- Email: [email protected]
New integral LEM formulae applied to the nonlinear bar
MECHANICS RESEARCH COMMUNICATIONS
Mechanics Research Communications 31 (2004) 161–168 www.elsevier.com/locate/mechrescom
New integral LEM formulae applied to the nonlinear bar P.P. Teodorescu
a,*
, Ileana Toma
b
a University of Bucharest, Academiei 14, 023984 Bucharest, Romania Department of Mathematics, Technical University of Constructions of Bucharest, Bd. Lacul Tei 124, 72302 Bucharest, Romania
b
Received 7 March 2003
Abstract The linear equivalence method (LEM), introduced by [Bull. Math. Soc. Sci. Math. de la Roumanie 24 (72) (1980) 4417; An. Univ. Bucuresßti, ser. Matematica 31 (1982) 75] to get solutions of nonlinear ODEs, was used so far to get differential type representations. New LEM representations of integral type are presented here and used for the study of the nonlinear elastic bar; a good approximating formula for the rotation of the cross-section at the bar end is also obtained, in case of a simply supported bar. A parallel old–new results is made by means of a programming code. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Linear equivalence method; Nonlinear elastic bar
1. Introduction The linear equivalence method (LEM) was introduced by Toma (1980, 1982) to get a local inverse under matriceal form of a nonlinear differential operator. Thus, LEM may be used no matter the boundary conditions; moreover, in its frame one can use the common techniques of linear ODEs and systems. Teodorescu and Toma (1984, 1986) had the idea of applying LEM to get the solutions of the nonlinear Bernoulli–Euler (B–E) equation under arbitrary Cauchy data; this offered the possibility of solving in a common mathematical frame various nonlinear problems, mathematically distinct: the cantilever bar––a Cauchy problem, the simply supported bar––a two-point problem, the hyperstatic bar––a nonlinear problem depending on a parameter. For all these cases there were obtained normal LEM formulae by using recurrrences of differential type. In this paper, there are obtained normal LEM representations of a new kind, involving only integrals. As LEM is more efficient from the computing point of view when applied to polynomial operators, we associated to the B–E equation a polynomial equivalent. The second section deals with this. In Section 3, the general LEM method is applied to the polynomial system. The LEM matrices are written and Theorem 1 specifies the integral LEM representation of exponential type. Due to the special form of the LEM matrices, this representation is precisely the normal LEM representation. There are also
*
Corresponding author. Tel.: +40-213273125. E-mail address: [email protected] (P.P. Teodorescu).
0093-6413/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0093-6413(03)00087-9
162
P.P. Teodorescu, I. Toma / Mechanics Research Communications 31 (2004) 161–168
obtained good approximating formulae for the shortening and also for the rotation of the cross-section at the bar end, in case of a simply supported bar. Section 4 is devoted to a numerical comparison between the two types of LEM formulae.
2. Statement of the problem Consider an elastic, isotropic bar, of variable rigidity, in case of an arbitrary diagram of the bending moment. This problem was widely studied, e.g. by Barten (1945), Rhode (1953), Nikomarov (1965) and Lau (1981). To this physical problem, according to Teodorescu (1984) and Teodorescu and Ille (1979), one can associate the nonlinear B–E equation YðyÞ y 00 f ðxÞð1 þ y 02 Þ
3=2
;
f ðxÞ ¼
1 MðxÞ ¼ ; qðxÞ EðxÞIðxÞ
ð1Þ
where the unknown function y represents the displacement of the bar axis. In (1), q is the curvature, of opposite sign to the bending moment MðxÞ; the rigidity EðxÞIðxÞ may vary according to the material properties and with the geometry of the cross-section. All these problems, already difficult by their nonlinearity, are all the more complicated by the appearance of the shortening u in the data; as it is known, the shortening depends on the rotation y 0 of the crosssection, which is the derivative of the unknown function y. The LEM representations were deduced from the polynomial differential system equivalent to (1) 2 3 dw fzw2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 6 dx 7 Pðw; zÞ 4 ; w ¼ 1 þ y 02 ; z ¼ y 0 ð2Þ 5¼ dz 0 fw3 dx by using for the coefficients differential recurrence relationships. As mentioned above, we use here a general theorem deduced by Toma (1988, 1995) that emphasizes a new type of LEM representation, based on integrals, valid on large intervals; this allows pointwise and global estimations of the solution and leads to a new approximating formula for y 0 ð0Þ in the case of the simply supported bar. Also, we could get an accurate approximating formula for the shortening.
3. The integral LEM representation The main idea of Toma (1980) in introducing LEM consists of an exponential transform that associates a nonlinear polynomial differential system with two equivalents: a linear PDE and a linear, while infinite, ODSystem. Applying to these equivalents the well-known standard methods working for linear operators, Toma (1982, 1988) deduced a local inverse of a nonlinear differential operator in matrix form and also a normal LEM representation of the functions belonging to its kernel, suitable to a qualitative study. In the case discussed here, we took the exponential transform vðt; r; nÞ ¼ erwþnz ; which, applied to the polynomial system (2), led us to the first LEM equivalent ov o3 v o3 v Lv f ðxÞ r 2 þ n 3 ¼ 0: ox or on or
ð3Þ
ð4Þ
P.P. Teodorescu, I. Toma / Mechanics Research Communications 31 (2004) 161–168
P1
163
j
ri
Introducing the expansion vðt; r; nÞ ¼ 1 þ iþj¼1 vij ðxÞ i! nj! , we get the second LEM equivalent v0ij ðxÞ ¼ f ðxÞ iviþ1;jþ1 þ jviþ3;j1 ; ði þ jÞ 2 N
ð5Þ
or, in vector form, SV
dV f ðxÞAV ¼ 0; dx
The associated LEM matrix can 2 0 B1 0 0 6 0 0 B 0 3 6 6 0 0 0 B5 A¼6 6... ... ... ... 6 4 0 0 0 0 ... ... ... ... 2
B2j1
0 6 1 6 6 0 6 ¼6 6... 6 0 6 4 0 0
2j 1 0 2 ... 0 0 0
VðxÞ ¼ ½V2m1 ðxÞm2N ; V2m1 ðxÞ ¼ ½v2m1i;i ðxÞi¼0;2m1 :
ð6Þ
be written in the form ... ... ... ... ... ...
0 2j 2 0 ... 0 0 0
0 0 0 ... 0 ... ... ... ... ... ... ... ...
0 0 0 ... B2j1 ... 0 0 0 ... 0 2j 2 0
0 0 0 ... 0 ...
3 ... ...7 7 ...7 7; ...7 7 ...5 ...
0 0 0 ... 2 0 2j 1
3 0 0 7 7 0 7 7 ...7 7: 0 7 7 1 5 0
Remark. In fact, the direct form of A, deduced from (4), is 2 3 0 0 B1 0 0 ... 6 0 0 0 B2 0 ...7 6 7: 4 0 0 0 0 B3 . . . 5 ... ... ... ... ... ...
ð7Þ
ð8Þ
ð9Þ
But, as B2j 0, j 2 N , the polynomial operator in (2) being odd, one can ‘‘contract’’ A to the form (7). This obviously holds for any odd polynomial operator. If we associate to the B–E equation (1) the arbitrary Cauchy conditions yðx0 Þ ¼ a;
y 0 ðx0 Þ ¼ b;
ð10Þ
they become, for the system (2), pffiffiffiffiffiffiffiffiffiffiffiffiffi wðx0 Þ ¼ 1 þ k2 ; zðx0 Þ ¼ k
ð11Þ
and (6) should be solved under the corresponding for the infinite vector V conditions h i 2m1i Vðx0 Þ ¼ ½V2m1 ðx0 Þm2N ; V2m1 ðx0 Þ ¼ ð1 þ k2 Þ 2 ki : i¼0;2m1
ð12Þ
Applying to this case the general LEM theorem established by Toma (1995) we obtain Theorem 1. The solution of the Cauchy problem (2), (11) is formally represented as the first two components of the vector
164
P.P. Teodorescu, I. Toma / Mechanics Research Communications 31 (2004) 161–168
VðxÞ ¼ eAuðxÞ Vðx0 Þ;
uðxÞ ¼
Z
x
f ðtÞ dt:
ð13Þ
x0
Using this theorem and also taking (7), (8), (12) and (13) into account, we get the (approximating) integral LEM representation for the solution of the Cauchy problems (1) and (10) pffiffiffiffiffiffiffiffiffiffiffiffiffi x3 ðxÞ x5 ðxÞ 2 2 2 2 2 3 2 3 2 4 y ðxÞ ffi 1 þ k xðxÞ þ ð3 þ 2 :3l Þ þ 3 :5 þ 2 :3 :5l þ 2 :3 :5l 3! 5! x7 ðxÞ 2 2 8! 6 3 3 2 2 4 3 2 4 3 :5 :7 þ 2 :3 :5 :7l þ 2 :3 :5 :7l þ l þ 7! 2 x9 ðxÞ 4 2 2 10! 8 3 4 3 2 2 5 4 3 2 4 7 4 2 2 6 3 :5 :7 þ 2 :3 :5 :7 l þ 2 :3 :5 :7 l þ 2 :3 :5 :7 l þ l þ 9! 2 " 3x2 ðxÞ x4 ðxÞ 2 x6 ðxÞ 2 2 7! 4 2 2 2 2 2 2 3 :5 þ 2 :3:5l þ 3 :5 :7 þ 2 :3 :5 :7l þ l þk 1þ þ 2 4! 6! 2 x8 ðxÞ 4 2 2 9! 3 :5 :7 þ 23 :34 :52 :7l2 þ 24 :35 :5:72 l4 þ l6 : þ ð14Þ 8! 2 0
In (14) we used the notations xðxÞ ¼ ð1 þ k2 ÞuðxÞ;
k l ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi : 1 þ k2
ð15Þ
The representation (14) is true for all Cauchy problems, no matter the point x0 . An integral form of the solution of (1) and (10) is deduced by straightforward integration of (14) Z x yðxÞ ¼ a þ y 0 ðtÞ dt: ð16Þ x0
Yet in what follows, we shall only use the particular case x0 ¼ 0. So, from now on, we consider x0 ¼ 0 in (16) and in (14) and (15), the function u being given by Z x uðxÞ ¼ f ðtÞ dt: ð17Þ 0
4. Applications We shall compare the results we obtained in Teodorescu and Toma (1982, 1984) by using convergent algorithms based on iterative LEM differential formulae with those obtained by the present integral LEM formulae, straightforwardly applied. In this paper, we chose to compare only the results concerning the cantilever and the simply supported bars, in the case of a constant bending moment. Teodorescu and Toma (1982, 1984, 1986) considered elastic steel bars of rectangular cross-section, constant or of heights h or widths b varying linearly along the bar axis, and of constant bending moment M ¼ 40 kN cm. We took l ¼ 100 cm and E ¼ 21000 kN/cm2 and considered the following three cases: (a) a bar with a constant height, its width varying linearly along the bar axis, so that b0 ¼ 9 cm, b1 ¼ bc0 , c ¼ 1:5;
P.P. Teodorescu, I. Toma / Mechanics Research Communications 31 (2004) 161–168
165
Table 1 Case a f ðxÞ
b
1 , a ¼ 1:3125 að300 xÞ
c 1 a3 ð300 xÞ3
, a ¼ 0:032014 cm2=3
a, a ¼ 0:0038095 cm1
(b) a bar with constant width b ¼ 6 cm and with a linearly varying height along the axis, so that h0 ¼ 1:5 cm, h1 ¼ hc0 , c ¼ 1:5; (c) a bar with constant cross-section, of dimensions h ¼ 9 cm, b ¼ 1 cm. The expressions of the corresponding formulae for f were calculated by Teodorescu and Toma (1984) and are given in Table 1. In all the above cases we considered the following approximating formula for the shortening, valid for small and moderate deformations Z 1 l 02 uðxÞ ffi y ðxÞ dx: ð18Þ 2 0 4.1. The cantilever bar With the representation (14), written for k ¼ 0, we get for the shortening u the approximating formula: Z 1 l 2 uðxÞ ffi u ðxÞ½1 þ u2 ðxÞ dx; ð19Þ 2 0 with u defined in (17). For the up mentioned cases, the second term in the integrand is of about 105 . The maximum bar deflection was computed by means of formula (16), taken at l u. The comparison with our previous results from Teodorescu and Toma (1984) is given in Table 2. On the first row, for each magnitude, we inscribed the previous results, marked by LEM differential. The present results are marked by LEM integral. 4.2. The simply supported bar In this case, a serious difficulty is to get y 0 ð0Þ. Teodorescu and Toma (1984) found an approximating formula based on LEM by using derivatives. Here, by means of the representations (14) and (16), imposing the condition yðl uÞ ¼ 0, we get the approximating formula Rl 35 u þ 12 u3 þ 38 u5 þ 165 u7 þ 256 u9 dx 0 0 y ð0Þ ffi R l ð20Þ ; 1 þ 32 u2 þ 158 u4 þ 35 u6 þ 315 u8 dx 0 16 256 which can be immediately extended to the closed form Rl 1=2 uð1 u2 Þ dx 0 y ð0Þ ffi R0 l : 3=2 2 ð1 u Þ dx 0
ð21Þ
The following approximating formula still works for small or moderate deformations y 0 ð0Þ ffi
4pðl=2Þ þ pðlÞ ; 1 þ 4qðl=2Þ þ qðlÞ
ð22Þ
166
P.P. Teodorescu, I. Toma / Mechanics Research Communications 31 (2004) 161–168
Table 2 Case a
b
c
d
u, cm
1.65 1.43
0.61 0.54
2.87 2.40
LEM differential LEM integral
f , cm
14.21 14.28
8.39 8.43
18.63 18.82
LEM differential LEM integral
0 ymax
0.316863 0.309663
0.211890 0.211635
0.397949 0.371731
LEM differential LEM integral
hmax
17°340 5300 17°120 1400
11°570 4800 11°570 0600
21°420 20°230 2900
LEM differential LEM integral
where p and q are given by 3 15 pðxÞ ¼ uðxÞ þ u3 ðxÞ þ u5 ðxÞ; 2 8 1 2 3 4 qðxÞ ¼ 1 þ u ðxÞ þ u ðxÞ; 2 8
ð23Þ
with u defined in (17). For the shortening u we deduce the formula Z h i 1 l uffi ðuðxÞ þ lÞ2 1 þ ðuðxÞ þ lÞ2 dx; 2 0
ð24Þ
where u and l are given by (15) and (16). Table 3 gives a comparison between the values obtained in Teodorescu and Toma (1984) for the simply supported bar, using the same markers as in Table 2. We see that formula (21) immediately yields to the following estimation Table 3 Case a
b
c
d
y 0 ð0Þ
0.145109 0.140953
0.084690 0.084087
0.192953 0.183012
LEM differential LEM integral
h0
8°150 2300 8°10 2300
4°500 2700 4°480 2300
10°550 1600 10°220 1600
LEM differential LEM integral
u, cm
0.28 0.40
0.08 0.18
0.62 0.62
LEM differential LEM integral
y 0 ðl uÞ
)0.166191 )0.171229
)0.126134 )0.128212
)0.193720 )0.202784
LEM differential LEM integral
hðl uÞ
)9°260 0900 )9°420 5300
)7°110 2000 )7°180 2200
)10°570 4300 )11°270 0500
LEM differential LEM integral
x0 , cm
51.53 50.22
54.92 54.62
49.73 47.26
LEM differential LEM integral
f , cm
3.84 3.63
2.56 2.50
4.75 4.27
LEM differential LEM integral
P.P. Teodorescu, I. Toma / Mechanics Research Communications 31 (2004) 161–168
kuk jy 0 ð0Þj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 1 kuk
167
ð25Þ
where by k k we mean the ‘‘sup’’ norm on C 0 ð½0; lÞ kuk sup juðxÞj:
ð26Þ
x2½0;l
It should be mentioned that for all three cases the ratio in the right member of (25) has the same value kuk qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 0:412: 2 1 kuk
ð27Þ
5. Conclusions The paper makes a parallel between the differential type LEM formulae and the new integral LEM representations, obtained for a B–E bar subjected to arbitrary bending moments and of variable rigidity. The main advantages of the integral LEM representations are that they are valid on larger intervals and that they drop the restrictions of regularity for some involved physical magnitudes, continuity being sufficient. The numerical comparison points out a perfect concordance with the previous results, based on LEM representations of differential type. Using integral LEM formulae, there were obtained new approximations for the shortening, in case of a cantilever bar, and of y 0 ð0Þ in case of a simply supported bar were; in this last case, a bound for y 0 ð0Þ also. It should be mentioned that the comparison between the new and old LEM results pointed out a coincidence up to the sixth decimal place. Another conclusion concerns the polynomial system (2), that enabled an efficient application of LEM. In fact, instead of (2) we could write 2 3 dw 2 6 du zw 7 0 6 7 Pðw; zÞ 4 ¼ ; ð28Þ 5 dz 0 w3 du with u given by (13). The system (28) does not effectively depend on the physical data and it can be considered as a ‘‘mathematical core, governing from the abstract’’, the mechanical phenomenon. Actually, Theorem 1 can be extended to far more general cases than the one we studied here; thus, this expansion of LEM could be a useful instrument for the study of other nonlinear problems. Actually, we intend to apply it firstly to a hyperstatic bar, which already requires such an extension.
References Barten, H.J., 1945. On the deflection of a cantilever beam. Quart. Appl. Math. 3, 272. Lau, J.H., 1981. Large deflection of cantilever beam. J. Engng. Mech. Division, Proc. ASCE 107, 259. Nikomarov, M., 1965. Exact bending calculus of cantilever and simply supported beams. Azernestr, Baku. Rhode, F.V., 1953. Large deflection of a cantilever beam with uniformly distributed load. Quart. Appl. Math. 11, 337. Teodorescu, P.P., Mechanical systems, Classical models, t.1–4, Techn. Publ. House, Bucharest, 1984-88-97-2002 (in Romanian). Teodorescu, P.P., Ille, V., 1979. Theory of elasticity and introduction to mechanics of deformable solids. Dacia Publ. House, ClujNapoca (in Romanian). Teodorescu, P.P., Toma, I., 1982. On the Cauchy type problem in the non-linear bending of a straight bar. Mech. Res. Comm. 9, 151. Teodorescu, P.P., Toma, I., 1984. Two fundamental cases in the non-linear bending of a straight bar. Meccanica 19, 52.
168
P.P. Teodorescu, I. Toma / Mechanics Research Communications 31 (2004) 161–168
Teodorescu, P.P., Toma, I., 1986. On the non-linear bending of a hyperstatic bar. Int. J. Engng. Sci. 24, 1257. Toma, I., 1980. On polynomial differential equations. Bull. Math. Soc. Sci. Math. de la Roumanie 24 (72), 4,417. Toma, I., 1982. Local inversion of polynomial differential operators by linear equivalence. An. Univ. Bucuresßti, ser. Matematica 31, 75. Toma, I., 1988. Techniques of computations by linear equivalence. Bull. Math. Soc. Sci. Math. de la Roumanie 33 (81), 4,363. Toma, I., 1995. The linear equivalence method and its applications. Flores, Bucharest (in Romanian).
Mechanics Research Communications 31 (2004) 161–168 www.elsevier.com/locate/mechrescom
New integral LEM formulae applied to the nonlinear bar P.P. Teodorescu
a,*
, Ileana Toma
b
a University of Bucharest, Academiei 14, 023984 Bucharest, Romania Department of Mathematics, Technical University of Constructions of Bucharest, Bd. Lacul Tei 124, 72302 Bucharest, Romania
b
Received 7 March 2003
Abstract The linear equivalence method (LEM), introduced by [Bull. Math. Soc. Sci. Math. de la Roumanie 24 (72) (1980) 4417; An. Univ. Bucuresßti, ser. Matematica 31 (1982) 75] to get solutions of nonlinear ODEs, was used so far to get differential type representations. New LEM representations of integral type are presented here and used for the study of the nonlinear elastic bar; a good approximating formula for the rotation of the cross-section at the bar end is also obtained, in case of a simply supported bar. A parallel old–new results is made by means of a programming code. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Linear equivalence method; Nonlinear elastic bar
1. Introduction The linear equivalence method (LEM) was introduced by Toma (1980, 1982) to get a local inverse under matriceal form of a nonlinear differential operator. Thus, LEM may be used no matter the boundary conditions; moreover, in its frame one can use the common techniques of linear ODEs and systems. Teodorescu and Toma (1984, 1986) had the idea of applying LEM to get the solutions of the nonlinear Bernoulli–Euler (B–E) equation under arbitrary Cauchy data; this offered the possibility of solving in a common mathematical frame various nonlinear problems, mathematically distinct: the cantilever bar––a Cauchy problem, the simply supported bar––a two-point problem, the hyperstatic bar––a nonlinear problem depending on a parameter. For all these cases there were obtained normal LEM formulae by using recurrrences of differential type. In this paper, there are obtained normal LEM representations of a new kind, involving only integrals. As LEM is more efficient from the computing point of view when applied to polynomial operators, we associated to the B–E equation a polynomial equivalent. The second section deals with this. In Section 3, the general LEM method is applied to the polynomial system. The LEM matrices are written and Theorem 1 specifies the integral LEM representation of exponential type. Due to the special form of the LEM matrices, this representation is precisely the normal LEM representation. There are also
*
Corresponding author. Tel.: +40-213273125. E-mail address: [email protected] (P.P. Teodorescu).
0093-6413/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0093-6413(03)00087-9
162
P.P. Teodorescu, I. Toma / Mechanics Research Communications 31 (2004) 161–168
obtained good approximating formulae for the shortening and also for the rotation of the cross-section at the bar end, in case of a simply supported bar. Section 4 is devoted to a numerical comparison between the two types of LEM formulae.
2. Statement of the problem Consider an elastic, isotropic bar, of variable rigidity, in case of an arbitrary diagram of the bending moment. This problem was widely studied, e.g. by Barten (1945), Rhode (1953), Nikomarov (1965) and Lau (1981). To this physical problem, according to Teodorescu (1984) and Teodorescu and Ille (1979), one can associate the nonlinear B–E equation YðyÞ y 00 f ðxÞð1 þ y 02 Þ
3=2
;
f ðxÞ ¼
1 MðxÞ ¼ ; qðxÞ EðxÞIðxÞ
ð1Þ
where the unknown function y represents the displacement of the bar axis. In (1), q is the curvature, of opposite sign to the bending moment MðxÞ; the rigidity EðxÞIðxÞ may vary according to the material properties and with the geometry of the cross-section. All these problems, already difficult by their nonlinearity, are all the more complicated by the appearance of the shortening u in the data; as it is known, the shortening depends on the rotation y 0 of the crosssection, which is the derivative of the unknown function y. The LEM representations were deduced from the polynomial differential system equivalent to (1) 2 3 dw fzw2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 6 dx 7 Pðw; zÞ 4 ; w ¼ 1 þ y 02 ; z ¼ y 0 ð2Þ 5¼ dz 0 fw3 dx by using for the coefficients differential recurrence relationships. As mentioned above, we use here a general theorem deduced by Toma (1988, 1995) that emphasizes a new type of LEM representation, based on integrals, valid on large intervals; this allows pointwise and global estimations of the solution and leads to a new approximating formula for y 0 ð0Þ in the case of the simply supported bar. Also, we could get an accurate approximating formula for the shortening.
3. The integral LEM representation The main idea of Toma (1980) in introducing LEM consists of an exponential transform that associates a nonlinear polynomial differential system with two equivalents: a linear PDE and a linear, while infinite, ODSystem. Applying to these equivalents the well-known standard methods working for linear operators, Toma (1982, 1988) deduced a local inverse of a nonlinear differential operator in matrix form and also a normal LEM representation of the functions belonging to its kernel, suitable to a qualitative study. In the case discussed here, we took the exponential transform vðt; r; nÞ ¼ erwþnz ; which, applied to the polynomial system (2), led us to the first LEM equivalent ov o3 v o3 v Lv f ðxÞ r 2 þ n 3 ¼ 0: ox or on or
ð3Þ
ð4Þ
P.P. Teodorescu, I. Toma / Mechanics Research Communications 31 (2004) 161–168
P1
163
j
ri
Introducing the expansion vðt; r; nÞ ¼ 1 þ iþj¼1 vij ðxÞ i! nj! , we get the second LEM equivalent v0ij ðxÞ ¼ f ðxÞ iviþ1;jþ1 þ jviþ3;j1 ; ði þ jÞ 2 N
ð5Þ
or, in vector form, SV
dV f ðxÞAV ¼ 0; dx
The associated LEM matrix can 2 0 B1 0 0 6 0 0 B 0 3 6 6 0 0 0 B5 A¼6 6... ... ... ... 6 4 0 0 0 0 ... ... ... ... 2
B2j1
0 6 1 6 6 0 6 ¼6 6... 6 0 6 4 0 0
2j 1 0 2 ... 0 0 0
VðxÞ ¼ ½V2m1 ðxÞm2N ; V2m1 ðxÞ ¼ ½v2m1i;i ðxÞi¼0;2m1 :
ð6Þ
be written in the form ... ... ... ... ... ...
0 2j 2 0 ... 0 0 0
0 0 0 ... 0 ... ... ... ... ... ... ... ...
0 0 0 ... B2j1 ... 0 0 0 ... 0 2j 2 0
0 0 0 ... 0 ...
3 ... ...7 7 ...7 7; ...7 7 ...5 ...
0 0 0 ... 2 0 2j 1
3 0 0 7 7 0 7 7 ...7 7: 0 7 7 1 5 0
Remark. In fact, the direct form of A, deduced from (4), is 2 3 0 0 B1 0 0 ... 6 0 0 0 B2 0 ...7 6 7: 4 0 0 0 0 B3 . . . 5 ... ... ... ... ... ...
ð7Þ
ð8Þ
ð9Þ
But, as B2j 0, j 2 N , the polynomial operator in (2) being odd, one can ‘‘contract’’ A to the form (7). This obviously holds for any odd polynomial operator. If we associate to the B–E equation (1) the arbitrary Cauchy conditions yðx0 Þ ¼ a;
y 0 ðx0 Þ ¼ b;
ð10Þ
they become, for the system (2), pffiffiffiffiffiffiffiffiffiffiffiffiffi wðx0 Þ ¼ 1 þ k2 ; zðx0 Þ ¼ k
ð11Þ
and (6) should be solved under the corresponding for the infinite vector V conditions h i 2m1i Vðx0 Þ ¼ ½V2m1 ðx0 Þm2N ; V2m1 ðx0 Þ ¼ ð1 þ k2 Þ 2 ki : i¼0;2m1
ð12Þ
Applying to this case the general LEM theorem established by Toma (1995) we obtain Theorem 1. The solution of the Cauchy problem (2), (11) is formally represented as the first two components of the vector
164
P.P. Teodorescu, I. Toma / Mechanics Research Communications 31 (2004) 161–168
VðxÞ ¼ eAuðxÞ Vðx0 Þ;
uðxÞ ¼
Z
x
f ðtÞ dt:
ð13Þ
x0
Using this theorem and also taking (7), (8), (12) and (13) into account, we get the (approximating) integral LEM representation for the solution of the Cauchy problems (1) and (10) pffiffiffiffiffiffiffiffiffiffiffiffiffi x3 ðxÞ x5 ðxÞ 2 2 2 2 2 3 2 3 2 4 y ðxÞ ffi 1 þ k xðxÞ þ ð3 þ 2 :3l Þ þ 3 :5 þ 2 :3 :5l þ 2 :3 :5l 3! 5! x7 ðxÞ 2 2 8! 6 3 3 2 2 4 3 2 4 3 :5 :7 þ 2 :3 :5 :7l þ 2 :3 :5 :7l þ l þ 7! 2 x9 ðxÞ 4 2 2 10! 8 3 4 3 2 2 5 4 3 2 4 7 4 2 2 6 3 :5 :7 þ 2 :3 :5 :7 l þ 2 :3 :5 :7 l þ 2 :3 :5 :7 l þ l þ 9! 2 " 3x2 ðxÞ x4 ðxÞ 2 x6 ðxÞ 2 2 7! 4 2 2 2 2 2 2 3 :5 þ 2 :3:5l þ 3 :5 :7 þ 2 :3 :5 :7l þ l þk 1þ þ 2 4! 6! 2 x8 ðxÞ 4 2 2 9! 3 :5 :7 þ 23 :34 :52 :7l2 þ 24 :35 :5:72 l4 þ l6 : þ ð14Þ 8! 2 0
In (14) we used the notations xðxÞ ¼ ð1 þ k2 ÞuðxÞ;
k l ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi : 1 þ k2
ð15Þ
The representation (14) is true for all Cauchy problems, no matter the point x0 . An integral form of the solution of (1) and (10) is deduced by straightforward integration of (14) Z x yðxÞ ¼ a þ y 0 ðtÞ dt: ð16Þ x0
Yet in what follows, we shall only use the particular case x0 ¼ 0. So, from now on, we consider x0 ¼ 0 in (16) and in (14) and (15), the function u being given by Z x uðxÞ ¼ f ðtÞ dt: ð17Þ 0
4. Applications We shall compare the results we obtained in Teodorescu and Toma (1982, 1984) by using convergent algorithms based on iterative LEM differential formulae with those obtained by the present integral LEM formulae, straightforwardly applied. In this paper, we chose to compare only the results concerning the cantilever and the simply supported bars, in the case of a constant bending moment. Teodorescu and Toma (1982, 1984, 1986) considered elastic steel bars of rectangular cross-section, constant or of heights h or widths b varying linearly along the bar axis, and of constant bending moment M ¼ 40 kN cm. We took l ¼ 100 cm and E ¼ 21000 kN/cm2 and considered the following three cases: (a) a bar with a constant height, its width varying linearly along the bar axis, so that b0 ¼ 9 cm, b1 ¼ bc0 , c ¼ 1:5;
P.P. Teodorescu, I. Toma / Mechanics Research Communications 31 (2004) 161–168
165
Table 1 Case a f ðxÞ
b
1 , a ¼ 1:3125 að300 xÞ
c 1 a3 ð300 xÞ3
, a ¼ 0:032014 cm2=3
a, a ¼ 0:0038095 cm1
(b) a bar with constant width b ¼ 6 cm and with a linearly varying height along the axis, so that h0 ¼ 1:5 cm, h1 ¼ hc0 , c ¼ 1:5; (c) a bar with constant cross-section, of dimensions h ¼ 9 cm, b ¼ 1 cm. The expressions of the corresponding formulae for f were calculated by Teodorescu and Toma (1984) and are given in Table 1. In all the above cases we considered the following approximating formula for the shortening, valid for small and moderate deformations Z 1 l 02 uðxÞ ffi y ðxÞ dx: ð18Þ 2 0 4.1. The cantilever bar With the representation (14), written for k ¼ 0, we get for the shortening u the approximating formula: Z 1 l 2 uðxÞ ffi u ðxÞ½1 þ u2 ðxÞ dx; ð19Þ 2 0 with u defined in (17). For the up mentioned cases, the second term in the integrand is of about 105 . The maximum bar deflection was computed by means of formula (16), taken at l u. The comparison with our previous results from Teodorescu and Toma (1984) is given in Table 2. On the first row, for each magnitude, we inscribed the previous results, marked by LEM differential. The present results are marked by LEM integral. 4.2. The simply supported bar In this case, a serious difficulty is to get y 0 ð0Þ. Teodorescu and Toma (1984) found an approximating formula based on LEM by using derivatives. Here, by means of the representations (14) and (16), imposing the condition yðl uÞ ¼ 0, we get the approximating formula Rl 35 u þ 12 u3 þ 38 u5 þ 165 u7 þ 256 u9 dx 0 0 y ð0Þ ffi R l ð20Þ ; 1 þ 32 u2 þ 158 u4 þ 35 u6 þ 315 u8 dx 0 16 256 which can be immediately extended to the closed form Rl 1=2 uð1 u2 Þ dx 0 y ð0Þ ffi R0 l : 3=2 2 ð1 u Þ dx 0
ð21Þ
The following approximating formula still works for small or moderate deformations y 0 ð0Þ ffi
4pðl=2Þ þ pðlÞ ; 1 þ 4qðl=2Þ þ qðlÞ
ð22Þ
166
P.P. Teodorescu, I. Toma / Mechanics Research Communications 31 (2004) 161–168
Table 2 Case a
b
c
d
u, cm
1.65 1.43
0.61 0.54
2.87 2.40
LEM differential LEM integral
f , cm
14.21 14.28
8.39 8.43
18.63 18.82
LEM differential LEM integral
0 ymax
0.316863 0.309663
0.211890 0.211635
0.397949 0.371731
LEM differential LEM integral
hmax
17°340 5300 17°120 1400
11°570 4800 11°570 0600
21°420 20°230 2900
LEM differential LEM integral
where p and q are given by 3 15 pðxÞ ¼ uðxÞ þ u3 ðxÞ þ u5 ðxÞ; 2 8 1 2 3 4 qðxÞ ¼ 1 þ u ðxÞ þ u ðxÞ; 2 8
ð23Þ
with u defined in (17). For the shortening u we deduce the formula Z h i 1 l uffi ðuðxÞ þ lÞ2 1 þ ðuðxÞ þ lÞ2 dx; 2 0
ð24Þ
where u and l are given by (15) and (16). Table 3 gives a comparison between the values obtained in Teodorescu and Toma (1984) for the simply supported bar, using the same markers as in Table 2. We see that formula (21) immediately yields to the following estimation Table 3 Case a
b
c
d
y 0 ð0Þ
0.145109 0.140953
0.084690 0.084087
0.192953 0.183012
LEM differential LEM integral
h0
8°150 2300 8°10 2300
4°500 2700 4°480 2300
10°550 1600 10°220 1600
LEM differential LEM integral
u, cm
0.28 0.40
0.08 0.18
0.62 0.62
LEM differential LEM integral
y 0 ðl uÞ
)0.166191 )0.171229
)0.126134 )0.128212
)0.193720 )0.202784
LEM differential LEM integral
hðl uÞ
)9°260 0900 )9°420 5300
)7°110 2000 )7°180 2200
)10°570 4300 )11°270 0500
LEM differential LEM integral
x0 , cm
51.53 50.22
54.92 54.62
49.73 47.26
LEM differential LEM integral
f , cm
3.84 3.63
2.56 2.50
4.75 4.27
LEM differential LEM integral
P.P. Teodorescu, I. Toma / Mechanics Research Communications 31 (2004) 161–168
kuk jy 0 ð0Þj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 1 kuk
167
ð25Þ
where by k k we mean the ‘‘sup’’ norm on C 0 ð½0; lÞ kuk sup juðxÞj:
ð26Þ
x2½0;l
It should be mentioned that for all three cases the ratio in the right member of (25) has the same value kuk qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 0:412: 2 1 kuk
ð27Þ
5. Conclusions The paper makes a parallel between the differential type LEM formulae and the new integral LEM representations, obtained for a B–E bar subjected to arbitrary bending moments and of variable rigidity. The main advantages of the integral LEM representations are that they are valid on larger intervals and that they drop the restrictions of regularity for some involved physical magnitudes, continuity being sufficient. The numerical comparison points out a perfect concordance with the previous results, based on LEM representations of differential type. Using integral LEM formulae, there were obtained new approximations for the shortening, in case of a cantilever bar, and of y 0 ð0Þ in case of a simply supported bar were; in this last case, a bound for y 0 ð0Þ also. It should be mentioned that the comparison between the new and old LEM results pointed out a coincidence up to the sixth decimal place. Another conclusion concerns the polynomial system (2), that enabled an efficient application of LEM. In fact, instead of (2) we could write 2 3 dw 2 6 du zw 7 0 6 7 Pðw; zÞ 4 ¼ ; ð28Þ 5 dz 0 w3 du with u given by (13). The system (28) does not effectively depend on the physical data and it can be considered as a ‘‘mathematical core, governing from the abstract’’, the mechanical phenomenon. Actually, Theorem 1 can be extended to far more general cases than the one we studied here; thus, this expansion of LEM could be a useful instrument for the study of other nonlinear problems. Actually, we intend to apply it firstly to a hyperstatic bar, which already requires such an extension.
References Barten, H.J., 1945. On the deflection of a cantilever beam. Quart. Appl. Math. 3, 272. Lau, J.H., 1981. Large deflection of cantilever beam. J. Engng. Mech. Division, Proc. ASCE 107, 259. Nikomarov, M., 1965. Exact bending calculus of cantilever and simply supported beams. Azernestr, Baku. Rhode, F.V., 1953. Large deflection of a cantilever beam with uniformly distributed load. Quart. Appl. Math. 11, 337. Teodorescu, P.P., Mechanical systems, Classical models, t.1–4, Techn. Publ. House, Bucharest, 1984-88-97-2002 (in Romanian). Teodorescu, P.P., Ille, V., 1979. Theory of elasticity and introduction to mechanics of deformable solids. Dacia Publ. House, ClujNapoca (in Romanian). Teodorescu, P.P., Toma, I., 1982. On the Cauchy type problem in the non-linear bending of a straight bar. Mech. Res. Comm. 9, 151. Teodorescu, P.P., Toma, I., 1984. Two fundamental cases in the non-linear bending of a straight bar. Meccanica 19, 52.
168
P.P. Teodorescu, I. Toma / Mechanics Research Communications 31 (2004) 161–168
Teodorescu, P.P., Toma, I., 1986. On the non-linear bending of a hyperstatic bar. Int. J. Engng. Sci. 24, 1257. Toma, I., 1980. On polynomial differential equations. Bull. Math. Soc. Sci. Math. de la Roumanie 24 (72), 4,417. Toma, I., 1982. Local inversion of polynomial differential operators by linear equivalence. An. Univ. Bucuresßti, ser. Matematica 31, 75. Toma, I., 1988. Techniques of computations by linear equivalence. Bull. Math. Soc. Sci. Math. de la Roumanie 33 (81), 4,363. Toma, I., 1995. The linear equivalence method and its applications. Flores, Bucharest (in Romanian).