* Corresponding author, Tel: +234-805-741 FINITE ELEMENT ANALYSIS OF A FREEFINITE ELEMENT ANALYSIS OF A FREEFINITE ELEMENT ANALYSIS OF A FREEFINITE ELEMENT ANALYSIS OF A FREE W. O. AjagbeW. O. AjagbeW. O. AjagbeW. O. Ajagbe1,21,21,21,2 DEPARTMENT OF CIVIL 3333 SEGUN LABIRAN AND ASSOCIATES EEEE----mail addressesmail addressesmail addressesmail addresses: 1111 alpha_est1@yahoo.com ABSTRACTABSTRACTABSTRACTABSTRACT The The The The existingexistingexistingexisting appappappapproximate analytical roximate analytical roximate analytical roximate analytical methods methods methods methods stress resultant and the actual three dimensional stress resultant and the actual three dimensional stress resultant and the actual three dimensional stress resultant and the actual three dimensional accurate method of analysis based on accurate method of analysis based on accurate method of analysis based on accurate method of analysis based on finite element methodfinite element methodfinite element methodfinite element methodunknowunknowunknowunknownnnn displacements at each node of a rectangular plate element. displacements at each node of a rectangular plate element. displacements at each node of a rectangular plate element. displacements at each node of a rectangular plate element. the stairs and to compare the finite element analysis with the athe stairs and to compare the finite element analysis with the athe stairs and to compare the finite element analysis with the athe stairs and to compare the finite element analysis with the avariation of stress resultants across a section is nonvariation of stress resultants across a section is nonvariation of stress resultants across a section is nonvariation of stress resultants across a section is nonmethods. This indicates that the effects of axial forces in flights are more than offset by themethods. This indicates that the effects of axial forces in flights are more than offset by themethods. This indicates that the effects of axial forces in flights are more than offset by themethods. This indicates that the effects of axial forces in flights are more than offset by themoments which causes lateral sway of the whole stair towards the upper flight. moments which causes lateral sway of the whole stair towards the upper flight. moments which causes lateral sway of the whole stair towards the upper flight. moments which causes lateral sway of the whole stair towards the upper flight. Keywords:Keywords:Keywords:Keywords: Stairs, Free-Standing, Finite Element, Plate Flexu 1. 1. 1. 1. INTRODUCTIONINTRODUCTIONINTRODUCTIONINTRODUCTION A free-standing stair is unsupported by walat the intermediate landing and attached only to the floor systems at the top and bottom of the (Figure 1). Stairs not infrequently form one of the most prominent visual features of a building, and as such present a challenge to both engarchitects. Unlike the normal floor or roof slab where slight reduction in thickness can seldom be seen, the provision of stairs having the maximum possible slenderness is often visually desirable, making vast difference between clumsiness and gconsequent increase in the amount of reinforcement required to compensate for restricting the effective depth to the minimum possible value is insignificant in relation to that required for the building as a whole, and is clearly outweighed by the enhanced appearance achieved. In recent years free-standing and geometric staircases have become quite popular. Many variations of these staircases exist. A number of researchers have come forward with different concepts in the fields of analytical, numeand of experimental assessments [1However, due to the lack of a simple rational design code, designers are forced to make a conservative design resulting in an unnecessarily heavy looki 741-6361 FINITE ELEMENT ANALYSIS OF A FREEFINITE ELEMENT ANALYSIS OF A FREEFINITE ELEMENT ANALYSIS OF A FREEFINITE ELEMENT ANALYSIS OF A FREE----STANDING STAIRCASESTANDING STAIRCASESTANDING STAIRCASESTANDING STAIRCASE W. O. AjagbeW. O. AjagbeW. O. AjagbeW. O. Ajagbe1111,,,,*, O. A. Rufai*, O. A. Rufai*, O. A. Rufai*, O. A. Rufai2222 andandandand J. O. LabiranJ. O. LabiranJ. O. LabiranJ. O. Labiran3333 IVIL ENGINEERING, UNIVERSITY OF IBADAN, IBADAN, SSOCIATES CONSULTING CIVIL-STRUCTURAL ENGINEERS, Ialpha_est1@yahoo.com, 2222 olalekan.rufai@yahoo.co.uk, 3333 wolelabiran@yahoo.co.uk methods methods methods methods ofofofof analyzinganalyzinganalyzinganalyzing freefreefreefree----standing stairs standing stairs standing stairs standing stairs failfailfailfail to predict the to predict the to predict the to predict the stress resultant and the actual three dimensional stress resultant and the actual three dimensional stress resultant and the actual three dimensional stress resultant and the actual three dimensional behaviorbehaviorbehaviorbehavior ofofofof the stair slab system. the stair slab system. the stair slab system. the stair slab system. AAAA more rationale but simple and more rationale but simple and more rationale but simple and more rationale but simple and finite element methodfinite element methodfinite element methodfinite element method is presentedis presentedis presentedis presented.... PPPPlate flexural analysis late flexural analysis late flexural analysis late flexural analysis displacements at each node of a rectangular plate element. displacements at each node of a rectangular plate element. displacements at each node of a rectangular plate element. displacements at each node of a rectangular plate element. SSSSpreadsheetpreadsheetpreadsheetpreadsheet (FEM 2D)(FEM 2D)(FEM 2D)(FEM 2D)the stairs and to compare the finite element analysis with the athe stairs and to compare the finite element analysis with the athe stairs and to compare the finite element analysis with the athe stairs and to compare the finite element analysis with the analytical method.nalytical method.nalytical method.nalytical method. The study revealsThe study revealsThe study revealsThe study revealsvariation of stress resultants across a section is nonvariation of stress resultants across a section is nonvariation of stress resultants across a section is nonvariation of stress resultants across a section is non----uniform, which is otherwise not recognized by the analytical uniform, which is otherwise not recognized by the analytical uniform, which is otherwise not recognized by the analytical uniform, which is otherwise not recognized by the analytical methods. This indicates that the effects of axial forces in flights are more than offset by themethods. This indicates that the effects of axial forces in flights are more than offset by themethods. This indicates that the effects of axial forces in flights are more than offset by themethods. This indicates that the effects of axial forces in flights are more than offset by themoments which causes lateral sway of the whole stair towards the upper flight. moments which causes lateral sway of the whole stair towards the upper flight. moments which causes lateral sway of the whole stair towards the upper flight. moments which causes lateral sway of the whole stair towards the upper flight. Finite Element, Plate Flexure, Plate Flexure, Beam stiffness is unsupported by walls or beams at the intermediate landing and attached only to the floor systems at the top and bottom of the flight Stairs not infrequently form one of the most prominent visual features of a building, and as such present a challenge to both engineers and architects. Unlike the normal floor or roof slab where slight reduction in thickness can seldom be seen, the provision of stairs having the maximum possible slenderness is often visually desirable, making vast difference between clumsiness and grace. Any consequent increase in the amount of reinforcement required to compensate for restricting the effective depth to the minimum possible value is insignificant in relation to that required for the building as a whole, e enhanced appearance standing and geometric staircases have become quite popular. Many variations of these staircases exist. A number of researchers have come forward with different concepts in the fields of analytical, numerical, design 1, 2, 3, and 4]. However, due to the lack of a simple rational design code, designers are forced to make a conservative sarily heavy looking structure. Ahmed et al [5there are code provisions for ordinary stairs like straight flight, half-turn, dogstanding stairs are based on rigothere is no guideline regarding threinforcement design. 2. WORK DON2. WORK DON2. WORK DON2. WORK DONE ON STAIRCASE DESIGNSE ON STAIRCASE DESIGNSE ON STAIRCASE DESIGNSE ON STAIRCASE DESIGNSA brief review of previous studies on the analysis and design of free-standing stairs is presented in this section. Cusens and Kuang [6], assumed that the staircase can be analyzed by reducing the plate to beam elements. Thus the staspace frame consisting of beams located in a position coincident with their longitudinal axes. The analysis was based on the application of some assumptions and the method of least work. It is widely known that the principle of least work is a powerful tool in solving statically indeterminate structural problems. This is true especially when the structure is a threedimensional frame of which members are subjected to torsional stresses in addition to the conventional bending and axial stresses. Taleb [7] also used the principle of the least work using equations of equilibrium of the entire stair and hence obtaining expressions directly for all redundant Nigerian Journal of Technology (NIJOTECH) Vol. 33 No. 4, October 2014, pp. Copyright© Faculty of Engineering, University of Nigeria, Nsukka, ISSN: 1115 http://dx.doi.org/10.4314/njt.v33i4.2 STANDING STAIRCASESTANDING STAIRCASESTANDING STAIRCASESTANDING STAIRCASE NIGERIA BADAN, NIGERIA wolelabiran@yahoo.co.uk to predict the to predict the to predict the to predict the distributiondistributiondistributiondistribution of any of any of any of any more rationale but simple and more rationale but simple and more rationale but simple and more rationale but simple and late flexural analysis late flexural analysis late flexural analysis late flexural analysis isisisis used toused toused toused to evaluatevaluatevaluatevaluateeee (FEM 2D)(FEM 2D)(FEM 2D)(FEM 2D) is also is also is also is also used to analyze used to analyze used to analyze used to analyze The study revealsThe study revealsThe study revealsThe study reveals that the that the that the that the uniform, which is otherwise not recognized by the analytical uniform, which is otherwise not recognized by the analytical uniform, which is otherwise not recognized by the analytical uniform, which is otherwise not recognized by the analytical methods. This indicates that the effects of axial forces in flights are more than offset by themethods. This indicates that the effects of axial forces in flights are more than offset by themethods. This indicates that the effects of axial forces in flights are more than offset by themethods. This indicates that the effects of axial forces in flights are more than offset by the effect of ineffect of ineffect of ineffect of in----plane plane plane plane Plate Flexure, Beam stiffness 5] observed that although there are code provisions for ordinary stairs like turn, dog-legged and others; free-based on rigorous analysis and guideline regarding their analysis and E ON STAIRCASE DESIGNSE ON STAIRCASE DESIGNSE ON STAIRCASE DESIGNSE ON STAIRCASE DESIGNS A brief review of previous studies on the analysis and standing stairs is presented in this section. Cusens and Kuang [6], assumed that the staircase can be analyzed by reducing the plate to beam elements. Thus the stair will be in the form of a space frame consisting of beams located in a position coincident with their longitudinal axes. The analysis was based on the application of some assumptions and the method of least work. It is widely known that least work is a powerful tool in solving statically indeterminate structural problems. This is true especially when the structure is a three-dimensional frame of which members are subjected to torsional stresses in addition to the conventional Taleb [7] also used the principle of the least work using equations of equilibrium of the entire stair and hence obtaining expressions directly for all redundant Nigerian Journal of Technology (NIJOTECH) Vol. 33 No. 4, October 2014, pp. 610 – 617 Copyright© Faculty of Engineering, University of Nigeria, Nsukka, ISSN: 1115-8443 www.nijotech.com http://dx.doi.org/10.4314/njt.v33i4.23 UNIV ERSIT Y O F IB ADAN L IB RARY FFFFINITE INITE INITE INITE EEEELEMENT LEMENT LEMENT LEMENT AAAANALYSIS OF A NALYSIS OF A NALYSIS OF A NALYSIS OF A FFFFREEREEREEREE----SSSSTANDING TANDING TANDING TANDING Nigerian Journal of Technology, acting at the supports. In the plane of the flights shear, tension and compression are ignored. The load cases include symmetrically and unsymmetrical placed loads. Sieve approaches the problems of free straight multiply - flight staircase in a procedure similar to folded plate analysis. Sieve’s theory [3] predicts that the symmetry in loadings (moment about z and Y axis) are both equal to zero. At this stage all primary moment have been known, subsequently the secondary moment will be calculated and shown to be small. (a) (b) Figure 1: Stair slab geometry, (a) elevation, (b) p The method of analyzing the statistically indeterminate staircase formed by a series of bar elements was first developed by [8]. The main difference in his assumptions from those suggested by [6] is that the landing slab can be represented by a curved bar element when the moments in the frame are computed. It was previously found that the moments produced due to unsymmetrical loads at some sections of the staircase are only slightly greater than those for the symmetrical loads. From geometric and loading symmetry of the staircase, it is much simpler to solve this problem by cutting the whole frame at the mid-point of the landing into two equal TANDING TANDING TANDING TANDING SSSSTAIRCASETAIRCASETAIRCASETAIRCASE,,,, W. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. Rufai Nigerian Journal of Technology, Vol. 33, No. 4, October 2014 acting at the supports. In the plane of the flights shear, ssion are ignored. The load cases include symmetrically and unsymmetrical placed loads. Sieve approaches the problems of free straight flight staircase in a procedure similar to folded plate analysis. Sieve’s theory [3] predicts that y in loadings (moment about z and Y axis) are both equal to zero. At this stage all primary moment have been known, subsequently the secondary moment will be calculated and shown to be Figure 1: Stair slab geometry, (a) elevation, (b) plan The method of analyzing the statistically indeterminate staircase formed by a series of bar elements was first developed by [8]. The main difference in his assumptions from those suggested by [6] is that the landing slab can be represented by a bar element when the moments in the frame are computed. It was previously found that the moments produced due to unsymmetrical loads at some sections of the staircase are only slightly greater than those for the symmetrical loads. From geometric ing symmetry of the staircase, it is much simpler to solve this problem by cutting the whole point of the landing into two equal halves, which will be treated as two separate cantilever beams, than it is to solve the of original structure. Thus each half of the frame can be considered as a cantilever structure with only two unknown redundant. However, none of the approaches is readily suitable for practical design because of considerable calculations. Hence there ought to be a scope for further improvement in the analysis and design procedures of free–standing stairs based on rigorous finite element analysis. 3333. . . . METHODOLOGYMETHODOLOGYMETHODOLOGYMETHODOLOGY 3.13.13.13.1 Finite Element MethodFinite Element MethodFinite Element MethodFinite Element Method Nicholas [9] stated finite element method to be a numerical technique for solving problems wdescribed by partial differential equations or can be formulated as functional minimization. This method Finite Element Method (FEM) is regarded as relatively accurate and versatile numerical tool for solving differential equations that model physThe methodology is used in various areas of engineering in which the problems are modeled by partial differential equations. The method has found considerable application in structural engineering and related disciplines. A domain of intereas an assembly of finite elements. Approximating functions in finite elements are determined in terms of nodal values of a physical field which is sought. A continuous physical problem is transformed into a discretized finite element probnodal values. For a linear problem a system of linear algebraic equations should be solved and vfinite elements can be recovered using nodal values.The finite element method is closely related to the classical variational concept of the Rayleighmethod. The modern finite element technique can be traced back to a paper in 1950was dubbed as the “finite element[11] and was further developed by Argyris strong development of the method from the engineer's point of view was led by ZienkiwiczThe mathematical theory of the finite elements has been developed and promoted by many scientists. Among them one can mention Strang and Fix Babuska and Aziz [15], Odeplate-bending problem is one of the first problems where finite element was applied at early 1960s. Considerable effort has been devoted over the past two decades in devising efficient and accurate bending elements. W. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. Rufai &&&& J. O. LabiranJ. O. LabiranJ. O. LabiranJ. O. Labiran Vol. 33, No. 4, October 2014 611 halves, which will be treated as two separate cantilever beams, than it is to solve the of original Thus each half of the frame can be considered as a cantilever structure with only two However, none of the approaches is readily suitable for practical design because of considerable calculations. Hence there ought to be a scope for ther improvement in the analysis and design standing stairs based on rigorous stated finite element method to be a numerical technique for solving problems which are described by partial differential equations or can be formulated as functional minimization. This method Finite Element Method (FEM) is regarded as relatively accurate and versatile numerical tool for solving differential equations that model physical phenomena. The methodology is used in various areas of engineering in which the problems are modeled by partial differential equations. The method has found considerable application in structural engineering and related disciplines. A domain of interest is represented as an assembly of finite elements. Approximating functions in finite elements are determined in terms of nodal values of a physical field which is sought. A continuous physical problem is transformed into a discretized finite element problem with unknown nodal values. For a linear problem a system of linear aic equations should be solved and values inside finite elements can be recovered using nodal values. The finite element method is closely related to the cept of the Rayleigh-Ritz method. The modern finite element technique can be traced back to a paper in 1950 [10]. The technique was dubbed as the “finite element method” by Clough and was further developed by Argyris [12]. The e method from the engineer's led by Zienkiwicz and Taylor [13]. The mathematical theory of the finite elements has been developed and promoted by many scientists. can mention Strang and Fix [14], , Oden and Reddy [16]. The bending problem is one of the first problems where finite element was applied at early 1960s. Considerable effort has been devoted over the past two decades in devising efficient and accurate bending UNIV ERSIT Y O F IB ADAN L IB RARY FFFFINITE INITE INITE INITE EEEELEMENT LEMENT LEMENT LEMENT AAAANALYSIS OF A NALYSIS OF A NALYSIS OF A NALYSIS OF A FFFFREEREEREEREE----SSSSTANDING TANDING TANDING TANDING Nigerian Journal of Technology, 3.2 Application o3.2 Application o3.2 Application o3.2 Application of Finite Element to Flexural in Plates f Finite Element to Flexural in Plates f Finite Element to Flexural in Plates f Finite Element to Flexural in Plates The study of a free-standing stair is performedusing plate flexural finite element analysis. Other proposed analytical methods have been compared to assess the relative merits and demerits of using any of the methods. The finite element analysis was developed on the basis of taking the stairs as a plate flexural element and then finalized as using both two and three dimensional finite elements to reveal the differences between the body structurfor the analysis when loaded with certain load combinations. Investigation on the use of variousmesh sizes using sensitivity analysis was carried out to determine the range of the most appropriate mesh size to be used in analyzing the stairs (Plate flexural type of finite element analysis isdue to its versatility and complete generalityprinciple is based on small deflection elastic theory, i.e. the deformation of the structure is assumed to be linearly proportional to the applied load, and does not suffer permanent deformation, thus, making the structure behave like a simple spring. The coordinates and node numbering system shown in Figure 3 can be defined for the rectangular element. The dimensions of the plate (coordinates (x, y, and z) are in the Cartesian coordinate system. (a) (b) Figure 2: Finite element mesh of the stair way (a) elevation, (b) plan TANDING TANDING TANDING TANDING SSSSTAIRCASETAIRCASETAIRCASETAIRCASE,,,, W. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. Rufai Nigerian Journal of Technology, Vol. 33, No. 4, October 2014 f Finite Element to Flexural in Plates f Finite Element to Flexural in Plates f Finite Element to Flexural in Plates f Finite Element to Flexural in Plates performed herein using plate flexural finite element analysis. Other proposed analytical methods have been compared to assess the relative merits and demerits of using any of e methods. The finite element analysis was developed on the basis of taking the stairs as a plate a beam element three dimensional finite elements to reveal the differences between the body structures for the analysis when loaded with certain load tigation on the use of various mesh sizes using sensitivity analysis was carried out to determine the range of the most appropriate mesh (Figure 2). e of finite element analysis is adopted due to its versatility and complete generality. Its principle is based on small deflection elastic theory, i.e. the deformation of the structure is assumed to be ed load, and does not thus, making the behave like a simple spring. node numbering system as can be defined for the rectangular The dimensions of the plate (a, b, t) and the in the Cartesian Finite element mesh of the stair way n The nodes 1 to 4 have their respective rotationsQRS; nodal forces TURV, UWVX YZdisplacement [ \U] ^ _URV, UWV,Ù V, … . , Ù V,URand \b]_QRV, QWV, [`V, … , QRS, QWV The displacement [ is given by:[ ^ cV d cef d cgh d cSfecifgdcjfe d ckfhe d cVlhQR ^ m nonW , QW ^ nonR bV ^ TQRVQWV[VX \UV] ^ pURVUWVÙ V q The nodal loads are related to displacements as:\U] ^ [r]\s] Where Q ^ m nonW \U] ^ _URV, UWV, Ù V, … , URS, U\Q] ^ _QRV, QWV, Q`V, . . . , QRS, QIn mathematical terms they are given as:QR ^ nonW ^ m(cV d cef d cgcifgdcjfe d ckfhe d cVlhQR ^ m nonW ^ m(ce d 2cSf dckhe d 3cVVfgh d cVehg) For the edge 1 -2, along the plate dimension, constant and is equal to zerot ^ cV d cgh d cuhe d cVlQR ^ m(cg d 2cuh d 3cVlheQW ^ ce d cvh d ckhe d cVenodes 1 and 2, y ^ 0 at node 1 t ^ tV ^ c, QR ^ QRV ^ mat y^b at node 2 t ^ tV ^ cV d cgw d cuweQR ^ QRe m (cg d 2cuw d 3cQW ^ QWe ^ ce d cvw d ckwethe constants can be evaluated as:\s(f, h)] ^ [x(f, h)]\c] ^ [xwhere [x(f, h)] is given by: W. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. Rufai &&&& J. O. LabiranJ. O. LabiranJ. O. LabiranJ. O. Labiran Vol. 33, No. 4, October 2014 612 The nodes 1 to 4 have their respective rotations θx1, to X YZ TURS, UWSX and URS,Ù S,z{ (1) , [`S, z{ (2) is given by: e d cvfh d cuhe dhg d cVVfg d cVefhg (3) (4) (5) (6) The nodal loads are related to displacements as: (7) (8) UWS, Ù Sz{ (9) QWS, Q`Sz{ (10) In mathematical terms they are given as: gh d cSfe d cvfh d cuhe dhg d cVVfg d cVefhg) (11) d cvh d 3cife d 2cjfh d (12) 2, along the plate dimension, f is a equal to zero Vlhg (13) e) (14) Vehg (15) nodes 1 and 2, y ^ 0 at node 1 mcg, QW ^ QWV ^ mce (16) d cVlwg (17) cVlwe) (18) e d cVewg (19) e evaluated as: [x(f, h)][|]}V\s] (20) UNIV ERSIT Y O F IB ADAN L IB RARY FFFFINITE INITE INITE INITE EEEELEMENT LEMENT LEMENT LEMENT AAAANALYSIS OF A NALYSIS OF A NALYSIS OF A NALYSIS OF A FFFFREEREEREEREE----SSSSTANDING TANDING TANDING TANDING Nigerian Journal of Technology, Figure 3: Rectangular Finite Element for Plate Flexure [x(f, h)] ^ ~ 0 0 m 1 0 m f m 2h 0 1 0 2f h 0 1 f h fe fh he When the respective values of (f, h) substituted from equation (18) for all nodes, the matrix [A] is formed. \�(f, h)] ^ ��� �� m n�onR�m n�onW�m en�onRnW��� �� ^ p mT2cS dm2(2cu d2(cv d 2cjf or \�(f, h)] ^ ~ 0 0 0 m 2 0 0 0 0 0 0 m 0 0 0 0 2 For a rectangular plate the bending moments are written as: �R ^ m ��R n�onR� d �V n�onW� � �W ^ m ��W n�onW� d �V n�onR� � �RW ^ 2�RW nonRnW where � ^ �R ^ �W ^ ���Ve(V}��) ; �RW ^ Ve (1 m �The stresses [b(f, h)] are written as where [�] ^ [�][|]}V and the stiffness matrix K is written as [r] ^ � � [�]�l {�l [�][�]bfbh and the nodal forces are given as per equation ([U] ^ �� � [�]�l {�l [�][�]bfbh� \s] Where the particular element is considered, the above matrices are given as suffix ‘e’. This will be used on the various elements obtained from the mesh to give the unknowns. These unknowns include the shear stress, membrane stress, bending moment and torsional moment. The notations used are presented in the Figure 4. These results are displayed based on the following outlined notations: SQx, SQy: Shear stresses (Force/ unit length. / thickness.) TANDING TANDING TANDING TANDING SSSSTAIRCASETAIRCASETAIRCASETAIRCASE,,,, W. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. Rufai Nigerian Journal of Technology, Vol. 33, No. 4, October 2014 3: Rectangular Finite Element for Plate Flexure 0 mfe m 2fh m3fe mfg m3fhe m3fe 2fh he 0 3feh hg fg feh fhe hg fgh fhg � substituted from equation (18) for all nodes, the matrix [A] is formed. T d 6ciR d 2cjh d 6cVVRWXd 2ckf d 6cVlh d 6cVefh)d 2ckh d 3cVVfe d 3cVehe)q 0 6f m 2h 0 0 m 6fh 0m 2 0 0 m 2f m 6h 0 m 6fh 0 0 4f 4h 0 6f 6he � � cV⋮cVe he bending moments are (24) (25) (26) �)� (27) are written as [�]\�(f, h)] and the stiffness matrix K is (28) and the nodal forces are given as per equation (f) (29) Where the particular element is considered, the above is will be used on the obtained from the mesh to give the unknowns. These unknowns include the shear stress, membrane stress, bending moment and torsional The notations used are presented in the These results are displayed based on the following Shear stresses (Force/ unit length. / Sx, Sy, and Sxy: Membranelength. / thickness.) Mx, My, Mxy: Moments per unit width (Force x Length/length) Mx, the unit width is a unit distance parallel to the local Y axis. My, the unit width is a unit distance parallel to the local X axis. Mx and My cause bending, while Mxy causes the element to twist out-of-plane. Figure 4: Notations for plate outputs 3333.2.2.2.2 Stiffness Matrix for an Inclined Beam ElementStiffness Matrix for an Inclined Beam ElementStiffness Matrix for an Inclined Beam ElementStiffness Matrix for an Inclined Beam ElementHence stiffness matrix for an inclined beam element is obtained by combining the stiffness matrelement and beam element and arranging in proper locations. This gives rise to six disbeam element as indicated in W. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. Rufai &&&& J. O. LabiranJ. O. LabiranJ. O. LabiranJ. O. Labiran Vol. 33, No. 4, October 2014 613 (21) substituted from equation (18) for all nodes, the matrix [A] is formed. (22) V Ve� ≡ [�]\c] (23) mbrane stresses (Force/unit Moments per unit width (Force x the unit width is a unit distance parallel to the the unit width is a unit distance parallel to the y cause bending, while Mxy causes the plane. Figure 4: Notations for plate outputs Stiffness Matrix for an Inclined Beam ElementStiffness Matrix for an Inclined Beam ElementStiffness Matrix for an Inclined Beam ElementStiffness Matrix for an Inclined Beam Element Hence stiffness matrix for an inclined beam element is obtained by combining the stiffness matrices of bar element and beam element and arranging in proper locations. This gives rise to six displacements for the ment as indicated in (30). UNIV ERSIT Y O F IB ADAN L IB RARY FFFFINITE INITE INITE INITE EEEELEMENT LEMENT LEMENT LEMENT AAAANALYSIS OF A NALYSIS OF A NALYSIS OF A NALYSIS OF A FFFFREEREEREEREE----SSSSTANDING TANDING TANDING TANDING Nigerian Journal of Technology, ��� � ���  ¡¢¡�¡ £¢£�£��� � ��� ^ ¤¥ ��� � ��� 12¦e §g⁄ m12©¦ §g⁄ 6m12©¦ §g⁄ 12©e §g⁄ m6 ¦ §e⁄ m6© §e ⁄ m12¦e §g⁄ 12©¦ §g⁄ m12©¦ §g⁄ m12©e §g⁄ 6¦ §e⁄ 6© §e⁄ 2Where c ^ cos Once the stiffness matrix is generated based on the equation stated above and load acting on each member is known then the displacement acting on the member can be derived and kept in the moment equation to produce other unknowns. 3333.3.3.3.3 Investigation Investigation Investigation Investigation of Variousof Variousof Variousof Various Mesh SizesMesh SizesMesh SizesMesh Sizes Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis In order to develop the effects of varying mesh sizes, a sensitivity analysis was done. This involvesanalysis of the stairs with various mesh geometrical properties being same. The result of this analysis is shown in Table 1 Table 1: Sensitivity analysis parameters Plate Mesh Type 1 Plate Mesh Type Flights (m) 0.28 x 0.54 Landing (m) 0.26 x 0.56 0.0933∑ (members) 75 ∑ (nodes) 96 3333.4.4.4.4 ComparisonComparisonComparisonComparison of Finite Element and Traditionalof Finite Element and Traditionalof Finite Element and Traditionalof Finite Element and TraditionalMethods of AnalysisMethods of AnalysisMethods of AnalysisMethods of Analysis The results of the traditional method of analysis earlier suggested [6] is compared withfor finite element method obtained herein. method of finite element considered here is beam stiffness matrix method. This method idue to its close relation of output with the proposed analytical method compared to that of the plate flexural method. 3333.5 General Parametric Study Info.5 General Parametric Study Info.5 General Parametric Study Info.5 General Parametric Study InformationrmationrmationrmationParameters stated in Table 2 were the carry out most of the stress analyses on the stairs. 4. ANALYSIS AND RESULTS.4. ANALYSIS AND RESULTS.4. ANALYSIS AND RESULTS.4. ANALYSIS AND RESULTS. The results of the analyses were generated in regions dividing the stairs into bottom and top flights and then the landing for three dimensional element and plate flexure. The stiffness matrix is considered for only three displacements each at a node for taking the TANDING TANDING TANDING TANDING SSSSTAIRCASETAIRCASETAIRCASETAIRCASE,,,, W. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. Rufai Nigerian Journal of Technology, Vol. 33, No. 4, October 2014 6¦ §e⁄ m12¦e §g⁄ 12©¦ §g⁄ 6¦ §e⁄m6© §e⁄ 12©¦ §g⁄ m12©e §g⁄ 6© §e⁄ 4 §⁄ m6¦ §e⁄ 6© §e⁄ 2/§m6¦ §e⁄ 12¦e §g⁄ m12©¦ §g⁄ m6¦ §e⁄6© §e⁄ m12©¦ §g⁄ 12©e §g⁄ 6© §e⁄2 §⁄ m6¦ §e⁄ 6© §e⁄ 4 §⁄ ��� � ��� �� � �� «¡¬¡Q¡«£¬£Q£�� � �� Where c ^ cosα, s ^ sinα, and α is the inclined angle of the beam. Once the stiffness matrix is generated based on the equation stated above and load acting on each member is known then the displacement acting on the member can be derived and kept in the moment Using Using Using Using In order to develop the effects of varying mesh sizes, a This involves the mesh sizes but other The result of this arameters Plate Mesh Type 2 0.0933 x 0.18 0.0933 x 0.0933 900 977 of Finite Element and Traditionalof Finite Element and Traditionalof Finite Element and Traditionalof Finite Element and Traditional The results of the traditional method of analysis with that obtained method obtained herein. The method of finite element considered here is beam matrix method. This method is considered due to its close relation of output with the proposed analytical method compared to that of the plate rmationrmationrmationrmation the values used to on the stairs. The results of the analyses were generated in regions dividing the stairs into bottom and top flights and then anding for three dimensional element and plate flexure. The stiffness matrix is considered for only three displacements each at a node for taking the stairs as a plate element (Figure 5). This proves more accurate due to its sensitivity of producing the foacting at any point on the stairs.The results of the plate’s analysis at the supports that of the plate’s elements 4 respectively. Table 2: Values used to perform the S/N Parameters 1 flight length 2 width 3 depth of flight4 depth of landing5 angle 6 load at flight 7 load at landing (a) (b)Figure 5: Deflected shape of the staircase (a) vertical deflection along upper and lower flightdeflection W. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. Rufai &&&& J. O. LabiranJ. O. LabiranJ. O. LabiranJ. O. Labiran Vol. 33, No. 4, October 2014 614 (30) the inclined angle of the beam. stairs as a plate element (Figure 5). This proves more accurate due to its sensitivity of producing the forces acting at any point on the stairs. The results of the plate’s analysis at the supports and are shown in Tables 3 and erform the parametric study Values 2700mm 1400mm depth of flight 150mm landing 175mm 30 ̊ 16.9 kN/m2 load at landing 15 kN/m2 (a) (b) Figure 5: Deflected shape of the staircase (a) vertical deflection along upper and lower flight, (b) lateral UNIV ERSIT Y O F IB ADAN L IB RARY FFFFINITE INITE INITE INITE EEEELEMENT LEMENT LEMENT LEMENT AAAANALYSIS OF A NALYSIS OF A NALYSIS OF A NALYSIS OF A FFFFREEREEREEREE----SSSSTANDING TANDING TANDING TANDING SSSSTAIRCASETAIRCASETAIRCASETAIRCASE,,,, W. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. Rufai &&&& J. O. LabiranJ. O. LabiranJ. O. LabiranJ. O. Labiran Nigerian Journal of Technology, Vol. 33, No. 4, October 2014 615 Table 3: Maximum and minimum forces and moments at the fixed supports Node Horizontal Vertical Horizontal Moment Fx (kN) Fy (kN) Fz (kN) Mx (kNm) My (kNm) Mz (kNm) Max Fx 86 62.925 41.729 2.066 1.303 -1.449 0.625 Min Fx 23 -62.925 41.729 2.066 -1.303 -1.449 0.625 Max Fy 23 -62.925 41.729 2.066 -1.303 -1.449 0.625 Min Fy 15 18.986 -4.259 0.175 -0.37 0.781 1.995 Max Fz 1 50.759 31.147 9.228 -0.567 1.044 0.066 Min Fz 19 -10.65 12.998 -4.728 -1.255 -0.791 2.257 Max Mx 92 35.528 28.463 -2.196 1.429 -1.078 1.64 Min Mx 21 -35.528 28.463 -2.196 -1.429 -1.078 1.64 Max My 1 50.759 31.147 9.228 -0.567 1.044 0.066 Min My 86 62.925 41.729 2.066 1.303 -1.449 0.625 Max Mz 103 -15.767 0.85 -4.545 0.862 -0.729 3.33 Min Mz 6 -50.759 31.147 9.228 0.567 1.044 0.066 Table 4: Maximum and minimum stresses and moments of the plate elements Forces Plate Shear Membrane Bending Moment SQx (N/mm2) SQy (N/mm2) Sx (N/mm2) Sy (N/mm2) Sxy (N/mm2) Mx (kNm/m) My (kNm/m) Mxy (kNm/m) Max Qx 59 0.682 -0.095 0.143 0.394 -0.383 21.597 15.818 11.056 Min Qx 63 -0.682 -0.095 -0.143 -0.394 -0.383 21.597 15.818 -11.056 Max Qy 55 0.49 0.539 0.278 2.074 0.679 6.447 16.542 4.77 Min Qy 61 0 -0.721 0 0 0.925 49.501 16.062 0 Max Sx 129 -0.024 -0.069 1.229 -0.053 0.055 2.892 0.168 -1.603 Min Sx 105 -0.021 0.048 -3.362 -0.173 0.122 1.037 0.156 -0.857 Max Sy 26 -0.048 0.021 0.173 3.362 -0.122 0.156 1.037 -0.857 Min Sy 42 0.069 0.024 0.053 -1.229 -0.055 0.168 2.892 -1.603 Max Sxy 61 0 -0.721 0 0 0.925 49.501 16.062 0 Min Sxy 107 -0.539 -0.49 -2.074 -0.278 -0.679 16.542 6.447 4.77 Max Mx 61 0 -0.721 0 0 0.925 49.501 16.062 0 Min Mx 103 -0.068 -0.005 -3.257 -0.004 -0.003 -1.783 -0.109 -1.837 Max My 55 0.49 0.539 0.278 2.074 0.679 6.447 16.542 4.77 Min My 33 0.005 0.068 0.004 3.257 0.003 -0.109 -1.783 -1.837 Max Mxy 68 0.104 -0.112 -0.143 0.358 -0.073 20.414 8.705 11.162 Min Mxy 70 -0.104 -0.112 0.143 -0.358 -0.073 20.414 8.705 -11.162 From the results obtained it is observed that the torsional moment is more critical at the centroid of the slab landing; the torsional moment is maximum at the center of the landing towards the upper flight and minimal at the landing towards the bottom flight. This moment increases based on the horizontal spacing found at the landing across the median of successive flights. It is best not to consider any open well when analyzing or designing a free standing stairs. The bottom and top flight supports prove to be symmetrical. At the mid span of the landing the shear along the x-coordinate is zero. The membrane shear stress and the maximum moment about the ‘x’-coordinate are both maximum at the center of the landing slab corresponding to the centroid of both upper and bottom flights. It is noted that the deflection of this body depends on the end conditions or fixity of the support and the continuous formation of the whole plate structure. The deflection is more at the landing and at the region where both flights are connected with the landing. Observation of the induced moments and forces as obtained for the Traditional analysis [11] with that of finite element employed in this work yields the following results as shown in Figures 6 and 7. It should be noted that the analytical method should be considered to be sufficiently conservative only when the loading is symmetrical for design purposes. The stairway behaves as a three dimensional plate structure, which is clearly indicated by its deflected shape. Except at the mid-span of flights, bending moment at other critical sections is not distributed uniformly across the width of the section. UNIV ERSIT Y O F IB ADAN L IB RARY FFFFINITE INITE INITE INITE EEEELEMENT LEMENT LEMENT LEMENT AAAANALYSIS OF A NALYSIS OF A NALYSIS OF A NALYSIS OF A FFFFREEREEREEREE----SSSSTANDING TANDING TANDING TANDING SSSSTAIRCASETAIRCASETAIRCASETAIRCASE,,,, W. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. Rufai &&&& J. O. LabiranJ. O. LabiranJ. O. LabiranJ. O. Labiran Nigerian Journal of Technology, Vol. 33, No. 4, October 2014 616 Figure 7: Comparison of Bending Moments at Landing It can be observed from Figure 7 that the behavior of the landing as modeled in the traditional analytical method [17] is close to that of finite element analysis except that the behavior of the free edge length of the stairs should attain a zero moment. The critical point of the landing is found to be too expensive. 5. CON5. CON5. CON5. CONCLUSION AND RECOMMENDATIONSCLUSION AND RECOMMENDATIONSCLUSION AND RECOMMENDATIONSCLUSION AND RECOMMENDATIONS 5.1 Conclusion5.1 Conclusion5.1 Conclusion5.1 Conclusion It is evident that the Finite Element Method is a robust means of numerically solving free standing structural problems. The stairway behaves as a three dimensional plate structure, which is clearly indicated by its deflected shape. Except at the mid-span of flights, bending moment at other critical sections is not distributed uniformly across the width of the section. Moment is more concentrated near the outer edge at support and near the inner edge at kink and at mid landing section. Of course, the specimen used was a simple plate flexural member, but the observations made in this study can apply to even complex structures with minor changes but related elements and boundary conditions. 5.2 Recommendations5.2 Recommendations5.2 Recommendations5.2 Recommendations The results obtained in this study are based only on symmetrical geometry of both the loading and elemental structure; hence subsequent study would consider unsymmetrical structure especially in cases of unequal flight lengths. Also no open well space is considered between the lower and upper flight to limit the torsional moment at the mid landing. 6. ACKNOWLEDGEMENT6. ACKNOWLEDGEMENT6. ACKNOWLEDGEMENT6. ACKNOWLEDGEMENT The authors wish to thank Prof. Femi Bamiro of the Department of Mechanical Engineering, University of Ibadan for his support during the study. 7.7.7.7. REFERENCESREFERENCESREFERENCESREFERENCES [1] Bangash, M. Y. H. and Bangash T., Staircases: structural analysis and design. A.A. Balkema, Rotterdam, 1999. Figure 6: Comparison of Bending Moments of Flight -10 -5 0 5 10 15 20 0 1 2 3 M o m e n t (K N m ) Horizontal distance of flight (m) Finite Element Cusen and Kuang UNIV ERSIT Y O F IB ADAN L IB RARY FFFFINITE INITE INITE INITE EEEELEMENT LEMENT LEMENT LEMENT AAAANALYSIS OF A NALYSIS OF A NALYSIS OF A NALYSIS OF A FFFFREEREEREEREE----SSSSTANDING TANDING TANDING TANDING SSSSTAIRCASETAIRCASETAIRCASETAIRCASE,,,, W. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. RufaiW. O. Ajagbe, O. A. Rufai &&&& J. O. LabiranJ. O. LabiranJ. O. LabiranJ. O. Labiran Nigerian Journal of Technology, Vol. 33, No. 4, October 2014 617 [2] Ng, F. S. and Chetty, A. T.,”Study of Three Flight Free Standing Staircase,” Journal of the Structural Division, ASCE, Vol 101, No. ST7, 1975, pp. 1419-1433 [3] Sieve, A., “Analysis of Free Straight Multiflight Staircases,” Journal of the Structural Division, ASCE, Vol.88, ST3, Proc. Paper 3168, 1962, pp.207-232. [4] Smith, E. A.,” Restrained Warping in Free-Standing Staircase,” Journal of the Structural Division, ASCE, Vol. 106, No. 3, 1980, pp. 734-738. [5] Ahmed, I., Muqtadir, A. and Ahmad, S., “Design Basis for Stair Slabs Supported at Landing Level,” Journal of the Structural Engineering, ASCE, Vol. 121, No. 7, 1995, pp.1051-1057. [6] Cusens, A. R., and Kuang, J. G, “Analysis of Free-Standing Stairs under Symmetrical Loading,” Concrete and Constructional Engineering, Vol. 60, No.5, 1965, pp. 167-172 [7] Taleb, N. J.: ‘The Analysis of Stairs with Unsupported Intermediate Landings,” Concrete and Constructional Engineering, Vol. 59, no.9 pp.315-320. 1964. [8] Fuchsteiner, W.,“The self-supporting spiral staircase, "Benton and steel concrete, Berlin, Germany, Vol.59, No. 11, pp. 256-258. 1954. [9] Nicholas, D. W. “Finite element analysis” thermodynamics of solids. CRS Press, 2003. [10] Turner, M. J., Clough, R. W, Martin, H.C. and. Topp, L. P: 1956, 'Stiffness and deflection analysis of complex structures.’ J. aero. Sci., 23, Num. 9, 805-823. [11] Clough, R.W, 1960 'The finite element method in plane stress analysis', Proc. ASCE 2nd Conference on Electronic Computation (Pittsburgh, PA., Sept. 8-9), 345-378. [12] Argyris, J.H.: Recent advances in matrix methods of structural analysis, Pergamon Press, Elmsford, New York, 1963. [13] ZienKiewicz, O. C. and Taylor, R. L.: The finite element method, volume 1, Basic formulation and linear problems, McGraw-Hill, New York, 4th Edn, 1989 [14] Strang, G and Fix, G. J.: An analysis of the finite element methods, Prentice-Hall Inc., Englewood Cliffs, N. J., 1973 [15] Babuska, .I and Aziz, A. K.: Lectures on the mathematical foundations of the finite element method, mathematical foundations of the finite element method with applications to partial differential equations, A. K. Aziz (ed.), Academic Press, 1972. [16] Oden, J. T. and Reddy, J. N.: Introduction to mathematical theory of finite elements, John Wiley and sons, New York, 1976. [17] Reynolds, C. E. and Steedman, J. C., Reinforced Concrete Designer's Handbook Eleventh Edition, E & FN Spon, London, 2008. UNIV ERSIT Y O F IB ADAN L IB RARY