A new approach for elasto-plastic finite strain analysis of cantilever beams subjected to uniform bending moment

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1 Sādhanā Vol. 41, No. 4, April 216, pp DOI 1.17/ x Ó Indian Academy of Science A new approach for elato-platic finite train analyi of cantilever beam ubjected to uniform bending moment GÖKHAN T TAYYAR Department of Naval Architecture and Marine Engineering, Itanbul Technical Univerity, Sarıyer, Itanbul, Turkey tayyargo@itu.edu.tr MS received 24 January 213; revied 3 March 215; accepted 5 February 216 Abtract. The reliability and limit of olution for tatic tructural analyi depend on the accuracy of the curvature and deflection calculation. Even if the material model i cloe to the actual material behavior, phyically unrealitic deflection or divergence problem are unavoidable in the analyi if an appropriate fundamental kinematic theory i not choen. Moreover, accurate deflection calculation play an important role in ultimate trength analyi where in-plane tree are conidered. Therefore, a more powerful method i needed to achieve reliable deflection calculation and modeling. For thi purpoe, a new advanced tep wa developed by coupling the elato-platic material behavior with precie general planar kinematic analyi. The deflection i generated preciely without making geometric aumption or uing differential equation of the deflection curve. An analytical finite train olution wa derived for an elato-platic primatic/non-primatic rectangular cro-ectioned beam under a uniform moment ditribution. A comparion of the analytical reult with thoe from the Abaqu FEM oftware package reveal a coherent correlation. Keyword. Curvature; deflection curve; cantilever beam; elato-platic analyi; tapered beam ubjected to tip moment; kinematic deflection theory. 1. Introduction The objective in econd-order tatic tructural analyi i to determine the balance of internal and external force throughout the deformed configuration of a tructure until platic hinge formation occur. Such analyi i a trancendental problem that require an iterative olution. Second-order tatic tructural analyi ha three main part: tatic equilibrium, kinetic analyi, and kinematic analyi (figure 1). The tatic equilibrium part evaluate internal force uing the external force throughout the deflected geometry (the deflection curve and cro ection). The kinetic part i curvature calculation from internal force uing the material contitutive law and ection integration. The kinematic part determine the deflection by atifying the compatibility requirement between the train (curvature) and diplacement [1]. Kinematic analyi can be claified into two main area. The firt type of analyi depend on olution uing differential equation under geometric aumption. The econd type of analyi depend on geometric conideration a in elatica or a in curvature-baed deflection method, where geometrically precie deflection calculation are obtained in term of the curvature without geometric aumption [2]. Accurate deflection calculation in finite train play an important role in ultimate or in pot-collape analyi, where in-plane tree are conidered. The olution time, convergence, and accuracy will be problem if the fundamental kinematic theorie are not well elected according to the expected deflection. When a olution i obtained analytically, the reliability and limit of the olution and the deflection depend on geometric aumption in kinematic theory. Otherwie, phyically unrealitic deflection are unavoidable, particularly for large train, large deflection, and large rotation. For the econd type of analyi, the mot well-known analytical olution i baed on elliptical integration, which are not capable of analyzing ditributed load, variable tiffne member, or material nonlinearity [3]. Moreover, olution with elliptic integral are very enitive to even mall error in the calculation [4]. The finite element method (FEM) i the mot widely ued numerical tructural analyi method [1]. FEM i baed on conventional kinematic theorie with geometric aumption. Failure or convergence problem are unavoidable when conidering high deflection beyond a certain limit, even with a large number of meh element and high CPU time. Therefore, there i a need for more powerful method for more reliable deflection calculation and modeling to atify the increaing requirement of tructural analyi. 451

2 452 Gökhan T Tayyar Figure 1. analyi. Iterative flow chart of econd-order tatic tructural The analytical olution for beam that conider material and geometric nonlinearitie are limited [5]. Solution are available for nonlinear elatic material model, uch a Ludwick type [6] and Ramberg Ogood type material [7]. Some form of moment curvature model attempted to imulate elato-platic behavior with hyperbolic-tangent [] or logarithmic type nonlinearitie [9]. Gao [1] reported an analytical olution for elato-platic finite train in wide plate. Thi tudy examined a new advanced tep by adapting elato-platic behavior to curvature-baed kinematic diplacement theory (KDT) [11]. In KDT, deflection i generated preciely without making any geometric aumption or uing differential equation of the deflection curve. A new analytical olution i propoed for elato-platic primatic/non-primatic rectangular cro-ectioned beam ubjected to a tip moment. The curvature value are ued geometrically to form the deflection curve. The aim i to have a plain and comprehenible preentation. Therefore, the compatibility condition and lateral torional buckling are retrained by auming planar deflection. In addition, internal force are elected a a uniform moment ditribution to avoid the need for an iterative procedure for econdorder theory and governing equation for the hear effect. The analytical reult of the application were compared with reult from the Abaqu FEM oftware package, and there wa a coherent correlation within the limit of the oftware for large train. Figure 2. Planar deflection curve of a tructure. All cro ection of the tructure are in equilibrium with the external and internal force until fracture occur. Reultant force have to be in equilibrium with the tre ditribution over the cro ection. Contitutive law are ued to expre the train ditribution. Strain ditribution over a cro ection i alway linear and proportional to it curvature value, even for a nonlinear tre ditribution. Therefore, thi equilibrium can be repreented by the curvature value of the cro ection, regardle of elatic or inelatic behavior (figure 3). The phyical meaning of the curvature i the rate of change in the lope of the major axi, a expreed in (1). The curvature value of the cro ection of the beamcolumn are uniform for the egment length d. The reultant force on the egment are contant, or the egment i infiniteimal [11]. The hape of the egment for uniform curvature ditribution i indicated by the arc of a circle with radiu r (figure 4). The radiu i equal to the abolute invere ratio of the curvature, a expreed in (2). h and h?dh repectively denote the initial and terminal point of the lope angle of the egment of the deflection curve with the x-axi (figure 4). 2. Fundamental of curvature-baed kinematic planar deflection calculation Let the axe in curvilinear coordinate be the direction of the normal vector of the principal plane of a tructure. Thi make it poible to decribe the planar diplacement of the tructure with the deflection of it major reference axi a with a general regular kew curve [12]. Briefly, the tructure i generated by cro ection in which the centroid C move along reference axi a. The plane of the cro ection i normal to a, a hown in figure 2, where i the curve length [12]. Figure 3. Strain and tre ditribution over the cro ection.

3 A new approach for elato-platic finite train analyi 453 without any geometric aumption. The relation between curvature (or train) and diplacement in curvilinear coordinate i conidered a the exact olution: dx d ¼ co hðþ ð4þ dz d ¼ in hðþ: ð5þ Finally, the analytical expreion for a deflection vector between the curvilinear length away from the reference point on a deflection curve can be obtained by ubtituting (3) into (4) and (5) and integrating. The following expreion i obtained (figure 1) [15]: Figure 4. Segment of a deflection curve. j ¼ dh=d ð1þ r ¼ j1=jj: ð2þ The olution to (1) i very imple when curvilinear coordinate are ued. If the curvature value over a egment i given, the only unknown dh can be obtained from (1). Uing dh, the deflection curve calculation turn into a baic geometry problem. If the location and lope of the initial point, the length, and the radiu of the arc in (2) are known, the terminal point of the egment can be determined eaily uing geometric conideration [11]. Briefly, the egment i expreed with an arc with center angle, curve length, chord length, and radiu of dh, d, dc, and r, repectively. However, the curvature on the tructure i not alway contant. Therefore, the main quetion i how the different arc can be connected to compoe a deflection curve, or how the deflection curve can be repreented for a nonuniform curvature ditribution. The deflection curve i a regular kew curve that i differentiable and need to meet the continuity condition [13, 14]. If the deflection curve i differentiable, the lope angle of the deflection curve in curvilinear coordinate can be evaluated from (1) jut by integration. If the integration begin from a pecific reference point where the lope angle i known, the lope angle of the terminal point can be derived from (1) by integrating a follow [15]: hðþ¼h ðþþ Z jðþd: ð3þ Thi integration only require curvature value of the d egment. Therefore, if the curvature ditribution i known, the lope angle of the deflection curve can be obtained aðþ¼x ðþiþz ðþk¼ Z co hðþdi þ Z in hðþdk; ð6þ where i and k are the unit vector of the rectangular Carteian coordinate ytem. The deflection curve of the entire tructure can be contructed with thee circular arc egment according to curvature-baed kinematic theory [11]. The deflection calculation become jut a kinematic geometric problem if the curvature can be dicrete and expreed by a function or a ditribution. If the curvature ditribution can be formulated for any complicated tructure, the deflection curve can be evaluated uing the diplacement vector (figure 5). Fundamentally, deflection calculation i baed on the differential equation of the deflection curve, a given in (1) and figure 3. The main goal i to determine the value of dz and dx. Small deflection theory aume that lateral deflection i o mall that the difference between dx and d i zero. Additionally, the tangent angle at any point i aumed to be contant for an infiniteimal element length, which i o mall that it yield it tangent value. The main aumption in mall deflection theory are ummarized a follow (figure 4): Figure 5. Relative location according to the reference point of the deflection curve.

4 454 Gökhan T Tayyar d ¼ dx and h ¼ dz dx : Therefore, the diplacement baed on mall deflection theory can be determined a follow: j ¼ dh=d d2 z dx 2 : The following i alo aumed: p d 6¼ dc ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx 2 þ dz 2 and h 6¼ arctan dz=dx ¼ u: If we conider large deflection theory, the main aumption are that the curvilinear length of the infiniteimal element i equal to the length of the diplacement vector, and that the tangent angle i contant for an infiniteimal element length. Therefore, diplacement under large deflection theory can be determined a follow: j ¼ dh=d d 2 z=dx 2 3=2 : 1 þ ðdz=dxþ 2 From a geometric perpective, it i impoible to ay that curvilinear length i equal to the magnitude of the incremental diplacement vector dc between adjacent node. Therefore, there will alway be numerical error if the egment length i not infiniteimal. Additionally, the lope angle at the initial point of the egment i not contant during the diplacement of the egment. Thi mean the deflection i linear between two adjacent point. However, the change in the lope angle i already defined with the curvature value in (1). With thi imple KDT, it i poible to extend the analytical or numerical olution of elatica or exact geometric olution of beam to ditributed load, variable tiffne member, or material nonlinearity under full geometric nonlinearity. Even though Green or Jaumann train component are powerful with Lagrangian formulation in nonlinear FEA, they fundamentally till include geometric aumption. Therefore, a large number of meh element are neceary in the cae of large rotation and large diplacement. Another advantage of KDT i that the egment length i not important when the curvature value i contant over the egment, even if rotation and diplacement are large (figure 5). Particular olution with the geometric ue of the curvature in deflection tudie have been reported. The wellknown application in the bending of beam are thoe with uniform curvature ditribution, where the deflection curve form with part of a circle [16]. Tayyar and Bayraktarkatal [17] reported a numerical iterative method for the nonuniform curvature ditribution of tiffened plate conidering econd-order theory and local plate buckling while neglecting the hear effect. Thi method provide an opportunity to form the mot complex deflection curve eaily via curvature of the individual egment. Exact planar kinematic diplacement theory for a non-uniform curvature ditribution wa firt reported with the application of a tapered rectangular elatic cantilever beam ubjected to a tip moment [11]. An analytical method for the deflection calculation uing curvature value wa originally reported with the application of an elatic rectangular tapered beam ubjected to a tip moment [15]. Numerical analyi of an elatic perfectly platic tiffened panel with KDT wa preented by Bayraktarkatal and Tayyar [1] and Tayyar et al [19]. 3. Analytical method for elatic perfectly platic material behavior The analyi i compoed of two part. In the kinetic part, the moment curvature relationhip of the cro ection i determined by ection integration from the reultant force and material model. In the kinematic part, the deflection calculation i evaluated from the curvature function obtained in the kinetic part. Kinetic analyi i conidered for tapered and primatic condition. A uniform moment ditribution i preferred for comprehenibility, and it i poible to derive any function for moment ditribution over the deflection curve for ue in the curvature equation to achieve a more ophiticated calculation. 3.1 Moment curvature relationhip of elatoplatic rectangular cro ection Under the aumption of the Bernoulli Navier hypothei, the train ditribution over the cro ection i linear, even when material and geometric nonlinearitie take effect. All equilibrium equation are evaluated over the cro ection. The train ditribution can be expreed a follow: eðþ¼j z ðz N zþ; ð7þ where z N repreent the hift between the neutral axi and curve axi (figure 3). Figure 6 how the typical tre train diagram of an elatic perfect platic material, where r repreent the tenion yield tre, and e repreent the tenion yield train. The train tre relationhip for the elato-platic, homogenou, iotropic material aume that the abolute value for the tre and train in the tenion and compreion ide are the ame. The tre ditribution can be derived uing the following expreion, where E repreent the Young modulu, and z cr repreent the abolute critical ditance from the neutral axi, where an inelatic behavior i tarted: r ¼ Ej ð z N zþ jjj\e = jz N zj : ðþ r Signðjðz N zþþ jjj e = jz N zj

5 A new approach for elato-platic finite train analyi 455 >< j e ¼ M EIffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j ¼ br 3 >: j p ¼ 3E 2 ðm MÞ jmj\2r I=h jm : ð13þ j 2r I=h Figure 6. Stre/train diagram of the elato-platic material. Figure 7 how the elatic, primary platic, and econdary platic tage of the train and tre ditribution over a rectangular cro ection before platic hinge (fully platic) occur. h repreent the height of the cro ection, and b repreent the width. The equilibrium of the internal and external force at any cro ection i obtained from equilibrium equation over the cro-ection area A. The firt equation how the relationhip between the reultant normal force, and the econd equation how the relationhip between the moment at the cro ection: ZZ N ¼ rdydz ð9þ A A ZZ M ¼ rzdydz: ð1þ The maximum moment capacity can be expreed a follow when the cro ection i fully platic: M ¼ bh 2 r =4: ð11þ Equation (9) can be atified only if z N equal zero when the action of normal force and local buckling doe not exit. Therefore, the neutral axi and curve axi i fixed at the centroid of the cro ection when a homogeneou material i conidered. The moment equation can be expreed by ubtituting () into (1) a follow: < EIj M ¼ : M r3 b 3E 2 j 2 jjj\2e =h jjj2e =h ð12þ where I repreent econd moment area. The curvature function can be evaluated from (12) and expreed a follow, where j \, M [, and M \ M : 3.2 Diplacement equation of elato-platic cantilever primatic beam A cantilever beam ubjected to a tip moment M with total beam length L i conidered (figure ). The width of the rectangular-beam cro ection i b, and the height of the cro ection i h. The reference point i elected a the clamped edge, where the lope angle i zero. Equation (3) can be expreed a follow by ubtituting (13) under a uniform bending moment, where the curvature will be contant throughout the curve. hðþ¼ j e jjj\2e =h : ð14þ j p jjj 2e =h The diplacement vector can be expreed a follow by ubtituting (14) into (6): >< aðþ¼ >: R R coðj e Þd i þ R co j p inðj e Þd j jjj\2e =h R : d i þ in j p d j jjj 2e =h ð15þ 3.2a Application for cantilever primatic beam: The dimenion and main propertie are elected a L = 1 mm, E = 26, N/mm 2, r = 13 N/mm 2, b = 2 mm, and h = 5 mm. No train hardening effect after yield tre i taken into account. The analytical olution could be obtained eaily uing (15). Figure 9 how the deflection curve of the axi from the analytical olution of the elato-platic material propertie. The in-plane deformation reult were compared with reult from Abaqu FEM oftware. The FEM reult were obtained uing the four-node quadrilateral membrane element M3D4R, a hown in figure 1. The highet deviation i approximately 4.2% when the tip diplacement dz/l i.6155 and M/M p i.99. Unfortunately, the nonlinear FEM olution failed beyond thi range, and convergence Figure 7. Strain/tre ditribution of an elato-platic rectangular cro ection.

6 456 Gökhan T Tayyar Figure. Primatic cantilever beam. could not be achieved. In contrat, the KDT-baed nonlinear olution generate reult for ratio of up to M/M p = 1. The hape of the deflection curve turn into a circle, which become increaingly maller when it i cloe to a platic hinge, a hown in figure Diplacement equation of tapered beam A tapered cantilever beam under a uniform moment M and total beam length L i conidered (figure 11). The width of the rectangular beam cro ection i b, and height of the beam cro ection are H and H min at the clamped edge and free end, repectively (figure 11). The height of the cro ection at any arc length can be expreed by (16), where i the curve length. h ðþ¼h Dh L : ð16þ The effect of tip moment i different in each cro ection. The applied M moment can form a platic hinge by the lo of height of the cro ection. Therefore, the critical variable need to be the height or it curve length intead of the moment. The critical curve length for a given tip moment where platic hinge formation occur by a decreae in cro ection height i repreented by cr, which can be obtained by ubtituting (16) into (13): cr ¼ L rffiffiffiffiffiffiffi Dh H 6M : ð17þ br Equation (1) i repreented in parametric form. The reference point i elected a the clamped edge, where the lope angle i zero. Equation (3) can be expreed a follow by ubtituting (17) into (13) and then into (3) under a uniform bending moment: 6M H þ h >< h E ðþ¼ ð ðþþ beh 2 h ðþ 2 \ cr h P ðþ¼ 2Lr ; >: pffiffi ½GðÞ G ðþš cr \ 3 EDh vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 u where bhðþr þ t4br b h ðþ2 r M A: 4 hðþ¼ ð1þ The diplacement vector can be expreed by ubtituting (1) into (6): aðþ¼ >< >: R R cr R co h P ðþdi þ R in h P ðþdk cr co h E ðþdi þ R co h P ðþdi þ R cr in h E ðþdk þ R in h P ðþdk \ cr \ L: cr cr co h E ðþdi þ R in h E ðþdk cr [ L or cr ð19þ x/l z/l M/Mp=.9994 M/Mp=.9975 M/Mp=.9975 M/Mp=.99 M/Mp=.95 M/Mp=. M/Mp=.2 M/Mp= Figure 9. Deflection curve of the elato-platic cantilever beam.

7 A new approach for elato-platic finite train analyi 457 M/Mp Nonlinear FEM-Abaqu.6 KDT dz/l Figure 1. Tip deflection of the elato-platic cantilever beam. The in-plane deformation of a planar curvature calculation for rectangular cro-ection ha been explained, and kinematic analyi wa derived for a primatic/nonprimatic elato-platic cantilever beam under a uniform moment ditribution. Nonlinear material behavior wa coupled uing KDT. The analytical reult of the application were compared with thoe from Abaqu FEM oftware, revealing coherent correlation within the limit of the oftware. The advantage of precie modeling of the deflection curve provided high accuracy in the finite train. Thi preliminary concept can be extended to a range of complicated problem. Uniform moment were elected, but there wa no retriction in uing a moment ditribution over the deflection curve. For a non-uniform moment ditribution, the reultant force of each point on the deflection curve will depend on the deflected hape being evaluated. If needed, thi econd-order calculation can be determined iteratively. Reference z/l 3.3a Application to cantilever tapered beam: The dimenion are elected a L = 1 mm, b = 2 mm, H = 5 mm, and H min = 4 mm. The ame material propertie are applied to the tapered beam. M i taken a M p at the tip of the tapered beam. The analytical olution of the elatic and elato-platic material property are obtained uing (19). FEM reult were obtained uing Abaqu oftware, and figure 12 plot the reult of the deflected hape. The highet deviation wa found to be.1% at the free edge under the given condition. 4. Concluion Figure 11. Tapered rectangular beam. Ela c olu on with geometric nonlinearity Nonlinear FEM olu on Elato-Pla c olu on with geometric nonlinearity Figure 12. Deflection curve of the tapered elato-platic cantilever beam at the platic moment. x/l [1] Chen W F and Duan L ed 2 Bridge engineering handbook. Boca Raton: CRC Pre [2] Tayyar GT 211 Determination of ultimate trength of the hip girder (in Turkih). Ph.D. Itanbul Technical Univerity, Itanbul. [Acceed ] [3] Feti D G 26 Nonlinear tructural engineering with unique theorie and method to olve effectively complex nonlinear problem. Berlin: Springer [4] Bona F D and Zelenika S 1997 A generalized elatica-type approach to the analyi of large diplacement of pringtrip. Proc. Intn. Mech. Eng. 221(C): [5] Lee K 22 Large deflection of cantilever beam of nonlinear elatic material under a combined loading. Int. J. Nonlinear Mech. 37: [6] Lewi G and Monaa F 192 Large deflection of cantilever beam of non-linear material of the ludwick type ubjected to an end moment. Int. J. Nonlinear Mech. 17: 1 6 [7] Prathap G and Varadan T K 1976 The inelatic large deformation of beam. J. Appl. Mech. 43: [] Oden J T and Child S B 197 Finite deflection of a nonlinearly elatic bar. J. Appl. Mech. 37: 4 52 [9] Lo C C and Gupta S D 197 Bending of a nonlinear rectangular beam in large deflection. J. Appl. Mech. 45: [1] Gao X 1994 Finite deformation elato-platic olution for the pure bending problem of wide plate of elatic linear-hardening material. Int. J. Solid Struct. 31(1): [11] Tayyar G T and Bayraktarkatal E 212a Kinematic diplacement theory of planar tructure. Int. J. Ocean Syt. Eng. 2(2): 63 7 [12] Hay G E 1942 The finite diplacement of thin rod. Tran. Am. Math. Soc. 51: [13] Bolton K M 1975 Biarc curve. Comput. Aided De. 7(29): 9 92 [14] Meek D S 22 Coaxing a planar curve to comply. J. Comput. Appl. Math. 14(1): [15] Tayyar G T 212 A new analytical method with curvature baed kinematic deflection curve theory. Int. J. Ocean Syt. Eng. 2(3):

8 45 Gökhan T Tayyar [16] Timohenko S 194 Strength of material part I elementary theory and problem. New York: D. Van Notrand Company [17] Tayyar G T and Bayraktarkatal E 212b A new approximate method to evaluate the ultimate trength of hip hull girder. In: Rizzuto E and Soare C G (ed) Sutainable maritime tranportation and exploitation of ea reource. London: Taylor & Franci Group. pp [1] Bayraktarkatal E and Tayyar G T 214 Geometric olution in progreive collape analyi of hull girder. J. Mar. Sci. Technol. 22(4): [19] Tayyar G T, Nam J and Choung J 214 Prediction of hull girder moment-carrying capacity uing kinematic diplacement theory. Mar. Struct. 39: