Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

FEA Simulation of the Biomechanical Structure Overload in the University Campus Planting

FEA Simulation of the Biomechanical Structure Overload in the University Campus Planting Hindawi Applied Bionics and Biomechanics Volume 2020, Article ID 8845385, 13 pages https://doi.org/10.1155/2020/8845385 Research Article FEA Simulation of the Biomechanical Structure Overload in the University Campus Planting 1 1 1 Stanislau Dounar , Alexandre Iakimovitch , Katsiaryna Mishchanka , 2 2 Andrzej Jakubowski , and Leszek Chybowski Belarusian National Technical University, Nezalezhnosti 65, 220027 Minsk, Belarus Maritime University of Szczecin, Waly Chrobrego 1-2, 70-500 Szczecin, Poland Correspondence should be addressed to Leszek Chybowski; l.chybowski@am.szczecin.pl Received 19 May 2020; Revised 30 October 2020; Accepted 6 November 2020; Published 23 November 2020 Academic Editor: Guowu Wei Copyright © 2020 Stanislau Dounar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Research of breakage of the chestnut tree branch on the planting of university campus is provided. Collapse is caused by a severe accidental wind gust. Due to collapse in the student environment, the investigation has additional methodical value for the teaching of FEA simulation. The model includes roots, trunk, branch, and conditional crown, where the trunk-branch junction is steady enough. The load-bearing system of tree is taken as an example of an effective bionic design. The branch has grown with the implementation of the idea of “equal-strength console”—the change of sections along the branch provides constant stress level and near uniform dispensation of their without stress concentrators. Static simulation of the tree loading is provided both in the linear formulation and in the geometrically nonlinear one. It is proved that in the trunk-branch junction area the stresses are twice lower than the branch itself, and it is not the place for fracture. For the given wind pressure, the work stress in the branch has exceeded twice the allowable level under bending with some torsion. In such construction (of the tree), the breakage could happen even in the perfect branch condition due to her severe overloading. 1. Introduction the weather station (located 5 km from the campus), the wind speed was only 12 m/s. Weather is regarded stormy if the The work relates to the sphere of the simulation (CAE) by wind speed exceeds 15 m/s. the finite element analysis (FEA) [1, 2]. An investigation is University authorities decided to investigate the incident close to the biomechanics [3, 4] because the stress-strain from an engineering point of view. Two groups of specialists state of the tree branch and trunk is discussed [5]. The work were formed: experts in the field of computational flow focuses both on engineering situation of tree load-bearing dynamics (CFD-group) and analysts of load-bearing systems system [6] and on the methodical use of the results to teach (stress analysis group (SA-group)) [10, 11]. students the possibilities of bionic design [7] and creative The CFD-group has provided computer simulation of problem solving [8]. airflows nearby the tree (0.3 km vicinity [12]) and revealed On the border of the university campus of BNTU, there is strong local wind amplification. It turned out that the tree a group of trees (Figure 1). This is part of a two-row planting, is placed in the focus of the double-wedged air manifold. namely, chestnuts (Aesculus hippocastanum). The object of The slot between buildings is continued by the gap in the modeling is tree 1, whose huge branch collapsed on a windy double-row planting just before the tree. In sum, north-east day, causing material damage [9]. Breakage took place in the wind is speeding up above university stadium and creates healthy, quality wood grains (fibers) in the area of the trunk- in the manifold stormy flow with the velocity of 24–25 m/s. branch junction. The tree remains standing and continues to Post factum observations of such local wind flow point out grow (Figure 2). of its steady character—wind gusts last about 5–7 s without The branch had a developed crown opposing the wind, significant oscillations. Therefore, trees are bent in the near static mode. but there was no storm in the summer city. According to 2 Applied Bionics and Biomechanics in the rectangular version in Figure 3(c) (RectCrown; blown surface—30 m ). Both types of crowns reach a height of 14 m. The main technic of 3D-building was surface pulling on by sections. There were 5 sections for the trunk. Section dimension changes from Ø580 mm to Ø390 mm going from the ground to the trimming level. The branch is pulled on by four basic diameters (marks 2–5 in Figure 4(a)) from Ø380 to Ø240 mm. The height difference between point 2 and point 5 is equal to 4 m. Branch bend in the 3–4 span has a radius of 2.4 m. The branch moves away from the trunk at the 60 angle (mark A60 in Figure 4(b)). Trunk-branch junction is smoothed by fillet Figure 1: The simulated tree (1) long before breakage of branch with 70 mm radius. (2014). Model DoubleTree (Figure 4(c)) with two main branches, two crowns, and common trunk with roots was built for additional proving of simulation results. 2.2. Wood Material Models. For results of stability proving, three material models were accepted for a parallel manner using during simulation. It should bring more confidence in results and limit model uncertainties of all issues. In the first model (ChestISO), wood is considered an isotropic, fully elastic material obeying Hooke’s law. According to the construction codes and sources [15], for the chestnut wood, it was appointed: the elastic modulus (E = 8000 MPa), Poisson’s ratio (μ =0:42), density (ρ = 600 kg/m ), Figure 2: The trunk-branch junction just after branch sawing and allowable stress (σ =16 MPa) (it is taken the same both (2018). for tension and compression). As it is not reliable initial data about mechanical characteristics of the crown, they are appointed a little arbitrary. Crown rigidity is considered very The SA-group has simulated the tree as a load-bearing low, i.e., elastic modulus (E =2 MPa). Crown density is the system standing under wind pressure [13]. The pressure variable parameter to simulate different mass of leaves on value was extracted from CFD-group work results—normal the branches (see below). wind level is equal (p = 380 Pa). A wind pressure of 600 Pa norm Orthotropic representation of the chestnut wood is pro- was taken into account too, as possible limit level for vided in the parallel manner with the isotropic one. Model hurricane-like situation. With the aim to disclose stress- ChestTKP is based on the local civil engineering code [16]. strain state of the tree and to reveal issues of the breakage, Elasticity modulus along grain is taken 8000 MPa, transversal the FEA was accomplished by the SA-group. to grain (400 MPa) (no difference between radial and tangen- Simulation has shown interesting result in two directions: tial direction) and shear modulus (all three) (400 MPa), and the engineering of biomechanical load-bearing system and Poisson’s ratios should be taken as 0.5, 0.02, and 0.02 (XY, the methodical improvement of teaching students the FEA. YZ, and XZ instances). Other orthotropic model ChestWH is more detailed and scientific [16, 17]. Elasticity modulus along grain is equal 2. Geometry Model of the Tree Load- 9400 MPa, transversal to grain (358 MPa and 678 MPa) Bearing System (radial and tangential direction), shear modulus (all three) (544 MPa, 396 MPa, and 134 MPa). Poisson’s ratios should 2.1. Geometry Representation. The tree with a broken branch was both laser scanned and sketched by gardeners just after be taken as 0.495, 0.052, and 0.035 (XY, YZ, and XZ the breakage. The SA-group members have provided 3D- instances). Thus, using of three different models proves modeling of this tree to bring variability of shapes and natural scattering of wood properties while simulating. reduce subjectivity of simulation (Figure 3). The scope of simulation embraces tree’s trunk 1, huge branch 2, and 2.3. FEA Mesh Variations. Several FEA mesh models of dif- crown. Remained branches 3 and 4 are not included in the ferent structure were created for tree simulation. Looking simulation scope. They are shown as conditionally trimmed. ahead, note all of them have shown good correspondence Branches 5 and 6 hold the crown. in results and minimal level of computing artefacts. According to the idea of tree crown variability [14], just Objects named solids and parts are used in the meshing two crowns were imaginarily matched to a given tree. A freely procedure. Solid brings monolithic mesh. The part consists growing crown in the curled version is shown in Figure 3(a) of several solids touching each other. Mesher joins their local (CurlCrown; blown surface—51 m ). The crown grown in meshes by common nodes. So part mesh is one-piece too. the constrained conditions (neighboring trees) has been built Other variants to simulate interaction between solids or parts 180 R2400 Applied Bionics and Biomechanics 3 CurlCrown 14 m RectCrown 51 m 14 m 30 m 2 4 Z X (a) (b) (c) Figure 3: Tree geometry: (a) model with curled crown (CurlCrown) on the downwind side; (b) the internal dimensions of the branch; (c) elongated crown (RectCrown) from upwind. R 70 A 60 (a) (b) (c) Figure 4: Models of the branch: (a) the trunk-branch junction; (b) the tree with two branches; (c) two crowns and stylized roots (DoubleTree). contact pair creation (special surface elements on the inter- (Figure 5(c)) permanently transforms into the branch’s mesh face). For that work, contact pairs are always in the bonded 2. For all wood, massive finite elements are joined together by common nodes. The tree’s crown was represented by a sepa- state. They work as perfect thin rigid glue layers. One of the finite element meshes is shown in Figure 5 rated mesh of volume finite elements. Crown and tree meshes (name it R-mesh (rare element density)). Finite elements have were conjugated by contact pair. mostly tetrahedral shape. It relates to trunk 1 (Figure 5(c)) and Figure 6 depicts alternative mesh (D-mesh (finite element to junction 3 between trunk 1 and branch 2. The branch itself packed with higher density)). The trunk and main branch were split to the sets of small solids. It was done by planes is meshed by hexahedral elements. It brings better accuracy in the critical part of the model. At the same time, higher normal to trunk/branch axes of growing. Solids 1 and 3 smoothness of the stress fields is achieved. The trunk and belong to the part “Branch.” Such solids are joined together branch create single solid. Accordingly, the trunk’s mesh 1 by common nodes at the faces like 2 and 3. 90° 4000 4 Applied Bionics and Biomechanics Crown 1225 kg Crown 1225 kg Stem and branches 1231 kg Wood 1231 kg X 1 (a) (b) (c) Figure 5: Meshes for: (a) leeward side of crown; (b) windward side of crown; (c) trunk 1 with junction 3 to branch 2. 4 2 34,529 BC BC (a) (b) Figure 6: Dense mesh for split solids (D-mesh): (a) partial view at solids; (b) picture of equivalent stress (σ ) (MPa) for LiBC condition set (stated below). Solid 4 and underlying ones create monolithic part standard FEA approach, especially that bending domination is “Trunk.” Parts “Trunk” and “Branch” are glued (BC) by expected. Line 2–2 (Figure 6(b)) goes between tension and compression zones. Equivalent stress maximum (34.529 MPa) bonded contact pair. Stress state for this model is shown in Figure 6(b). Stress field near marker BC is smooth and con- here (D-mesh) relates well to analog simulation by R-mesh. tinuous. Also, interfaces between solids are not visible any- Thus,bothmeshmodelsare appropriateenough. where. It means precision and fidelity of the D-mesh model. Mesh in Figure 6 is denser compared with one in Figure 5. 2.4. Boundary Conditions. The simulation was provided in the static form. That assumption is based on the CFD-group Outer surfaces of the trunk and branch are covered in Figure 6 by set of thin finite element layers (3). It brings accuracy for conclusion as about smooth, long-time patterns of wind gusts representation of surface stress effects. Branch core is in the local natural manifold acting on the simulated tree. modelled relatively coarse finite elements (2). That is a Oscillations and resonant effects are out of modeling scope. Applied Bionics and Biomechanics 5 FX = 0 FY = 18434 N FZ = –11528 N (a) (b) (c) Figure 7: Tree fastened to ground A and loaded by B (wind pressure) and C (gravity force): (a) leeward windward; (b, c) windward. Deformation shapes and reaction force vectors are for LiBC and HeBC condition sets at (b, c), respectively: ×1. Crown is a conditional object of the plate’s shape. Simu- Variations during tree simulation pointed out two repre- sentative sets of boundary conditions. They are called “Light” lation has focused on the lower branch (1st order branch— cite of breakage). Branches of the 2nd order are placed above (LiBC) and “Heavy” (HeBC) and are marked by color in and are built approximately. Branches of the 3rd order are Table 1. “Fork” space creates between them for other variants not regarded. of the model parameters. LiBC set refers to the simple, isotro- Interaction between crowns is not simulated. Leaves are pic, linear model of the tree under storm-like wind pressure. considered inner components of the crown. The mass of all HeBC set gives possibility to estimate ultimate deflection of leaves on the main branch (crown mass) is a really uncertain the heavy orthotropic tree in the near hurricane situation. parameter. It was taken at three levels—750, 1225, and 1550 kg—marked below as L-leaves, M-leaves, and H- 2.5. Nonlinearity and Orthotropy Checks. Meshes R-mesh leaves. Crown mass governs the gravity force. Simulation and D-mesh were used for simulation as three wood material pointed out that gravity force starts to play a role only at models. Loading was provided up to 600 Pa wind pressure. It hurricane-like wind pressure (600 Pa), where strong sloping was revealed (Figure 8(a)) that large deformation simulation of the crown occurs (Figure 7(c)). (NonL) brings higher levels of stresses and displacements in the tree compared to geometrically linear model (Lin). Non- The ground is simulated as a rigid base (mark A in Figure 7(a)). Wind pressure (mark B) is uniformly distrib- linear solution points out rise of branch top displacement on uted upon the windward side of the crown. Gravity force 33%. Maximal equivalent stress rises on 35%. It relates to the (mark C) is dispensed through all materials according to tree with the heavy crown (H-leaves). In the case of light their densities. crown (L-leaves), the nonlinear curve passes lower. Here, the difference between nonlinear and linear results does not Parallel modelling by different models and various condi- tions is the feature of that work. Intentional variation of exceed 18%. It is obvious that deflection of heavy crone by model factors was provided by different authors to control wind stimulates growing of the gravity force moment. So, uncertainties. The aim of parallel simulations was to ensure crown hanging-off additionally grows. Nonlinear simulation result stability and to estimate the sensibility of tree stress- is the way to disclose that interaction. The comparison of curves for the isotropic wood model strain state to the chatter of the entering factors. Table 1 depicts the scope of varied factors. Near full (“ISO”) and orthotropic models (“TKP” and “WH”) is given crossing of all steps was achieved. Geometrical linearity/non- in Figure 8(b). Orthotropic lines are placed near each other linearity of the tree model was investigated. That is a single with the difference below 7%, whereas the isotropic model kind of nonlinearity into the FEA model. Friction is not turns up much more rigid. Displacements for TKP-tree are 58% stronger than those for the ISO-tree (both models included, and wood is taken as fully elastic. If the model was simulated as linear (Lin), only one step possess the same elasticity modulus at 8000 MPa). of loading is provided. The model undergoes stepped loading However, stress levels for all three materials are placed in (30 steps) when large deformations are counted in the stiff- vicinity to each other (with the range of only 13%—despite of ness matrix of the tree, so geometrical nonlinearity (NonLin) displacements). It relates to the nonlinear simulation of heavy crown trees. became observable. 6 Applied Bionics and Biomechanics Table 1: Steps of model factors to vary. Leading sets of boundary conditions (BC) Model variation factors LiBC HeBC Trunk-branch material ChestISO (isometric) ChestTKP (orthotr.) ChestWH (orthotr.) Mesh R-mesh D-mesh Crown shape CurlCrown RectCrown DoubleCrown Leaves mass L-leaves (750 kg) M-leaves (1225 kg) H-leaves (1550 kg) Wind pressure 380 Pa 600 Pa Geometrical nonlinearity Lin (1 step) NonLin (30 steps) In the “light-crown” case, wood material variation causes green arrows, which means that the principal middle stress adifference of 15% for displacements and 6% for stresses (σ ) is near zero in the whole tree. Therefore, exactly, SSTe (linear solutions). is the place of one-axis tension, and at the same time, SSCo As a result, there are no principal differences between is the place of one-axis compression. Both σ and σ vectors 1 3 linear and nonlinear solutions concerning the shape of are oriented along the branch. This is a clear picture of bend- deflection and stress state features. ing. Some vector’s winding around the branch axis points out the presence of the small torsion moment (in a moderate proportion to the bending one). 3. Results and Discussion The conclusion about bending dominance in the stress- 3.1. Depiction of the Tree Stress-Strain State for Isotropic strain state of the branch is proved by distributions of the Model. Figure 9(a) shows natural scale deformational principal stresses (Figure 12). The fields of tension on wind- displacements of the tree. The crown significantly deflects ward side are shown in Figure 12(a) (almost completely coin- on its top (above 2 m). The branch is much more rigid, and cident with Figure 10(a)), where principal maximum stress the displacement (below the crown) is less than 100 mm. (σ ) creates SSTe (marks “30.729”–“34.069”–“30.177”). It is The distribution of the equivalent stress (σ ) for the the single place of high tension, but longitudinal gradients DoubleTree model is smooth enough (Figure 9(b)). There is are very low here, because tension stress is near the same in not just local, sharp stress concentration. The trunk is it. Therefore, SSTe should be taken into account as ridge- stressed moderately (14.5 MPa). Some stress increasing is like increase, not just a point of stress concentration, and visible at the trunk-root junction (29.6 MPa). The main wood breakage could start spontaneously in any place of SSTe. attention should be paid to the strips “34.448 MPa” and The picture of the principal minimum stress (σ ) shows “34.077 MPa.” The first marker precisely relates to the place smooth focusing of compression with small gradients along of the branch breakage. the branch from leeward (marks “-31.058”–“-34.103”–“30.5” Figure 10 depicts the concentration of equivalent stress in Figure 12(b)) and points out the SSCo. (σ ) (von Mises stress) in the basic tree model, which The SSTe and SSCo features received elongated shape. discloses both one-axis tension regions (indicates principal Large length is causes by branch section changing. The branch maximal stress (σ )) and one-axis compression regions as the kind of beam very close to the ideal “equal-strength con- (principal minimal stress (σ )). The tree surface has no local sole” is rising in diameter from leaves to the trunk. The bending stress concentrators, discontinuities, and high-gradient moment is enhancing in this direction at the same time. Branch regions. The bottom part of the branch is the only placed thickening (inertia moment enhancing) effectively counteracts with relatively high stresses. Here, Strip of Strong Tension to growing bending moment. The quick increase of branch (SSTe) is shown (between marks 1–2 in Figure 10(a)), where diameter in the trunk vicinity is relating the reinforcement of equivalent stress reaches level σ =34:181 MPa. This is the the trunk-branch junction. It results in stresses stabilizing and most tensioned part of the tree on the windward side far away is an example of self-organized wood growth to limit and level from the trunk-branch junction—“tensioned fiber”—by a the stresses. This is the bionic stresses stabilization (BiSS) or classic theory of bending. Equivalent stress (σ ) near the “ironing” of stress concentrators. trunk-branch junction is equal only to 13.614 MPa. The trunk is a slightly stressed object with σ =6:509 MPa. 3.2. Stress Distribution for Orthotropic Wood Model. Wood Strip of Strong Compression (SSCo) lays between 3 grain (fiber) orientation is always known for a living tree and 4 in Figure 10(b). Equivalent stress (σ ) here reaches only approximately. Thus, three simple, different variants 34.08 MPa level. That is so-called “compressed fiber” by of orientation were simulated (Figure 13) on the D-mesh classic theory of bending. base. It is a geometrical model assembled from split solids. Figure 11 demonstrates the direction of the principal Wood grain vector (WGV) was oriented inside every solid stress vectors. On the leeward side, one could see dominance normally to its bottom face. It caused (Figure 13(a)) an of the principal minimum stress (σ ) (blue arrows in uneven shape of stress isolines. Transitions between solids Figure 11(a)) as manifestation of SSCo feature. On the wind- are clearly visible. ward side, we can see the principal maximum stress (σ ) (red Nevertheless, stress picture, described above for the iso- arrows in Figure 11(b)) as SSTe feature. There are not visible tropic model (Figure 10(b)), is preserved. One can see stress Applied Bionics and Biomechanics 7 0 100 200 300 400 500 600 Wind pressure (Pa) B Lin H S Lin H S NonL H B NonL H S NonL L B NonL L (a) 0 100 300 400 500 600 Wind pressure (Pa) B ISO C WH B TKP S ISO B WH S TKP C ISO S WH C TKP (b) Figure 8: Curves of “along-wind” crown top displacement (mm) (mark “C” in the curve name), branch top displacement (mark “B”), and −1 maximal equivalent stress on the branch surface (10 MPa, mark “S”): (a) linear (“Lin”) and nonlinear (“NonL”) loading for the heavy (“H”) crown (means H-leaves) and for the light (“L”) one (means L-leaves) in the case of ChestWH material; (b) nonlinear tree loading for materials ChestISO, ChestTKP, and ChestWH (marks “ISO,”“TKP,” and “WH”, respectively); H-leaves: HeBC. Displacement (mm) Stress (MPa/10) 8 Applied Bionics and Biomechanics Type: total deformation 2215.5 Unit: mm Time: 1 26.03.2019 19:21 1054.3 34.977 2249.8 max 602.78 34.448 293.65 0 min 94.522 14.515 29.682 4.4073 (a) (b) wind Figure 9: Total displacement (mm) of the tree with the crown of RectCrown type under wind pressure (p = 380 Pa) (a) and the picture of norm equivalent stress (σ ) (MPa) for the DoubleTree model (b). A: Static structural A: Static structural Equivalent stress Figure Type: equivalent (von-Mises) stress Type: equivalent (von-Mises) Unit: MPa Unit: MPa Time: 1 Time: 1 26.03.2019 18:02 26.03.2019 18:08 34.595 max 34.181 34.595 max 30 30 34.089 26 26 22 22 18 18 14 14 10 10 13.614 6 6 14.039 9.9204e-5 min 9.9204e-5 min 6.8025 6.5091 (a) (b) Figure 10: Distribution of the equivalent stress (σ ) (MPa) through the surfaces of the branch and trunk on the windward (a) and leeward (b) wind sides. Pressure ðp Þ = 380 Pa; RectCrown, ×1. norm concentrator (SSCo), marked as A (38.24 MPa). Additionally, of WGV) and in Figure 13(c) (WGV orients along the branch two local extremums (B: 25.27 MPa) and (C: 24.58 MPa) are and smoothly extends that orientation into the trunk). founded at ends of branch-trunk junction. It may be stated that the orthotropic model of the tree is The system consisting of stress spots A, B, and C is more tangible to local geometry unevenness than isotropic revealed again in Figure 13(b) (vertical-dominant orientation one. Spots B and C probably are tied with some kind of that Applied Bionics and Biomechanics 9 A: Static structural Vector principal stress Type: vector principal stress Unit: MPa Time: 1 26.03.2019 20:11 Maximum principal Middle principal Minimum principal (a) (b) wind Figure 11: Vectors of principal stresses on the leeward (a) and windward (b) sides. Pressure ðp Þ = 380; RectCrown, ×1. norm ess 30.729 –30.5 34.069 –34.103 30.177 –31.058 13.503 6.5046 –6.7152 (a) (b) Figure 12: Distributions of the principal maximum stress (σ ) (a) from windward and minimum stress (σ ) (b) from leeward. Pressure 1 3 wind ðp Þ = 380 Pa; RectCrown, ×1. norm wind effect. Orientation vector variations are not crucial for the branch breakage under wind pressure (p = 380 Pa) is norm stress state of a tree branch. Main stress spots and stress levels highly likely possible. Nonlinear geometry effects amplify remain the same for both isotropic and orthotropic wood deformation and overloading of the branch through displa- material representations. cing of the crown’s mass center to leeward. In its turn, the gravity force starts to create a bending moment relative to 3.3. Nonlinear Estimation of the Branch Overloading. The the trunk’s rest (eccentrically compression) and increase stress-strain state pictures, shown above, point out that even more deviation of the branch from the vertical axis. 10 Applied Bionics and Biomechanics A - 38.24 34.291 37.148 B - 25.27 13.854 24.555 C - 24.58 19.148 12.091 (a) (b) (c) Figure 13: Distributions of the equivalent stress (σ ) for different wood orthotropy models: “normal-to-split” wood grain (a), vertical- wind dominant wood grain (b), and “along main branch” wood grain (c). Pressure ðp Þ = 380 Pa, leeward; (a, b) ChestWH and (c) norm ChestTKP; Lin; ×1. A: Static structural A: Static structural Equivalent stress 4.625 m Equivalent stress Type: equivalent (von-Mises) stress Type: equivalent (von-Mises) stress Unit: MPa Unit: MPa Time: 380 Time: 1 24.03.2019 15:33 24.03.2019 11:10 69.634 max 53.048 54.897 max 54.202 7.873e-5 min 8.8375e-5 min 67.134 12.321 12.479 (a) (b) Figure 14: Equivalent stress distribution (σ ) (MPa) for the linear solution (a) and for the geometrically nonlinear one (b; stepped loading). wind Peak wind pressure ðp Þ = 600 Pa; ×1. peak The comparison of the linear and nonlinear solutions is 67.1 MPa (Figure 14(b)). For the trunk part of the tree, the given in Figure 14. Figure 14(a) shows the picture of equiva- nonlinear effects are not so strong. lent stress (σ ), calculated for fully linear assumptions and Thus, the pressure of a stormy wind overloads the tree one-step loading. Figure 14(b) gives the distribution of equiv- branch up to fracture. It happens above the allowable stress alent stress, when the large deformation effects are accounted level for wood. Therefore, there is no need to look for a con- and the stepped loading solution is achieved. In the second centrator or damaged place along the branch to explain the case, the crown’s top displacement has risen about twice. event of destruction [18]—the branch should fall under the The stresses along SSTe and SSCo have grown approximately influence of strong bending and torsion moments. Our task in a quarter. Equivalent stress on the windward side of the was to point out the fact of severe overloading in healthy wood material possibility, but details of the cracking model branch (SSTe) is increasing from 54.2 MPa (Figure 14(a)) to Applied Bionics and Biomechanics 11 A: Static structural Vector principal stress A: Static structural Type: vector principal stress Static structural Unit: MPa Time: 1. s Time: 1 31.03.2019 12:59 31.03.2019 13:05 A Fixed support Maximum principal Fixed support 2 Middle principal Fixed support 3 Minimum principal D Pressure: 3.8e-004 MPa Pressure 2: 3.8e-004 MPa A X (a) (b) A: Static structural Maximum principal stress Type: maximum principal stress A: Static structural Unit: MPa Equivalent stress Time: 1 Type: equivalent (von-Mises) stress 31.03.2019 12:53 Unit: MPa 34.977 35.675 max Time: 1 31.03.2019 13:00 36.154 max 34.448 8 16 4 12 –3.7038 min 3.8512e-9 min 34.409 36.037 14.515 29.682 14.574 X 22.146 (c) (d) wind Figure 15: Simulation of the tree with two big branches (DoubleTree model and LiBC) loading by wind pressure ðp Þ = 380 Pa: (a) norm double fastening root; (b) vectors of principal stresses; (c) principal maximum stress (σ ) (MPa); (d) equivalent stress distribution (σ ) 1 e (MPa); ×1. may be the topic for the further investigation [19–21]. In the are placed far enough from the trunk. The trunk itself is future research, the uncertainty analysis is planned to be stressed stronger (σ =14:57 MPa in Figure 15(d)) due to big- done [22, 23]. ger blown surface of both crowns. It should pay attention to the underground stress concentrator (σ =29:68 MPa) in 3.4. Variations for Sensitivity Checks: Stability of “Ironed” Figure 15(c). Stress Concentrators during Wind Rotation. The DoubleTree Wind direction influence on the branch stress-strain state model (Figure 15) approves earlier conclusions. Both branches is estimated in Figure 16. The tree crone is built as a kind of have ribbon-like tensed and compressed fields. Stress peaks sail in that work. Four trees with identic, parallel crones were 12 Applied Bionics and Biomechanics 34.48 34.21 33.42 6.30 34.13 ° ° Figure 16: Stability of peak stress (σ ) (MPa) on the branch regardless of the wind direction (from left to right, trunk rotates at angles 0 ,30 , ° ° 60 , and 90 relatively to crown). LiBC, ×1. included in the model. Wind assumes acting normally to the flat crones. Every trunk-branch system was rotated at its own angle around the vertical axis relative to the crone. The angle changes at 30 from tree to tree. So, left tree orientation (Figure 16) is similar to the one from Figure 10. Right tree (Figure 16) has accumulated an angle of rotation equal to 90 . Its loading represents blowing off from the perpendicu- lar direction. Four markers (34.13–34.48 MPa) show high indepen- dence of tension stress strips (SSTe) from the wind direction. Compression stresses (SSCo) from the other side of the branch are at the constant level too. Bending (paired tension – compression system) is the dominating feature of the branch stress-strain state. Some moderate torsion is present for every branch in Figure 16. Thus, even large changes in Figure 17: Obliquely growing campus tree in the need for the wind direction remain, and the branch stress state is the assessment and FEA simulation. same. The pair of ironed stress concentrators on the healthy branch should be taken as steady BiSS effect. Let us pay attention to a whole lot of similar trees, planted SSCo is revealed. Both of the strips form a picture of bending in university campuses. Some of them need serious assess- of the console beam. ment and possibly help (Figure 17). The branch is a beam similar to the ideal “equal-strength Each problematical tree is the object for FEA analysis. console.” The main stresses (σ and σ ) (values near the 1 3 The simulation could predict eventual collapse. In addition, constant) appeared on the SSTe and SSCo along the branch. simultaneously, a lively and directly learning process the On those areas, it is not any stress concentrator, but the uni- mechanical students may be provided. The spheres of 3D- form stress state only. It is provided by spontaneous BiSS scanning, recovery of geometry, dynamics of fluid, and 3D- and effective self-reinforcement through the self-organized printing may be involved. wood growth. The trunk-branch junction is steady and developed. In the junction area, the stresses are twice lower than the branch itself. 4. Conclusions Thus, trunk-branch junction is formed as a region of the signif- The investigated branch during a storm undergoes mainly icant reinforcement, and it is not the place for fracture. the bending and some torsion. Gravity compression does High stresses rising in a smooth and uniform manner only not take a significant part in the stress state. along SSTe and SSCo are reached at 30-34 MPa at the moder- The SSTe is formed on the windward side in the bottom ate crown (RectCrown model). But it significantly exceeds the third of the branch. On this level, on the leeward side, the allowable stress for chestnut wood (16 MPa). When the crown Applied Bionics and Biomechanics 13 was developed (CurlCrown model), a tree deformation data,” Agricultural and Forest Meteorology, vol. 265, pp. 137– 144, 2019. becomes nonlinear, and the stress at all rises up to 67 MPa due to partially eccentric action of the gravity force. [6] K. R. James, G. A. Dahle, J. Grabosky, B. Kane, and A. Detter, “Tree biomechanics literature review: dynamics,” Arboricul- In such tree construction for the given wind pressure, the ture & Urban Forestry, vol. 40, pp. 1–15, 2014. breakage of branch could happen even in the perfect branch [7] A. Samek, Bionika. Wiedza Przyrodnicza dla Inżynierów, condition and without the stress concentrator due to severe Wydawnictwa AGH, Kraków, 2010. overloading, because the predicted stress exceeds twice allow- [8] D. Chybowska, L. Chybowski, B. Wiśnicki, V. Souchkov, and able stress for the chestnut tree. S. Krile, “Analysis of the opportunities to implement the Inside almost every university campus, we can find BIZ-TRIZ mechanism,” Engineering Management in Produc- appropriate plants and trees as the investigation object. tion and Services, vol. 11, no. 2, pp. 19–30, 2019. Because the campus is a part of the student’s environment, [9] M. J. Mortimer and B. Kane, “Hazard tree liability in the United then modeling of the tree attracts interest of students. The States: uncertain risks for owners and professionals,” Urban load-bearing system of the tree serves as a complex and at Forestry & Urban Greening,vol. 2,no.3, pp.159–165, 2004. the same time understandable example to study both FEA [10] S. Downar, A. Jakimowicz, C. Z. Jakubowski, and simulation and bionic principles of the design. A. Jakubowski, “Conception of simultaneous teaching the stu- The tree branch became a good illustration of the “equal dents of direction “machine design” to three-dimensional strength console” idea. We can see the rational changing of modeling and virtual testing by FEM-analysis,” General and branch sections—stresses are leveled along the main part of Professional Education, vol. 1, pp. 26–32, 2016. the branch. It makes students see the bionic design sense. [11] S. Downar, A. Jakimowicz, and A. Jakubowski, “Methodology Tree simulation teaches students to create models of of mechanical student quick involvement into CAD- and load-bearing systems without stress concentrators. Mechani- CAE-area simultaneously,” General and Professional Educa- cal students generally know that different junctions are tion, vol. 3, pp. 11–17, 2017. usually the most stressed places into machines. The trunk- [12] S. E. Hale, B. A. Gardiner, A. Wellpott, B. C. Nicoll, and branch junction is the counterexample. It shows the potential A. Achim, “Wind loading of trees: influence of tree size and of bionic-style reinforcements. competition,” European Journal of Forest Research, vol. 131, The task on tree theme teaches the students a lot of no. 1, pp. 203–217, 2012. modeling technics. There are flexible system simulation, [13] E. de Langre, “Effects of wind on plants,” Annual Review of geometrical nonlinearity, and branch-crown contact inter- Fluid Mechanics, vol. 40, no. 1, pp. 141–168, 2008. action. Thus, student gets acquainted with the complex [14] C. Ciftci, S. Brena, B. Kane, and S. Arwade, “The effect of stress state of the branch, including bending, torsion, and crown architecture on dynamic amplification factor of an eccentrical compression. open-grown sugar maple (Acer saccharum L.),” Trees, vol. 27, no. 4, pp. 1175–1189, 2013. [15] G. V. Lavers and G. L. Moore, The Strength Properties of Tim- Data Availability bers, Building Research Establishment, London, UK, 1983. [16] Republic Belarus Wood structures, “Design code,” 2013, TKP All results are provided in the paper. 45-5. 05-275-2012. [17] Department of Agriculture, Wood handbook—Wood as An Conflicts of Interest Engineering Material. Gen. Tech. Rep. FPL–GTR–113, Forest Service, Forest Products Laboratory, Madison, WI: USA, 1999. The authors declare no conflict of interest. [18] B. Kane, Y. Modarres-Sadeghi, K. R. James, and M. Reiland, “Effects of crown structure on the sway characteristics of large Authors’ Contributions decurrent trees,” Trees, vol. 28, no. 1, pp. 151–159, 2014. [19] P. Areias, J. Reinoso, P. P. Camanho, J. César de Sá, and All authors have contributed equally. T. Rabczuk, “Effective 2D and 3D crack propagation with local mesh refinement and the screened Poisson equation,” Engi- neering Fracture Mechanics, vol. 189, pp. 339–360, 2018. References [20] P. Areias and T. Rabczuk, “Steiner-point free edge cutting of [1] O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method tetrahedral meshes with applications in fracture,” Finite Ele- Vol. 1: Basic Formulation and Linear Problems, Butterworth- ments in Analysis and Design, vol. 132, pp. 27–41, 2017. Heinemann, Oxford, UK, 2000. [21] P. Baranowski, Ł. Mazurkiewicz, J. Małachowski, and [2] J. Karliński, M. Ptak, and L. Chybowski, “Simulation-based M. Pytlik, “Experimental testing and numerical simulations methodology for determining the dynamic strength of tire infla- of blast-induced fracture of dolomite rock,” Meccanica, 2020. tion restraining devices,” Energies, vol. 13, no. 4, p. 991, 2020. [22] N. Vu-Bac, T. Lahmer, X. Zhuang, T. Nguyen-Thoi, and [3] C. Mattheck, Design in Nature, Springer-Verlag, Berlin, T. Rabczuk, “A software framework for probabilistic sensitivity Germany, 1998. analysis for computationally expensive models,” Advances in [4] M. Ptak, M. Ratajczak, A. Kwiatkowski et al., “Investigation of Engineering Software, vol. 100, pp. 19–31, 2016. biomechanics of skull structures damages caused by dynamic [23] L. Chybowski, M. Twardochleb, and B. Wiśnicki, “Odlučivanje loads,” Acta of Bioengineering and Biomechanics, vol. 21, 2019. na temelju multikriterijske analize značajnosti komponenti u [5] T. Jacksona, A. Shenkina, A. Wellpottb et al., “Finite element kompleksnom pomorskom sustavu,” Naše More, vol. 63, analysis of trees in the wind based on terrestrial laser scanning no. 4, pp. 264–270, 2016. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Bionics and Biomechanics Hindawi Publishing Corporation

FEA Simulation of the Biomechanical Structure Overload in the University Campus Planting

Loading next page...
 
/lp/hindawi-publishing-corporation/fea-simulation-of-the-biomechanical-structure-overload-in-the-10deMmTbVp

References (27)

Publisher
Hindawi Publishing Corporation
Copyright
Copyright © 2020 Stanislau Dounar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ISSN
1176-2322
eISSN
1754-2103
DOI
10.1155/2020/8845385
Publisher site
See Article on Publisher Site

Abstract

Hindawi Applied Bionics and Biomechanics Volume 2020, Article ID 8845385, 13 pages https://doi.org/10.1155/2020/8845385 Research Article FEA Simulation of the Biomechanical Structure Overload in the University Campus Planting 1 1 1 Stanislau Dounar , Alexandre Iakimovitch , Katsiaryna Mishchanka , 2 2 Andrzej Jakubowski , and Leszek Chybowski Belarusian National Technical University, Nezalezhnosti 65, 220027 Minsk, Belarus Maritime University of Szczecin, Waly Chrobrego 1-2, 70-500 Szczecin, Poland Correspondence should be addressed to Leszek Chybowski; l.chybowski@am.szczecin.pl Received 19 May 2020; Revised 30 October 2020; Accepted 6 November 2020; Published 23 November 2020 Academic Editor: Guowu Wei Copyright © 2020 Stanislau Dounar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Research of breakage of the chestnut tree branch on the planting of university campus is provided. Collapse is caused by a severe accidental wind gust. Due to collapse in the student environment, the investigation has additional methodical value for the teaching of FEA simulation. The model includes roots, trunk, branch, and conditional crown, where the trunk-branch junction is steady enough. The load-bearing system of tree is taken as an example of an effective bionic design. The branch has grown with the implementation of the idea of “equal-strength console”—the change of sections along the branch provides constant stress level and near uniform dispensation of their without stress concentrators. Static simulation of the tree loading is provided both in the linear formulation and in the geometrically nonlinear one. It is proved that in the trunk-branch junction area the stresses are twice lower than the branch itself, and it is not the place for fracture. For the given wind pressure, the work stress in the branch has exceeded twice the allowable level under bending with some torsion. In such construction (of the tree), the breakage could happen even in the perfect branch condition due to her severe overloading. 1. Introduction the weather station (located 5 km from the campus), the wind speed was only 12 m/s. Weather is regarded stormy if the The work relates to the sphere of the simulation (CAE) by wind speed exceeds 15 m/s. the finite element analysis (FEA) [1, 2]. An investigation is University authorities decided to investigate the incident close to the biomechanics [3, 4] because the stress-strain from an engineering point of view. Two groups of specialists state of the tree branch and trunk is discussed [5]. The work were formed: experts in the field of computational flow focuses both on engineering situation of tree load-bearing dynamics (CFD-group) and analysts of load-bearing systems system [6] and on the methodical use of the results to teach (stress analysis group (SA-group)) [10, 11]. students the possibilities of bionic design [7] and creative The CFD-group has provided computer simulation of problem solving [8]. airflows nearby the tree (0.3 km vicinity [12]) and revealed On the border of the university campus of BNTU, there is strong local wind amplification. It turned out that the tree a group of trees (Figure 1). This is part of a two-row planting, is placed in the focus of the double-wedged air manifold. namely, chestnuts (Aesculus hippocastanum). The object of The slot between buildings is continued by the gap in the modeling is tree 1, whose huge branch collapsed on a windy double-row planting just before the tree. In sum, north-east day, causing material damage [9]. Breakage took place in the wind is speeding up above university stadium and creates healthy, quality wood grains (fibers) in the area of the trunk- in the manifold stormy flow with the velocity of 24–25 m/s. branch junction. The tree remains standing and continues to Post factum observations of such local wind flow point out grow (Figure 2). of its steady character—wind gusts last about 5–7 s without The branch had a developed crown opposing the wind, significant oscillations. Therefore, trees are bent in the near static mode. but there was no storm in the summer city. According to 2 Applied Bionics and Biomechanics in the rectangular version in Figure 3(c) (RectCrown; blown surface—30 m ). Both types of crowns reach a height of 14 m. The main technic of 3D-building was surface pulling on by sections. There were 5 sections for the trunk. Section dimension changes from Ø580 mm to Ø390 mm going from the ground to the trimming level. The branch is pulled on by four basic diameters (marks 2–5 in Figure 4(a)) from Ø380 to Ø240 mm. The height difference between point 2 and point 5 is equal to 4 m. Branch bend in the 3–4 span has a radius of 2.4 m. The branch moves away from the trunk at the 60 angle (mark A60 in Figure 4(b)). Trunk-branch junction is smoothed by fillet Figure 1: The simulated tree (1) long before breakage of branch with 70 mm radius. (2014). Model DoubleTree (Figure 4(c)) with two main branches, two crowns, and common trunk with roots was built for additional proving of simulation results. 2.2. Wood Material Models. For results of stability proving, three material models were accepted for a parallel manner using during simulation. It should bring more confidence in results and limit model uncertainties of all issues. In the first model (ChestISO), wood is considered an isotropic, fully elastic material obeying Hooke’s law. According to the construction codes and sources [15], for the chestnut wood, it was appointed: the elastic modulus (E = 8000 MPa), Poisson’s ratio (μ =0:42), density (ρ = 600 kg/m ), Figure 2: The trunk-branch junction just after branch sawing and allowable stress (σ =16 MPa) (it is taken the same both (2018). for tension and compression). As it is not reliable initial data about mechanical characteristics of the crown, they are appointed a little arbitrary. Crown rigidity is considered very The SA-group has simulated the tree as a load-bearing low, i.e., elastic modulus (E =2 MPa). Crown density is the system standing under wind pressure [13]. The pressure variable parameter to simulate different mass of leaves on value was extracted from CFD-group work results—normal the branches (see below). wind level is equal (p = 380 Pa). A wind pressure of 600 Pa norm Orthotropic representation of the chestnut wood is pro- was taken into account too, as possible limit level for vided in the parallel manner with the isotropic one. Model hurricane-like situation. With the aim to disclose stress- ChestTKP is based on the local civil engineering code [16]. strain state of the tree and to reveal issues of the breakage, Elasticity modulus along grain is taken 8000 MPa, transversal the FEA was accomplished by the SA-group. to grain (400 MPa) (no difference between radial and tangen- Simulation has shown interesting result in two directions: tial direction) and shear modulus (all three) (400 MPa), and the engineering of biomechanical load-bearing system and Poisson’s ratios should be taken as 0.5, 0.02, and 0.02 (XY, the methodical improvement of teaching students the FEA. YZ, and XZ instances). Other orthotropic model ChestWH is more detailed and scientific [16, 17]. Elasticity modulus along grain is equal 2. Geometry Model of the Tree Load- 9400 MPa, transversal to grain (358 MPa and 678 MPa) Bearing System (radial and tangential direction), shear modulus (all three) (544 MPa, 396 MPa, and 134 MPa). Poisson’s ratios should 2.1. Geometry Representation. The tree with a broken branch was both laser scanned and sketched by gardeners just after be taken as 0.495, 0.052, and 0.035 (XY, YZ, and XZ the breakage. The SA-group members have provided 3D- instances). Thus, using of three different models proves modeling of this tree to bring variability of shapes and natural scattering of wood properties while simulating. reduce subjectivity of simulation (Figure 3). The scope of simulation embraces tree’s trunk 1, huge branch 2, and 2.3. FEA Mesh Variations. Several FEA mesh models of dif- crown. Remained branches 3 and 4 are not included in the ferent structure were created for tree simulation. Looking simulation scope. They are shown as conditionally trimmed. ahead, note all of them have shown good correspondence Branches 5 and 6 hold the crown. in results and minimal level of computing artefacts. According to the idea of tree crown variability [14], just Objects named solids and parts are used in the meshing two crowns were imaginarily matched to a given tree. A freely procedure. Solid brings monolithic mesh. The part consists growing crown in the curled version is shown in Figure 3(a) of several solids touching each other. Mesher joins their local (CurlCrown; blown surface—51 m ). The crown grown in meshes by common nodes. So part mesh is one-piece too. the constrained conditions (neighboring trees) has been built Other variants to simulate interaction between solids or parts 180 R2400 Applied Bionics and Biomechanics 3 CurlCrown 14 m RectCrown 51 m 14 m 30 m 2 4 Z X (a) (b) (c) Figure 3: Tree geometry: (a) model with curled crown (CurlCrown) on the downwind side; (b) the internal dimensions of the branch; (c) elongated crown (RectCrown) from upwind. R 70 A 60 (a) (b) (c) Figure 4: Models of the branch: (a) the trunk-branch junction; (b) the tree with two branches; (c) two crowns and stylized roots (DoubleTree). contact pair creation (special surface elements on the inter- (Figure 5(c)) permanently transforms into the branch’s mesh face). For that work, contact pairs are always in the bonded 2. For all wood, massive finite elements are joined together by common nodes. The tree’s crown was represented by a sepa- state. They work as perfect thin rigid glue layers. One of the finite element meshes is shown in Figure 5 rated mesh of volume finite elements. Crown and tree meshes (name it R-mesh (rare element density)). Finite elements have were conjugated by contact pair. mostly tetrahedral shape. It relates to trunk 1 (Figure 5(c)) and Figure 6 depicts alternative mesh (D-mesh (finite element to junction 3 between trunk 1 and branch 2. The branch itself packed with higher density)). The trunk and main branch were split to the sets of small solids. It was done by planes is meshed by hexahedral elements. It brings better accuracy in the critical part of the model. At the same time, higher normal to trunk/branch axes of growing. Solids 1 and 3 smoothness of the stress fields is achieved. The trunk and belong to the part “Branch.” Such solids are joined together branch create single solid. Accordingly, the trunk’s mesh 1 by common nodes at the faces like 2 and 3. 90° 4000 4 Applied Bionics and Biomechanics Crown 1225 kg Crown 1225 kg Stem and branches 1231 kg Wood 1231 kg X 1 (a) (b) (c) Figure 5: Meshes for: (a) leeward side of crown; (b) windward side of crown; (c) trunk 1 with junction 3 to branch 2. 4 2 34,529 BC BC (a) (b) Figure 6: Dense mesh for split solids (D-mesh): (a) partial view at solids; (b) picture of equivalent stress (σ ) (MPa) for LiBC condition set (stated below). Solid 4 and underlying ones create monolithic part standard FEA approach, especially that bending domination is “Trunk.” Parts “Trunk” and “Branch” are glued (BC) by expected. Line 2–2 (Figure 6(b)) goes between tension and compression zones. Equivalent stress maximum (34.529 MPa) bonded contact pair. Stress state for this model is shown in Figure 6(b). Stress field near marker BC is smooth and con- here (D-mesh) relates well to analog simulation by R-mesh. tinuous. Also, interfaces between solids are not visible any- Thus,bothmeshmodelsare appropriateenough. where. It means precision and fidelity of the D-mesh model. Mesh in Figure 6 is denser compared with one in Figure 5. 2.4. Boundary Conditions. The simulation was provided in the static form. That assumption is based on the CFD-group Outer surfaces of the trunk and branch are covered in Figure 6 by set of thin finite element layers (3). It brings accuracy for conclusion as about smooth, long-time patterns of wind gusts representation of surface stress effects. Branch core is in the local natural manifold acting on the simulated tree. modelled relatively coarse finite elements (2). That is a Oscillations and resonant effects are out of modeling scope. Applied Bionics and Biomechanics 5 FX = 0 FY = 18434 N FZ = –11528 N (a) (b) (c) Figure 7: Tree fastened to ground A and loaded by B (wind pressure) and C (gravity force): (a) leeward windward; (b, c) windward. Deformation shapes and reaction force vectors are for LiBC and HeBC condition sets at (b, c), respectively: ×1. Crown is a conditional object of the plate’s shape. Simu- Variations during tree simulation pointed out two repre- sentative sets of boundary conditions. They are called “Light” lation has focused on the lower branch (1st order branch— cite of breakage). Branches of the 2nd order are placed above (LiBC) and “Heavy” (HeBC) and are marked by color in and are built approximately. Branches of the 3rd order are Table 1. “Fork” space creates between them for other variants not regarded. of the model parameters. LiBC set refers to the simple, isotro- Interaction between crowns is not simulated. Leaves are pic, linear model of the tree under storm-like wind pressure. considered inner components of the crown. The mass of all HeBC set gives possibility to estimate ultimate deflection of leaves on the main branch (crown mass) is a really uncertain the heavy orthotropic tree in the near hurricane situation. parameter. It was taken at three levels—750, 1225, and 1550 kg—marked below as L-leaves, M-leaves, and H- 2.5. Nonlinearity and Orthotropy Checks. Meshes R-mesh leaves. Crown mass governs the gravity force. Simulation and D-mesh were used for simulation as three wood material pointed out that gravity force starts to play a role only at models. Loading was provided up to 600 Pa wind pressure. It hurricane-like wind pressure (600 Pa), where strong sloping was revealed (Figure 8(a)) that large deformation simulation of the crown occurs (Figure 7(c)). (NonL) brings higher levels of stresses and displacements in the tree compared to geometrically linear model (Lin). Non- The ground is simulated as a rigid base (mark A in Figure 7(a)). Wind pressure (mark B) is uniformly distrib- linear solution points out rise of branch top displacement on uted upon the windward side of the crown. Gravity force 33%. Maximal equivalent stress rises on 35%. It relates to the (mark C) is dispensed through all materials according to tree with the heavy crown (H-leaves). In the case of light their densities. crown (L-leaves), the nonlinear curve passes lower. Here, the difference between nonlinear and linear results does not Parallel modelling by different models and various condi- tions is the feature of that work. Intentional variation of exceed 18%. It is obvious that deflection of heavy crone by model factors was provided by different authors to control wind stimulates growing of the gravity force moment. So, uncertainties. The aim of parallel simulations was to ensure crown hanging-off additionally grows. Nonlinear simulation result stability and to estimate the sensibility of tree stress- is the way to disclose that interaction. The comparison of curves for the isotropic wood model strain state to the chatter of the entering factors. Table 1 depicts the scope of varied factors. Near full (“ISO”) and orthotropic models (“TKP” and “WH”) is given crossing of all steps was achieved. Geometrical linearity/non- in Figure 8(b). Orthotropic lines are placed near each other linearity of the tree model was investigated. That is a single with the difference below 7%, whereas the isotropic model kind of nonlinearity into the FEA model. Friction is not turns up much more rigid. Displacements for TKP-tree are 58% stronger than those for the ISO-tree (both models included, and wood is taken as fully elastic. If the model was simulated as linear (Lin), only one step possess the same elasticity modulus at 8000 MPa). of loading is provided. The model undergoes stepped loading However, stress levels for all three materials are placed in (30 steps) when large deformations are counted in the stiff- vicinity to each other (with the range of only 13%—despite of ness matrix of the tree, so geometrical nonlinearity (NonLin) displacements). It relates to the nonlinear simulation of heavy crown trees. became observable. 6 Applied Bionics and Biomechanics Table 1: Steps of model factors to vary. Leading sets of boundary conditions (BC) Model variation factors LiBC HeBC Trunk-branch material ChestISO (isometric) ChestTKP (orthotr.) ChestWH (orthotr.) Mesh R-mesh D-mesh Crown shape CurlCrown RectCrown DoubleCrown Leaves mass L-leaves (750 kg) M-leaves (1225 kg) H-leaves (1550 kg) Wind pressure 380 Pa 600 Pa Geometrical nonlinearity Lin (1 step) NonLin (30 steps) In the “light-crown” case, wood material variation causes green arrows, which means that the principal middle stress adifference of 15% for displacements and 6% for stresses (σ ) is near zero in the whole tree. Therefore, exactly, SSTe (linear solutions). is the place of one-axis tension, and at the same time, SSCo As a result, there are no principal differences between is the place of one-axis compression. Both σ and σ vectors 1 3 linear and nonlinear solutions concerning the shape of are oriented along the branch. This is a clear picture of bend- deflection and stress state features. ing. Some vector’s winding around the branch axis points out the presence of the small torsion moment (in a moderate proportion to the bending one). 3. Results and Discussion The conclusion about bending dominance in the stress- 3.1. Depiction of the Tree Stress-Strain State for Isotropic strain state of the branch is proved by distributions of the Model. Figure 9(a) shows natural scale deformational principal stresses (Figure 12). The fields of tension on wind- displacements of the tree. The crown significantly deflects ward side are shown in Figure 12(a) (almost completely coin- on its top (above 2 m). The branch is much more rigid, and cident with Figure 10(a)), where principal maximum stress the displacement (below the crown) is less than 100 mm. (σ ) creates SSTe (marks “30.729”–“34.069”–“30.177”). It is The distribution of the equivalent stress (σ ) for the the single place of high tension, but longitudinal gradients DoubleTree model is smooth enough (Figure 9(b)). There is are very low here, because tension stress is near the same in not just local, sharp stress concentration. The trunk is it. Therefore, SSTe should be taken into account as ridge- stressed moderately (14.5 MPa). Some stress increasing is like increase, not just a point of stress concentration, and visible at the trunk-root junction (29.6 MPa). The main wood breakage could start spontaneously in any place of SSTe. attention should be paid to the strips “34.448 MPa” and The picture of the principal minimum stress (σ ) shows “34.077 MPa.” The first marker precisely relates to the place smooth focusing of compression with small gradients along of the branch breakage. the branch from leeward (marks “-31.058”–“-34.103”–“30.5” Figure 10 depicts the concentration of equivalent stress in Figure 12(b)) and points out the SSCo. (σ ) (von Mises stress) in the basic tree model, which The SSTe and SSCo features received elongated shape. discloses both one-axis tension regions (indicates principal Large length is causes by branch section changing. The branch maximal stress (σ )) and one-axis compression regions as the kind of beam very close to the ideal “equal-strength con- (principal minimal stress (σ )). The tree surface has no local sole” is rising in diameter from leaves to the trunk. The bending stress concentrators, discontinuities, and high-gradient moment is enhancing in this direction at the same time. Branch regions. The bottom part of the branch is the only placed thickening (inertia moment enhancing) effectively counteracts with relatively high stresses. Here, Strip of Strong Tension to growing bending moment. The quick increase of branch (SSTe) is shown (between marks 1–2 in Figure 10(a)), where diameter in the trunk vicinity is relating the reinforcement of equivalent stress reaches level σ =34:181 MPa. This is the the trunk-branch junction. It results in stresses stabilizing and most tensioned part of the tree on the windward side far away is an example of self-organized wood growth to limit and level from the trunk-branch junction—“tensioned fiber”—by a the stresses. This is the bionic stresses stabilization (BiSS) or classic theory of bending. Equivalent stress (σ ) near the “ironing” of stress concentrators. trunk-branch junction is equal only to 13.614 MPa. The trunk is a slightly stressed object with σ =6:509 MPa. 3.2. Stress Distribution for Orthotropic Wood Model. Wood Strip of Strong Compression (SSCo) lays between 3 grain (fiber) orientation is always known for a living tree and 4 in Figure 10(b). Equivalent stress (σ ) here reaches only approximately. Thus, three simple, different variants 34.08 MPa level. That is so-called “compressed fiber” by of orientation were simulated (Figure 13) on the D-mesh classic theory of bending. base. It is a geometrical model assembled from split solids. Figure 11 demonstrates the direction of the principal Wood grain vector (WGV) was oriented inside every solid stress vectors. On the leeward side, one could see dominance normally to its bottom face. It caused (Figure 13(a)) an of the principal minimum stress (σ ) (blue arrows in uneven shape of stress isolines. Transitions between solids Figure 11(a)) as manifestation of SSCo feature. On the wind- are clearly visible. ward side, we can see the principal maximum stress (σ ) (red Nevertheless, stress picture, described above for the iso- arrows in Figure 11(b)) as SSTe feature. There are not visible tropic model (Figure 10(b)), is preserved. One can see stress Applied Bionics and Biomechanics 7 0 100 200 300 400 500 600 Wind pressure (Pa) B Lin H S Lin H S NonL H B NonL H S NonL L B NonL L (a) 0 100 300 400 500 600 Wind pressure (Pa) B ISO C WH B TKP S ISO B WH S TKP C ISO S WH C TKP (b) Figure 8: Curves of “along-wind” crown top displacement (mm) (mark “C” in the curve name), branch top displacement (mark “B”), and −1 maximal equivalent stress on the branch surface (10 MPa, mark “S”): (a) linear (“Lin”) and nonlinear (“NonL”) loading for the heavy (“H”) crown (means H-leaves) and for the light (“L”) one (means L-leaves) in the case of ChestWH material; (b) nonlinear tree loading for materials ChestISO, ChestTKP, and ChestWH (marks “ISO,”“TKP,” and “WH”, respectively); H-leaves: HeBC. Displacement (mm) Stress (MPa/10) 8 Applied Bionics and Biomechanics Type: total deformation 2215.5 Unit: mm Time: 1 26.03.2019 19:21 1054.3 34.977 2249.8 max 602.78 34.448 293.65 0 min 94.522 14.515 29.682 4.4073 (a) (b) wind Figure 9: Total displacement (mm) of the tree with the crown of RectCrown type under wind pressure (p = 380 Pa) (a) and the picture of norm equivalent stress (σ ) (MPa) for the DoubleTree model (b). A: Static structural A: Static structural Equivalent stress Figure Type: equivalent (von-Mises) stress Type: equivalent (von-Mises) Unit: MPa Unit: MPa Time: 1 Time: 1 26.03.2019 18:02 26.03.2019 18:08 34.595 max 34.181 34.595 max 30 30 34.089 26 26 22 22 18 18 14 14 10 10 13.614 6 6 14.039 9.9204e-5 min 9.9204e-5 min 6.8025 6.5091 (a) (b) Figure 10: Distribution of the equivalent stress (σ ) (MPa) through the surfaces of the branch and trunk on the windward (a) and leeward (b) wind sides. Pressure ðp Þ = 380 Pa; RectCrown, ×1. norm concentrator (SSCo), marked as A (38.24 MPa). Additionally, of WGV) and in Figure 13(c) (WGV orients along the branch two local extremums (B: 25.27 MPa) and (C: 24.58 MPa) are and smoothly extends that orientation into the trunk). founded at ends of branch-trunk junction. It may be stated that the orthotropic model of the tree is The system consisting of stress spots A, B, and C is more tangible to local geometry unevenness than isotropic revealed again in Figure 13(b) (vertical-dominant orientation one. Spots B and C probably are tied with some kind of that Applied Bionics and Biomechanics 9 A: Static structural Vector principal stress Type: vector principal stress Unit: MPa Time: 1 26.03.2019 20:11 Maximum principal Middle principal Minimum principal (a) (b) wind Figure 11: Vectors of principal stresses on the leeward (a) and windward (b) sides. Pressure ðp Þ = 380; RectCrown, ×1. norm ess 30.729 –30.5 34.069 –34.103 30.177 –31.058 13.503 6.5046 –6.7152 (a) (b) Figure 12: Distributions of the principal maximum stress (σ ) (a) from windward and minimum stress (σ ) (b) from leeward. Pressure 1 3 wind ðp Þ = 380 Pa; RectCrown, ×1. norm wind effect. Orientation vector variations are not crucial for the branch breakage under wind pressure (p = 380 Pa) is norm stress state of a tree branch. Main stress spots and stress levels highly likely possible. Nonlinear geometry effects amplify remain the same for both isotropic and orthotropic wood deformation and overloading of the branch through displa- material representations. cing of the crown’s mass center to leeward. In its turn, the gravity force starts to create a bending moment relative to 3.3. Nonlinear Estimation of the Branch Overloading. The the trunk’s rest (eccentrically compression) and increase stress-strain state pictures, shown above, point out that even more deviation of the branch from the vertical axis. 10 Applied Bionics and Biomechanics A - 38.24 34.291 37.148 B - 25.27 13.854 24.555 C - 24.58 19.148 12.091 (a) (b) (c) Figure 13: Distributions of the equivalent stress (σ ) for different wood orthotropy models: “normal-to-split” wood grain (a), vertical- wind dominant wood grain (b), and “along main branch” wood grain (c). Pressure ðp Þ = 380 Pa, leeward; (a, b) ChestWH and (c) norm ChestTKP; Lin; ×1. A: Static structural A: Static structural Equivalent stress 4.625 m Equivalent stress Type: equivalent (von-Mises) stress Type: equivalent (von-Mises) stress Unit: MPa Unit: MPa Time: 380 Time: 1 24.03.2019 15:33 24.03.2019 11:10 69.634 max 53.048 54.897 max 54.202 7.873e-5 min 8.8375e-5 min 67.134 12.321 12.479 (a) (b) Figure 14: Equivalent stress distribution (σ ) (MPa) for the linear solution (a) and for the geometrically nonlinear one (b; stepped loading). wind Peak wind pressure ðp Þ = 600 Pa; ×1. peak The comparison of the linear and nonlinear solutions is 67.1 MPa (Figure 14(b)). For the trunk part of the tree, the given in Figure 14. Figure 14(a) shows the picture of equiva- nonlinear effects are not so strong. lent stress (σ ), calculated for fully linear assumptions and Thus, the pressure of a stormy wind overloads the tree one-step loading. Figure 14(b) gives the distribution of equiv- branch up to fracture. It happens above the allowable stress alent stress, when the large deformation effects are accounted level for wood. Therefore, there is no need to look for a con- and the stepped loading solution is achieved. In the second centrator or damaged place along the branch to explain the case, the crown’s top displacement has risen about twice. event of destruction [18]—the branch should fall under the The stresses along SSTe and SSCo have grown approximately influence of strong bending and torsion moments. Our task in a quarter. Equivalent stress on the windward side of the was to point out the fact of severe overloading in healthy wood material possibility, but details of the cracking model branch (SSTe) is increasing from 54.2 MPa (Figure 14(a)) to Applied Bionics and Biomechanics 11 A: Static structural Vector principal stress A: Static structural Type: vector principal stress Static structural Unit: MPa Time: 1. s Time: 1 31.03.2019 12:59 31.03.2019 13:05 A Fixed support Maximum principal Fixed support 2 Middle principal Fixed support 3 Minimum principal D Pressure: 3.8e-004 MPa Pressure 2: 3.8e-004 MPa A X (a) (b) A: Static structural Maximum principal stress Type: maximum principal stress A: Static structural Unit: MPa Equivalent stress Time: 1 Type: equivalent (von-Mises) stress 31.03.2019 12:53 Unit: MPa 34.977 35.675 max Time: 1 31.03.2019 13:00 36.154 max 34.448 8 16 4 12 –3.7038 min 3.8512e-9 min 34.409 36.037 14.515 29.682 14.574 X 22.146 (c) (d) wind Figure 15: Simulation of the tree with two big branches (DoubleTree model and LiBC) loading by wind pressure ðp Þ = 380 Pa: (a) norm double fastening root; (b) vectors of principal stresses; (c) principal maximum stress (σ ) (MPa); (d) equivalent stress distribution (σ ) 1 e (MPa); ×1. may be the topic for the further investigation [19–21]. In the are placed far enough from the trunk. The trunk itself is future research, the uncertainty analysis is planned to be stressed stronger (σ =14:57 MPa in Figure 15(d)) due to big- done [22, 23]. ger blown surface of both crowns. It should pay attention to the underground stress concentrator (σ =29:68 MPa) in 3.4. Variations for Sensitivity Checks: Stability of “Ironed” Figure 15(c). Stress Concentrators during Wind Rotation. The DoubleTree Wind direction influence on the branch stress-strain state model (Figure 15) approves earlier conclusions. Both branches is estimated in Figure 16. The tree crone is built as a kind of have ribbon-like tensed and compressed fields. Stress peaks sail in that work. Four trees with identic, parallel crones were 12 Applied Bionics and Biomechanics 34.48 34.21 33.42 6.30 34.13 ° ° Figure 16: Stability of peak stress (σ ) (MPa) on the branch regardless of the wind direction (from left to right, trunk rotates at angles 0 ,30 , ° ° 60 , and 90 relatively to crown). LiBC, ×1. included in the model. Wind assumes acting normally to the flat crones. Every trunk-branch system was rotated at its own angle around the vertical axis relative to the crone. The angle changes at 30 from tree to tree. So, left tree orientation (Figure 16) is similar to the one from Figure 10. Right tree (Figure 16) has accumulated an angle of rotation equal to 90 . Its loading represents blowing off from the perpendicu- lar direction. Four markers (34.13–34.48 MPa) show high indepen- dence of tension stress strips (SSTe) from the wind direction. Compression stresses (SSCo) from the other side of the branch are at the constant level too. Bending (paired tension – compression system) is the dominating feature of the branch stress-strain state. Some moderate torsion is present for every branch in Figure 16. Thus, even large changes in Figure 17: Obliquely growing campus tree in the need for the wind direction remain, and the branch stress state is the assessment and FEA simulation. same. The pair of ironed stress concentrators on the healthy branch should be taken as steady BiSS effect. Let us pay attention to a whole lot of similar trees, planted SSCo is revealed. Both of the strips form a picture of bending in university campuses. Some of them need serious assess- of the console beam. ment and possibly help (Figure 17). The branch is a beam similar to the ideal “equal-strength Each problematical tree is the object for FEA analysis. console.” The main stresses (σ and σ ) (values near the 1 3 The simulation could predict eventual collapse. In addition, constant) appeared on the SSTe and SSCo along the branch. simultaneously, a lively and directly learning process the On those areas, it is not any stress concentrator, but the uni- mechanical students may be provided. The spheres of 3D- form stress state only. It is provided by spontaneous BiSS scanning, recovery of geometry, dynamics of fluid, and 3D- and effective self-reinforcement through the self-organized printing may be involved. wood growth. The trunk-branch junction is steady and developed. In the junction area, the stresses are twice lower than the branch itself. 4. Conclusions Thus, trunk-branch junction is formed as a region of the signif- The investigated branch during a storm undergoes mainly icant reinforcement, and it is not the place for fracture. the bending and some torsion. Gravity compression does High stresses rising in a smooth and uniform manner only not take a significant part in the stress state. along SSTe and SSCo are reached at 30-34 MPa at the moder- The SSTe is formed on the windward side in the bottom ate crown (RectCrown model). But it significantly exceeds the third of the branch. On this level, on the leeward side, the allowable stress for chestnut wood (16 MPa). When the crown Applied Bionics and Biomechanics 13 was developed (CurlCrown model), a tree deformation data,” Agricultural and Forest Meteorology, vol. 265, pp. 137– 144, 2019. becomes nonlinear, and the stress at all rises up to 67 MPa due to partially eccentric action of the gravity force. [6] K. R. James, G. A. Dahle, J. Grabosky, B. Kane, and A. Detter, “Tree biomechanics literature review: dynamics,” Arboricul- In such tree construction for the given wind pressure, the ture & Urban Forestry, vol. 40, pp. 1–15, 2014. breakage of branch could happen even in the perfect branch [7] A. Samek, Bionika. Wiedza Przyrodnicza dla Inżynierów, condition and without the stress concentrator due to severe Wydawnictwa AGH, Kraków, 2010. overloading, because the predicted stress exceeds twice allow- [8] D. Chybowska, L. Chybowski, B. Wiśnicki, V. Souchkov, and able stress for the chestnut tree. S. Krile, “Analysis of the opportunities to implement the Inside almost every university campus, we can find BIZ-TRIZ mechanism,” Engineering Management in Produc- appropriate plants and trees as the investigation object. tion and Services, vol. 11, no. 2, pp. 19–30, 2019. Because the campus is a part of the student’s environment, [9] M. J. Mortimer and B. Kane, “Hazard tree liability in the United then modeling of the tree attracts interest of students. The States: uncertain risks for owners and professionals,” Urban load-bearing system of the tree serves as a complex and at Forestry & Urban Greening,vol. 2,no.3, pp.159–165, 2004. the same time understandable example to study both FEA [10] S. Downar, A. Jakimowicz, C. Z. Jakubowski, and simulation and bionic principles of the design. A. Jakubowski, “Conception of simultaneous teaching the stu- The tree branch became a good illustration of the “equal dents of direction “machine design” to three-dimensional strength console” idea. We can see the rational changing of modeling and virtual testing by FEM-analysis,” General and branch sections—stresses are leveled along the main part of Professional Education, vol. 1, pp. 26–32, 2016. the branch. It makes students see the bionic design sense. [11] S. Downar, A. Jakimowicz, and A. Jakubowski, “Methodology Tree simulation teaches students to create models of of mechanical student quick involvement into CAD- and load-bearing systems without stress concentrators. Mechani- CAE-area simultaneously,” General and Professional Educa- cal students generally know that different junctions are tion, vol. 3, pp. 11–17, 2017. usually the most stressed places into machines. The trunk- [12] S. E. Hale, B. A. Gardiner, A. Wellpott, B. C. Nicoll, and branch junction is the counterexample. It shows the potential A. Achim, “Wind loading of trees: influence of tree size and of bionic-style reinforcements. competition,” European Journal of Forest Research, vol. 131, The task on tree theme teaches the students a lot of no. 1, pp. 203–217, 2012. modeling technics. There are flexible system simulation, [13] E. de Langre, “Effects of wind on plants,” Annual Review of geometrical nonlinearity, and branch-crown contact inter- Fluid Mechanics, vol. 40, no. 1, pp. 141–168, 2008. action. Thus, student gets acquainted with the complex [14] C. Ciftci, S. Brena, B. Kane, and S. Arwade, “The effect of stress state of the branch, including bending, torsion, and crown architecture on dynamic amplification factor of an eccentrical compression. open-grown sugar maple (Acer saccharum L.),” Trees, vol. 27, no. 4, pp. 1175–1189, 2013. [15] G. V. Lavers and G. L. Moore, The Strength Properties of Tim- Data Availability bers, Building Research Establishment, London, UK, 1983. [16] Republic Belarus Wood structures, “Design code,” 2013, TKP All results are provided in the paper. 45-5. 05-275-2012. [17] Department of Agriculture, Wood handbook—Wood as An Conflicts of Interest Engineering Material. Gen. Tech. Rep. FPL–GTR–113, Forest Service, Forest Products Laboratory, Madison, WI: USA, 1999. The authors declare no conflict of interest. [18] B. Kane, Y. Modarres-Sadeghi, K. R. James, and M. Reiland, “Effects of crown structure on the sway characteristics of large Authors’ Contributions decurrent trees,” Trees, vol. 28, no. 1, pp. 151–159, 2014. [19] P. Areias, J. Reinoso, P. P. Camanho, J. César de Sá, and All authors have contributed equally. T. Rabczuk, “Effective 2D and 3D crack propagation with local mesh refinement and the screened Poisson equation,” Engi- neering Fracture Mechanics, vol. 189, pp. 339–360, 2018. References [20] P. Areias and T. Rabczuk, “Steiner-point free edge cutting of [1] O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method tetrahedral meshes with applications in fracture,” Finite Ele- Vol. 1: Basic Formulation and Linear Problems, Butterworth- ments in Analysis and Design, vol. 132, pp. 27–41, 2017. Heinemann, Oxford, UK, 2000. [21] P. Baranowski, Ł. Mazurkiewicz, J. Małachowski, and [2] J. Karliński, M. Ptak, and L. Chybowski, “Simulation-based M. Pytlik, “Experimental testing and numerical simulations methodology for determining the dynamic strength of tire infla- of blast-induced fracture of dolomite rock,” Meccanica, 2020. tion restraining devices,” Energies, vol. 13, no. 4, p. 991, 2020. [22] N. Vu-Bac, T. Lahmer, X. Zhuang, T. Nguyen-Thoi, and [3] C. Mattheck, Design in Nature, Springer-Verlag, Berlin, T. Rabczuk, “A software framework for probabilistic sensitivity Germany, 1998. analysis for computationally expensive models,” Advances in [4] M. Ptak, M. Ratajczak, A. Kwiatkowski et al., “Investigation of Engineering Software, vol. 100, pp. 19–31, 2016. biomechanics of skull structures damages caused by dynamic [23] L. Chybowski, M. Twardochleb, and B. Wiśnicki, “Odlučivanje loads,” Acta of Bioengineering and Biomechanics, vol. 21, 2019. na temelju multikriterijske analize značajnosti komponenti u [5] T. Jacksona, A. Shenkina, A. Wellpottb et al., “Finite element kompleksnom pomorskom sustavu,” Naše More, vol. 63, analysis of trees in the wind based on terrestrial laser scanning no. 4, pp. 264–270, 2016.

Journal

Applied Bionics and BiomechanicsHindawi Publishing Corporation

Published: Nov 23, 2020

There are no references for this article.