6.5 Plate Problems in Rectangular Coordinates

Transcription

1 6.5 lte rolems in Rectngulr Coordintes In this section numer of importnt plte prolems ill e emined ug Crte coordintes Uniform ressure producing Bending in One irection Consider first the cse of plte hich ends in one direction onl. rom 6.. the deflection nd moments re The differentil eution 6..9 reds d d f ( ) ( ) ( ) (6.5.) d d d ( ) (6.5.) d The corresponding eution for em is d / d p( ) / EI. If p( ) / ( ) ith the depth of the em ith I h / the plte ill respond more stiffl thn the em fctor of /( ) fctor of out % for. ce Eh EI (6.5.) The etr stiffness is due to the constrining effect of em. hich is not present in the 6.5. eflection of Circulr lte Uniform Lterl Lod A solution for circulr plte prolem is presented net. This prolem ill e emined gin in the section hich follos ug the more nturl polr coordintes. Consider circulr plte ith oundr (6.5.) clmped t its edges nd sujected to uniform lterl lod ig Solid echnics rt II 5 Kell

2 Solid echnics rt II Kell 5 igure 6.5.: clmped circulr plte sujected to uniform lterl lod The differentil eution for the prolem is given The oundr conditions re tht the slope nd deflection re zero t the oundr: long (6.5.5) It ill e shon tht the deflection ) ( c (6.5.6) is solution to the prolem. irst this function certinl stisfies urther letting ) ( f (6.5.7) the relevnt prtil derivtives re c c c c c c c f c c f c cf cf (6.5.8) Sustituting these into the differentil eution no ields c 6 (6.5.9) so the deflection is ) ( 6 (6.5.)

3 This is plotted in ig The mimum deflection occurs t the plte centre here m. (6.5.) 6 igure 6.5.: mid-plne deflection of the clmped circulr plte The curvture / long rdil line is displed in ig The curvture is positive tord the centre of the plte (the plte curves uprd) nd is negtive tords the edge of the plte (the plte curves donrd). ( ) () ( ) ( ) igure 6.5.: curvture in the clmped circulr plte The moments occurring in the plte re from the moment-curvture eutions 6.. nd ( ) ( ) 6 ( ) ( ) 6 ( ) 8 ( ) ( ) (6.5.) The moment long rdil line is of the sme chrcter s the curvture displed in ig Solid echnics rt II 5 Kell

4 The out-of-plne sher forces re from 6..5 (6.5.) At the plte centre the epressions ecome ( ) (6.5.) 6 Stresses in the lte rom nd the stresses in the plte re z z z ( ) ( ) h z ( ) ( ) h z ( ) h z h h / z h h / ( ) ( ) (6.5.5) Converting to polr coordintes ( r ) through r cos r (6.5.6) nd ug stress trnsformtion rr r cos cos cos cos (6.5.7) leds to the ismmetric stress field { rolem } rr r z ( ) r ( ) h z ( ) r ( ) h (6.5.8) Solid echnics rt II 5 Kell

5 At the plte centre z rr (6.5.9) h At the plte edge r z z (6.5.) h h rr or the sher stress the trction cting on surfce prllel to the epressed s (see ig. 6.5.) plne cn e t e e zr z z r e z e z cose e e cose r z r (6.5.) here e i is unit vector in the direction i. Thus zr z cos z z z cos z r z h h / (6.5.) e z er zr e z e z igure 6.5.: stress components cting on surfce Note tht the mimum stress in the plte is m rr h / (6.5.) h The mimum sher stress on the other hnd is ( ) / h stress is of n order h / smller thn the norml stress. zr /. Thus the sher Solid echnics rt II 5 Kell

6 6.5. An Infinite lte ith Sinusoidl eflection Consider net the clssic plte prolem ddressed Nvier in 8. It consists of n infinite plte ith n undulting up/don usoidl deflection ig ( ) (6.5.) igure 6.5.5: A plte ith usoidl deflection ifferentition of the deflection leds to the curvtures cos cos (6.5.5) nd hence the pressure ( ) ( ) (6.5.6) The pressure thus vries like the deflection. There is no need for supports for the plte ce the up lods lnce the don lods. rom the moment-curvture reltions Solid echnics rt II 55 Kell

7 cos cos (6.5.7) nd from 6.. the sher forces re cos cos (6.5.8) Note tht oth / nd / re constnt throughout the plte A Simpl Supported lte ith Sinusoidl eflection olloing on from the previous emple consider no finite plte of dimensions nd ith the sme usoidl deflection 6.5. simpl supported long the edges. In ht follos tke in 6.5. to e negtive so tht the plte is pushed don tords the centre. According to 6.5. nd the deflection nd slope is zero long the supported edges s reuired. The verticl rections t the supports re given Hoever ccording to En c there re vring non-zero tisting moments over the ends of the plte. Thus the solution given is not uite the solution to the simpl supported finite-plte prolem unless one cn someho ppl the ect reuired tisting moments over the edges of the plte. It turns out hoever tht the solution is correct solution ecept in region close to the edges of the plte. This is eplined in ht follos. Tisting oments over ree Surfces Consider n element of mteril of idth d ig The element is sujected to tisting moment d ig This tisting moment is due to sher stresses cting prllel to the plte surfce (see ig. 6..8). This sstem of horizontl forces cn e replced the stticll euivlent sstem of verticl forces shon in ig to forces of mgnitude seprte distnce d. Reclling Sint-ennt s principle the difference eteen the stticll euivlent sstems of forces of ig nd ill led to differences in the stress field ithin the plte onl in smll region ver close to the plte-edges. Solid echnics rt II 56 Kell

8 d d () () igure 6.5.6: Euivlent sstems of forces leding to the sme tisting moment; () horizontl forces () verticl forces Consider net distriution of tisting moment long the plte edge ig As cn e seen this distriution is euivlent to distriution of shering forces (per unit length) of mgnitude ( ) (6.5.9) d d d igure 6.5.7: A distriution of tisting moments long plte edge The totl verticl rection long the edges cn no e tken to e (6.5.) (nd / long the other edges) nd this gives correct solution to the prolem. rom these rections re Solid echnics rt II 57 Kell

9 Solid echnics rt II Kell 58 ) ( ) ( ) ( ) ( (6.5.) Corner orces Integrting 6.5. over the four edges the resultnt uprd forces on the four edges (ith the re ll four uprd) re (6.5.) nd the resultnt of these m e epressed s up (6.5.) The resultnt donrd force is ug don ) ( dd dd (6.5.) The difference eteen up nd don is due to the re-distriuted tisting moment nd is eplined follos: consider gin ig here the edge tisting moments hve een replced ith stticll euivlent distriution of sher forces. It cn e seen tht there results sher forces t the ends of the plte-edge (the corners ) here the sher forces hve no neighouring sher force of opposite sign ith hich to cncel out. There re concentrted forces (per unit length) t the plte-corners of mgnitude. Emining ig hich shos the edge the force ) ( is positive up heres the force ) ( is positive don. There re lso contriutions to the corner

10 forces t () nd ( ) from the djcent edges shon in ig One finds tht the donrd concentrted forces t the corner re () ( ) ( ) ( ) (6.5.5) Adding these to don of En no gives the up of En hsicll if one pplies pressure to simpl supported plte the plte ill tend to rise t the four corners in tisting ction. The corner forces re necessr to keep the corners don nd so produce the deflection () ( ) () ( ) igure 6.5.8: corner forces in the simpl supported plte The rtio of the resultnt donrd corner force to the donrd force due to the pplied pressure don is (6.5.6) or sure plte this is ( ) / ; ith. this is 5% A Rectngulr lte Simpl Supported t the Edges The ove solution cn e used to solve the prolem of simpl supported plte loded n ritrr pressure distriution through the use of ourier series. Solid echnics rt II 59 Kell

11 Consider gin this plte hose displcement oundr conditions re Assume the deflection to e of the form ( ) m n A m n (6.5.7) (6.5.8) ith A coefficients to e determined. It cn e seen tht this function stisfies the oundr conditions. Tking the derivtives of this function m n m A m n (6.5.9) etc. nd sustituting into the differentil eution 6..9 gives m n m n A ( ) (6.5.) m n This cn e ritten compctl in the form m n C m n ( ) (6.5.) here C m n A (6.5.) It remins to choose the coefficients of the series so s to stisf the eution identicll over the hole re of the plte. One cn evlute the coefficients s one does for ordinr ourier series lthough here one hs doule series nd so one proceeds s follos: first multipl oth sides of (6.5.) k / here k is n integer nd integrte over eteen the limits [ ] so tht C m n k d m n k ( ) d (6.5.) Ug the orthogonlit condition Solid echnics rt II 6 Kell

12 n k d / n k (6.5.) n k leds to C m mk m k ( ) d (6.5.5) No there re functions of onl so multipling oth sides ( j / ) nd folloing the sme procedure one hs C jk k ( ) d j d (6.5.6) nd hence the coefficients C re (replcing the dumm suscripts j k ith m n ) C m n ( ) dd (6.5.7) Thus the coefficients A of the originl epression for the deflection ( ) re A m n m n ( ) dd (6.5.8) It is no possile to solve for the coefficients given n loding ( ) over the plte nd hence evlute the deflection moments nd stresses in the plte tking the derivtives of the infinite series for. This solution is due to Nvier nd is clled Nvier s solution to the rectngulr plte prolem. A similr solution method hs een used Lév to solve more generl prolem tht of rectngulr plte simpl supported on to opposite sides nd n one of the conditions free simpl-supported or clmped long the other to opposite sides. or emple considering sure plte this involves ug tril function for the deflection of the form (compre ith 6.5.8) ( ) n n n ( ) (6.5.9) nd then ttempting to determine the functions n (). A Uniform Lod In the cse of uniform lod ( ) one hs Solid echnics rt II 6 Kell

13 A m m 6 m 6 n n n m d m cos( m ) cos( n ) m n d n n m n 5 (6.5.5) The resulting series in converges rpidl. The deflection t the centre of the plte is then or sure plte m5 n A m5 n 5 m n m ( / ) n ( ) / (6.5.5) 6 6 m5 n 5.6 m n ( ) / (6.5.5) enoting the re A this is clmped circulr plte; denoting the re there 6.5. gives.6a /. Corner orces The tisting moment is.a /. This cn e compred ith the A the mimum deflection En. ( ) v v m n A m n cos cos (6.5.5) nd the four corner forces reuired to hold the plte don re no Solid echnics rt II 6 Kell

14 () v ( ) v ( ) v ( ) v m n m n m n m n A A A A cos m cos n cos m cos n (6.5.5) or uniform lod over sure plte ug the corner forces reduce to v v v.6 m5 n 5 m n.85 m n m n (6.5.55) (for. ) here is the resultnt pplied force rolems. erive the epressions for the stress components in polr form for the clmped circulr plte under uniform lterl lod En Solid echnics rt II 6 Kell