Pltes on elstic foundtion Circulr elstic plte, xil-symmetric lod, Winkler soil (fter Timoshenko & Woinowsky-Krieger (1959) - Chpter 8) Prepred by Enzo Mrtinelli Drft version ( April 016)
Introduction Winkler model: modulus of subgrde 0 σ s = k0w N/mm 0.071 kg f /cm.76 0.007.15 0.05 5.5 0.0679 6.9 0.16 1.8
Introduction Circulr elstic (thin) pltes The Kirchhoff Love theory of pltes is two-dimensionl mthemticl model tht is used to determine the stresses nd deformtions in thin pltes subjected to forces nd moments. The following kinemtic ssumptions tht re mde in this theory: stright lines norml to the mid-surfce remin stright fter deformtion stright lines norml to the mid-surfce remin norml to the mid-surfce fter deformtion the thickness of the plte does not chnge during deformtion. (fter Wikipedi) Germin-Lgrnge eqution (crtesin coordintes) w w w q x x y y D Prepred by Enzo Mrtinelli Drft version ( April 016)
Introduction Circulr elstic (thin) pltes Germin-Lgrnge eqution (crtesin coordintes) q ww D x y Conversion to polr (cylindricl) coordintes y r x r cos y r sin x w w 1 w cos sin x r r w w 1 w sin cos y r r 1 1 r r r r Prepred by Enzo Mrtinelli Drft version ( April 016)
Introduction Circulr elstic (thin) pltes under xil-symmetric lod Germin-Lgrnge eqution (polr coordintes) q w w D 1 1 r r r r 1 1 w 1 w 1 w q w r r r r r r r r D d 1 d d w 1 dw q dr r dr dr r dr D Prepred by Enzo Mrtinelli Drft version ( April 016)
Introduction Circulr elstic (thin) pltes on Winkler soil under xil-symmetric lod d 1 d d w 1 dw q k0w dr r dr dr r dr D q k w d w d w 1 d w 1 dw 0 dr r dr r dr r dr D Since n uniformly distributed lod q pplied on n elstic pltes (or elstic bems) on Winkler soils with uniform k 0 does not determine ny stress within the plte, the sole effect of point lod P pplied in the centre of the plte is considered herefter. dy q dx k 0 w q=0 P R Prepred by Enzo Mrtinelli Drft version ( April 016)
Conversion to dimensionless form - Chrcteristic length: k0 1 D l [F ][L] 1 [F ][L] [L] dr r dr dr r dr l d 1 d d w 1 dw w 0 - Dimensionless quntities: w Dimensionless displcement z l r Dimensionless rdius x 1 d 1 d d z 1 dz z l 0 l dx x dx dx x dx l Dimensionless derivtives: dw l dz dz d 1 d d z 1 dz z 0 dr l dx dx dx x dx dx x dx d w d dw d dz 1 d z dr dr dr l dx dx l dx i i d w d dw 1 d w dr dr dr l dx i i1 i d z d z 1 d z 1 dz dx x dx x dx x dx z 0 Prepred by Enzo Mrtinelli Drft version ( April 016)
Generl solution The proposed mthemticl trnsformtions led to the following homogeneous th -order liner differentil eqution with vrible coefficients: d z d z 1 d z 1 dz dx x dx x dx x dx z 0 Therefore, in principle the generl solution of this differentil eqution might be written s follows: z A X x A X x A X x A X x 1 1 where X i re four independent solutions of the differentil eqution under considertion nd A i re four integrtion constnts depending on the ctul boundry conditions. Note: d 1 d 1 dx x dx dx x dx d Xi dxi i X 0 i 1.. Prepred by Enzo Mrtinelli Drft version ( April 016)
Approximtion by power series Since X i re unknown (s the differentil eqution under considertion includes vrible coefficients), n pproximtion bsed on power series my be firstly ssumed. Hence, the generl expression of Xi cn be tken s follows: NT n 1 n NT Xi nx 0 1x x... nx... N x T n0 where N T is the highest monomil in the series. Since X i should be solution of the generl differentil eqution, the nd order lplcin of ech term n x n should find similr monomil such tht: X i X i 0 x d 1 d d x 1 d x dx x dx dx x dx n n n n n n n n x x 0 n n n n n n d nx 1 dnx n n 1 x n x n x dx x dx dx n n n n n n d 1 d n n n n n n x n n n n x n n n x n n n x x dx n n n n Recursive definition of n
Approximtion by power series: first independed solution X 1 n n n n z A X x A X x A X x A X x 1 1 Bsed on the recursive definition of the n coefficients, first independent solution X 1 cn be built up by ssuming 0 s the first nonzero coefficient: n 0 : n : n 8 : 0 1 1 0 1 8 8 8 6 8 1 0 n 1 : 1 1 1 1 10 8 6 8 1 1 1 8 1 1 X1 x 1 x x x... 6 1756 16600 Prepred by Enzo Mrtinelli Drft version ( April 016)
Approximtion by power series: second independed solution X n n n n Before proceeding with constructing the second independent solution X of z, it should be noted tht the ssumption of 1 =1 is not dmissible. In fct, if one ssumes 1 =1 (nd, t the sme time, 0 = = =0) the resulting solution X of z would hve first term x: X x... which implies nonzero first derivtive of X (nd, hence, of the generl solution z) which cnnot be ccepted becuse of the fct tht rottions (j=dx /dx=1+ ) should be zero t x=0. Therefore, the second solution X should be built by ssuming =1 nd, t the sme time, 0 = 1 = =0: X x... this leds to n expression of X whose first nonzero monomil is of nd order nd, hence, the second derivtive of X is non zero (c=d X /dx =1+ ) ): this is cceptble becuse the curvture is nonzero t the plte centre (x=0). Prepred by Enzo Mrtinelli Drft version ( April 016)
Approximtion by power series: second independed solution X n n n n Bsed on the recursive definition of the n coefficients, first independent solution X cn be built up by ssuming s the first nonzero coefficient: n : n 6 : n 10 : 1 6 6 6 6 1 10 10 10 8 6 6 10 0 1 0 n 1 : 1 1 1 1 1 10 8 6 10 1 1 6 1 10 1 1 X x x x x x... 576 68600 10095600 Prepred by Enzo Mrtinelli Drft version ( April 016)
Approximtion by power series: third independed solution X n n n n The ssumption =1 (nd, t the sme time, 0 = 1 = =0) is not even dmissible, s it would led to the function hving the following expression: X x... This is not dmissible becuse it would led to finite vlue of sher for x=0: d d X 1 dx V... dx dx x dx wheres sher should be divergent (it should tend to infinity) for x0, s finite force P should be in equilibrium with sher stresses on n infinitesiml circumference of rdius e. P x=e V Prepred by Enzo Mrtinelli Drft version ( April 016)
Approximtion by power series: third independed solution X Since, on the one hnd, sher V for x0, nd on the other one hnd, both X 1 nd X would led to finite vlue of sher for x=0, the solutions X nd X should be cpble to reproduce the condition V for x0. Therefore, different solution, including the term log x (which diverges for x0 long with ll its derivtives) d n new unknown function F, is preliminrily defined: X X1 log x F The function X should be, in turn, solution of the generl differentil eqution: X X 0 i First of ll, the nd lplcin of X is determined i d 1 d 1 dx x dx dx x dx d Xi dxi i X 0 d X1 1 X log x X F x dx Prepred by Enzo Mrtinelli Drft version ( April 016)
Approximtion by power series: third independed solution X Then: The following condition hs to be met: X X 0 d X1 1 1 X X log x X F X log x F 0 x dx nd, fter collecting the term log x, d X1 x dx the following reltionship is derived for F : log x X X F F 0 1 1 d X1 F F x dx 6 7 8 10 111 6 1756 16600 8 F F x x... Prepred by Enzo Mrtinelli Drft version ( April 016)
Approximtion by power series: third independed solution X 6 7 8 10 111 8 F F x x... 8 6 1 10 8 6 n (n 1) (n ) F F ( 1) x The function F my lso be pproximted by the following power series: F b x b x b x... 8 1 8 1 whose nd lplcin cn be expressed s follows: n T n 1 n n (n ).. n b x n n b x n n n n n (n 1) (n ) n n b x b x ( 1) x n (n ).. n 1 n n n n n n n b 1 1 n (n 1) (n ) n n b ( 1) n n n (n ).. Prepred by Enzo Mrtinelli Drft version ( April 016)
Approximtion by power series: third independed solution X n b 1 1 n (n 1) (n ) n n b ( 1) n n n (n ).. Prepred by Enzo Mrtinelli Drft version ( April 016)
Approximtion by power series: fourth independed solution X A similr procedure leds to defining the fourth independent solution: X X log x F nd, similr considertions, cn be done in defining F for X is solution of the generl differentil eqution. F c x c x c x... 6 10 1 6 10 1 Prepred by Enzo Mrtinelli Drft version ( April 016)
Approximtion by power series: fourth independed solution X Prepred by Enzo Mrtinelli Drft version ( April 016)
Boundry conditions The generl solution, whose expression ws defined by th pproximte solutions, is reported below: z A X x A X x A X x A X x 1 1 Four boundry conditions should be written for determining the four constnts A 1 A. The first two ones re imposed for r=r (nmely, for x=r/l): Dimensionl expression Dimensionless expression P R r R x R l dr d w dw r dr rr 0 dx d z dz x dx R x l 0 d d w 1 dw dr dr r dr rr 0 d d z 1 dz dx dx x dx R x l 0 Prepred by Enzo Mrtinelli Drft version ( April 016)
Boundry conditions The generl solution, whose expression ws defined by th pproximte solutions, is reported below: r 0 dw dr rr z A X x A X x A X x A X x 0 1 1 Other two boundry conditions re imposed for r=0 (nmely, for x=0): Dimensionl expression Dimensionless expression x R l dz dx x0 0 P R P el V d d w 1 dw lim e l D P 0 dr dr r dr re l e 0 d d z 1 dz lim e k0l P 0 dx dx x dx xe e 0 Prepred by Enzo Mrtinelli Drft version ( April 016)
Boundry conditions: expressions of constnts First of ll, let s consider the third boundry condition: Since X 1 nd X re polynomils with even order power terms (i.e. x 0, x, x 8 nd x, x 6, x 10, respectively), their first derivtive is zero for x=0. Moreover, the first derivtive of X is lso zero for x=0 (or, better, in the limit of x 0): dx d x lim X log x F lim x log x...... 0 dx dx x e0 e0 x0 xe xe Therefore, the first derivtive of z in x=0 only tkes nonzero contribution from X : dz dx d 1 A A lim X log x F A lim... 0 dx dx dx x 1 e0 e0 x0 x0 xe xe which is only stisfied for A =0. dz dx x0 0 Prepred by Enzo Mrtinelli Drft version ( April 016)
Boundry conditions: expressions of constnts Then, the fourth condition cn be considered: nd, once gin, no contribution comes from X 1 nd X, but the only nonzero terms is given by the third derivtives of X : d d z 1 dz d d X 1 dx d d 1 d A A X log x F dx dx x dx dx dx x dx dx dx x dx xe d d z 1 dz lim e k0l P 0 dx dx x dx xe e 0 d d 1 d d d 1 d A X log x F A x log x... dx dx x dx dx dx x dx Therefore: A xe xe A x x e e x xe xe lim e k0l A P 0 e e 0 P 8 k0l A P 0 A 8 k l 0 Prepred by Enzo Mrtinelli Drft version ( April 016)
Boundry conditions: expressions of constnts Finlly, the first two equtions cn be considered for determining the two constnts A 1 nd A : dx d z dz x dx R x l 0 d d z 1 dz dx dx x dx R x l 0 z x A X x A X x A X x 1 1 w r r l z l d w Mr r D dr r dr dw M r D r dr dr 1 dw d w d d w 1 dw Vr r D dr dr r dr Prepred by Enzo Mrtinelli Drft version ( April 016)
Comprisons: proposed procedure vs FEM solution N T = N T =1 0.06 0.06 0.0 0.0 0.0 0.0 1000 000 000 000 5000 1000 000 000 000 5000-0.0-0.0-0.0-0.0 Prepred by Enzo Mrtinelli Drft version ( April 016)