Available online at ScienceDirect. Procedia Engineering 172 (2017 )

Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 172 (2017 ) 1093 1101 Modern Building Materials, Structures and Techniques, MBMST 2016 Iterative Methods of Beam-Structure Analysis Maciej Szumigała Institute of Structural Engineering, Faculty of Civil and Environmental Engineering, Poznan University of Technology, Piotrowo 5, Poland Abstract The article presents the concept of computational analyses of steel beam, composite and reinforced concrete structures accounting for physical and geometric nonlinearities. Due to the magnitude of the task and ease of constructing a calculation model for the static analysis of building structures (especially frame constructions), beam models are still applied. In the beam element, due to limitations connected with discretization, which are based only on a single dimension (length - along the x-axis), not many parameters in terms of the remaining two measurements of the cross-section (y, z) can be accounted for. Most often, the crosssection and moments of inertia are introduced to calculate the rigidity of axial compression/tension and flexural rigidity as elastic parameters. The problem arises of how to account for nonlinear laws of physics, local yielding the lack of uniformity among crosssections, local instability, the changeability of physical laws due to high temperatures, etc. The authors of the present work attempt to answer these questions and present a synthetic calculation model. 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license 2016 The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of MBMST 2016. Peer-review under responsibility of the organizing committee of MBMST 2016 Keywords: non-linear global structural analysis; shear lag effect; local instability; nonlinear constitutive laws. 1. Introduction In accordance with the guidelines of all three Eurocode Building Codes, it is recommended that static calculations of structures be carried out in the form of a global analysis. Global analysis relies on indicating an adequate set of internal forces (M, N, V) in the structure, which are in equilibrium with the determined set of external impacts, and makes it possible to account for additional factors such as the sensitivity of a structure to imperfections and the effects of general instability, the influence of deformations on the static stability of the structure (second order effect), susceptibility of joints, cracking of reinforced concrete structures and the interactions of the structures with the ground. Corresponding author. Tel.: +48 61 665-2401; fax: +48 61 876-6116. E-mail address: maciej.szumigala@put.poznan.pl 1877-7058 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of MBMST 2016 doi:10.1016/j.proeng.2017.02.169

1094 Maciej Szumigała / Procedia Engineering 172 ( 2017 ) 1093 1101 Additional issues, such as the shear lag effect in plate girders, effective width in reinforced concrete and composite T-sections, local instability, varied rigidity of reinforced concrete and composite cross-sections caused by cracking, as well as nonlinear physical properties of concrete and plasticity of steel ought to be accounted for (sometimes simultaneously) in addition to the general factors. The present work presents the concept of static analyses accounting for these additional factors. Nomenclature b eff effective width b o flange width according to Figure 1 I gr gross moment of cross-sectional inertia I eff ( com,ed,ser) moment of effective cross-sectional inertia according to E.1 for maximum stress com,ed,ser > gr in the analyzed span gr max. stress from bending in serviceability limit state calculated based on gross cross-sectional features x(y) value of maximum normal stress in flange owing to the shear lag effect x,max normal stresses in flange K stiffness matrix curvature B rigidity 2. Types of static analyses In accordance with the concepts contained in Eurocodes, a structure can be subjected to the following analyses. Due to the conside geometry and applied static equations, a division into the following is made: first order (I) analysis second order (II) analysis. In the case of first order analysis, the principle of rigidification is fully applied and the initial geometry of the structure is accounted for in the beginning and final phase of the calculations. It is assumed that deformations are small enough to where they can be overlooked when calculating displacements and internal forces. In second order analysis, the initial geometry is also assumed in the initial phase of the calculations but the deformations and displacements occurring as a result of the calculations influence the final geometry of the structure as well as the size and field of the distribution of internal forces. A modification of the rigidity matrix by the geometric matrix occurs (1), and the calculations are usually carried out iteratively to obtain the necessary convergence. Second order analysis is usually applied in tilting systems with high axial forces with slender bars and significant imperfections. K u=p, K= KE + KG, KG = K(N) (1) In the case of calculation algorithms according to the second order theory, the modification of elements of the rigidity matrix due to normal forces (1) are applied in beam models, or the correction of rigidity matrix elements is accounted for due to the current deformation field. There is an application for incremental methods in the calculation process. The majority of commonly applied engineering software for calculating beam constructions has such calculation options. Analyses of this kind do not allow for the superposition of loads, but "packets" of standard combinations of loads ought to be compiled and independent calculations using nonlinear methods carried out each time. One must keep in mind to add combinations connected with global and local imperfections to the existing sets. Structural analysis is also dependent on the properties of the material, or more precisely, the accepted physical model: elastic analysis, plastic analysis. In this case we can talk of global elastic analysis (GEA) and global plastic analysis (GPA). Elastic analysis can be

Maciej Szumigała / Procedia Engineering 172 (2017 ) 1093 1101 1095 used in every case (regardless of the order of the cross-section). The linear relationship s-e is assumed for the entire load range. Plastic load-bearing capacity of a cross-section can also be determined for the beams of a structure in which the external forces (M, N, V) have been established in accordance to the principles of elastic analysis. Elastic analysis is used when the load-bearing capacity of the cross-sections is determined by local instability. Plastic analysis is applied in structures whose material is characterized by a non-linear s-e nature. The structure in areas of possible plastic hinges has to contain adequate capacity to turn (for steel, reinforced concrete and composite structures) and, at the same time, stability has to be ensu. Thus, it is usually requi that the steel or composite cross-sections be Class 1 (possibly 2). The presented division of the structure according to the above criteria is simplified, and the individual analyses occur as interlinked. Thus, we are ultimately presented with: an elastic model and first order analysis, an elastic model and second order analysis, an elastic-plastic model and first order analysis, an elastic-plastic model and second order analysis, a rigid-plastic model and first order analysis, a rigid-plastic model and second order analysis. Eurocodes describe the principles and rules according to which the structure can be classified as requiring elastic analysis alone, and only of the first or second order ( cr ), and when global and/or local imperfections ought to be accounted for. Certain simplified methods of accounting for second order effects and the influence of imperfections (equivalent load method, amplification of loads or internal forces) are also provided. 3. Other factors influencing structural analysis 3.1. Shear lag effect PN-EN 1993-1-5: 2008, as compa to the formerly binding in Poland PN-B-03200: 1990 standard, introduced extended guidelines for calculating the load-bearing capacity of plate girder structures and calls for accounting for the shear lag effect. According to the classic beam theory, the value of normal stress at points with (x, y, z) coordinates is determined using the following simple dependency: M y σx z (2) I y This means that the stress distribution in direction y is constant for the entire flange width. However, if the flange of the beam is wider, this assumption becomes incorrect. In beams for which the flange width is large (b o >L e/50) as compa to their length, the effect of non-linear stress distribution along the width of the flange occurs - this phenomenon is refer to as the shear lag effect. The highest value of stress in the flange occurs at points located directly over the centre, and the further away from the centre, the more the value of stress decreases (Fig. 1). Fig. 1. Schematic stress distribution in wide flange [1].

1096 Maciej Szumigała / Procedia Engineering 172 ( 2017 ) 1093 1101 This means that in the case of steel plate girders (especially in the flanges), Bernoulli's principle does not apply and normal stress distribution is not even on the width of the walls, and especially the flanges. This is the result of cross-section deformation. Due to the described phenomenon, in the case of wide flange beams, calculations according to the elementary beam bending theory are not accurate. The simplest method according to [1], proposed in some project standards (including in Eurocode 3), is ucing flange width to so-called effective width. It is recommended that the effective width be determined according to the following dependence: b eff 1 x,max b 0 0 x y dy (3) In order to determine the extent of the influence of the shear lag effect, a substitute (effective) flange width is sought, for which an equal normal stress distribution can be assumed in accordance with the classic bending theory (Formula 2). The resultant of flange stress in both cases (for b o and b eff) is the same. In practice, however, the value of the effective width coefficient is determined, which expresses the ratio of effective width to the actual flange width according to [1]: b eff (4) b 0 If the distribution of normal stress in the flange is close to constant (the shear lag effect is very small), the value of the coefficient will be close to unity. Along with a change in the level of flange stress (increase in stress - Fig. 2), the shear lag effect increases, and the value of the coefficient decreases. In engineering practice, the distribution of stress in a flange x(y) is an unknown which ought to be indicated. In such a case, the effective width cannot be calculated using Formula 2. It can, however, be easily determined transforming Formula 3 to take the following form: b eff b 0 (5) The shear lag effect is strongly dependent on the stress level, as has been demonstrated in the figure below. low stress high stress Fig. 2. Distribution of normal stress in plate girder flange in the area of an intermediate beam support [1]. The uced flange width ought to be accounted for in statistical analyses [1] accordingly to item 3.1 of the PN- EN 1993-1-5 standard, and generally, b eff is not a constant value, but is a function of the stress level.

Maciej Szumigała / Procedia Engineering 172 (2017 ) 1093 1101 1097 Fig. 3. Changes in stress distribution in concrete slab along with increasing loads [2]. 3.2. Local stability In accordance with PN-EN 1993-1-5, class 4 steel cross-sections subjected to the local loss of stability are calculated in according to the theory of supercritical load-bearing capacity. According to item 4.2.1 of the above standard, one can indicate the load-bearing capacity of plate girders in which compressed walls with a class 4 crosssection occur assuming the geometric features of an effective cross-section (A eff, I eff, W eff), both when assessing the load-bearing capacity of the cross-section as well as the stability of the element due to buckling or lateral torsional buckling. The effective cross-section ought to be determined iteratively, as the effective width is dependent on the stress distribution (parameter), which changes along with the geometry of the cross-section. The load-carrying capacity is determined at the "borderline state", i.e. for = f y. Attachment F of standard PN-EN 1993-1-5 allows for another, more precise means of ucing the cross-section dependant on the current stress level: A c,eff = A c (6) 1 0,055(3 ) / p, ( p p, ) 1 0,188/ p, ( p p, ) 0,18 1,0 0,18 1, 0 (7) ( 0,6) ( 0,6) p, p p, p span walls cantilever walls p, p f com, Ed y / M 0 _ p f y cr b / t 28,4 k com, Ed, ser p, ser (8) p f y / M 0 at a borderline state of use I eff gr I gr ( Igr Ieff ( com, Ed, ser)) (9) com, Ed, ser Based on the above, it turns out that effective width b eff and moments of inertia I eff depend on the stress level com, thus on the internal force distribution field, which signifies an influence of the structure on the static response [3] and deformations.

1098 Maciej Szumigała / Procedia Engineering 172 ( 2017 ) 1093 1101 3.3. Fire engineering Fig. 4. Influence of local instability on changes in the curvature. European standards place great emphasis on the fire safety of structures as early as at the design stage, and hold the building designer responsible for ensuring such safety. This is why Parts 1-2 of the individual standards are titled "Structural fire design...". The destructive effects of fire on non-combustible materials such as steel and concrete are based on the detrimental change of mechanical parameters leading to decreased load-bearing capacity and rigidity, and the loss of stability. Moreover, under fire exposure conditions, additional deformations and tensions connected with the thermal expansion of materials take appear. The picture below (Fig. 7) presents how dependencies for steel and concrete change depending on the temperature of the fire. Above all, the yield strength, strength of materials, and the module of elasticity is decreased [4]. Concrete responds to fire temperatures with similar changes. Its compressive strength decreases, as does the deformation module. This means that in addition to the decrease in the load-bearing capacity of the structure in the event of a fire, its rigidity changes drastically (falls) (Fig. 8), and thus the static response of the structure is changed. Fig. 5. Physical properties of steel and concrete depending on fire temperatures [5]. Eurocodes recommend using advanced fire models and applying global structural analysis. This usually implies the necessity of using the beam model in static analysis as well as accounting for the decreasing rigidity and changing displacement field and internal force field connected with it.

Maciej Szumigała / Procedia Engineering 172 (2017 ) 1093 1101 1099 Fig. 6. Change in curvature of beam bent under the influence of fire temperatures. 3.4. Varied rigidity of composite and reinforced concrete cross-section Concrete as a building material, as opposed to homogeneous and isotropic steel, is characterized by nonlinear physics (Fig. 9). What is more, it is a brittle material. Its tensile strength is approximately 10% of its compressive strength. This means that the direction (sign) of stresses in concrete cross-sections is important, and according to EC2 and EC4 recommendations, concrete subjected to tensile strain is sometimes overlooked in calculations regarding load-bearing capacity and rigidity. In more precise numeric models, steel is also described by nonlinear relationships Nonlinear physics determines the changeability of the fundamental physical constant applied in static calculations (nowadays commonly automated), i.e. Young's modulus. Young's tangent modulus is a derivative at the point to the curve, i.e. E()=d/d, and is changeable. The changeability of the modulus of elasticity requires special algorithms to be applied and statistical analyses should be carried out in line with nonlinear procedures, usually iterative. Generally speaking, physical nonlinearity (and also geometric nonlinearity) can be accounted for by applying appropriate incremental procedures, e.g. the Newton-Raphson method. Global analysis, however, is usually carried out on beam structural models and thus the application of the modulus of elasticity, even in the form of a deformation function E=E() does not lead to an accurate result, as the beam is treated as a single-dimension element and it is difficult to establish deformations for determining a reliable modus of elasticity, even when possessing information regarding the parameters of the cross-section (I, A). a b c Fig. 7. Idealized graphs of nonlinear physical properties of concrete: (a) compressed, (b) in tension and (c) steel [6]. The issue is even more complicated when we are dealing with reinforced concrete or composite cross-sections [2]. Modern day software makes it possible to numerically analyze very complex details and elements by using the finite element method of surface or spatial elements [6]. It is, however, difficult to model entire building structures in such a way. Due to the magnitude of the numeric task, practical beam models are still commonly applied. Tools for the analysis of beam models are, in the majority of cases, constructed based on the elastic model and make it possible to account for geometric nonlinearity. There is the problem of physical nonlinearity, with the complexity of the issue illustrated by dependency below. =f() N = B N(x,y, N) N(N, B N), M = B M(x,y, M) (M, B M) (10) 4. Global analysis of construction using method of generalized constitutive law The author is familiar with previous attempts, all introducing physical nonlinearity to the beam model. For example, in the Frame application [7], in addition to the general discretization of structures into individual beams, the discretization of the cross-section into layers was simultaneously applied. The general method (generalized cross-sectional rigidity B) was proposed by the author of this article. This method is takes into account the nonlinear relationship (M = B ) [2] instead of the linear one ( = E ). In the traditional calculation procedure K u=p K= KE + KG, KE = f(e, A, I) (11)

1100 Maciej Szumigała / Procedia Engineering 172 ( 2017 ) 1093 1101 was suggested modification of the stiffness matrix to form K= KB + KG, KB = f(b) = f(b N, B M) B N=f(N, e N) B M = f(m, ) (12) The method is intended for steel-concrete composite structures and relies on cross-sectional homogenization and calculating one generalized cross-sectional rigidity of a composite beam B(M) instead of separate rigidities for the steel E a()i a and concrete E c()i c parts. In the proposed procedure, the generalized rigidity is changeable and dependant on the distribution field of bending moments or, more precisely, curvatures. A special algorithm was constructed (i.e. distribution field of internal forces and displacements) by the iterative selection of cross-sectional rigidity. For a loaded structure (steel, reinforced concrete or composite), distribution fields of internal forces and displacement fields are determined iteratively accounting for physical as well as geometrical nonlinearity. Structural calculations in the presented procedure take place in a two-stage process. In the first stage, the physical properties for beam cross-sections are established in the form of (B κ M) rigidity curvature moment. A separate subprogram based on the layer model is used for this purpose. The cross-section is first discretized (modelled) with elementary layers (rectangles). At the same time, two materials can be accounted for in the cross-section, i.e. concrete and steel, providing their physical and mechanical properties. We can, therefore, account for steel, concrete, reinforced concrete and steelconcrete composite cross-sections. Optionally, the influence of welding stresses and concrete shrinkage can be accounted for. Calculations are based on gradual cross-sectional rotation input (assigning extreme deformations ). The deformations result in stresses established in accordance with the physical parameters assigned to steel and concrete. Applying Bernoulli's principle and basic equilibrium equations (by integrating on the cross-sectional surface), the iterative location of the neutral axis is determined. Upon indicating the state of equilibrium (13), the bending moment is calculated. In this way, the location of the neutral axis, curvature, bending moment and generalize rigidity established for subsequent values of deformations: i c A ic + i a A i a = N (lub 0) = ( g+ d)/h M= i iz i B=M/ Results are recorded in the form of matrices and ente into the main program. For each of the cross-sections of the structure, a separate B M κ matrix ought to be established. The generalized physical law can be determined in another way, e.g. by using 3D models and applying more advanced programming (e.g. the Abaqus program [6]). The elementary detail of the cross-section is subjected to further kinematic inputs (rotations), and its static response calculated. In calculations of the beam model, the functional dependency B is used in practice as opposed to B M. The rigidity-curvature relationship is a monotonic function and is suitable for iterative calculations seeking convergence in subsequent steps of the calculations. The extrapolation of rigidity takes place iteratively until obtaining the assumed accuracy, with the iterative loops being embedded within each other due to geometric nonlinearities. N=N/B N M=M/B M i= N(N, B N( N))+ M(M, B M()) z i i = f(m, N, B M, B N) (14) Due to the axial forces in frame structures, which may be significant, generalized constitutive law is established as expanded to include axial deformations and the rigidity of axial shortening or lengthening (Equation 12). Normal forces occurring in the beams of a structure fundamentally affect the form of the generalized physical law, which has been presented in the graph below (Fig. 12).

Maciej Szumigała / Procedia Engineering 172 (2017 ) 1093 1101 1101 M- M-K 30000 140000 M 25000 20000 N=0 N=500 15000 N=1000 N=1500 N=2000 10000 N=2500 N=3000 5000 N=3500 N=4000 N=4400 0 0,0E+00 5,0E-04 1,0E-03 1,5E-03 2,0E-03 2,5E-03 3,0E-03 M 120000 100000 80000 60000 40000 20000 0 0,00E+00 5,00E-04 1,00E-03 1,50E-03 2,00E-03 K IPE450 HKS300 HEB260 IPE400 Fig. 8. Influence of axial forces and cross-sectional geometry on the form of the generalized constitutive law M- The presented algorithm makes it possible to account for, in addition to physical and geometric nonlinearity, the effects connected with the cooperating width of the plate in reinforced concrete and steel-concrete composite T- sections (Fig. 3). 4. Summary The presented algorithm for global steel, reinforced concrete and steel-concrete composite beam structures, based on the concept of generalized cross-sectional rigidity, appears to be a synthetic and universal calculation approach. Applying this type of non-standard understanding of the constitutive law, many different factors can be accounted for, such as: physical nonlinearity of materials, geometric nonlinearity, various types of cross-sectional materials, crosssectional uction due to the shear lag effect, class 4 cross-section uction, the change in the rigidity of beams in fire conditions and changeable effective width in reinforced concrete and composite T-sections. The presented algorithm also allows for taking into account the susceptibility of connections of any given characteristics and calculating frame structures using the method of rigid-plastic analysis (plastic hinges). References [1] M. Szumigała, K. Ciesielczyk, Shear lag effect in the numerical experiment, Archives of Civil Engineering 3 (2015) 31-50. [2] M. Szumigała, Composite steel-concrete beam and frame structures under actual state of loading, Publishing House of Poznan University of Technology, Poznan, 2007. [3] M. Szumigała, K. Ciesielczyk, The impact of local instability on the stiffness in structures, XIV-th Symposium Stability of Structures, Zakopane 2015. [4] M. Szumigała, R. Studziński, The impact of fire on the durability of steel structures, In: M. Kamiński, J. Jasiczak, W. Buczkowski, T. Błaszczyński editors, Permanent repair solutions in buildings, DWE, Wrocław, 2010. [5] Dissemination of Fire Safety Engineering Knowledge throughout Europe DIFISEK+, RFS2-CT-2007_00030, 2008. [6] ABAQUS User s Manual, version 6.4, Hibbitt, Karlson & Sorensen, Inc 2003. [7] T. Łodygowski, Geometrically nonlinear analysis of rigid-plastic and elastic-plastic beams and frames, IPPT PAN, Warszawa 1982.