Shallow Spherical Shell Rectangular Finite Element for Analysis of Cross Shaped Shell Roof

Transcription

1 Electronic Journal of Structural Engineering, 7(7) Shallow Spherical Shell Rectangular Finite Element for Analsis of Cross Shaped Shell Roof A.I. Mousa Associated Professor of Civil Engineering, The Universit of Gaza, Palestine M.H. El Naggar Professor of Civil Engineering, The Universit of Western Ontario, London, Canada ABSTRACT: A new spherical rectangular finite element based on shallow shell formulation is developed in this paper. The element has si degrees of freedom at each corner node, five of which are the essential eternal degrees of freedom and the additional sith is associated with the in-plane shell rotation. The displacement fields of the element satisf the eact requirement of rigid bod modes of motion. The element is based on independent strain assumption insofar as it is allowed b the compatibilit equations. The element developed herein is first validated b appling it to the analsis of a benchmark problem involving a standard spherical shell with simpl supported edges. A reliable finite element package program, ANSYS, is used for structural analsis. The results of the analsis showed that reasonabl accurate results were obtained even when modeling the shells using fewer elements compared to other shell element tpes. The element is then used in a finite element model to analze cross shaped spherical roof structures. The distribution of the various components of deflection and stress is obtained. Furthermore, the effect of introducing circular arched beams as stiffeners spanning the two diagonall opposite end corners is investigated. It is found that the stiffeners reduced the deflections and the stresses in the roof structure b considerable value. INTRODUCTION Considerable attention has been given to appling the finite element method in the analsis of curved structures. Grafton and Strome [] developed conical segments for the analsis of shell of revolution. Later Jones and Strome [] modified the method and used meridional elements which were found to lead to considerabl improved results for the stresses. Curved rectangular and clindrical shell elements were also developed (Connor and Brebbia [] Cantin and Clough [] and Sabir and lock [5]). However, to model a shell of spherical shape using the finite element method triangular and rectangular spherical shell elements are needed. Several spherical shell elements have been developed to analze shells of spherical shape. Most notabl, the higher order elements of Lindberg et al [], Yang [7] and Dawe [8] have resulted in an improvement of the accurac of the results. However this improvement is achieved at the epense of increased computational efforts and storage. Meanwhile, a simple alternative strainbased approach has been used to develop curved elements. In this approach, the eact terms representing all the rigid bod modes and the displacement functions representing the element strain are determined b assuming independent strain functions insofar as it is allowed b the compatibilit equations (Ashwell and Sabir [9]). This approach has been emploed successfull in the development of different clindrical, hperbolic and conical shell elements b Sabir et al

2 Electronic Journal of Structural Engineering, 7(7) [9-] and b Mousa et at [-5]. These elements were found to ield faster convergence when compared with other available finite elements. The strain-based approach was also used to develop rectangular and triangular spherical shell elements (Sabir [], Sabir and Djoudi [7] and Mousa [8]). These spherical elements possess onl the five essential eternal nodal degrees of freedom at each corner node and were found to have ecellent convergence. Most spherical shell structures are supported b circular arched beams. In finite element analsis, these beams are usuall modeled using finite elements having si degrees of freedom at each end node to represent their general three dimensional forms of deformation. These si degrees of freedom are the usual five essential eternal degrees used in shell analsis together with a sith representing the rotation about the normal to the shell surface. It is therefore of interest to introduce in the shell an additional degree of freedom (as a sith degree) representing this in plane (as drilling) rotation in order to enhance the compatibilit between the spherical shell and curved beam elements. The strain based approach, also known as the Cardiff Approach, is emploed in the present stud to develop a rectangular strain-based spherical shell element has an in-plane rotation as a sith degree of freedom. A shallow shell formulation is used to obtain the displacement fields. The element has si degrees of freedom at each of the four corners, the rigid modes are eactl represented and the straining is based on independent strains enforcing the elastic compatibilit equations. The element developed herein is first tested b appling it to the solution of a benchmark shell problem, and is then used to analze a four-corner cross shaped roof structure. A reliable finite element package program, ANSYS, is used for structural analsis. The distribution of the various components of stress and deflection is obtained. The stiffening effect of circular arched central beams on the performance of the roof is also investigated. DEVELOPMENT OF DISPLACEMENT FUNCTIONS FOR THE NEW RECTANGULAR SPHERICAL SHELL ELEMENT. Theoretical considerations A rectangular shallow spherical shell element and the associated curvilinear coordinates are shown in Figure. Fig. Coordinate Aes for Rectangular Spherical Shell Element For the shown sstem of curvilinear coordinates, the simplified strain displacement relationship for the spherical shell elements can be written as: u ε = + k w r w =, v w u v, ε = +, ε = + r w w k =, k = Where u, v and ε w are the displacements in the, ε ε, and z aes; and are the in-plane direct and shearing strains; k, k, and k are the changes in-direct and twisting curvatures and r is the principal radii of curvatures. The above si components of strain can be considered independent, as the are a function of the three displacements u, v and w and must satisf three additional compatibilit equations. These compatibilit equations are derived b eliminating u, v and w from Equation, hence, the are: ε k k + ε / k / = / k / = ε + k r + k () r = ()

3 Electronic Journal of Structural Engineering, 7(7) In order to keep the element as simple as possible and to avoid the difficulties associated with internal non-geometric degrees of freedom, the developed element should possess si degrees of freedom at each of the four corner nodes: u, v, w, θ, θ and φ. Thus, the shape functions for a rectangular element should contain twent four independent constants. To obtain the displacement fields due to rigid bod movements, all the si strains given b Equation are equated to zero and the resulting partial differential equations are integrated to ield: u = a r r r + r v = a a5 a r r r r + w = a These displacement fields are due to the si components of the rigid bod displacements and are represented in terms of the constants a to a. If the element has si degrees of freedom for each of the four corner nodes, the displacement fields should be represented b twent four independent constants. Having used si constants for the representation of the rigid bod modes, the remaining eighteen constants are available for epressing the displacements due to the strains within the element. These constants can be apportioned among the strains in several was. For the present element, the following is proposed: ε ε = a 7 a r = a a r ε = a + a = a a 8 k + 5 k = a k = a + [a ] () () Equation is derived b first assuming the unbracketed terms and adding the terms between brackets to satisf the compatibilit condition (Equation ). It is then equated to the corresponding epressions in terms of u, v and w from Equation and the resulting equations are integrated to obtain: u = a7 8 5 r r r r r v = a r r r r r r w = a r The complete displacement functions for the element are obtained b adding the corresponding epressions for u, v and w from Equations and 5. The translational degrees of freedom for the element are u, v, w. The three rotations about the, and z aes are given b: w θ = = a w θ = = a () (5)

4 Electronic Journal of Structural Engineering, 7(7) v u φ = = a r r 8 5 r 7 9 r The Stiffness matri [K] for the shell element is then calculated in the usual manner, i.e., K T T [ C ] { B DBdv }[ ] = C where B and D are the strain and elasticit matrices, respectivel, and C is the matri relating the nodal displacements to the constant a to a. CONSISTENT LOAD VECTOR The simplest method to establish an equivalent set of nodal forces is the lumping process. An alternative and more accurate approach for dealing with distributed loads is the use of a consistent load vector which is derived b equating the work done b the distributed load through the displacement of the element to the work done b the nodal generalized loads through the nodal displacements. If a rectangular shell element is acted upon b a distributed load q per unit area in the direction of w, the work done b this load is given b: P = a b a b qwdd where a and b are the projected half length of the sides of the rectangular element in the and directions, respectivel. If w is taken to be represented b: {w} = [N T ]{a}=[n T ][C - ]{d} where N T for the present element is given b (see Equations and 5): (7) (8) (9) T N = [,,,,,,,,,,,,,, ] {a} is the vector of the independent constants, [ C ] is the inverse of the transformation matri and {d} is a vector of the nodal degrees of freedom. The work done b the consistent nodal generalized force through the nodal displacements {d} is given b: P = {F} T {d} Hence, from Equations 8, 9 and, the nodal forces are obtained, i.e. a b T T {} F = q N [ C ]{}[ a N] dd a b Equation gives the nodal forces for a single element; and the nodal forces for the whole structure are obtained b assembling the elements nodal forces.. PROBLEMS CONSIDERED. Spherical cap subjected to a point load: The geometr of the spherical shell considered in this section is shown in Figure. The shell is simpl supported on the boundaries of a square plan form such that the normal displacement is zero along the edges, while the in-plane displacements normal to the edges are freel allowed to take place. The shell has the following dimensions and elastic properties: a = b = mm, r = mm, t =.5 mm, E = 77 kg/mm and ν =.. It is subjected to a point central load p = 5. kg. The smmetr of the shell and loading allows consideration of a quadrant of the shell to be analzed. This spherical cap was analzed previousl b Gallagher [9] using a rectangular shell finite element with linear membrane displacement function for u and v and w. () ()

5 Electronic Journal of Structural Engineering, 7(7) z b a (a) Shell Geometr (b) 8 8 mesh Fig. Spherical Cap that both elements require onl mesh size to converge to acceptable results with a difference of less than % in the case of the triangular element developed b Sabir and Djoudi [7] and less that.5% in the case of present element compared with the series solution, while it was reported that Gallagher element [9] gives an error of more than % for the same mesh size. Further investigations on the deflection are shown in Figure, which indicates ecellent agreement between the results obtained from the present element and the series solution for the variation for normal deflecting along a centre line. The direct stress resultant, N, and the bending moment, M, are computed along the centre line "o" in the direction. The results obtained from the (8 8) meshes are presented in Figures 5 and for N and M, respectivel. The results obtained from the series solution are also shown in the figures for comparison. It is noted that the results for the membrane stress resultants agree well with those obtained from the series solution. Yang [7] has solved this problem b using a rectangular finite element, where all the displacement components are represented b cubic polnomials and also proposed a series (closed form) solution. Later on, Dawe [8] used a quintet order triangular element, having 5 degrees of freedom to analze the same shell. He reported that the results obtained were more accurate than those given b Yang [7] for the same number of elements. However, this element has an ecessivel large number of degrees of freedom, which would require large computational effort to perform the analsis using this element. In comparison, the present element includes onl si degrees of freedom. The spherical shell described above is analzed here using: the triangular spherical shell element that has onl five degrees of freedom at each corner node, which is given in reference [7]; and the rectangular spherical shell element developed in the current stud. The results from both analses are compared together and with those obtained form the series (closed form) solution given b Yang [7]. Convergence tests were carried out for the normal deflection at the centre of the shell. Figure shows W(cm) W(cm) Present elem. Ref (7). Series solution 8 Number of elements Fig. Convergence of W at the Center. Present elem.. Ref(7).8 Series solution... 5 Fig. Variation of W Along "o" 5

6 Electronic Journal of Structural Engineering, 7(7) M(kg.cm/cm) N(kg/cm) Present elem. Ref(7) -5 Series solution - Fig. 5 Variation of N Along the Center Line.5 Present elem Ref(7).5 Series solution Fig. Variation of M Along the Center Line mm. The Young's modulus of the shell material, E = 9 kg/cm and its Poisson's ratio, ν =.. The dimensions a and b are taken to be. m and. m, respectivel. The cross shaped roof is loaded b a normal uniforml distributed load of 5 kg/m and it is assumed to be fied to rigid supports at all four end corners. Due to smmetr and uniform loading, onl one quadrant of the shell is analzed. The conditions for smmetr along the centre line require that the in-plane displacements and rotations normal to the plane of smmetr are zero. C = m r Roof Geometr r These results clearl show that the response of spherical shell can be adequatel analzed using a reasonable number of elements to obtain convergence of deflection and stresses. The results also confirm the suitabilit of the proposed rectangular shell element for the analsis of spherical shell roof Structures in the form of a cross shaped. This form of structure is often used b architects to roof ehibition halls and public buildings. The authors found that one of this tpe of roof was implemented in the field. To the knowledge of the authors, no known solution to this problem is available in the published literature. Fig. 7a Cross Shaped Spherical Shell Roof a b b quadrant a a. Analsis of Spherical Cross Shaped Shell Roof: The spherical shell roof structure considered is shown in Figure 7. It has a central rise above the cornered supports of. m and a total horizontal span between each of the two diagonall opposite supports of 8 m. it is spherical in form having a radius of curvature of.5 m in each of the principle directions and and a thickness of Plan View Fig. 7b Cross Shaped Spherical Shell Roof b

7 Electronic Journal of Structural Engineering, 7(7) Two models (shown in Figure 8) are used to eamine the effect of the mesh size needed for the analsis of one quarter of the shell on the calculated deflections and stresses. In the first model, a moderate mesh comprising eight elements is considered (Figure 8b), while the second model considered a finer mesh of hundred eight elements (Figure 8c). (c) Fine Mesh Fig. 8 Configuration of Mesh Used in the Analses (a) Meshed Roof The analsis was performed and the quantities necessar for design are calculated including the normal deflections, the direct membrane stress resultants in the and directions as well as the bending stress resultant M along the central line o. Figure 9 shows the distribution of the normal deflections of the roof. The hoop stress, N, and the direct stress, N, are presented in Figures and, respectivel. Figure displas the bending stress resultant, M. (b) A Moderate Mesh Deflection(cm ).E Fig. 9 Radial Deflection along o 7

8 Electronic Journal of Structural Engineering, 7(7) N(kg/m).E+ N(kg/m).E+ M(kg.m/m) Fig. Hoop Stress N along o 88 8 Fig. Direct Stress N along o Fig. Bending Moment M along o Figure 9 shows that the normal deflection along o is almost constant and downward throughout the central portion, a, of the shell. Thereafter, the deflection decreases and becomes upward reaching a maimum of almost the same magnitude as the downward deflection at the centre of the shell. The deflection decrease over the last portion, a, of the shell until it becomes zero at the corner support. support along o, but becomes tensile over a small length between the two regions. On the other hand, the direct stress resultant N increases steadil at a small rate along o over the central region but increases rapidl over the region near the support, as shown in Figure. This behavior is consistent with the membrane theor, which gives an almost constant stress resultant N in the central region (near the crown) and then increases rapidl towards the support. It is also noted from Figures and that the magnitude of the maimum stress resultant N is much larger (one order of magnitude) than the maimum of the hoop stress, N. Figure shows that the bending stress resultant M along o is generall small in the central region, for a length of about two thirds of the span. Afterwards, it changes sign for a small length but increases sharpl towards the support with a maimum value at the fied support, as epected. All the above figures show clearl that, for all practical purposes, the results obtained from the two models do not differ significantl, confirming again the efficienc of the developed element. However, the fine mesh was used in the following section.. Cross shaped shell with circular arched beam stiffeners: In practice, shells having large spans are stiffened b beams. To investigate the effect of such stiffening beams on the deflections and stresses, the case of cross shaped roof structure stiffened b arched beams spanning between two diagonall opposite supports is considered. To investigate the effect of the depth of the diagonal arched beam stiffeners on the resulting deflections and stresses, two cases were considered d = and mm, while the width, B is kept at 5 mm. The laout and geometr of the problem considered are shown in Figure. Figure shows that the hoop stress resultant N is compressive over the central part and at the 8

9 Electronic Journal of Structural Engineering, 7(7) A The hoop stress, N, and the direct stress, N, are presented in Figures 5 and, respectivel. Figure 7 displas the bending stress resultant, M. A Stiffeners (a) Deflection (m).e No stiffner cm stiffner cm stiffner 8 t d 5 mm beam roof Fig. The Effect of Stiffeners On the Deflection Along the Center Line "o" Section A-A (b) Fig. Cross Shaped Spherical Shell with Stiffening Beams, a) laout; b) geometr The developed strain based rectangular spherical element is used in the analsis of the spherical shell and the diagonal arched beams are idealized b a si nodal degrees of freedom beam element. The stiffness matri of the beam is based on cubic variation of both the vertical and horizontal displacements and linear variation of the aial displacement and the angle of twist. Thus, the beam element and the strain-based rectangular element have the same degrees of freedom and complete compatibilit is ensured when these elements are used in the assembl of the overall structural matri. After assembling the global stiffness matri and establishing the load vector using Equation, the derived equilibrium equations were solved and the response was obtained. The results reported here include the normal deflections, the direct membrane stress resultants in the and directions as well as the bending stress resultant M along the central line o. Figure shows the distribution of the normal deflections of the roof. N(kg/m).E+ N(kg/m).E No stiffner cm stiffner cm stiffner 8 No stiffner cm stiffner cm stiffner Fig. 5 The Effect of Stiffeners On (N) Along the Center Line "o" 8 Distance from center Fig. The Effect of Stiffeners On (N) Along the Center Line "o" 9

10 Electronic Journal of Structural Engineering, 7(7) M(kg/m).E No stiffner cm stiffner cm stiffner Fig. 7 The Effect of Stiffeners On (M) Along the Center Line "o" Figure shows that the stiffeners resulted in significant reduction of roof deflections. For eample, the maimum deflection (at the centre) decreased b 5% for the shallow beam and 8% for the deeper beam case, compared with the case of the shell having no stiffeners. In addition, the shell stress resultants decreased significantl. For eample, Figure 5 shows the direct stress resultant N has decreased b about % and %, for the cases of d = mm and d= mm, respectivel. Similar observations can be made from Figure with regard to the direct stress resultant N along o. Likewise, the bending stress, M, has been reduced significantl due to the use of the stiffeners as shown in Figure 7. A reduction of about % is achieved with the thinner beam and about % with the thicker beam. 5. CONCLUSIONS: A new spherical shell rectangular strain-based finite element was developed using shallow shell formulations. The element has si degrees of freedom at each corner node, the essential five eternal degrees of freedom as well as an additional sith degree of freedom representing the in-plane rotation. The element is therefore full compatible with and can be used in conjunction with arched beam elements having all the essential si degrees of freedom. The element was verified and validated through the analsis of a simpl supported square spherical shell problem subjected to concentrated load, and 5 comparing the results with the predictions of other approaches. The results of the analsis showed that reasonabl accurate results were obtained even modeling the shells using fewer elements compared to other shell element tpes. The developed element was then used in a finite element model to analze the response of cross shaped spherical roof structure. The distribution of normal deflection and the various components of stresses are evaluated. The results showed that this tpe of shell roof ehibits large deflections at the centre and large stresses at the corners because of its non uniform shape and supporting conditions. The response of the roof stiffened using diagonal arched beams was also analzed. The diagonall lined stiffeners were found to considerabl reduce the deflections and stresses of cross shaped spherical roof structures.. REFERENCES:. Grafton, P.E and Strome, D.R. "Analsis of a smmetric shells b the direct stiffness method" AIAA.Vol., 9, PP-7.. Jones, R.E and Strome, D.R "Direct stiffness method analsis of shells of revolution utilizing curved elements "AIAA J., Vol., 9, PP Brebbia, C and Connor, j.j. "Stiffness matri for shallow rectangular shell element" J. Eng. Mech. Div. ASCE 9 EMS, 97, PP-5.. Cantin, G. and Clough, R.W. "A curved clindrical shell finite element". AIAA Journal, vol, 98, PP Sabir, A.B. and lock, A.C "A curved clindrical shell finite element". Int. J. Mech. Sci., 97, P5.. Cowper. G.R lindberg. G.M and Olson, M.D. "A shallow shell finite element of triangular shape". Int. J. solids and structures, Vol., 97, PP Yang, T.Y. "high order rectangular shallow shell finite element" J. Eng. Mech. Div. ASCE 99 EM, 97, P Dawe. D.J. "High order triangular finite element for shell analsis" Int. J. Solids Structures,Vol., 975, PP Ashwell, D.G. and Sabir, A.B "A new clindrical shell finite element based on simple independent strain functions". Int. J. Mech. Sci.,Vol., 97, P7-8.. Sabir A.B. and Charchaechi, T.A. "Curved rectangular and general quadrilateral shell

11 Electronic Journal of Structural Engineering, 7(7) finite elements for clindrical shells ".The Math. of Finite Element and Appli. IV Academic Press 98 P-9.. Sabir A.B and Ramadhani, F. "A shallow shell finite element for general shell analsis" Variation Method In Engineering Proceedings of The nd International Conference, Universit of Southampton. England, 985.Sabir, A.B and Djoudi, M.S "A shallow shell triangular finite element for the analsis of the analsis of hperbolic parabolic shell roof", FEMCAD, Struct. Eng. and Optimization, 99, pp9-5.. Mousa, A.I "Finite Element Analsis of a gable roof" Computational Structural Engineering in Practice, Civil Comp. Press. 998, P5-8.. Mousa. A. I "Finite Element Analsis of groined vault clindrical in plan" AL- Azhar Engineering Journal, Vol.,, pp7-5.. Mousa, A.I, and Sabir, A.B "Finite element analsis of fluted conical shell roof structures", Computational Structural Engineering in Practice, Civil Comp. press- ISRN O , 99, PP Sabir, A.B "Strain based shallow spherical shell element". Proc. Int. Conf on the Math. Finite elements and application, Brunel Universit,997.. Sabir A.B and Djoudi M.S "A shallow shell Triangular Finite Element for the Analsis of spherical shells". Structural Analsis J., 998. PP Mousa, A.I, "Triangular finite element for the analsis of spherical shell structures" UWCC Publications, Internal Report, Universit of Wales, college Cardiff, U.K Gallagher. R.H "The development and evaluation of matri methods for thin shell structural Analsis" PhD thesis, StateUniversit of New York Buffalo, 9. 5