By Peng Wei, Sun Bingnan, Tang Jinchun

According to analytical equations, a catenary aspect is gifted for thefinite point research of cable buildings. in comparison with frequently used point (3-node point, 5-node element), a software with the proposed point is of lesscomputer time and higher accuracy.

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Proof Let un ! u in L1(0 L) and supn En(un ) < +1. Then we have sup n ZL 0 (junj + ju00njp ) dt < +1: By interpolation, we deduce that supn kunkW 2 p (0 L) < +1 hence un * u weakly in u 2 W 2 p (0 L). In particular u00n * u00 in Lp (0 L), so that F (u) = Z (0 l) f(u00 ) dt limninf Z (0 l) f(u00n ) dt = limninf En(un ): If u 2 C 2( 0 L]) then, upon choosing (un )i = u(xni) we have un ! u and En(un) = Z (0 l) f(u00 + o(1)) dt so that limn En(un ) = F(u). For a general u 2 W 2 p (0 L) it su ces to use an approximation argument.

57) Lennard Jones potentials 51 Proof Again, note preliminarily that F will be nite only on increasing functions so that we need to identify it only on functions u satisfying u(t+) > u(t;) on S(u). The liminfinequality will be obtained by comparison. Let supn En (un) < +1 and un ! u in L1(0 L). Let vn(xni ) = un (xni) ; Mxni: Note that vn ! v = u ; Mx, and that n X vn (xni) ; vn (xni;1) E (u ) = E~ (v ) = n n where n n n(z) = i=1 n n n 1 (J(z + M) ; minJ): n Note that n ! +1 n minJ : n With xed k 2 N let E~nK be de ned by n n vn(xni) ; vn(xni;1) X E~nK (w) = n min k n i=1 2 1 (min J ; 1 )o: k n By the results of the previous chapter E~nK ;-converge to F K de ned by 8 Z

Now, consider the sequence (wn)n de ned by Z 8 > v (a) + v_ n (t) dt if x < x0 n > < (a x) Z wn(x) = > X v (a) + v _ (t) dt + vn](t) if x x0. 15) n n n n n 0 n 0 t2S (vn ) vn](t). e. Indeed, since x0 is the limit point of the sets S(vn ), for any > 0 xed we can nd n0 ( ) 2 N such that for any n n0( ) and for any i 2 In jx0 ; xin+1j < . Hence, by construction, for any n n0( ) and for any x 2 (a b) n x0 ; x0 + ], wn (x) = vn (x), that is, the two sequences (vn ) and (wn) have the same pointwise limit.

### A catenary element for the analysis of cable structures by Peng Wei, Sun Bingnan, Tang Jinchun

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