 q_ s h 2 f l __ ~y(fl + s h f l c h f l ) ]
f
2#Ux =
(12)
ch{y} d i
Es(i) sin i x - Ea(i) cos i x ~(f12 _ sh2fl) {[(1 - 2v) (fl + shflchfl) - ~ysh2fl] shiy
0
_ [f12 + (1 - 2v) sh2fi - Cy(fl + shflchfl)] chiy} di
2#u, =
Es(~) cos ~x + EA(~) sin ~X
f
~fi~---sh~
(13)
{[fi2 _ 2(1 - v) shZfl - ~y(fi + shflchfl)] shay
0
+ [2(1 -- v)(fl + shflchfl) + ~ysh2fl] ch~y} d~
(14)
where fl = ~h, and # and v are the shear modulus and Poisson's ratio of the layer material, respectively. The functions Es(i) and EA(~) correspond to the symmetric and anti-symmetric components of the upper surface loading. It is seen that on y = h the normal and shear stresses vanish automatically, and that on y = 0 the shear stresses vanish. The normal stress on y = 0 is given as:
ary(x, 0) = S [Es(~) cos ix + EA(~) sin ix] d~.
(15)
0
As in [4], we let
Es(~) = i r
Jo(i) dt
(16)
0
EA(I) = i O(t) Jl(~t) &.
(17)
0
Then Eq. (15) becomes
f
v,
~(t) dt
+ x
Ixl
f Ixl
4)(t) dt t l/(t 2 -
"
(181
x 2)
The additional boundary conditions that need to be satisfied by the finite elastic layer can be written as follows:
O(x) = uy(x) = O,
x = -Lo,
M(x) = V(x) = O,
x = L - Lo,
(19, 20) (21, 22)
where M and V are the moment and shear, and 0 is the average value of the slope measured through the thickness given by h
0 = ~1 [ ~guy dy. 0
(23)
Rotation effects in the global/local analysis of cantilever beam contact problems
53
Thus, we also need to calculate the moment, shear and average slope due to the stresses given by Eqs. (10)-(12). These quantities are given by the following expressions:
Y~x dy = -
ME(X) = f
[Es(~_)cos {x + Ea({) sin {x] }5 d~
0
f
V~(x) =
(24)
0
~y dy =
0
[Es(~)sin ix - EA({) cos ix] d~
(25)
0
1 f f l - ( 3 - 2 v ) shfl [Es({) sin {x - EA({) cOS ix] fl@ shfl)
O~(x) = ~
d{.
(26)
0
We note that Eqs. (24) and (25) satisfy the relationship V = dM/dx.
2.3 Beam theory solution For the beam theory solution, the displacement is taken to be of the form
uf(x) = ao + alx + a2X2 + a3x3
(27)
where the constants ao, ..., a3 are determined by superposing the infinite layer and beam theory solutions and enforcing the conditions at the ends of the beam. Assuming that the hypotheses of Euler-Bernoulli beam theory are valid, we have
uf(x) -
aUf ( y _
(28)
and
aL(x)-
2#
~?u~~
1 - v c~x
(29)
Using Eqs. (27)-(29), the moment, shear and average slope are calculated to be h
MB(x) = ~ ya~x dy = - 2 D ( a 2 + 3a3x)
(30)
0 h
V~(x) = ~ a~y dy = --6Da3
(31)
0 h
1f
O~(x) = ~
~Our x dy = al + 2a2x + 3a3x2
(32)
0
where D = #h3/6(1 - v). Thus, the solution sought will be a superposition of the elasticity solution given by Eqs. (10)-(15) and (24)-(26), and the beam theory solution given by Eqs. (27)-(32). The two solutions are matched properly when the boundary conditions, Eqs. (1)-(5) and Eqs. (19)--(22), are all satisfied.
54
M. Zhou and W. R Schonberg
2.4 Contact problem solution
The next step in the solution of the static contact problem for the cantilever beam is to solve for the constants al, a2 and a3. This is achieved by superposing the beam theory and elasticity solutions and then applying the beam theory boundary conditions in superposed form. Thus, in superposed form, Eqs. (19), (21) and (22) become M e ( L -- Lo) + Mn(L - Lo) = 0
(33)
VE(L -- Lo) + VB(L - Lo) = 0
(34)
0~(-Lo) + 0B(-Lo) = 0.
(35)
As in [4], applying Eqs. (2)-(5), (27)-(32) to Eqs. (33)-(35) yields the following:
al = - ~
lf{Lo(L-Lo/2) ~
[Es(~) sin ~(L - Lo) - EA(~) cos ~(L - Lo)]
0
}
+ ~-~ [Es(~) cos ~(L - Lo) + EA(~) sin ~(L - Lo)] d~
1 ~ f l - ( 3 - 2 v ) shfl
+2-~# d
fl~ -~ sh-fl)
[Es(~)sin ~Lo + EA(~) cos ~Lo] d~
(36)
o
a 2 =
1 2D
[Es(~) cos ~(L - Lo) + EA(~) sin ~(L - Lo)] ~-
_ - -
o
L - 2D Lo
f
[Es(r sin ~(L - Lo) - Ea(~) cos ~(L - Lo)] d~
(37)
0
1 l" d~ a3 = 6D j [Es(~) sin ~(L - Lo) - EA(~) sin ~(L - Lo)] ~-.
(38)
0
The two remaining boundary conditions given by Eqs. (5) and (20) still need to the satisfied. To be consistent with the use of small deflection theory thus far, it is assumed that the difference ]uy(x, 0) - uy(x, h)[ is negligibly small. This allows the beam theory expression for transverse displacement to be superposed with the elasticity solution expression at the beam upper surface, that is, at y = 0. Thus, we first consider Eq. (5) in the following superposed form: uye(x, O) + uff(x) = A + Oox -- x2/2R.
(39)
Differentiating with respect to x yields ~ u f (x, 0) +
,~x
~Uy B
~
X
(x) = 0o - --
R
(40)
Rotation effects in the global/local analysis of cantilever beam contact problems
55
Separating Eq. (40) into a symmetric and an antisymmetric part with respect to x yields 0 u f (x, 0)[A + 2a2x = _.X_ 3X R
(4])
OUyE
(42)
~X (X, O)ls + al + 3aax 2 = 0o.
Consider first Eq. (41). Substituting for uf(x, 0) according to Eq. (14), combining terms in
Es({) and Ea(~), and making use of Eqs. (16) and (17) yields f 0(t) f [ h [-6 3 ( f i \+ ~s h3 f- l-c s-~-/~ h f l ~ ] sin {x + ~x cos ~(L - L0) + x(L-Lo) r sin { ( L 0
- Lo)
1
Jo(~t) dr
dt
0
cos {(L - Lo) a,(~t) d~ dt = __OXR 0
(43)
0
An asymptotic evaluation of the kernels in Eq. (43) shows that they a,e all convergent at the lower limit of integration. However, at the upper limit, the first term in the first kernel is divergent, This is adjusted by adding and subtracting the term h3 f ~O(t) -~ 0
Jo((t) sin (x d{ dt
0
in Eq, (43) as follows:
ha 6
; ; $(t)
0
Jo({t) sin ~x d~ dt +
0
x
+ ~ cos ~(L - Lo) + + ~ q~(t) ~d 0
i
0
x(L -- Lo) ~
(fl + shflchfi
tp(t).
\ ~33 - - ~
0
)
+ 1 sin ix
]
sin ~(L .- Lo)j Jo(~t) c;g &
x(L - Lo) cos {(L - Lo) Jl(~t) d{ dt = Dx -~
x sin
(44)
0
Noting that for x > 0 Oro(~t)sin ~x d{ at =
O(t) 0
0
V~
- tz) at
(45)
0
allows us to re-write Eq. (44) as follows: h3 6
O(t) -de + / x 2 _ t 2) 0
-}-.
+
h (fl + sh3ch3
0(t) 0
V
\ ?~~ ~ -
)
x
+ 1 sin ~x + ~ cos ~(L - Co)
0
x(L -- 1-,o) sin ~(L -- Lo)] Jo(~t) d~ dt
; ;E x
~ sin ~(L - Lo)
qb(t)
0
A
0
x(L - Lo)
cos ~(L - Lo)] Jl(~t) dd dt = Dx " R A
(46)
56
M. Zhou and W. E Schonberg
Equation (46) can be viewed as an Abel integral equation of the form
f
O(t)dt - g(x)
(47)
0 with the solution
2 di
xg(x)
(48)
0
Thus, we can multiply Eq. (46) by the operator
d f
xdx
0
so that after simplification Eq. (46) reduces to
O(x) --
r
O(t) Kl(x, t) dt -
0
S
Dx
(2(0 K2(x, t) dt = - - -R
(49)
0
where
K~(x,t)= i [h3{fl+shflchfl l~-\-~g- ~
+1
) ~XJo(~X)+ ~icos~(Lx Lo3] Jo(~t)dZ + ~x(L- Lo)
o
(50)
7C K~(x, t) = ~tx.
(51)
Returning to Eq. (42) and noting that by definition Oo = ~~uf - x (0, O)]s +
~-x~Uf(O)[s = ~h3 3~ BfEa(~)7 7s (fl + shflchfl) d~ + a~
(52)
0
we have h 3 [" EA(~) cos (~x)
h3 f EA(~) ([3 + shflchfl) d~ + al + 3a3x 2 = - ~ flf ~-~zfl (fl + shflchfi) d~ + al.
0
o
(53)
Collecting terms yields
h3 f fl+shflchfl
6D
~ _-sh2~ (1 - cos ix) EA~) d~
0 X'2 i 2D o
[Es(~) sin ((L -- Lo) -- Ea(~) cos ((L -- Lo)] d~ = 0.
(54)
Rotation effects in the global/local analysis of cantilever beam contact problems Substituting for
f
~(t)
0
Es({) and
Ea({) according to Eqs. (16) and (17), wo obtain the following:
; { h3 ~-[-Shflchfl ~ - - sl-~
(1 - cos i x ) +
x 2 cos
((L -- Lo)'~Jt(~t) d{ dt
2
3
0
2
qJ(t) 0
57
~
Jo({t) d{ dt = O.
(55)
0
An asymptotic analysis reveals that all the terms are bounded as ~ ~ 0, but as ~ ~ o% the first term of the first kernel is divergent. This is corrected by adding and subtracting the term
7
qS(t) 0
[1 -- cos ({x)]
Jl({t) d~ dt
0
in Eq. (55) as follows:
6
qS(t) 0
+
f
d~ dt
[1 - - c o s ( ~ x ) ] J l ( ~ t ) 0
f {~(fi+shfichfl \fl-/---s/~
)
qS(t)
0
0
2
O(t) 0
~
x2cos{(L-Lo)} ~
+ 1 (1 -- cos ix) + 2
Jo({t) d~ dt =
dl(~t)
d~ dt
0.
(56)
0
F r o m [7] we have
f (~ f(1-c~ 0
Jl(~t)d~dt=f ~(t)
0
0
t2) x + ~ 1
dt-x
O (57)
and
fo(of sin~(L-L~Jo(~t) d~ dt = f~~ O(t)dt. 0
o
(58)
0
Substituting Eqs. (57) and (58) into Eq. (56) and multiplying both sides with x yields
h3 f t(o(t) dt + h3 f t(o(t) dt 6 ~ 6 x + ~ ( ~ _ t z) 0
- x
0
t o
\ ~--W/r/~ 0
+ 1
(1 - cos ~x)
0
rCx3 f O(t) dt = O. + xZ 2 c~ ~(L-~ L~ Jl(~t) d~ dt - -4 0
(59)
58
M. Zhou and W. R Schonberg Again, Eq. (59) can be viewed as an Abel integral equation
f ( ~ t4~(t) ~ t2) dt= g(x)
(60)
0
with the solution 2 d f
dx.
xg(x)
(61)
0
After simplification, Eq. (59) reduces to h3
h3 f
+
~)(t~-~)dt+t
0
i o(t)K3(x't)dt+f~(t) K4(x,t)dt=O 0
(62)
0
where 3
K3(x, t) = ~- 7rx2
(63)
h3ffl+shflchfl ) K4(X, t) = f ~\ 75-- s~ + 1 [J0(~x) - ~xJt(~x) - 1] Jl(~t) d~.
(64)
0
Equations (49) and (62) are two coupled integral equations for the unknown auxiliary functions 0(x) and qS(x).These equations are solved numerically for a given value of semi-contact length c. While Eq. (49) is identical to the corresponding equation in [4], Eq. (62) is significantly different because of the inclusion of the 0o in the boundary condition at the contact site. Once 0(x) and ~b(x) are obtained, all necessary physical quantities may be calculated for the specified semi-contact length. Stress may be calculated using Eq. (18), while the resultant load due to the symmetric stresses is obtained as follows:
P = - +c ~ or,, dx = -re i 0(t) at. --c
(65)
0
The resultant moment due to the antisymmetric stresses is obtained as follows:
M = -
Xayy dx = - 2 --C
t4)(t) dt.
(66)
0
In order to evaluate the deflection under the indenter A that would cause the contact region to extend over the given contact length, the constant ao must be determined. As in [4], superposing Eqs. (14) and (27) yields
u,(x, O)
1- v ~ /2
-
J 0
fi + shfichfl [Es(~) cos ~x + EA(~) sin ~x] d~ + ao + alx
~2(fi2 _ sh2fi)
+ a 2 x 2 + a 3 X3 .
(67)
Applying Eq. (20) yields ao
l -- v i fl)~+ 2shfich s~ fl [Es(~) cos ~Lo -- EA(Q sin ~ L o ]d_~ - + alLo -- aaLo 2 + aaLo 3 #
0
(68)
Rotation effects in the global/local analysis of cantilever beam contact problems
59
where a~, a2 and aa are given by Eqs. (36) - (38). Substituting Eqs. (36) - (38) and (68) into Eq. (67), and recalling that uy(0, 0) = A, yields
1-vfl+shfichfi
1-cos~Lo
+--
sin~Lo--fl(3 - shfi)
2D
o
cos , ( L - Lo! ~2 j Jo(~t) d~ - ~-~ Lo
(3L - Lo)
dt + o
x
sin ~Lo
~
7 shfl } nt ~(fl - shfl) cos ~Co Jl(~t) d~ - ~ Lo2 dt. 3
Lo fi --(3 -- 2v)
+ 21~
(69)
An asymptotic analysis reveals that both kernels are bounded as ~ ~ 0, but the first two terms in each kernel are divergent as ~ ~ oc. This is adjusted by adding and subtracting the appropriate terms to yield
= i *(0 ~(t) d~ + i 4(t) I(~(t)dt o
(70)
o
where
K~(t)-
1 - v f [,(fi+shfichfl ~
( \ ~
)(1-cos~Lo)Losin~Lo
+1
~
fi-~h~
3Lo2cos~(L-Lo)} h~ ~
o
Vcosh_ ~
1--v f ~(fi+shflchfl
+ 4
#h
)sin~Lo
Lo- I~L~176
(71)
Locos~Lo~
o
1 -- v
t
nt
Lo + (1/~o2 - t 2)
8D
(72)
Lo 2 .
To assess the validity of the elasticity solution presented herein, the predictions of the elasticity solution for the beam displacement are compared with the predictions of a beam theory solution that uses as input the contact pressure generated by the elasticity solution. Such a beam theory solution for displacement can be expressed as follows: +e
Zl B = ~
0
p(x') Lo2(2Lo + 3x') dx'-
~
--c
Noting that
--c
p(x) = -ayy(x, 0) and using Eqs. (18), (65) and (66), we can write Eq. (73) as follows:
AB= ~ 3 p + Lo ~ 2 M_
tatp(t)dt + ~
1 o
t3dp(t)&.
(74)
o
It is noted that while the value of c may not be known a priori, in practice one can develop a family of curves for a given geometry relating P, A, and c using the equations presented herein. These curves can then be used to determine A for a given P (or vice versa) without first having to know the value of c.
60
M. Zhou and W. R Schonberg
3 Results and discussion In the solution for the auxiliary functions ~p(x) and q~(x), Eqs. (49)-(51) and (62)-(64) are first non-dimensionalized. This is accomplished by the use of the non-dimensional parameters c/h, Lo/h, L/h, t/c, x/c, and qJ = Rh3~/Dc and ~b = Rh3~/Dc. This non-dimensional scheme results in the following relationships between dimensional and non-dimensional stresses, loads, moments, and displacements:
rryy = P•yy/C
(75.1)
P = hZ#ff/(1 - v) R
(75.2)
M = h3#M/(1 - v) R
(75.3)
A = h2A/R.
(75.4)
Solutions to the problem were obtained for c/h = 0.2, 0.5, and 1.0, L/h = 5.0, 10.0, 15.0 and 20.0, and for each c/h, Lo/h = 0.25 L/h, 0.5 L/h, and 0.75 L/h. These are the same parameter values as in the original isotropic cantilever beam indentation study [4]. The results of this parametric study are discussed in detail in the following paragraphs. It was found that the stress distributions for different values of L/h remained virtually identical for the same value of c/h. This is illustrated in Tables 1 a, b, which compare peak nondimensional symmetric stress component values and total stress values for different values of c/h and Lo/h. These tables also compare corresponding stress values obtained in this study to values obtained in the previous study that did not include upper-surface rotation effects [4]. As is evident in Tables l a and 1 b, the peak stress values calculated using the analytical model developed in this paper begin to differ significantly from the results of the original model as c/h approaches unity. Specifically, for small values of c/h, the stresses predicted by
Table 1 a. Comparison of non-dimensional maximum surface stress values, revised and original analytical models, symmetric component only
Lo/L = 0.25
LolL = 0.50
Lo/L = 0.75
L/h
New model
Old model
New model
Old model
New model
Old model
c/h
10.0
0.628 0.615 0.494
0.635 0.622 0.560
0.628 0.615 0.494
0.635 0.622 0.561
0.628 0.615 0.494
0.635 0.617 0.561
0.2 0.5 1.0
20.0
0.628 0.615 0.494
0.635 0.625 0.560
0.628 0.615 0.494
0.635 0.621 0.564
0.628 0.615 0.494
0.635 0.622 0.562
0.2 0.5 1.0
Table 1 b. Comparison of non-dimensional maximum surface stress values, revised and original analytical models, total stress values
Lo/L = 0.25
LolL = 0.50
Lo/L = 0.75
L/h
New model
Old model
New model
Old model
New model
Old model
c/h
10.0
0.628 0.624 0.908
0.636 0.648 0.982
0.628 0.624 0.908
0.636 0.649 0.983
0.628 0.624 0.908
0.636 0.645 0.981
0.2 0.5 1.0
20.0
0.628 0.624 0.908
0.636 0.650 0.982
0.628 0.624 0.908
0.636 0.649 0.988
0.628 0.624 0.908
0.636 0.649 0.983
0.2 0.5 1.0
Rotation effects in the global/local analysis of cantilever beam contact problems
61
the revised model are approximately 1.5% smaller than those predicted by the original model; the gulf widens to approximately 10% when c/h was equal to one. This is due to the fact that when c/h < 1, the lateral motion of the indenter is minimal and indenter rotation effects are negligible. However, as c/h -~ 1, the magnitude of the lateral motion is apparently significant enough so that the attendent indenter rotation effects become prominent. Table 2 shows a comparison between the predictions of the elasticity solution developed herein and the predictions of the beam theory solution for transverse displacement. The results of the elasticity solution are seen to agree very well with those of the beam theory solution. It is noted that the agreement improves as Lo/h increases and as c/h decreases. This can be explained by the following considerations. First, as c/h decreases, both the beam theory solution and the elasticity solution more accurately model a beam subjected to a concentrated load rather than a beam subjected to a distributed load. Second, as Lo/h increases, the effects of shear deformation on beam response become negligible. While the elasticity solution incorporates those effects, the beam theory solution does not. Therefore, these two solutions will match more closely for indenter locations which result in minimal shear deformations. Table 3 shows a comparison between the revised elasticity solution developed herein, which includes beam upper-surface rotation effects, and the original elasticity solution [4], which does not. We see that for practically all cases considered, the two solutions agree quite well. However, in two instances, the two solutions differ significantly. Specifically, the displacement values for c/h = 1, L/h = 10.0 and Lo/L = 0.75, and for c/h = 1, L/h =- 20.0, and Lo/L = 0.5 generated with the original model are not consistent with other data from the original model and differ significantly from the corresponding data generated with the revised model. A careful examination of these two sets of data presented in Table 3 reveals that there must be something wrong with the two data points generated by the original model. Except for these two data points, the maximum difference between two solutions for central displacement is on average approximately 5%. Table 2. Non-dimensional displacement comparisons: revised analytical model vs. beam theory
Lo/L = 0.25 L/h
Analytical model
Lo/L = 0.50 Beam theory
Analytical model 16.1 126,3 3 374.5 128.7 1019.4 28669.8
10.0
2.0 15.6 378.4
2.0 15.4 387.4
20,0
16.1 126.4 3473,6
16.1 126.6 3485.5
Lo/L = 0.75 Analytical model
Beam theory
c/h
16.1 126.7 3 386.2
54.3 429.2 11784.0
54.3 429.5 11799,1
0,2 0.5 1.0
128.7 1019.8 28687.1
434.4 3 435.1 98185.0
434.5 3445.6 98208.3
0.2 0.5 1.0
Beam theory
Table 3. Non-dimensional displacement comparison: revised analytical model vs, original analytical model
Lo/L = 0.25 L/h
New model
LolL = 0.50 Original model
New model
Lo/L = 0.75 Original model
New model
Original model
c/h
10.0
2.0 15.6 378.4
2.0 15.3 383.5
16.1 126.6 3374.5
16.1 127.5 3487.5
54.3 429.2 11784.0
54.3 438.6 8705.0
0.2 0.5 1.0
20.0
16.1 126.4 3 473.6
16.2 128.5 3 327.0
128.7 1019.4 28 669.8
127.1 1001,0 35 890.0
434.4 3435.1 98185.0
435.7 3466.0 101570.0
0.2 0.5 1,0
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M. Zhou and W. P. Schonberg: Rotation effects in cantilever beam problems
4 Conclusions The global-local method presented in [4] is quite successful in obtaining a solution which can model the local behavior as well as the global response of a cantilever beam indented by a rigid cylindrical punch. An parametric study was performed to determine the effects of including beam rotation at the contact site in the problem formulation. It was found that the effects are negligible for small contact lengths, but become significant when the contact length is large or when the indenter is relatively close to the fixed end of the cantilever beam. The implication of this result is that the rotation effects addressed in this paper must also be included in contact and low velocity impact problems involving beams that are elastically supported at both ends but in which the indentation or impact is not symmetric, that is, it does not occur at mid-span. Naturally, whether or not the revised analytical model is more correct than the original model cannot be inferred from the results presented in this paper. Ultimately, some experiments will have to be performed to determine the ability of the analytical models to replicate actual stress and deformation values. However, we can conclude that the revised analytical model is more self-consistent than the original analytical model. This is based on the fact that the predictions of the revised model are closer to those obtained using a beam theory solution that uses information provided by the model than the predictions of the original model. Despite its apparent success, the concept of superposing an infinte elastic layer solution with a beam theory solution does have its limitations. Future efforts in this field should involve extending the method to include a general, two-dimensional elasticity solution for the finite layer portion of the total solution. This will provide a more general form of the transverse displacement, rather than just an expression for the displacement of the beam's upper surface.
Acknowledgement The authors would like to acknowledge the University of Alabama in Huntsville Civil and Mechanical Engineering Departments for support and for providing the computer facilities for this work.
References [i] Keer, L. M., Miller, G. R.: Smooth indentation of a finite layer. J. Eng. Mech. 109, 7 0 6 - 717 (1983). [2] Keer, L. M., Lee, J. C.: Dynamic impact of an elastically supported beam-large area contact. Int. J. Eng. Sci. 23, 987-997 (1985). [3] Schonberg, W. P., Keer, L. M., Woo, T. K.: Low velocity impact of transversely isotropic beams and plates. Int. J. Solids Struct. 23, 871 - 896 (1987). [4] Keer, L. M., Schonberg, W. E: Smooth indentation of anisotropic cantilever beam. Int. J. Solids Struct. 22, 8 7 - 106 (1986). [5] Keer, L. M., Schonberg, W. R: Smooth indentation of a transversely isotropic cantilever beam. Int. J. Solids Struct. 22, 1033 - 1053 (1986). [6] Sneddon, 1. N.: Fourier transforms. New York: McGraw-Hill (1951). [7] Watson, G. N.: A treatise on the theory of Bessel functions, 2nd ed. Cambridge: University Press 1966. Authors' address: Grad. Res. Ass. M. Zhou and Prof. W. R Schonberg, Department of Civil and Environmental Engineering, University of Alabama in Huntsville, Huntsville, AL 35899, U.S.A.
