Step 1 Define Joint Coordinates

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2 Contents 1. Introduction to the SABRE2 Software System Tutorial Problem Definition and Analysis of a Tapered I Section Member Overview Using the SABRE2 Calculator Problem Definition Elastic Linear Buckling Analysis and Results Inelastic NonLinear Buckling Analysis and Comparison to DG 25 Results Using SABRE Step 1 Define Joint Coordinates Step 2 Define Member to Joint Connectivity and Member Cross Section Dimensions Step 3 Define Essential Nodes and the Cross Section Dimensions at these Nodes Step 4 Subdivide Member Segments and Assign Material Step 5 Define Boundary Conditions Step 6 Define Load Conditions Step 7 Check Structural Analysis Parameters Step 8 Conduct Structural Analysis Step 9 Inspect Results Tutorial Adding a Step in the Flange Width to the Tapered Member Overview Adding a Step to the Outside Flange in SABRE Adding a Step to the Outside Flange in the SABRE2 Calculator Tutorial Clear Span Frame Overview Notes on Modeling of the Clear Span frame in SABRE Inspection of Results for the Clear Span Frame Demonstration Problems Overview W21x44 Beam Lateral Torsional Buckling Curves W21x44 Beam Column Strength Envelopes i

3 5.4 Roof Girder Example Elastic Lateral Torsional Buckling of Web Tapered Beams Elastic Buckling of a Stepped Web Tapered Column Validation Case Studies Overview Doubly Symmetric Beams Singly Symmetric Beams with a Larger Compression Flange Singly Symmetric Beams with a Smaller Compression Flange Doubly Symmetric Beam Columns Singly Symmetric Beam Columns with a Larger Compression Flange, Subjected to Positive Bending Singly Symmetric Beam Columns with a Smaller Compression Flange, Subjected to Positive Bending Experimental Database Employed in the Development of the AISC and AASHTO I Section Flexural Resistance Provisions Potential Directions for Future Work References ii

4 1. Introduction to the SABRE2 Software System SABRE2 is a structural analysis and design software system focused on efficient, rigorous assessment of the strength of frames composed of web tapered and general nonprismatic steel I section members. SABRE2 achieves these capabilities via innovative computational buckling analysis techniques in conjunction with frame finite elements based on open section thin walled beam theory. Specifically, SABRE2 handles: Single and multiple web taper, Steps in the cross section geometry, Double and single symmetry of the member cross sections, and Any combination of compact, noncompact and slender flanges and webs. In addition, SABRE2 can be used to assess compact/nonslender doubly symmetric prismatic members. SABRE2 addresses all the AISC member strength limit states within its buckling calculations, and it can be used to model general lateral and/or torsional bracing as well as member continuity across braced points and any type of end restraints. In this regard, SABRE2 represents a fundamental advancement in the design of steel I section column, beam and beam column members. With SABRE2, the consideration of attributes such as moment gradient, load height, end restraint, member continuity effects and beamcolumn strength interactions is handled via a rigorous computational framework, removing the need for tedious and relatively inaccurate C b, K and beam column strength interaction calculations. Although SABRE2 can handle 3D systems subjected to general spatial loading, its primary emphasis is on the strength (in plane and out of plane) of frames, or their component members, subjected to loads that are nominally in the plane of the frame. Where desired, SABRE2 can capture the influence of offaxis loads, including the position of transverse loads through the cross section depth (i.e., load height ). SABRE2 displays a surface rendering of the deformed (pre buckled or buckled) member geometries under load, allowing for a more complete understanding of the warping (i.e., cross bending) response of I section member flanges in resisting torsion. SABRE2 provides four types of buckling analysis: 1) Elastic Linear Buckling Analysis (ELBA) buckling of the ideal elastic structure neglecting the influence of pre buckling displacements. 2) Elastic Nonlinear Buckling Analysis (ENBA) buckling of the ideal elastic structure considering pre buckling displacement effects. This is accomplished by a second order elastic load deflection solution to determine the pre buckling deformed geometry and internal forces. Conversely, linear buckling analysis uses a first order elastic load deflection solution to estimate the underlying internal forces. The elastic buckling of the structural system is evaluated based on these forces. 3) Inelastic Linear Buckling Analysis (ILBA) buckling of the elastic or inelastic structure neglecting the influence of pre buckling displacements. To obtain the inelastic buckling load, SABRE2 uses an extension and generalization of the traditional AISC column inelastic Stiffness Reduction 1

5 Factor (SRF) method. This approach provides for the calculation of the buckling resistance of any type of I section column, beam or beam column member. The SABRE2 inelastic buckling algorithm can be used to produce a rigorous direct calculation of the AISC member axial resistance, c P n, the AISC member flexural resistance, b M n, and/or the based beam column resistance under combined axial compression (or tension) and flexure. SABRE2 can account for the influence of any type or combination of bracing, member end translational, rotational and/or warping restraint, and continuity with adjacent framing within its calculation of the inelastic buckling resistance. 4) Inelastic Nonlinear Buckling Analysis (INBA) buckling of the elastic or inelastic structure considering pre buckling displacement effects. For beam type members, inelastic linear buckling analysis usually is sufficient to determine the flexural resistance b M n. In addition, column axial resistances, c P n, can be obtained accurately from an inelastic linear buckling analysis in problems where the pre buckling flexural and torsional displacements are relatively small. However, for general members and frames, the solution must track the changes in the structure s geometry under the applied load to satisfy the AISC design requirement that equilibrium must be considered on the deformed geometry of the structure. The SABRE2 INBA algorithm calculates the prebuckling load deflection response based on either the AISC Effective Length or Direct Analysis method rules. However, SABRE2 replaces the traditional checks of the member resistances via Specification algebraic resistance equations, which use various approximations such as C b, K x, K y, K z, etc., by the direct calculation of the buckling resistance of the structure with its members having reduced stiffnesses derived from the Specification resistance equations. As a result, SABRE2 provides a rigorous calculation of c P n, b M n, and/or the beam column resistances within the context of the AISC Effective Length or Direct Analysis methods. The need to determine resistances from the AISC Specification strength equations is replaced by a more general buckling analysis calculation. This allows for a more accurate implementation of the Specification provisions. SABRE2 achieves its solution efficiency by calculating the member inelastic stiffnesses using Stiffness Reduction Factors (SRFs) obtained from the Specification resistance equations, as well as by modeling the member behavior using a Thin Walled Open Section (TWOS) frame element. This frame element uses seven (7) degrees of freedom at each joint or nodal location three translations, three rotations, and one warping degree of freedom. The SABRE2 software system provides two user interfaces for the above calculations: 1) The SABRE2 Calculator, and 2) The main SABRE2 analysis and design engine. The SABRE2 Calculator is a streamlined version of the more general SABRE2 engine aimed at the design of individual fully braced members. The Calculator can handle any doubly or singly symmetric, prismatic, tapered and/or stepped I section member subjected to loads within the plane of the member web. However, it assumes flexurally and torsionally simply supported member end conditions, which is a routine practice commonly employed for the calculation of column c P n and beam b M n values in 2

6 design. Multiple member unbraced lengths can be considered by the Calculator; however, the recommended routine practice is to focus on the strength of unbraced lengths between locations where the member is braced against both twist and out of plane lateral displacement. A given problem can be defined using the graphical user interface of the Calculator; however, problem definition via a streamlined text (.inp) file is also available. The general SABRE2 engine may be used for design solutions where it is desired to consider (or assess) the demands on the bracing system. In addition, the general SABRE2 engine addresses members with any end conditions, multiple members and/or overall framing systems. The SABRE2 Calculator can write out a binary.mat file, which can be read subsequently into the general SABRE2 engine. The general SABRE2 engine then can be employed to include the additional modeling capabilities not available in the Calculator. Problems defined, or modified, using the more general SABRE2 engine cannot be input into the SABRE2 Calculator. SABRE2 is focused only on AISC Load and Resistance Factor Design (LRFD) at the present time. Allowable Strength Design (ASD) can be accommodated within the context of all the procedures implemented in SABRE2. The restriction to LRFD is simply a matter of focusing limited development resources. In addition, the current SABRE2 system is not aimed at production design. Only one load case or load combination is accommodated at any one time. Similar to MASTAN2 (Ziemian 2016), SABRE2 is focused on teaching, demonstration, and fundamental validation of structural analysis and design concepts. For further details of the inelastic buckling based design calculations in SABRE2, the user is encouraged to study the paper by White et al. (2015). For details of the frame element formulation in SABRE2, the user is encouraged to see the paper by Jeong and White (2015). Similar methods to those implemented in SABRE2 have been developed in the prior research by Trahair and Hancock (2004) and Trahair (2009 and 2010). More recently, Kuculer, Gardner and Marcorini (2015a, b, c and d) and Gardner (2015) have developed comparable procedures in the context of design to Eurocode 3. The following sections present various tutorial, demonstration and validation cases. Sample problem definition files corresponding to these cases are provided with the software. 2. Tutorial Problem Definition and Analysis of a Tapered I Section Member 2.1 Overview This tutorial evaluates the design resistance of the linearly tapered member shown in Fig This problem is from Examples 5.6 through 5.8 of the AISC/MBMA Design Guide 25 (Kaehler et al. 2011), Frame Design Using Web Tapered Members (DG 25). 3

7 The member in Fig is representative of a column in a metal building frame, subjected to combined axial compression and an applied moment at its top. In the figure, the member is rotated 90 o clockwise from its actual orientation for ease of display. The column has a single point brace to its outside flange at 90 inches above the column base, and it is assumed to have flexurally and torsionally simplysupported end conditions. That is, the member is assumed to be rigidly restrained against twist, in plane lateral displacement, and out of plane lateral displacement at its ends. However, the flanges are free to warp and bend laterally at the member ends. The intermediate point lateral brace has an elastic stiffness of 2.2 kip/in. The member cross section is singly symmetric. Its outside (top) flange, which is subjected to flexural tension and is restrained laterally at the intermediate braced point, has a smaller thickness. Its larger inside flange is subjected to flexural compression and does not have any intermediate braced point. Point brace, br = 2.2 kip/in x Flange in kip 11.3 kip Flange 1 90 in 144 in d w = h = 12 in at left hand end d w = h = 24 in at right hand end t w = 1/8 in b f = 6 in, both flanges Fig Simply supported web tapered member. E = ksi F y = 55 ksi t ft = 7/32 in, outside or top flange, Flange 2, subjected to flexural tension in this problem t fc = 5/16 in, inside or bottom flange, Flange 1, subjected to flexural compression in this problem 2.2 Using the SABRE2 Calculator The SABRE2 Calculator does not address finite bracing stiffness. The Calculator assumes rigid bracing (zero displacement) constraints at the defined braced points. In this section, the above member is evaluated using this assumption. It should be noted that the use of elastic buckling load ratios, e, in AISC/MBMA DG 25 (Kaehler et al. 2011) and in general stability design, is valid only for cases involving full bracing, that is, situations where the bracing stiffness and strength are sufficient to develop close to the rigidly braced structural capacity. The mapping from the elastic buckling resistance to the member design resistance, captured by the AISC Specification equations (as represented in DG 25), is not appropriate for the general case of 4

8 an inelastic member restrained by a flexible elastic bracing system. This is because finite stiffness elastic braces are not as effective at the higher load levels associated with the theoretical elastic buckling of the member. However, as elastic components, they may provide substantially larger restraint relative to the inelastic member. AISC Appendix 6 addresses this issue by substituting P u < c P n and M u < b M n for the elastic critical loads P cr and M cr in its bracing strength and stiffness requirements. In contrast, SABRE2 directly models the underlying inelastic Stiffness Reduction Factors (SRFs) associated with the AISC strength limit states. This allows for a rigorous capture of the interaction between inelastic members and their bracing systems Problem Definition Figure provides the SABRE2 Calculator text (.inp) file problem definition for the member shown in Fig The particulars of this file are as follows: Lines starting with * are comment lines. Comment lines are used to label the required sequential definition of the input parameters. The line labeled ** Design Axis defines the position of the design axis for the member. The design axis is a simple reference axis used for defining the member geometry. A value of 1 indicates that the web mid depth is employed as the design axis, a value of 2 indicates that the extreme web fiber at the top flange, Flange 2, is used, and a value of 3 indicates that the extreme web fiber at the bottom flange, Flange 1, is used as the design axis. The design axis, located either at the web mid depth, the top flange or the bottom flange, is always a straight horizontal line in the Calculator. Members having a general orientation can be defined using the main SABRE2 engine. The top or bottom of the web are used when the Design Axis is specified at the top or bottom flange since, in cases where there is a step in the flange thickness, there typically is no step in the web depth unless an end plate type splice connection is employed. The top or bottom of the web is often a straight line even if the flange thickness is stepped at some position along the member length. The ** Design Axis data line is followed by the ** Units data line. A value of 1 on this line indicates that US units (inches and kips) are employed. A value of 2 indicates that SI units (kn and cm) are employed. The ** Units data line is followed by group of data lines defining the joint and essential node geometry, loads and boundary conditions. Joints are specified at the member ends. Essential nodes are specified at any locations where there is a step or a change in the taper of the member cross section, and at any intermediate load and/or bracing locations. The line following ** X Coordinates defines the X coordinates of the joints and essential nodes in the order from left to right. At a minimum, one must specify the X coordinates for the member s joints (i.e., the nodes at the member ends). In this tutorial, one essential node is defined at the intermediate brace location. All the input values are separated by white space. 5

9 **************************************************** * SABRE2Calculator * **************************************************** ** Design Axis 1 ** Units 1 * Define Joint & Essential Node Geometry, Loads & Boundary Conditions ** X Coordinates ** bf ** tf ** bf ** tf ** dw ** tw ** Afillets ** Px ** Py ** Mz ** Boundary Conditions ** Load Height ** Step * Subdivide Segment(s)and Assign Yield Stresses ** # of Elements 8 8 ** Fyf ** Fyw ** Fyf * Specify Analysis Parameters ** Use J = 0 for Slender Web Sections in EBA (Yes = 1, No = 0) 1 ** Use the AISC (2016) Provisions (Yes = 1, No = 0) 1 ** Use the DM SRF for GNA in INBA (Yes = 1, No = 0) 1 Figure SABRE2 Calculator text input (.inp) file for the tutorial problem. 6

10 The next six lines define the cross section dimensions of the I shape at each of the joints and essential nodes, starting with the width and thickness b f1 and t f1 of Flange 1, where Flange 1 is the bottom flange in the elevation view for a member defined by specifying the joint and essential node coordinates from left to right. The next two input lines give the width and thickness b f2 and t f2 of Flange 2, the top flange in the above elevation view. Next, the web depth between the insides of the flanges, d w, and the web thickness, t w, are specified for each of the joints and essential nodes. For welded I sections, the web depth d w is denoted by the symbol h in the AISC Specification. However, for rolled sections, d w is different than the AISC definition of h. The above input lines are followed by the line labeled ** Afillets. This line gives the total area of the four web to flange fillets at each of the member joints and essential nodes. For welded sections, this area is commonly taken equal to 0.0. For rolled sections, A fillets can be calculated as A 2b f t f d w t w, where A is the nominal cross section area specified in the AISC Manual. The next three lines define any concentrated loads (total factored LRFD required loads) applied at the member joints and essential nodes. The line labeled ** Px defines any concentrated loads directed along the design axis of the member. The next line, labeled ** Py, defines any loads applied perpendicular to the design axis within the plane of the web. (The position of these loads through the cross section depth is defined by an additional input parameter, discussed below.) Lastly, the data line labeled ** Mz defines any concentrated moments applied at the joints or essential nodes, causing bending within the plane of the problem. The data corresponding to the member joints under the ** Boundary Conditions label gives the location of the displacement constraint at the member ends. This is the shear center if the corresponding value is 1, the top flange (Flange 2) if the value is 2, the bottom flange (Flange 1) if the value is 3, and the cross section centroid if the value is 4. The joints at the ends of the member in the Calculator are always torsionally and flexurally simply supported. This means that the twist rotation and all the translational displacement degrees of freedom are fixed at the left hand end of the member, and the twist rotation and the two lateral displacement degrees of freedom are fixed at the right hand end of the member in all cases. One can address more general end conditions by writing a.mat file from the Calculator and then opening this file in the general SABRE2 engine. The data corresponding to the member essential nodes under the ** Boundary Conditions label indicates any rigid lateral bracing constraints in the out of plane direction at these nodes. A value of 1 indicates that there is no lateral bracing at the nodal location, a value of 2 means that the top of the top flange (Flange 2) is braced laterally, and a value of 3 indicates that both the top and bottom flange are braced laterally (or alternately put, both lateral displacement and twisting of the cross section are prevented at the corresponding node). The case where only the bottom flange (Flange 1) is braced laterally is not accommodated in the Calculator. Generally, it is expected that the member is defined with the top flange, Flange 2, being the outside flange, which may be braced directly by a roof or wall system. Correspondingly, the bottom flange, Flange 1, is always taken as the inside flange, i.e., the flange that is braced typically by diagonal bracing extending from the roof or wall system. 7

11 The next line, denoted by the comment ** Load Height, indicates the position of the loads at the member joints and essential nodes through the depth of the cross section. For this data line, the value 1 indicates that the corresponding loads are applied at the cross section shear center, 2 indicates that the loads are applied at the top flange level, 3 indicates that the loads are applied at the bottom flange level, and 4 indicates that the loads are applied at the centroid of the cross section. A.mat file can be written from the Calculator and then read into the main SABRE2 engine if the user wishes to define the applied loads at a more general or arbitrary height through the depth of the cross section. The line labeled ** Step indicates whether there is a step in one or more of the crosssection dimensions at the essential nodes. The value 1 indicates that there is no step, whereas the value 2 denotes a step. In cases where no step is defined, the cross section dimensions are varied linearly between the adjacent joints and/or essential nodes. When a step is defined at a given essential node, the change(s) in the cross section dimensions between the previous joint or node and the essential node under consideration are implemented abruptly at this essential node. Several attributes of the Step function are as follows: 1) The web depth is always varied linearly between adjacent essential nodes and/or joints in the Calculator (and in SABRE2), regardless of whether the geometry is stepped or not at a given essential node. 2) The Calculator (and SABRE2) model steps in the cross section geometry by tapering the section dimensions over a short length. This is necessary to maintain continuity of the flange warping displacements. For cases where only one of the cross section dimensions is stepped, the taper transition is always placed on the side of the splice that has the larger cross section dimension. For cases where multiple cross section dimensions are stepped, the SABRE2 Calculator (and SABRE2) place the taper transition on the side of the step where the sum of the flange areas is larger. If the sum of the flange areas is the same, the taper transition is located on the side of the step where the total cross section area is larger. This practice approximates the fact that, if a higher order analysis were conducted (say with the flange and web plates modeled by shell elements), the section with the larger plate areas would tend to be only partially effective at the physical step in the cross section. 3) Figure shows several example graphs of how a given dimension, plotted on the vertical axis, varies along the length of the member when there is no step and when there is a step. 4) Steps do not have any meaning at the member end joints. The Calculator reads the data values corresponding to the joints simply as place holders, to align the Step data with other joint and essential node data in the.inp file. 5) If desired, the user can model any taper transition explicitly. Taper transitions are defined by inserting essential nodes along the length of the member, and by providing the cross section dimensions at each essential node. The Calculator (and SABRE2) vary the cross section dimensions linearly between the adjacent joint and/or essential node values as long as the Step parameter is specified as 1. The next group of data lines in the input file specify the number of frame elements to be employed in each segment between the member joints and/or essential nodes, as well as the 8

12 yield strengths to be used within each of these lengths. In this tutorial, the input file first specifies that 8 elements are to be used within each of the two unbraced lengths of our member. This is followed by the specification of the Flange 1, web and Flange 2 yield strengths respectively for each length. The steel elastic modulus is set automatically to 29,000 ksi or 20,000 kn/cm 2 depending on whether US or SI units are employed. No Step Stepped Figure Example variations in a given cross section dimension when there is no step versus when there is a step at an intermediate essential node. The last group of data lines defines several broad characteristics of the analysis and design solutions: 1) The first of these data lines defines whether the St. Venant torsional constant should be taken as zero or not for slender web members when an Elastic Buckling Analysis (EBA) is employed. If the data value is equal to 1, then J is taken equal to zero in the buckling analysis, if the member web is slender under flexure per the AISC Specification Table B4.1b. When an inelastic Linear or Nonlinear Buckling Analysis is employed, J is always taken equal to zero for slender web members (within the member length where the web is slender). This is because Section F5 of AISC uses J = 0 for these member types. 2) The second of these data lines defines whether the calculation of the member stiffness reduction factors should be based on the current AISC (2016) Specifications or on recommended updated member strength equations developed in recent research by Subramanian 9

13 and White (2016a & b). If the data value is equal to 1, the AISC (2016) provisions are employed. 3) The third data line specifies whether the stiffness reduction factors, 0.8 on all elastic stiffness contributions and 0.8 b on the flexural rigidities EI x and EI y per the AISC Direct Analysis Method, should be applied to the elastic stiffnesses of the geometric nonlinear (pre buckling load displacement) analysis when an Inelastic Nonlinear Buckling Analysis (INBA) is employed. When an Elastic Nonlinear Buckling Analysis (ENBA) is employed, the software always uses the specified nominal elastic member stiffnesses. Upon starting the SABRE2Calculator, a blank screen appears with the menu shown in Fig at its top. Selecting the File menu button displays the pull down menu shown in Fig One can select Open from this menu to open the.inp file corresponding to the above problem. The Calculator then displays the elevation view of the member shown in Fig An isometric view of the member can be obtained, if desired, by selecting View > Defined Views > Isometric (XYZ) View (Fig ). This produces the image of the member shown in Fig Note that Fig also shows that the option White Background is checked, which is the option employed to produce the figures of the Tutorial. One can observe from Fig that the displacement constraints are shown as magenta arrows at their corresponding locations. The applied loads are shown by green arrows. In addition, joints 1 and 2 and the member flanges 1 and 2 are labeled. (The member flanges are always labeled at the starting end of the member.) The design axis is denoted by the solid blue line, and the cross section shear center is indicated by the yellow line. The FEA nodes in the Calculator (and in SABRE2) are actually positioned along the shear center, which is a curve along the member length for singly symmetric tapered cases. The FEA nodes are shown by the blue dots on the shear center curve. If one clicks on the Modeling menu button at the top of the screen, the dialog panels shown in Fig appear on the right hand side of the screen. The Define Geometry and Loads panel can be used to create all the parameters defined in the.inp file from scratch, and then the.inp file can be written from the Calculator using the File menu. In addition, these panels may be used to edit the parameters of a problem that have been read into the Calculator from any.inp file, including adding and deleting of essential nodes. When entering the above data in these and other panels of the graphical user interface, it should be noted that one can select text already entered in a given cell, hold down the Ctrl key on the keyboard and type the C key (for copy), and then select any other cell, hold down the Ctrl key and type the V key, to paste the text into that cell. By clicking on Analysis Parameters in the main menu of the Calculator, the solution parameters discussed above in the context of the.inp file can be inspected and modified if desired. Furthermore, the Analysis Parameters dialog panel allows the user to select the root finding algorithm used for inelastic buckling analysis solutions. However, it is rarely necessary to use anything other than the default Brent s method (Brent 1973). Once the Analysis Parameters are finalized, the problem can be analyzed and the results viewed via the Analysis pull down menu shown in Fig

14 Fig Main menu of the SABRE2 Calculator. Fig File pull down menu of the SABRE2 Calculator. Fig Elevation view of the member defined in the.inp file from Fig Fig View pull down menu of the SABRE2 Calculator showing the Defined Views selections. 11

15 Fig Isometric view of the tutorial web tapered member in the SABRE2 Calculator. Fig The Modeling dialog panels of the SABRE2 Calculator. 12

16 Fig SABRE2 Calculator pull down menu options under Analysis Elastic Linear Buckling Analysis and Results If Elastic Linear Buckling Analysis is selected from the menu shown in Fig , the Calculator runs this type of analysis, displays the buckled mode shape on top of the undeformed geometry, and displays the Results dialog panel on the right hand side of the screen. The eigenvalue, which is the scalar multiple of the applied load at incipient buckling, denoted by the symbol e in DG 25, is displayed at the top of the main viewing window as well as within the Results dialog panel. Figure shows the display of the buckling mode on top of a semi transparent rendering of the undeformed geometry. Figure shows the corresponding Results dialog panel. One can observe that in this problem, the member buckles elastically at e = of the factored LRFD applied load (i.e., the load specified in the.inp file). It should be noted that the displacements are undetermined in the buckling mode obtained from a linear buckling analysis. That is, the buckling mode is simply the pattern of the buckling displacements. It can be scaled by a positive or negative value to best convey the mode shape. The Calculator automatically scales the buckling mode by 1.2 times the member s largest flange width as an estimated optimum scale factor for viewing. Depending on the computer system, the default Scale Factor may need to be negated to obtain the view shown in Fig Fig Member elastic linear buckling mode corresponding to the specified combined bending and axial compression specified in the tutorial problem. 13

17 Fig Elastic Linear Buckling results dialog panel in the SABRE2 Calculator. In addition, by using the pull down menu within the Results dialog panel, one can view the buckling mode without showing the undeformed geometry, as well as diagrams of the member internal axial force or major axis moment at incipient elastic buckling. Figure shows the internal moment diagram for the tutorial problem. The diagram ordinates for a given element along the member length can be displayed within the Diagram Data dialog panel by clicking on the desired element. The element corresponding to the values in the dialog panel is highlighted. In addition, one can select the Data Labels option under the View menu to label the values of the response at all the nodes in the model. The Data Labels option has been turned on in the view shown in Fig Fig Member moment diagram with selected element values displayed using the Diagram Data dialog panel and the Data Labels option turned on. Comparison to DG25 Results If the tutorial problem is modified to consider only the applied bending moment, by changing the applied axial load P x from 11.3 kip to 0.0 kip in the Modeling menu, the Elastic Linear Buckling Analysis (ELBA) algorithm gives e = This can be compared to the estimate C b ( eltb ) Cb=1 = 1.38 x 1.08 =

18 from pages 125 and 126 of DG 25. If the applied moment is set to zero such that only the required axial load of P x = 11.3 is considered, the ELBA algorithm gives e = if J is taken equal to zero within the partial length of the member where the web is slender in flexural compression, which is the default in the Calculator (and in SABRE2). If the parameter is set to consider the non zero J at all cross sections, e = is obtained. This can be compared to the governing Applied Load Ratio for Constrained Axis Torsional Buckling of e = P ecat /P u = 157 / 11.3 = 13.9 from page 116 of DG Inelastic NonLinear Buckling Analysis and Comparison to DG 25 Results By selecting Analysis > Inelastic Nonlinear Buckling Analysis (see Fig ), an Inelastic Nonlinear Buckling Analysis (INBA) can be run to determine directly the AISC Specification based beam column capacity in the tutorial example. The program displays the member critical buckling mode along with information about the strength solution at the top of the viewing window (Fig ). The phrase ( Strength Eqs.) indicates that the solution is based on the current beam, column and beamcolumn equations in the AISC Specification. Under Analysis Parameters, the program also provides a solution option that is based on recommended modifications that provide an improved characterization of member strengths (Subramanian and White, 2016a & b). The phrase Applied Load Ratio = indicates that buckling is occurring at n = In other words, the factored design strength of this member is reached at of the specified applied loads. Interestingly, the inverse of the corresponding beam column Unity Check value in DG 25, which is a reasonable estimate of n from the extensive DG 25 manual calculations, is 1/0.852 = for Solution A (page 134 of DG 25) and 1/0.879 = for Solution C (page 137 of DG 25). The SABRE2 Calculator result is more accurate, and avoids all the separate and tedious column, beam and beam column calculations associated with the application of the AISC design strength provisions to web tapered members. Fig Member inelastic buckling mode and corresponding solution information. 15

19 Figure shows a diagram of the beam column Stiffness Reduction Factor (SRF) along the length of the member at the maximum design strength limit. The right most element is selected to display its SRF values in the figure. Points P1 through P5 are the sampling points for the five point Gauss Labatto numerical integration employed by the frame elements in SABRE2. The values of the SRF at each of these stations, marked within the elements from left to right, are displayed in the Diagram Data panel. The Data Labels option is turned off in this view, since the spacing of the integration points is too close to display the labels on a view of the full member. Similarly, various other responses and design resource parameters (e.g., the internal axial force, R pg, R pc, R pt, M yc, M yt, J, the cross section factored plateau strengths b M max, the cross section factored effective yield load b P ye, and the cross section unity check values may be plotted along the member length and inspected to understand the member design characteristics. White et al. (2015) discuss the variables associated with the inelastic buckling solutions. Fig Beam column SRF diagram with values displayed using the Diagram Data dialog panel. Interestingly, if the tutorial problem is modified to consider only the applied bending moment, by changing the applied axial load P x from 11.3 kip to 0.0 kip in the Modeling menu, one obtains the result shown in Fig at the top of the viewing window. In this case, the Applied Load Ratio at the maximum design strength condition is n = If an Inelastic Linear Buckling Analysis (ILBA) is employed for this problem the corresponding n is The difference in the results is in the 5 th significant digit. This illustrates a common result that the capacity of beams often is not sensitive to prebuckling displacement effects. In addition, the results message in Fig shows (TFY) after it lists n. This is an indication that the member strength is actually governed by Tension Flange Yielding. Fig Results from an Inelastic Nonlinear Buckling Analysis (INBA) of the tutorial member, considering only applied bending moment (P x = 0.0 kip). AISC/MBMA DG 25 gives a LRFD flexural strength ratio of M r /M c = 0.80 on its page 132, after 10 pages of laborious calculations. This matches with the factor n = , i.e., 1.25 = 1/0.80. The last phrase in the results summary shown in Fig , Applied Load Ratio = (Buckling), indicates that the Lateral Torsional Buckling (LTB) of the tutorial member (as a beam) occurs at of the specified required moment of 1800 in kip. That is, the rigorous b M n for the LTB limit state obtained from the Calculator is 16

20 1.316 x 1800 in kip = 2369 in kip. This compares to the estimate b M n = 0.9 x 2580 in kip = 2332 in kip on page 126 of DG 25. If the applied moment is set to zero in the tutorial problem and only a required applied axial load of P x = 11.3 is considered, an Applied Load Ratio of n = is obtained. This value is based on a calculation that incorporates the new AISC Section E7 Unified Effective Width approach for the effective width of the I section web within a computational Inelastic Linear Buckling Analysis (ILBA) (INBA gives n = ). The corresponding result in DG 25 is P r / b P n = 0.103, or n = 1/0.103 = The solution from the SABRE2 Calculator is more accurate and reflects the advances achieved in Section E7 of AISC (2016). It should be noted that the number of pages in AISC Section E7 has been reduced substantially, and the associated manual calculations are significantly more streamlined compared to the prior Q factor based calculations. The axial resistance calculations for this problem take up 16 pages in DG 25 but can be substantially shortened via the AISC Section E7 improvements. 2.3 Using SABRE2 The main SABRE2 engine provides for the analysis and design of general members and frames, including the consideration of finite stiffness bracing. The SABRE2 problem definition, analysis, and design checking of the member shown in Fig is explained below. As an alternative to the problem definition using SABRE2 s graphical user interface, if a version of the problem has been defined for the Calculator, one can write out a.mat file from the Calculator, then open this file in SABRE2 and modify the model as needed. Step 1 Define Joint Coordinates Figure shows the SABRE2 pull down menu for defining the joint coordinates. Upon selecting Define Joint(s), the Coordinates dialog panel (Fig ) is displayed on the right hand side of the screen. Each of the joint coordinates need to be entered using this dialog panel. When the Apply button at the bottom right corner of the screen is clicked, the corresponding joint appears in SABRE2 s main viewing window (see Fig ). In SABRE2, the joints are defined as the points at the ends of any lengths that are defined as members. In this tutorial, we will consider the problem in Fig as a single member with end joint coordinates (0, 0, 0) and (144, 0, 0). Therefore, one can enter these coordinates using the Joint Coordinates dialog panel (Fig ) and click the Apply button after each definition to obtain the result shown in Fig Note that consistent units should be used throughout all problem definition steps in SABRE2. The default consistent units are US (inches and kips). Fig Joint coordinate definition pull down menu. 17

21 Fig Joint coordinate definition dialog panel. Fig SABRE2 main viewing window after the definition of joints 1 and 2. 18

22 Step 2 Define Member to Joint Connectivity and Member Cross Section Dimensions The next step of the problem definition in SABRE2 is to define our single member by specifying its connectivity to joints 1 and 2, and specifying its cross section dimensions at each of these joints. This is accomplished using the Define Member(s) & Section(s) dialog, which is accessed via the pull down menu shown in Fig Upon selecting Define Member(s) & Section(s), the dialog panel shown in Fig appears at the top of the right hand side of the screen. One can either type the joint numbers directly into the dialog panel cells for the start and end joints of the member, joints i and j, or click on the desired joint numbers in the main viewing window to indicate the start and end joints. If desired, the user can select a new definition for the member Design Axis. The Design Axis is simply a reference axis used to define the member geometry. The Design Axis is by default taken as the Mid web depth. However, in certain cases for metal building frames, it can be simpler to define the geometry based on the lines along the outside (top) of the web at the outside (top) flange of the members. The Design Axis also may be set as the bottom of the web at the bottom flange. The cross section depth is always measured perpendicular to the Design Axis. In this tutorial, we will accept the default Mid web depth as the Design Axis. Next, the user needs to input the section dimensions at the start and end joints. SABRE2 provides a Section Database dialog, located just below the Define Member panel, which can be used to select any rolled wide flange section from the AISC Section Database. However, for general welded I sections, the section dimensions at each of the joints must be entered using the Define Section dialog panel (Fig ), located near the bottom right corner of the screen. Figure shows this panel with the section dimensions entered for the member from Fig It should be noted that one can select text that has already been entered in a given cell of this or other dialog panels, hold down the Ctrl key on the keyboard and type the C key (for copy), and then select any other cell, hold down the Ctrl key and type the V key, to paste the text into that cell. Once Joints i and j and the cross section dimensions at these joints are specified, the user can click on Apply at the bottom right corner of the screen to create the member. The elevation view of the member shown in Fig then appears in the SABRE2 main viewing window. Fig Member to Joint Connectivity dialog panel. Figure Define Section dialog panel. 19

23 Fig Elevation view of member after applying the member to joint connectivity and the Section definitions for the start and end joints. Step 3 Define Essential Nodes and the Cross Section Dimensions at these Nodes The next step in the problem definition of the tutorial beam column in SABRE2 is to define an Essential Node at the braced location, 90 inches from the column base. This is accomplished by selecting the Add node(s) menu item in Fig The user can then select the M1 member label in the main viewing window to indicate that M1 is indeed the member for which it is desired to add a node. The distance from joint i is then entered within the dialog panel at the top right corner of the screen as shown in Fig When the Apply button next to this input cell is clicked, SABRE2 shows the cross section dimensions at this essential node associated with a linear variation in values between joints jn1 and jn2 (or in general, between the adjacent joints and/or essential nodes) within the Define Section dialog panel (Fig ). If these dimensions are the desired ones, which they are in this case, the user simply clicks Apply at the bottom right corner of the screen to add the essential node. This additional node is highlighted by an x symbol and lines representing the cross section are shown on the member at the position of the essential node, as illustrated in Fig Fig Add Nodal Coordinates dialog panel. Fig Add Section dialog panel. 20

24 Fig Elevation view of member after adding the Essential Node at the braced location. Step 4 Subdivide Member Segments and Assign Material The fourth step of the problem definition in SABRE2 is to select the appropriate Subdivide Segment(s) & Assign Material menu from Fig Since the member in Fig is homogeneous (i.e., the same material is used for both flanges and for the web plate), we will select the Homogeneous Member(s) menu for this step (see Fig ). This brings up the Member and Segment dialog panel (Fig ) at the top right corner of the screen. Since we have two member segments in the tutorial problem, one on each side of the intermediate Essential Node at the braced point, we need to define the material and the number of elements to be used in each of these segments. This can be accomplished by clicking on the label M1S1 for the first segment, then entering 8 in the cell labeled # of elements and changing the default 50 to 55 in the cell labeled Fy = in the Homogeneous Member(s) dialog panel (see Fig ). Once all the values are as desired in this dialog panel, the user can click Apply at the bottom right corner of the screen to apply these values to Member 1 Segment 1, i.e., M1S1. The user can then click on the label M1S2, and perform the same operations for this segment. SABRE2 indicates Member 1 Segment = 2 Matl. & Elem. Assigned at the top of the viewing window and changes the color shade on the segments where the assignments have been made. Alternatively, if all the member segments have the same number of elements and the same material properties, one can simply click the Apply All button in the Homogeneous Member(s) dialog to apply the specified values to all the member segments. SABRE2 indicates All Member Material & Elem. Assigned at the top of the viewing window. It should be noted that the element discretization is not displayed until one initiates the subsequent steps of defining either the Boundary Conditions or the Loading Conditions of the problem being considered. One can confirm the definitions for each of the member segments, as well as change the definitions as appropriate, by clicking on the labels for each member segment. Note that the material definition in Fig includes the weight density of the steel. The default weight density in SABRE2 is 1.2 times the nominal weight density of steel, equal to 1.2 x kcf / 12 3 = kip/in 3 in US units. If the self weight of the steel is being considered in a load combination 21

25 other than one corresponding to a dead load factor of 1.2, this value should be updated to the appropriate factored weight density of the steel. Fig Assign Material & Subdivide Segments pull down menu. Fig Member and Segment dialog after after selecting the label M1S1. Fig Homogeneous Members dialog after inputting the desired number of elements. Step 5 Define Boundary Conditions The fifth step of the problem definition in SABRE2 is to define the member displacement boundary conditions. One can define the loads as the fifth step, if desired, but we will address the displacement boundary conditions next in this tutorial. The Define Fixities dialog panel is accessed by selecting Define Fixities from the pull down menu shown in Fig This brings up the Node and Coordinates panel at the top right corner of the screen, the Constraint Location panel in the middle on the right hand side of the screen, and the Define Fixities panel at the bottom right corner of the screen. SABRE2 also displays a view of the structure showing the element discretization along the reference axis for the structural analysis (the shear center axis is used as the reference axis for the structural analysis in SABRE2). This is shown in Fig Fig Boundary Conditions pull down menu. 22

26 Fig Elevation view of the member upon selecting the Define Fixities dialog. One can assign the desired displacement constraints at any of the nodes by clicking on the node symbol (i.e., the desired blue dot on the shear center axis of the member), which causes SABRE2 to display the corresponding node number and nodal coordinates in the top right hand dialog panel. One then defines the location of the displacement constraint (shear center, Flange 1, Flange 2 or the cross section centroid), and finally uses the radio buttons in the Define Fixities dialog to indicate the degrees of freedom where the displacements are fixed. The default location for the displacement constraint is the shear center, labeled as S.C. Figure shows the three dialog panels after selecting Joint 1 and specifying the required displacement constraints at this joint, and Fig shows these panels after doing the same tasks for Joint 2. Figure shows an isometric view of our member, obtained by selecting the pull down menu View > Defined Views > Isometric (XYZ) View after the above definitions are applied. Note that each of the displacement constraints is represented by a magenta colored arrow. Fig Node and Coordinates, Constraint Location and Define Fixities dialog panels corresponding to Joint 1 (Node 1). Fig Node and Coordinates, Constraint Location, and Define Fixities dialog panels corresponding to Joint 2 (Node 17). 23

27 Fig Isometric view of our member after the displacement boundary conditions are specified. The intermediate brace in the tutorial problem is defined as a grounded spring having a stiffness of 2.2 kip/in. This provided brace stiffness is assumed to be developed by wall panels and/or wall bracing connected at the level of an outset girt located at 90 inches above the column base. For the assessment of the effectiveness of this bracing stiffness by the 2016 AISC Appendix 6 procedures, this stiffness is divided by 2/ = 2/0.75 = for the buckling analysis. That is, a value of 2.2 kip/in / = kip/in is entered into the Calculator for the buckling analysis. If the buckling analysis shows that this reduced bracing stiffness is sufficient to develop the required loads, the physical bracing stiffness of br = 2.2 kip/in is sufficient. The point bracing at the Essential Node is defined in SABRE2 by selecting the Define Discrete Grounded Spring option in Fig and then clicking on the node symbol at the Essential Node location in the main viewing window. SABRE2 then shows the FEA nodal (shear center) coordinates of this node in the dialog panel at the top right corner of the screen, the Spring Location dialog panel with the default S.C. shown as its pull down selection in the middle on the right, and the Define Stiffness panel at the bottom right side of the screen. Figure shows these dialog panels with the appropriate input values for our problem. Upon clicking Apply in the bottom right corner of the screen, a discrete nodal brace is displayed at the Essential Node location as illustrated in Fig

28 Fig Node Coordinates, Spring Location, and Define Stiffness dialog panels associated with the definition of a discrete nodal brace (grounded spring) stiffness at the intermediate Essential Node of the tutorial problem. Fig Isometric view of our member after the intermediate nodal brace has been defined. Step 6 Define Load Conditions Figure shows the pull down menu corresponding to the applied load definitions in SABRE2. In our tutorial problem, these loads are the axial force of 11.3 kip and the applied moment of 1800 in kip at the top of the member. Upon selecting Define Point Loads in Fig , the dialog panels shown in Fig appear on the right hand side of the screen. To define the applied loads, the user clicks on the blue dot at the top of the column (the node on the shear center axis at the right hand side of the display), changes the location for the load height from the default S.C. (for shear center) to Cent. (for centroid) as shown in the Define Load Height dialog panel, and then inserts the above values into the Define Loads dialog panel as shown in Fig

29 Fig Loads pull down menu. Fig Load definition dialog panels. Step 7 Check Structural Analysis Parameters Before running the structural analysis, it is always a good idea to check that the various analysis parameter options available in SABRE2 are set to the desired values. Figure shows the SABRE2 window obtained when Analysis Parameters is selected from the menu at the top of the screen. The dialog panels on the right hand side of the screen include the options available in the Calculator, discussed previously in Section 2.2, as well as several additional options available in the main SABRE2 engine: The member self weight may be included. The y direction is taken as the gravity load direction if this option is turned on. The self weight option is set to off by default. Note that when the selfweight is included, the appropriate factored weight density should be entered in the dialog shown in Fig The increment size and maximum applied load ratio may be specified for first and second order load deflection analyses. A value of 1.0 corresponds to application of the full specified load in one increment. Multiple buckling modes, rather than just the critical mode, may be requested for Elastic Linear Buckling Analysis. Step 8 Conduct Structural Analysis Given the completed problem definition, including the modification of any of the analysis parameters as appropriate, various types of analysis may be applied to the model of the member or structure in the main SABRE2 engine. The available analysis types are listed in the pull down menu shown in Fig Two types of load deflection analysis may be conducted, first and second order elastic. In addition, the four types of buckling analysis introduced in Section 1 are available within the main SABRE2 engine. Furthermore, for the ILBA and INBA algorithms there are streamlined option referred to as an Inelastic 26

30 Linear Buckling Check and an Inelastic Nonlinear Buckling Check. The difference between the Check and the Analysis for these procedures is as follows. The Analysis options involve an iterative solution to determine the capacity of the structure, e.g., c P n for a column, b M n for a beam, the corresponding beam column resistance for a beam column member, or the general capacity for a framing system. Conversely, the Check options involve a streamlined calculation to determine whether the structure has sufficient capacity to resist the specified loads. If the Buckling Applied Load Ratio (ALR) from a Check is greater than 1.0, the structure has the capacity to resist the specified loads whereas if the Buckling ALR from the Check is less than 1.0, the structure cannot support the specified loads. If the ALR corresponding to the actual load capacity, n, is desired. The user needs to run an Analysis. However, if the user simply needs to evaluate if the structure has sufficient capacity to resist the applied loads, a Check is sufficient. Fig Analysis Parameters dialog panels shown with the isometric view showing the tutorial beamcolumn after the applied loads have been defined. Given the selection of the analysis method, SABRE2 executes the structural analysis, shows the corresponding pre buckling or buckling deflections, and opens the corresponding Results dialog panel for further inspection of the responses. 27

31 Fig Analysis pull down menu, showing the various types of analysis available within the main SABRE2 engine. Step 9 Inspect Results Figure shows the buckling mode for the tutorial beam column obtained from an Inelastic Nonlinear Buckling Analysis (INBA) using the main SABRE2 engine. The only difference between this model and the one discussed previously in Section 2.2 is that the zero displacement constraint on Flange 2 located at 90 inches from the left hand end of the member is replaced here by a flexible nodal brace having a stiffness of 2.2 kip/in / (2/0.75) = kip/in. As one might expect, since this nodal brace is located on the flange subjected to flexural tension, and since the applied bending moment causes significantly larger stresses than the applied axial load, the impact of considering the elastic stiffness of the brace is small. The n of is the same as that summarized in Fig and the SRF values are essentially the same as those in Fig In addition, a wide range of other responses are available for plotting via pull down menus provided in the Results dialog panels. Fig Member inelastic buckling mode and corresponding solution information at the top of the screen for the tutorial member with an elastic intermediate brace. 28

32 If the applied moment is set to zero such that the above member is subjected solely to concentric axial compression, the result shown in Fig is obtained. The member s inelastic buckling load is n = times the specified applied axial load. This result can be contrasted with the corresponding result discussed at the end of Section 2.2, where the lateral brace at Flange 2 was taken as rigid (zero displacement) and the inelastic buckling load was obtained as times the specified applied load. The small lateral brace stiffness is sufficient to develop a resistance of the member under uniform axial compression equal to / x 100 = 96.5 % of the resistance corresponding to rigid bracing. From Fig , the member failure mode is clearly Constrained Axis Torsional buckling (CATB). That is, the member is buckling by twisting about the top flange. One can observe that the CATB mode is very similar in form to the LTB mode for the tutorial beam column. The above result is from an Inelastic Nonlinear Buckling Analysis (INBA). One should expect that the effect of the pre buckling bending displacements of the member are nil in this problem, since the prebuckling bending displacements of the member are close to zero (these displacements are not exactly equal to zero since the centroidal axis of a tapered singly symmetric member is slightly curved). If an Inelastic Linear Buckling Analysis (ILBA) is employed instead, n = This result is with 0.01 % of the above result from an INBA. Fig Constrained Axis Torsional buckling of tutorial member under concentric axial compression. 29

33 3. Tutorial Adding a Step in the Flange Width to the Tapered Member 3.1 Overview Suppose that it is desired to step the width of the outside flange in the tutorial beam column down to a 4 inch width over a length of 72 inches from the column base. This section explains how the corresponding step in the cross section is defined and modeled in the main SABRE2 engine as well as in the Calculator. 3.2 Adding a Step to the Outside Flange in SABRE2 One can reduce the size of the outside flange in the bottom 72 inches of the beam column, given the above definition of the member in SABRE2, using the following steps: 1) First, modify the flange width at joint jn1 by selecting Properties > Define Geometry > Define Member(s) and Section(s), selecting the member label M1 in the main viewing window, and then modifying the parameters in the Define Section dialog panel as shown in Fig Upon clicking Apply at the bottom right corner of the screen, a linear taper is inserted from b f = 4 inches at the column base to b f = 6 inches at the intermediate essential node. Fig Define Section dialog panel showing the change in b f2 for Joint i. 2) Next, the user needs to insert an additional essential node at 72 inches above the base. This is accomplished via the Properties > Define Geometry > Add node(s) menu. Upon entering this menu, one can click on the member M1 label in the main viewing window to indicate that we indeed would like to add a node to member M1. The user then enters the Position from joint i as 72 and clicks Apply to fill the Define Sections dialog panel with the cross section dimensions corresponding to a linear variation between the adjacent joints and/or nodes. 3) In this case, we want the outside flange dimension to be 6 inches above the step in the crosssection. Therefore, the user needs to set the bf2 cell in the Define Section dialog panel to 6 inches, as shown in Fig

34 Fig Dialog panels corresponding to the addition of an essential node at 72 inches above the column base. 4) Next, the user can select Step in the Step dialog panel and click Apply at the bottom right corner of the screen to insert the new essential node, including the desired step. Figure shows an isometric view of the resulting member geometry. The * symbol, marks an additional intermediate node that SABRE2 has automatically inserted. This symbol is slightly offset from the member design axis to facilitate the ability of the user to select and inspect the corresponding cross section. Fig Member geometry after adding a new essential node with a step in the cross section. SABRE2 does not currently have the sophistication to retain the number of elements, member section properties, and load and displacement boundary conditions in regions or locations where the member is unaffected by the above changes, nor to infer what these characteristics should be within the changed portions of the member. As such the definitions discussed previously in Section 2.3 Steps 4, 5 and 6 need to be re entered once the above change is made. Upon going to Properties > Assign Material and 31

35 Subdivide Segment(s) > Homogeneous Member(s), SABRE2 displays the view shown in Fig The user can select the labels for each of the member segments, M1S1, M1S2, etc. to define the number of elements and set the member yield stress within each segment. In this case, 6 elements are specified in section M1S1, and one element is specified in the short segments M1S2 and M1S3. The same number of elements as before (8) is specified in the top segment M1S4. A yield strength of F y = 55 ksi is input for each of these segments. Fig View of member within Properties > Assign Material and Subdivide Segment(s) > Homogeneous Member(s). Upon selecting Conditions > Boundary Conditions > Define Fixities and selecting the node (located on the shear center axis) at the starting end of the member, one can observe that the shear center coordinates of this node are substantially changed from that of the previous non stepped member (compare Fig to Fig ). Fig Coordinates dialog panel for Node 1 within Conditions > Boundary Conditions > Define Fixities. 32

36 Upon completing the definition of all the load and displacement boundary conditions, one can run the various analyses to evaluate the modified member. Figure shows the result from the Inelastic Nonlinear Buckling Analysis (INBA). Figures through show the corresponding beam column inelastic Stiffness Reduction Factor (SRF), the cross section factored plateau resistances b M max in flexure, the cross section effective yield resistances c P ye under axial compression, and the cross section unity check values at the strength limit respectively. Fig Buckling mode for tapered and stepped beam column member obtained from Inelastic Nonlinear Buckling Analysis (INBA). Fig Tapered and stepped beam column SRF at the strength limit. Fig Tapered and stepped beam column b M max values. 33

37 Fig Tapered and stepped beam column c P ye values. Fig Tapered and stepped beam column cross section unity check values. 3.3 Adding a Step to the Outside Flange in the SABRE2 Calculator Figure shows the SABRE2 Calculator.inp file corresponding to the tapered and stepped beamcolumn problem. Similar to the results for the tutorial problem in Section 2, the brace at the girt location on the flange subjected to flexural tension has a negligible influence on the beam column resistance (although there is a minor but measurable effect of the non rigid point brace on the resistance of the member under pure axial compression). 4. Tutorial Clear Span Frame 4.1 Overview Figure shows the elevation view of a clear span frame that has been studied extensively in prior research related to the development of DG 25 (White and Kim 2006; Kim 2010). This is an interior frame within a 300 ft long building (out of the plane of the frame) that has the following characteristics: 90 ft clear span from outside to outside of its 8 inch outset girts, 19 ft eave height, ½ : 12 roof slope, 25 ft frame spacing, Steel minimum specified yield strength, F y = 55 ksi. The frame is symmetric about its ridge and has simply supported base conditions. Its span to eave height ratio (4.74) is relatively large. This in turn generates relatively large thrust in the roof girders. The frame was designed by Mr. Duane Becker of Chief Industries for conditions in Reading, CA, using ASCE 7 05 and an extension of the AISC (1989) ASD provisions. 34

38 **************************************************** * SABRE2Calculator * **************************************************** ** Design Axis 1 ** Units 1 *Define Joint & Essential Node Geometry, Loads and Boundary Conditions ** X Coordinates ** bf ** tf ** bf ** tf ** dw ** tw ** Afillets ** Px ** Py ** Mz ** Boundary Conditions ** Load Height ** Step * Subdivide Segment(s) and Assign Material ** # of Elements ** Fyf ** Fyw ** Fyf * Specify Analysis Parameters ** Use J = 0 for Slender Web Sections in EBA (Yes = 1, No = 0) 1 ** Use the AISC (2016) Provisions (Yes = 1, No = 0) 1 ** Use the DM SRF for GNA in INBA (Yes = 1, No = 0) 1 Figure SABRE2 Calculator text (.inp) file for the tapered and stepped beam column. 35

39 9 purlins spaced at 5' cc 1.00' 10.00' 10.00' 21.11' 1/2 12 d = 40.75" c4 B r1 C r2 r3 D r4 r5 r6 E r7 r8 r9 r10 c ' A c2 6.00' 15.10' 8" girt & purlin outsets 7.50' d = 10" c ' CL Fig Elevation view of clear span frame. The load parameters and magnitudes for this frame are as follows: Dead load: 1.96 psf along the slope of the roof, plus the self weight of the members. Collateral load: 3 psf along the slope of the roof. Live load: 12.0 psf along the roof slope. Wind load: o Basic wind speed: 85 mph o Exposure category C o q = psf Snow load: o Ground snow load: 30 psf o Exposure factor: C e = 1.0 o Thermal factor: C t = 1.0 o Importance factor: I = 1.0 o p f = p s = 21 psf on the horizontal projection of the roof area. In the following, the above frame is modeled and analyzed in SABRE2 for the LRFD 1.2 (Dead + Collateral) Uniform Snow load combination. This load combination is the critical one governing the overall strength of the frame. Given the above parameters, the factored uniform snow load is 36

40 1.6 x 21 psf x 25 ft = 840 plf along the horizontal projection of the frame and 840 plf x 12 / = plf along the slope of the roof girders. The factored Dead + Collateral load is 1.2 x (1.96 psf psf) x 25 ft = plf along the slope of the roof girders, in addition to the factored self weight of the frame. Table summarizes the web and flange geometry for the clear span frame. The columns have a larger inside flange, but the roof girders are all doubly symmetric. There are several steps in the web thickness along the length of the roof girders. The continuity and connection plates at the knees of the frame are assumed to have the same dimensions as the corresponding column and roof girder flanges. The outside flanges of the columns and roof girders are assumed to be fully braced in the out of plane direction at each of the girt and purlin locations shown in Fig Diagonal braces to the inside flanges are indicated by the double dashed lines in this figure, and are assumed to provide full bracing to the inside flange at these locations. The purlins are spaced at 5 ft on center except at the ridge and the knee of the frame, and the girts are located at 7.5 and 6 ft spacings starting at the base of the frame. Both of the column flanges are braced laterally at these girt locations. The bottom flange of the roof girders is unsupported at the purlin locations 20 ft and 30 ft from the inside of the knee, but otherwise, both flanges are braced at each purlin location. Table Summary of web and flange geometry for the clear span frame. Length Location Web Inside Flange Outside Flange d (in) t w (in) h/t w h c /t w b f (in) t f (in) b f /2t f b f (in) t f (in) b f /2t f A c / / /8 8.0 c c c B 7/32 C r / / /8 8.0 r r D r / / /8 8.0 r r E r / / /8 8.0 r r r r r Table shows the x and y coordinates of points along the outside edge of the web (at the inside of the outside flange) of the left hand column of the clear span frame as well as the cross section dimensions at these locations. The outside edge of the web works well as the design axis for specifying 37

41 the cross section geometry of this frame in SABRE2. The point at the base at the outside edge of the web of the left hand column is taken as the origin of the rectangular Cartesian coordinate system. Table shows the corresponding data for the roof girders. Similarly, the edge of the web at the outside flange is taken as the design axis for these members. The origin of the on slope coordinate for the roof girders is taken as the upper left corner of the frame at the outside edge of the column web. Table Column design axis coordinates and cross section dimensions (in). Location x y bfi tfi h tw bfo tfo c c c c c The locations designated with a prime symbol (') in Table correspond to physical transitions in the web thickness. At r1, the web is explicitly tapered over a 0.5 inch length starting at the location where the roof girder is welded to the connection plate at the panel zone and continuing into the panel zone. This models a step in the web thickness from 0.25 inch within the roof girder in region C (see Fig ) to inches within the panel zone. SABRE2 inserts a taper over a short length within the section having the larger cross section dimensions or area, when a step is specified by the user. Since the critical cross section for the strength calculations is located at position r1', and since the taper is essentially within the thickness of the end connection plates, it is preferred to explicitly taper the web in this way to model the step at r1 in this problem. In addition, as a simplification, the location of the purlin and the flange diagonal brace at r1 is modeled at position r1' rather than at the physical position, which is 1.71 inches from this location. In addition, the cross section transitions at r3 and r5 are assumed to be located at the bracing positions, although Fig shows that these transitions are slightly offset from the bracing locations. Table Roof girder design axis coordinates and cross section dimensions (in). Location x y bfi tfi h tw bfo tfo x on slope r r r1' r r r3' r r r5' r r r r r Within the knee joints, the column and roof girder cross sections overlap. SABRE2 does not currently address the direct modeling of the response within the finite size panel zone regions. To avoid the limit 38

42 of resistance of the members occurring within the panel zone, the yield strength of the panel zone is set fictitiously to 110 ksi. Given the above loading parameters, the total factored uniform snow load applied to the frame is 840 plf x 90 ft = 75.6 kip the total factored applied Dead + Collateral load is plf x / 12 x 90 ft = 13.4 kip and the total factored self weight (neglecting any additional miscellaneous steel) is 1.2 x 4.22 kip = 5.1 kip. Therefore, neglecting any non symmetry due to the modeling of out of plumbness or the corresponding notional lateral load required for gravity load cases by the AISC Specification, the column factored gravity load vertical reactions should be approximately (75.6 kip kip kip) / 2 = 47.0 kip The factored self weight is modeled in SABRE2, based on the cross section dimensions and the element lengths in the structural analysis model. The other factored loads are assumed to be applied to the frame at the purlin locations at the top of the roof girder flanges, as well as a statically equivalent loading at the intersection of the column and roof girder frame element axes at the frame s knees. Based on the tributary on slope lengths, the corresponding concentrated loads at these locations are: and (839.3 plf plf) x 5 ft = 4.94 kip at locations r2 through r8 (839.3 plf plf) x (2.5 ft ft) = 3.46 kip at location r9 (839.3 plf plf) x (2.5 ft ft / 2) = 4.47 kip at r1 (75.6 kip kip) / 2 7 x 4.94 kip 3.46 kip 4.47 kip = 1.99 kip at the juncture of the column and roof girder element reference axes (the element shear center axes) within the panel zones. where ft is the distance from the upper left corner of the building envelope to the purlin location r1 along the roof slope (= 45 ft x / 12 5 ft x 8 spaces 1 ft). In addition to the above vertical load at the juncture of the column and roof girder element reference axes, the statically equivalent moment due to the eccentricity of this load, which comes from the eave strut, needs to be included at this location. This moment is taken as 1.99 kip x ( in in) = 54.0 in kip 39

43 where inches is the x coordinate of the column and roof girder reference axes at the outside edge of the webs in SABRE2 and inches is one half of the outset girt depth plus the outside column flange thickness, which is taken as the location of the resultant force (1.99 kip) from the eave strut. 4.2 Notes on Modeling of the Clear Span frame in SABRE2 The creation of the SABRE2 model of the above clear span frame is similar to that for the above beamcolumn examples. However, there are some additional details. Specific attributes pertaining to the creation of the clear span frame model are as follows: The model is constructed by defining two members, the column and the roof girder on the lefthand side of the ridge, and then mirroring this geometry about the ridge, taking advantage of the symmetry of the frame. The mirror tool is located at Properties > Define Geometry > Mirror w.r.t. y axis. The design axis of the members is taken as the line along the outside edge of the web (at the inside of the corresponding flange) for both the above column and roof girder. The joint and essential node x and y coordinates, and the cross section geometry at each of these positions for each of the members are listed in Tables and The member end joints are defined first along with the cross section geometries at these joints. The end joint at the top of the column, labeled as location c5 in Table 4.1 2, is at the eave height. The end joint at the left hand end of the roof girder, labeled as r0 in Table 4.1 3, is at the outside edge of the column web. The cross section depths are defined as the depth of the panel zone perpendicular to the design axis at both of these positions. The member essential nodes are defined after the member end joints are specified. When the essential nodes are defined initially, these nodes are specified without any step and thus SABRE2 assumes a linear variation in the cross section dimensions between the adjacent nodes that have already been defined. These cross section dimensions are edited where this variation does not apply. If the cross sections at r1' and r5', where we have a discrete change in the taper angle, are defined first, the web depths for the other cross sections are determined automatically by SABRE2 when the Apply button in the Add Node dialog panel is clicked (see Fig ). At the cross sections of the roof girder where the web thickness is to be stepped, the crosssections dimensions on the side of the step toward the ridge of the frame (i.e., toward the end node of Member 2) should be specified. Once all the essential nodes are defined, the user can click on the nodes corresponding to the steps in the web thickness, then specify Step in the Step dialog and click Apply at the bottom of the screen, to have SABRE2 insert the desired step in the cross section geometry at these positions. Once the number of elements are specified for each of the member segments (only one element should be assigned for the short segments corresponding to the cross section steps), SABRE2 automatically generates the FEA nodes and elements for the frame model. This includes the automatic calculation of the juncture points for the element reference axes in the vicinity of the 40

44 member end joints (the cross section shear center is employed as the element reference axis in SABRE2). It is recommended that an essential node should be placed in the column and in the roof girder at the panel zone edges of the knee. In addition, it is recommended that the length of the members within the knee should be modeled using only one element corresponding to each of the members. As noted above, SABRE2 automatically calculates the juncture of the element reference axes within the panel. At the present time, if multiple elements are defined within the panel, the juncture between the column and roof girder reference axes must occur within the first element starting from the end joint of each of these members. All the displacement boundary conditions and all the loads on the left hand side of the frame are defined. Then the mirror tool, mentioned above, is employed with Mirror Type > All to mirror all attributes of the model about the ridge of the frame. Figure shows the resulting SABRE2 model. Fig SABRE2 model of the clear span frame. When defining the displacement boundary conditions at the base of the columns, the Position of the Displacement Constraint is set to Cent., such that the column bases are modeled using the common practice of simply supported at the centroid of the column base cross sections. The Position of the Loads transferred from the roof purlins to the roof girders is set to Flg2, such that these loads are applied to the girder model including an offset to the top flange. Once the above symmetric model is defined, the nominal out of plumbness required by the AISC Specification for gravity only load combinations is entered by going to Properties > Define Geometry > Define Joint(s), clicking on joints jn2, jn3 and jn5 in succession, and for each of these 41

45 joints, adding inches x = inches to the Coord X value and clicking Apply. Since all the essential nodes are defined relative to the joint position at the start of the member, the global coordinates of the essential nodes are shifted automatically by this change. The modeling of the response at the knee joints of the clear span frame is relatively coarse, but is similar to common frame analysis design approximations used in current practice. An essential node is placed in the columns at the bottom of the panel zones and another essential node is placed at the inside edge of the panel zones in the roof girders. One frame element is employed for the column and one for the roof girder within the panel zone as noted above. At the juncture of these elements, SABRE2 automatically assumes continuity of all the displacement degrees of freedom unless a release is specified. Continuity of the displacements and the rotations is certainly correct; however, the warping of the roof girder flanges and column flanges is physically different at this juncture point. This fact is represented by going to Conditions > Boundary Conditions > Define Element Flexural &/or Warping Releases, selecting the label for the top element in the column, and then setting the Warping Release for Node j to Free. As noted in Section 4.1, to avoid the limit of resistance of the members occurring within the panel zone, the yield strength of the elements with the panel zone is set fictitiously to 110 ksi. This is accomplished under Properties > Subdivide Elements & Assign Material > Homogeneous Member(s). 4.3 Inspection of Results for the Clear Span Frame Figure shows the deflected geometry of the clear span frame from a second order elastic analysis at the required 1.2 (Dead + Collateral + Self Weight) Uniform Snow load level. Figure shows the buckling mode and the controlling strength limit state information for this load combination from Inelastic Nonlinear Buckling Analysis (INBA). The first part of the strength limit state information shown at the top of the main viewing window indicates the type of buckling analysis (INBA) and that this analysis is based on the current AISC (2016) Specification strength equations. The second part, Applied Load Ratio = (CFY), indicates that the governing limit state for this frame and load combination is associated with reaching the beamcolumn cross section strength limit associated with the plateau strength in flexure in combination with the axial loading (at of the required loading). As will be shown below, this corresponds to reaching a cross section unity check of 1.0 in the roof girder at the inside edge of the knee joint on the right hand side of the frame (considering the influence of an initial out of plumbness to the right of 1/500 of the height above the base as required by the AISC Specification for gravity only load combinations). The limit of resistance obtained from the above INBA differs little from that obtained an Inelastic Linear Buckling Analysis (ILBA) for this structure. The CFY limit state is attained at the same location in the frame at an applied load ratio of based on an ILBA. The above smaller value is due to the second order effects of the gravity loads acting through the displacements within the plane of the frame. 42

46 Figure Deflected geometry of the clear span frame from second order elastic analysis under 1.2 (Dead + Collateral + Self Weight) Uniform Snow. Fig Bucking mode and controlling strength limit state information for the load combination 1.2 (Dead + Collateral + Self Weight) Uniform Snow on the clear span frame. 43

47 The third part of the strength limit state information shown at the top of the main viewing window, Applied Load Ratio = (Buckling) gives the multiple of the required design load at which buckling occurs, given the SRF values at the above load level where the governing Compression Flange Yielding (CFY) limit state is reached. The buckling mode shown in Fig corresponds to this load level. The governing CFY limit state does not involve buckling. Figure shows a diagram of the axial forces in the clear span frame at the governing CFY strength limit, along with the axial force at the bottom of the right hand column in the Diagram Data panel. These and all the subsequent diagrams shown below are drawn on the design axis of the members, located at their outside flanges. Figure shows the corresponding internal moment diagrams, highlighting the moment of 7026 in kip at the critical roof girder cross section at the inside edge of the righthand knee. Fig Clear span frame axial forces at the CFY strength limit with the axial force values shown for the bottom element of the right hand column. Fig Clear span frame moments at the CFY strength limit, with the moments shown for the critical roof girder element just to the inside of the right hand knee. Figure shows a plot of the frame member SRF values at the governing CFY strength limit. Note that the SRF values for the elements within the frame panel zones are larger due to the use of the artificial F y = 110 ksi in these elements. This artificial increase in the yield strength for these elements is just a simple way of avoiding a condition where the strength of the frame is limited by the over simplified representation of the response by the frame elements employed within the panel zones. Figure highlights the SRF values in the critical roof girder frame element adjacent to the right hand knee. 44

48 Figure shows the AISC R pg values for all the frame member cross sections associated with the above strength limit state calculations. The R pg values in the roof girder frame element adjacent to the right hand knee are highlighted in the Diagram Data panel. Figure shows the St. Venant torsion constant (J) values utilized in the strength limit state calculations. Per the AISC Specification and AISC/MBMA DG 25, J is taken equal to zero for the flexural resistance calculations at cross sections having slender webs. For the shallower cross sections in the clear span frame and/or at locations in the frame where the webs are thicker, the webs are noncompact; therefore, the non zero St. Venant torsion constant is employed at these locations. Fig Clear span frame SRF values at the CFY strength limit, with the SRF values shown for the critical roof girder element just to the inside of the right hand knee. Fig Clear span frame R pg corresponding to the strength limit state calculations, with the R pg values shown for the critical roof girder element just to the inside of the right hand knee. 45

49 Fig Clear span frame J values used in the strength limit state calculations, with the J values at the bottom of the right hand column highlighted. Figure shows the diagrams of the maximum potential resistance, b M max, at all of the crosssections of the clear span frame, with the cross sections in the critical element just inside the right hand kneed highlighted. The b M max values for the frame elements within the knee are artificially high, to prevent these elements from limiting the resistance of the frame as discussed previously. Figure shows the cross section effective yield strengths c P ye throughout the clear span frame. Again, the critical roof girder element adjacent to the right hand knee is highlighted. Fig Clear span frame b M max values used in the strength limit state calculations, with the b M max values highlighted for the critical roof girder element adjacent to the right hand knee. Fig Clear span frame c P ye values used in the strength limit state calculations, with the c P ye values highlighted for the critical roof girder element adjacent to the right hand knee. Lastly, Figure shows the cross section unity check values throughout the clear span frame. As noted previously, the critical cross section is at the inside edge of the right hand panel zone. This crosssection has a unity check of 1.0, which corresponds to this cross section having reached the Compression Flange Yielding (CFY) limit state. 46

50 Fig Clear span frame cross section Unity Check values, with the values highlighted for the critical roof girder element adjacent to the right hand knee. 5. Demonstration Problems 5.1 Overview The following subsections present a number of basic demonstration problems highlighting the capabilities of the elastic and inelastic buckling procedures implemented within the SABRE2 software system. Sample SABRE2 Calculator and/or main SABRE2 engine problem definition files are provided for these problems as well as the tutorial problems from the above sections with the download of the SABRE2 software. 5.2 W21x44 Beam Lateral Torsional Buckling Curves Consider the LTB resistance of a suite of W21x44 beams, Grade 50 steel, having torsionally and flexurally simply supported end conditions and unbraced lengths ranging from zero to 4.5L p = 20.0 ft. Figure shows the results for the uniform bending case as well as a basic moment gradient case involving an applied moment at one end and zero moment at the other end of the beams. Various Inelastic Linear Buckling Analyses (ILBA) are conducted to generate the strength curves shown in this figure. The following observations can be made from this LTB study: 1. The ILBA results for the LTB resistance under uniform bending match exactly with the calculations from the AISC Specification Section F2 equations. Therefore, only one curve is shown for the uniform bending case in Fig The exact match with the AISC Specification LTB strength comes from the use of the LTB Stiffness Reduction Factor (SRF) derived directly from the AISC equations. Various details of this SRF calculation are explained in White et al. (2015). 2. The buckling analysis results for the LTB resistance under the moment gradient fit closely with the calculations from the AISC Specification Section F2 using a moment gradient factor C b = However, there are some important differences between the rigorous LTB curve obtained from SABRE2 and the one obtained using the Section F2 LTB equations. These differences may be explained as follows: 47

51 a. For longer unbraced lengths, where the beam is elastic and the corresponding LTB stiffness reduction factor is simply b = 0.9, the buckling load determined from SABER2 is approximately 6 % larger than the capacity determined from the AISC Section F2 equations with C b taken as The SABRE2 solution is the more accurate assessment. The 1.75 value for C b is a lower bound approximation developed by Salvadori (1955). The SABRE2 solution is approximately 11 % larger than the solution with C b = 1.67 obtained from AISC Eq. (F1 1) for this problem. AISC Eq. (F1 1), originally developed by Kirby and Nethercot (1979), gives a lower lower bound solution than Professor Salvadori s equation for this problem. b. For intermediate unbraced lengths at which the maximum moment at incipient buckling is larger than b F L S xc = 0.9(0.7F y S xc ), the inelastic buckling analysis solution is again fully consistent with the AISC Section F2 equations, but is a more accurate assessment of the LTB resistance than the direct use of the AISC Section F2 equations. In this case, as the buckling resistance increases above b F L S xc, some reduction in the LTB resistance occurs due to the onset of yielding at locations where the internal moment is largest. The approach taken in AISC Chapter F is to simply scale the uniform bending LTB resistance by C b, but with a cap of b M max (= b M p for the W21x44 cross section). As discussed by Yura et al. (1978), this approach tends to over predict the true response to some extent in the vicinity of the point where the elastic or inelastic LTB design strength curve intersects b M max, although this approximation is considered to be acceptable. The LTB resistances obtained from the buckling analyses are slightly smaller than those obtained directly from the AISC Section F2 equations in the vicinity of the location where the LTB resistance reaches the plateau resistance b M max. This reflects the more rigorous accounting for inelastic stiffness reduction effects on the LTB resistance in the SABRE2 solution M u / b M p Uniform Moment AISC & SABRE2 Moment Gradient AISC Moment Gradient SABRE L b /L p Fig Lateral torsional buckling design resistances for W21x44 beams (F y = 50 ksi), calculated using the AISC Specification equations and using ILBA with the corresponding LTB stiffness reduction factor. 48

52 5.3 W21x44 Beam Column Strength Envelopes Figures and show the beam column strength envelopes determined from INBA for several suites of flexurally and torsionally simply supported W21x44 members, Grade 50 steel, subjected to uniform primary bending moment and a linear primary moment gradient loading respectively. Similar to the results for the moment gradient case in the previous example, SABRE2 provides a more rigorous solution than available via the ordinary application of the Chapter E, F and H (column, beam and beamcolumn interaction) provisions of the AISC Specification. Again, details of the SRF calculation are explained in White et al. (2015). The following observations can be made from these plots: 1. For the shorter members, the strength envelopes are essentially equal to the fully effective cross section plastic strength curves (with the effective yield load P ye equal to the yield load P y = F y A g ) at smaller axial load values. This result corresponds to the mechanics of the beam column strength problem reported by Cuk et al. (1986) and approximated by a combination of Eqs. (H1 1) and (H1 2) in the AISC (2016) Specification. 2. For larger axial load values, the strength envelopes peel away from the above in plane strengths at a particular axial load level and approach the AISC out of plane column strength of these simply supported members in the limit of zero bending and pure axial compression. The strength envelopes in these regions are slightly convex due to the loss of effectiveness of the W21x44 webs with increasing axial force. That is, the W21x44 webs are slender under uniform axial compression. 3. The members with longer lengths are not able to develop b M n = b M p for the case of zero axial load; rather, their bending resistance is governed by out of plane lateral torsional buckling. The moment gradient loadings allow the development of b M n = b M p for larger member unbraced lengths in the case of zero axial load. 4. For the longest W21x44 members considered (i.e., L = 15 ft), subjected to uniform primary bending moment, the failure is entirely due to elastic beam and beam column lateral torsional buckling. In this case, the resulting strength envelope ranges between 0.9 x P e for pure axial compression and zero bending and 0.9 R b M e for pure bending and zero axial compression, where P e is the theoretical out of plane flexural buckling load and M e is the theoretical lateral torsional buckling moment for pure bending. The shape of the strength curve in this case matches exactly with the theoretical beam column elastic LTB resistance, scaled by an interpolated reduction factor ranging from 0.9 x to 0.9 R b. 49

53 Fig Beam column strength envelope curves obtained from SABRE2 for several suites of flexurally and torsionally simply supported W21x44 members subjected to uniform primary bending moment and uniform axial compression. 50

54 Fig Beam column strength curves obtained from SABRE2 for several suites of flexurally and torsionally simply supported W21x44 members subjected to uniform axial compression and moment gradient loading from an applied moment at one end, zero moment at the opposite end. It is not possible to perform beam column strength calculations with this level of rigor by a second order load deflection analysis combined with the traditional application of separate manual strength interaction equations. SABRE2 incorporates the AISC member strength equations ubiquitously within a buckling analysis, via calculated net stiffness reduction factors (SRFs), to provide a more rigorous characterization of the member resistances. 5.4 Roof Girder Example Figure shows a roof girder design example adapted from a suite of example problems developed by the AISC Ad hoc Committee on Stability Bracing (AISC 2002). The girder has a 70 ft span and is subjected to gravity loading applied from outset roof purlins connected to its top flange and spaced at 5 ft on center. End negative moments are transmitted to the girders due to the rotational restraint from the columns in a clear span portal frame that this member is a component of, and end axial compression loads are applied to the girder from the thrusts at the foundation level of the clear span frame. The girder is considered as a subassembly isolated from the rest of the frame in this example, and consistent with common practice, the ends of the subassembly are assumed to be flexurally and torsionally simply supported. That is, the girder flanges are assumed to be free to rotate about the axis of the web (i.e., both the warping or cross bending of the flanges and the out of plane lateral bending of the flanges is unrestrained at the girder ends), and the girder major axis bending rotations are assumed to be unrestrained at the member ends, but the vertical and out of plane lateral deflections and the twist rotations at the ends of the girder are assumed to be rigidly restrained. 51

55 Fig Roof girder example, adapted from (AISC 2002). The top flange in this problem is assumed to be braced at the purlin locations by light weight roof deck panels, having a shear panel stiffness of 5 kip/in (G' = 1 kip/in). Flange diagonal braces are provided from the purlins to the bottom flange at the mid span of the girder plus at two additional locations on each side of the mid span with a spacing of 10 ft between each of these positions. These diagonal braces restrain the lateral movement of the bottom flange relative to the top flange, and therefore they are classified as torsional braces. The provided elastic torsional bracing stiffness from the combination of these components and the roof purlins is taken as T = 6400 in kip/rad. This estimate of the provided torsional bracing stiffness is outlined in (AISC 2002). These torsional braces combine with the panel lateral bracing from the roof deck to provide out of plane lateral stability to the roof girder. The above bracing stiffnesses are divided by 2/ and the corresponding reduced stiffnesses are employed for inelastic buckling analysis, per AISC Appendix 6 (AISC 2016). That is, given the calculation of the ideal bracing stiffnesses i, i.e., the bracing stiffness necessary to develop the required load in a buckling analysis model, the design philosophy of AISC Appendix 6 is that the required stability bracing stiffnesses are 2/ times the i values. This practice is derived from Winter (1960), who showed that in general, bracing stiffnesses larger than I are necessary to avoid excessive second order amplification of the bracing system deformations and internal forces. When the nominal bracing stiffnesses are specified as in the above, they are divided by 2/ to allow for an inelastic buckling analysis assessment consistent with this practice. The AISC (2002) calculations are based on the assumption that the roof diaphragm is effectively rigid. In addition, the axial compression in the roof girder is assumed at zero in the original calculations. Given the above adaptations, there are multiple attributes of this example that present substantial unknowns and/or difficulties for assessment of the roof girder by traditional methods: 1) If the panel bracing system required stiffness is checked just for the demands from the flexural loading using the AISC (2016) Appendix 6 rules, i.e., 2( Mu.max / ho) CdCt 2 (236 ft kip / in) x 1 x 2 kip br 10.3 L ft in br one can conclude that the panel bracing system does not provide full bracing. This does not preclude the consideration of partial bracing from the roof panels, but such considerations are beyond the scope of Appendix 6. 2) The roof girder is braced by a combination of panel bracing at its top flange and torsional bracing from the purlins and flange diagonal braces at selected locations. In addition, the roof girder is subjected to combined axial compression and major axis bending. AISC (2016) Appendix 6 does not provide any direct guidance for assessment of combined bracing systems and/or general bracing of beam column members, other than generally permitting buckling analysis methods such as in SABRE2 for assessment of the bracing stiffness. Some new guidance regarding these considerations is provided in the AISC (2016) Appendix 6 Commentary. 52

56 3) The torsional braces are not uniformly spaced along the length of the roof girder. The torsional brace spacing is 15 ft at the girder ends in the above design, and 10 ft in four interior segments. The Appendix 6 torsional bracing rules are based on an underlying model involving lateral torsional buckling of an elastic I section member braced by continuous torsional bracing, and the application of this model to discrete torsional bracing by effectively summing the torsional bracing stiffnesses and dividing by the total member length. It is anticipated that this approximation is slightly conservative in the maximum positive moment region of the above roof girder and slightly optimistic in the negative moment regions at the girder ends, but the specific magnitudes of the approximation are unknown. 4) Since the girder qualifies as being only partially braced by the roof panels per AISC Appendix 6, at least for the internal moments at the mid span of the girder, the specific unbraced lengths KL y and KL b that should be employed in the traditional calculation of the member axial and flexural resistances c P n and b M n are unknown. This issue precludes the use of either the Effective Length Method (ELM) or the Direct Analysis Method (DM) provisions of the AISC Specification to assess the above problem. It also precludes the use of the new AISC (2016) Appendix 1, Section 1.2 approach of Design by Elastic Advanced Analysis, since the Appendix 1, Section 1.2 provisions are based effectively on the assumption of full bracing in the calculation of b M n. Furthermore, as noted in in Section 2.2 of this document, elastic analysis models cannot be used to assess the strength of inelastic members restrained by flexible elastic bracing systems. The only methods within the scope of the AISC (2016) provisions that can be employed to assess the strength of the roof girder, addressing all of the above considerations, are: 1) Inelastic Nonlinear Buckling Analysis (INBA) as implemented in SABRE2, which is derived from and expands the AISC (2016) Chapter C, E, F and H and Appendix 6 provisions to allow for the overall assessment of the roof girder strength and the adequacy of the stiffness of the bracing system, combined with simple member force percentage rules from Appendix 6 for the assessment of the bracing system strength requirements, and 2) Design assessment via rigorous test simulation procedures, which is permitted by AISC (2016) Appendix 1, Section 1.3. This approach provides for a direct calculation of the bracing system strength requirements, in addition to checking of the adequacy of the bracing system stiffnesses. However, the amount of labor associated with defining appropriate geometric imperfections for this direct assessment is prohibitive for routine design, unless all of this labor can be automated within software in some fashion. Furthermore, the computational demands are much greater compared to the approach in SABRE2. These issues are discussed at greater length by Jeong et al. (2016). Figure shows the governing overall lateral torsional buckling (LTB) mode for the above example roof girder, determined using the INBA algorithm in SABRE2. The lines shown with a diamond symbol in the horizontal plan at the top flange level represent the shear panel bracing from the roof deck, and the circular symbols at the mid span and at the two locations on each side of the mid span, denote the torsional bracing from the roof purlins and the framing of a flange diagonal to the bottom flange of the girder. The magenta arrow symbols indicate zero displacement constraints, and the green arrows 53

57 represent the applied loads, as discussed in the prior examples. The banner at the top of the main viewing window confirms that the result is indeed from an INBA based on the current (AISC (2016)) resistance equations, and that the strength limit state is indeed governed by LTB. Fig Governing overall lateral torsional buckling mode for the roof girder. Figure shows the variation in the net SRF along the length of the girder obtained from the SABRE2 solution. This plot is generated by using the Create Report option under Results > Diagrams & Deflected Shape, writing out a report text file for the SRF diagram, and then reading the data from this report into Excel. Net SRF Position along girder length (ft) Fig Variation of the net Stiffness Reduction Factor (SRF) along the length of the roof girder at its maximum design resistance corresponding to n =

58 The applied load ratio at incipient inelastic buckling of the roof girder under the required gravity load is n = Therefore, the girder and its bracing system are not quite sufficient to support the required LRFD loading. It should be noted that if the axial load is assumed to be zero, a n of is obtained from an ILBA, indicating that the roof girder and its bracing system are sufficient to support the required loading neglecting the axial loading effects. A n of is obtained from an INBA, illustrating the fact that pre buckling displacement effects are negligible for in this problem when only considering the flexural loading. A n of is obtained from an ILBA with the required axial loading included, showing that there is a minor second order effect of the axial load acting through the girder vertical displacements in the plane of the web in this problem. One can observe a noticeable lateral deflection within the adjacent shear panels on each side of the mid span torsional brace in Fig This indicates that the light roof panel bracing is indeed providing less than full bracing at the first braced point on each side of the mid span (based on the idealization of the bracing stiffness as the nominal stiffness divided by 2/ ). Nevertheless, the overall design strength is slightly larger than the LRFD required strength if the axial loading is neglected. Figure shows that significant yielding is developed both at the mid span and at the girder ends when the system maximum design resistance is reached. Also, there is a slight decrease in the net SRF in the vicinity of the girder inflection points. This decrease is due to the use of a net SRF equal to 0.9 x x a at locations where the moment approaches zero, where a is the traditional AISC column stiffness reduction factor, whereas at locations dominated by bending actions, the net SRF approaches 0.9 x ltb, where ltb is the basic stiffness reduction factor derived from the AISC (2016) lateral torsional buckling resistance equations. The development of stiffness reduction factors that match rigorously with the AISC (2016) resistance equations is discussed in detail by White et al. (2015). It is important to emphasize that the accurate assessment of the combined bracing stiffnesses is somewhat challenging for this problem using any method other than the SABRE2 buckling analysis. The basic requirements specified in AISC Appendix 6 do not address combined lateral and torsional partial bracing. The AISC (2016) Appendix 1 provisions provide guidance for the use of advanced load deflection analysis methods for the general stability design. However, the application of these methods necessitates the modeling of an appropriate initial out of alignment of the girder braced points as well as out ofstraightness of the girder flanges between the braced points. The geometric imperfections needed to evaluate the different bracing components are in general different for each of the bracing components, and the geometric imperfections necessary to evaluate the maximum strength of the girder are in general different from those necessary to evaluate the bracing components. One can rule out the need to perform many of these analyses by identifying the girder critical unbraced lengths as well as the critical bracing components. However, short of the type of buckling analysis provided by SABRE2, it can be difficult to assess which unbraced lengths and which bracing components are indeed the critical ones. SABRE2 provides not only an assessment of the adequacy of the bracing system stiffnesses, but it also provides a direct check of the member design resistance given the member s bracing restraints and end boundary conditions. 55

59 The only shortcoming of the above buckling analysis approach in the context of the above type of design problem is that this approach does not provide any direct estimate of the bracing strength requirements. However, based on numerous results from experimental testing and from refined FEA simulation of experimental tests, it is recommended that the simple member force percentage rules of AISC (2016) Appendix 6 can be used to specify the minimum required strengths for the different bracing components. 5.5 Elastic Lateral Torsional Buckling of Web Tapered Beams Figure shows the generic geometry for a suite of web tapered I section beams studied originally by Yang and Yau (1987). The beams are torsionally and flexurally simply supported and are subjected to a concentrated transverse load at their mid span. The beam cross sections are all doubly symmetric, and the mid span load is applied at the shear center depth. This example varies the ratio of the smallest depth at the ends of the beam to the largest depth at the beam mid span,, and studies the influence of the corresponding taper change on the elastic lateral torsional buckling resistance of the beams. A similar suite of web tapered beams has been studied by Andrade and Camotim (2005) and by Chang (2006). Jeong (2014) provides an updated assessment of the problem with comparisons to shell finite element results. P d w.max d w.max L/2 L/2 E = ksi G = 11,150 ksi b f = 6 in, t f = 0.5 in d w.max = 24 in, t w = in L = 240 in Fig Simply supported web tapered I beam subjected to concentrated mid span loading. This example demonstrates the importance of properly addressing the mechanics associated with the web taper in the formulation of corresponding thin walled open section beam elements. Beam elements formulated assuming only prismatic geometry cannot be employed in a stepped discretization to capture the lateral torsional buckling resistance of tapered beams. For planar problems, a stepped discretization using prismatic elements converges to the correct result when enough elements are employed. However, the changes in the warping response as a function of the taper have to be addressed explicitly in the element formulation for thin walled open section beam elements to produce results that converge to the correct theoretical solution in the limit that a large number of elements is employed. 56

60 Figure compares the SABRE2 solution for the elastic buckling load of the beams versus the taper ratio, using 10 beam elements to model the full beam, to the 10 element solution provided originally by Yang and Yau (1987). It can be observed that the results from the two different solutions are practically identical. Figure shows the elastic buckling mode of the beam for = P cr (kip) SABRE2 (10 elem) Yang and Yau (10 elem) α Fig Comparison of SABRE2 results to results from Yang and Yau (1987) for the web tapered beam. Fig SABRE2 depiction of the Elastic buckling mode of the tapered beam for = 0.4. Figure compares two SABRE2 solutions for the elastic buckling load of the beams versus the taper ratio to several other solutions. In this plot, the 10 element solution from Fig is presented along with a second solution in which a very large number of elements is employed to ensure that the results are fully converged for all practical purposes. These solutions are compared to the results from a converged ABAQUS geometrically nonlinear shell finite element model of the beam developed by Jeong (2014). It can be observed that the SABRE2 10 element solution matches well with the ABAQUS shell 57

61 FEA solution for the full range of the parameter. The converged SABRE2 solution deviates from the ABAQUS solution by 3.1 % at = 0.1. However, in this case the web is only 2.4 inches deep at the member ends. The flanges of the beam are 0.5 inches thick throughout the length. In the shell FEA discretization, the web is modeled between the mid thickness of the flanges. Therefore, there is a small overlap between the shell elements in the web and the shell elements in the flanges at the web flange juncture. In most cases, the influence of this overlap is negligible. However, for the cases with small, the flange thicknesses and this overlap start to become a significant fraction of the depth at the ends of the beam. This is the primary source of the difference between SABRE2 and the ABAQUS shell FEA solution for small values. 34 P cr (kip) SABRE2 (10 elem) SABRE2 (Converged) 3DShell (ABAQUS) Mastan (10 elem) Stepped avg depth Mastan (10 elem) Stepped smallest depth α Fig Comparison of SABRE2 results to 3D Shell FEA and Mastan2 solutions for the web tapered beam from Yang and Yau (1987). Figure also compares the SABRE2 solutions to the result obtained from Mastan2 (Ziemian 2016) using 10 elements. The thin walled open section beam formulation in Mastan2 is based on the assumption of prismatic member geometry. Therefore, the Mastan2 solution can be used to demonstrate the effect of using a stepped discretization on the elastic LTB predictions. Two Mastan solutions are shown, one in which the average beam depth is used within the length of each element and one in which the smallest beam depth is used within the length of each element. One can observe that using stepped prismatic elements clearly gives incorrect results for the elastic LTB resistance of the tapered beam. Jeong (2014) shows that the Mastan2 solution using the smallest beam depth within the length of each element is identical to the 10 element solution obtained using the software StabLab (2016), which is advertised to provide solutions for tapered beams. 5.6 Elastic Buckling of a Stepped Web Tapered Column Figure shows an adaptation of the stepped web tapered column example from Section C.3.2 of Design Guide 25 (Kaehler et al. 2011). Section C.3.2 of DG 25 addresses the calculation of the in plane elastic buckling load of this member via the Method of Successive Approximations. This same member is considered here via SABRE2, but both the in plane and out of plane elastic buckling are considered. This 58

62 facilitates a discussion of the best practices for modeling of such a member in SABRE2. Similar to the previous examples, the column is rotated 90 o clockwise in Fig for ease of display. The in plane attributes of the member, illustrated in Fig , are exactly the same as in DG 25. However, DG 25 does not define the member s out of plane attributes. These are as follows: The member is torsionally simply supported at both of its ends. Both the left and right flanges of the member (the top and bottom flanges in the figure) are braced at 8.75 ft from the column base and at the step, located at 17.5 ft from the column base. Braced point (Typ.) in x x x 45 kip x in 30 kip in 8.75 ft = 17.5 ft 8 ft E = ksi Top web height = 12.0 in Step web height = 33.0 in Bottom web height = 12.0 in Left (Top) flange = 0.5 in x 8 in Right (Bottom) flange above step = in x 8 in Right (Bottom) flange below step = 0.75 in x 8 in Web thickness above step = in Web thickness below step = 0.25 in Fig Stepped column. The simplest approach to model the step in this type of member in SABRE2 is to use a taper over a very short length. As such, the bottom length of the member is assumed to vary from the bottom web depth of 12.0 inches to the step web depth of 33.0 inches over a length of inches = 17.5 ft. A 0.1 inch long element is then included at the step that varies the web depth from 33.0 back to 12.0 inches in the top length. The top length is then modeled as a prismatic segment. The very sharp taper within the above 0.1 inch long element at the step invalidates the assumption of cross section warping continuity. The tapered element formation in SABRE2 is based on the assumption that the taper angle is relatively small such that the longitudinal directions of the flanges and of the member are effectively all the same. This assumption is typically assumed to be satisfied sufficiently for taper angles of 15 o or less. The above idealization of the member is created by first defining the member based on its overall length and the cross sections at its joints, i.e., at its end nodes. Then additional essential nodes are added at inches from the base, and at inches from the base along with the cross sections on each side of the step. The step modeling capability in SABRE2 is aimed at handling steps in cross section dimensions other than the depth. Since there is a major change in the depth for the step considered in this problem, the physical step is modeled explicitly as a sharp web taper in the member. The member is modeled with 20 total elements in its segment 1 below the step, one element within the step, and 10 elements in the top segment 3 of the step. 59

63 Of course, the out of plane lateral bending of the physical right hand (bottom) flanges in the above member is not likely to be the same on each side of the step although the left hand (top) flange is continuous at this location. The only way to rigorously model the interaction of the flange plates in the vicinity of the step would be to use a higher order FEA model such as shell finite elements in this region. Nevertheless, the behavior associated with the step can be approximated in a reasonable and conservative way by releasing the weak axis rotational and warping continuity in the member either just above or just below the step. This is achieved by going to Conditions > Boundary Conditions > Define Element Flexural &/or Warping Releases. For the model of this member included with the download of SABRE2, the Flexural Release M y is set to Pinned at Node j of Element 20, just below the step, and the Warping Release is set to Free at Node j for this same element. Given the above idealization, the elastic buckling of the member in the out of plane and in plane directions can be assessed by requesting a sufficient number of buckling modes for Elastic Linear Buckling Analysis (ELBA) from the Analysis Parameters menu. For this problem, the user should request 6 buckling modes for ELBA on this page to obtain the key out of plane and in plane buckling results. Upon executing the ELBA, the first buckling mode is as shown in Fig Due to the statics of this problem, there is some additional compression in the left (top) flange throughout the member length due the eccentricity of the axial loadings relative to the non prismatic member centroidal axes. Also, the left flange is the smaller flange. As such, the buckling mode in Fig is a torsional flexural type buckling mode with the predominant out of plane displacements occurring in the left flange. One should note the rotational discontinuity in the flanges at the step. This is due to the minor axis rotational and warping releases at the top of the element just below the step. SABRE2 does not actually show a break in the continuity of the longitudinal displacements at the release, as a simplification; however, clearly, there is a rotational discontinuity in the lateral displacement curve for the left hand flange. Fig First elastic buckling mode for the stepped column. 60

64 Figure shows the 4 th elastic buckling mode for the above member, which involves the torsionalflexural buckling of the top segment of the column. Again, the left hand (top) flange is the smaller flange, and this flange also is subjected to an additional small flexural compression. Therefore, the predominant out of plane displacements are in this flange. Also, the break in the rotational continuity at the step is again apparent. Due to this break in the member weak axis rotational and warping continuity, and since the member is braced in the out of plane direction at the step, the out of plane buckling of the bottom and top segments of the member are uncoupled from one another. Fig Fourth elastic buckling mode for the stepped column. Finally, the 6 th elastic buckling mode, which involves the in plane elastic buckling of the member is shown in Fig This is the buckling mode that is evaluated in DG 25. DG 25 reports an eigenvalue of e = 62.8 from its method of successive approximations solution, as well as an eigenvalue of e = 64.2 from GT SABRE (Chang, 2006). These results compare to an eigenvalue of e = from SABRE2. 61

65 Fig Sixth elastic buckling mode for the stepped column. 6. Validation Case Studies 6.1 Overview SABRE2 has been tested using a wide range of case studies, in addition to the above demonstration problems, to verify the general correctness of its solutions. The following sections explain these validation studies. Sample SABRE2 Calculator and SABRE2 files from these validations are provided with the download of the software. 6.2 Doubly Symmetric Beams This set of case studies focuses on the flexural resistance in uniform bending, as governed by a combination of the AISC Lateral Torsional Buckling (LTB) and Flange Local Buckling (FLB) limit states, for several suites of doubly symmetric I section beams. All the beams are flexurally and torsionally simply supported. Nine different doubly symmetric cross sections are considered, and for each cross section the beam lengths are varied from less than and equal to length that the AISC Manual denotes as L p ' to a length greater than and equal to L r. The length L p ' corresponds to the plateau resistance for a given cross section, that is the unbraced length corresponding to the largest potential flexural resistance for a given web and flange slenderness. The length L r is the unbraced length at the transition between the inelastic and elastic LTB limit states. That is, L r is the shortest length at which the flexural resistance in uniform bending may be characterized by the theoretical elastic LTB resistance. The reader is referred to White and Chang (2007) and White (2008) for an overview of how the AISC flexural resistance equations work for cross sections with different flange and web slenderness values. 62