Elasto-Plastic Behaviour of Thick Plates with a Higher-Order Shear Deformation Theory

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1 Proc Indian Natn Sci Acad 79 N o. 4 D e c e m b e r S l. I s s u e, P a r t A, Printed in India. DOI: /tinsa/2013/v79i4/47993 Research Paer Elasto-Plastic Behaviour o Thick Plates with a Higher-Order Shear Deormation Theor T K A N T 1, R K TRIPATHI 2 and V RODE 3 1 Deartment o Civil Engineering, Indian Institute o Technolog Bomba, Mumbai , India 2 Deartment o Civil Engineering, National Institute o Technolog, Raiur , India 3 D e a r t m e n t o Civil Engineering, Shri G S Institute o Technolog and Science, Indore , India (Received 18 Februar 2013; Revised 21 Ma 2013; Acceted 25 Ma 2013) A H i g h e r O r d e r S h e a r D e o r m a t i o n T h e o r ( H O S T ) i s u t i l i z e d o r e l a s t o - l a s t i c a n a l s i s o l a t e b e n d i n g u s i n g i n c r e m e n t a l inite element o r m u l a t i o n. M o d i i e d N e w t o n - R a h s o n m e t h o d h a s b e e n u s e d t o s o l v e t h e n o n - l i n e a r e q u a t i o n s. Y i e l d i n g o the material has been modelled using von Mises ield criterion, associated low rule and isotroic hardening. Results are comared with available benchmark and other solutions. Comarisons clearl demonstrate better erormance o the theor or inelastic resonse. Ke Words: Plate-Bending; Higher Order Theor; Elasto-Plastic Resonse; Incremental Finite Element Method Introduction Bending behaviour o lates is usuall described b three mathematical models: Poisson-Kirchho late t h e o r [ 1 ], R e i s s n e r - M i n d l i n i r s t o r d e r l a t e t h e o r i e s [2-4] and higher order late theories [5-9]. Poisson- Kirchho theor, which is also known as Classical Plate Theor (CPT), is based on assumtion that straight lines erendicular to mid-surace (i.e., transverse normals) beore deormation remain straight and normal to the mid-surace and undergo no change in length during deormation. These assumtions iml that the transverse normal strain z and transverse shear strains and are zero. It can also be easil shown that CPT is comutationall ineicient (requires C 1 continuit o transverse dislacement) rom the simle inite element ormulation s oint o view. The irst order shear deormation theories (FOSTs ), which include transverse shear deormation, can be classiied on the basis o the assumed ield as (i) stress-based theories and (ii) dislacement-based theories. Reissner [2, 4] and Mindlin [3] are the two ioneers to rovide irst order shear deormation theories based on the assumed stress and assumed dislacement variations through the thickness o the l a t e, r e s e c t i v e l. T h e s e t h e o r i e s r o v i d e a i r s t - o r d e r basis or the consideration o the transverse shear d e o r m a t i o n e e c t a n d r e q u i r e d a C o c o n t i n u o u s i n i t e element ormulation or the numerical analsis. However, FOSTs have ollowing limitation : The transverse shearing strains/stresses turn out to be constant through the late thickness and a ictitious shear correction coeicient is introduced to correct the shear energ. Lo et al. [5] have resented a theor or lates based on assumed higher-order dislacement ield. Kant [6] has derived an isotroic version o the comlete governing equations o such a theor based Author or Corresondence: s: tkant@civil.iitb.ac.in, rajesh_triathi64@ahoo.co.in, vrode@sgsits.ac.in

2 564 Tarun Kant et al. on the minimum otential energ rincile. A C o inite element ormulation o this higher order theor is resented, or the irst time, b Kant et al. [7]. In this theor, the in-lane dislacements are eanded in the owers o the thickness coordinate (z) b Talor series, which allows: Quadratic variation o the transverse shearing strains through the late thickness avoiding the introduction o a shear correction coeicient. Waring o the cross-section is automaticall incororated. When the late is loaded beond elastic limit, the lastic strain occurs, which causes a redistribution o stresses. The comutation o this redistribution was not an eas task beore digital comuters arrived. Hence modelling o lastic behaviour started as limit analsis and then switched over to incremental nonlinear analsis. Initiall or a long san o time, limit analsis was used or the lastic analsis o beams and rames due to their linear idealisation. Simlicit o their ormulation leads to the collase loads. Plates, due to their two-dimensional idealisation, make the roblem comlicated and in most o the cases lead to the uer and lower bounds to the collase loads instead o collase loads themselves. The aismmetricall loaded, circular lates were the simlest roblem to begin with, and thereore attracted the attention o man researchers. Sokolovsk [10] was considered to be the ioneer amongst them. He dealt with the non-linear bending o siml suorted circular lates with small dislacements and thin late assumtions. He used the Henck s deormation theor or a strain-hardening material obeing von Mises ield criterion. Pell and Prager [11], Hokins and Prager[12] and Hokins and Wang[13] redicted the load-carring caacities o circular lates o a erectl lastic material and subjected to rotationall smmetric loads or various suort conditions. The initial works were conined to the circular late case. Hodge [14] took the research out o this barrier b resenting the rocedure o inding the bounds o noncircular lates. The revious researches mostl used Tresca ield criterion since the directions o rincial moments were alread known or circular lates due to radial smmetr. This was not the case with the rectangular lates and thereore the von Mises ield criterion was used or thin lates o erectl lastic material and resented the bounds or a siml suorted square late under uniorm load as a secial case. Hodge and Beltschko [15] ormulated uer and lower bound roblem as a mathematical rogramming roblem, but b using inite element reresentations o velocit and moment ields insuring the satisaction o ield condition throughout the late. The lastic limit loads o the clamed circular late have been resented b Guowel et al. [16] based on uniied ield criterion. The limit analsis is alicable to rigid erectl lastic behaviour while almost all the materials are elastic beore ielding. The current state o stress in a ielded material again deends uon the histor o loading. Thereore, an analsis starting rom the loading in the elastic range and graduall increasing to lastic range till ailure would be closer to the true behaviour o the late. Hathornthwaite [17] was the irst to use an incremental method or the elasto-lastic analsis o lates. He comuted the delections o lates with circular smmetr or an elastic-lastic material. Comarison was made with redicted limit loads o Hokins and Prager[12]. Armen et al.[18] develoed a inite element technique or lastic bending b additionall considering in-lane stresses and geometric nonlinearit. The develoed the ormulation b interreting lastic strains as initial strains and using stress-strain relations rom incremental lasticit theor. Material behaviour was based on von Mises ield criterion with Prager-Zieglar kinematic hardening. Dinis et al. [19] considered the semiloo shell element or elasto-lastic large dislacement analsis o lates and shells. An initial stiness lasticit algorithm and elasto-lastic model was considered or the material non-linear analsis. Owen and Figueiras[20] etended the alication o semiloo element analsis to include the eect o transverse shear b deriving a shear correction coeicient or clindrical bending with the assumtion o constant transverse shear strains. Redd and Chandrashekhara [21] derived a non-linear version o Sander s irst-order shear-deormation shell theor

3 Elasto-Plastic Behaviour o Thick Plates with a Higher-Order Shear Deormation Theor 565 accounting or von Karmann s strains and Hill s anisotroic ield criterion or the elasto-lastic bending analsis o lates and shells. Watanabe and Kondo [22] urther consolidated their earlier (Kondo and Watanabe [23]) work b considering more numerical eamles and better comarisons. The derived the elastic-lastic incremental tangent stiness matri without numerical integration over the area o the elements using the nodal dislacements and nodal orces as undamental variables instead o generalised strains and stresses resectivel. Paadooulos and Talor [24] develoed an inelastic inite element analsis o lates, which includes the eects o transverse shear-deormation. The used the Reissner-Mindlin s late bending theor and an elasto-lastic constitutive model in the stressresultant orm based on generalisation o Von Mises lane stress criterion and associated low rule. The lastic constitutive rate equations were integrated b a return-maing algorithm and solved incrementall using a Newton method. Prasad and Sridhar[25] have develoed an elasto-lastic inite element rocedure using degenerated shell element with assumed strain ield technique, von Mises ield criteria and isotroic hardening. The have demonstrated the correctness and alicabilit o the method b numerical eamles. The deendenc o incremental elasto-lastic analsis on loading histor required the use o an accurate elastic late bending theor. In the resent stud a secial HOST model which includes distortion o the transverse normal is utilized or elasto-lastic analsis o late bending. von Mises criterion [26], associated low rule and isotroic hardening have been used in the ormulation. The modiied Newton-Rahson s [27] method has been used or solution o non-linear equations. A Higher-Order Shear Deormation Theor The develoment o the resent theor starts with the assumtion o the dislacement ield in the ollowing orm[6, 7]: u(,,)( z,)( z,) z v(,,)( z,)( z,) z w(,,)( z,) w (1) The terms have usual meaning ecet the terms and which are the corresonding higher order terms in the Talor s series eansion. The transverse dislacement comonent w has onl one term w 0 and thus disregards transverse normal deormation. Strain-Dislacement Relationshis The linear relationshis between these dislacements and strains can be obtained b using the deinitions o strains rom the theor o elasticit: where, u v z z 0 3 z z 3 z 3 u v z z 2 z v w z z u w 2 2 z z z,,,,,,,, w0 w0,,,,, 3, 3 (2) The leural and transverse shear strains in the late can be written in the concise matri orm as: 3 3 z z z z z 2 2 z z z (3) and s

4 566 Tarun Kant et al. Stress-Strain Relations Assuming normal stress z to be negligibl small comared to other normal stresses, the constitutive matri takes the orm: and 1 1 sm C E (4) The stress-strain relationshi in the matri orm, Q sm. 11 Q21 Q22 z Q 31 Q32 Q 33 Q11 sm. 3 Q21 Q22 z Q 31 Q32 Q 33 z Q44 Q45 Q44 Q45 z 2 z Q45 Q55 Q 45 Q55 (5a) (5b) where, Q11 E Q 2 22, Q21 E, Q 2 31 Q32 0, 1 1 Q G Q Q and Q Energ Eression and Plate Constitutive Relations The total otential energ o the late with volume V and surace area A can be written as: 1 T T U W dv da 2 P V A (6) Here U is the strain energ o the late, W is the work done b the eternal orces and P is the vector o orce intensities corresonding to generalised dislacement vector deined at the mid-surace. Substituting the eressions or strain comonents in the above equation and integrating through the late thickness h, one gets: in which, 1 ˆ Τ ˆ Τ 2 da A P A da T 0 T T ( w,,,,) ˆ (,,,,,,,,,) ˆ ( M, M, M, M, M, M, Q, Q, Q,) Q (7) (8) Constitutive relations in terms o stressresultants can be written in the matri orm as: M M Q11 H3 Q21H3 Q22H3 sm. M Q 31H3 Q32 H3 Q 33H3 M Q 11H5 Q21H5 Q31H 5 Q11 H 7 Q M 21H5 Q22H5 Q32 H5 Q21H 7 Q22H 7 Q31H 5 Q32 H5 Q33H 5 Q31H 7 Q32 H7 Q33H 7 M and, (9a) Q Q55H1 Q Q45H1 Q44H1 sm. Q Q55H3 Q45H3 Q55H 5 (9b) Q Q 45H3 Q44H3 Q45H5 Q44H 5 i i 1 in which, H h h i, i 1, 3, 5, 7 i 2 2 Writing in a more concise orm, we have M D 0 χ M χ Q 0 s Q D (10) or, σˆ D ε ˆ (10a) Elasto-Plastic Analsis Ater establishing an accurate basis o the overall elasto-lastic constitutive relations or the late in the

5 Elasto-Plastic Behaviour o Thick Plates with a Higher-Order Shear Deormation Theor 567 elastic range b including the higher-order terms or stress-resultants, curvatures etc., and the start o the lasticit is marked b the ielding o the material. An initial ield condition, a hardening rule and a low rule seci the lastic resonse o a strain-hardening material. The initial ield seciies the states o stress at which lastic deormation irst occurs and ma be considered a multi-aial generalisation o the ield oint in a simle tension test. The hardening rule describes the modiication o the ield condition due to strain hardening during lastic low. The low rule ermits the determination o lastic strain rates at each oint in the rogressive loading histor. Yield Criteria or Bending o Plates Assuming the ield unction ˆF to be a unction o bending moments ˆ but not o shear orces ˆ s the late bending ield criterion [27] is given b h / 2 Fˆ ( ˆ,)( ˆ,) F, z dz (11) h / 2 Neglecting the through-the-thickness stresses z, z and z or the late, the Von-Mises criterion or the initial ield is: Relacing b (, and ) (M, M b M ) 2 0 h and 0 b M in the above equation, Von- 4 Mises criterion in terms o stress-resultants becomes: M M M M 3M M (12) Hardening Rules The Isotroic Hardening [28], in which the initial ield surace uniorml eands without distortion and translation, has been considered. The equation or the subsequent ield surace can thus be written as: ˆ 2 ()() ˆ ˆ ˆ (13) in which 2 ˆ, whose value deends uon lastic midlane strain histor, governs the size o the ield surace. Flow Rule The dierentiation o elastic otential unction (i.e., the comlimentar energ densit unction) with resect to stress-resultants ˆ gives the elastic mid- lane strains, a lastic otential unction g( ˆ, ˆ,) ˆ ma be similarl considered or the lastic mid-lane strain increments. Thus the lastic low [27, 28] rule ma be deined as: dˆ g d ˆ (14) in which d is a ositive scalar unction, which is nonzero onl when lastic deormations, occur and which varies throughout the histor o lastic midlane straining. The loading arameter d gives the length or the magnitude o the lastic mid-lane strain increment vector g otential surace d ˆ, while the gradient o the lastic ˆ gives the direction. Looking at the similarit o the roerties o lastic otential and ield unction, the ma be assumed to be actuall identical. ˆ ˆ ˆ g( ˆ,,) F( ˆ, ˆ,) ˆ and thereore, dˆ d ˆF ˆ (15) ˆF i.e., the lastic low develos along the normal to the current ield surace (called the normalit condition). The above relation (15) associates the lastic low with the current ield surace hence it is called the associated low rule and it has been used in the resent ormulation. ˆ

6 568 Tarun Kant et al. Elasto-Plastic Incremental Stress-Resultants/ Mid- Plane Strains Relationshi The material behaves artl elastic and artl lastic ater the initial ield. In case o late bending, as discussed earlier, the stresses can be relaced b stress-resultants and strains can be relaced b the mid-lane strains. The change o mid-lane-strain during an increment in the stress-resultant can be divided into elastic and lastic comonents such that: e dˆ dˆ dˆ (16) The elastic mid-lane strain increments and the stress-resultant increments are related as: or, dσˆ ˆ 0 ˆ D d d ˆ ε σ ˆ dˆ s εs dˆ 0 ˆ Ds (17) 1 ε D dσ (18) The lastic stress-strain relations are given b the low rule in the orm o a relation between lastic mid-lane strain increment and the current ield unction in case o late bending as given in equation (15). Thus, the comlete incremental relationshi between the stress-resultants and the corresonding mid-lane-strains or the elasto-lastic deormation is: ˆ dεˆ D 1 dσˆ d F σˆ (19) Ater roer substitution one gets the comlete elasto-lastic leural stress-resultant curvature relationshi as: and dσ Dˆ e dε ˆ (20) Dˆ where, e T D aˆ aˆ D D T H aˆ D aˆ F F F aˆ,,, M M M h / 2 h / 2 Dˆ D z dz and Dˆ D h / 2 h / 2 Solution o Nonlinear Equilibrium Equation The incremental equilibrium equation [27, 28] or the late can be written at some stage in the solution (i.e., at an iteration during a load increment) as (d ) + K T (d ) d = 0 (21) where, (d ) is residual orce and K T (d ) is the tangential stiness matri. s T s dz T ˆ e T ˆ da T s s s K [ [B ] [D ] B [B ] D B ] (22) A Finite Element Formulation In the resent inite element ormulation [28-30] 9- node heterosis [31] element is used. The heterosis element is ormulated using 9-node Lagrangian shae unctions or rotations, and 8-node Serendiit shae unctions or lateral dislacements. The heterosis element has the imroved characteristics and gives a better consistenc in erormance as comared to the arent elements in a wide range o roblems. In case o ver thin lates with selective integration[32, 33] shear locking is avoided. The Incremental Finite Element Solution Procedure The incremental iterative inite element solution[28] with modiied Newton-Rahson rocedure has been adoted. While ielding, the normalit condition or the increments in stress-resultants has been reserved. The convergence norms have been alied or both residual orces and dislacements. Numerical Results A convergence stud is irst conducted on square and circular lates to decide the aroriate late discretisation or urther elasto-lastic stud. The

7 Elasto-Plastic Behaviour o Thick Plates with a Higher-Order Shear Deormation Theor 569 convergence results are resented in Tables 1-3. For circular lates, discretisation scheme used b Liu et al. [34] has been used to roduce least shaedistorted elements. Based on the convergence stud, 16 elements and 32 elements meshes have been used or one quarter o square and circular lates resectivel. The validit and accurac o the HOST ormulation has been tested b erorming the linear elastic analsis and comaring the resent results with the available eact three-dimensional elasticit and other benchmark solutions or siml suorted and Table 1: Convergence o elastic dislacements or a clamed square late subjected to uniorml distributed load (E = 10.92, = 0.3, G = 4.2, h = 0.1 and a = 10.00) Mesh Normalised central dislacement ( w = w c c D/q 0 a 4 ) Paadooulos & HOST Eact solution Talor [35] Liu et al. [34] Table 2: Convergence o elastic dislacements or a siml suorted square late subjected to a oint load at centre (E = 10.92, = 0.3, G = 4.2, h = 0.1, and a = 10.00) Mesh Normalised central dislacement ( w = w c c D / P 0 a 2 ) Ibrahimbegovic Wanji & HOST Eact solution & Fre [36] Cheung [37] Liu et al. [34] Table 3: Convergence o elastic dislacements o a clamed circular late subjected to uniorml distributed load (E = 10.92, = 0.3, G = 4.2, h = 1.0, and R = 5.00) No. o Normalised central dislacement ( w = w c c D/q 0 a 4 ) elements clamed square lates. The central dislacements, or comarison o lates with dierent thicknesses, have been normalised as: w c 3 E h wc 4 q a 0 where, w c is the actual transverse dislacement at the centre o the late, E is the modulus o elasticit o the late material, q o is the uniorml distributed transverse load over the late, a is the length o the side and h is the thickness o the late. Results resented in Tables 4 and 5 clearl indicate that the HOST results are ver close to threedimensional results. Hence, it is suitable or urther investigation in the elasto-lastic range. In elasto-lastic stud all results have been comared with the eisting analtical solutions (either eact or in the orm o uer and lower bounds). The nondimensional eressions used or the normalised dislacements, normalised uniorml distributed loads and normalised oint loads are 2 0 Paadooulos & HOST Eact solution Talor [35] w c 2 w, cd a M q q a M and 0 M resectivel. Where, 0 is the oint load at centre, D[=Eh 3 /12(1-2 )] is the leural rigidit o the late and M [= 0 h 2 /4] is the ull lastic moment. For circular lates these eressions remain same ecet radius R relaces

8 570 Tarun Kant et al. Table 4: Elastic delection o a siml suorted square late subjected to uniorml distributed load a/h Eact 3D Srinivas Redd [9] Savithri [39] FOST HOST Elastic nondimensional central and Rao [38] dislacement w c 3 Eh wc 4 q a 0 Table 5: Elastic delection o a clamed square late subjected to uniorml distributed load a/h Eact 3D Srinivas Savithri [39] FOST HOST Elastic nondimensional central and Rao [38] dislacement w c 3 Eh wc 4 q a 0 Fig. 1: Normalised load versus normalised central dislacement or clamed circular late with uniorml distributed load: Comarisons o dierent ormulations or d/h=100 the side o late a. A series o roblems rom thick late to thin late are analsed to demonstrate versatilit o the HOST. FOST results are based on Reissner-Mindlin late bending which include transverse shear deormation without waring o transverse cross

9 Elasto-Plastic Behaviour o Thick Plates with a Higher-Order Shear Deormation Theor 571 sections. A shear correction coeicient o 5/6 is used here. Clamed Circular Plate with Uniorml Distributed Load Fine discretization or converged solution were irst established. In Table 6, normalised collase loads have been resented or thick to thin lates and comared with the other results in the thin late range. It is observed that, the HOST redicts normalised collase load higher than the other inite element results as well as limit analsis results or all values o d/h. Fig. 1 shows comarison o normalised dislacements versus normalised uniorml distributed loads or dierent ormulations. In the elastic range the variation is almost similar or all ormulations. Deviations are observed in the normalised load-dislacement curves, b dierent ormulations, ater incetion o ielding. HOST redicts highest collase load as comared to other available results. The other ormulations are based on either FOST or CPT. Normalised load-dislacement results have been reresented, or dierent d/h ratios, in Fig. 2. It is observed that, the dierence between moderatel thick to thin lates (d/h = 10, 15, 20, 50, 80 and 100) is less in comarison to the dierence between thick (d/h = 5) and thin (d/h = 100) lates. This dierence graduall reduces in lastic region and the curves or all thicknesses aear to be converging to almost the same collase load. Initiation and sread o lastic zones with resect to load increments have been resented in Fig. 3 or d/h = 20. In this case, ormation o lastic zone starts at the clamed edge ollowed b the ormation o another zone at the centre with urther load increments. The central lastic zone then sreads aster than the edge lastic zone. It is also observed that, a number o elements remain unielded u to re-collase and then suddenl ield and lead to collase. Thus, the inal collase takes lace onl ater ielding o all the Gauss oints. Clamed Square Plate with Uniorml Distributed Load Normalised collase loads have been resented or thick to thin lates and comared with the other results in the thin late range in Table 7. It is observed that HOST redicts normalised collase load slightl higher than the uer bound or thin late range. In Fig. 4 normalised uniorml distributed loads versus normalised central dislacements have been lotted and comared with the results given in Owen Fig. 2: Normalised load versus normalised central dislacement or clamed circular late with uniorml distributed load: Comarisons o dierent d/h ratios (HOST)

10 572 Tarun Kant et al. Fig. 3: Progress o lastic zone with load increments or d/h=20: Clamed circular late with uniorml distributed load (HOST) Fig. 4: Normalised load versus normalised central dislacement or clamed circular late with uniorml distributed load: Comarisons o dierent ormulations or a/h=30 Fig. 5: Normalised load versus normalised central dislacement or clamed circular late with uniorml distributed load: Comarisons o dierent a/h ratios (HOST)

11 Elasto-Plastic Behaviour o Thick Plates with a Higher-Order Shear Deormation Theor 573 Fig. 6: Progress o lastic zone with load increments or a/h=30: Clamed circular late with uniorml distributed load (HOST) Table 6: Collase loads or clamed circular late with uniorml distributed load (E = 10.92, = 0.3, 0 = and R = 5.00) d/h h Initial UDL M 0 FOST HOST Other Finite Element Analsis: (Watanabe and Kondo [22]); (Paadooulos and Talor [24]) d/h = 100 Limit Analsis: (Hokins and Wang [13]) (Guowel et al. [16]) Table 7: Collase loads or clamed square late with uniorml distributed load (E = , = 0.3, 0 = and a =6.00) d/h h Initial UDL M 0 FOST HOST Other Finite Element Analsis: (Watanabe and Kondo [22]), (Paadooulos and Talor [24]) a/h = 100 Limit Analsis: Uer bound (Hodge and Beltschko [15]), Lower bound (Johnson [40]) and Figueiras [20] or a/h = 30. In the elastic range the values are almost same but ater incetion o ielding there are deviations and HOST redicts higher collase load. The eect o thickness on normalised loaddislacement behaviour is shown in Fig. 5. All loaddislacement curves are ductile in nature. It is observed that, the dierence between thin to moderatel thick lates (a/h = 10, 15, 20, 50, 80 and 100) is ver less in comarison to the dierence between thick (a/h = 5) and thin (a/h = 100) lates. This dierence graduall reduces in lastic region and the curves or all thicknesses aear to be converging to almost the same collase load. Sread o lastic zones during the incremental loading rocess or late a/h =30 using HOST is resented in Fig. 6. Initiall, lastic zones start rom mid suort and sread towards the corners. Net lastic zone starts rom the centre o late and sreads towards the corners o the late. Even ater meeting o these two zones, lates still ehibit resistance to loads because o artial ielding o corner elements and mid suort elements. Final collase takes lace onl ater ielding o all the Gauss oints. Conclusions A reined theor, based on a higher-order dislacement model incororating the waring o the transverse cross-sections, which is signiicant or thick lates, has been emloed or the elasto-lastic analsis o thick to thin lates. This ormulation is irst o its kind or or elastolastic analsis o thick lates. The C o isoarametric ormulation adoted is ver simle and eicient as comared to other inite element ormulations. HOST has redicted higher collase load in general

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