# Introduction to Modeling-Dr Temsah

December 14, 2017 | Author: Ayman Trad | Category: Structural Analysis, Finite Element Method, Beam (Structure), Stiffness, Stress (Mechanics)

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Descripción: Modeling techniques using finite element method and tools using softwares like Etabs...

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STRUCTURAL MODELING

SIMPLIFIED GUIDELINES OF MODELING TECHNIQUES BY DR YEHYA TEMSAH

Preface Structural analysis is a key part of design in civil structural engineering. The structural analysis is the procedure that enables the determination of the structural response (the internal forces and the movement components) considering the applied external effects (loads, displacements, thermal…) and the boundary conditions. It was not too long ago that structural analysis methods were performed manually using the various conventional theory of structures methods such as the moment distribution method, the slope deflection method, the matrix method…. The use of these conventional manual methods is commonly accompanied with difficulties when conducting complex structural analysis e.g. the study of three dimensional structures, the dynamic analysis, the non-linear behavior…Moreover, these methods require long time for calculation and may result in inaccuracy of the obtained results. Another method of calculation is the finite element method (FEM). The FEM, developed long time ago, provides high level of accuracy when used in the structural analysis, but the mathematical complexity of the method made it impractical for manual analysis. The development of the computers and the evolution of their capacity in the previous decades allowed for the integration of the FEM as numerical method for the use in the structural analysis. Accordingly, the FEM became typically the base for the modern structural analysis. Performing adequate numerical structural studies requires basic knowledge in the FEM and strong theoretical structural background (Theory of Structures, structural behavior of Tall Buildings…). In other words, The behavior of a structure ,when subjected to more or less complex effects (dynamic, nonlinearity…), should be predictable by the user based on his former theoretical knowledge and experience in the structural domain. The importance of the above concept was highlighted by ACI President's Memo José M. IzquierdoEncarnación 2003:

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“As a rule, a program should be used only if engineers can predict the general deflection and distribution of moments in the structure prior to obtaining a solution. The computed solution is used to verify the results previously predicted by the engineers. If the solution is significantly different from the prediction, engineers should use the results only if they can satisfactorily explain the reason for the discrepancy and find it acceptable.” The type of expected results may include:  Approximate values of some structural response components such as the slabs/beams deflections or the sway/drifts buildings. The approximate values may be obtained from simplified theory of structures methods (moment distributions...)  Recognizing the deformed shape of the whole structure and the structural elements under the applied loads Since most of the software, manuals provide guidelines for the use of specific software considering the user as already having the required theoretical knowledge and adequate experience, the intent of this manual to provide simplified basic guidelines of the structural modeling techniques combining:  The complexity of numerical analysis by using Finite Element Method(FEM)  The systematic procedures to use these software that are stated in the software help manuals The aim is to enable the user to construct a numerical model that properly generate the expected responses of a structure In other words, it is a simplified framework to provide guidelines for all structural engineers including fresh graduates and undergraduate students. It presents clarifications and answers that help the user comprehend the different aspects of structural modeling by understanding the concepts of analysis of the structural elements and the various ways to reflect this analysis as given inputs within the software. It tackles the structural elements as separate subjects clarifying the different ways to deal with these elements based on given criteria. Hoping this manual to serve its purpose, it is only the first edition. Your comments, feedbacks, suggestions and queries are all welcomed to bring out the best of and enhance the editions yet to come. 3

Contents 1. Introduction 2. Degrees of Freedom 3. Finite Elements for Structural Modeling 4. Local Axis System 5. Meshing of Area Elements (Slabs, Walls, Domes…) 6. Modeling of Columns 7. Vertical Alignment 7.1 Elements of centerlines along the same vertical axis 7.2. Use stiff rigid elements 7.3. Use of shell FE 8. Modeling of Structural Walls and Core Walls 8.1 Meshing of Walls 8.2 Vertical Discontinuity in Walls 8.3 Openings in the Structural Walls 8.4 Pier assignment of shear walls and core Walls 8.5 Modeling of Walls and Core-Walls with Frame FE 9. Horizontal Alignment 10. Modeling of Beams 11. Deep Beam (Wall-Beam) 12. Modeling of Floor Slabs 13. Modeling of Ramps and Stairs 14. Story Data 15. Basic Assumptions 16. Lateral Earth Pressure on Basement Walls 17. Stiffness Modifiers 17.1 Stiffness modifiers for the FE direct results 17.2 Stiffness modifiers for the FE indirect shell results 4

18. Diaphragm behavior of floor slabs 19. Connectivity of Vertical to Horizontal Structural Elements 20. P-Delta analysis of buildings 21. Seismic additional eccentricity 22. Modeling of Foundations 23. Modeling of Piles 24. Modeling of Pile-Raft foundation 25. Import of geometrical data from AutoCAD Files

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1.

Introduction

The structural study of new or existing structures aims to design or assess the structures capacity to support the effects of the straining effects as applied external loads, imposed movements, temperature, acceleration transmitted from supports…. The structural analysis constitutes a major step in the structural study. It consists of determining the structural responses (movements and internal forces) resulting from the straining effects and boundary conditions (supporting systems). This may be performed manually using the conventional methods of the theory of structures yet will be approximate, or by more accurate methods such as the use of Finite Elements Method (FEM). To carry out the analysis of a given structure by using FE, its structural elements are divided into finite number of small elements of shapes like lines (frame elements) or areas (triangles or quadrangle elements). FE forms interconnected with their boundary nodes.

FE models can be created using one-dimensional (frame elements), twodimensional (2D shell elements) or three-dimensional (3D solid elements).

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2.

Degrees of Freedom

The degrees of freedoms (D.O.F) represent the components of movement (translations and rotations) of an element from an initial to a final position. The movement components for an element in space consist of:

► 3 displacement components: UX, UY, and UZ ► 3 rotational components: RX, RY, and RZ Where “Ui” represents the displacement parallel to “i” axis, and “Rj” represents the rotation around the “j” axis. 3. Finite Elements for Structural Modeling The finite elements that are commonly used for the structural analysis are: ► The Frame (or bar) elements with various D.O.F, such as the frame elements with 1 D.O.F (along translation D.O.F direction) to represent tie beams, or frame elements with 3 D.O.F (one translation and 2 rotation D.O.F) to represent beams subjected to non-axial loads within one plane (local x-z or local x-y), or with 6 D.O.F to represent columns subjected to axial forces and shear forces along both (X & Y) horizontal directions

Frame Element 7

Frame elements may be also used to represent structural walls (column with equivalent section), or slabs (the grillage method) where the slab is represented by a grid of 3 D.O.F frame elements along the length and the width of the slab. ► Shell elements with various D.O.F, as example: The plate elements (shell elements with 3 D.O.F UZ, RX, & RY) are used to represent slabs The membrane elements (shell elements with 3 D.O.F UX, UY, and RZ) are used to represent structural elements in the case where the axial forces represent the major internal force components (as the domes and arched roof) The shell elements with 6 D.O.F to represent structural elements where the generated internal forces include axial forces and flexural moments (shear walls, prestressed concrete floor slabs, water tank walls and slabs,…)

Quadrangle and Triangle Shell Elements ► Volumetric elements of 3, 6, or 8 nodes of 6 D.O.F, used for the modeling of volumetric structures such as dams, thick transfer slabs....

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4. Global and Local Axis System The geometry of a structure (joint coordinates) is defined in an user defined global axis system. The structural response is not affected by the location of the global axis origin (0,0,0).

Frame Local Axis

Shell Local Axis

With the finite element method, each element is allocated an independent local axis system (independent from the global system and the local systems of the other elements), where typically:  The local x (or 1) axis is parallel to the element neutral axis, from origin joint to end joint  The local y ( 2 or 3) axis is the second planar axis, perpendicular to x axis  The local z (2 or 3) axis is perpendicular to xy (1-2 or 1-3) plane The right hand rule may also helps in defining the local axis system shown in the next figure.

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The analysis results for FE are obtained according to the local axis systems. For example the internal forces of the frame element parallel to local axis 1 (or x) represents the axial force (FX), V2 (or FZ) the vertical shear, V3 (or FY) the horizontal shear, T (Mx) the torsion moment, M2 (MZ) the horizontal flexural moment, M3 (MY)the vertical flexural moment.

For the shell elements F11, F12 and F22 represent the in-plane internal forces, whereas V11, V12, V22, M11, M22, & M12 represent the out-ofplane internal forces.

F11→ FXX; F22→FYY; F12→FXY S11→ SXX; S22→SYY; S12→SXY M11→ MXX; M22→MYY; M12→MXY

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5. Basic Assumptions The modeling of structures with finite elements software is commonly performed taking into consideration the following: 1. Rigid connectivity between FE, which generates complex behavior of the rigidly interconnected structural systems (framing systems, dual system, coupled system….). Rigid default connectivity type may however be modified with total or partial releases of one or more than one D.O.F 2. Linear behavior of materials (Hook's law). However, an enhancement of the material behavior may be induced with the modification of the elements stiffness 3. Assignment by default of support type as in Etabs where all joints at the base level are assigned pin support type. The type of supports may be modified to a fixed type in order to enable a transmission of moments to the foundations. More realistic system of foundation may be adopted with plate elements (representing the foundations) supported by springs (elastic foundations) 4. The dimensions of the FE (frame elements and shell elements) are geometrically extended to the connectivity joint or line of these elements.

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In the case of large difference between the clear length and the FE length, rigid elements may be added to account for this difference.

6. Meshing of Area Elements (Slabs, Walls, Domes…) Meshing is the operation that transforms a contour area (wall, slab, dome...) into a set of FE (shell, membrane, or plate) type triangles, quadrangles, or combination of both.

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The regularity of the FE shapes and their size affect the accuracy of the analysis results. The most regular FE shapes are the square for quadrangle FE and the equilateral triangle for the triangular FE. However, it is recommended to consider the ratio of shape not more than 1:2 (the minimum length to maximum length). The "mesh refinement" transforms the FE into smaller element sizes and therefore increases the FE number in a contour area. The next figures show a rectangular (8x8m) solid slab, 20 cm thick meshed with regular rectangular mesh than with an irregular mesh respectively. The moment value at the center of area for the regularly meshed slab was 32.7 KNm whereas the irregular mesh generated 30.4 KNm moment at the same point.

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Theoretically, the use of regular elements shape and the mesh refinement (decreases of the FE size) results in results that are more accurate. However very fine mesh results in large size of the generated model and higher run time without significant increase of the accuracy of the obtained results. It is recommended to use refined mesh in the zones where concentration of stresses are expected i.e. zones of the slab near supports or where subjected to concentrated loads or moments. It is advisable to use mesh size with the ratio 2:2:1 (the size of the element in its plane directions is 2 times the thickness). Another recommendation is to consider the elements size not greater than 0.5m, or the span divided by 10. The below figure 1 shows the layout of 20cm concrete solid slab supported by 9 columns with 8.0 m spacing, whereas figures 2, 3, and 4 show the moment maps due to the self weight of the slab meshed with elements size (1.6x1.6m), (0.4x0.4 m), and (0.1x0.1 m) respectively.

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Figure 3

Figure 4

It can be noticed that:  The moment value is increased by 6.76 % with the coarse mesh (1.6x1.6m) for the fine mesh (0.4x0.4m) -from 68.669 KNm/m to 73.313 KNm/m-.The fine mesh represents the recommended ratio (2:2:1).  The use of very fine mesh (0.1x0.1m) generates moment (73.216 KNm/m) slightly different than the moment of the fine mesh. (variation of 0.13%). In brief, when the structural study targets the global behavior of the building -as in Etabs- and the design of the structural elements (except the floor slabs design), the refinement of the floor slabs mesh is not of major importance since the results will not be used for the design of the slabs. The refinement becomes of importance when the study targets the slabs analysis and design (as in SAFE). 7 . Modeling of Columns Columns as previously mentioned are commonly represented by frame F.E. The frame FE analysis results, namely the internal forces, are directly used for the design of sections, or the determination of the capacity, of the columns.

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8. Vertical Alignment As the dimensions of columns and/or walls elements may be reduced in the upper floors as the internal forces decrease, the center lines (CL) of these elements become non-vertically aligned; therefore the frame FE will be disconnected at the floor levels. The vertical continuity of the vertical elements is realized throughout the following methods: 8.1. Elements of center lines along the same vertical axis Vertical elements can be considered aligned if their center lines are at the same vertical "Z" axis or if their center lines are slightly shifted. Slight difference in the vertical alignment of the center lines may be ignored in order to avoid the complexity related to the use of short rigid elements to connect the CL at the floor levels

Architectural layout (eccentric CL of Columns) Vertical Alignment 16

8.2. Use of Rigid Elements The differences between the columns center lines may be considered with the use of stiff rigid elements at the floor levels (applicable for both columns and walls). Rigid Element is a weightless frame FE with extremely high flexural and shear stiffness.

8.3. Use of shell FE Compared to the frame FE, higher geometrical and analytical accuracy are generated with the use of shell elements because more joints are used to define the column (4 FE instead of 2). However, the design of the column in this case should not be performed in the same software since shell elements are considered as shear walls in the design process (as in ETABS).

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9. Horizontal Alignment of walls The walls and core-wall may have different thicknesses of their parts as shown in the next figures. These differences lead to discontinuity of the neutral axis of the wall and the core wall parts and therefore disconnected wall section and core-wall section when shell elements are used to represent these structural elements.

The automatic conversion of sections including thickness variations from AutoCAD Polyline to structural wall elements may result in improper connections of the shell elements. To avoid the complexity of connectivity by rigid elements and to enable the continuity between the parts, it is advisable to consider an idealized wall or core wall sections as shown in the next figures.

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10. Columns embedded or connected to Structural Walls Columns may be embedded in the structural walls in the basement floors as for the cases shown in the below figures.

The following assumptions may be adopted for the numerical study:  The part of the core-wall going along the retaining wall may be considered same as a part of the retaining wall with different thickness (as previously explained in paragraph 9).  The implanted column need not to be assigned within the shell core-wall element, as the internal forces in the embedded columns decrease due to the shell stiffness. The internal forces of the column section above the shell element commonly govern the design of the column. Typically, the reinforcement of the critical section at the bottom of the column (at the link with the shell element) is extended to the floor below in the shell element.  The columns linked to a structural wall may be shifted to the wall center line (or the wall center line shifted to the columns c.g ), or a rigid stiff beam element may be used to connect the column c.g to the wall center line. 19

11. Structural Walls and Core Walls Structural (shear) Walls and core-walls are commonly represented by shell FE. The internal forces generated in these elements include in-plane and out-ofplane components (Axial and shear forces, torsion and biaxial flexural moments). Considering shear wall as a membrane FE results in an underestimation of the wall capacity since in the membrane FE 3 in-plane DOF are involved in the analysis whereas in the shell FE 6 DOF are involved →→ no generation of the out of-plane internal forces in the shear wall modeled as membrane FE. 10.1 Meshing of Walls A rectangular wall may be represented with minimum one rectangular shell element in each floor. In Etabs, it has by default the local axis system as shown in the next figure.

For a higher accuracy of the analysis results, it’s highly advisable to mesh wall elements as previously recommended (the single wall shell element is not automatically meshed by default in Etabs). 10.2 Vertical Discontinuity in Walls A wall varying in length along the building height results in discontinuous vertical shell elements.

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Deformed Shapes due to Lateral Forces To ensure the continuity of the wall, shell elements may be assigned auto edge constraints (Etabs assigns by default auto-edge constraints). A better approach of this case may be obtained by subdividing the longer element into several elements as shown in the next figure.

Deformed Shapes due to Lateral Forces 10.3 Openings in the Structural Walls Openings in the structural walls may be ignored if the area of the openings is smaller than 15% of the wall area at the same floor (an exception is to be considered for the case of longitudinal or transversal strip openings). Larger openings shall be included within the wall shell elements.

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10.4 Pier assignment of shear wall and core Walls When shell elements are used for the walls and core-walls, the generated results are distributed per unit length of the element (forces/ unit length, moments/unit length...). For design purposes it may be preferable to obtain the resultants of wall internal forces as concentrated along the neutral axis, similarly to the frame FE results. This may be achieved with different ways in the software (as advanced "reduced results" in Robot Millennium or as pier results in Etabs....) The pier function (or the reduced results) generates internal forces including an important component for the design of the wall section that is the in-plane moment of the wall (this component is not directly obtained as FE shell result). The in-plane moment (M3 or MY) is calculated from the summation of the couples of axial forces generated from the normal stresses of the shell element (s22 or syy).

Walls and core walls(1) may be assigned same pier label at all floor levels except for the case were the wall is subdivided in more than one shell within the same floor (case for large wall opening(2) ), in such case each shell is assigned a different pier label. 22

The core wall maybe assigned several pier labels for each shell element. Whatever is the assignment method of the core wall, the design results will be the same (the area of reinforcement of the whole pier section = summation of the reinforcement area of the different piers) since the pier

results -or the reduced results- are generated from the same FE shell analysis. (1)

When a wall is assigned an opening, the shell element shall be subdivided into elements connected at boundary joints as illustrated in the next figures.

Wall (shell element) with opening

Subdivision into several shell elements 23

10.5 Walls and Core-Walls with Frame FE Structural walls and core walls may be represented by frame FE for a simplified analysis as follow:  The structural wall is assigned a frame FE -wide section- at the center of gravity1 (CG) line of the wall sections.

 The Core-wall is assigned 2 frames FE: o The first frame element is going along the neutral axis of the corewall, and assigned the section properties related to axial stiffness (namely the area and the modulus of elasticity). o The second frame is located along the vertical axis of the shear center and assigned the section properties related to the flexural and torsion stiffness properties (Ix, Iy, and Iz) 2,3 .

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The main advantage of using the Frame FE approach is the direct use of the results (the internal forces) in the design of the wall/ core-wall sections, whereas the use of shell elements requires followed by a pier assignment (or reduced results) to generate results similar to the frame. The main disadvantage is related to the disregard of the warping stresses of the core-wall section behaving as thin-walled. (1)

The Center of Gravity (CG) -or the center of area- is structurally defined as the point of the cross section of an element that causes uniform stresses and shortening -or elongation- of the element when subjected to normal forces. When the applied normal force is eccentric with respect to the CG, it generates different axial shortening (or elongation) of the section points.

(2)

The Shear Center (SC) is structurally defined as the point of the cross section of an element that causes lateral displacement for the case of vertical elements (in-plane displacement of the element cross sections), when a lateral force is applied at the SC. When the lateral force is applied eccentrically to the SC the element, sections displace horizontally and rotate (twist).

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Shear Force along the Shear center

Torsion moment generated from eccentric shear force

10.4 Boundary zones of shear walls and Core-Walls with Shear walls are assigned boundary zone of higher ratio of reinforcement when the normal stresses at the wall boundaries exceed certain limit. The boundary zones length from the of shear wall edge is varying from 0.15Lw to 0.25Lw as shown in the next figure.

For the core wall sections the boundary zones may be defined as the corner zones of the walls intersection as the sections type U-shape, L26

shape, box-shape ... However in the case of complex section shape not all walls intersection constitutes boundary zones.

The end zone in core-wall sections may be identified with as the zones maximum normal stress values within the simplified normal stress equation for a section subjected to axial force (N), and biaxial moments (Mx and My) as explained in the following procedure. - The normal stress equation: (for an elastic linear distribution of stresses) = N/A ± (Mx/Ix)*x ± (My/Iy)*y - The datum of the stress diagram (line 0f zero stress) is obtained for = 0, which lead to: = N/A ± (Mx/Ix)*x ± (My/Iy)*y → y= a*x+b (line equation) - The extreme values of stresses for a given (N, Mx ,and My) are obtained for the points at the far points of the section on the datum line as shown in the next figure. As it can be concluded, the external corner edges of the section generate the extreme stresses, whereas the stresses at all section parts (including the interior walls intersections) are within the extreme values.

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11. Modeling of Beams Similarly to the columns, beams are commonly represented in the numerical model as frame FE or with shell elements as for the cases listed in the next paragraph. Due to the rigid -by default- connectivity type between beam frame elements to the vertical (columns or walls) frame or shell elements, the numerical analysis generates framing (beam-columns) and coupling (beam-walls) behaviors. Due to the complexity of the design, detailing, and execution of the beamcolumn and the beam-wall connections, the beam elements may not be included in the building numerical. The beams role as supporting elements of the floor slabs may be included in the study of the slabs (using SAFE software for example). When beams are included in the structural model in order to support gravity loads, rotational releases should be assigned to the beam-column and beam-wall connection to prevent framing or coupling behavior. 28

However releases of all structure beams may lead to instability warning messages due to excessive releases.

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12. Deep Beam (Wall-Beam) Beams may be represented by shell FE (that is considered as the more accurate modeling approach) for the following cases of beams:  Deep beams (or the wall beams) where the beam depth may cover a whole floor height. The deep beams may be used to support the loads of implanted columns  The coupling beams which are beams connecting walls along their strongest axis of inertia  Beams connecting slab parts of different levels within the same floor.

Similar to the case of pier assignment of shear walls, when beams are represented by shell elements, they are assigned spandrel label to generate design forces (or as reduced results) at the centerline of the beam.

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The internal forces include the in-plane flexural moments (M3 or My) that are calculated from the summation of the couple of forces generated from the normal stresses (s11 or sxx), in a way similar to the determination of the in-plane moment of shear wall assigned as pier. 13. Floor Slabs Floors slabs are represented by thin 6 DOF shell FE. However slabs may be represented by either:  Membrane FE to prevent the slab flexural stiffness from generating framing behavior ,with columns, or coupling behavior (with walls) due to lateral forces.  Plate FE which includes the flexural out of-plane slab inertia in the numerical analysis. In this condition, no temperature analysis can be performed since no in-plane D.O.F are involved.  Shell FE, where all degrees of freedom are used to generate in-plane and out of-plane internal forces (as the PT slabs). 14. Ramps and Stairs Ramps and Stairs are type of inclined slabs between story levels. However, and since they do not affect significantly the gravity loads distribution or the diaphragm behavior of the floor slabs when buildings are subjected to lateral forces, an approximation may be considered as flat ramps and stairs at each floor level. 15. Story Data The story data for the numerical model may be summarized with the following points:  Stories number: is the same as the number of floors or number of slabs.  The story height: is the distance between the floor slabs midthickness, except for the floor directly above foundation where the 31

story height is considered as the distance from the top of the foundation to the mid-thickness of the first slab.

 When the foundation system in not included in the numerical model, the first floor slab is the cover slab of the foundation level, i.e. if the building consists of 2 basement floors, the first floor slab assigned is the basement 1 architectural slab.  Unlike the architectural drawings of floor slabs that shows the slab geometry ,including shafts, recessed zones…, and the vertical elements above this slab, in some software (as ETABS), slabs are assigned the vertical elements below -supporting- the slab, supporting it. 16. Lateral Earth Pressure on Basement Walls The basement Walls constitute important stiffeners for the buildings when subjected to lateral forces due to wind pressure or earthquakes. The earth pressure on the basement walls may be ignored due to the following reasons: 32

 In the case of completely embedded basement floors - The basement walls are subjected to equal and opposite lateral earth pressure. Thus, the corresponding resultant is equal to null. - The lateral earth pressure generates compressive stresses in the floor slabs and commonly minor effects on the vertical elements. However, in the case of non-rigid diaphragm of the floor slabs, the effect of the lateral pressure on the vertical elements should be taken into consideration.  In the case of partially embedded basement floors (no basement peripheral walls at one or more than one sides of the basement floors), part of the lateral earth pressure is transmitted as story shear to the vertical elements. In such case, the earth pressure should be assigned to the basement walls.

17. Stiffness Modifiers When the structural analysis is conducted considering elastic behavior of concrete material (Hook’s low:  = E) as in the case of most engineering software, the flexural cracking of concrete and the corresponding reduction of the flexural stiffness is not taken into consideration. Since the vertical elements (columns and walls) are mostly subjected to compressive axial forces, they crack less than horizontal elements (beams and slabs) which are subjected mostly to flexural moments. The elastic analysis results in:  Underestimation of the internal forces in the vertical elements  Overestimation of the internal forces in the horizontal elements.  Underestimation of the building lateral displacements (sway and drifts), and deflections of slabs and beams.

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The effects of concrete cracking can be considered with the ACI318 (6.6.3.1.1) reduced inertia for vertical and horizontal elements as follow:

The reduction of inertia may be assigned to the different elements to affect the direct FE analysis results, or to affect the indirect FE shell results as explained hereafter:

17.1 Stiffness modifiers for the FE direct results  The frame FE (columns, beams, walls as wide columns...): the reduction of stiffness is assigned for the moments of inertia about axis 2 (or z) and axis 3 (or y).  The slabs (represented by plate of shell FE) → the reduction of stiffness is assigned for the flexural movements m11 (or mxx) and m22 (or myy). 17.1 Stiffness modifiers for the FE indirect shell results When shell FE represent walls and wall-beams, the modification of the stiffness modifier, the out of-plane bending along m11 (mxx) and m22 (myy), will not affect the in-plane main in-plane flexural moment M3 (My) generated from normal stresses s22 (syy) for the walls, and s11 (sxx) for the wall-beam. The modification of the stiffness modifier components should therefore be as follow:

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 For The wall-beam: the reduction is assigned to membrane f11 (sxx) direction.  For The structural walls and the core-walls: the reduction is assigned to membrane f22 (syy) direction (1), bending m22 (myy) direction, and bending m11 (mxx) direction (2). (1)

It is to note that when reducing the membrane f22 (syy) modifier in order to reduce the flexural stiffness for the in-plane moments, this reduces also the axial stiffness of the walls; thus resulting in an underestimation of design axial forces of the walls due to the gravity (dead and live) loads.

(2)

Where the floor slabs are assigned rigid diaphragm behavior (no in-plane moments are generated), m11 (mxx) has no significant value and may be omitted.

18. Fixity level for seismic forces When subjected to lateral earthquake forces, The fixity level of the building may be defined as the level of maximum internal forces, or the level of minimum displacements. Based on this definition, the fixity level may not be necessarily the same as the foundations level, especially for buildings with basement floors connected to the basement peripheral walls. In most situations, when the basement floors are connected to the peripheral basement walls ( the retaining walls) the ground level represents the fixity level of the building with the reduction of the earthquake moments on the structural resisting system of the building, due to the contribution of the basement wall flexural stiffness.

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However, during the design phase of the vertical elements, it should be taken into account the fact that as the earthquake moment decreases below the ground level, the axial gravity forces increase, and therefore the design may be governed by maximum moment at ground level or maximum axial at lower basement levels. 18. Diaphragm behavior of floor slabs The effects of the lateral forces are commonly related to location of the applied forces. This in turn is related to specific geometrical points such as the center of mass (COM), and the center of rotation (COR). The Center Of Mass represents the location of the resultant of floor mass i.e. the center of slab area when the mass is uniformly distributed, and the related vertical elements masses. The Vertical element masses include half the vertical elements height below the slab, and half the height above slab as shown in the next figure. The vertical element mass is considered as the center of gravity of the section.

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Floors –lumped- masses

Masses at a floor level

The Center of Rotation represents the center of inertia when the resisting system to lateral forces consists of shear walls, core walls, cantilevered columns…) or the center of shear rigidity for the frame resisting system. The COR location is defined as the resultant of vertical elements inertia (or shear rigidity). The inertia of vertical elements is located at the center area of solid sections (rectangular, circular….) or the shear center for the thin walled sections (core walls). The position of the COM or the COI may be determined considering arbitrary origin as shown in the next figures for the determination of the COR.

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For the study of the building response to lateral forces as wind pressure, earthquakes, earth pressure…. (Except the temperature effects), diaphragm behavior is assigned to the floor slabs. The rigid diaphragm behavior may be explained by the following figures.

Points “A” and “B” have original coordinates (xA, yA), (xB, yB) with respect to a reference point “O”, and “” the angel AOB. When the floor slab is subjected to horizontal force and torsion (in-plane) moment, the new coordinates of A' (xA’, yA’),B' (xB’, yB’) and ’ are defined in the deformed slab. If: xA= xA’; yA= yA’; xB= xB’; yB= yB’; and =’, than “A”, “B”, “O” (and all slab points) are moved with the same degrees of freedom (2 displacements UX, UY, and one rotation RZ) and the slab is behaving as rigid diaphragm. The assignment of diaphragm to floor slab has the main advantage of reducing the numbers of unknowns (degrees of freedom) for each slab point from 6 times the number of joints to 3, which results in reduction of the runtime of the analysis. In addition, the diaphragm extent is used in Etabs to evaluate the wind forces at the different story slabs levels. The diaphragm behavior cannot be assigned to floor slabs for the analysis of loads that generate in-plane forces or deformation such as the temperature gradient and the prestressed forces. The diaphragm represents an infinitely in-plane (horizontal) stiffness of the floor slab. When the in-plane stiffness is reduced by any factor, semirigid or flexible diaphragm behavior may be assigned to the floor slab. 38

Rigid diaphragm is not applicable for the following cases.  The opening area in slab exceeds 50% of the floor area  The strip slab shape

Another criteria for the use of rigid diaphragm behavior, is when the maximum lateral displacement in a floor slab exceeds 20 percent the average displacement at the same floor slab. 19. Connectivity of Vertical to Horizontal Structural Elements By default, the connectivity of the structural finite elements in the numerical models is of rigid type, and may be transformed to released or partially released connection types. However, these types of connectivity do not effectively represent the accurate behavior of connections in the executed structures (namely the concrete structures) due to the following reasons: a) The generated forces within the connecting joint are based on perfect elastic behavior of the material (Hook’s low) and gross section properties of the connected elements. Yet at certain level of stresses, the material behaves non-linearly, and cracks of concrete sections start to appear leading to reduction of the sections inertia. Same phenomena may be generated in “plastic” zones of the elements, away from the connectivity joint, due to excessive reversal shear or normal stresses. 39

Moreover, the use of stiffness modifiers to account for the reduction of elements stiffness constitutes also an approximation since it is based on the consideration of same –uniform- reduction of the stiffness along the element length. The realistic behavior generates reduction of inertia (due to cracking) that varies within the same element from section to another in accordance to the stresses magnitude. b) The common execution detailing practice of the horizontal-tovertical connection where no continuity in reinforcement is provided. The floor slabs and beams are typically detailed as “pin” supported by the columns and walls, as shown in the next figures. A quasi hinge behavior is developed for the shown typical detail unless the development length of rebar is enough extended to allow for fixed or partially fixed joint behavior.

The rigid -by default- connectivity of the vertical elements to the horizontal elements at top floor level generates moments that govern the design of the vertical elements at this level. The ratio of reinforcement of the vertical elements, at this level, commonly exceeds the ratio at several floors below. The vertical element can 40

be assigned end release to prevent the generation of moments at the connecting joint. In the case where direct end release for walls (or slab) is not available in the software as in Etabs for example, a reduction of the out-of plane stiffness modifier (m22) leads to same release results.

c) The release of rotation at an element edge cannot be achieved in the common practice of construction unless special element, like bearing pads is used. This is due to the requirement of real hinges to reduce the section moment of inertia of connected element edge to enable free rotational movements. 20. P-Delta analysis of buildings The P-delta analysis of building accounts for the geometrical nonlinearity effects on the vertical elements. The next sketches show the generated forces (Diagrams of normal forces, shear forces, and moments respectively) of a vertical cantilever without geometrical nonlinearity and with geometrical P-Delta analysis.

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The effect of P-delta analysis may not have important effects on building unless the stability coefficient ratio "" of the cumulative secondary to main moment exceeds 0.1. According to the IBC 1617.4.6.2 "" is determined by the equation:  = Px/VxhsxCd Where: Px = total unfactored vertical design load above level x = design story drift occurring simultaneously with Vx Vx = seismic shear force acting between levels x and x-1 hsx = story height below level x Cd = deflection amplification factor

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21. Seismic additional eccentricity The torsion additional eccentricity for the study of seismic forces, given as minimum of 5% of the floor projected length perpendicular to the story seismic shear force, should be considered as positive and negative respectively. By considering two cases of loading (positive and negative eccentricities) the analysis generates an envelope of internal forces for the resisting system elements. When the structural study includes dynamic analysis cases of loading, no additional eccentricity need to be taken into consideration for the static seismic analysis since the value of the base shear, which is used for the scaling of the dynamic base shear, is not affected by the additional eccentricity 22. Modeling of Foundations The modeling of the foundations is performed as thick plates set on elastic supports. However, the following should be taken into consideration The soil stiffness is determined with the soil subgrade modulus "Ksub" that is commonly provided by the geotechnical study or by approximate methods as equal to 120*qallowable or Es/B(1-2),..... Where:  qallowable = soil bearing capacity (t/m2)  Es= soil modulus of elasticity  B = the least foundation planar size   = soil poisson's ratio  By using the subgrade modulus, the software generates springs (representing the soil stiffness) with single partially restricted degree of freedom parallel to the vertical global direction.  Lateral supports (pin, roller, or spring) should be assigned to the foundation to prevent instability of the structure  The above motioned methods represent an approximation to the soil-structure interaction behavior, since it does not account for the main important factors: 43

o The non-linear behavior of the soil o the mutual interaction behavior of the soil-structure (redistribution of stresses in the upper structural system and the soil) o The effects of load from adjacent structures 23. Modeling of Piles The bearing pile is commonly represented by spring with one -or multiD.O.F. The spring is assigned stiffness along the vertical and horizontal directions for pile-foundation hinge connectivity, or springs along the vertical, horizontal, and rotational directions for the pile-foundation rigid connectivity type. The vertical stiffness Kz of the spring represents the least value between the reinforced concrete capacity and the geotechnical capacity (skin capacity + end bearing capacity if exist). The best evaluation of Kz value is determined from the load bearing test, which is commonly performed at the early stage of structure execution. Therefore, a preliminary value of Kz should be adopted for the structural study phase. The preliminary value of Kz can be assumed as: o Similar to the values from already tested piles of a nearby project o From recommended geotechnical studies. o Conservatively evaluated as the axial stiffness of the concrete pile as: F= Kz allowable→ Kz = F/ allowable where: F= Apile*conc Apile = Cross section area of pile = πd2/4 d = Pile diameter conc = Allowable normal stress of concrete assumed limited to 0.25f'c f'c = Compressive strength of concrete allowable = Allowable settlement of pile that may be assumed as 0.01d Kz = πd2* 0.25 f'c/(4*0.01d)≈ 20*f'c*d 44

In the absence of accurate method to evaluate the horizontal stiffness (Kx and Ky) it may be assigned an approximate value of 0.1 Kz. A better modeling of the pile may be obtained by rigidly connecting points of the foundation on the circumference of the pile to the point spring.

Higher accuracy of pile behavior may be obtained with non-linear Kz spring.

24. Modeling of Pile-Raft foundation The modeling of the pile-raft foundation may be performed with springs in accordance to the above-mentioned approaches. However, the large difference between the pile and the soil vertical spring stiffness should be taken into consideration. Otherwise, the analysis results in overestimation of the soil spring contribution to support the loads transmitted to the soil. There are several approximate ways to account for this difference, such as:  Neglecting the soil vertical stiffness in the case where the piles are grid-closely-type distributed 45

 Assigning reduced vertical stiffness to the soil between piles  Neglecting the soil stiffness in the zones surrounding the piles and considering full soil stiffness outside these zones in the case of spaced distribution of piles 25. Warnings Warnings in structural numerical analysis may be generated by many reasons such as the boundary conditions, the F.E mesh (size, shape irregularity, connectivity). Warning may be ignored in some cases or they may importantly affect the analysis results. 25.1 Boundary conditions The boundary conditions may generate geometrical instability of the structure or the supporting joints, and therefore warnings, as illustrated in the following cases. Case1

Case2

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Case3

25.2 Loss of accuracy- Negative stiffness Most software (as Etabs) performs arithmetic calculations with 15 digits of accuracy. 3 cases might be encountered: 1. If the loss of accuracy is less than 6 digits, it is considered negligible by the software and won't be reported 2. If the loss of accuracy is above 11 digits, the software will give an error massage and the running is aborted. 3. If the loss of accuracy is between 6 and 11 digits, the global force balance relative error should be checked for each load case to be relatively small (within 1% for example). this can be checked in the *.out file, the *.log file, or equivalent. 25.2 Negative stiffness The warning due to negative stiffness results commonly for unstable structure. The factors related to the stability (as the boundary conditions and the stiffness) should be revised. 47

25. Import of geometrical data from AutoCAD Files The geometrical structural data that are used for the numerical model may be prepared from the architectural drawings following the below recommended steps:  Four new layers are to be created in the AutoCAD architectural files for th2e: slabs (S-Slab), Walls (S-Wall), Columns (S-Column), and Openings (S-Opening).  The new structure layers are used to create the contour of the corresponding elements i.e. Slabs, Walls, Columns, and Openings. All contour lines are to be closed polyline type.  Each floor slab is exported into a new file with a reference point (same horizontal coordinates point at all levels) at origin (coordinates 0, 0, 0). The reference point should be architecturally fixed such as an inner corner of the lift shaft or staircase.  The floor slab components are to be scaled to 1 unit length= 1m  The new file is to be saved as dxf file (preferably an old AutoCAD version) with the same architectural floor slab label  Same above procedure is repeated for the floor slabs of different geometry

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Will be added to this guidelines: 1. Including Mxy to Mxx and Myy by Wood and Armer or other 2. Caisson behavior of basement floor slabs that better represents the interaction soil-foundation-structure 3. to ensure the orientation of the local axis of the wall prior to assign pier behavior 4. Water tank case of study 5. wall moment diagram in dual system 6. level of scaling dynamic to static 7. check of mass source 8. axial forces in vertical elements manual versus numerical 9. to ask najib kasty about unconnected mesh Upcoming:  Tutorial movies for Buildings using Etabs and Robot  Tutorial movies for reinforced concrete Slabs and foundations using Safe  Tutorial movies for prestressed concrete slabs using Adapt  Advanced modeling of structures using Abaqus

Contact mail: [email protected]

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