Plate Theory. Section 13: PLATE BENDING ELEMENTS

Transcription

1 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering Plte Theor A plte is structurl element hose mid surfce lies in flt plne. The dimension in the direction norml to the plne is referred to s the thickness of the plte. A plte is chrcteried b smll thickness reltive to the other dimensions of the plte. Pltes m be clssified b the reltive thickness of the structurl element. We cn spek of:. Reltivel thick pltes ith smll displcements. Reltivel thin pltes ith smll displcements. Ver thin pltes ith lrge deflections. Etremel thin pltes (membrnes) ith lrge or smll deflections If plte is curved s opposed to flt, e refer to these structurl elements s shells. We strt the discussion ith clssicl thin plte theor ttributed to Kirchoff

2 Wshkeic College of Engineering Section : PLATE BENDING ELEENTS Plte Geometr nd Deformtion The bsic derivtion of the field equtions for thin pltes strts ith the geometr shon in the figure belo: The plte surfces re t nd the mid-surfce is t t

3 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering Note tht t t b c nd if t is more thn bout one-tenth the spn of the plte, then trnsverse sher deformtions must be ccounted for nd the plte is sid to be thick. Also note tht the trnsverse deflection is much smller thn the thickness t, i.e., t These re things tht should be checked hen running finite element nlsis ith thin plte elements. Anlogous to bems there re severl tpes of plte theories ssocited ith geometric ssumptions.

4 Wshkeic College of Engineering Section : PLATE BENDING ELEENTS Kirchhoff Assumptions nd Theoreticl Formultion Consider differentil slice from plte ith cutting plnes perpendiculr to the -is s shon belo. The lod q from the previous figure cuses the plte to deform lterll in the positive direction. The deflection t n point P in the plte is functionll dependent on its position reltive to the - plne, i.e.,, nd the plte does not stretch in the -direction. The line -b drn perpendiculr to the mid-surfce before loding remins perpendiculr to the mid-surfce fter loding.

5 Wshkeic College of Engineering Section : PLATE BENDING ELEENTS This is consistent ith the Kirchoff ssumptions hich re stted s follos:. Stright lines norml to the mid surfce remin norml. This implies tht Hoever tht is right ngles in the plne of the plte do not necessril remin right ngles fter the plte is loded.. Thickness chnges re neglected nd lines norml to the mid surfce do not undergo etension. This implies Too thin to hve resonble Norml stresses re considered negligible. = Vrition in. embrne forces re neglected. Plne stress solutions re superimposed lter in the plne of the membrne. In-plne deformtions in the nd directions t the mid surfce re ero. u Free surfce,, v,, 5

6 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering The ssumption regrding: cn be interpretted grphicll in the folloing to figures: b rigid bod rottion One cn esil see in the second figure tht normls do not remin norml in the presence of strin.

7 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering The sme grphicl interprettion cn be mde regrding 7

8 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering u,, For Kirchoff pltes displcements in the -direction re ssumed to vr linerl from the mid surfce: As result strins in the direction re chrcteried b the prtil differenctil eqution to the right: u 8

9 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering v,, Simrlrl for displcements in the direction: hich leds to: Finll, the in plne sher strin is given b the epression v v u 9

10 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering The curvture of the pltes re defined s Thus the strins cn be epressed s

11 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering For pltes the ssumption of plne stress is mde in the direction. Thus the constitutive reltionship is here D E D

12 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering Note tht the stresses vr linerl from the middle surfce, just like bending stresses in bems. Also note tht the sher stresses ( ) produced b bending lso vr linerl from the middle surfce. The sher stresses nd re present nd required for equilibrium, lthough the corresponding strins re ssumed negligible. Prbolic vritions of the stresses re ssumed s in bem theor. The bending stresses cn be simplified to resultnt moments (,, ). These moments re resultnts of the liner stress vritions through the thickness

13 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering The moments,, d ( torsionl moment) s ell s d the sher forces d nd shon to the d right re per unitl d length quntities. d

14 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering E E net pge t t d d d d d d d Here here

15 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering t t t t Et d E d Thus 5

16 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering In similr fshion If e sum forces in the -direction nd sum moments bout the nd es ( nd re lloed to vr cross the differentil element, hence the use of grdients in the figure belo nd on the net pge) ssuming pressure q is pplied to the plte t t d q qdd dd dd qda d d d d d d F

17 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering In similr fshion summing moments bout the -is leds to Similrl, summing moments bout the -is ields (sho for homeork) d d d d d d d qdd d dd d d dd d d d 7

18 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering Thus the equtions of equilibrium t point for plte re These equilibrium equtions long ith the moment-curvture reltionships leds to the folloing non-homogenous fourth order, mied prtil differentil eqution in tht must be solved for pltes: If is knon, the strins re knon. If the strins re knon the stresses re knon. q q Et 8

19 Wshkeic College of Engineering Section : PLATE BENDING ELEENTS Plte Element Formultion bsed on Kirchoff Theor In pper published in 98 Hrbok nd Hrudle cited 88 tpes of plte element formultions. In this section one is presented nd issues ssocited ith the formultion re discussed. Consider the bsic rectngulr plte element ith four nodes ech ith three degrees of freedom, i.e., one out of plne displcement () nd to in plne rottions ( nd ). This element is depicted belo. Shon in the figure re possible nodl forces (f i ) nd nodl moments (f i nd f i ): Keep in mind tht unknon deformtions (displcements nd rottions) re temed up ith pplied ctions (forces nd moments). 9

20 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering The nodl deformtion vector t n node i is identified s here The negtive sign on is due to the fct tht negtive displcement () is required to produce positive rottion bout the -is (recll the right hnd rule). The totl element deformtion vector is i i i i d T d

21 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering Becuse there re degrees of freedom for the rectngulr element term polnomil in nd is selected. A complete cubic polnomil hs terms., to other terms We need to more terms from the qurtic polnomil line of Pscl s polnomil tringle. To understnd ho to pick these terms e need to discuss Continuit Conditions. To tpes on continuit re considered. The re clssified in the folloing mnner: C Continuit - The element is C comptible if long n side of n element is completel specified b the degrees of freedom on long tht side. Thus no gps pper long the edge here to elements meet. C Continuit - The element is C comptible if it is C comptible, nd the slope the side is completel specified b the degrees of freedom on long tht side.

22 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering With C continuit the slopes from to elements ill mtch long n edge here the elements meet. This hppens in the folloing depiction of elements vieed on edge: Where in the follo figure slopes do not mtch long the edges of elements here the meet. With Kirchoff plte theor, the slopes must mtch, giving the smoother description of the deformed surfce in the first figure.

23 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering If e dd nd terms s the other terms in the polnomil for then e get to the required terms:, 7 8 The element stisfies the trnsverse deflection continuit. Along n edge beteen to nodes, s the edge beteen nodes i nd j in the figure belo 9 5 the deflection ill vr in cubic mnner nd, derivtive of, ill vr qudrticll.

24 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering Tht is for reltive to the locl element coordinte is, then The constnts,, nd 7 cn be uniquel determined long this edge of the element b four degrees of freedom long tht edge: i, j, θ i nd θ j. Consequentl C continuit ill hold nd pprentl C is stisfied. But 7, 7, 8 5,

25 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering is cubic in, hich requires more boundr conditions to find, 5, 8 nd. But onl to more deformtion conditions eist long this edge of the element, i.e., i nd j. This slope is not uniquel defined nd slope discontinuit cn occur long this edge. Hence this element does not stisf C continuit conditions. This does not men tht the constnts through cn not be determined. To see this consider tht 5

26 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering With or C d

27 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering Finding the inverse of this reltionship cn be trivill stted s With here d C P P 7

28 Section : PLATE BENDING ELEENTS Then Wshkeic College of Engineering or PC d here N d N PC is the mtri of shpe funtions. 8

29 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering Returning the ttention to the strins in plte element We cn rite d C B B 9

30 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering here nd The element stiffness mtri is B C B B d d B D B t k c b T T

31 Section : PLATE BENDING ELEENTS Wshkeic College of Engineering There re mn different s of interpolting deformtions over plte, leding to mn different tpes of plte elements hving been proposed for Kirchoff s plte theor. The sme issue comes up for indlin-reisner plte theor. Some proposed elements for Kirchoff s theor hve encountered problems ith locking - refusl of the element to deform in some circumstnces, s the set of equtions becomes over-constrined, due to inconsistent ssumptions being mde. This is referred to s sher locking. Other elements hve hd the opposite problem of lcking stiffness to resist certin eroenerg modes of deformtion. These spurious modes of deformtion then dominte the solution. Another nme for this is hourglss deformtion, s the unnted in-plne deformtion of qudrtic element resembles this shpe.