1

Abstract This paper presents a panel element that is purposely designed to be made large in size without compromising the fidelity of analysis results. The design is to help overcome a common trade-off present in practicing finite element analysis: where finite element theory requires that element sizes remain sufficiently small, practitioners are often motivated to use larger element sizes that simplify the modelling process and reduce excesses in generated results. A related conflict arises in the practical requirement for shell elements with a ‘drilling’ stiffness (a rotational stiffness about the normal to the element plane). Typical shell elements either make no admission for a drilling stiffness or when present, element size may likewise place additional modelling constraints on the engineer. To satisfy these trade-offs, we propose a panel element that is purposely designed to be large in size and whose method includes the accurate prediction of drilling stiffness in a convenient way. The method is based on the principle of substructuring, internally meshing of panel. Finally we identify a number of performance benefits that become available as a consequence of modelling at the panel’s slightly higher level of abstraction. Keywords: panel element, drilling stiffness, large element, substructuring, floor & wall. 1 Introduction A common conflict in practicing finite element analysis is the trade-off between the practical benefits of using larger elements against the numerical benefits of finer meshing. We have found one solution to this problem by introducing an intermediate modelling abstraction which we refer to here as a ‘panel’ element.

A Large Panel Element for Practical Finite Element Modelling

T Q Li and T Ward Arup. London, UK.

2

The proposed element is based on the principle of substructuring: the large panel is meshed internally into a set of smaller shell elements. From the consideration of a user, the panel element is a four node shell element capable of modelling large physical objects such as a shear wall or a floor slab with a single element. From the consideration of the finite element analysis solver, the panel element is a ‘condensed’ mesh of smaller shell elements. The panel element handles the intermediate detail between the two interfaces. This paper is based upon the work in reference [1], but further verification and validation works have been conducted to demonstrate the practical and time saving benefits of using this panel element. A ‘drilling’ stiffness applied to a shell element can become problematic when a shell element becomes small. We commonly observe in practical cases where a drilling moment can lead to poor results if applied directly to a small shell element. One typical example is a moment applied via a beam over a single torsional connection. The larger size of a panel element makes it possible to more conveniently accommodate drilling moments over a wider area and predict drilling stiffness more accurately. This paper recommends a method to calculate the drilling stiffness that makes use of the larger area available to a panel element. We have found this to retain greater accuracy than previously proposed methods [2-7] in which the drilling stiffness is more sensitive to individual element size. To account for when a drilling stiffness is not required we make the inclusion of drilling stiffness optional by taking the choice as a user defined parameter. Finally, a panel element can make use of the performance benefits available by modelling at a slightly higher level of abstraction. Three benefits arise. The data associated with each panel element is independent and so the processing for any panel can be executed independently in a parallel way. We also find a reduction in the repetitive calculation of any individual panel since it is only necessary to process a single shell element for each panel. Finally, we commonly observe a second reduction possible at the structure level where we have found it very likely that in a large model, many identical panels exist but where only one panel needs to be processed. 2 The panel element 2.1 Meshing of the panel element The panel element represents a substructure with internal meshing of shell elements at an arbitrary degree of refinement. Example refinements may represent a single panel by 4×4, 4×6 or 6×6 etc. shell elements. We restrict the discussion in this paper to a meshing scheme of 4×4 Quad8 shell elements, whose ‘internal’ assembly is shown in Figure 1(a). We refer to all nodes in the panel interior as ‘internal’, and reserve the name ‘external’ for all remaining nodes that exist on the element edges

3

and corners. Figure 1(b) shows the active ‘external’ nodes of the panel element as given to the global assembly. Figure 1(c) shows the panel element as seen by the end users, a typical 4 node element. All degrees of freedom (d.o.f.) associated with internal nodes are ‘condensed’ during the panel element construction and do not appear in the global analysis. As a consequence, the total number of d.o.f. for a model using panel elements will be significantly less than for an equivalent model using smaller shell elements. The derivation of any individual shell element stiffness matrix [Ke] can be found from a variety of sources e.g. [8] and is not discussed further. This paper concentrates on the idea of the panel element as a larger entity: the assembly of the panel element stiffness matrix, the assembly of a corresponding load vector and a method to construct drilling stiffness from this internal arrangement.

(a) Meshing of panel element

4

(b) Panel element to the solver in global analysis

(c) Panel element to the users

Figure 1 Panel element and the meshing scheme

5

2.2 Stiffness matrix of the panel element Given a known stiffness matrix of shell elements [Ke] and the element numbering as labelled in Figure 1(a), the stiffness matrix of the mesh in Figure 1(a) can be assembled using the C++ routine shown in Figure 2. Five d.o.f. are used at each node, except for each corner node that has six d.o.f. void AssembleK(double** K) { for(int iE = 0; iE < 16; ++iE) { for(int iR = 0; iR < 40; ++iR) { for(int iC = 0; iC < 40; ++iC) { const int iTR = vTopo[iE][iR/5]; const int iTC = vTopo[iE][iC/5]; int iRG = 5*(iTR-1) + iR%5; int iCG = 5*(iTC-1) + iC%5; if(iTR >= 62) iRG = 6*(iTR-1) + iR%5 - 61; if(iTC >= 62) iCG = 6*(iTC-1) + iC%5 - 61; K[iRG][iCG] += Ke[iR][iC]; } } } } where:

vTopo: A array holding the topologies (node numbers) of each of the small shell elements as shown in Figure 1(a), e.g. vTopo[1][2] gives the third node number of element 2.

Ke: A shell element stiffness matrix with size 40×40. K: An uncondensed stiffness matrix of the meshed model.

Figure 2 Assemble of stiffness matrix for the meshed model

Using the numbering system shown in Figure 1(a), the internal and external degrees of freedom are separated automatically in the stiffness matrix [K], with internal d.o.f. followed by external d.o.f. The total number of d.o.f. for the meshed model is Nt = 329 and the number of internal d.o.f. Ni = 165. The uncondensed stiffness matrix [K] and the corresponding load vectors can be condensed using the C++ routine in Figure 3. void CondenseMatrix(std::vector<std::vector<double>>& K,

std::vector<std::vector<double>>& P, int Nt, int Np, int Ni)

{

6

for(int k = 0; k < Ni ; ++k) { for(int i = k + 1; i < Nt; ++i) { const double dFactor = K[i][k]/K[k][k]; for(int j = k; j < Nt; ++j) K[i][j] = K[i][j] - dFactor*K[k][j]; for(int j = 0; j < Np; ++j) P[i][j] = P[i][j] - dFactor*P[k][j]; } } for(int k = Ni-1; k > 0 ; --k) { for(int i = k-1; i >= 0; --i) { double dFactor = K[i][k]/K[k][k]; for(int j = 0; j < Nt; ++j) K[i][j] = K[i][j] - dFactor*K[k][j]; for(int j = 0; j < Np; j++) P[i][j] = P[i][j] - dFactor*P[k][j]; } } for(int i = 0; i < Ni; ++i) { const double dFactor = K[i][i]; for(int j = 0; j < Nt; j++) K[i][j] = K[i][j]/dFactor; for(int j = 0; j < Np; j++) P[i][j] = P[i][j]/dFactor; } } where:

Nt: The number of d.o.f. in meshed model (329 in this example). Ni: The number of internal d.o.f. (165 in this example). Np: The number of load cases considered. K: The stiffness matrix. At the routine start, K contains the stiffness

matrix of the meshed model, assembled using the routine in Figure 2. At the routine end, K contains

�[1] [K]ib′

[0] [K]bb′� = �

[1] [K]ii−1[K]ib[0] [K]bb − [K]bi[K]ii−1[K]ib

� (1)

P: The load matrix. Each column represents a load case. At the routine

start, P holds the load matrix for the meshed model, at the routine end, P contains

�[u]ii

[P]b′� = �

[K]ii−1[P]i[P]b − [K]bi[K]ii−1[P]i

� (2)

7

[K]ib′ : An external node displacement to internal node displacement conversion matrix.

[K]bb′ : A condensed stiffness matrix of the panel element containing external d.o.f. only.

[u]ii: An internal node displacement generated by internal node loads with external nodes being fixed.

[P]b′ : A condensed load matrix corresponding to the condensed stiffness matrix [K]bb′ for global analysis.

Figure 3 Condense of panel element stiffness matrix

All useful matrices for both global and panel element analyses can be obtained from the above operations. For this example, the number of external d.o.f. Ne = Nt - Ni = 164 and represents a saving of about half the total number of d.o.f. when compared to the meshed model that would otherwise contribute to the global structure stiffness matrix.

2.3 Load vector for the panel element To represent a face load as equivalent nodal loads, all nodes of the mesh in Figure 1(a) must be considered. These are to be subsequently condensed to external nodes and so be compatible with the condensed stiffness matrix. Given a known nodal load vector for the meshed model {P}, we partition the vector into an internal block {P}i followed by an external block {P}b. The condensed nodal load vector {P}b′ at external nodes for global analysis becomes

{P}b′ = {P}b−[K]bi[K]ii−1{P}i (3) where:

{P}b The external block of an equivalent nodal load vector to the face load. The length is 164 for our example.

{P}i The internal block of an equivalent nodal load vector to the face load. The length is 165 for our example.

{P}b′ A condensed load vector at external nodes for global analysis. The length is 164 for the example discussed.

In practice {P}b

′ is not derived directly from Equation (3), better efficiency can be gained by calculating {P}b

′ during the same operation as the stiffness matrix condensation of Figure 3 as given by Equation (2). If multiple load cases exist, each load vector may be represented as a single column of a load matrix. The code routine given in Figure 3 considers such a scenario. 3 Analysis results of panel element 3.1 Nodal displacements

8

Given a condensed stiffness matrix [K]bb

′ and a condensed load vector {P}b′ , a global stiffness matrix and global load vectors can be assembled in the usual way [8]. The final external node displacements {u}b are given directly from a global analysis. The final displacements for the internal nodes {u}i′ can be determined from the sum

{u}i′ = [K]ii−1[P]i + [K]ii−1[K]ib{u}b (4) where:

{u}i′ A final nodal displacement vector for internal nodes. {u}b A final nodal displacement vector for external nodes.

The first term of the sum in Equation (4) represents the contribution to internal node displacement as generated by loading at internal nodes while in the absence of any external node displacement. The second term represents the contribution to internal node displacement as generated by displacements at external nodes in the absence of any other loads. 3.2 Nodal forces/moments & total forces/moments at edges Once a global solution gives nodal displacements for all external nodes, forces and moments at external nodes can be calculated from

{F}b = [K]bb′ {u}b (5) where:

{F}b A force vector representing the forces/moments at external node. {u}b An external node displacement vector.

{F}b can be used to calculate the total forces and moments at each of the four edges of the panel element. In engineering design, it may often arise that a single reported value for the total force or moment at a section is more useful than a detailed stress distribution across the section. One example may be while calculating the required reinforcement for a wall or a slab. A large panel element is a natural fit, where should a single panel element represent a wall or slab, the total forces and moments at a panel edge immediately represent the total forces and moments for a section through the structural object. Edge forces and moments can be calculated from the known forces and moments at external nodes given by Equation (5). Figure 4 shows one option for a sign convention.

9

(a) Total edge forces

(b) Total edge moments

Figure 4 Sign convention of total forces and moments at panel element edges 3.3 Forces, moments, stresses & strains within the element When a panel element is used to model a shear wall, the total forces and moments at a panel edge is typically sufficient for the purposes of engineering design. When a panel element is used to model floors and other structural objects subjected to out-of-plane bending, greater detail as to the forces, moments, stresses and strains within a panel element will typically be required. Given that a panel element is represented internally as a mesh, such information is known as part of the internal solution and can be used. For example, strains within the panel may be calculated from known internal and external displacements. 4 Drilling stiffness 4.1 Introduction

10

A typical shell element has five degrees of freedom at each node: three translations and two rotations. Rotations normal to the element plane, commonly referred to as ‘drilling d.o.f.’ are not included. If such shell elements are used in an otherwise unrestrained global assembly with six free degrees of freedom (three translations and three rotations) some additional method of constraint will be necessary. If a connected element is exactly co-planar, it is straightforward to automatically suppress the presumably unwanted drilling d.o.f. at the connected nodes from the global analysis. Alternatively, a small artificial drilling stiffness may be assumed and so keep all six d.o.f. in the global assembly. Should two connecting elements be perpendicular, all six degrees of freedom at the connecting nodes will have an associated stiffness. A greater problem occurs when shell elements are connected in planes that are nearly coplanar. If the angle between element normals is small, but large enough to be justified in the analysis, the small stiffness contribution in an otherwise unrestrained drilling direction is likely to lead to poor conditioning in the resulting structure matrix. The alternative approach to suppress the drilling stiffness will likewise lead to unsatisfactory results. Solutions introduced by Allman and Cook include the drilling d.o.f. as a part of the element formulation [2-7]. The technique is to include four mid-side nodes with transverse side displacements that are instead expressed as four rotational d.o.f. at element corners. The method is effective under certain conditions: elements are reasonably coplanar [9], spurious modes are handled effectively [7] and where the element size is sufficiently large with respect to the effective area of an applied moment. In practice however we observe a conflict between the requirement to sufficiently refine element mesh sizes and yet keep element sizes large enough to accommodate the desire to apply drilling moments over single connections for practical modelling. We have found that overly small elements can be a poor choice of compromise in practical cases. For an Allman-Cook element [7], Figure 5 demonstrates the effect element mesh size has on the observed value of rotation at a node when a moment is applied only at that node. For typical shell formulations, the size of a connecting beam is not known during the shell element formulation. A panel element, working at a higher level of model abstraction, has a greater opportunity to consider the size of a connecting beam section when deriving the drilling stiffness. In this paper a beam section size is taken as an additional parameter to the panel element.

11

(a) The model (thickness = 0.5 m, E = 2.8×1010 Pa, Moment = 1×106 N.m)

(b) Rotation versus element size for a constant applied moment

Figure 5 Effect of element size on drilling stiffness using Allman-Cook method

12

4.2 Derivation of drilling stiffness Figure 6 demonstrates the significance that an area over which a drilling moment is applied has on drilling stiffness. In this example, a square 6×6 meter panel is modelled by a finer mesh of Quad8 shell elements. A set of drilling moments are applied to the top right hand corner, varying with respect to a rigid area over which the moment is applied. The relationship between the rotational stiffness of the panel and the area of the applied moment is drawn in Figure 6 (b).

(a) The model (thickness = 0.5 m, E = 2.8×1010 Pa)

13

(b) Drilling stiffness versus moment application area

Figure 6 Effect of moment application area to the drilling stiffness

The derivation of the drilling stiffness for a panel element is illustrated in Figure 7. A ‘dummy’ node lying within the element plane and containing two translational d.o.f. is placed near each corner. In this way, an area over which a drilling moment is to act can be flexibly specified by defining the distance between the dummy node and the element edges. For example, this area may be set as the equivalent section area of a connecting beam. The displacement of each dummy node is constrained by the linear interpolation of adjacent edge node displacements. For example, the displacements of dummy node 3 from Figure 7 are constrained by the following equations:

ux = u1.x + (u2.x−u1.x)b

(𝑑 − 𝑏)

uy = u3.x + (u4.x−u3.x)a

(𝑐 − 𝑎) (6)

where:

a, b The nodal spacing at horizontal and vertical edges respectively. c, d The width and height of the rectangular moment application area. ux, uy Horizontal and vertical displacements of the dummy node

(constrained by the edge displacements and they are not independent). u1.x, u2.x Horizontal displacements at nodes 1 & 2 as shown in Figure 7. u3.y, u4.y Vertical displacements at nodes 3 & 4 as shown in Figure 7.

14

Each dummy node is linked with its respective corner node by a rigid beam in order that each corner node has a drilling stiffness equivalent to that of the moment applied through the area cd. In practice, a beam element with a large in-plane bending stiffness and no axial or out-of-plane bending stiffness is used to connect the corner node with the dummy node. We have found this simpler approach to be sufficient in practice, keeping corner nodes as independent d.o.f. and avoiding the need for additional rigid constraints. Since dummy nodes are dependent constraints, no additional d.o.f. will appear in global stiffness assembly, and their introduction does not change the total count of d.o.f. in the panel element.

(a) The panel element with dummy nodes and rigid bars

15

(b) Generation of drilling stiffness

Figure 7 Generation of drilling stiffness

5 Worked examples The proposed panel element has been implemented in the structural finite element software Oasys GSA [10]. Two models using the software are presented here: a linked shear wall and a high rise building core. The geometry for each example has been separately constructed using both Panel and Quad8 shell elements, illustrated in Figures 8 and 9 respectively. The displacements at each floor level are summarised in Table 1. Both alternatives presented similar results, while use of the Panel element retained many of the practical benefits of using large elements. The time taken to analyse both examples on a single core PC is given in Table 2. For both examples, the difference between the Panel and Quad8 equivalent was roughly a factor of two. For larger models, or if a parallel process was used, the potential benefits may be higher.

16

(a) Panel element model (b) shell element model

total height = 40 m; total width = 14 m; link beam length = 2 m thickness = 200 mm; E = 2.8x1010 N/m2 point load = 30 kN

Figure 8 Linked shear wall modelled by panel and shell elements

17

(a) Panel element model (b) shell element model

total height = 40 m; 6x6 meter square in plan; wall thickness = 200 mm E = 2.8x1010N/m2 point load (2 at each level) = 2 × 30 kN

Figure 9 Elevation of square core modelled by panel and shell elements

Table 1 Comparison of horizontal displacements of panel and shell element models

Level Linked Shear Wall Square Core panel shell % panel shell %

1 0.192 0.197 -2.49% 0.230 0.236 -2.65% 2 0.549 0.550 -0.32% 0.654 0.661 -1.14% 3 1.025 1.020 0.43% 1.240 1.249 -0.71% 4 1.573 1.560 0.82% 1.951 1.961 -0.52% 5 2.156 2.133 1.07% 2.753 2.764 -0.42% 6 2.750 2.716 1.24% 3.617 3.630 -0.35% 7 3.335 3.290 1.36% 4.520 4.534 -0.31% 8 3.902 3.846 1.47% 5.443 5.458 -0.28% 9 4.448 4.380 1.56% 6.371 6.388 -0.26% 10 5.008 4.923 1.71% 7.314 7.329 -0.21%

Note: see Figures 8 & 9 for the dimensions and material properties etc. of the models

18

Table 2 Computing time of panel and shell element models (second) panel shell %

Linked shear wall model 0.245 0.405 60.5% Core model 0.452 0.827 54.7%

6 Conclusion For the engineer, being able to work at a higher level of abstraction presents the immediate benefit that the model is easier to construct, manipulate and maintain. Results may be presented at a level that matches structural objects (walls, floors) and total forces and moments are immediately available at panel edges that may more naturally reflect sections of structural objects. The size of saved result data is also reduced. Other benefits from working with larger objects also become apparent: a large element can better accommodate the possible sizes of a connecting beam and so retain the benefits of being able to do so over a single connection. To make use of this advantage, the proposed method can accurately predict drilling stiffness within the panel element. The method is able to make greater use of the true area over which an applied drilling moment may act. For the finite element solver, the use of larger, repeated structural objects provides a natural opportunity for performance improvements. An immediate observation is that while panel elements are larger, resulting accuracy is not compromised. A panel element handles the intermediate detail and as a part of this handling, passes a reduced set of d.o.f. for global assembly. In our example the reduction was approximately a factor of two. Some of this reduction is traded for the additional work incurred in the construction of the panel element, but it is here where the repeated use of independent panel elements provides opportunities for both deduplication of work and parallelism. References [1] T.Q. Li, “A panel element for parallel finite element analysis”, in Proceedings of

the Third International Conference on Parallel, Distributed, Grid and Cloud Computing for Engineering, B.H.V. Topping, P. Iványi, (Editors), Civil-Comp Press, Stirlingshire, United Kingdom, paper 37, 2013. doi:10.4203/ccp.101.37

[2] D.J. Allman, “A compatible triangular element including vertex rotation for plane elasticity analysis” Computers and Structures, 19:1-8, 1984

19

[3] D.J. Allman, “A Quadrilateral finite element including vertex rotation for plane elasticity analysis”, International Journal for Numerical Methods in Engineering. 26:717-730, 1988

[4] D.J. Allman, “Evaluation of the constant triangle with drilling rotations”, International Journal for Numerical Methods in Engineering. 26:2645-2655, 1988

[5] R.D. Cook, “On the Allman triangle and a related quadrilateral element”, Computers and Structures, 2:1065-1067, 1986

[6] R.D. Cook, “A plane hybrid element with rotational d.o.f. and adjustable stiffness”, International Journal for Numerical methods in Engineering, 24:1499-1508, 1987

[7] H Richard et al, “A refined four-noded membrane element with rotational degrees of freedom”, Computers and Structures, 1:75-84, 1986

[8] O.C. Zienkiewicz & R.L. Taylor, “The Finite Element Method for Solid and Structural mechanics”, Sixth edition, 2005

[9] R.D. Cook, D.S. Malkus, M.E. Plesha & R.J. Witt, “Concepts and Applications of Finite Element Analysis”, Fourth Edition, 2001

[10] Oasys Software. GSA Version 8.6. Arup, 13 Fitzroy Street London W1T 4BQ, 2012