Kirchhoff nd Mindlin Pltes A plte significntly longer in two directions compred with the third, nd it crries lod perpendiculr to tht plne. The theory for pltes cn be regrded s n extension of bem theory, in the sense tht bem is 1D speciliztion of 2D pltes. In fct, the Euler- Bernoulli nd Timoshenko bem theories both hve its counterprt in plte theory: Kirchhoff theory for pltes Euler-Bernoulli theory for bems Mindlin theory for pltes Timoshenko theory for bems The Kirchhoff theory ssumes tht verticl line remins stright nd perpendiculr to the neutrl plne of the plte during bending. In contrst, Mindlin theory retins the ssumption tht the line remins stright, but no longer perpendiculr to the neutrl plne. Tht mens tht Kirchhoff theory pplies to thin pltes, while Mindlin theory pplies to thick pltes where sher deformtion my be significnt. Figure 1 shows the nottion for the bending moments nd sher forces in pltes. The documents posted on this website use different convention for bems nd pltes. The BEAM quntities re formulted with ONE index with the following mening: M y is the bending moment bout the y-xis (leding to the nottion I y for the corresponding moment of inerti) V z is the sher force in the direction prllel to the z-xis Conversely, the PLATE quntities re formulted with TWO indices with the following mening: M yy is the bending moment obtined by integrting the xil stress σ yy V yz is the sher force obtined by integrting the sher stress τ yz M xy is the twisting moment obtined by integrting the sher stress τ xy The first index of stress denotes the surfce norml of the plne the stress cts on, while the second index is the direction of the stress. Also notice tht in plte theory, the moments nd sher forces re mesured per unit length long the plte edge. τ xz τ yz σ xx τ τ yx xy σ yy V xz z, w Vyz M xx M yy M yx M xy x, u y, v θ y θ x Figure 1: Stresses nd stress resultnt in plte element. Kirchhoff nd Mindlin Pltes Updted Februry 9, 2018 Pge 1
Section Integrtion Letting the z-xis hve its origin t the neutrl plne of the plte, the moments re defined by the integrls M xx M yy M xy M yx z σ xx dz (1) z σ yy dz (2) z τ xy dz (3) z τ yx dz (4) From bsic solid mechnics it is known tht τ xy τ yx, which implies tht the twisting moments re equl: M xy M yx. In the Kirchhoff plte theory the postultion from Euler- Bernoulli bems is mde tht the sher deformtion is zero, hence the sher strin, sher stress, nd sher forces re omitted from the theory. Of course, the sher forces cn lter be recovered by equilibrium considertions, but t first V xz nd V yz re left out of the Kirchhoff theory. However, in Mindlin theory the sher forces re obtined by integrtion of the sher stresses: V yz V zx τ yz dz (5) τ xz dz (6) Equilibrium Consider the infinitesiml plte element shown in Figure 2. It extends dx in x-direction nd dy in y-direction. Its thickness is h nd it is subjected to distributed lod of intensity q(x,y) in the z-direction. Equilibrium in the z-direction yields q dx dy + V yz dy dy dx + V xz x dx dy 0 q + V yz y + V xz x 0 (7) Moment equilibrium bout the y-xis, t the left front edge in Figure 2 yields M xx x dx dy + M yx y dy dx V xz dy dx 0 V xz M xx x + M yx y Moment equilibrium bout the x-xis, t the right front edge in Figure 2 yields (8) Kirchhoff nd Mindlin Pltes Updted Februry 9, 2018 Pge 2
M yy y dy dx M xy x dx dy + V yz dx dy 0 V yz M yy y + M xy x The three equilibrium equtions cn be combined into one. Prtil differentition of Eq. (8) with respect to x nd prtil differentition of Eq. (9) with respect to y, followed by substitution of those equtions into Eq. (7) yields (9) q + 2 M xx x 2 + 2 M yy y 2 + 2 2 M xy x y 0 (10) z M yx V yy V xx M xy M yy dx dy M xx x M yy + M yy y dy y M xy + M M + M xx xx xy x dx x dx V xz + V xz x dx Figure 2: Infinitesiml plte element. V yz + V yz y dy M yx + M yx y dy Mteril Lw For the reltively thin pltes it is pproprite to consider the plne stress version of Hooke s lw to model the in-plne behviour: σ xx σ yy τ xy E 1 ν 2 1 ν 0 ν 1 0 0 0 1 ν 2 Mindlin theory lso requires the reltionship between the other sher stresses nd strins: ε xx ε yy γ xy (11) τ xz G γ xz (12) τ yz G γ yz (13) Kirchhoff nd Mindlin Pltes Updted Februry 9, 2018 Pge 3
Kinemtics It is the kinemtic reltionships tht best revel the difference between the Kirchhoff nd Mindlin theories. In Euler-Bernoulli theory for bems nd Kirchhoff theory for pltes, rottion is equted with the derivtive of the lterl displcement. Tht implies tht stright lines remin stright nd perpendiculr to the neutrl xis during bending. This ssumption is relxed in Timoshenko bem theory nd Mindlin plte theory. To understnd the implictions of these ssumptions, the fundmentl kinemtic equtions from solid mechnics re first listed: ε ε xx ε yy ε zz γ xy γ yz γ zx x 0 0 0 y 0 0 0 z y x 0 0 z y z 0 x Among those six equtions, the third is irrelevnt here becuse ε zz is ssumed to be zero. To spell out the other equtions, it is recognized from Figure 1 tht nd u v w (14) u z θ y (15) v z θ x (16) Substitution of Eqs. (15) nd (16) into Eq. (14) yields the following five remining equtions: ε xx z θ y,x (17) ε yy z θ x,y (18) γ xy z θ y,y z θ x,x (19) γ yz θ x + w,y (20) γ zx θ y + w,x (21) Similr to the Euler-Bernoulli bem theory, in Kirchhoff plte theory it is ssumed tht θ x w,y (22) nd θ y w,x (23) Tht implies tht in Kirchhoff theory, Eqs. (17) through (19) reds: ε xx z w,xx (24) Kirchhoff nd Mindlin Pltes Updted Februry 9, 2018 Pge 4
while the sher strins γ yz nd γ zx re zero. ε yy z w yy (25) γ xy z w yx z w xy 2 z w xy (26) Differentil Eqution For Kirchhoff pltes, the combintion of stress resultnt, mteril lw, nd kinemtic equtions, s well s integrtion long z, yields the following governing equtions: 2 w M xx D x 2 2 w M yy D y 2 + ν 2 w y 2 + ν 2 w x 2 M xy D (1 ν) 2 w x y where the plte stiffness, D, comprble with EI for bem, is defined s: (27) (28) (29) D E h 3 12 (1 ν 2 ) Substitution of Eqs. (27), (28), (29) into the equilibrium eqution from Eq. (10) yields the fourth order differentil eqution for plte bending: 4 w x + 4 w 4 y + 2 4 4 w x 2 y q 2 D Eqs. (27) nd (28) llow useful interprettion for pltes tht spn in one direction only. A strip of such plte cn be considered s bem. In other words, when one of the curvtures in the prentheses in Eqs. (27) nd (28) is zero then the equtions tke the form (30) (31) M xx D 2 w x 2 (32) This does indeed led to the conclusion tht D tke the plce of the bending stiffness EI from bem bending, when unit-width strip of the plte is considered. The use of D in plce of EI essentilly ccounts for the constrined strin, i.e., plne strin, in the plte continuing on the sides of the plte strip. Recovery of Sher Forces: Kirchhoff s Sher Force The nomly of bem theory due to Nvier s hypothesis crries over to plte theory. Insted of including sher stresses nd strins in the theory, the sher forces re recovered only fter solving the differentil eqution. The equilibrium in Eqs. (8) nd (9) re Kirchhoff nd Mindlin Pltes Updted Februry 9, 2018 Pge 5
employed for this purpose. However, these two equtions re only one prt of the totl sher force in plte theory, s explined next. According to the theory bove, three stress resultnts ct long the edge of plte: bending moment, sher force, nd twisting moment. The number of unknowns in the generl solution to the differentil eqution is insufficient to prescribe tht mny boundry conditions. This leds to closer exmintion of the twisting moment, nd subsequently its inclusion into the totl sher force. To this end, consider the twisting moment M xy nd its vrition in the y-direction. Figure 1 my be of help in visulizing this. Next, imgine tht within ech infinitesiml segment of length dy the twisting moment M xy gives rise to force pir. Let the forces be dy prt; becuse the moment within length dy is M. xy dy, ech force is M xy. When the twisting moment vries long y, then there will be surplus of the force M xy within ech infinitesiml segment. This surplus is sme s the chnge of M xy within length dy, nmely M xy y. The totl sher forces re then: V xz + M xy y D 3 w x + (2 ν) 3 w 3 x y 2 (33) V yz + M yx y D 3 w y + (2 ν) 3 w 3 x 2 y (34) The force-interprettion of the twisting moment leds to nother conclusion. When M xy nd M yx nd vries long the plte edge there is net sher force t the corner of the plte. For exmple, when squre plte is bending under uniform downwrd loding then the corners will experience uplift. This is due to the net unblnced concentrted force equl to 2M xy t the corner. This sher force is known s Kirchhoff s sher force nd the corner uplift is referred to s the Kirchhoff effect. Nvier s Solution This solution for thin pltes ws presented to the French Acdemy in 1820 by Clude- Louis Nvier nd is explined in detil in one of Timoshenko s books (Timoshenko nd Woinowsky-Krieger 1959). A simply supported rectngulr plte with length in the x- direction nd length b in the y-direction is considered. Nvier s solution stems from the preliminry considertion of one sine pillow s loding on the plte: q(x, y) α sin π x sin π y b (35) where α is the mximum mplitude of the lod, t the middle of the plte. Furthermore, the following tril solution stisfies the boundry conditions tht require zero displcement nd bending moment on the edges: w(x, y) C sin π x sin π y b (36) Substitution of Eqs. (35) nd (36) into the differentition eqution in Eq. (31) yields: Kirchhoff nd Mindlin Pltes Updted Februry 9, 2018 Pge 6
C π 4 sin π x 4 sin π y b + C π 4 b sin π x 4 sin π y b π 4 +2 C 2 b sin π x 2 sin π y b α D sin π x sin π y b (37) The sme sine product ppers in ll terms; hence, they cncel, nd rerrnging yields the unknown constnt in the solution: C α (38) 1 D π 4 + 1 2 2 b 2 Nvier extended this pproch by using multiple sine functions to describe the lod, expressed s series expnsion: q(x, y) q mn sin mπ x nπ y sin b (39) m1 n1 The coefficient q mn is in generl different for ech n nd m nd essentilly describes the mgnitude of the lod. To determine q mn, i.e., to link Eq. (39) with ctul lod, e.g., uniformly distributed lod, it is useful to express Eq. (39) s n integrl insted of sum. This is cleverly done by multiplying Eq. (39) by n identicl sine product, only with different counters n nd m, nd integrting the result: Becuse b 0 0 q(x, y) sin!mπ x sin!nπ y b dx dy b q mn sin mπ x nπ y sin b sin 0 0 m1 n1 Eq. (40) simplifies to!mπ x sin sin mπ x sin!mπ x dx 0 when m!m 0 sin mπ x sin!mπ x dx 2 0 when m!m!nπ y b dx dy (40) (41) b 0 0 q(x, y) sin!mπ x sin!nπ y b dx dy b 4 q mn (42) from which q mn is solved for specific lod distributions q(x,y). For the cse of uniformly distributed lod, q(x,y)q o, Eq. (42) yields: Kirchhoff nd Mindlin Pltes Updted Februry 9, 2018 Pge 7
q mn 4q o b!mπ x sin sin!nπ y b b dx dy 0 0 16q o m n odd π 2 mn 0 otherwise For the cse of point lod with vlue P positioned t x ξ nd yη Eq. (42) yields sum over ll m nd n, now both odd nd even: q mn 4P b (43) mπξ nπη sin sin b (44) Now to the displcement solution. The solution in Eqs. (36) nd (38) contined one sine term. Nvier s solution contins mny: q mn mπ x nπ y w(x, y) sin m 2 2 D π 4 + n2 sin b (45) m1 n1 2 b 2 with the coefficient q mn determined bove. Lévy s Solution Nvier s solution is conceptully strightforwrd ppliction of double trigonometric series. However, the series does not converge fst; thus, high-order derivtives of w my be inccurte. Around 1899 Levy suggested nother pproch (Timoshenko nd Woinowsky-Krieger 1959). Agin simply supported rectngulr plte with length in the x-direction nd length b in the y-direction is considered, but now the coordinte system is shifted s shown in Figure 3. y y b Coordinte system for Nvier s solution b/2 Coordinte system for Levy s solution x Origin x b/2 Origin Figure 3: Coordinte systems for plte solutions. Levy formulted solution tht first focuses on the x-direction spn from x0 to x consisting of homogeneous solution, w h, nd prticulr solution, w p : Kirchhoff nd Mindlin Pltes Updted Februry 9, 2018 Pge 8
w(x, y) h m ( y) sin mπ x q(x, y) + m1,3,5... 24D x4 2x 3 + 3 x!#### "#### $!#### "#### $ w h ( ) w p (46) where h m is function tht depends on y only, nd m must be odd due to symmetry. The prticulr solution, w p, stisfies the differentil eqution in Eq. (31) nd lso the boundry conditions t the two edges x0 nd x, nmely zero displcement nd moment/curvture. Now h m (y) must be formulted such tht it stisfies the homogeneous version of the differentil eqution in Eq. (31), nd such tht ww h +w p stisfies the full differentil eqution nd ll boundry conditions. Applying those two conditions, nd reformulting the prticulr solution s the series expnsion w p q 24D ( x4 2x 3 + 3 x) 4q4 π 5 D 1 mπ x sin 5 m (47) m1,3,5... yields the solution (Timoshenko nd Woinowsky-Krieger 1959) w(x, y) 4q4 π 5 D m 5 m1,3,5... mπb mπb tnh 2 2 + 2 1 2 cosh mπb cosh mπ y + 1 2 sin mπ x mπb (48) 2!+ 2 cosh mπb 2y mπ y sinh b 2 Assuming b the mximum deflection t the middle of the plte, i.e., t x/2 nd y0, cn be expressed reltive to the mximum deflection of comprble bem: w mx 5q4 384D 4q4 π 5 D mπ x mπ x m 1 2 ( 1) tnh + 2 m 5 2 cosh mπ x (49) m1,3,5... Edge Moments The solutions presented bove re for simply supported pltes. Those solutions cn be superimposed with the solution for plte with distributed moments long the edges to enforce other boundry conditions. Using Levy s coordinte system in Figure 3, Timoshenko presents in Article 41 of his book on Pltes nd Shells (Timoshenko nd Woinowsky-Krieger 1959) solution for simply supported plte subjected to uniformly distributed moment, M 0, bout the x-xis long the edges y±b/2: Kirchhoff nd Mindlin Pltes Updted Februry 9, 2018 Pge 9
where w(x, y) 2M 0 2 π 3 D γ m mπb 2 1 m 3 cosh mπb γ m sin mπ x (50) 2 m1,3,5... mπb mπ y tnh 2 cosh mπ y mπ y sinh (51) A more generl solution with vrying intensity of the edge moments is sin mπ x w(x, y) 2 2π 2 D m 2 cosh mπb E m γ m (52) m1 2 where the edge moment, now vrying in the x-direction, is expressed s f (x) E m sin mπ x (53) m1,3,5... Two edges clmped, two edges simply supported For uniformly loded plte, simply supported long the two edges x(0,) nd clmped long the two edges y(±b/2), Timoshenko finds the following edge moment fctors to be substituted into Eq. (52): E m 4q2 π 3 m 3 mπb 2 mπb 2 mπb mπb mπb tnh 2 1+ tnh 2 2 mπb tnh 2 mπb mπb tnh 2 2 1 The totl solution is the sum of Eq. (48) nd Eq. (52). The solution for other boundry conditions cn be obtined in similr mnner, lthough it cn get bit mthemticlly messy. (54) References Timoshenko, S. P., nd Woinowsky-Krieger, S. (1959). Theory of Pltes nd Shells. McGrw-Hill. Kirchhoff nd Mindlin Pltes Updted Februry 9, 2018 Pge 10