AN EXPERIMENTAL STUDY OF I N S T A B I L I T Y IN SQUARE PLATES TWISTED BY CORNER FORCES BY GORDON C. WILLIAMS B.Sc, Queen's U n i v e r s i t y , 1984 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE Department We accept to of Mechanical this STUDIES Engineering t h e s i s as c o n f o r m i n g the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA June 1988 • G o r d o n C. W i l l i a m s , 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of rtee-U.*^! Ev^^gcr'in^ The University of British Columbia Vancouver, Canada Date DE-6 (2/88) ABSTRACT An experimental twisting of A of study plates is with self-equilibrating simple were twisted sides to The apparatus at the unwanted while free edges, corner loads. was designed measuring centre. of method between the pre- post-bifurcation the experimental strains are and found to midsurface to the various sizes was results, with of plates on both encountered its own determine of application strains under strains linearly non-linear through regions analytical vary the surface plate used relationship and on d i f f i c u l t y the is and the I n i t i a l deflection Rayleigh-Ritz presented and weight. an analytical curvatures twisting. the due In in both midsurface the Gaussian which collapse curvature. Non-dimensional experimental and curvature groups non-linear bifurcation are load-strain, strain-Gaussian dimensional groups is identified load-curvature and relationships. These collapse regions. The identified as the results curvature a i i at function in of midsurface non- both the the the point plate linear of dimensional parameters. surface The Also strain are literature. resolved The and experimental bifurcation the shown for present analytical the corner the expressions load at load-curvature compared w i t h A large results, form and which a are relationships point in of for important using the c r i t i c a l bifurcation analytical basis for relationship discrepancy theoretical results constitutive are occurs. and point of results found in the literature is bifurcation. verification in the twisted of future measurement plate test. of TABLE OF CONTENTS Abstract Table of ii Contents List of Figures List of Tables iv v i i x Nomenclature xi Acknowledgements 1. 1 INTRODUCTION 1.1 1.2 1.3 2. x i i i Background 1 1.1.1 Description 1.1.2 Determination Literature of the of Instability Constitutive 1 Coefficients Review 4 1.2.1 Material Testing 5 1.2.2 Point Instability 6 Purpose and of Scope of the Study 7 10 PLATE BENDING THEORY 2.1 Linear 2.1.1 2.2 3 Theory Limitations Non-Linear 2.2.1 10 the Linear Theory Governing 2.2.1.1 of 16 17 Equations Geometric Theory Strain iv 18 18 2.2.1.2 Constitutive 2.2.1.3 Compatibility 20 2.2.1.4 Equilibrium 21 2.2.1.5 Stress 23 2.2.2 Boundary 2.3 A p p r o x i m a t e 19 Function Conditions 24 Solutions 2.3.1 A n a l y t i c a l Plates 3. Relations 26 Solution with Unequal for Midsurface Strain Curvatures 3.1 A p p a r a t u s 37 3.1.1 Design 37 3.1.2 Instrumentation 42 Strain 3.1.2.1 3.1.2.2 S t r a i n 3.1.3 Plate Selection Indicator 42 and B r i d g e Configuration 44 Design 45 3.1.3.2 S i z e Experimental 3.2.1 Initial 3.2.2 Test EXPERIMENTAL Gauge Estimate 3.1.3.1 4. 27 37 DESCRIPTION OF EXPERIMENT 3.2 in of Plate and M a t e r i a l Procedure Procedure Principal 4.2 Plate Curvature 49 51 52 55 RESULTS Plate 46 51 Preparation 4.1 Non-Linearity Strain 56 61 v 5. C O M P A R I S O N OF RESULTS WITH THEORETICAL AND NUMERICAL MODELS 5.1 72 Collapsing Data into Form 72 5.1.1 Principal Strains 73 5.1.2 Principal Curvature 79 5.1.3 Midsurface Strain 5.1.4 Comparison of Approximate 81 Load-Curvature Results With Solutions 84 5.1.4.1 Chandra[5] 86 5.1.4.2 Reissner[9] 87 5.1.4.3 Ramberg 88 5.1.4.4 L e e a n d Hsu[7] 5.2 Plate 5.3 Comparison 5.4 Non-Dimensional Curvature of Under Experimentally 5.3.2 Analytical Load Measurement of i t s 88 Own W e i g h t the Curvature 5.3.1 Limiting and Miller[2] Determined Bifurcation and Surface Shear Bifurcation Point Point During 91 92 Points Strain Modulus 89 95 the 96 6. CONCLUSIONS 7. BIBLIOGRAPHY 105 8. APPENDICES 107 99 A. Instrumentation B. Finite Difference 107 Analysis vi 113 LIST Figure 1.1, A n t i c l a s t i c curvatures Figure 1.2, shape along Deflection cylindrical OF the of FIGURES with equal and opposite diagonals. the surface, plate a) 2 after bifurcation Concave up. b) into a Concave down. Figure 2.1, M 2 Corner loads forming couples of magnitude x ' y 1 Figure 2.2, Constant twisting Figure 2.3, Midsurface moment forces N x N M , 11 x N applied to a small element. 22 Figure.3.1, Schematic Figure Details 3.2, Support 3.3, Quarter Figure 3.4, Design Figure 3.5, Graph Figure 4.1a, Figure 4.1b, Plate weight. Figure 4.1c, of 6" 8" Plate weight. 10" the the b) plate two twisting support Rolling bridge of Plate of of Post, Figure weight. 1 apparatus posts, Support a) Pinned Post. 40 configuration. plate and non-linear principal location ratio strains, vs. 45 of strain plate gauges.45 size. uncorrected for plate. principal 49 plate 57 strains, uncorrected for plate. principal 38 plate 58 strains, plate. uncorrected for plate 59 v i i Figure 4.1d, Plate weight. Figure 4.2a, Plate weight. Figure 4.2b, 4.2c, 4.2d, 4.3a, 6" Figure Figure 4.3c, 4.3d, 12" Figure Plate uncorrected for plate plate. principal 60 strains, corrected for plate plate. principal 62 strains, corrected for plate plate. principal 63 strains, corrected for plate plate. principal 64 strains, corrected for plate plate. curvature, 65 corrected for plate weight. Plate 67 curvature, corrected for plate weight. plate. 10" Figure 12" strains, plate. 4.3b, 8" 10" Plate weight. Figure 8" Plate weight. Figure 6" Plate weight. Figure 12" principal 5.1a, Plate 68 curvature, corrected for plate weight. plate. Plate 69 curvature, corrected for plate weight. plate. Non-dimensional 70 principal strain. Top Gl direction. Figure 5.1b, Non-dimensional 74 principal strain. Top G3 direction. Figure 5.1c, Non-dimensional 75 principal strain. Bottom Gl direction. Figure 5.1d, Non-dimensional 76 principal direction. Figure 5.2, Non-dimensional strain. Bottom G3 77 curvature. v i i i 80 Figure 5.3, Non-dimensional Figure 5.4, Comparison of midsurface approximate strains. and 82 experimental results. 85 Figure 5.5, Curvature Figure 5.6, Bifurcation Figure B . l , Deflection supported Figure B.2, mesh Figure B.3, by due varying of of between Convergence mesh plate weight. 90 point. of sizes 93 plate opposite Convergence sizes to under load and corners. centre 2 uniform and deflection for varying 10. centre between ix 115 116 bending 2 and moment 10. for 117 LIST OF Table 3.1, 6061-T6 Table 3.2, Plate size Table 3.3, Plate design Table A . l a , Strain gauge data sheet, Lot 1. 109 Table A . l b , Strain Gauge Data Sheet, Lot 2. 110 Table A.2, Vishay Material TABLES and properties. non-linearity 50 ratio. parameters. P-350A Digital 51 Strain Indicator specifications. Table A.3, Vishay SB-1 50 Ill Switch and specifications. Balance Unit 112 x NOMENCLATURE a -simplifying variable e x £y Y y -bending e x e r y -midsurface e* e y Y strain x' a a relative s t r i a n n T x ' y ' _ edge ^ij C s t i D j flexural e -plate dimension E - e l a s t i c f -plate F -Airy stress instability axes axes r e s s a n < ^ strain relative to x'-y' axes position coefficients r i g i d i t y function h thickness -plate k y x-y axes dimension modulus K of axes modulus G -shear x x-y to x-y x-y curvature "displacement -plate K point relative to to to length D j relative relative at ratio ratio -non-dimensional i j i s Poisson's surface diagonal xy y' -plate c along strains ~ at strains Py - n o n - d i m e n s i o n a l x £ strains Y.^y - s u r f a c e -Poisson's P CT e* x v X -surface e* e X containing K 1 -indices xy "lending y curvatures axes xi and twist relative to the x- K -absolute value of plate along M M x -bending the and N x N N y xy -midsurface -non-linear P -corner P c u -load v load, at z' uniform twist axis moment per the x-y axes forces, per ^ unit positive point of unit length length 3 in z direction i n s t a b i l i t y w -displacements y' to during ratio the x' x-y twisting relative N curvature x, -coordinate of the y, z axes plate midsurface parallel to axes parallel to the edge of the plate x y z -coordinate axes parallel to the diagonals of the plate V Q -midsurface strain energy stored in the plate, energy stored in the plate, per unit volume V -midsurface strain x i i total ACKNOWLEDGEMENTS I would l i k e to thank P r o f e s s o r Ramsey f o r h i s guidance during this project and h i s comments i n the final p r e p a r a t i o n of t h i s thesis. I would l i k e to thank the t e c h n i c i a n s of the Department of Mechanical Engineering for their h e l p . In p a r t i c u l a r I would l i k e to thank Tony B e s i c for machining the p l a t e s t h a t were tested. L a s t , but most of a l l , I t h e i r many l o n g d i s t a n c e much a p p r e c i a t e d . would phone l i k e t o t h a n k my p a r e n t s for c a l l s . Their support is very S u p p o r t f o r t h i s work was p r o v i d e d by the N a t i o n a l and E n g i n e e r i n g R e s e a r c h C o u n c i l of Canada. x i i i Science 1 1. 1.1 BACKGROUND 1.1.1 DESCRIPTION Shown in loaded loads a Figure on OF THE 1.1 l a t e r a l l y uniform is with adjacent twist, or curvatures parallel to the edge the deflection the the can load is of and be the plate with remains of the free edges self-equilibrating theory equal diagonals. through constraints with linear shape, plate verified plate opposite Classical the increased significant becomes stiffer. anticlastic takes equal square predicts and Any straight experiment linear line during as long theory as are to. noticeable, plate thin across This adhered a anticlastic twisting. small INSTABILITY corners. opposite As INTRODUCTION place section. Due to to shape a to point non-linear At is a no cylindrical the a symmetry where deviations particular longer the load stable surface of deflections with plate occur and and a and the curvature transition circular and become cross- loading, the INTRODUCTION Figure 1.1, A n t i c l a s t i c curvatures plate either along with the deflection can a r b i t r a r i l y concave ( F i g . 1.2a), a) Figure shape 1.2, up Concave cylindrical and take or one of concave the plate surface, a) two down b) of opposite diagonals. up, Deflection down. equal after Concave forms: (Fig. Concave bifurcation up. b) 1.2b) down, into Concave a INTRODUCTION The bi-stable phenomenon can easily cardboard and applying small sheet of one's fingertips. cylindrical The form lateral load mode continues it forces the number of more working stresses factors are analysis tensile to be by the corner forced the Once it in that deflect from taken mode using load a with one application has twisting OF C O N S T I T U T I V E light are be of a a particular under corner of the relationship, the plate are shear from the sections the no can modulus can modern sheet safety stress material from sufficient. determined of The the deduced application or sheet. longer be structures while of those are material the thin use that determined; by strength increasing requires load-deflection the high through This heavier square measuring strains accurately and COEFFICIENTS fabricated reduced modulus a weight being and being testing shear simple other centre. techniques. properties By the can demonstrated alone. increases, The the DETERMINATION 1.1.2 As at to sheet be 3 through corner loads. load-curvature be calculated using theory. Presently, to obtain deflection must be reliable small in test results, comparison to the the corner thickness of INTRODUCTION the plate testing to thin exceeded the the self-weight with phenomenon, bending. in the twist, the isotropic as wood or composites as part of their layered 1.2 what is uniform the testing be is range identified. behavior is the and is In the i n s t a b i l i t y produces is the no unsymmetrical longer maximum orthotropic tested procedure Testing easier linear In in a load state and test. materials, can is range. account. plate limiting coefficients. to by in a materials twisting thin determining single fabricate ply, than a such sheet their or lamina, multi- laminate. LITERATURE Kelvin which the during as and in the point non-linear above, thereby plate material size, yield taken occurs well saves be deflection non-linearities As constitutive linear reasonable range, must this the material described Once uniform strain of deflection Associated of within plates before large plate stay 4 and REVIEW Tait[l] expected from a n t i c l a s t i c occurring during noted in the shape, large 1883 linear and the discrepancy between theory, the deflections. predicting synclastic a deflection the INTRODUCTION 1.2.1 Many MATERIAL methods modulus, discuss G, 10 twisting length to between square in Miyagawa, their corner the This They due strains One They (a/h) determining and of in plates order shear briefly method small 10 the Miller[2] convenient used show the would be Shibuya square ranging results and clearly is in plates from the displays another large plot for various a/h ratios the plate Ueda[3] with 1.0 to is the with a reduce the the at mm. the 26.6 Of of the centre of of plate the synclastic estimated used, from relationship the sizes This 3.2 transition into done ratios mm t o strain the have a/h typical shape considerable. therefore, and principal anticlastic bifurcation to measured; of load uniform curvature plate. ratio for Ramberg methods. thickness plate. shape. But, a suggested material. testing and interest from a recently, 240 the in been effects. extensive to have thickness non-linear More TESTING different of 5 point plate tested. midsurface non-linearity curvature of can was not not be calculated. The find idea its of measuring constitutive the deflection coefficients of was a twisted extended by plate to Tsai[4] INTRODUCTION to orthotropic two twist tests independent results region used analysed is equation. to not The 1.2.2 Lee based OF square region on critical uniform with of the to incorporating determine orthotropic the 4 materials. The verified. plates inplane corner into the An A i r y stresses. large stress The respect to minimum potential deflection. inplane shear the The deflection function error in is the compatibility energy stress is used function requirement at the used plate INSTABILITY Hsu[6,7] twisted linear the tests Chandra[5]. the method exactly. POINT and by principal satisfy boundary bending for minimized calculate does two proposed a orthotropic represent function He experimentally square is to with constants were Twisting plates. 6 the synclastic dimensional plate using with a of von after takes moment at displacement free edges place. the the which a field its plate Up assumes results non- analysis to the an into give bifurcation a unstable, transformation of of difference equations. Their point into finite Karman deflection, shape, shape the numerical non-linear value anticlastic calculate as the M the non= 21. INTRODUCTION Ramsey[8] also non-linear twist to shell in which rectangular free edge Associated of with an order beyond show that a free edge in conditions of of the the same bifurcation, c r i t i c a l M c r = but value uses of 1.42 was magnitude less a than the stresses 1.3 PURPOSE A N D SCOPE The purpose of resolve for the short edge. He shows instability. shape, which Fung i s develops the occurs adjacent develop shape. to after boundary and W i t t r i c k [ 1 0 ] . They to the plate satisfy the boundary They do n o t , however, OF THE STUDY a i s two rather results fold: large for discrepancy the point of in a the itself. study theoretical and by cylindrical the of equations he a p p r o x i m a t e s instability, zone i n s t a b i l i t y to of which along cylindrical point von Karman in type described which for plate conditions transition the 1) of the the non-linear the phenomenon examine He f i n d s twisted boundary existence layer i s solves long loading point [7]. Reissner[9] the theory. the be Ka= 3 . 2 9 h / a . A v a l u e calculated, result calculates 7 some bifurcation, INTRODUCTION 2) to generate into the experimental non-linear understanding The following 1) To is the design a) the of the apparatus further twisted the to: and loads which deflect opposite the incremental plate into corner the non- range; only no allow plates project. equal with to for phenomenon. apply apply c) range this scope linear b) of data 8 lateral moment for or plates point loads inplane of at the forces; various corners and sizes to be tested. 2) To determine using a linear deflection range finite and of test plate differences, non-linear a/h ratios, which effects limit under self- we i g h t . 3) To measure and To a) principal b) midsurface To for increments in load strains, strains, and curvatures. find strains 5) strains calculate: c) 4) surface non-dimensional and develop curvatures an groups which approximate midsurface strain curvatures and in a twisted the collapse analytical plate by for with corner calculated the results. solution unequal forces. for INTRODUCTION To compare the experimental analytical and numerical conclusions. results models and with derive 10 PLATE 2. In this section develop 2.1 An both the the THEORY elastic square throughout surface with diagonals. This equal in 1.1. Fig. and the to equivalent along the two +P/2 These plate are to loads applied l o a d s ,of at P/2 have when plate a uniform bent into used twisting an state stress be adjacent opposing a uniform corner corner twisting At each corner, of P/2. Statically small interval couples as can the shown in as can plate be by shown shown distributed P is equivalent along moment produced loads load the loads moment the theory. anticlastic along of to bending curvatures the form is [12] opposite opposite edge edge[12]. corner and of non-linear will plate simple Along be the THEORY convention and plate equal applying by linear LINEAR M , , x y sign BENDING free Fig. replaced loads edges. 2.1. of PLATE Figure Along w i l l 2.1, the be which Corner x' loads edge, subject results each to in a a forming plate constant couples element twisting BENDING moment of of THEORY magnitude length of 11 M , , x Ay, (P/2)Ay , r twisting moment f P per unit length of M x'y' Similarly, of the Fig. the plate same = method producing - Ax' can uniform be ...(2.1) 2 Ax' applied twisting 2.2, Constant twisting moment M the moments 2.2. Figure to , other as edges shown in PLATE From be equilibrium, calculated moment can be the internal using Mohr's shown to occur in the sides of the plate, value of the moments in M = xy moment-curvature K are y respectively, Poisson's given is ratio. The direction M = D(K M = D(K of the of the can bending ±45° to the diagonals. The diagonals are (2.2) are + vK ) x y + \)K ) y . . . (2.3) x M = D(1-\0K the curvatures xy the twist flexure in in the rigidity the x and y directions plate and v is of the 1 plate, D, is by D where of relationships[12] xy and angle direction y x an maximum plate 0 x K The the 12 1 M where at within THEORY = - M = M , , = - | y x'y 2 x The moments c i r c l e . the BENDING E is the Young's = E modulus h (2.4) 3 and h is the plate thickness. PLATE Combining Eq. curvatures, (2.2), we (2.3) = -K X K therefore have direction of x-y these If axes, the plate are also the plate the and solving for 13 the = -6(1+M) y t equal and opposite diagonals. are and the Because principal the deflection ...(2.5) h =0 xy deflections small (2.4) THEORY find K We and BENDING are there in the is no twist curvatures of the small, curvatures curvatures the and associated twist are in the plate. slopes related to by K x ax y = - f-f 8x 2 ...(2.6) J 8 w 9x3y 2 K We consider downward Using the plate in in the curvature positive the direction of curvature x-y xy along coordinates the is = the plate is convex z. x-axis the deflection of the then -K v if (x -y ) J 2 ...(2.7) PLATE The of distribution the plate assumption. the can of stress be calculated Strains midsurface of vary the and strain using linearly plate and x e 8x Using the Eq. (2.6) plate and (z=-h/2) (2.8), Eq. circumflex we (2.5) curvatures, _ distance z from by ...(2.8) 3 w 3x3y 2 9 the surface = - ^ K 2 x y = - ^ K 2 y = +h K signifies demonstrated with given plate 2 x r 'xy In thickness Kirchhoff the 14 strains on the top of are e the the 2 xy e where the THEORY = -z |~7 8x y = r through with are BENDING no twist, ...(2.9) xy that that occur it is equal along a and the bending strain. opposite x-y axes. Therefore, A e A = -e x y .(2.10) = o A r 'xy The strains, and lie along e x and the e y are the diagonals. principal strains PLATE By measuring load, the the shear materials the constants is principal modulus surface can relationship be BENDING strain found. between for For the THEORY a given Eq. (2.5), corner isotropic three elastic G Combining 15 ...(2.11) (2.9) and (2.11), and rearranging, we find G ...(2.12) x Another method the plate the x-axis of along the determining one of the deflection G is diagonals W c , using -K c w Combining with Eq. small deflections calculate the strain deflection. or shear in Eq. a deflection distance (2.7), c. and an modulus of Along is 2 ...(2.13) (2.11), l ^ S i y 2 w h c For at the = —2— 2 c (2.5) through we get ...(2.14) 3 elastic from Conversely, the we material, we can load and surface predict the strain corner can PLATE and deflection a given for material BENDING knowing 16 THEORY the load corner appli ed. 2.1.1 In LIMITATIONS the into linear OF THE L I N E A R theory consideration neglects 1) the certain before limiting using Edge not this effects is are caused resisted by T vary along the section. Saint Venant's rectangular of theory. T , , x'y' and edges overall of if Midsurface deflected stresses by x , must The length the the of stresses will a be the be taken theory shows shear biggest x ratio occur the and the for a distribution , , y'z ' over but the occurs is at negligible large. because non-developable negligible cross- torsion deviation is twisting does plate of cross-section, a/h into alone, theory transverse The assuming , cross-section cross-section. the factors following. moment 2) THEORY if the plate surface. the a/h is The ratio is small. 3) The deformation neglected. large. This due is to transverse negligible if shear the a/h stress is ratio is PLATE In [2] the these total ratio 2.2 In a of effect NON-LINEAR surface the a The with Plates with be was strain large increase taken to is the in load The The to for the based on is of these is shown with twisted surface, curvature, plate linear an that a/h that i s , stretching of theory, the if the thickness of the as diverge the when of the deriving is and used. no had surface. deflections are plate. from non-linear There are of we middle the midsurface stretching problem equations the negligible coupled complicated. these it minimized stretching consideration equations. equations Gaussian no differential solution is non-developable deflections effect. into a discussion there considerable errors and occurs. respect with the into surface that discussed 17 10. non-zero midsurface small these plate previous assumed are THEORY THEORY , with middle the of approximately bending In limitations BENDING linear forces have midsurface the theory a must governing differential closed approximate form solution solutions PLATE 2.2.1 GOVERNING These 1910 f i r s t are and terms. Below for developed use future 2.2.1.1 The a It can of displacements directions [12]. the the where derived These problem most of of von equations Karman are the of the in the non-linearity significant summary by in non-linear equations involved Strain of be the shown the u, plate from midsurface v and originally differential is w which The l i e s in the midsurface the geometry dependent are in on the x, y strain x-y that the midsurface and z equations are au 3x . . . (2.15) y r.x y strains. to respectively. ex the in brief Geometric e 18 reference. stretching where only is midsurface plane. equations approximation plates THEORY EQUATIONS non-linear and BENDING au a y overbar signifies that these are midsurface PLATE We expect the inplane displacements, comparison with the lateral therefore, drop the second produce the BENDING u and displacement, order terms v, w. in THEORY to We u and be 19 small in can, v to strains x = 9u 8x y - | Y 3y I i 'xy Typically, when the midsurface become i 2 3u 3^ = i + ( 2 + ; | w ay ) ...(2.16) a , 3v ^ , 3 w . , 3 w . 3x 3x 3y° + slopes, large, aw ^3x ( } ( 3w/3x the and 3w/9y, midsurface of the strains also Hooke's law become large. 2.2.1.2 When Constitutive the assumption relations for an Relations a = z 0 isotropic is made, material a o e x = x E = -v reduce to y v =r L E a I-y the a =ZE + E ...(2.17) T 'xy where the midsurface overbar on the quantities. G stresses signifies that these are PLATE 2.2.1.3 The and the are in subject second adding, we + midsurface terms of to 3x^ u compatibility of v " 3x3y ( c , ) forces, N x = ho N = ho y (2.17) 1_ Eh from into Eq. Eq. ,_3_i ^3x 2 3 2 Eq. By (2.16), ( T x N 3x y / N y 2 } ( 3y ..-(2.18) 2 } can be N by xy expressed J x ...(2.19) y N xy Substitution condition. strains, x Eq. 20 producing and x midsurface the and " ~3x3y stresses, the a derivatives eliminate 3yT" The THEORY Compatibility strains taking BENDING = hT ((2.19) (2.18), xy into Eq. (2.17), yields the result V ,3 w x 3x3y and 2 k ; v 3x* ^3y* ; ; then from (2.20) PLATE 2.2.1.4 For in the A exact solution, unit no in the equilibrium small N x forces element of N each x, w i l l rectangle length body THEORY 21 Equilibrium equilibrium (x-y) BENDING y be and z size dx by the the plate should x be The inplane f i r s t . dy applied applied, of directions. examined and are element has as and midsurface shown y in forces Fig. 2.3. equilibrium of per If the requires 3N 3N 8 X 3 3N ...(2.21) y 3N - 3x T 2 + IT* 3y = 0 respectively. The higher order due to element not being equilibrium, ^ the midsurface the terms that might be planar, included are in Eq. small and N N (2.21), have been neglected. For z projected we find onto the z axis. forces Neglecting the x N y higher xy are order terms PLATE BENDING THEORY Figure 2.3, M i d s u r f a c e small forces element. 3w and N N , N^, + N §-7) d x d y x x y applied 22 to a , 2 XF (N x 3x 2 3N The quantities therefore z in they direction + 2N x y 3x3y 3N the are y 3y . 3 .(2.22) 3N square zero. 2 This brackets leaves 3N match the Eq. (2.21); equilibrium in the as (2.23) The sum o f the can be added to midsurface the load forces on the projected element, onto the q(dx)(dy), z axis which is PLATE supported edges a* by the a y ^ + " Substituting 8*w ^ ax^ Eq. there 3 v a ^ + four " = N x along the element's + N x a ^ + 2 Haf N x y + N y • • • relationships, Eq. ( 2 ' 2 through a A ) (2.3) D ( q to , .. 8 w N ax^ + 2 N 8w x y a ^ familiar 8 w. 2 . J y N ^ -..(2.25) bi-harmonic equation when forces. Function equations From x the no m i d s u r f a c e problem . J + are that the these must be three satisfied equilibrium and the to the the equations, compatibility we c a n s o l v e solve four Eq. equation, unknowns N non- Eq. , N , x ' y ' and w. y Theses If q 1 , = (2.21) a n d E q . ( 2 . 2 5 ) (2.20). ( moment-curvature reduces Stress 2.2.1.5 linear i^T ^ 8*w A are 2 the a x ^ 2 (2.25) The shears 23 gives (2.6), + and the THEORY giving - and moments BENDING equations the c a n be reduced introduction function, F(x,y), i s of down an A i r y found such to two stress that equations function[11]. PLATE BENDING THEORY 24 8 F 3y 2 x N 2 =hg y ...(2.26) 3 F = -h 3 x 3 y 2 Nx y The x-y equilibrium satisfied into 3*w 3x" the , by the stress remaining , 3 i w S^v 3x 3y 2 3y" 2 Eq.(2.21), equations, function. equations, D ^h 3y 3x 2 identically Substituting this function yields h , f l ^ F ^ w ^ F ^ w _ . B are 3x 2 3y 2 2 3 F 3w 3x3y 3x3y 2 . 2 . ; and a? Equations + 2 (2.27) a n d differential solution d i f f i c u l t to ' + a3?8F for in practical The the boundary conditions ( a^ V and the ••• - )] (2 coupled As F. } non-linear mentioned boundary 28) earlier, conditions the is problems. CONDITIONS to w and equations 2.2.2. B O U N D A R Y solution W (2.28) f o r m t w o equations these E [ ( plate bending imposed along problem the must edges of satisfy the the plate. PLATE BENDING THEORY A plate with to f r e e e d g e s h a s i s no m i d s u r f a c e forces 25 applied t h e e d g e s . The s t r e s s f u n c t i o n must s a t i s f y t h e following: x=0,a - z e r o normal force = o i ^ a y -zero shear 2 force ^ 3x3y y=0,a - z e r o normal = 0 ...(2.29) force a F 2 ax -zero shear 2 = 0 = 0 force a F 2 3x3y A l o n g t h e f r e e e d g e s t h e b e n d i n g moment a n d K i r c h h o f f are zero. x=0,a From t h e s e r e q u i r e m e n t s we g e t : - z e r o b e n d i n g moment a w +^v ix^ -zero Kirchhoff a w7= 2 axT y=0,a shear + j aT n 0 shear ( 2 " ^ a^yT " 0 ...(2.30) - z e r o b e n d i n g moment aw W aw _ w ~ 2 2 + v -zero Kirchhoff aw 3 W n 0 shear ,0 "" (2 . } aw 3 n = 0 PLATE The last applied boundary to the plate found for a along the edges. 2.3 A corner APPROXIMATE number of using In the and Of P, the a to variety the twisting From moment Eq. moment 26 (2.1) was we + 2M , between curvatures along the midsurface have developed twisted plate methods. opposite an midsurface a n t i c l a s t i c the strain plate diagonals. The strain varied with the midsurface strain changes and with equal results square of curvature. interest the twisting non-linear for plate load. [2,5,7,9] of relationship that the THEORY SOLUTIONS calculated plate in corner is showed the relates the load, solutions problem curvature by investigators approximate [2], condition BENDING after plate is how the are the point no of bifurcation, longer equal in where the magnitude. in the curvatures PLATE ANALYTICAL 2.3.1 WITH The UNEQUAL approach used in SOLUTION used [2], but is similar the the This degree of the loss forms UBC a of more Reduce, general a the curvatures freedom increases symmetry to displacement allows substantially FOR M I D S U R F A C E 27 THEORY STRAIN IN PLATES CURVATURES deflection extra BENDING in in function to be the the displacements algebraic approach the in magnitude. displacement number To for unequal the solution. symbolic Rayleigh-Ritz of field coefficients, u and manipulate v. the manipulation due to This equations, program, was used. Using the midsurface deflections be of estimated strain plates, by Eq. applying equations (2.16), the for the principle the large midsurface of strain minimum can potential energy. It is Fig. K x of assumed 1.1, is and Ky the plate that bent along is the into the then plate, an w the anticlastic diagonals given with of the coordinates shape plate. with The shown in curvatures deflection by ...(2.31) PLATE The midsurface strains, displacements in deflection The by the w. the Eq. u (2.16), and v in-plane BENDING depend directions on as displacement THEORY the well can 28 as be the approximated polynomials 00 u ^ = 2 . 1 , ...(2.32) J _ = . 2 d , x k,l where The i, j , k, conditions 1 are of symmetry y IJ 00 v 1 1 C.. x . K positive and k 1 y i integers. antisymmetry u(x,y) = u(x,-y) u(x,y) = are (2.33) -u(-x,y) and v(x,y) = v(x,-y) (2.34) v(x,y) These c^j conditions and d ^ are are zero = satisfied v(-x,y) if some of the coefficients giving i = 1,3,5,... j = 0,2,4,... ...(2.35) and PLATE k = 0,2,4,... 1 = 1,3,5,... BENDING THEORY 29 ...(2.36) Expanding power the series terms gives u = c 1 0 x + c 1 2 v = d 0 1 y + d 2 1 xy of + c Eq. (2.32) xy* + c 3 0 x x*y + d 0 3 y 1 A 3 up to and + c x y 2 3 2 + d x y 3 2 3 3 including + C + d 5 q X f i f t h 5 ...(2.37) Substituting Eq. (2.16) = c x = d y x y = 2 d Using £ io + oi + 2 ! the c d y t l 3 displacements, Eq. : y 2 2i^ ^ + the + c A i d i,y" d + J x 3 + 4 1 Y x* + 2 following . = x/a, rt 3c 3 0 + 3d d 23 y x x y 0 3 3 + + 3c 2 + 3d 2 2C 1 2 notation = y/a, x y 2 3 J xy x y c 0 5 ys and (2.37) into = K a , c d 2 1 = D / a , d „ = D /a* d 2 3 = D,/a\ d = D /a* c 3 0 = C /a , c d 0 1 = D , 0 3 = D /a , 2 0 2 3 x = C /a*. c 2 i a 4 2 3 + y* 0 5 + 2C |(Kx) ( K^ y ) (K + \ x y 3 3 J 2 2 + K K ...(2.38) xy p = K b y y = C /a* ltt 5 xy x" convenience 3 2 = 3 1 A c 1 0 5 0 + 5d 2 + 4C for P + 5c = C^/a , c d (2.31) 2 2 2 3 x c 2 gives 2 x x=y + d 2 0 0 5 = C 5 / a * 2 6 ...(2.39) PLATE leads BENDING 30 THEORY to ...(2.AO) = To 2Dtin + 4D &*n x 2 satisfy plate, these edges For along w i l l related T xy of 3 one of imposed have zero + 2c^ along stresses along then t h e edge to to also we w i l l equilibrium parallel and shear stresses Ac^n 2 c^n + conditions t h e symmetry A r b i t r a r i l y zero 3 the boundary due to other + 2D,c:n the normal setting then + 3 n + P P ^n x y the edges must be z e r o . the edges to of the By be zero, on t h e d i s p l a c e m e n t s , normal s e t ~o , and shear and T , . the stresses. to x'= a / 2 . a triangular the x and y axes, the midsurface edge element the boundary stresses ~a , x a y with sides stresses are and by o T Substituting strains, Hooke's E q . (2.40) = X x'y' o +o V x xy •2 2 law, Eq. (2.17), into ...(2.41) o -o x y E q . (2.41), and the and using generalized the notation PLATE gives the midsurface coefficients polynomials 2(l-v) + D and position i n E x' {C + D^ 0 stresses i n terms BENDING of i n the plate. THEORY t h e 12 The f o r t h + C ^ 0 2 + C n * + 3C-,cy» + 3 C A ^ n 3 J 2 + D,£* + 3 D 3 n 2 jC 2 1 + 5C c;* + 5 8 2 + 3 D £ n » ' + 5D 5 n" + ^ P n + | 2 - + 2D En + 3 4 2C lr\ + 4C2£n x 3 + ^,V^ . (D 0 + D ; + D E 2 2 IC 4 + 3D3n 2 + 3D,c;'n 2 • • • (2.43) + 5D5n" n )] 2 P y Along the edge, zero, the non-dimensional x'=a/2, w h e r e CT , x n Substituting from 2 a 4 + PxPy^nl ) p^ order £, a n d n a r e + cxUD^r, + A D : n + | 31 this into the expression and f , x coordinates , are are expressed by = ±z - I V2 E q . (2.43) gives ...(2.44) and factoring the I PLATE = gTJZ^y + D ["2aD, - 4aD - 2aC, - 4aC 2 + 5 C + 3 C , + C ] + 4V2 V 2 5 - 1 0 D - 3 D , - 3 C , - 2 C ] + 2V - 12aC 5 x + p + 2Dj + 2 C + p + 30D x y 2 2 2 5 - A a D , - AaC 3 2 0 2 - 6aD, 1 y 3 10D - 6D - 2 C ] + [4C + 2 C + 4D + p 5 + a C , + 6aC 2 + 2aDj + 2aCj + a D , + 2aC x t 32 6 + 3D, + 6D + 3C, + 6C 2 a + 6 C ] + 272 £ [ a p p - [~2ap P 2 THEORY + 5D + 3D, 2 [3aD, + 2aD 8 BENDING ...(2.45) 3 - 2 C - p.2 2 X + 5D + 6D + C,]} 2 0 5 3 and y' = 8(THO tlOD [ " 5 ° " 5 3 D * " ° + 3D, - 3 C , - 2C ] + 2 E g 2 + 5 C S + 3 " C + C ' ] + 4 [p£ - 2 D + 2 C - p 2 2 a 2 V L V - y 30D 5 (2.46) - 3D, - 6D + 3 C , + 6 C + 6 C ] + 2V2 ll-lC^ 3 3 + 6D - 2C ] + [4C + 2C - 4D 3 For 2 0 the normal boundary, each must (2.46) X 0 and shear term i n be z e r o . P y 5 stresses form Solving these becomes unwieldily at point. assumed the y , b e 1/3 constants other p to constants, we g e t this making be z e r o equations i=l along of equations 1/2. to 5, with in them, CQ a n d DQ a n d t h e c u r v a t u r e s , P v for terms of the x and t h e 12 a general Solving i n the E q . (2.45) To s i m p l i f y a equal and D ^ , where 5 2 brackets 10 constants. 6 to the square These y - 5D - 6D + C ] ) unknown is + P + 10D 2 and form PLATE Yk 12 C = ^ 3 ( " 1 1 6 (68C C (-4C k ^ Eq. (2.47) coefficients Placing Eq. a as - ° ( 2 0 0 certain might 0 y + 3p (-4p -p x x )] - ..(2.47) ^ 116D - 0 » C x + + 68D (AC ) 3p p ) 20D 2 0 D 0 1_ [-20C 72 In 44 + y p x 33 0 ^2 to 3 THEORY + 4D ) + (44C ° ~ P 4 4 D - 0 0 ° ( A C + - 0 [-52C j ° BENDING 0 - + 0 52D 0 3p p ) x 3p p ) v v x' y + 3 P y above results (2.40), gives the x 0 4 ° o + 3 p x y p } antisymmetry the (-p 4D ) + be y expected back strains from into as exists a the this in type strain function the of of problem. equations, position as PLATE i [2q* ( 3 p p = J7 x + n x 2 ("3P P x " y + E (-3pxPy - = y [2n* X 2 V I = W ("3P P y + AAC (3p + 3r,£(p P x To solve 20C xy P - y + energy energy per - 20D ) 0 0 0 52D ) - 116D ) terms V of o The elastic the volume + 20D ) 0 0 2 (C - 0 D ) 0 (3pxPy - + 2 n E in volume, 0 ' (3p xy " p 52C ° + 7 6 D o ) the V C and Q plate be D , we Q a require minimum. The that the strain is Q =2 - i (xoxe + yoye + xy'xy ? r ) only the + 68D ) 0 0 coefficients energy AAC 12D )] (2.A9) strains 2(1+\J) of + AC y 0 5 2 D o ) = T 7 T T - T [^r- 0 0 + 2AD ] 0 stored unit + 2i> 0 ° " n in 2 + 96n E 0 - V0 or x + D ) 0 0 + AD ) 0 (3p p 21" + (-C + 2AC ] 0 - 0 the strain + AAD ) 75C 12C for 0 2 + 96n S 0 34 ...(2.A8) " L 2 AAD ) THEORY y y x - 0 + 20C (-3P P x + V 0 (3p p ^ + n H6C 52C 2 i + 68C y BENDING 1-M ( stored, plate £ 2 x V + 2\)e , x is e + e) 2 y y found + \ 2 by r 2 'xy )] ...(2.50) integration over PLATE = A J" m J" x=0 In terms of THEORY 35 (a//2)-x a/V2 V BENDING V h dydx 0 . . . (2.5.1) y=0 non-dimensional coordinates 1//2 V . = Aha m J 2 J" 1=0 Upon substitution equations blossom Eq. in size. h a J E 2 minimize this cj + by dndE ...(2.52) =o (2.48) and After (2.50) 2 101152 CD + 0 0 into integration, 0 0 r + 1386A0 0 n [333 p p - 7776 p p C 60A.800 . x y "x y m We of V - we 7776 p p D "x y 1386A r (2.52), the get 0 D*] ...(2.53) setting 9V —52 8 C = ...(2.5A) av —51 When we carry equations for out these CQ a n d DQ, C 0 0 ° = 0 operations, we and 0 the two find = 3/1A6 pp ...(2.55) 7 D solve = 3/1A6 p p x y PLATE which we are equal. find Placing the following e x midsurface = [17011" - 109n + 1701" - 109c? 2 = [i70n3E x y relationship no v + + eirtf] n conclusions with v= n)/ are c a n be made of point XY strains strains, at a l l points curvature surfaces have the centre (3/146)p ' p x x The amount the and curvatures e in x surface the and e y , equal. Gaussian At the on the p l a t e the diagonals, The m i d s u r f a c e Y , about strains e x 4) ...(2.56) 1/3. the midsurface direction 3) P P 12] TIT^ + ^ the midsurface For any p a r t i c u l a r (£, 2) E q . (2.48), jgf- + 12] + 170&* " 2 between materials 1) into 36 JOH few i n t e r e s t i n g for THEORY strains 109n y A back = [17CV " e r the constants BENDING vary of , ~z a n d y linearly the plate. no m i d s u r f a c e the plate * a n d Y = 0. xy of thickness of midsurface of the the Cylindrical strain. e = "e = x y strain plate. with i s independent of 37 3. This section is DESCRIPTION OF EXPERIMENT divided into two subsections, apparatus and procedure. The It f i r s t section covers: the design with loads the plate used; and the design numerical analysis The section It other covers the operations 3.1 describes of the apparatus base plate, corner forces; of plates used the in describes i n i t i a l completed the the the in the tests. supports and instrumentation under test the before and plate experimental preparation during which the determining used the the size. procedure tests and used. the test. APPARATUS 3.1.1 DESIGN Shown in consists Fig. 3.1 of base, a is a two schematic supports of and the a apparatus. loading It system. The Figure 3.1, Schematic of the plate twisting apparatus DESCRIPTION apparatus of equal two is height, base is diagonal support Along two of cables heads straddle with the from a to test the the loading gap system of between f i t t i n g levelling test its is variable two tables between of the plate. For a set corners, attaching the and of holes the Along of The stabilized using the vertically hanger rests posts the different the support of allowing holes base one plates. each allowing restrictions. leveled sizes series has plate. positioning a base free from bolts and on the from the two screws. plate is supported with two 6" posts at either end diagonal. Pinned Support hemispherical aligns the steel different are through plate free the allowing diagonal plate, be 30"x30"xl/4" accommodate pass test to slots to other underneath The two posts the below made are cables size of to 39 EXPERIMENT tables. The The designed OF a function providing seat. matching is Post, to point A ball seat stop shown in the support. in Fig. bearing the plate plate from sits 3.2a, has in the above. The moving a seat and b a l l ' s laterally while DESCRIPTION OF E X P E R I M E N T PLATE CORNER •BALL BEARING PINNED SUPPORT POST PLATE CORNER BALL BEARING ROLLING SUPPORT POST Figure 3.2, Details Support of the Post, b) two support Rolling posts, Support a) Post. Pinned 40 DESCRIPTION The of Rolling circular with to Support the Post, cross-section. matching groove move inward during induced during bending. The plate Each is cable corners loaded is of shown the test Cable is in ball the bending, via passed A two plate. thus a and Fig. 3.2b, bearing 1/16" through plate in OF E X P E R I M E N T aligns This no has allows inplane then hole in clamped groove this groove the plate forces multi-stranded small a 41 are cables. opposite to prevent its prevents the removal. The Fixed test plate loaded. It through a corner runs hole protruding longer the loop the cable The Hanger to the base hanger, on hanger to plate plate, plate. plate hole the hole the achieve long and the other corner ending The a Anchor preventing is vertically in diameter, corner loop Rod, is the down just which placed is through extraction of hole. the carries weights 6" while base connects which placing test base through Cable by weight. the l i f t i n g the the across back loaded the in the and from from below than approximately of the other the 1, corner weights. 2 and desired 5 of The lb. load up the test plate sizes to 30 plate is in lb. series in DESCRIPTION With this method condition 3.1.2 with of plate uniform curvature of the principal surface the linear sides be measured were plate. the diagonals described the in free edge Section 2.1. on in be either of side of surface range, order calculated to the knowing plate. strains surface In is strains eliminate the on effect strains, mounted at gauges of the the were plate. rotational two rectangular centre, aligned A one with Mohr's on strain either side the 90° arms circle can then placement accuracy and numbering is in to of along be find used the strains. placement Strain following suitable easily non-linear surface The the gauge 3.1.2.1 The achieve strain. the the The as measurement must measure principal one the To can strain in midsurface check plate but of rosettes the range, sufficient, to twisting we 42 INSTRUMENTATION The both loading, OF E X P E R I M E N T Gauge items strain and shown Fig. 3.4. Selection were gauge: considered in the selection of a DESCRIPTION 1) strain-sensitive 2) backing 3) gauge length, 4) gauge pattern, 5) self 6) grid 7) The material, compensation, resistance, selected was It a is temperature completely general corrected in terminals following Micro-Measurements a encapsulated copper-coated the alloy, temperature CEA-13-125UR-350. is 43 encapsulation. gauge which OF E X P E R I M E N T purpose for easy number constantan aluminium. polyimide, for gauge with It is large soldering. The rosette a gauge, integral, gauge has characteristics: 1) gauge length of 2) gauge pattern .125", is a 45° in co-planar orientation, 3) resistance 4) gauge The data lots of The sheets strain strain of factor are of in 2.0. Appendix A . l following the for the two used. were mounted instructions[13]. The wires prescribed manner for 1/4 loops formed in were and approximately included gauges gauges 3508, the were bridge wires soldered on operation. adjacent to manufactures in the Strain the gauge relief and the DESCRIPTION wires were taped firmly to the plate to OF E X P E R I M E N T prevent 44 induced strain. 3.1.2.2 A Strain Vishay was Indicator Instruments used with a model Vishay specifications appear The gauges, in six a quarter Unit. and be strain This bridge allowed switched read The to from three 3.3, was with high quality compared used. of because to the from the gauges strain digital Fig. neglected Appendix It the gauge and two to Strain Balance Indicator Unit. Their A.2. rosettes, to be the were Switch where the connected and independently indicator Balance balanced strain could display. bridge is both temperature the Configuration Digital Switch configuration quarter desensitization in Bridge P-350A SB-1 the the the wire and gauge, wire configuration, simple and shown accurate in when compensated gauges. due resistance, to resistance resistance. wire was used The insignificant was when DESCRIPTION STRAIN OF E X P E R I M E N T GAUGE — DUMMY Figure 3.3, 3.1.3 PLATE Shown in bridge 3.4 is the dimensional at "A" at description configuration. DESIGN associated applied V RESISTOR Quarter Fig. corners 45 by the corners of design of parameters. support "B" by the loading is in the plate Upward posts and cables. Section tested loads were downward The with applied loads detailed 3.1.1. a TOP the B VIEW c- 2 (5) a NUMBERS IN BRACKETS D E S I G N A T E G A U G E N U M B E R S ON B O T T O M S I D E . Figure 3.4, Design of plate and location of strain gauges. DESCRIPTION It was that found the from i n i t i a l considerable; plate 3.1.3.1 for by The amount strain large Eq. we if at z= the was loads self weight was approximately clearly its found plate +h/2. plate small, neglect (2.10), own plate were not applied, 1/2 the linear. weight proper 1 prompted lengths and effect. Non-Linearity was midsurface the (2.16), its unwanted non-linearity the terms versa, to ascertain Plate loaded was 24 " x 2 4 " x . 1 2 5 " 46 at with the free time edges of the and research supported corners. of then of a corner by this solution a uniformly opposite the to reduce Estimate comparing If to of deflection caused investigation analytical due relationship non-linearity thicknesses No centre t h i c k n e s s e s . When further tests deflection the load-curvature This i n i t i a l OF E X P E R I M E N T it the and w i l l f i r s t of the be maximum to the If ratio of these the term the can terms be by plate strain deflection divide ratio in the w i l l be estimated bending terms non-linear, is and vice linear. of the midsurface surface maximum bending strains w i l l strain, strain, be by Eq. DESCRIPTION (w = .' hw N* where the A the comma amount linear of the midsurface The in shown With deflection equal difference Appendix the ... (3.1) ,xx program using to Eq. must to the be zero. The procedure From boundary to the conditions midsurface explained and force. the results (w, ) 'x into and (w, max Eq. (3.1) to )' 'xx'max can determine that measure the used. in so method The far used value as a of is N larger very w i l l rough give number B.3, the maximum slope and curvature due a represents non-linearity. Appendix are ratio. approximations greater developed zero is determined, substituted c l a r i f i e d quantitative a and non-linear It represents B. deflections calculated the N (2.25), w i t h ' be and was 2.2.2, a g a i n w i t h Section in 2 X differentiation difference forces f i n i t e 47 non-linearity. f i n i t e calculate shown denotes ) OF E X P E R I M E N T are DESCRIPTION (w (w where is q the is the flexural ) , x max = ) , x x max = plate self-load, rigidity. a ^ ...(3.2) Sii D is the Substituting N* edge Eq. length, and into Eq. (3.2) plate self-load = .119 ^ D where p is the Substituting material this ...(3.3) is q and = ...(3.4) ph density. flexural rigidity, Eq. (2.4), into Eq. gives (3.3) N* Eq. .357 D 48 gives (3.1) The .206 OF E X P E R I M E N T is (3.5) values of increases (l-\>») and graphed rearranged a/h, self-weight. = 1.43 As there the is l i t t l e ratio exponentially. ...(3.5) in Fig. non-linearity increases, the 3.5. due For to small its non-linearity DESCRIPTION OF EXPERIMENT 49 CN 0 0 0.0 Figure The Graph 3.5, i n i t i a l where 24" the non-linear plate pronounced, reduce of 40 .0 tested, had a/h= non-linearity 80.0 PLATE SIZE, a/h ratio in v s . plate which the By r e d u c i n g 192. ratio 120.0 by over size. non-linearities a / h by a half an order of magni tude. 3.1.3.2 S i z e A l l plates thickness used in material and were of a l l Material made .125". the from T h e same specimens properties 6061-T6 material due t o a r e shown aluminum in i t s with a nominal and t h i c k n e s s a v a i l a b i l i t y . Table 3.1. were The we DESCRIPTION Density - Ultimate Strength, Strength, Tensile Tensile Shear Elastic Moduli, Poisson's Ratio displayed i s in Table p s i G - 3.75 x 1 0 6 p s i 0.33 - properties for • a n d 6061 T6 varying properties, sizes of Length, plates a 0.125" Non - l i n e a r * Ratio, 24" 2.16 18 0.683 12 0.1350 10 0.0651 8 0.0262 6 0.0084 3.2, Plate the A l u m i n i u m 6061-T6 Thickness: Edge x 3.2. Material: Table 20 K s i - 6 .125" calculated 35 K s i - * N 24 K s i - 10 the of 38 K s i - 10.0 Using value 3 - 3.1, 6061-T6 M a t e r i a l of size 50 E Table thickness l b / i n .100 Shear Yield OF E X P E R I M E N T and n o n - l i n e a r i t y ratio. N and DESCRIPTION Based on were 3.4 this analysis the and are chosen The variation were f 6" .125" 0.6" 0.3" 8 .125 0.8 0.4 10 .125 1.0 0.5 12 .125 0.9 0.5 from examined. to design on the For the and 0.1% c r i t i c a l dimensions respectively. 2% stock largest manufacturing in Fig. parameters. nominal selected shown 51 3.3. e approximately machined Table h Plate 3.3, parameters listed in a Table design OF E X P E R I M E N T where plate the was a p p r o x i m a t e l y the The flatness i n i t i a l a and plates was +6% of were f i r s t deflection +0.007" o r h, due the thickness. 3.2 3.2.1 The EXPERIMENTAL INITIAL i n i t i a l PROCEDURE PREPARATION preparation calibration check the and weights of the hanger. prior to strain testing, indicator consisted and of weighing a of DESCRIPTION The strain indicator internal calibration external precision described The Switch Indicator The on 0.1 and an TEST strain Unit. Each quarter Switch check ensure were resistor. Balance Unit the and electronic the The instruction system was done indicator using and calibration the an method is manual[14]. was connected schematics weights balance. on were the to the Unit. calibrated Each Strain by was measured the Switch measuring to within PROCEDURE gauges gauge corresponded A in check 52 gram. 3.2.2 The the circuit following hanger them in calibration OF E X P E R I M E N T to bridge and the recorded. is to identified channel hookup Balance same connected number, l i s t / d a t a the were to by which described by Fig. it was the and Balance 3.4, connected. schematic The on the Unit. recording procedure sheet was was used followed. The for each test following to items DESCRIPTION 1) Date. 2) Plate size. 3) Gauge factor for the OF E X P E R I M E N T particular lot of 53 gauges used. 4) Strain 5) The six controls: a) gauge b) calibration c) balance switch off, d) quarter bridge switch Room strain Indicator factor set to 2.00, switch off, and temperature. gauges must be balanced unstrained position. To bending- of the plate under its vertical plane during the on test held The ball The in the plate was bearings ball positioned centre The then of at the strain the the which end plate sits of the strain own weight, the the plate the plate induced the shown Rolling groove an the plate was process. apparatus as in by balancing corners on the with with in Fig. Support nearest the 3.2. Post to was the plate. with zero cables were Strain measurements lb. corner of negate mounted under bearing, on. then corner attached load. were load and made was the at measured. strain 2 lb. was The again increments two recorded. up to 30 DESCRIPTION During were the taken unloading of periodically the to plate check the for OF strain EXPERIMENT measurements measurement d r i f t . 55 4. In this It is broken strains the section on into both amount Each of section been for i n i t i a l the In of plate sections, of plate, along the under the raw results. In and strains and curvatures the or have the principal other in an describing no uncorrected correction curvatures its described. diagonals. form the are containing displayed strains plate results plate uncorrected i n i t i a l the One the results the RESULTS own corrected been has caused weight. and by These form subtracted are the from a l l results. The to the the deflection essentially two sides has form. the individual curvature corrected made the EXPERIMENTAL Gl the and plate corners and comparison curvature when directions G3 diagonals between are negative. graphed on the following graphs running between the hanger corners respectively. the purposes, shown the absolute with the value sign of shown support the in post strain the refer For and legend box EXPERIMENTAL 4.1 PLATE The data PRINCIPAL from computer. the Using lot 2) calculation 3) The gauge 4.1a-d. The be by seen increases G3 The in size in entered following into was a done: correction, strains in the axes of the strain at due zero Gl results to. the corner offsets larger. strains. strain strains the the in are plate load. both shown the Fig. self-weight As Gl in the and plate G3 Under the self-weight of direction are larger those than can the plate in the direction. corrected results, strains figures a the due to progression plate strain, one principal become strains the principal offset i n i t i a l to of were and i n i t i a l the directions the factor calculation uncorrected sheet spreadsheet 1) gauge, 56 STRAIN recording a RESULTS size depending can on half the value symbols. The greater change in slope of shown in the be seen. the principal The have can the of be in the black the these relation midsurface found strain, representing had Comparing strains amount between midsurface line 4.2a-d, removed. direction, strain the Fig. self-weight the of in by or the taking white greater the non-linearity. EXPERIMENTAL RESULTS 57 PLATE PRINCIPAL STRAINS UNCORRECTED TOR PLATE WEIGHT MATERIAL: ALUMINIUM SIZE: 6'X6'X.125' 0.0 100.0 200.0 300.0 -300.0 500.0 600.0 700.0 800.0 PRINCIPAL STRAIN (MICRO-STRAIN) Figure 4 . 1 a ,P l a t e weight. principal 6" plate. strains, uncorrected for plate EXPERIMENTAL RESULTS 58 PLATE PRINCIPAL STRAINS UNCORRECTED FOR PLATE WEIGHT MATERIAL: ALUMINIUM SIZE: 8"X8'X.125" LD O • rO / LD • CM CO G O o CE r\j O f LJ 2 o or • o £ o y LJ CE 5?° • o • n LEGEND (NEG.) - Gl DIR.,TOP = G2 D I R . , T O P - G l DIR.,BOTTOM •= R ' D I R . , B O T T O M ( N E G . ) 2 O 0.0 100.0 200.0 300.0 PRINCIPAL Figure 4 . 1 b ,P l a t e weight. STRAIN principal 8" plate 400.0 500.0 600.0 700.0 (MICRO-STRAIN) strains, uncorrected f o r plate EXPERIMENTAL RESULTS 59 PLATE PRINCIPRL STRAINS UNCORRECTED FOR PLATE WEIGHT MATERIAL: ALUMINIUM SIZE: 1•'XI••X.125' o in m • I I j I I 200.0 300.0 400.0 500.0 I CD 0.0 100.0 600.0 700.0 PRINCIPAL STRAIN (MICRO-STRAIN) Figure 4.1c, Plate principal weight. 10" plate. strains, uncorrected for plate EXPERIMENTAL RESULTS 60 PLATE PRINCIPAL 5TRRIN5 UNCORRECTED TOR PLATE WEIGHT MATERIAL: ALUMINIUM SIZE: 12"X12'X.125" o m i i ro 0.0 Figure I 100.0 200.0 i i 300.0 i 400.0 500.0 PRINCIPAL STRAIN (MICRO-STRAIN) 4.1d, Plate weight. principal 12" plate. strains, uncorrected i 600.0 for 700.0 plate EXPERIMENTAL The 6" that by plate, diverge the Fig. with increase two sets the curvatures equal, The values of both diverge. The the the same. The and 10" are plates are in which which the same, the fact for a where the 6" i n i t i a l l y caused that these that strains given characteristics to is indicates midsurface similar lines divergence The 61 both are load. at small plate. After coincident also strains along the diagonals are the curvatures along the diagonals are no plates, between has coincident formed diagonals are midsurface longer which the The strain. are and 4.2d, lines therefore, 8" of almost load. lines equal, Fig. the two midsurface are strain point same, in plate, has increasing coincident along 12" this of 4.2a, RESULTS Fig. the two showing 4.2b,c, have extremes. there is characteristics The a thickness dependence of on the edge length. 4.2 PLATE CURVATURE The curvature due to is bending. following was related To done: directly calculate the to the strain curvature in in the the plate, plate the 62 EXPERIMENTAL RESULTS PLATE PRINCIPAL STRAINS CORRECTED FOR PLATE WEIGHT MATERIAL: ALUMINIUM SIZE: 6'X6'X.125' D.O 100.0 200.0 300.0 400.0 500.0 600.0 700.0 BDO.O PRINCIPAL STRAIN (MICRO-STRAIN) Figure 4.2a, Plate weight. principal 6" plate. strains, corrected for plate EXPERIMENTAL RESULTS 63 PLRTE PRINCIPAL STRRINS CORRECTED FOR PLRTE WEIGHT MATERIAL: ALUMINIUM SIZE: 8'X8'X.125" 0.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 PRINCIPAL STRAIN (MICRO-STRAIN) Figure 4 . 2 b ,Plate weight. principal 8" plate. strains, corrected for plate EXPERIMENTAL RESULTS 64 PLATE PRINCIPAL 5TRRIN5 CORRECTED FOR PLATE WEIGHT MATERIAL: ALUMINIUM SIZE: 10•X]•'X.125" 0.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 PRINCIPAL STRAIN (MICRO-STRAIN) Figure 4.2c, Plate principal weight. 10" plate. strains, corrected f o r plate EXPERIMENTAL RESULTS 65 PLATE PRINCIPAL STRRINS CORRECTED FOR PLATE WEIGHT MATERIAL: ALUMINIUM SIZE: 12'XI2'X.125' o Ln ro i i 0.0 i i 100.0 200.0 | 300.0 100.0 | 500.0 I 600.0 700.0 PRINCIPAL STRAIN (MICRO-STRAIN) Figure 4.2d, Plate principal weight. 12" plate. strains, corrected for plate EXPERIMENTAL 1) Find the the bending magnitude diagonal 2) Use Eq. 3) Subtract and strain of on (2.9) the taking principal opposite to off by the i n i t i a l the of the 66 average strains sides calculate RESULTS the of along the plate. curvature. curvatures due to plate wei ght. Shown in Fig. the different the i n i t i a l negative due 6" and opposite along order With the of other along the after which the The curvature slight is the that are the Only only expected maximum diverge for are corrected for the G3 d i r e c t i o n is curvatures at the are equal largest showing the onset only of slightly curved producing from linear theory the curvature. 4.3b,c equal relationship used. divergence is Fig. in system shows line lines results diagonals. plates, diagonals the a what 10% a t The 4.3a, The from load-curvature coordinate there i n s t a b i l i t y . the plates. the Fig. is deviation the to plate, curvatures a sized is curvature. The the 4.3a-d and d, for small showing the curvatures curvatures, bifurcation. in EXPERIMENTAL RESULTS 67 PLATE CURVATURE CORRECTED FOR PLATE WEIGHT MATERIAL: ALUMINIUM SIZE: 6'X 6•X.125" in ro / o ro LD « 1 r CO CD o £ CE C°M- O LJ J* G3 HIDDEN 2 o ct: • o o L J cr / / LEGEND • - GR UGE 1 DIR. o - GA UGE 3 DIR. (NEG.) D LD " / D D. 0.000 0. D02 0. D04 0.006 0. 008 0.010 0.012 0. CURVATURE (I /IN. ) Figure 4.3a, P l a t e plate. curvature, corrected for plate weight. 6" EXPERIMENTAL RESULTS 68 PLRTE CURVATURE CORRECTED FOR PLRTE WEIGHT MATERIAL: ALUMINIUM SIZE: 8'X8."X.125" Ul o• in • (M CO CD o / O LJ • CD O / LJ f— cr o LEGEND • - GF1UGE 1 DIR. o ^ - o r1UGE 3 DIR. (NEG.) o in o D 0.000 0.002 0.004 0.006 0.0DB 0.010 0.012 0.014 CURVATURE (1/IN.) Figure 4.3b, Plate plate. curvature, corrected for plate weight. 8" EXPERIMENTAL 69 RESULTS PLRTE CURVRTURE CORRECTED FOR PLRTE WEIGHT MRTER]RL: RLUM1NIUM SIZE: 1•'XIO•X.125 * 0.000 0.004 0.002 0.006 0.012 0.008 0.014 CURVATURE ( 1 / 1 N . ) Figure 4.3c, Plate 10" curvature, plate. corrected for plate weight. EXPERIMENTAL RESULTS 70 PLATE CURVATURE CORRECTED FOR PLATE WEIGHT MATERIAL: ALUMINIUM SIZE: ]2•XI2'X.125* 0.000 0.004 0.002 CURVATURE Figure 4.3d, Plate 12" curvature, plate. 0.014 0.008 0.006 ll/IN, corrected for plate weight. EXPERIMENTAL The effect plates. of For curvature non-linearity small values relationship become stiffer Due to self-weight, the Gl direction expected Gl 8" after direction would in the after did is the linear, along curvature the of of the manufacturing. The slight large enough a to the corner which larger load- the larger plates curvature G3 d i r e c t i o n . plates found in 71 region. It while the exhibited that this i n i t i a l trigger G3 the other the curvature was curvature be that this, plate in would bifurcation increase was was the curvature would seen after have It direction not. curvature plates point Three of clearly non-linear than decrease. plate the is RESULTS but not in mode the flat the G3 after bi furcation. When the lateral could plates load be its was created cardboard. to where In i n i t i a l a l l loaded applied as can cases shape. to be past see done this the if the with failed point a and other of stable twisted the bifurcation mode sheet plate of returned a 72 5. COMPARISON OF RESULTS WITH THEORETICAL AND NUMERICAL MODELS The results, shown non-dimensional be made about limited graphs to the then Section form. plates are in This 4, allows deformation of of particular compared are to compiled more and general the plate size. The theoretical shown in statements which are to not non-dimensional and numerical results. 5.1 COLLAPSING The principal corrected to a which strain for form from curvature form coincident this in F i g . in which curves. collapse graphs, 4.2 The come FORM the and which 4.3, results were are from non-dimensional partly from theory reduced the four groupings and observation. non-dimensional the and NON-DIMENSIONAL self-weight achieve partly from INTO non-dimensional plates The DATA analysis in groups for Section the 2.3.1. midsurface strain come COMPARISON For to convenience, their 5.1.1 From the terms non-dimensional PRINCIPAL E q . (2.12) curvature quantities i n this we g e t t h e p r o p o r t i o n a l G with and strain relationship - h T G E q . (2.11) To keep ••• for isotropic the non-dimensional by any non-dimensional to multiply 2 by (a/h) a l l the materials dimensional Shown vs. i n load principal sides of the tested was taken F i g .5.1a-d strain plate. form materials, ( 5 - 1 ) we g e t ...(5.2) we c a n m u l t i p l y quantity. hT" As refer section. e « (1+v) £ 7 ^ ' need 73 STRAINS e Replacing load, OF RESULTS To c o l l a p s e both sides the results we giving tt have h^E a \>=.33, ...(5.3) the n o n - 2 4 as Pa / E h . are the non-dimensional graphs along and on the two diagonals for both load COMPARISON OF RESULTS P R I N C I P A L STRAIN TOP GAUGE 1 DIRECTION CORRECTED TOR PLATE WEIGHT MATERIAL: ALUMINIUM o o p / x x LJ \ OO LEGEND • - 12 X12 X.125' o - ]0"X]0'X.]25" A8'X 8"X. 125" + - 6"X 6"X.]25' LD , X X D D_ , / / O — CE O / LJ -z. or o o / LINE/01 THEORY 9 y CE t\ y O CO ZZL j y LJ o tn I CD o o 0.0 ].0 2.0 3.0 4.0 5.0 NON-DIMENSIONAL PRINCIPAL STRAIN, £ a x x 2 / h * K 2 Figure 5 . 1 a ,Non-dimensional di rection. principal strain. TopG l 6.0 COMPARISON OF R E S U L T S P R I N C I P A L STRAIN TOP GAUGE 3 DIRECTION CORRECTED FOR PLATE WEIGHT MATERIAL: ALUMINIUM D LD r\j I I j LD x x LEGEND 12 X12"X.125' 10 XIO'X.125' 8 X B'X.125" 6 X 6'X.125' 3 9VX3.94X _c LJ \ ro x X D 0.0 Figure 0.5 1.0 1.5 2.0 NON-DIMENSIONAL PRINCIPAL STRAIN, 5.1b, Non-dimensional di rection. principal strain. 2.5 £a**2/b**2 T o p G3 75 COMPARISON OF RESULTS 76 P R I N C I P A L STRRIN BOTTOM GAUGE 1 DIRECTION CORRECTED TOR PLATE WEIGHT MATERIAL: ALUMINIUM o in — 0.0 0.5 1.0 1.5 2.0 2.5 3.0 NON-DIMENSIONAL PRINCIPAL STRAIN, £ax*2/hxK2 Figure 5.1c, Non-dimensional principal strain. Bottom G l 3.5 COMPARISON OF 77 RESULTS P R I N C I P A L STRRIN BOTTOM GAUGE 3 DIRECTION CORRECTED FOR PLATE WEIGHT MATERIAL: ALUMINIUM o o 0.0 3.0 2.0 3.0 4.0 NON-DIMENSIONAL PRINCIPAL STRAIN, Figure 5.1d, Non-dimensional principal strain. 5.0 6.0 £o**2/h**2 Bottom G3 COMPARISON There is excellent the In results collapse onto one coincident In 5.1b 5.1c the results and linear between plates. Fig. both agreement and non-dimensionalized and the edge thickness the present This length comparison results groups. from to this the agreement thinner Up a to This of of the sections four the curve. from the tests Fig. 4 plate of has smallest check strain plate done identified of are [3] are about plate here. a radical plates the change occurs at a the the good After half used in 1.5, the in takes point, [3], rate as is the For shape the with which place. anticlastic greater non- agreement this point departure from on approximately in bifurcation and shape further a smaller ends synclastic non-linear plotted. provides experimental close results 78 experiment. dimensional the and the OF R E S U L T S to load the is increased. The experimental principal while up agreement to the the strain to is 2 on due bending second. results agree of the the to 1 on Gl the strains Top G3 and with the linear Top and Gl G3 B o t t o m midsurface in the f i r s t Bottom graphs. strains case line are and up to a graphs, This being added subtracted in COMPARISON 5.1.2 The PRINCIPAL 79 CURVATURE non-dimensional relationship OF R E S U L T S come quantities from Eq. for 2.5. the By load-curvature multiplying both sides by 2 a /h we get the proportional Ka IT which collapse As the in have a v=.33, Using these directions quantities are shown result shown approximately and on case, the ••• a l l the materials non-dimensional curvature of directions the 3.0 c ( 5 - 4 ) load tested was bifurcate linear Up linear the showing a n t i c l a s t i c the to a theory Gl shape in curvature load-curvature 4.0 in and G3 5.2. the graph. the and curvatures Fig. using the to the in the 3.0 corresponds between " strain therefore expected also 2 2 4 Pa / E h . as is , ,x , , Pa iVE J results. principal taken The the relationship theory closely. in the the of the start the 2.1 of relationship curvature and Section is Between Gl and linear a G3 transition synclastic shape. COMPARISON OF R E S U L T S 80 CURVATURE CORRECTED TOR PLATE WEIGHT MATERIAL: ALUMINIUM SIZE: .125' THICK o 12 o = 12 A = 10 + = 10 X = 8 o = 8 V = 6 B = 6 • = LD D a tr o 0.0 Figure 5.2, ] .0 LEGEND X12" GAUGE XI2' GAUGE XI •" GAUGE X10' GAUGE X 8' GAUGE X 8' GAUGE X 6' GAUGE X 6' GAUGE 2.0 3.0 1 3 1 3 1 3 1 3 DIR. DIR. DIR. DIR. DIR. DIR. DIR. DIR. 4.0 • • 5.0 6.0 NON-DIMENSIONAL CURVATURE Non-dimensional curvature. 7.0 8.0 COMPARISON It is within which the the the in the thickness From from react in lines to of the non-linear the theory region, same the After linear the plates manner as region bifurcation increases with the non-dimensional the 81 in point quickly. same curvatures form form. STRAIN midsurface non-dimensional found the in MIDSURFACE the occurs. non-linear coincident 5.1.3 section bifurcation deviation Even f i r s t OF R E S U L T S strain analysis relationship at the in Section centre of 2.3.1 the the plate was be e = x e y « p K x p K ...(5.5) y whe r e p x = K a x p y = K a y K K The midsurface Gaussian curvature relationship amount of strain for in a l l scatter. is plotted Fig. the 5.3. plates (5.6) against The non-dimensional results tested with show a linear just a small COMPARISON OF R E S U L T S 82 MIDPLRNE STRRINS UNCORRECTED FOR PLATE WEIGHT MATERIAL: ALUMINIUM SIZE: .125' THICK o o D. I O CD. o o• cn i CE LEGEND 12'X12 GAUGE o = 12'X12 GAUGE A = 10'XIO GAUGE + = 10X10 GAUGE X = 8'X 8 GAUGE O = 8'X 8 GAUGE V -= 6 X 6 GAUGE 6 X 6 GAUGE • = DIR. DIR. DIR. DIR. DIR.. DIR. DIR. DIR. o• ZS 7° CO I co i o oc o i +/ XL- CD- f CE f— CO • X o• "0>5 3 X" I LJ ZZ. <^v ° CE o- x vx CL Q _ 1 O • ro CD CDCM I CD CD I O CD 0.0000.000 -0.002 -0.003 -0.004 -0.005 -0.006 -0.007 -0.008 -0.009 -0.0)0 NON-DIMENSIONAL GAUSSIAN CURVATURE Figure 5.3, Non-dimensional midsurface strains. COMPARISON Linear Using regression this line was we get e The linear one explained analytical The of to and not is the but 2 field in through shown and the of in u v less The used than line. the discrepancy in freedom [2] this which the is number of displacement field more increased, decrease, in exact can developing the an decrease, the overall unequal degrees of would As of in was only freedom the number plate plate sense the calculated true displacement of curvatures strain the the of strain true the for represent the opposite it (i.e. degrees freedom the calculated on degrees problem equal on converging normal the 12 solution. f i e l d converging of had requirements closely. the in are number the the i n i t i a l l y symmetries not with fit . . . (5.7) 40% analysis therefore allowing shape of equilibrium Therefore best 2 (2.55). the displacement also Eq. is satisfy problem, would .0124 in type degrees increasing freedom of the equation = .0124 K K a x y y 3/146 calculate 83 result. DOF). As does By by freedom, one By of displacement reduced = e x to the coefficient analytical be used OF R E S U L T S true degrees of energy energy. strains would strains. the as plate in to Section assume 2.3.1, a COMPARISON the point of the total potential the degree have to As f i r s t a good. off, 5.1.4 be To predicted achieve an of the inplane approximation the theoretical by 84 minimizing accurate result displacements would increased. showed the magnitude of the calculation linear there was a curvature, which was experimentally this plate relationship w i l l COMPARISON be used is linear relationship independent was coefficient that that the could energy. freedom Although Gaussian of of be it fact bifurcation OF R E S U L T S with v e r i f i e d . of the was the The thickness later. OF L O A D - C U R V A T U R E RESULTS WITH APPROXIMATE best fit SOLUTIONS The dotted line load-curvature experimental in Fig. results results 5.4 in are Chandra[5], Reissner[9], Hsu[7). The result dashed line. All the results curvature from are approaches represents Fig. the 5.2. and linear asymptotic zero. Compared approximate Ramberg the the with to theory is and of Lee plotted linear the these solutions Miller[2] the of result and with a as the COMPARISON OF R E S U L T S 85 COMPRRISON OF RESULTS CORRECTED FOR PLRTE WEIGHT MATERIAL: ALUMINIUM o o LEGEND EXPERIMENTAL Gl DIR. EXPERIMENTAL G3 DIR. LINEAR THEORY CHANDRA REISSNER RAMBERG AND MILLER v - LEE AND HSU o o A + x o in O Ln O CE O _} _ or or o o CE - 9 II I 4 // i / / OJ ' /* o o Ii > / / / s £ * D I Z. /. I- / CO O /p 1 J Ln o UJ 0 I i * /* • ' '• o in A y y > in o o 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 NON-DIMENSIONAL CURVATURE Figure 5.4, Comparison results. of approximate and experimental 9.0 COMPARISON 5.1.4.1 Taking RESULTS 86 Chandra[5] the load-deflection converting we OF it to the form relationship to be in graphed, [5], with and v=.33, get Pa Eh" 1 Ka 2 The f i r s t part of term the the equation 2 square and ...(5.8) [1 + .0270 (^-) ] h 8 in 2 2 brackets the second represents term the the linear non-linear part. As can be seen experimental in Fig. 5.4, data very well. corresponding In Chandra's represent The vary This stresses but over the along the only extra strain imposed midsurface strain energy plate results. Airy the stress of fit of 6, function the are the the plate are not. and satisfied in not curvature boundary edge on does the the high. stresses along is a conditions shear condition (5.8) At 40% the boundary periodically boundary is analysis, the normal exactly, load Eq. in zero an boundary plate and very not well. satisfied The the does shear shear overall distorts an overly stresses stress sense. the s t i f f COMPARISON Reissner solves into the non-linear this solution the E q . (31) form with the of [9] \>=.33, w e 2 equation is strongly from Reissner solves he along he of It a long is rectangular interesting experimental two of results to plate compare from twisting the the of boundary conditions are constraint is not although a non-dimensional short the the plate. the compared a deviates square are better equations. Along in the to the plate than To do conditions the other an o v e r a l l results imposed with met w i t h it boundary conditions when they where von Karman principal sides ...(5.9) results. satisfy sides if not F i g . 5.4 non-linear Venant's the in exactly opposite into [1 + .0222 ( M i ) ' ] h I Mi 8 h experimental meaningful match, rearranged get = the Saint is graphed can only satisfies view region. with Pa Eh* this twisting plates. When This 87 RESULTS Reissner[9] 5.1.4.2 square OF w i l l latter sides sense. only be boundary former. This and the results [5]. In do COMPARISON 5.1.4.3 By Ramberg taking Eq. dimensional and (67) form, RESULTS [2] we get = | ^ and p u t t i n g it into non- 2 This to equation due as to the the the 5.1.4.4 When the Figure theory. Taking dimensional Eq. missing. (14) of midsurface strain of freedom the section, close results up 6. to the Lee and Hsu were an e r r o r shows the a line slope quantities of for respectively, Either experimental of the is incorrect displacements, energy actual as stored a on good produced. results [7] 2 Hsu[7] results 8 of is is of ...(5.10) (~-) ] of previous strain f i t the curvature degree the Lee and experimental and in midsurface experimental with distribution limited discussed [1 + .00917 corresponds a non-dimensional Although 88 Miller[2] of Pa Eh* OF both the compared was found in representing this line deflection we find graphs the their work. linear plate and e q u a t i n g and moment, there axes, to is a the nonEq. factor F i g . 8 and 9, (5) of have two COMPARISON been labeled wrong or has been incorrectly the 4 the This in change one of typed. denominator the diraensionless Most of Eq. OF R E S U L T S l i k e l y (14) it provides the closest fit to correction the results are shown quantities was should the have the 89 latter; been a 2. experimental results. With this coincides with dimensional showing Fig. 8 seen in As seen for a could 5.2 A grid Fig. and scheme in 9 of for the CURVATURE was those [7], of 12. B. then point at a to a starts of much up to It nondeviate bifurcation higher 5.4. load shown than in what is results. the moment This ITS between predicted Appendix results Fig. has along not with yet converged machine errors discrepancy. UNDER made and The occurs division comparison weight also 5, experimental account PLATE of stiffness. [7] the in experimental curvature less of the in OWN W E I G H T the from curvatures the linear due finite to selfdifference COMPARISON OF RESULTS 90 CURVATURE DUE TO PLATE WEIGHT COMPARISON OP EXPERIMENTAL AND PD RESULTS MATERIAL: ALUMINIUM SIZE: .325' THICK « 'o o in to. • o A + = = - LEGEND EXPERIMENTAL DATA, EXPERIMENTAL DATA, FINITE DIFFERENCE, FINITE DIFFERENCE, Gl G3 Gl G3 • DIR. DIR. DIR. DIR. / / > or ZD CD _J CE CDCM zz o c n LJ ^ Q I O 2 o o d: o O in' o d. 1 0.0 2.0 4.0 6.0 8.0 10.0 NON-DIMENSIONAL PLATE LOAD, Figure 5.5, Curvature due to plate 12.0 hqoxK2/D weight. 14.0 *10 -5 COMPARISON The deflection at the plate, used diagonals values at the Gl and is experimental 10% the less unreasonable 1) the for The 5.3 In as and section Hsu[7] and experimentally are strain COMPARISON this the the were as the is in external gauge be determined shown to finite 5.5. the The results error the Fig. posts, is the The in expected. result This converging Fig. are error closely. approximately is not as the grid is B.2.1. very small making comparatively factors such as the large. the weight of wires. theoretical Ramsey[8] along of reasons: strains reading are . 91 portion compared numerical s t i l l shown curvature supporting numerical following centre measurements. would the the then results OF THE C U R V A T U R E the in strain follow measured There the the between solution instrument 3) These experimental. the refined 2) calculate largest results The points, experimental the than to from direction G3 grid centre. direction, curvature In the calculated difference In was the OF R E S U L T S w i l l be point. BIFURCATION points of compared POINT bifurcation with the of Lee COMPARISON 5.3.1 The EXPERIMENTALLY bifurcation experiment curvature points were found graphs, dimensional DETERMINED form of by Fig. of the [8] plates examining in 92 POINT four 4.3a-d. OF R E S U L T S the They Fig. in corrected are 5.6. tested plotted Square this plate in the symbols non- mark the points. Using the curvature experimentally determined and strain, information work It of was the deduced was strains are of in the Up the point in along had opposite 2.3.1 edge of sign. the and is the was further extracted length, bifurcation, Eq. 5.1.3 that of by only not the Eq. on the the (5.7) of of given but strain results direction are their diagonals and dependent Using (5.7), point placed Section in Eq. between from the [3]. with strain and opposite they plate plate to al results equal the midsurface et and their shown bifurcation that sides comparing centre on Miyagawa opposite It midsurface relationship the this plate the by midsurface diagonals (5.7). on experiment. the at the The curvature of the thickness. curvatures gives gauges are equal and COMPARISON OF RESULTS 93 CURVATURE BIFURCATION POINT MATERIAL: ALUMINIUM LD CD • oA + x- oo O o — o - LJ v •= O LEGEND .125" THICK .118' .098' .079' .059' .039' BEST FIT LINE o + y > or ZD o o CO o 2: o A o CO LJ n ° . r: o CD X X 1 / < > > s o * N/ y ^ i . A A > o A "t X o • X x A -<r y o / I O + A A A A • • • CO o 0.000 0.005 0.010 0.015 0.020 0.025 THICKNESS-LENGTH RATIO, h/o- Figure 5 . 6 , Bifurcation point. 0.030 COMPARISON i where K is the Substituting Eq. (2.9) the curvature surface strain where the total surface the diagonal. strain relationship of ' . . . (5.12) the strain Eq. = £ + i circumflex strain, with (^rr-) = -.0124 is asterisk, midsurface Combining along gives e* the 94 ...(5.11) 2 curvature-bending i The = -.0124 K a positive in OF R E S U L T S ' and bending ...(5.13) overbar strain represent at the the surface and respectively. (5.12), we get the non-linear relationship e* Shown in Fig. bifurcation strain the at 5 for the bending curvatures, of = e - [3] various point strain, using of is the sizes ( ^ ^ ) Eq. (2.10). ...(5.14) surface of strain plates. bifurcation, using Eq. .0124 we (5.14), at the Taking the can and then then point surface solve the of for COMPARISON This The wascompleted solid line andthe results represents OF RESULTS a r e shown the least square i n Fig. f i t 9 5 5.6. giving the equation Ka The scatter i s moderately with estimating Fig. 4.3, slowly. that the the lines error bifurcation with from the errors representing would point. . . . (5.15) of bifurcation. i n strain calculated, associated high the point A small curvature = 3.64 h/a A certain c a nbe seen the curvatures measurement, produce t h e measurement It associated diverge a n dhence t h e significant amount i n changes of error of the points i s from i n also Fig. 5 i n [3]. By taking Eq. (5.15) andrearranging i t into the non- 2 dimensional curvature noted 5.2, occurs Eq. 5.1.2, K a / h , we s e e t h e n o n - d i m e n s i o n a l at the point of bifurcation. As the bifurcation at a curvature o f between point, 3.0 shown a n d 4.0, i n Fig. which (5.15). ANALYTICAL Ramsey[8] form i s a constant i n Section v e r i f i e s 5.3.2 curvature found BIFURCATION that POINTS the point of bifurcation occurred at COMPARISON Lee and Hsu[7], factor of 2 noted after i n Section is an extrapolated a factor Comparing Eq. 5.4 In more c r 8 times we f i n d =21.6, i n F i g .9 of between to the difference these [7]. results. the experimental i s 10% l e s s There for result, [8] and for [7]. LOAD AND SURFACE OF MODULUS SHEAR the determination relationship through test, t h e method Using the results strain calculated. test. ...(5.17) found results LIMITING surface of a 5.1.4.4, has difference the analytical (5.15), 700% of M the correction =24.9 h/a Ka from making 96 ...(5.16) =3.29 h/a Ka while OF R E S U L T S i s of limited obtained values DURING G and the shear the application at which These STRAIN of by the onset in this stress-strain the twisted of plate bifurcation. experiment, the i n s t a b i l i t y should T H E MEASUREMENT occurs the load and c a nbe n o t be exceeded d u r i n g the COMPARISON The point strain of at bifurcation this i s given curvature OF R E S U L T S by E q . (5.15). c a n be found using 97 The bending E q . (2.9). This gives -2^*4 2 c Substituting this into 2 the (h/a) , factoring To i s the surface calculate 2 the surface strain, E q . (5.14), and we g e t e* c This ...(5.18) a = 1.66 (h/a) strain the load at ...(5.19) 2 at which the point relationship between corner Substituting the curvatures load at bifurcation of occurs. bifurcation and curvature, the point of we u s e t h e E q . (2.5). bifurcation, E q . (5.15), P Replacing = 2D(l-\0 (3.64 h / a ) the flexural rigidity, c We now have surface the limiting strain, ...(5.20) 2 c which = D, gives ^ 6 0 7 Eh* (1+v) a ...(5.21) 2 values should of the corner load and n o t be e x c e e d e d d u r i n g the COMPARISON test. These limiting values taken to apply to any OF R E S U L T S size of 98 isotropic plate. Care the be instability. with [2] must . edge effects not With and make ratios the larger transverse h/a ratio than shear large about will be 1/10 to avoid problems encountered 99 6. In the analysis dimensional more deformation particular The experimental were results. general of the For found to which be a number collapse sizes are conclusions of made not made plate 1) The principal strains l i e along diagonals The the be the linear in can of 2) which different statements plate, results of non- the plate about this the limited to plates of size. following strain the quantities experimental allows of CONCLUSIONS the and linear non-linear non-dimensional about and at principal non-linear the of the centre the regions: of plate the in both regions. quantities, which collapse 2 load-strain plate relationship, are (l+v)Pa the 4 /Eh 2 and 3) e(a/h) These . non-dimensional results for the same quantities plate collapse thickness in the both the CONCLUSIONS linear and non-linear regions with very 100 l i t t l e scatter. 4) The non-dimensional thickness of the point of bifurcation, thickness the the plate the the The linear a and of the of the plate the linear and But, after rate load with strain be the for at to made linear the the The of thinner it changes synclastic shape increased. relationship the of different plates which the is load-strain can in of collapse. shape coincides conclusions plates results not the non-dimensional non-dimensional following curvatures higher of region. the do anticlastic start in non-linear different as The collapse start from 5) also results linear theory approximately about and is the up to 1.5. principal non-linear regions: 1) The of principal the plate curvatures in both the l i e along linear the diagonals and non-linear which collapse regions. 2) The non-dimensional load-curvature thickness, in quantities relationship both the for linear plates and of the non-linear the same CONCLUSIONS regions, of the bi are same linear The in thickness of the 4 2 and Ka / h . behave Plates similarly load-curvature and c o i n c i d e s non-dimensional following strain / E h after furcation. 3) T h e s t a r t a 2 (l+v)Pa 101 conclusions the plate in with the curvature c a n b e made both the relationship linear of theory up approximately about p r e - and the is to 3.0. midsurface post-bifurcation regions. 1) At any p a r t i c u l a r midsurface point strains, e on the and e x equal 2) in the direction The m i d s u r f a c e strains of , y' the vary plate, the normal are diagonals. linearly with the non- 2 dimensional 3) curvature no midsurface strains At the of t = x e l i t t l e 4) Gaussian The centre the .0124 K K a x y y scatter in experimentally shown above determined the in the There experimental surfaces. with results. determined midsurface than the i n c r e a s i n g the are relationship was found w a s 40% l e s s one. By developable plate, 2 K^K^a . strain analytically degree of CONCLUSIONS freedom of the analytically converge 5) The non-dimensional and solutions The in a did results f a i r l y the load-curvature not good satisfy Ramberg in-plane displacements field bifurcate. I n i t i a l the of due In of and and was would of Rayleigh-Ritz 6 by the error or the methods, method, of in could as of are up to The of the displacement [7] might use deflections of be of plate finite have the plate non-linearity design in process. elements achieved in linear calculating amount such Hsu[7] results the large The the closely. freedom of their errors. with satisfactory of because and results machine during fit Lee inability the results respectively. lack the experimental conditions 5 self-weight. sizes poor and and in the experimental of the estimating other the encountered their different retrospect, simple to a to The boundary with convergence were compared Miller[2] limited The differences deflections, plates was problems plates f i n i t e [2] lack independent results. the curvature in to are produced and agreement solution due were Reissner[9] of strain strains. strains solutions non-dimensional to true midsurface the thickness. approximate Chandra[5] displacements, determined midsurface plate Four on in-plane 102 or a comparable CONCLUSIONS or better less results time I n i t i a l and curvatures due deflection after applying a curvature The analytic less and The to at of was the [7] point of corner load, which * e = c Three 1.66 areas involves the less point a are scatter with the point of of f a i r l y The Once of a particular not be was Ka Lee and Hsu[7] Although the in and need of the used method the P = o bifurcation formed in the The results. h/a. are 10% load-curvature is point of d i f f i c u l t determination be The only surface exceeded during 4 2 E h / ( l + \>)a the to second is be the used strain up and test, respectively, plate accurately of the consideration. measuring more can c r i t i c a l further can 3.64 results. .607 of = predicted relationship not of and good, point should the or mode could bifurcation Ramsey[8] method 2 the other flatness load. bifurcation in influence to process bifurcation. assumed, is in better of of bifurcation. (h/a) due manufacturing stress-strain the are the scatter plate shear non-linearity whether respectively. not. causing elastic plate plate, point lateral is twisted in the 700% more bifurcation the self-weight, results relationship measure to mode The in formed deflection displacement estimating effort. imperfections by in 103 The curvature determined the f i r s t so with development CONCLUSIONS of a in the the of testing vertical plate under orthotropic The al new work [3]. both apparatus plane, its In this sides of weight. And l a s t l y , own here into goes plate were enabled demonstration for midsurface results curvature, occurs; The the a l l of results using the and and which to of in future the twisted of the the the principal of twisting Miyagawa allowing strains on This dimensionless quantities determination of at et the curvatures. strain of region. work and plate which expressions which bifurcation novel. this thesis analytical measurement plate is the measured, strains test deflection non-linear beyond surface are presented verification importance load the experiment, of collapse the unwanted calculation the hold the present the would negating materials presented which 104 test. of form models a basis and are constitutive for of the direct relationships 105 7. [1] Kelvin andTait P . G . Treatise on Natural Philosophy, Vol. 1, P a r t 2, C a m b r i d g e U n i v e r s i t y Press, 1883, pp. [2] BIBLIOGRAPHY 203-204. Ramberg W. a n d M i l l e r J . A . " T w i s t e d S q u a r e P l a t e Method and Other Methods f o r D e t e r m i n i n g t h e Shear Stress-Strain Relation of Flat Sheet", J . R e s . N a t ' l . B u r . S t a n d a r d s , V o l . 50, N o . 2, 1953, p p . 111-123. [3] Miyagawa M . , S h i b u y a Y . , a n dUeda M. "Measurement o f Shear Modulus o f E l a s t i c i t y b y Means o f T w i s t e d Square P l a t e " , TheTwelfth Japan Congress on M a t e r i a l s R e s e a r c h - T e s t i n g Method and A p p a r a t u s , pp.261-264. [4] Tsai S.W. "Experimental Determination of the Elastic Behavior of Orthotropic Plates", Journal of E n g i n e e r i n g f o r I n d u s t r y , A u g . 1965, p p . 315-318. [5] Chandra R. "Ont w i s t i n g o f O r t h o t r o p i c Plates Large Deflection Regime", AIAA J o u r n a l , No. [6] i n the V o l . 14, 8, 1976, p p . 1130-1131. H s uC S . a n d L e e S . S . " S t a b i l i t y of Doubly Periodic Deformed C o n f i g u r a t i o n s of Plates and Shallow Shells", J . A p p l . M e c h . , V o l . 37, 1970, p p . 641- 650. [7] LeeS . S . andHsu C S . " S t a b i l i t y of Saddle-like Deformed Configurations of Plates andShallow Shells", Int. J . N o n - L i n e a r M e c h a n i c s , V o l . 6, 1971, pp.221-236. [8] Ramsey H. "A Rayleigh Quotient f o r the Instability of a Rectangular P l a t e w i t h Free Edges Twisted by Corner F o r c e s " , Journal de Mecanique Theorique e t Appliquee, [9] N o . 2, 1985, pp.243-256. Reissner E. "Finite Twisting andBending of t h i n Rectangular Elastic Plates", J . Appl. Mech., V o l . 24, [10] V o l . 4, Fung 1957, p p . 3 9 1 - 3 9 6 . Y . C a n dW i t t r i c k W . H . " A Boundary Layer Phenomenon i n t h e Large D e f l e c t i o n of Thin Plates", Quart. J . M e c h . A p p l . M a t h . , V o l . 8, 1955, p p . 353-355. BIBLIOGRAPHY Goodier J . N . McGraw-Hill, [11] T i m o s h e n k o S . and 3rd e d i t i o n , [12] Timoshenko S. and W o i n o w s k y - K r i e g e r Theory and S h e l l s , 2nd e d i t i o n , McGraw-Hill, 1959. [13] Measurements Group 200 A d h e s i v e , V i s h a y , USA. [14] Measurements Group I n s t r u c t i o n Strain Indicator, Vishay, [15] S z i l a r d R. T h e o r y a n d A n a l y s i s o f P l a t e s , C l a s s i c a l and Numerical Methods, P r e n t i c e - H a l l Inc., New J e r s e y , 1974, pp. 158-185. [16] James S t r a i n Gage Instruction T h e o r y of New Y o r k , 106 E l a s t i c i t y , 1970. of Plates New Y o r k , Installation with Bulletin B-127-9, Manual P-350A USA, 1980. M-Bond Digital M.L., S m i t h G . M . , a n d W o l f o r d J . C . A p p l i e d Numerical Methods for D i g i t a l Computation, Harper a n d Row I n c . , New Y o r k , 1 9 7 7 , p p . 343-348. 107 8. A. INSTRUMENTATION In this are appendix described. gauges used, The and and balance A . l S T R A I N GAUGE Two lots were the the of sub-section second DATA strain The in the gauges by A the were with the indicator general A . l a and for these strain and switch was and indicator using the is experiment. Group is and were given in strain gauges They of type Section are A.lb. strain Indicator the description The measured in Measurements specifications Tables used INSTRUMENT S P E C I F I C A T I O N S The instrumentation SHEET VISHAY Unit. the deals strain A.2 Strain for unit. CEA-13-125UR-350. listed specifications f i r s t manufactured 3.1.2.1. APPENDICES the Vishay designed Vishay SB-1 for P-350A Switch and resistance Digital Balance type strain APPENDICES gauges and strain, is allows from designed output of channel simplify A.3 to be read digital display. The to provide a of channels be data specifications and strain a 10 can the on method a single independently reduction for these and two directly, Switch and balanced to interpretation. units are shown Unit reading indicator. zero micro- Balance sequentially strain in 108 the Each output to The in Tables A.2 APPENDICES SMSM5 ENGINEERING DATA SHEET u THE INFORMATION APPEARING ON THIS SHEET HAS BEEN COMPILED SPECtFICALLV FOR THE GAGES CONTAINED IN THIS PACKAGE. THIS FORM IS PRODUCEO WITH ADVANCED EQUIPMENT ft PROCEDURES WHICH PERMIT COMPREHENSIVE QUALITY ASSURANCE VERIFICATION OF ALL DATA SUPPLIED HEREIN, SHOULD ANV QUESTIONS Aftis{ RELATIVE TO THESE GAGES. PLEASE MENTION GAGE TYPf. ITEM NUMBER. AND LOT NUMBER. 2.065 cr © to 50 sen • S-* Z ~~ So. cn cn •0.9 I* ft ft 3 2 1 " " o- o -o •0.7 *© a SECl PRECISION STRAIN GAGES > a SEC1 coot HALE IOH. NORTH CAROLINA 951516 MEASUREMENTS GROUP s 2.135 M N • . . jTZ Mlcro-M»a»ur»m»nt» Division IP »o fi S o ; *»m Z if | . 5-n o O ro 2 . cn &** 1 30 Cd cn IP GENERAL INFORMATION: SERIES CEA FEATURE GAGES GENERAL DESCRIPTION: CCA M M «r* • a>n»wa CXXPOBI family of L W P I fpgs am H»lid with • fully ancapajhatae and and a • TEMPERATURE RANGE; - 10CTF f-»*C7 B *«0*F 1*208*0fart> FATIGUE LIFE: Faiigua lit* « t ma«*ad functnn af aMoar tottn Hrmmxr. WW 3D' ANG a lif*«<llb* 10* t > o - « 1150ai<n/in f j*n/tr»/ M. Una MIA Sotaar. CEMENTS: &*r»pat*«» with MM Canitiad M-Bond 200 but ii irill normally W pro*.* tt» araam arrwn Iwntt M«r» I' M Bond AE-10/15. M Bond GA 2, M Bono 600. and M-Bond 610 an M Bond 610 a Bw KMC* ovar rht •ntinj opanrtinB njnpt, R«*» to M M Cratog A 110 tor information on bond-TO. anann. M l n B 127. 6 IX. and B-137 f<V iraxaJlatnn prandura*. •300'F I'tSO'O, H i n t aMdw tvo* 961A 163-37) tin wad Attar ma\ M SOLDER: it oparating twrxtntum **.<• no, I U M U*M tor Mad *nae*v*ant. U-bm a>ioar tvo* «50 IBS-SI tm• w w w v t M n t K w y tn -*00*F i*30S'CJ. Kw*w ro MM Cauiog A 110fartunnar intormrion on vhjarv aid BulMrun TT 606 for was marfimant ntftruQuaa. BACKING: Th» backing of CEA Saia Gaga* na. baari (pacialN traatadfaroptimum bond formation wrth aJI appropnan Knj gaga tetmwm. No rurrhar ctMninrj a nacaaaarv if oamsmtnation of tna prarjwafl ajrtaca it aroMMd dwm) Handling. T E M P I F U T U R E INDUCED APPARENT S T R A I N LOT NO.. TEMPERATURE IN'CELSIUS 4- <^i.Btiio^4.i7i>pV3.a tti^*e.ooiiDV-.i.eatio^' fn €,^.64110V4.17iloS^.g2>l0^^32ilO^-4.Btt[10"V f*Q M -ion TESTED ON: •>» o ™T>A--TA ii lUTitk ag«-T4 ALlMDUl *m> •» TEMPERATURE IN'FAHRENHEIT MOO _,,„,„ cOOE. KM 16 !• — *W0 ENG. £ c J - N TEST P R O C E D U R E S U8EO BY M K ^ O - M E A S U R E M E P / T t FOR STRAIN OAOE PERFCMUatAMCE EVALUAT)OM OPTICAL DEFECT AMALYSU GAGE FACTOR AT TFf a f VARIATION WITH TEMPEFU.TURE APPARENT S T R A I N VERSUS TEMPCRATUNC THAMSVERH SEWmVITY INITIAL RESIST AMCC FATIGUE LIFE STRAIN LIMITS GAGE THICKNESS CREEP AND DPUFT Table A . l a , S t r a i n gauge data ASTM ESS 147 « AJTM E3S1-67 Q A 6 T M CJS.1-67 Bfcw I W M Raw. C a i w — ^ R i u iM< ASTM Uftl-*? M-MJ P*ajBMaa% OtMai M l TnaaWaarf aa RaaaBnea kaWrA MA« M l (MetMaW N-M PtaaaWaa (SkwS*> aj MAS 642II sheet, Lot 1. 109 APPENDICES =M=M= ENGINEERING DATA SHEET I® TXl INFORMATION A m A R I N G ON THIS SHEET HAS K I M COMPILED S"*fClFl CALL V 'OR THE CAGES CONTAINED IN THIS PACKAGE THIS FOAM IT PRODUCED WITH ADVANCED EQUIPMENT • PROCEDURE* WHICH RERMIT COM* A€ HI NIL VE OUALITV ASSURANCE VERIFICATION OF ALL DATA SULLIED HEREIN SHOULD ANY QUESTIONS ARIU RELATIVE TO THEM OASES, FLEAM MENTION OAGE TYFE.MTCH N U M B E R , AMD LOT N U M l t R MMUC SETTMI r ~ : m f-""i • Mlcro-Mt»iur*m«nU Division MEASUREMENTS GROUP, INC. PRECISION STRAIN GAGES GENERAL INFORMATION: CEA-SERIES STRAIN GAGES O C H C R A l D C S C A i m O N . C E A OROta V* • e a n a * » i - p u r p o w lamih/ of C O n M n U n ttr»m paoaa WKH^y i M ••> asparmwmai R W • n a i f t i i Tha o a o M TUmMATimc auopiM0 wit* • rvl'y arKAWKriatao gnO *na t x p o — c o p o a t - c o a w t i i m a g i V aoMMr* ( A M H A N O C : -100* to *400*F ( - W K> 'ZOT Q ten O O W A X K J I in t a r e w m u w u n n M I ^ - T V M R f PtATUftf C O H W A A T I O M : S M d m cutwa EMMw. gmgm 1/1 iJiMnjmd • T W A I N LIMITI: ApproxmaMrty S** KK o a o a langtrti m k v o a r . and W O K M W I 9% tor Rjngtm U-MMI 1/8 m (3 2 mm) M T K l l I t LRU F a n g u a M a a a * W f » « l w » ^ 0 « 0< W N W pO»nt l o n r W O " WHFi SQ-AWG H M i O t r a c t t y MRCPaMi IDQAQa t A M MmgiM ^ M« w<c M 10* CYO*» at i l S C O y i n / i n (tvn/m/ ua»ng. W-Dna 3 S i A ao*0»< C C M E H T t : ComtMt.&ic wtm M - M C4KUfwd M - B o n o TOC but ft v i H normally not proncM tha p r a a m : a w . IM*II MrcroM a « u u r » m a n a M - B o r > g A E - i O n S . i * - p o o t f GA-2.M-tk>f>aacC.aA0U-tAonfl6i0araa»o>ii»r>l M - B o o c 610 r» iha M P W C * Of*t h a » " i > r a o p a ' i l > n c ranoa r W a r U> M - U Cai**og A-110*0* mrormatjon o n eonaiftO affaou. «nfl Bwi»trna B-12?. B - I X ano 6-13* t w tfttiaiianori procaouraa EfcCH-Dfft « ecwfStine, tomoaratura «<H not a*OMd *300*F f l S C C j . U-Ltm aotoar M l A (63-37) bn-MaC ao*0»« mar » m a d for MAC artaehmanl U - L m a l o U a * 450 (B6-5) ur-*ntimo-> a a t t s l t c t o r y to **0C* » r*2O0*C; to M - M Catalog A-110 Io' • u n r w information o n ao>0*"V a^e Tacn Tip T T 4 0 6 tor «nacnm#rTt t a c t i n ^ u a a T • V A C * I N 0 T h * b a m r t g Of C t A - S w M paoaa Naa ba*n wwciaitv M a t M '©* Optimum b O M formation wrrri »li Bopropnata airain Oaot aonawvaa No l u r t n c ctaamng • n i o i n i r y rl ooniartwoaijon of (ft* prafMrafl aurtaoa • avo>xMd Ounng n*no>.r»Q QMS ROM. anM-oc THtHMAi. OUTTUT LOT NO.. TtaanRATURE IN *ciL»LS • j ; | ! ; i FACTOR ! "} /: i • WD i 0 111 Hi / / / ; ; ! ^-lMi ,/i M'C 1 1 I i i n ; ; il ; ; i ! i i i ! i i /. / - :* ft- i TW*RJMU.OUW t f ! 1 5 1» '«?.<3i 10 *1 -2.3b S i ¥ * . » ! C * t ' - 3 . T9i It*! € -l.Eilo'il.S7ilo'l-6.»<i*: T t3.«il0^ -3.».ldV I'd £ i ! 3 w •aoo X£*-U ALLK14> •aa 1fiATU*U IN'FAHRENHEIT CODE JEHL_«« £ E A 2 _ T U T mOCCDUfUU USED BT KIOTO EMail WF.MEHTt ORTtCAL O f F I C T ANAL T i l l O A 0 I R l t l t T A N C E AT M * C AMD fr*** K H Eft-* R r r* O i r v c l M B S T n OAOI r » C T O « • T N ' C 4 R R > * M M M w n « H f « * - R M B J O n U T « 4 J R ..UTTil-ll T 1 M » l « 4 T L > R t C O f " P ( C « l f T Of OAOE P A C T O R **TFJ | » 1 DvrwctW* feMttXHI) T H E R M A L OUTRUT A»T1I I-IS1 ( I M H*MMfl • * * • - C — a n m u . r , R * « ^ a M ) T R A N t V f R | E » I M i r T t v m r AT EM* C • 90% H H AtTN I « i • A T I C U E LIFE • * » (M»«n.»«l • TRAIN LIMITS «M» ( M . a . i . . S | OAOC THICKNESS »'•«•••"• C R E E F A N D DRIFT F i a M w n ( I t w W i M N A t M l M*in»«i l Table A . l b , S t r a i n Gauge Data Sheet, Lot 2. 110 2.0 RANGE & DISPLAY SENSITIVITY READABILITY ACCURACY CALIBRATION INPUT CURCUITS SPECIFICATIONS ±50,000 microstrain (ue): 10,000uc on the primary balance knob. P o l a r i t y and range extension on the range extender knob. Total reading i n single d i g i t a l display. A l l bridge completion r e s i s t ors are standard self-temperature-compensated s t r a i n gages bonded to 2024 aluminum. BALANCE 10-turn lock knob provides approximately ±2000 micros t r a i n (GF=2, quarter and half bridge or 1200 f u l l bridge) for zeroing the d i g i t a l readout prior to load a p p l i c a t i o n . On-off switch provided. BRIDGE EXCITATION 1.5 v o l t s RMS, 1000 Hz square wave. Variable: Null meter deflects from zero to f u l l scale with 40 to 4000ue (nominal at GF=2). lue. 10.11 of reading or Sue, whichever i s greater, for R-120B and GF-2. •0.31 of reading or 5ut, whichever i s greater, for R=120n, GF-1.5-4.5. ±0.5» of reading or 5ue, whichever i s greater, for R=50-2000(J, GF=1.5-4.5. Self-contained shunt-calibration across internal 120fi and 350(1 dummy gages. Simulates +5000ut (±0.1%) on quarter bridge operation regardless of lead-wire resistance (GF=2). 50-20000 half or f u l l bridge internal dummy gage provided for 1200 and 3S0fi 3-wire quarter bridges. LEAD WIRE CAPACITANCE T y p i c a l l y less than .05% loss of accuracy with .005 EFFECT microfarads. Negligible e f f e c t with 500 feet of lead wire. AMPLIFIER AC t r a n s i s t o r i z e d . GAGE FACTOR Continuously variable from 0.10 to 10.00 (calibrated between 1.50 and 4.50 only). OSCILLOSCOPE OUTPUT Linear range ±250 mv DC. S e n s i t i v i t y variable from approximately 0.2 to 20 ue/mv Bandpass approx. DC to 60 Hz (±51) . GALVANOMETER OUTPUT F u l l scale ±1/2 ma unf i l t e r e d DC. Sensitivity variable from approximately 80 to 8000iiE/ma. > PJ o n m APPENDICES 2.0 SPECIFICATIONS 10 CAPACITY channels plus OPEN position. EXTERNAL Will CIRCUITS or accept full any inputs 3-wire RESISTANCE 50 to or Quarter may be 2- or circuits. 2000 ohms. Compatible OUTPUT half, circuits combination. bridge INPUT BRIDGE quarter, bridge with any strain indicator. BALANCE RANGE Quarter (GF=2) and Half Bridge: uc with 120-ohm +2000 half Full bridge RESOLUTION SWITCHING (GF=2) REPEATABILITY ±2000 U E for Range proportional 1 pc. Better than 1 ye. SIZE 9"w. [230 WEIGHT 5-1/2 A . 3 , Vishay SB-1 Switch specifications. - with x x bridge. to resistance. than tight Table 120-ohm Better Aluminum CASE indicator. Bridge: bridge BALANCE in dust and spray- detachable 6"h. x 6"d. 1 5 0 x 150mm] pounds 12.4 and Balance kg] Unit cover. 112 APPENDICES B. FINITE Linear DIFFERENCE f i n i t e deflection corners. were of were a loaded uniformly these calculated non-linearity The f i n i t e linear mesh and The are difference to calculate plate supported deflections, used difference points used in the Section the by curvatures 3.1.3.1 to opposite and slopes calculate the ratio. form. equations ANALYSIS differences Having 113 method is quite derivatives in the replaced along method and is by the discussed its basic differential quantities plate in in governing difference inside simple at boundary. selected The finite [12,15,16]. 2 A central was used difference to approximate equations. The midsurface forces, plate and corners. were A solved program was its was imposed the accuracy. a at set to equations. error equation, at a l l their of Eq. mesh points Eq. simultaneous generate and to (2.27), were , with zero on the (2.30) boundaries equations and which plate. solve made verify Ax (2.29), respective the Tests order differential conditions, of exist, of governing deflection written solutions with applied boundary formed for simultaneous analytical The were This the bi-harmonic surface. (2.1), scheme, on the these problems program where and test APPENDICES B.l DEFLECTION The non-dimensional supported The by RESULTS deflection opposite supported • 114 corners corners are of is shown a uniformly shown with in loaded Fig. zero plate B . l . deflection. At the 4 centre and the free corners the deflections are .0703 qa /D 4 and a .0893 qa the edge is assumes of almost and D the cylindrical curvature where q is the flexural shape present loading, r i g i d i t y . with under plate only this a The small plate amount loading. CONVERGENCE determine reasonable parallel series to respectively, length, a n t i c l a s t i c B.2 To an /D 10 by the tests 10, in The deflection B.2 and size, B.3 the mesh. the plate were mesh and .2% edge moment there and was examined varying of the the moment a and for mesh are last change .1% r e s p e c t i v e l y . only made to obtain bending moment convergence. size from 2 A by 2. convergence For be deflection was made, and should centre increments deflection show the respectively. approximately figures fine results, to of how small plotted increment was in Fig. plate small, The error in for convergence the 10 by 10 2 APPENDICES DEFLECTIONS PLATE SUPPORTED BY TWO CORNER POSTS Figure B . l ,Deflection supported of plate by opposite under uniform corners. load and 115 APPENDICES EFFECT OF GRID REFINEMENT -o CENTRE DEFLECTION LO > 1 O ><^. '—'r- Q X • d C 3 — <M CT ZZ. LU 1 I L_ • LxJr^ CD o 2.0 Figure B.2, 4.0 GRID Convergence mesh sizes of ~~1 6.0 8.0 (N BY N) centre between r 2 deflection and 10. 10.0 for varying 116 APPENDICES 117 EFFECT OF GRID REFINEMENT CM © o CNTR. MOMENT CM I o „ X CD 2.0 6.0 8.0 centre bending GRID (N BY N) Figure B.3, Convergence mesh sizes of between 2 and 10. 10.0 moment for varying APPENDICES B.3 In MAXIMUM V A L U E S Section required design At each using of 3.1.3.1 for of the the mesh axes the test maximum the SLOPE curvature of curvatures with and twist plate edges. calculated using a ,xx max= ) 3 5 7 <3 both the plate diagonals. slope was found using maximum the points the The calculated maximum values c i r c l e . and the . . . ( B . l ) was equal geometric in the relationship giving ( w in in /° of This corners a 4 direction [12] supported - the in used were curvature at maximum slope where Mohr's found The and non-linearity the ( w was AND plate. point were maximum estimation parallel curvature The OF C U R V A T U R E 118 was direction of maximum ,x max= ) also of found the slope - 2 at 0 6 be 3 the diagonal can <J* / ...(B.2) D supported between clearly the seen corners posts. in F i g . and The B . l . was
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An experimental study of instability in square plates twisted by corner forces Williams, Gordon Colin 1988
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Title | An experimental study of instability in square plates twisted by corner forces |
Creator |
Williams, Gordon Colin |
Publisher | University of British Columbia |
Date Issued | 1988 |
Description | An experimental study is presented on the non-linear twisting of plates with free edges, through the application of self-equilibrating corner loads. A simple apparatus was designed and various sizes of plates were twisted while measuring the surface strains on both sides at the centre. Initial difficulty was encountered due to unwanted deflection of the plate under its own weight. The Rayleigh-Ritz method is used to determine an analytical relationship between midsurface strains and curvatures in the pre- and post-bifurcation regions of twisting. In both the experimental and analytical results, the midsurface strains are found to vary linearly with the Gaussian curvature. Non-dimensional groups are identified which collapse the experimental load-strain, load-curvature and midsurface strain-Gaussian curvature relationships. These non-dimensional groups collapse the results in both the linear and non-linear regions. The curvature at the point of bifurcation is identified as a function of plate dimensional parameters. Also shown are the expressions for critical surface strain and corner load at which bifurcation occurs. The experimental load-curvature relationship and point of bifurcation are compared with analytical results found in the literature. A large discrepancy in the literature is resolved for the theoretical point of bifurcation. The present results form a basis for verification of future analytical results, and are important in the measurement of constitutive relationships using the twisted plate test. |
Subject |
Twisted plate structures -- Stability Plates (Engineering) -- Stability Strains and stresses |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-09-16 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0097910 |
URI | http://hdl.handle.net/2429/28527 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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