finiteelementfailureanalysisofgfrplaminatesin plate...

Research ArticleFinite Element Failure Analysis of GFRP Laminates inPlate-Cone Reticulated Shell

Xing Wang,1 Yu Jiang,1 Yonghui Huang ,2 Yue Huang ,2,3 and Fan Wang4

1School of Civil Engineering, Guangzhou University, Guangzhou 510006, China2Guangzhou University-Tamkang University Joint Research Center for Engineering Structure Disaster Prevention and Control,Guangzhou University, Guangzhou 510006, China3School of Civil Engineering, Qingdao University of Technology, Qingdao 266033, China4Guangzhou University Library, Guangzhou University, Guangzhou 510006, China

Correspondence should be addressed to Yonghui Huang; [email protected] and Yue Huang; [email protected]

Received 28 October 2019; Accepted 29 February 2020; Published 19 June 2020

Guest Editor: Guangming Chen

Copyright © 2020 Xing Wang et al. ,is 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.

Plate-cone reticulated shell is a new type of spatial structures with good mechanical behavior, technical economy, and architecturalappearance. In this paper, usingANSYS software, the strength failure analysismodel of composite laminates is established in cooperationwith the Strength Criterion ofHoffman.,e effects of layer number, laying direction, and thickness of laminates on the ultimate strengthof laminates are studied by detailed parametric analysis, which provides a theoretical basis for the design of composite plate-conereticulated shell and GFRP laminated plates. Some important conclusions are obtained and can be applied to engineering practice.

1. Introduction

Plate-cone reticulated shell is an emerging spatial structurein recent years, which is developed based on Kaiser alu-minum stressed-skin dome. ,e structure is assembled withcone elements and truss members, which are connected atthe joints using bolts [1] as shown in Figures 1 and 2. Plate-cone reticulated shell is a special type stress-skin structuresof half continuity and half lattice since ventral members (alsoventral members and bottommembers) of common double-layer reticulated shell are replaced by cone elements. ,econe plates can be made of conventional materials such asaluminum alloy, and steel or light composite materials suchas carbon fiber reinforced polymer (CFRP) or glass fiberreinforced polymer (GFRP). Plate-cone reticulated shell isan effective structure that can make full use of the strengthand stiffness of the plates. At the same time, it integratesload-bearing, enclosure and decoration into a whole. Be-cause of the high strength-to-weight ratio, good technicaland economic benefits, and distinctive architectural visualeffect, the plate-cone reticulated shell has been widely usedas a long-span spatial structure [2, 3]. ,e idea of plate-cone

reticulated shell was originated from the Kaiser aluminumstressed-skin dome. ,e first Kaiser aluminum stressed-skindome was built on the Hawaiian village of Honolulu, USA, in1957 with a span of 44.2m [4]. To date, there are many otheraluminum stressed-skin domes which have been built inschools, banks, city centers, conference halls etc., such asthree Temcor aluminum dome stadiums with 71m span onElmira College, New York, and the airlines dome with 60mspan on Schipol Airport of Amsterdam, Holland [1]. Sincethe 1990s, composite materials such as FRP have been in-creasingly used in civil engineering [5]. Because FRP is muchlighter and stronger than ordinary steel and aluminum alloy,it is more preferred to be used as the prefabricated coneelement in plate-cone reticulated shells.

Many studies have been carried out in relation toconventional single-layer or double-layer reticulated dome.Among them, Fan et al. [6] studied the elasto-plastic stabilityof seven types of commonly used single-layer reticulatedshells. Xiong et al. [7] investigated the elasto-plastic stabilityof single-layer latticed shells with aluminum alloy gussetjoints. Hiyama et al. [8] investigated the global bucklingbehaviors of an aluminum alloy double-layer spatial latticed

HindawiAdvances in Polymer TechnologyVolume 2020, Article ID 2809302, 12 pageshttps://doi.org/10.1155/2020/2809302

mailto:[email protected]:[email protected]://orcid.org/0000-0001-5153-7877https://orcid.org/0000-0003-2426-6842https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2020/2809302

structure with tubular pipes, ball connections, and joiningbolts via experimentation and analysis. Xie and Li [9] studiedthe natural vibration characteristics of an aluminum alloydouble-layer reticulated shell with various structural di-mensions. Zhi et al. [10] examined the failure mechanisms ofsingle-layer reticulated domes subjected to seismic loads.Zhai et al. [11] carried out the dynamic response analysis ofthe reticulated domes under blast loading using the finiteelement (FE) software ANSYS/LS-DYNA for studyingdamage model and damage assessment. Lin et al. [12]studied the failure modes of a reticulated dome in a smallairplane.

Most of the aforementioned studies are focused on thestructural behavior of reticulated domes of homogeneousmaterials. In contrast, the studies on the GFRP laminates inplate-cone reticulated shell are limited, among which Robak[13] studied the structural use of plastics pyramids indouble-layer space grids, Wang and Wang [14] researchedthe preliminary application of GFRP on plate-cone reticu-lated shells and analyzed the plate-cone reticulated shell as awhole, material design and structure design have not beendone. Composite materials possess some distinct mechanicalproperties, such as anisotropy, nonhomogeneity, low in-terlaminar shear modulus and low interlaminar shear tensilestrength, geometric nonlinearity, and material nonlinearity.,ese mechanical characteristics make the problem morecomplicated and challenging than the conventional mate-rials that is homogeneous, continuous, linear elastic, andisotropic. Hence, it is necessary to investigate the behaviorand strength of the composite plate-cone reticulated shellwith FRP laminates.

In this paper, based on the composite mechanics and thetheory of plate and shell, the elastic stress in the principaldirection of single-layer plate of plate-cone reticulated shellis calculated by using finite element analysis. ,en, theultimate strength of each layer under each load step isobtained by the Strength Criterion of Hoffman, and thefailure load of the first failed layer of laminated plate (definedas the first layer strength) and the failure load of the lastfailed layer (defined as the last layer strength) are obtainedand evaluated. ,e strength and main influencing factors ofcomposite laminates are studied comprehensively anddeeply which provides theoretical foundation for the designof composite plate-cone reticulated shells with laminatedplates.

2. Failure Analysis of GFRPComposite Laminates

Generally, in composite laminates, the fiber orientation ineach layer is different. In some cases, even the materialproperties and thickness at different layers are different.Hence, the resistance of each layer of fibers to external load isdifferent. It is highly unlikely for all layers of fibers to reachthe ultimate strength and fail at the same time under certainexternal loads. Instead, the failure of the laminates ofteninitiates from the weakest layer and propagate layer by layer.

,e different strength criteria of single-layer compositeplate has been proposed for the different situations. StrengthCriteria of Tsai–Hill did not consider the influence of dif-ference with tensile strength and compressive strength onmaterial failure. Hoffman considered the factor of difference

Truss members

Cone elements

Top joint

Bottom joint

Figure 1: Segment of the plate-cone reticulated shell.

Cone elements

Ball

Truss members

Bolt

(a)

Cone elements

Bolt

(b)

Figure 2: Sketch of connecting joints: (a) top joint and (b) bottom joint.

2 Advances in Polymer Technology

with tensile strength and compressive strength and suppliedsome linear terms shown as follows [15]:

C1 σ2 − σ3( 2

+ C2 σ3 − σ1( 2

+ C3 σ1 − σ2( 2

+ C4σ1+ C5σ2 + C6σ3 + C7τ

223 + C8τ

231 + C9τ

212 < 1.

(1)

Considering the orthotropy and plane stress state of thesingle-layer plate, the Strength Criteria of Robert [15] whichis shown in equation (2) was derived from equation (1):

F �σ21 − σ1σ2

XtXc+

σ22YtYc

+Xc − Xt

XtXcσ1 +

Yc − Yt

YtYcσ2 +

τ212S2< 1,

(2)

where F is the failure value and F≥ 1.0 means material failureoccurs. Xt, Xc, Yt, Yc, and S are the basic strength of thesingle-layer FRP laminate which can be obtained from the

material property test, σ1 and σ2 are the principal stressesalong the principal directions, and τ12 is the maximum shearstress.

To perform the failure analysis, the normal stressσx, σy, τxy of each layer of laminates along the referenceaxis of element’s coordinate x, y, and z are calculated firstly.Because the layering angle of each layer is different, thecalculated normal stress of each layer may not follow theprincipal direction of elasticity. In order to carry out strengthfailure analysis, the normal stress σx, σy, τxy of laminatesunder the element’s coordinate system should be trans-formed into the principal stress σ1, σ2, τ12 along theprincipal direction of elasticity of the layer by using equation(3). ,en, the strength of each layer is calculated accordingto the Strength Criterion of Hoffman which is shown inequation (2).

σ1σ2τ12

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭�

cos2 θ sin2 θ 2 sin θ cos θ

sin2 θ cos2 θ −2 sin θ cos θ

−sin θ cos θ sin θ cos θ cos2 θ − sin2 θ

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

σxσyτxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

. (3)

In order to obtain the normal stress of each layer oflaminates, nonlinear static analysis of the plate-cone retic-ulated shell under step-by-step loading was carried out usingcommercial finite element software ANSYS. ,en, theprincipal stress of the single-layer plate is obtained accordingto equation (3). ,e ultimate strength of each layer at eachload step is obtained by Strength Criterion of Hoffman, andthe ultimate strength of the first failed layer (defined as thefirst layer strength) and the ultimate strength of the lastfailed layer (defined as the last layer strength) are analyzedand evaluated.

It should be point out that the strength of the first failedlayer is generally considered as the ultimate strength of thecomposite laminates in engineering design, so the stiffnessreduction of the laminate plate caused by the failure of acertain layer is generally not considered in the structuralanalysis. Nevertheless, the ultimate strength of the last failedlayer of the laminate is also calculated in this paper in orderto evaluate the interval between the ultimate strength of thefirst layer and that of the last layer. ,is is critical and es-sential to identify a reasonable design of the laminatecomposite, which is characterized by the full use of thepotential of composite materials.

3. Finite Element Analysis

3.1. Geometry and Material Properties of the Model. A plate-cone cylindrical reticulated shell with quadrangular pyramidsis studied using finite element analysis, as shown in Figure 3.,e span of structure S� 30m, length L� 45m, vector heightF� 10m, and thickness h� 1.5m. ,e top connecting trussmembers are steel pipes of Φ108× 5.0mm, and the triangleplates of cones are orthogonal symmetric GFRP laminatedplates that consist of four layers of laminates with total

thickness equal to 8mm.,ematrix of each layer is composedof glass/epoxy resin, and the laying mode of laminated platesis [0/90]°S, i.e., the orientation of fibers are 0° in the first andthe fourth layers and 90° in the second and the third layers.,e material properties of the GFRP plates are listed inTable 1. ,e steel pipes are considered as ideal elastic plasticmaterial with elastic modulus E� 206GPa, yield strengthfy � 235MPa, and Poisson ratio μ� 0.3.

3.2. Finite Element Model. For the composite plate-conereticulated shell, the FEM analysis model only with the topmember, without bottom members and middle members, isadopted in this paper. ,e model considers the joints of thetop members as hinged and the bottom joints as rigid (asshown in Figure 2). In the ANSYS model, the spatial trusselement Link8 with 2 nodes and 6 degrees of freedom is usedto model the top truss members, and the finite strain shellelement Shell181 with 4 nodes and 24 degrees of freedom isused to model the triangular plate elements. ,is elementtype was demonstrated to suit for analyzing layered com-posite structures where material properties were varied atdifferent layers [16–18].

,e translational displacements of bottom nodes alongtwo longitudinal edges are restricted in three directions.Uniformly distributed vertical loads of 2 kN/m2 is appliedperpendicular to the shell surface. Self-weight of thestructure is also considered. Figure 4 shows the finite ele-ment model of the plate-cone cylindrical reticulated shell,which contains 326 nodes, 600 shell elements, and 275 linkelements.

3.3. Numerical Results. ,e displacements and internalforces of the plate-cone cylindrical reticulated shell are

Advances in Polymer Technology 3

obtained. Figure 5 shows the vertical displacement of theplate-cone reticulated shell. Figures 6 and 7 present the axialforce of the top member and the laminates. Figures 8 and 9show the stresses in each layer of laminates.

From Figure 5, it can be seen that the maximum dis-placement of the plate-cone reticulated shell is 6.18mm.,eratio of the maximum displacement to the span of thestructure is 1/4854, which reveals that the composite plate-cone reticulated shell has large stiffness similar to theconventional steel plate-cone reticulated shell.

From Figures 6 and 7, it can be seen that the compositeplate-cone reticulated shell has the same rules for staticinternal force distribution as the common steel plate-conereticulated shell [19, 20]. ,e maximum axial compressionforce of top members is −32.732 kN, and the maximum axialcompression force of plates is -51.055 kN. ,e strength ofeach layer of laminates was calculated according to equa-tions (2) and (3)fd2.,e results show that the strength of thefirst and fourth layer of composite laminates is 0.007029, andthe strength of the second and third layer of composite

30m

45m

(a) (b)

Figure 4: FE model of the plate-cone cylindrical reticulated shell.

L

C

(a)

SF

h

(b)

Figure 3: Geometric parameters of the plate-cone cylindrical reticulated shell.

Table 1: Material properties of the GFRP laminate.

EX (GPa) EY � EZ (GPa) GXY �GXZ (GPa) GYZ (GPa) PRXY � PRXZ53.74 17.95 8.63 5.98 0.25PRYZ Xt �Xc (GPa) Yt (GPa) Yc (GPa) S (GPa)0.49 1.034 0.027 0.138 0.041


laminates is 0.071455. ,erefore, the composite plate-conereticulated shell has notable residual strength. However, theinternal force distribution is not uniform among layers oflaminates, so it is necessary to analyze the reasonable layingdesign for laminates of the composite plate-cone reticulatedshell.

4. Parametric Analysis

4.1. Effect of the Number of Laminate Layers. In order toevaluate the effects of various design parameters on thebehavior of the GFRP laminates, a parametric analysis isconducted using the FE model established above. Firstly, theeffect of the number of laminate layers on the strength ofeach layer is studied. ,ere are two groups of laying modeused in this plate-cone cylindrical reticulated shell. ,e firstgroup is composed of laminates with fibers oriented in 90°and 0°: [90/0/90] (mode 1), [90/0]°S (mode 2), [90/0/90/0/90](mode 3), [90/0/90]°S (mode 4), and the number of thelaminate layers for modes 1 to 4 is 3 to 6, respectively. ,esecond group is composed of laminates with fibers orientedin −45° and +45°: [−45/+45/−45] (mode 5), [−45/+45]°S(mode 6), [−45/+45/−45/+45/−45] (mode 7), and [+45/−45/+45]°S (mode 8), and the number of laminate layers formodes 5 to 8 is also 3 to 6, respectively. A schematic diagramshowing the different modes is presented in Figure 10. ,e

total thickness of the composite laminates is 8mm in allcases, and the thickness of each layer is equal.

Figures 11 and 12 show the ultimate strength of thereticulated shell composed with the first group (modes 1∼4)and the second group (modes 5∼8) of laminates,respectively.

It can be seen from the figures that the number oflaminate layers has insignificant effect on the failure load ofthe composite plate-cone reticulated shells with qua-drangular cones for the same laying mode and totalthickness. Taking the first group as an example, it is foundthat the first failed layer of laminates is the layer with fibersoriented in 90° occurring in the direction of 90° fibers, whilethe failure of the last layer occurs in the direction of 0°fibers. ,e first layer strength and the last layer strength arealmost the same even though the number of layers isdifferent. ,e first layer strength and the last layer strengthfor the mode with four layers of laminates (i.e., laying mode2) are only 8.16% and 4.75% greater than that of the modewith three layers of laminates (i.e., laying mode 1), re-spectively. ,erefore, it can be concluded that the influenceof the number of layers on the ultimate strength and failuresequence of each layer of laminates can be ignored if thetotal thickness of the GFRP laminates is the same. Con-sidering the material cost and time consumption, it isrecommended that the layer number of laminates shouldnot be too much, and the total layer number of laminates issuitable to be 4 layers.

4.2. Effect of Laying Direction. ,e laying direction oflaminates is another critical parameter of laminates. Becausethe laying structures are different, that is to say, if the orderof each layer in the laminate is different, the ultimatestrength of the laminate may be totally different even for thesame material system. ,erefore, for the composite plate-cone reticulated shell, studying the influence of laying di-rection on the strength of the laminate is very necessary andof significant importance in practice.

In order to discuss the influence of laying direction onthe strength of the structure, the mechanical properties ofeight kinds of laying directions (total thickness is 8mm, andthe thickness of each layer is uniform) are calculated usingfinite element analysis. Combining with the Strength Cri-terion of Hoffman, the ultimate strength and the failuresequence of layers of laminates are obtained. ,e detailedresults are shown in Table 2.

It can be found from Table 2 that the laying direction ofthe laminate has significant influence on the strength of thestructure. It can also be observed that

(1) ,e ultimate strength of the plate-cone reticulatedshell with different laying modes are different. ,efirst layer strength of laminates with laying mode of[0/90]°S and [90/0]°S are 26.6 kN/m2, while the lastlayer strength is about 130 kN/m2. However, forlaminates with laying modes of [−45/+45]°S and[+45/−45]°S, all of four layers almost fail concur-rently, and the failure load is about 44.5 kN/m2.

–0.00618 –0.00475 –0.003321 –0.001891 –0.461E – 030.254E – 03–0.005465 –0.004035 –0.002606 –0.001176

MN

Figure 5: Vertical displacement of the plate-cone reticulated shell(m).


(2) ,e failure sequence of layers is also different. ,efirst failed layers of laminates with laying modes of[90/0]°S, [+45/0]°S, and [90/+45]°S are the first andfourth layers, while the last failed layers are thesecond and third layers. However, for laminates withlayingmodes of [0/90]°S, [0/−45]°S, and [+45/0]°S, thefailure sequence of layers are just the opposite, thatis, the first failed layers are the second and third

layers, and the last failed layers are the first andfourth layers. For laminates with laying modes of[−45/+45]°S and [+45/−45]°S, all four layers failedsimultaneously.

(3) ,e strength interval between the first layer strengthand the last layer strength is also different. Forlaminates with laying modes of [0/90]°S and [90/0]°S,there is the largest strength interval between the first

10.321 2.0683 –13.774 –27.449 –32.732

10.835 3.0259 –11.829 –24.730 –29.725

10.580 3.2284 –11.065 –23.468 –28.272

10.184 3.0243 –10.913 –23.009 –27.692

9.8917 2.7595 –10.994 –22.902 –27.511

9.7321 2.5733 –11.105 –22.915 –27.480

9.6628 2.4795 –11.178 –22.946 –27.491

9.6447 2.4525 –11.201 –22.959 –27.498

–0.0

3911

–0.3

193

–0.4

417

–0.3

957

–0.2

814

–0.1

838

–0.1

312

–0.5

719

–0.1

468

–0.3

668

–0.5

161

–0.4

335

–0.2

956

–0.2

069

2.02

391.

5211

–0.7

293

–0.2

289

–0.0

491

–0.0

354

–0.0

600

3.77

713.

0684

2.03

141.

2473

–0.7

977

–0.5

877

–0.5

111

4.85

304.

0786

2.89

041.

9424

1.33

260.

9951

–0.8

491

Figure 6: Axial force of top members of the plate-cone reticulated shell (kN).


strength and the last strength. ,e ratio between thelast layer strength and the first layer strength forlaminates with laying modes of [0/90]°S and [90/0]°S,

[+45/90]°S and [+90/+45]°S, and [+45/0]°S and [+0/+45]°S are 4.89, 2.95, and 1.27, respectively. And forlaminates with laying modes of [−45/+45]°S and

–5105510290

–1690.7

–1858.6–14846721.73

–4156.7–190701236.8

–472809523

–56.963

–450468621.81510.5

–4389–20925382.83

–412847413.9

–2540.7

–2905.5–9354.3772.01

–4154–19331202.8

–308785814.42856.7

–1952.8–1898.4237.58

–270943330.5

–3249.4

–3334–12935282.94

–12683–638.41–2573.1

–1455.81351

–127.14

–164253399.52433.8

–7635.81975.7936.93

–2320.3–5554.2355.05

–3910.6–3011.7–966.7

–1598.5–127.78–320.94

–472579713.6–1699

–1667.5–13320705.98

–448299485.177.367

3869.2179891111.5

–4058–19499–245.2

–389587684.6

–2375.9

–2655.8–8318695.85

–412537662.61536.4

–3870–18019103.79

–270483696.72606.3

–1603.3–1582.1524.86

–250469815.8524.86

–2956–12385–80.17

–1094823.813

–2496.3

–944.291104.3354.44

–12647–352.162251.5

–1790–5649–167.6

–3875.4–2730.8863.73

–1021.8–524.87253.73

–2379.8–2262

–940.42

–1618.8–12930577.41

–448659203.2

–1435.1 –3045.6–17476768.6

–437499115.4234.12

–3228–18937164.21

–390247153.61605.5

–1918.7–8141.9466.35

–378757436.7–2168

–3089–17488–27.56

–251253184.22681.1

–896.33–1514.2304.85

–2196–11899–132.86

–11037–693.172262.6

–2478–3046.1870.81

–273.931171.5241.99

–1090–5344

–175.58

–341.29–377.66183.51

–242793784

–2928.4

–10519187.61

–2335.8

–2158.9–1987.7–877.92

–437529094

–1136.6

–379137132.21706.3

–243503214.92739.2

–10619–614.572297.9

–2274.1–2907.4877.53

–1609.8–12859399.54

–2548.3–17088443.88

–2748–18682118.51

–1362.5–8069.2224.04

–2514–17266–72.63

–283.15–1624.396.649

–1560–11830–155.57

366.02962.2952.866

–432.6–5417.1–138.73

309.84–562.7888.232

–433408968.6384.14

–374577193.7

–2011.9

–240883636.8

–2760.7

–10608119.46

–2206.3

–2429.6–1992.2–832.56

–2514.3–2668.5880.89

–10675–411.72303.6

–241253344.72772.8

–374597177.91793.3

–433309046.6

–917.59

–1615.5–12904240.19

–2322.1–16892226.16

–2537–18607730195

–1071.5–8027.269.121

–2220–17204–81.18

70.404–1734

–29.703

–1197–11855–140.07

751.14763.17–39.33

–40.226–5558.1–105.12

712.46–770.822.576

–432138915.8501.53

–432009016.5

–785.24

–373347077.3

–1920.3

–373217185.71847.1

–240953507.5

–2673.9

–241053433.12771.4

–1079529.95

–2143.5

–10831–254.872284.6

–2741–2057.6–807.31

–2792.6–2469.9869.89

–1621.4–12951126.86

–2240.3–16801103.43

–2466–186.434.302

–959.61–8004.1–0.828

–2110–17195–64.89

225.06–1797.1–66.389

–1039–11895–104.58

935.25632.45–64.33

147.31–5661.3–69.187

907.82–914.87–3.428

–431808919

576.79

–431708992

–712.6

–373117054

–1876.9

–372977162.91866.8

–10935–40.171–2133.9

–10950–163.762246.6

–2951.8–2126

–805.15

–29.85–2339853.63

–1624.3–1297452.831

–2219.7–1676338.646

–2451–1862111.821

–930.72–7991

–13.375

–2083–17201–36.18

276.15–1824.3–46.698

–987.7–11920–54.94

1005.2567.55–42.40

218.29–5718.5–35.065

986.45–990.18–7.241

–431708941.6630.45

–431678966.3

–668.31

–373097081.5

–1865.4

–373047124.11867.5

–241643443.1

–2673.3

–241623456.12709.9

–11003–83.032–2159.6

–11007–116.32201.8

–3051.9–2189.2–815.9

–3060.1–2254.1

833.9

–16252–129810.000

–22164167540.000

–2449–186160.000

–92717–7986.6

0.000

–2081–172030.000

28612–1831.2

0.000

–978–119280.000

1022.1549.10.000

235.31–5735.6–0.000

1006–1013–0.000

–241383450.8–2654.3

–241373461.82747.3

Figure 7: Axial force of laminates of the plate-cone reticulated shell (N).


–9.710.232

–0.213

–8.570.1730.188

–5.870.1120.355

–3.110.0750.303

–1.440.0557

0.11–0.23

–0.9350.0904

–0.63–1.22

–0.153

–0.66–1.330.0477

–0.471–0.6030.0961

–0.63–1.230.025

–0.356–0.1320.0295

–0.535–0.8240.0341

–0.2930.0751–0.015

–0.406–0.3580.0429

–0.312–0.0163–0.0389

–0.701–0.2160.121

–0.696–0.1980.108

–2.38–0.140–0.323

–2.37–0.1220.282

–5.14–0.011–0.408

–5.130.0120.326

–7.850.128

–0.318

–7.850.1430.192

–9.000.208

–0.0076

–9.000.223

–0.214–0.21

–0.8380.0877

–0.59–1.150.139

–0.61–1.240.0308

–0.437–0.5380.0848

–0.59–1.140.066

–0.291–0.1110.326

–0.459–0.786

–0.0093

–0.1880.06380.0446

–0.293–0.358

–0.0204

–0.189–0.03560.0316

–0.419–0.158–0.118

–0.431–0.2060.109

–2.06–0.0859–0.313

–2.07–0.1300.283

–4.75–0.035–0.397

–4.76–0.0040.336

–7.410.163

–0.297

–7.420.1300.201

–8.540.227

0.0102

–8.540.210

–0.180–0.203–0.8130.0721

–0.437–1.110.096

–0.46–1.200.0205

–0.298–0.5210.0585

–0.44–1.11

–0.003

–0.158–0.1000.0383

–0.32–0.75

–0.0164

–0.06130.07350.0270

–0.163–0.334

–0.0218

–0.0616–0.02120.0229

–8.330.2130.030

–8.330.212

–0.143

–7.210.156

–0.271

–7.210.1370.214

–4.610.040

–0.367

–4.620.00430.343

–1.98–0.072–0.293

–1.99–0.1220.288

–0.380–0.1390.110

–0.394–0.1960.110

–0.202–0.8090.0498

–0.346–1.080.055

–0.37–1.180.0148

–0.194–0.5120.0281

–0.34–1.09

–0.009

–0.041–0.1020.0122

–0.20–0.741

–0.0193

0.06140.06560.0067

–0.0384–0.334

–0.0172

0.0632–0.02750.0110

–8.250.2070.049

–8.250.212

–0.115

–7.130.145

–0.252

–7.130.1440.225

–4.570.032

–0.346

–4.580.01410.347

–1.99–0.077–0.277

–2.00–0.1100.288

–0.431–0.141–0.104

–0.442–0.1830.110

–0.203–0.8120.0299

–0.305–1.070.028

–0.33–1.170.0091

–0.139–0.5070.0097

–0.28–1.08

–0.010

0.0269–0.106

–0.0036

–0.131–0.74

–0.0174

0.1360.0564

–0.0049

0.0371–0.399

–0.0131

0.141–0.03710.0028

–8.230.2050.063

–8.230.211

–0.099

–7.100.139

–0.240

–7.100.1450.231

–4.570.024

–0.335

–4.570.01980.347

–2.03–0.0835–0.269

–2.03–0.1010.286

–0.489–0.148–0.101

–0.496–0.1730.109

–0.204–0.8150.0158

–0.29–1.060.013

–0.32–1.170.0043

–0.118–0.505

–0.0001

–0.206–1.08

–0.0081

0.0567–0.109

–0.0083

–0.101–0.741

–0.0130

0.1720.0499

–0.0080

0.0734–0.344

–0.0086

0.176–0.0445–0.0004

–8.220.2050.073

–8.220.210

–0.090

–7.100.137

–0.235

–7.100.1440.234

–4.580.021

–0.334

–4.580.02130.344

–2.05–0.0889–0.267

–2.06–0.0966

0.281

–0.528–0.154–0.101

–0.532–0.1670.107

–0.204–0.8160.0659

–0.29–1.060.005

–0.32–1.170.0015

–0.113–0.504

–0.0017

–0.26–1.08

–0.0045

0.0666–0.110

–0.0058

–0.0914–0.742

–0.0069

0.1860.0464

–0.0053

0.0874–0.347

–0.0044

0.915–0.0485–0.0009

–8.220.2070.079

–8.220.208

–0.084

–7.100.139

–0.234

–7.100.142

–0.234

–4.580.020

–0.335

–4.580.02080.339

–2.07–0.092–0.271

–2.07–0.0941

0.276

–0.547–0.158–0.102

–0.548–0.1620.104

–0.2040.8170.000

–0.29–1.060.000

–0.32–1.170.000

–0.1120.503

0.0000

–0.26–1.080.000

0.0685–0.1100.000

–0.0895–0.7420.000

0.1890.04540.000

0.0907–0.3480.000

0.199–0.0497

0.000

Figure 8: Normal and shear stress (σx, σy, τxy) of the first and fourth layers of laminates (MPa).


2.35–3.060.212

1.98–2.71–0.189

1.34–1.86–0.357

0.766–0.984–0.304

0.427–0.455–0.117

–2.79–0.232–0.0903

–3.57–0.408–0.154

–3.91–0.437–0.048

–1.74–0.253–0.0964

–3.61–0.41–0.025

–0.342–0.136–0.0030

–2.41–0.309–0.0350

0.27–0.07960.0157

–1.02–0.188–0.0439

–0.0064–0.01010.0397

–0.536–0.2630.121

–0.483–0.259–0.108

–0.858–0.0669–0.0317

–1.05–0.155–0.0208

–2.19–0.0494–0.0444

–0.0197–0.7860.322

0.0343–0.781–0.282

0.842–1.640.407

0.911–1.64–0.326

1.72–2.490.318

1.77–2.48–0.192

2.17–2.840.0073

2.21–2.840.213

–2.50–0.209–0.0881

2.140.269–0.0099

2.09–2.690.180

–3.37–0.382–0.139

–3.65–0.405–0.0307

1.76–2.340.297

1.66–2.35–0.201

–1.55–0.23–0.0871

–3.37–0.382–0.0131

0.971–1.510.396

0.799–1.52–0.335

–0.286–0.112–0.066

–2.31–0.280.0098

0.0906–0.6740.312

–0.04300.685–0.283

–0.408–0.1630.118

–0.554–0.176–0.109

–2.43–0.203–0.0721

–3.28–0.327–0.096

–3.55–0.35–0.0205

–1.52–0.183–0.0583

–3.27–0.3290.0034

–0.278–0.067–0.0381

–2.22–0.230.0166

–0.226–0.0742–0.0269

–0.995–0.1090.0219

–0.0636–0.0238–0.0229

–0.358–0.1480.110

–0.53–0.162–0.110

0.0119–0.6460.292

–0.0315–0.659–0.287

0.905–1.470.366

0.799–0.1480.343

1.70–2.280.271

1.64–2.28–0.213

2.06–2.62–0.030

2.06–2.620.142

–2.41–0.202–0.0499

–3.21–0.293–0.055

–3.50–0.318–0.0148

–1.51–0.148–0.028

–3.42–0.2920.0091

–0.304–0.0304–0.0121

–2.22–0.190.0194

0.1820.0304–0.0066

–1.01–0.0.6940.0173

–0.1040.0149–0.011

2.03–2.60–0.048

2.05–2.600.115

1.65–2.25–0.224

1.65–2.250.252

0.876–1.460.345

0.821–1.46–0.347

–0.107–0.6520.276

0.0072–0.66–0.288

–0.356–0.1460.104

–0.483–0.175–0.110

–0.366–0.1840.101

–0.1460.0381–0.0028

–0.443–0.190–0.109

–1.04–0.04630.0131

0.1410.05260.0049

0.914–0.6640.268

0.0381–0.669–0.286

–2.22–0.1680.0175

–0.327–0.09430.0037

0.852–1.460.334

0.939–1.46–0.347

–3.23–0.2740.0101

–1.51–0.130–0.0087

1.63–2.250.240

1.65–2.24–0.231

–3.49–0.304–0.0091

–3.17–0.278–0.028

2.02–2.59–0.063

2.04–2.590.098

–2.42–0.202–0.030

–2.43–0.203–0.0158

2.02–2.59–0.073

2.04–2.59–0.089

–3.15–0.272–0.0129

–3.49–0.30–0.0043

1.62–2.240.235

1.65–2.24–0.233

–1.50–0.1230.0001

–3.23–0.2670.0081

0.842–1.460.332

0.844–1.46–0.344

–0.34–0.0040.0083

–2.23–0.1580.0131

0.0793–0.6730.267

0.0562–0.675–0.281

0.1150.06290.0080

–1.06–0.03560.0086

0.377–0.1980.101

–0.417–0.201–0.107

–0.1750.0490.0004

–2.44–0.203–0.066

–3.15–0.27–0.005

–3.49–0.299–0.0015

1.63–2.240.234

1.64–2.24–0.234

0.840–1.460.334

0.843–1.46–0.339

–3.23–0.2650.0045

–0.3460.00250.0058

–1.50–0.1210.0017

–2.24–1.550.0069

0.0718–0.6780.270

0.0656–0.0678–0.275

0.1020.06670.0053

–1.08–0.03170.0044

–0.388–0.2040.102

–0.400–0.2050.104

–0.1940.0543–0.000

–1.08–0.307–0.000

0.9880.676–0.000

–2.240.155–0.000

–0.3470.0031–0.000

–3.23–0.265–0.000

–1.50–0.121–0.000

–3.49–0.299–0.000

–3.15–0.27–0.000

–2.44–0.203–0.000

–0.1900.05320.0009

2.03–2.59–0.079

2.03–2.590.084

Figure 9: Normal and shear stress (σx, σy, τxy) of the second and third layers of laminates (MPa).


[+45/−45]°S, all four layers fail at the same time, sothere is no strength interval between the last layerstrength and the first layer strength.

Based on the above analysis, it can be concluded that thelaying direction of laminate is a key factor affecting theultimate strength of GFRP laminate of the composite plate-cone reticulated shell.

4.3. Effect of :ickness of Laminate. In order to study theinfluence of the thickness of laminates on the ultimatestrength of laminates, the parametric analysis for laminateswith different total thickness and different thickness of

individual layer is conducted. ,e triangular plates of coneelements are adopted with different laying design, i.e., thelaying directions are all [90/0]°S, but the total thickness of thelaminates and the thickness of each layer are different. In thispaper, the mechanical properties of six kinds of laminateswith different laying modes (seen in Table 3) are calculated,and the ultimate strength of the structure under uniformlydistributed loads are obtained by combining the StrengthCriterion of Hoffman; more detailed results are shown inTable 3.

From Table 3, it can be seen that the thickness ofcomposite laminates has a significant effect on the ultimatestrength of laminates in plate-cone reticulated shells. With

x

y

o90°

0°

90°

[90/0/90]

2

1

z

x

y

0°

90°

(1) (2) (3) (4)

z

o

3

2

1

[90/0]°s [90/0/90/0/90] [90/0/90]°s

z

x

y

0°

90°

0°

90°

z

o

4

3

2

1

z

x

y

90°

0°

90°

o3

2

1

0°

90°3

4

90°

590°

0°

90°

6

5

4

(a)

z

x

y

–45°+45°

–45°

x

y

+45°

z

x

y

+45°

–45°

+45°

z

x

y

+45°

–45°+45°

z

(8)(7)(6)(5)

o

3

2

1o

2

1o

4

3

2

1 o3

2

1

[–45/+45/–45] [–45/+45]°s [–45/+45/–45/+45/–45] [+45/–45/+45]°s

–45°

–45°

+45°

4

–45°3

z

5

–45°+45°

–45°

+45°

6

5

4

(b)

Figure 10: Composition of FRP laminates with fibers oriented in different directions. (a) Group 1. (b) Group 2.

1 2 3 40

20

40

60

80

100

120

140

Laying mode

Ulti

mat

e str

engt

h (k

N/m

2 )

First layer strengthLast layer strength

Figure 11: Ultimate strength of the reticulated shell composed with first group laminates.


the change of the thickness of each layer, the first layerstrength and last layer strength of the structure vary re-markably. Generally, the first layer and last layer strengthswill increase with the increase of the total thickness and thethickness of the second and fourth layers. It can also befound that with the change of total thickness, the strengthinterval between the first layer strength and last layerstrength of the laminates changes slightly. ,e ratios be-tween the last layer ultimate strength Q2 and the first layerultimate strength Q1 are about 5.

5. Conclusions

,e strength of each layer in FRP laminate is one of the maincontrolling factors in the design of composite plate-cone

reticulated shell. In this paper, using ANSYS software, astrength model of composite laminate is established usingthe Strength Criterion of Hoffman without considering thestiffness degradation of laminates. ,e effects of number oflayers, laying direction, and thickness of laminates on theultimate strength of laminates are studied in the parametricanalysis. Based on the results of the numerical analysis, thefollowing conclusions can be drawn:

(1) In the composite plate-cone reticulated shell, theinfluence of the number of laminate layers on theultimate strength and failure sequence of each layerof laminated plates is negligible. Taking into accountother factors, such as material cost and time con-sumption, the total layer number of laminate isrecommended as four layers.

Table 3: Comparison of ultimate strength of laminated plates with different thicknesses.

,ickness of each layer Total thickness (mm) First layer strength Q1 (kN/m2) Last layer strength Q2 (kN/m2) Q2/Q1[2mm/2mm] S 8 26.6 132.3 4.99[1mm/1mm] S 4 14.0 76.8 5.48[3mm/1mm] S 8 23.2 121.3 5.23[1mm/3mm] S 8 28.8 134.0 4.66[2mm/1mm] S 6 18.5 100.0 5.40[1mm/2mm] S 6 21.7 108.5 5.01Note: [3mm/1mm] S indicates that the thickness of the first and fourth layers is 3mm, and that of the second and third layers is 1mm.

5 6 7 842

43

44

45

46

47

48

Laying mode

Ulti

mat

e str

engt

h (k

N/m

2 )

First layer strengthLast layer strength

Figure 12: Ultimate strength of the reticulated shell composed with second group laminates.

Table 2: Uniformly distributed load when layer-by-layer failure of laminated plates.

Laying mode First layer strength (kN/m2) Layers failed at first Last layer strength (kN/m2) Layers failed at last[0/90]°S 26.6 2nd and 3rd layers 130.0 1st and 4th layers[90/0]°S 26.5 1st and 4th layers 132.3 2nd and 3rd layers[−45/+45]°S 44.5 All of four layers 44.5 All of four layers[+45/−45]°S 44.4 All of four layers 44.5 All of four layers[+45/0]°S 65.4 1st and 4th layers 83.0 2nd and 3rd layers[0/+45]°S 65.5 2nd and 3rd layers 83.0 1st and 4th layers[+45/90]°S 14.8 2nd and 3rd layers 43.6 1st and 4th layers[90/+45]°S 14.8 1st and 4th layers 43.8 2nd and 3rd layers


(2) ,e laying direction is one of the key factors affectingthe ultimate strength of the composite laminate inthe plate-cone reticulated shell. ,e influence oflaying direction on the ultimate strength andstrength interval is significant. It should be fully usedin design of practical engineering to achieve rea-sonable strength interval of laminate, for making fulluse of potential and advantages of compositematerial.

(3) ,e influence of the thickness of composite laminateon the ultimate strength of laminate is significant.,e first layer strength and last layer strength willincrease with the total thickness increase, and thethickness of the second and fourth layers increase.But the influence of the total thickness on thestrength interval is negligible.

Data Availability

,e data used to support the findings of this study are in-cluded within the article.

Conflicts of Interest

,e authors declare there are no conflicts of interest.

Acknowledgments

Financial support from the National Natural ScienceFoundation of China is gratefully acknowledged. ,is re-search was funded by the National Natural Science Foun-dation of China, grant nos. 51678313 and 51608135, andTaishan Scholars Program of Shandong Province, China,grant no. tsqn201909127.

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