materials
Article Cross-Section Deformation and Bending Moment of a Steel Square Tubular Section
Stanisław Kut * and Feliks Stachowicz Department of Materials Forming and Processing, Rzeszow University of Technology, al. Powst. Warszawy 8, 35-959 Rzeszów, Poland; [email protected] * Correspondence: [email protected]; Tel.: +48-17-743-2527
Received: 22 October 2020; Accepted: 13 November 2020; Published: 16 November 2020
Abstract: When bending thin-walled profiles, significant distortion of the cross-section occurs, which has a significant impact on the course of the bending moment characteristics and on the value of allowable bending curvatures. This paper presents the results of experimental and numerical modeling of the box profile bending process, which was carried out in order to determine the dependence of the cross-sectional shape and bending moment of bending curvature. Extensive numerical calculations were used to model the process of shaping a square pipe from a circular tube and to model the bending process, especially when taking into account the effects of such a deformation path. The pure bending moment characteristics and the deformation of the cross-section were performed for a 25 25 2 mm × × square tube made of S235JR structural steel. The innovative approach for determining the parameters of cold bending square tubes pertained to considering the stress state in the preserved material in individual areas of their cross-section. The results of numerical modeling—after considering the history of deformation (i.e., the process of forming a square pipe from a pipe with a circular cross-section)—gave a satisfactory agreement with the results of experimental tests, both in terms of the degree of pipe wall deflection and the characteristics of the bending moment.
Keywords: shaping; bending; box profile; constructional S235JR steel; deformation; FEM
1. Introduction Rectangular steel tubes are widely used as structural elements in numerous applications. The advantages of rectangular pipes over circular tubing include the fact that they are easy-to-fit regarding the direction, produce a high torque when twisting, and reduce the weight of structured parts [1]. Rectangular pipes are usually produced from a welded round tube using a cold drawing trough and a head, which is comprised of idle rolls [2,3]. However, many experimental works and theoretical studies, including numerical calculations [4,5], are devoted to the phenomena that occur during the formation of rectangular pipes, but many issues still need to be worked out. For example, differences in mechanical properties result from the formation process observed around the periphery of rectangular pipes [6], especially in their corners, which may probably affect the metal flow process in the production of bent products. During the elastoplastic bending process of thin-walled profiles, their cross-sections are subjected to distortions, which strongly affect the bending moment characteristics, the value of the allowable (minimum) bending radius and the amount of the springback e.g., [7–10]. In the case of bending pipes with a circular cross-section, the parameters that describe the deformation degree of the cross-section is described by an indicator called ovalization. In the case of bending pipes with a rectangular cross-section (box profiles), it is not possible to apply a similar unique parameter due to the complex shape of the cross-section after bending, which is significantly different from the initial cross-section. The cross-sectional deformation that occurs during bending leads to the pipe
Materials 2020, 13, 5170; doi:10.3390/ma13225170 www.mdpi.com/journal/materials Materials 2020, 13, x FOR PEER REVIEW 2 of 10
after exceeding its critical curvature value [7]. The observed changes in the geometry of the cross- section of rectangular tubes and the determination of the impact of these changes on the value of other parameters characterizing the bending process have been the subject of many studies. Based on the numerical modelling results of fine element method (FEM), Chen and Masuda [9] found that the flattening ratio can be expressed by a sole function of nondimensional curvature, independent of tube thickness in elastic bending. During plastic bending, flattening is influenced by tube thickness and the material parameters of yield stress and strain hardening; however, these effects are very small. MaterialsThe issues2020, 13related, 5170 to the bending process of box profiles are presented in numerous publications,2 of 10 including experimental studies [8,11], theoretical considerations based on the deformation theory of collapsingplasticity after[12–15], exceeding numerical its critical modelling curvature with value the [use7]. The of observedFEM [16–19], changes and in the the geometryuse of artificial of the cross-sectionintelligence identification of rectangular [20]. tubes Using and thethe determinationdeformation theory of the of impact plasticity of these and changes the energy on themethod value that of otherPoulsen parameters et al. [14] characterizing built, the suck-in the bending of the exte processrnal haveflange been of the the profile subject was of many almost studies. independent Based on of thestrain numerical hardening modelling and the results material’s of fine yield element stress. method Observations (FEM), Chenand measurements and Masuda [ 9of] foundthe bent that cross- the flatteningsection shape ratio of can the be box-section expressedby enabled a sole functionus to establish of nondimensional relationships curvature,that determine independent the dependence of tube thicknessof the cross-sectional in elastic bending. when forming During the plastic bending bending, curvature flattening [8]. The is influenced following indices by tube are thickness assumed and for thethe material description parameters of the shape of yield of a cross-section stress and strain after hardening; bending (Figure however, 1): these effects are very small. The issuesUzw: deflection related to of the the bending inner horizontal process ofwall, box profiles are presented in numerous publications, includingUzz: deflection experimental of the studies outer horizontal [8,11], theoretical wall, considerations based on the deformation theory of plasticity [12–15], numerical modelling with the use of FEM [16–19], and the use of artificial Uyw: deflection of the inner profile corners, intelligence identification [20]. Using the deformation theory of plasticity and the energy method that Uyz: deflection of the outer profile corners, Poulsen et al. [14] built, the suck-in of the external flange of the profile was almost independent of strain hardeningUym: maximum and the material’s lateral displacement yield stress. of Observations points of the and wall measurements beyond the original of thebent cross-section. cross-section shapeThe of the horizontal box-section walls enabled were those us to that establish normalized relationships to the thatbending determine plane; thethe dependencevertical walls of were the cross-sectionalthose parallel to when the bending forming plane. the bending ‘Inner’ curvature and ‘outer’ [8 ].refer The to following the direction indices forward are assumed and away for from the descriptionthe bending of axis, the shaperespectively. of a cross-section after bending (Figure1):
Figure 1. Characteristic tube dimensions and displacements of the deformed cross section [8]. Figure 1. Characteristic tube dimensions and displacements of the deformed cross section [8].
Uzw: deflection of the inner horizontal wall, The purpose of this work was to verify the correctness of the numerical model used to determine Uzz: deflection of the outer horizontal wall, the changes in the geometry of the box profiles’ cross-section, as well as the relationship between the Uyw: deflection of the inner profile corners, bending moment and the bending curvature based on the results of experimental studies. An Uyz: deflection of the outer profile corners, additional purpose of the research was to determine the effect of technological deformations and Uym: maximum lateral displacement of points of the wall beyond the original cross-section. internalThe horizontalstresses created walls werein the those plastic that shaping normalized of a round to the bendingtube into plane; a square the verticalsection wallson selected were thosebending parallel parameters to the bending of a 25 × plane. 25 × 2 ‘Inner’mm square and ‘outer’steel pipe. refer The to thescope direction of the research forward work and away included: from the1. bendingFEM numerical axis, respectively. modelling during the cold plastic shaping process of a round tube with an Theexternal purpose diameter of this (d work = 30 was mm) to and verify a wall the correctnessthickness (g of = the2 mm) numerical into a square model cross-section used to determine pipe. the2. changesFEM numerical in the geometry modelling of the of the box process profiles’ of cross-section, forming a square as well pipe as with the relationship a pure bending between moment. the bendingModel moment A took and into the account bending the curvature technological based on defo thermations results of and experimental residual stresses studies. created An additional during purposeshaping of the researchof a square was topipe. determine Model theB did effect not of technologicaltake into account deformations the impact and internalof technological stresses createddeformations in the plastic and shaping internal of astresses round tubecreated into during a square shaping section of on a square selected pipe. bending parameters of a 25 25 2 mm square steel pipe. The scope of the research work included: × × 1. FEM numerical modelling during the cold plastic shaping process of a round tube with an external diameter (d = 30 mm) and a wall thickness (g = 2 mm) into a square cross-section pipe. 2. FEM numerical modelling of the process of forming a square pipe with a pure bending moment. Model A took into account the technological deformations and residual stresses created during shaping of a square pipe. Model B did not take into account the impact of technological deformations and internal stresses created during shaping of a square pipe. Materials 2020, 13, 5170 3 of 10
Materials 2020, 13, x FOR PEER REVIEW 3 of 10 3. Experimental four point bending of a 25 25 2 mm square pipe was tested to verify the results × × 2. Materialof numerical and Experimental calculations. Procedure 2. MaterialThe study and was Experimental carried out Procedureon a 25 × 25 × 2 mm box profile, which was cold shaped from a welded round tube made of constructional S235JR steel. The samples prepared from the profile wall were The study was carried out on a 25 25 2 mm box profile, which was cold shaped from a welded tested under uniaxial tension using × the Hollomon× strain–stress relationship in the form of 휎 = round tube made of constructional S235JR steel. The samples prepared from the profile wall were 530휀0.044, which showed significant reproducibility. Other tensile parameters were as follows: yield tested under uniaxial tension using the Hollomon strain–stress relationship in the form of σ = 530ε0.044, stress Re = 388 MPa; ultimate strength Rm = 486 MPa; Young modulus E = 199 GPa; Poisson’s ratio ν = which showed significant reproducibility. Other tensile parameters were as follows: yield stress 0.3. Re = 388 MPa; ultimate strength Rm = 486 MPa; Young modulus E = 199 GPa; Poisson’s ratio ν = 0.3. The experimental tests were carried out with the help of a device mounted on a testing machine The experimental tests were carried out with the help of a device mounted on a testing machine (Figure 2). It should be noted that the sample was placed in the holders in such a way that the wall (Figure2). It should be noted that the sample was placed in the holders in such a way that the wall with with the weld was a vertical wall, indicating that the joint material was in the neutral layer. The the weld was a vertical wall, indicating that the joint material was in the neutral layer. The bending bending on four supports ensured that the pipes were loaded with a pure bending moment, without on four supports ensured that the pipes were loaded with a pure bending moment, without the use the use of lateral forces. During bending, the deflection of horizontal (inner and outer) and vertical of lateral forces. During bending, the deflection of horizontal (inner and outer) and vertical profile profile walls, as well as the bending force, were measured continuously. Then, the bending moment walls, as well as the bending force, were measured continuously. Then, the bending moment value value was calculated. For several selected bending steps, measurements were made of the tube was calculated. For several selected bending steps, measurements were made of the tube curvature curvature and the displacement of characteristic points of individual walls of the cross-section, i.e. and the displacement of characteristic points of individual walls of the cross-section, i.e., the value the value of the section widening factor 2Uym, the value of the deflection of the internal horizontal of the section widening factor 2Uym, the value of the deflection of the internal horizontal wall Uzw, wall Uzw, and the external horizontal wall Uzz. and the external horizontal wall Uzz.
Figure 2. LaboratoryLaboratory four four point point bending bending test test device device [8]: [18-]: lever, 1—lever, 2 - clamping 2—clamping jaws, 3 jaws, - sample, 3—sample, 4—connector,4 - connector, 5—bending5 - bending arm, arm, and and 6 6—traverse - traverse of of the the testing testing machine. machine. 3. Numerical Calculations 3. Numerical Calculations In order to determine the distributions and values of plastic deformations and residual stresses In order to determine the distributions and values of plastic deformations and residual stresses prevailing in the pipe after the cold shaping process, we constructed a numerical model of the cold prevailing in the pipe after the cold shaping process, we constructed a numerical model of the cold profiling process. The starting material was a round pipe with an external diameter (d = 30 mm) profiling process. The starting material was a round pipe with an external diameter (d = 30 mm) and and a wall thickness of 2 mm, made of S235JR constructional steel, which was profiled to obtain a wall thickness of 2 mm, made of S235JR constructional steel, which was profiled to obtain a 25x25x2 a 25 25 2 mm square pipe. Due to the fact that the pipe length did not change during profiling, mm ×square× pipe. Due to the fact that the pipe length did not change during profiling, the profiling the profiling analysis was carried out in a flat deformation state. Due to the occurrence of plane analysis was carried out in a flat deformation state. Due to the occurrence of plane symmetry, the symmetry, the numerical model was made for half of the cross-section. The shaping tools were modelled numerical model was made for half of the cross-section. The shaping tools were modelled as perfectly as perfectly rigid. Quad 4 elements type 11 were used to discretize the deformable body [21]. In order rigid. Quad 4 elements type 11 were used to discretize the deformable body [21]. In order to improve to improve the behavior of these elements during bending, an additional interpolation function was the behavior of these elements during bending, an additional interpolation function was applied applied during the assumed strain procedure [21]. An elastic–plastic model of the pipe material with during the assumed strain procedure [21]. An elastic–plastic model of the pipe material with non- non-linear strengthening described by the Hollomon power equation was adopted. Isotropic properties linear strengthening described by the Hollomon power equation was adopted. Isotropic properties of the pipe material were assumed for both elastic and plastic deformations. The strain curve was of the pipe material were assumed for both elastic and plastic deformations. The strain curve was introduced into the program in a tabular form. The entered coordinates of the strain hardening curve introduced into the program in a tabular form. The entered coordinates of the strain hardening curve were calculated on the basis of the pipe material parameters, which were determined on the basis of the were calculated on the basis of the pipe material parameters, which were determined on the basis of uniaxial tensile test: Re = 388 MPa, Rm = 486 MPa, C = 530 MPa, and n = 0.044. To describe the friction the uniaxial tensile test: Re = 388 MPa, Rm = 486 MPa, C = 530 MPa, and n = 0.044. To describe the friction between the tools and the profiled pipe, we used the Coulomb friction model with a friction coefficient of 0.1. Numerical modelling was performed using a commercial MSC.MARC/Mentat system, which is considered one of the most advanced programs for FEM analysis, and particularly
MaterialsMaterials2020 2020, 13, ,13 5170, x FOR PEER REVIEW 4 of4 of 10 10
the uniaxial tensile test: Re = 388 MPa, Rm = 486 MPa, C = 530 MPa, and n = 0.044. To describe the betweenfriction the between tools and the thetools profiled and the pipe, profiled we used pipe, the we Coulomb used the friction Coulomb model friction with amodel friction with coe affi frictioncient ofcoefficient 0.1. Numerical of 0.1. modelling Numerical was modelling performed was using performed a commercial using MSC.MARC a commercial/Mentat MSC.MARC/Mentat system, which is consideredsystem, which one ofis theconsidered most advanced one of the programs most adva fornced FEM programs analysis, for and FEM particularly analysis, and for modellingparticularly nonlinearfor modelling and contact nonlinear issues. and contact The change issues. in The the shapechange of in the the cross-section shape of the cross-section during pipe shapingduring pipe is shownshaping in Figureis shown3. The in Figure input material 3. The input is a pipe material with ais circulara pipe with cross-section a circular (Figure cross-section3a), which (Figure in the 3a), technologicalwhich in the process technological is constructed process by is longitudinalconstructed rolling.by longitudinal Such a pipe rolling. is also Such formed a pipe by is longitudinal also formed rollingby longitudinal and, in subsequent rolling and, stages, in subsequent it takes intermediate stages, it takes shapes intermediate (Figure3b–d) shapes until (Figure the required 3b–d)square until the sectionrequired is obtained square section (Figure is3e). obtained (Figure 3e).
FigureFigure 3. 3.Change Change in in the the shape shape of of the the cross cross sectionsection ofof thethe pipe in the subsequent subsequent stages stages of of profiling: profiling: (a) (a)round round pipe, pipe, (b (b––dd) )intermediate intermediate shapes; shapes; (e (e) )square square pipe. pipe.
AsAs is is commonly commonly known, known, the the properties properties of of elements elements formed formed via via cold cold working working are are significantly significantly affaffectedected by by the the strain strain hardening hardening phenomenon, phenomenon, which which is is when when a a material’s material’s formability formability is is reduced. reduced. FigureFigure4 shows 4 shows the the distribution distribution and and values values of of the the total total equivalent equivalent plastic plastic strain strain in in the the pipe pipe material material afterafter shaping. shaping. The The highest highest values values of of plastic plastic strain strain occur occur in in the the corners, corners, especially especially in in the the area area of of the the internalinternal radii, radii, where where their their calculated calculated values values reach 0.55 reach (Figure 0.554 a).(Figure Quite significant4a). Quite materialsignificant hardening material pointshardening are also points deformations are also in deformations other areas of thein other material. areas The of inequality the material. of the distributionThe inequality of plastic of the strainsdistribution on the cross-sectionof plastic strains of the on pipethe cross-section is even more of evident the pipe when is even deformations more evident are when smaller deformations than 0.1, Materials 2020, 13, x FOR PEER REVIEW 5 of 10 asare shown smaller in Figure than 0.1,4b. as shown in Figure 4b. The second important parameter that has a significant impact on the properties of components shaped via cold forming are residual stresses that prevail in the material after the technological process [16]. Residual stresses are the result of heterogeneous plastic deformations in the material after its formation. The greater the heterogeneity and gradient of plastic deformations, the higher the material stress values, which may be close to the yield point in accordance with the course of the strain curve. Figure 5a shows the distribution and values of Huber–Mises equivalent stresses in the final shaping phase are shown in Figure 5a, while Figure 5b shows the distribution of these stresses that prevail in the material after the technological process. As can be seen, the largest values of equivalent stresses occur in the areas of the largest deformation gradients and amount to about 590 MPa in the final shaping phase. In turn, the residual stresses in this area reach the highest value of about 563 MPa, i.e., only slightly lower (about 27 MPa) than the yield point. Both plastic deformation and own stress have a significant impact on the material after the technological process. As demonstrated after the cold pipe profiling process, their share is significant. For this reason, we
carried out an analysis of the impact of these parameters for a square pipe bending process. Figure 4. Distribution and values of the total equivalent plastic deformations in the pipe material after Figure 4. Distribution and values of the total equivalent plastic deformations in the pipe material after cold shaping; value scale: (a) strain range (0–0.55) and (b) strain range (0–0.10). cold shaping; value scale: (a) strain range (0–0.55) and (b) strain range (0–0.10). The second important parameter that has a significant impact on the properties of components shaped via cold forming are residual stresses that prevail in the material after the technological
Figure 5. Distribution and values of the Huber–Mises equivalent stresses in the material of the shaped pipe: (a) in the final shaping phase and (b) after the shaping process (internal stress).
Figure 3e shows the FEM analysis of the 25 × 25 × 2 beam bending process, which was carried out in two variants. In the first variant (Model A), both residual stresses and plastic deformation that prevail in the beam material after the technological process were taken into account (Figures 4 and 5b), whereas in the second variant (Model B), only the previously calculated cross-sectional shape was taken into account without considering residual stresses and deformations. Other parameters of the spatial stress state used in both models were the same. 3D models of the beam bending were made on the basis of bending device geometry, which was used in the experimental research described later in the article. Due to the occurrence of two plane symmetries while bending the tested beam along the beam axis and perpendicular to the cross-section in the middle of the distance between the supports, the numerical models were simplified to ¼ of the experimental model. The beam was defined as a deformable body, while the elements of the device in contact with it were perfectly rigid. To discretize the deformable body, 8-node hex-shaped elements of type 7 were used [21]. The size of the elements on the cross-section of the beam was 0.5 × 0.5 mm, while on its length it was 1 mm, which gave a total of 68,400 finite elements. The 3D model during bending, together with the mesh and defined bodies in contact, as well as planes of symmetry, is shown in Figure 6. An elastic–plastic model for beam material with a non-linear strain hardening curve described by the Hollomon power equation was adopted. The material parameters were the same and defined in the same way as during the simulation of the square pipe shaping process. In all models, the evolution of the plastic surface came as a result of the strain hardening phenomenon, which was described
Materials 2020, 13, x FOR PEER REVIEW 5 of 10
Materials 2020, 13, 5170 5 of 10 process [16]. Residual stresses are the result of heterogeneous plastic deformations in the material after its formation. The greater the heterogeneity and gradient of plastic deformations, the higher the material stress values, which may be close to the yield point in accordance with the course of the strain curve. Figure5a shows the distribution and values of Huber–Mises equivalent stresses in the final shaping phase are shown in Figure5a, while Figure5b shows the distribution of these stresses that prevail in the material after the technological process. As can be seen, the largest values of equivalent stresses occur in the areas of the largest deformation gradients and amount to about 590 MPa in the final shaping phase. In turn, the residual stresses in this area reach the highest value of about 563 MPa, i.e., only slightly lower (about 27 MPa) than the yield point. Both plastic deformation and own stress have a significant impact on the material after the technological process. As demonstrated after the cold pipeFigure profiling 4. Distribution process, and their values share of the is total significant. equivalent For plastic this reason, deformations we carried in the outpipe an material analysis after of the impactcold of shaping; these parameters value scale: for (a) astrain square range pipe (0–0.55) bending and process.(b) strain range (0–0.10).
Figure 5. Distribution and values of the Huber–Mises equivalent stresses in the material of the shaped pipe:Figure (a 5.) in Distribution the final shaping and values phase of and the (Huber–Misesb) after the shaping equivalent process stresses (internal in the stress). material of the shaped pipe: (a) in the final shaping phase and (b) after the shaping process (internal stress). Figure3e shows the FEM analysis of the 25 25 2 beam bending process, which was carried × × out inFigure two variants. 3e shows In the the FEM first variantanalysis (Model of the A),25 × both 25 × residual 2 beam stressesbending and process, plastic which deformation was carried that prevailout in two in the variants. beam material In the first after variant the technological (Model A), both process residual were takenstresses into and account plastic (Figures deformation4 and5 thatb), whereasprevail in in the the beam second material variant after (Model the technologica B), only thel previously process were calculated taken into cross-sectional account (Figures shape 4 wasand taken5b), whereas into account in the without second consideringvariant (Model residual B), only stresses the previously and deformations. calculated Other cross-sectional parameters ofshape the spatialwas taken stress into state account used without in both modelsconsidering were residual the same. stresses 3D models and deformations. of the beam bendingOther parameters were made of onthethe spatial basis stress of bending state used device in both geometry, models which were was the usedsame. in 3D the models experimental of the beam research bending described were latermade in on the the article. basis Due of bending to the occurrence device geometry, of two plane which symmetries was used while in the bending experimental the tested research beam alongdescribed the beamlater in axis the andarticle. perpendicular Due to the occurrence to the cross-section of two plane in symmetries the middle ofwhile the bending distance the between tested beam along the beam axis and perpendicular to the 1cross-section in the middle of the distance the supports, the numerical models were simplified to 4 of the experimental model. The beam was definedbetween as the a deformablesupports, the body, numerical while the models elements were of simplified the device into contact¼ of the with experimental it were perfectly model. rigid. The Tobeam discretize was defined the deformable as a deformable body, 8-nodebody, while hex-shaped the elements elements of ofthe type device 7 were in contact used [ 21with]. The it were size ofperfectly the elements rigid. To on discretize the cross-section the deformable of the beam body was, 8-node 0.5 hex-shaped0.5 mm, while elements on its lengthof type it 7 waswere 1 used mm, × which[21]. The gave size a totalof the of elements 68,400 finite on the elements. cross-section The 3Dof the model beam during was 0.5 bending, × 0.5 mm, together while withon its the length mesh it andwas defined1 mm, which bodies gave in contact, a total of as 68,400 well as finite planes elem of symmetry,ents. The 3D is model shown during in Figure bending,6. An elastic–plastic together with modelthe mesh for beamand defined material bodies with ain non-linear contact, as strain well hardening as planes curveof symmetry, described isby shown the Hollomon in Figure power 6. An equationelastic–plastic was adopted.model for Thebeam material material parameters with a non-linear were the strain same hardening and defined curve in thedescribed same way by the as duringHollomon the power simulation equation of the was square adopted. pipe shapingThe material process. parameters In all models, were the the same evolution and defined of the plastic in the surfacesame way came as asduring a result the of simulation the strain hardeningof the square phenomenon, pipe shaping which process. was described In all models, using the the evolution isotropic hardeningof the plastic model. surface The came calculations as a result used of the th Huber–Mises–Henckye strain hardening phenom plasticityenon, condition, which was the associateddescribed Prandtl–Russ plastic flow law, and the implicit integration of differential equations over time with the
Newton–Raphson method. Materials 2020, 13, x FOR PEER REVIEW 6 of 10
using the isotropic hardening model. The calculations used the Huber–Mises–Hencky plasticity Materialscondition,2020 the, 13, 5170associated Prandtl–Russ plastic flow law, and the implicit integration of differential6 of 10 equations over time with the Newton–Raphson method.
Figure 6. 3D model of the analyzed bending process: 1, bent beam with a fine element (FE) mesh; Figure 6. 3D model of the analyzed bending process: 1, bent beam with a fine element (FE) mesh; 2 2 and 3, surfaces of the clamping jaw; 4 and 5, longitudinal and transverse symmetry planes. and 3, surfaces of the clamping jaw; 4 and 5, longitudinal and transverse symmetry planes. Figures7 and8 show a comparison of the changing cross-sectional shape of the bent beam and Figures 7 and 8 show a comparison of the changing cross-sectional shape of the bent beam and the distribution and values of plastic deformations for Model A and Model B respectively. Taking into the distribution and values of plastic deformations for Model A and Model B respectively. Taking account technological deformations and residual stresses created when cold shaping a square pipe into account technological deformations and residual stresses created when cold shaping a square (model A), results had a much greater degree of deformation in the pipe’s cross-section than when pipe (model A), results had a much greater degree of deformation in the pipe’s cross-section than we did not take into account the impact of technological deformations and internal stresses created when we did not take into account the impact of technological deformations and internal stresses during the shaping (model B). Photographs of the pipe cross-sections before and after the bending created during the shaping (model B). Photographs of the pipe cross-sections before and after the process (Figure7d) confirmed the above mentioned statement. The shape and size of the deflection of bending process (Figure 7d) confirmed the above mentioned statement. The shape and size of the the cross-section walls observed in the experimental sample photograph were similar to the image of deflection of the cross-section walls observed in the experimental sample photograph were similar to the cross-section obtained via numerical modeling, which considered technological deformations and the image of the cross-section obtained via numerical modeling, which considered technological residual stresses created when cold shaping a square pipe (model A). deformations and residual stresses created when cold shaping a square pipe (model A). With the measurements results carried out during the experimental studies, we compared numerical calculations for the displacement values of selected points in the profile cross-section, whicha) occurred duringb) a curvature increase. Ifc) the horizontal walls deflectiond) degree internally and externally changed, good compliance of the calculation and experimental results were obtained (Figure9). However, this was only when we considered technological deformations and residual stresses created when cold shaping a square pipe (model A). Similarly, large differences between the numerical calculation results and experimental results were found when we analyzed the degree of the vertical wall bending (Figure 10). This was because it did not take into account the technological deformations and stresses created when cold shaping a square pipe (model B). During experimental bending tests, we observed that bending the pipe above the curvature κ > 6.0 caused a localization of the deformation on the internal horizontal wall and led to the pipe’s local collapse. The above observation was confirmed by the numerical calculation results carried out according to model A, which indicated an intensive increase in the deflection of the internal horizontal wall for large bending 0.00 0.35 0.14 0.07 0.28 0.70 0.63 0.49 0.42 0.21 curves. The location0.56 of the deformation in the pipe cross-section resulted in its buckling and caused a significant share of compressive stresses. It should be noted that deflecting the vertical walls in the profile cross-section slightly aff ected the moment of inertia value, which is what determined the bending moment value. Figure 7. The change in the cross-sectional shape of a bent beam (model A) and the distribution of total substitute plastic deformations for: (a) 1/R = 0 m−1, (b) 1/R = 4 m−1, (c) 1/R = 6 m−1, and (d) cross- section templets before (upper) and after bending the local break (lower).
Materials 2020, 13, x FOR PEER REVIEW 6 of 10
using the isotropic hardening model. The calculations used the Huber–Mises–Hencky plasticity condition, the associated Prandtl–Russ plastic flow law, and the implicit integration of differential equations over time with the Newton–Raphson method.
Figure 6. 3D model of the analyzed bending process: 1, bent beam with a fine element (FE) mesh; 2 and 3, surfaces of the clamping jaw; 4 and 5, longitudinal and transverse symmetry planes.
Figures 7 and 8 show a comparison of the changing cross-sectional shape of the bent beam and the distribution and values of plastic deformations for Model A and Model B respectively. Taking into account technological deformations and residual stresses created when cold shaping a square pipe (model A), results had a much greater degree of deformation in the pipe’s cross-section than when we did not take into account the impact of technological deformations and internal stresses created during the shaping (model B). Photographs of the pipe cross-sections before and after the bending process (Figure 7d) confirmed the above mentioned statement. The shape and size of the deflection of the cross-section walls observed in the experimental sample photograph were similar to Materialsthe 2020image, 13 ,of 5170 the cross-section obtained via numerical modeling, which considered technological7 of 10 deformations and residual stresses created when cold shaping a square pipe (model A). Materials 2020, 13, x FOR PEER REVIEW 7 of 10
a) b) c) d) 0.00 0.35 0.14 0.07 0.28 0.70 0.63 0.49 0.42 0.21 0.56
Figure 7. The change in the cross-sectional shape of a bent beam (model A) and the distribution of total Figure 8. Change in cross-sectional shape of 1the bent beam (model1 B) and 1distribution of total substitute plastic deformations for: (a) 1/R = 0 m− ,(b) 1/R = 4 m− ,(c) 1/R = 6 m− , and (d) cross-section Materialssubstitute 2020Figure, 13 ,7. x plastic FORThe changePEER deformations REVIEW in the cross-sectional for: (a) 1/R = 0 shapem−1, (b of) 1/R a be = nt4 mbeam−1, and (model (c) 1/R A) = 6and m− 1the. distribution of 7 of 10 templets before (upper) and after bending the local break (lower). total substitute plastic deformations for: (a) 1/R = 0 m−1, (b) 1/R = 4 m−1, (c) 1/R = 6 m−1, and (d) cross- Withsection the templets measurements before (upper) results and carriedafter bending out duri the localng the break experimental (lower). studies, we compared numerical calculations for the displacement values of selected points in the profile cross-section, which occurred during a curvature increase. If the horizontal walls deflection degree internally and externally changed, good compliance of the calculation and experimental results were obtained (Figure 9). However, this was only when we considered technological deformations and residual stresses created when cold shaping a square pipe (model A). Similarly, large differences between the numerical calculation results and experimental results were found when we analyzed the degree of the vertical wall bending (Figure 10). This was because it did not take into account the technological deformations and stresses created when cold shaping a square pipe (model B). During experimental bending tests, we observed that bending the pipe above the curvature κ > 6.0 caused a localization of the deformation on the internal horizontal wall and led to the pipe’s local collapse. The above observation was confirmed by the numerical calculation results carried out according to model A, which indicated an intensive increase in the deflection of the internal horizontal wall for large bending curves. The location of the deformation in the pipe cross-section resulted in its buckling and causedFigure a significant 8. Change in share cross-sectional of compressive shape ofstresses. the bent It beam should (model be noted B) and that distribution deflecting of the total vertical substitute walls Figure 8. Change in cross-sectional shape of the bent beam (model B) and distribution of total in the profile cross-section slightly affected1 the moment1 of inertia value, which1 is what determined plastic deformations for: (a) 1/R = 0 m− ,(b) 1/R =−1 4 m− , and (c)−1 1/R = 6 m− . −1 thesubstitute bending momentplastic deformations value. for: (a) 1/R = 0 m , (b) 1/R = 4 m , and (c) 1/R = 6 m .
With the measurements results carried out during the experimental studies, we compared numerical calculations for the displacement values of selected points in the profile cross-section, which occurred during a curvature increase. If the horizontal walls deflection degree internally and externally changed, good compliance of the calculation and experimental results were obtained (Figure 9). However, this was only when we considered technological deformations and residual stresses created when cold shaping a square pipe (model A). Similarly, large differences between the numerical calculation results and experimental results were found when we analyzed the degree of the vertical wall bending (Figure 10). This was because it did not take into account the technological deformations and stresses created when cold shaping a square pipe (model B). During experimental bending tests, we observed that bending the pipe above the curvature κ > 6.0 caused a localization of the deformation on the internal horizontal wall and led to the pipe’s local collapse. The above observation was confirmed by the numerical calculation results carried out according to model A, which indicated an intensive increase in the deflection of the internal horizontal wall for large bending curves. The location of the deformation in the pipe cross-section resulted in its buckling and caused a significant share of compressive stresses. It should be noted that deflecting the vertical walls in the profile cross-section slightly affected the moment of inertia value, which is what determined FigureFigure 9. 9.The The eff effectect of of bending bending curvature curvature when when deflectingdeflecting the horizontal walls walls of of the the 25 25 × 2525 × 2 square2 square the bending moment value. × × steelsteel pipe. pipe.
Figure 9. The effect of bending curvature when deflecting the horizontal walls of the 25 × 25 × 2 square steel pipe. Materials 2020, 13, 5170 8 of 10 Materials 2020, 13, x FOR PEER REVIEW 8 of 10
Materials 2020, 13, x FOR PEER REVIEW 8 of 10
FigureFigure 10. The 10. The effect effect of bendingof bending curvature curvature when when deflectingdeflecting the the vertical vertical walls walls of ofthe the 25 25× 25 ×25 2 square2 square × × steel pipe.steel pipe. Figure 10. The effect of bending curvature when deflecting the vertical walls of the 25 × 25 × 2 square As expected,steelAs expected, pipe. the the improved improved modelling modelling ofof thethe numerical degree degree of ofdistortion distortion for forthe thebent bent square square tube’stube’s cross-section cross-section had goodhad compliancegood compliance with thewith bending the bending curve, whichcurve, waswhich experimentally was experimentally determined As expected, the improved modelling of the numerical degree of distortion for the bent square via numericaldetermined calculations via numerical (Figure calculations 11). This (Figure occurred 11). This during occurred technological during technological deformations deformations and residual tube’sand residual cross-section stresses hadafter good making compliance a square tubewith frtheom bendinga circular curve, tube viawhich the coldwas formingexperimentally process stresses after making a square tube from a circular tube via the cold forming process (model A). determined(model A). Thevia numericalbending characteristics calculations (Figure that were 11). Thisdetermined occurred numerically during technological in accordance deformations with the The bending characteristics that were determined numerically in accordance with the assumptions for andassumptions residual stressesfor model after B makingdiffered a fromsquare the tube bend froming acharacteristics circular tube viadetermined the cold formingexperimentally, process model(modelespecially B diff A).ered in The the from bending small the bending characteristics curvature characteristics that area. were Only determined determined in the bending experimentally,numerically curvature in accordance range especially of κ with> in3.0 the mthe−1 small 1 bendingassumptionswere curvature the bending for area. modelmoment Only B values indiffered the numerically bending from the curvature calculated.bending range characteristics Over of theκ >wide3.0 rangedetermined m− were of bending the experimentally, bending curvatures, moment valuesespeciallythe numerically bending in momentthe calculated. small values bending Over did curvaturenot the change wide area. rangemuch, Only ofwhich bendingin the was bending partially curvatures, curvature a result the of bendingrange the pipe of momentκ material’s > 3.0 m values−1 did notwereslight change the strain bending much, hardening. moment which On was values the partially other numerically hand, a result even calculated. ofa slight the pipe Overstrain material’s the hardening wide range slight should of strain bending increase hardening. curvatures, the value On the othertheof hand, the bending bending even moment a moment. slight values strain At thedid hardening samenot change time, should themuch, decre increasewhichase thewas thecross-section’s partially value ofa result the moment bending of the ofpipe inertia moment. material’s value At the slight strain hardening. On the other hand, even a slight strain hardening should increase the value samewas time, a result the decrease of the walls the cross-section’sdeflecting, especially moment the horizontal of inertia valueones, which was a completely result of the eliminated walls deflecting, the of the bending moment. At the same time, the decrease the cross-section’s−1 moment of inertia value especiallydeformation the horizontal strengthening ones, effect. which For completely bending curvatures eliminated κ > the6.0 deformationm , the pipe collapsed strengthening locally, e ffect. was a result of the walls deflecting, especially the horizontal ones, which completely eliminated the which was accompaniedκ by a sharp1 drop in the bending moment value. For bendingdeformation curvatures strengthening> 6.0 effect. m− ,For the bending pipe collapsed curvatures locally, κ > 6.0 which m−1, wasthe pipe accompanied collapsed locally, by a sharp dropwhich in the was bending accompanied moment by value. a sharp drop in the bending moment value.
Figure 11. The effect of bending curvature on the bending moment of the 25 25 2 square steel pipe. × × Materials 2020, 13, 5170 9 of 10
4. Conclusions Our results made it possible to describe changes in the shape of a square tube’s cross-section as a function of bending curvature. Further, our research determined the impact of these changes on force characteristics, i.e., the value of the bending moment on bending curvature. As a result of cross-section 1 distortion for a curvature of κ > 2.0 m− in the initial bending phase, we observed a decrease in the 1 bending moment value. For a curvature of κ > 6.0 m− , the pipe collapsed locally. Satisfactory agreement of the experimental results and numerical calculations was obtained by considering the technological deformations and residual stresses created during the cold plastic shaping process of a round tube into a square cross-section pipe.
Author Contributions: Conceptualization, S.K. and F.S.; methodology, S.K. and F.S.; experimental investigation, F.S.; numerical calculation, S.K.; analytical and formal analysis, S.K. and F.S.; validation, S.K. and F.S.; writing—original draft preparation, F.S.; review and editing, F.S. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest.
References
1. Leu, D.K.; Wu, J.Y. Finite element simulation of the squaring of circular tube. Int. J. Adv. Manuf. Technol. 2005, 25, 691–699. [CrossRef] 2. Thomsen, E.G.; Yang, C.T.; Kobayashi, S. Plastic Deformation in Metal. Processing; Macmillan: New York, NY, USA, 1965. 3. Bayoumi, L.S. Cold drawing of regular polygonal tubular sections from round tubes. Int. J. Mech. Sci. 2001, 43, 241–253. [CrossRef] 4. Bayoumi, L.S.; Attia, A.S. Determination of the forming tool load in plastic shaping of a round tube into a square tubular section. J. Mater. Proc. Technol. 2009, 209, 1835–1842. [CrossRef] 5. Hosseinzadeh, M.; Mouziraji, M.G. An analysis of tube drawing process used to produce squared sections from round tubes through FE simulation and response surface methodology. Int. J. Adv. Manuf. Technol. 2016, 87, 2179–2194. [CrossRef] 6. Oczkowicz, T.; Cywinski, M. Hardness characteristics of square tube drawn from a circular tube. Bänder Bleche Rohre 1989, 14, 17–20. (In German) 7. Kecman, D. Bending collapse of rectangular and square section tubes. Int. J. Mech. Sci. 1983, 25, 623–636. [CrossRef] 8. Stachowicz, F.; Kut, S. Bending moment and cross-section deformation of a box profile. Adv. Sci. Technol. Res. J. 2020, 14, 85–93. [CrossRef] 9. Chen, D.H.; Masuda, K. Rectangular hollow section in bending: Part I—Cross-sectional flattening deformation. Thin-Walled Struct. 2016, 106, 495–507. [CrossRef] 10. Paulsen, F.; Welo, T.; Sovik, O.P. A design method for rectangular hollow sections in bending. J. Mater. Proc. Technol. 2001, 113, 699–704. [CrossRef] 11. Miller, J.E.; Kyriakides, S.; Bastard, A.H. On bend-stretch forming of aluminum extruded tubes—I Experiments. Int. J. Mech. Sci. 2001, 43, 1283–1317. [CrossRef] 12. Corona, E.; Vaze, S. Buckling of elastic-plastic square tubes under bending. Int. J. Mech. Sci. 1996, 38, 753–775. [CrossRef] 13. Kim, T.H.; Reid, S.R. Bending collapse of thin-walled rectangular section columns. Comput. Struct. 2001, 79, 1897–1911. [CrossRef] 14. Paulsen, F.; Welo, T. A design method for prediction of dimensions of rectangular hollow sections formed in stretch bending. J. Mater. Proc. Technol. 2002, 128, 48–66. [CrossRef] 15. Petrone, F.; Monti, G. Unified code-compliant equations for bending and ductility capacity of full and hollow rectangular RC sections. Eng. Struct. 2019, 141, 805–815. [CrossRef] 16. Chiew, S.P.; Jin, Y.F.; Lee, C.K. Residual stress distribution of roller bending of steel rectangular structural hollow sections. J. Constr. Steel Res. 2016, 119, 85–97. [CrossRef] 17. Shim, D.S.; Kim, K.P.; Lee, K.Y. Double-stage forming using critical pre-bending radius in roll bending of pipe with rectangular cross-section. J. Mater. Proc. Technol. 2016, 236, 189–203. [CrossRef] Materials 2020, 13, 5170 10 of 10
18. Lan, X.; Chen, J.; Chan, T.K.; Young, B. The continuous strength method for the design of high strength steel tubular sections in bending. J. Constr. Steel Res. 2019, 160, 499–509. [CrossRef] 19. Zhu, H.; Stelson, K.A. Modeling and closed-loop control of stretch bending of aluminum rectangular tubes. J. Manuf. Sci. Eng. 2003, 125, 113–119. [CrossRef] 20. Litwin, P.; Stachowicz, F. Artificial intelligence identification of box profile bending parameters. J. Artif. Intell. Study 2004, 1, 99–106. 21. Rojek, J. Modelling and Computer Simulation of Complex Problems of Nonlinear Mechanics by Methods of Finite and Discrete Elements; IPPT PAN: Warszawa, Poland, 2007. (In Polish)
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.
© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).