Ponding refers to a phenomenon of accumulation of water on top of a structure. Even though
most light weight membrane structures are designed to prevent its occurrence, it can be initiated
in some cases during rainfall by an event such as drifted snow settling on the surface of the
structure causing a local depression of the membrane structure. The present work proposes
a method to calculate the static deformation of a membrane structure due to a given volume
of ponding water. The method involves coupling of a structural solver for the membrane and
a volume conserving solver representing the static behavior of an incompressible fluid. The
coupling is performed in a partitioned manner with the linearized behavior of the incompressible
fluid incorporated in the structural equations to accelerate the coupling iterations. Using this
method, the final deformation of the structure due to ponding is calculated by applying loads
due to a fixed volume of water

Degroote, Joris

Kodunthirappully Narayanan, Navaneeth

Wüchner, Roland

Ghent University Academic Bibliography

Proceedings of the IASS Annual Symposium 2019 — Structural Membranes 2019Form and Force7—10 October 2019, Barcelona, SpainC. La´zaro, K.-U. Bletzinger, E. On˜ate (eds.)Calculation of static deformation of membranestructures under the load of ponding waterNavaneeth K. Narayanan∗,b, Roland Wu¨chnerb and Joris Degrootea,c∗ Department of Flow, Heat and Combustion Mechanics, Ghent UniversitySint-Pietersnieuwstraat 41, 9000 Ghent, Belgiumnavaneeth.kodunthirappullynarayanan@ugent.bea Department of Flow, Heat and Combustion Mechanics, Ghent Universityb Chair of Structural Analysis, Technical University of MunichcFlanders Make, BelgiumAbstractPonding refers to a phenomenon of accumulation of water on top of a structure. Even thoughmost light weight membrane structures are designed to prevent its occurrence, it can be initiatedin some cases during rainfall by an event such as drifted snow settling on the surface of thestructure causing a local depression of the membrane structure. The present work proposesa method to calculate the static deformation of a membrane structure due to a given volumeof ponding water. The method involves coupling of a structural solver for the membrane anda volume conserving solver representing the static behavior of an incompressible fluid. Thecoupling is performed in a partitioned manner with the linearized behavior of the incompressiblefluid incorporated in the structural equations to accelerate the coupling iterations. Using thismethod, the final deformation of the structure due to ponding is calculated by applying loadsdue to a fixed volume of water.Keywords: ponding, partitioned coupling, volume conserving solver, hydrostatic load1 IntroductionMembrane structures are very efficient in carrying loads compared to the materials usage, butthese structures undergo significant surface movement when loaded, making them vulnerable tothe formation of a water pond. Rainfall along with an event such as snow settling or some deadloads on the structure can trigger accumulation of water in local depression. If the structure isnot stiff enough this can cause more deflection leading to more ponding and so on. This cycleof increasing deformation and water accumulation may increase the load indefinitely resultingin structural collapse or failure. For the cases where the structure is much stiffer with gooddraining characteristics, the excess water will drain, resulting in a stable pond. However, evenin this case if the rain is accompanied by strong winds, the wind flow around the structure mayinduce large oscillations. In 2011, during the Pukkelpop festival held in Kiewit (Belgium), astrong wind interacting with ponding water led to huge swaying of tents eventually resulting inthe collapse of the structure. Studying such cases will involve fluid structure interaction (FSI)simulation between the membrane structure, the ponding water and the wind flow.Copyright c© 2019 by Navaneeth K. Narayanan, Roland Wu¨chner and Joris DegrootePublished by the International Association for Shell and Spatial Structures (IASS) with permission.1Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019Form and ForceIn order to simulate such systems it is necessary to find the static deformation of the membranestructure under the load of the ponding water. This will be the pre-processing step for the FSIsimulation. The other applications of this analysis include floating caps of oil storage tanks [9],and optical reflector forming using ponding loads [7]. Some of the older works on studying theponding on membranes include the work of Szyszkowski and Glockner [8] who studied pondingstability and deformation on spherical inflatables, where they solved axi-symmetric membraneequations with the hydrostatic loads. Tuan [7] focused on large deformations and strains ofinitially flat, simply supported circular membranes under gradually accumulated fluid pressure.He used fourth-order Runge-Kutta numerical integration with an iterative finite element analysisusing shell elements to calculate the deformation due to the ponding. However, these studiesonly involved axi-symmetric geometries. One example of a general approach is presented in thework of Bown et al. [3], where an in-house structural code inTENS is coupled with a shallowwater solver in a partitioned method to analyse ponding on tensioned membrane structures.The idea presented in the current work is similar to Bown et al., except here a volume conservingsolver is used instead of a transient shallow water solver. The volume conserving solver conservesthe given volume of incompressible fluid with a constraint that the top surface is normal to grav-ity, thus representing the free surface of the fluid in static condition. The coupling iterations areaccelerated by adding the linearized static behaviour of the fluid under gravity in the structuralsolver. In terms of implementation this will translate into adding the load stiffness matrices tothe tangent stiffness matrix of the membrane elements and a linear update of the top surfacein the non-linear iterations of the structural solver. The required load stiffness matrices arementioned shortly in the later sections but this has been extensively discussed in [1, 2], wherethe authors derive all the tangent stiffness matrices from the principle of virtual work. In eachcoupling iteration, hydrostatic load is applied on the structure based on the plane’s height andthe volume-conserving solver updates the plane’s height depending on the deformation of thestructure in order to conserve a given volume of water. When the convergence is achieved in thepartitioned iterations, the deformed shape of the structure is determined. The following sectionsdiscuss the proposed volume-conserving solver and how it is coupled to a structural solver inKRATOS [4], an open-source finite element framework. At the end, an example is presentedwhere the static deformation of the membrane structures is calculated due to the accumulationof an incompressible fluid.2 Mathematical models2.1 Hydrostatic loadConsider a membrane surface denoted by ∂Ωs containing a certain volume of incompressiblefluid. The free surface of the fluid, denoted by ∂Ωf is perpendicular to gravity. The wettedregion of the membrane, ∂Ωfs experiences hydrostatic pressure proportional to the height of thefree surface from the fluid above. If the gravity is along the z-direction then mathematically,the traction at a point P with z-coordinate, z on the structure can be stated as:t = −γf (z − zf )n ∂Ωfs (1)t = 0 ∂Ωs \ Ωfswhere, zf is the z-coordinate of the free surface, γf is the specific weight of the fluid, n isthe outward unit normal vector at the point P. Figure 1 clearly indicates the symbols andterminologies used.2Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019Form and ForceYZXPFigure 1: Hydrostatic loading by a fluid on a membrane.2.2 Structural equationsthe variational formulation of the static non-linear membrane in its reference configuration canbe written as−∫∂Ω0sh S : δE dS +∫∂Ω0st0 · δu dS = 0, (2)where S and E are the second Piola-Kirchhoff stress tensor and the Green-Lagrange strain tensor,respectively. The first term in the equation 2 is the virtual internal work (δWint) and the secondterm is the external virtual work (δWext), corresponding to the virtual displacement δu. Withthe assumptions of large displacement and small strains, we only consider the Saint-VenantKirchhoff material law, equation 3, in the numerical example presented in this paper.S = 2µE+ λE : I (3)The tangent stiffness matrix of the membrane elements (Kmem) can be obtained by discretisingequation 2 and deriving δWint by the nodal displacement vector uˆ. For more details on thederivation of membrane elements, please refer to [10]. For the case of hydrostatic loading, δWextcan be written in the current configuration by using equation 1 givingδWext =∫∂Ωfsδu · pnds= −∫∂Ωfsδu · γf (z − zf )nds. (4)In the equation 4, the external work term is calculated based on zf determined by the volumeconserving solver. By linearizing the discretized form of equation 4 and subtracting it from themembrane tangent stiffness matrix the convergence speed of the coupling iterations between thestructural and the volume conserving solver can be improved. Even though, the free surface ismoved by the volume conserving solver in every coupling iteration to conserve a given volumeof water, the water level should also change during the non-linear iterations of the structural3Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019Form and Forcesolver. This is not directly performed by the volume conserving solver as it was found to beunstable in some numerical experiments. Actually, if the volume conserving solver was used instructural iterations, the external coupling iterations is not required and the convergence will bemuch faster. But for the algorithm to be more robust, it was decided to perform the couplingexternal to the structural solver with the linear update of the free surface inside the non-lineariterations of the structure, given byDzf [u] = ∆zf =− ∫Ωfs ∆u · n dsAf. (5)The linearization of the external work term from the hydrostatic pressure in the equation 4 isgiven byDδWext[u] =− γf∫∂Ωfsδu ·Dz[u] n ds+ γf∫∂Ωfsδu ·Dzf [u] n ds− γf∫∂Ωfsδu · (z − zf ) Dn[u] ds. (6)Substituting Dz[u] from equation 5 we getDδWext[u] =− γf∫∂Ωfsδu ·Dz[u] n ds− γfAf(∫∂Ωfsδu · n ds)(∫∂Ωfs∆u · n ds)− γf∫∂Ωfsδu · (z − zf ) Dn[u] ds. (7)Discretizing equation 7 using the shape functions matrix N and the nodal displacement vectoruˆ, we can write the equation 7 in terms of the load stiffness matrix Kp, the incremental nodaldisplacement vector ∆uˆ and the virtual displacement δuˆ. The discretized equation for eachmembrane element inside the wetted region Ωfs is given in equation 8, which is assembled in aglobal matrix to give Kp.DδW eext[u] =δuˆeT−γf∫ξ∫ηNT n˜eTz N dξdη −γfAf∫ξ∫ηNT n˜ dξdη∫ξ∫ηn˜TN dξdη−γf∫ξ∫η(z − zf )(gξ × ∂N∂η− gη × ∂N∂ξ)dξdη∆uˆe=δu˜TKep∆u˜e (8)with the corresponding virtual external work for each element,δW eext = −δuˆeTγf∫ξ∫ηNT (z − zf )n˜ dξdη= δuˆeTFep (9)4Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019Form and ForceIn equation 8 and 9, the integration is performed in the parametric domain, where the unitnormal vector n is replaced by the normal vector n˜ using the relation, n ds = n˜ dξdη. Inequation 8, gξ and gη are the basis vectors at any point in the element. In the current work,only constant strain triangle (CST) elements are considered with the numerical integrationbased on Gauss quadrature. For the elements that are cut by the free surface the integration isperformed by dividing the cut triangles and performing Gauss integration on the triangles thatare below the free surface, as shown in the figure 2. The final equation after the assembly forarbitrary δu˜ is solved in every non-linear iteration of the structure, until convergence, givingδu˜T (Kmem −Kp) ∆u˜ = δu˜T (Fp − Fmem)=⇒ (Kmem −Kp) ∆u˜ = Fp − Fmem (10)where Fmem is the global internal force vector of the membrane. In each non-linear structuraliteration the global hydrostatic force vector Fp is calculated based on the position of the freesurface which is updated according to equation 5.3 Volume conserving solverThe volume conserving solver consists of two components: a volume calculation algorithm and aiterative algorithm to conserve a given volume. The volume of ∂Ωfs in figure 1 can be calculatedusing the relation,Vf =∫∂Ωs(z − zf )ez · nds (11)and its derivative with respect to zf ,∂Vf∂zf=∫∂Ωs−ez · nds = Af (12)where Af is the area of the free surface. The integration in equation 11 and 12 is performednumerically using Gauss quadrature, in the same way as explained in the previous section. Inthe current work, leap-frogging Newton’s method is used for conserving a given volume. Thismethod is discussed in detail in [6]. It consists of a Newton step followed by a pseudo secantstep. The main advantage of this method is that it only involves one derivative evaluation,like Newton’s method and yet it can attain cubic convergence. The Newton method and leap-frogging method were tested for volume conservation with a complicated geometry and it wasfound that the leap-frogging Newton was much more robust and had faster convergence ratethan Newton’s method. Hence, it was chosen over the other. The equation used for iterationto conserve the volume is given in equation 13, with the function f and xn being the differencebetween the current volume and the target volume (V nf − Vt), and the z-coordinate of the freesurface at the nth iteration, respectively. The derivative of function, f ′ is given by equation 12,which is the area of the free surface at the nth iteration.x˜n = xn − f(xn)f ′(xn)xn+1 = xn − f(xn)2f ′(xn) (f(xn)− f(x˜n)) (13)5Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019Form and ForceGauss pointzyFigure 2: Gauss integration in the wetted region.4 Partitioned couplingThe static structural deformation under the load of a given volume of fluid, Vt is obtainedwhen the variational equation of the structure, equation 2, and boundary conditions given inequation 1 are satisfied with fluid volume V = Vt. The solution of the problem is obtained byiterating between structural solver and volume-conserving solver until the norm of the displace-ment change between the iterations is under a certain tolerance. Because the linearized part ofthe fluid is added in the structural system, there was no need to use any convergence acceleratorslike Aitken or IQNLS [5]. This claim is also supported by the numerical experiments that werecarried out. For example, in the numerical example presented in the paper, for most of the fluidvolume the number of coupling iterations was around 3 or 4.5 Numerical example: Ponding on a pnuematic membrane structureBefore performing the simulation of this case, the volume conserving solver was validated witha known case where the enclosed volume was already known. The partitioned coupling betweenthe solvers was also validated with the numerical results presented in table 2 of [7]. Here, themaximum vertical displacements corresponding to four different fluid volumes were comparedand were found to be in very close agreement. In the numerical example presented in thispaper, we consider an inflated thin-walled membrane hemisphere with Young’s modulus E =7 × 106 N/m2, Poisson ratio ν = 0.45 and thickness t = 0.002 m. A difference in pressure ofp = 0.5 kPa compared to the atmospheric pressure is applied at the internal surface. To get theperfect hemisphere of diameter D = 10 m after the application of pressure, an isotropic normalpre-stress calculated from the formula σmem = pD/4t = 1.2MPa is applied on the membranewith zero shear stress. The hemisphere is clamped at the bottom boundary and for reducingthe computational time only a quarter section of the hemisphere is simulated considering thesymmetry of the problem. To start the ponding process, we first apply a dead load of w =1 kPa on the top surface of the membrane enclosed by a circle of radius Rdead = 1.736 m asa seed event. Due to the dead load, there will be a local depression in the hemisphere. In6Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019Form and Forcethe created depression, a hydrostatic load due to different volumes of water was applied. Theresults are obtained by running the partition coupling iterations with various target volumesas input in the volume conserving solver. In all the simulations considered in the example, thevolume conserving solver is initialized from the topmost point of the undeformed hemisphere.The vertical displacements of top most point of the hemisphere corresponding to the differentfluid volumes are given in table 1. Figure 3 shows the deformed shape of a cut section of thehemisphere for Vt = 1.6 m3. The distance from the free surface is scaled to positive and negativevalues very close to zero to show the wetted surface of the hemisphere.Table 1: Vertical displacement of the top most point of the hemisphere (uz) and vertical positionof the free surface (zf ) for different volume of water (Vt).Vt [m3] 0.0 0.8 1.6 2.4 3.2 4.0uz (m) -0.972 -2.442 -3.709 -4.950 -6.214 -7.531zf (m) N.A. 7.986 6.950 5.887 4.774 3.590(a) Distance from the free surface. (b) Deformed geometry with mesh.Figure 3: Deformed geometry of inflated the hemispherical membrane due to the ponding ofwater with Vt = 1.6 m3.6 Conclusion and OutlookAn algorithm for calculating the static deformation of a structure due to ponding load from afixed volume of fluid was developed. The algorithm involved partitioned coupling of a structuralsolver and a volume conserving algorithm, responsible for maintaining a constant volume of fluidon the structure. The linearized contribution of the fluid behavior was added in the structuraliterations to accelerate the coupling. The obtained deformation will be used as the initialcondition for the FSI simulation of the membrane structure, ponding water and wind flow.7Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019Form and Force7 AcknowledgementsThe authors gratefully acknowledge the funding of the Research Foundation – Flanders (FWO)for this work (project number G086517N).References[1] T. Rumpel and K. Schweizerhof, ”Hydrostatic fluid loading in non-linear finite elementanalysis,” International Journal for Numerical Methods in Engineering, vol. 59(6), pp. 849-870, 2004.[2] C. Hoareau and J.-F Deu¨. ”Non-linear finite element analysis of an elastic structure loadedby hydrostatic follower forces,” Procedia Engineering, Elsevier, vol. 199, pp. 1302-1307,2017.[3] A. Bown, T. Makin and D. Wakefield, ”Beyond Static Analysis: Investigation of MembraneStructure Performance Using Time Stepping, Transient and Progressive Analyses,” ProcediaEngineering, vol. 155, pp. 1877-7058, 2016.[4] P. Dadvand, R. Rossi and E. On˜ate, ”An object-oriented environment for developing finiteelement codes for multi-disciplinary applications, ”Archives of computational methods inengineering,” vol. 17(3), pp. 253–297, 2010.[5] J. Degroote, ”Partitioned Simulation of Fluid-Structure Interaction.” Archives of Compu-tational Methods in Engineering, vol. 20(3), pp. 185-238, 2013.[6] J.B. Kasturiarachi, ”Leap-frogging Newton’s method,” International Journal of Mathemat-ical Education in Science and Technology, vol. 33(4), pp. 521-527, 2002.[7] C. Y. Tuan, ”Ponding on circular membranes” International journal of solids and structures,vol. 35, pp. 269-283, 1998.[8] W. Szyszkowski and P. Glockner ”Finite deformation and stability behaviour of sphericalinflatables subjected to axi-symmetric hydrostatic loading” International journal of solidsand structures, vol. 20, pp. 1021-1036, 1984.[9] H. I. Epstein and T. J. Strnad, ”Liquid-filled, liquid-supported, circular structural mem-branes,” Computers & structures, vol. 21, pp. 443-451, 1985.[10] R. L. Taylor, E. Onate, P. -A. Ubach, ”Finite element analysis of membrane structures”Textile composites and inflatable structures, pp. 47-68, 2005.8

Calculation of static deformation of membrane structures under the load of ponding water

https://biblio.ugent.be/publication/8639258/file/8639261.pdf

Calculation of static deformation of membrane structures under the load of ponding water

Abstract

Similar works

Full text

Available Versions

Ghent University Academic Bibliography