Steel benchmark frames for structural analysis and validation studies: Finite element models and numerical simulation data

A sample of twenty-two steel benchmark frames was formed and used to investigate the accuracy of a single increment predictor-corrector (SIPC) solution scheme; an approximate method of second-order elastic analysis. Each of the frames is based on a structure from the literature and, as a collection, the set includes a diverse assortment of practical planar geometries and a wide range of sensitivities to second-order effects. This data article presents the details of these frames, including finite element models, relevant nodal coordinates and element connectivity, and detailed information regarding member sizes, support conditions, and applied loading. In addition, this article presents simulation data obtained from testing the SIPC method using the benchmark frames, and assessing its accuracy and precision. Error analysis results, based on comparisons of joint displacements and member design moments simulated using the SIPC method with those obtained using the more exact and computationally expensive work-control (WC) method, are summarized. The finite element modelling and subsequent structural analysis utilized the software package MASTAN2, which provides user-friendly features to execute both SIPC and WC methods. A detailed description of these analysis methods and the algorithms used to generate data is provided in “Efficient geometric nonlinear elastic analysis for design of steel structures: Benchmark studies” Ziemian and Ziemian [1]. The benchmark frame models are more generally useful for any researcher interested in testing and validating structural analysis and design methods, and the simulation data allow for comparisons with the results of other proposed solution schemes.


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A sample of twenty-two steel benchmark frames was formed and used to investigate the accuracy of a single increment predictor-corrector (SIPC) solution scheme; an approximate method of second-order elastic analysis. Each of the frames is based on a structure from the literature and, as a collection, the set includes a diverse assortment of practical planar geometries and a wide range of sensitivities to secondorder effects. This data article presents the details of these frames, including finite element models, relevant nodal coordinates and element connectivity, and detailed information regarding member sizes, support conditions, and applied loading. In addition, this article presents simulation data obtained from testing the SIPC method using the benchmark frames, and assessing its accuracy and precision. Error analysis results, based on comparisons of joint displacements and member design moments simulated using the SIPC method with those obtained using the more exact and computationally expensive work-control (WC) method, are summarized. The finite element modelling and subsequent structural analysis utilized the software package MASTAN2, which provides user-friendly features to execute both SIPC and WC methods. A detailed description of these analysis methods and the algorithms used to generate data is provided in "Efficient geometric nonlinear elastic analysis for design of steel structures: Benchmark studies" Ziemian and Ziemian [1] . The benchmark frame models are more generally useful for any researcher interested in testing and validating structural analysis and design methods, and the simulation data allow for comparisons with the results of other proposed solution schemes.
© 2021 The Author(s

Value of the Data
• The data provides a variety of useful resources for investigating structural behaviour and analysis methods using finite element software. The set of twenty-two benchmark frame models is important for testing/validating computational analysis methods and comparing results with those from commercial software and/or other works in the literature. • Researchers and practicing engineers interested in studying or validating structural design and analysis methods and/or newly proposed solution schemes can use the developed set of benchmark frame models, which includes a variety of structural systems with a range of sensitivities to second order effects, realistic geometries, and boundary conditions that satisfy current design codes. Researchers and practicing engineers can also use the provided frame models/data and the associated analysis results to compare the performance of other proposed solution schemes to that of the SIPC method.
• The finite element models of the perfect and/or imperfect frames can be directly opened in MASTAN2 and further studied using any of the analyses and graphical visualization features available in the software. The frame nodal and connectivity data can be used to create the same benchmark structures (perfect and/or imperfect geometries) with other finite element software packages. The numerical simulation data can be used and/or reproduced as a basis for comparison with the results of other second-order elastic analysis methods or different approximate solution schemes.

Data Description
This data article presents the computational models and details associated with a collection of twenty-two planar steel frames, as well as the numerical simulation data produced by elastic analyses performed on the frames.
The benchmark frames are introduced in Fig. 1 , which displays the structure's geometry, load case investigated, and critical buckling load ratio α cr . For each frame in Fig. 1 , the following data are provided in the Mendeley data repository [3] : • Description of principal characteristics, including geometry, member sizes, and loading details (PDF file). A sample of this data is presented in Fig. 2 . • Finite element models of both perfect and imperfect (with global sway) geometries (MAS-TAN2 [2] files). • Finite element model data for both perfect and imperfect geometries (Excel file), including nodal coordinates and element connectivity, useful for creating the same frame models with other analysis programs. Fig. 2 illustrates the detailed descriptive information that is provided as supplementary data [3] for each frame.
This set of benchmark frames has been used to test and validate a proposed solution scheme for the approximate second-order elastic analysis of structures; the single increment predictorcorrector (SIPC) method [1] . Table 1 presents a summary of the error analyses completed to assess the accuracy and precision of the SIPC method, including comparisons of simulation data produced using SIPC with those produced using the more exact and computationally expensive work-control (WC) method [13] . Table 1 also includes two indicators used to assess each frame's sensitivity to instability. The first is an amplification factor proposed by Merchant [14] to estimate second-order forces from first-order LA results. Merchant's amplifier is based on a frame's critical buckling load factor, α cr , as follows: The second indicator is the maximum ratio of second-order (WC) to first-order (LA) lateral joint displacements at any joint in the frame, or ( δ WC / δ LA ) . A comparison of the two indicators is presented in Fig. 3 .

Table 2
Details of finite element modeling of benchmark frames.

Geometry Members Modeled as 4 planar line (6 dof) elements Multiple lean-on columns
Modeled as single column on each side of frame Imperfections Global sway; Magnitude: H/500; Direction: same as lateral load (or natural lean, for gravity-only frames). Imperfections were modeled by horizontally translating all nodes by an amount equal to their vertical coordinate divided by 500.
Material Steel E = 200 GPa (29,000 ksi); F y = 345 MPa (50 ksi) or 250 MPa (36 ksi); Elastic modulus used Stiffness reduced to 0.8E, per AISC's direct analysis method (DAM) [16] , for all elastic analyses; Stiffness reduced to 0.9E for geometric and material nonlinear analyses with imperfections (GMNIA); Yield strength used F y (for DAM), 0.9F y (for GMNIA); Loading Gravity load Distributed loads modeled by equivalent concentrated loads applied at the beam quarter points. Gravity loading includes self-weight of components.

Lateral load
Applied as concentrated loads in direction as shown ( Fig. 1 ) Live/Dead load ratios Equal to L/D of original frame in literature (where applicable) LRFD load combinations [15] Multi-story frames without wind: 1.2D + 1.6L + 0.5L r Single-story frames with and without wind: 1.2D + 1.6 L r + (L or 0.5W) Single-and multi-story frames with wind: 1.2 D + 1.0L + 0.5L r + 1.0W Load application In all analyses, gravity and lateral loads are applied proportionally (simultaneously) Load magnitudes Determined such that the applied load ratio ≈ 1.0 in GMNIA, using appropriate factored load combination and initial global sway imperfection in the direction of lateral loading or in direction shown in Fig. 1 (for gravity-only frames: 3, 9, and 10).

Experimental Design, Materials and Methods
The finite element models of the benchmark frames were created in MASTAN2 as described in Table 2 below.
Several different analyses were performed on each frame model in order to validate the proposed SIPC method and examine its accuracy and precision as a function of the frame's sensitivity to second order effects. The parameter settings used for each analysis type are described in Table 3 . Fig. 4 also provides an illustration of how the analysis options are selected within MASTAN2.
The described error analysis is based on comparisons of the lateral displacement of each joint and the design moment in each member in the frame, as simulated by SIPC and WC analyses. As described in [1] , the WC data was first filtered in order to avoid error calculations on extremely small and insignificant joint displacements and/or member design moments. Specifically, a joint displacement that was less than 1/10 0 0 of the minimum member length in the frame was removed from consideration. Similarly, each member moment was first divided by the product of the member's plastic section modulus and material yield strength, resulting in the associated proportion of the member's plastic moment capacity. These normalized moments from the WC data were then filtered, for the same reasons described above. Specifically, a normalized member moment that was less than 0.10, or 10% of the amount of moment required to form a flexural plastic hinge in the member, was removed from consideration. Table 3 Details of the computational analyses performed in MASTAN2 on each frame.

Frame analyses
Elastic, first-order Linear (eigenvalue) buckling analysis (LBA) Used to compute critical buckling load ratio, αcr , corresponding to lowest sway buckling mode. Performed on perfect geometry, using stiffness 0.8E.
Linear analysis (LA) Results used for computations of second-to first-order ratios and/or amplification factors. Performed on imperfect geometry, using stiffness 0.8E.
Elastic, second-order Note: MASTAN2 accounts for second-order effects using an updated Langrangian formulation and geometric stiffness matrices [13] . Geometric nonlinear analysis with imperfections (GNIA) Single increment predictor-corrector (SIPC) analysis, a proposed approximate GNIA method; Increment size = 1; Performed on imperfect geometry, using stiffness 0.8E. GNIA Work-control (WC) analysis with step-size = 0.001. Used as the 'expected' or 'exact' elastic solution for comparisons when assessing the accuracy of the SIPC method; Performed on imperfect geometry, using stiffness 0.8E.
Inelastic, second-order Geometric and material nonlinear analysis with imperfections (GMNIA) Results used only to determine magnitude of loading placed on the frames; Performed using modified tangent stiffness method [1] with imperfect geometry, stiffness 0.9E, and yield stress 0.9F y ; Objective is to have structure fail at 1.0 * load factor. All error analyses were based on percent error calculations. For example, the comparison of the lateral displacements of joint j was made by computing the percent error as,