Plane Frames Research Articles

A rcGAN-based surrogate model for nonlinear seismic response analysis and optimization of steel frames

The combination of the surrogate model and optimization algorithm to solve structural optimization problems is an efficient way to lower computational costs and reduce time consumption. However, the development of surrogate models for structural analysis frequently faces challenges due to limited datasets. To tackle this issue, this paper presents a surrogate model capable of training on limited datasets while simultaneously predicting multiple concerned indicators, and demonstrates its effectiveness in performance assessment and design optimization through two seismic design case studies. Specifically, an improved model architecture based on the conditional Generative Adversarial Network (cGAN) is proposed. The feasibility of this surrogate model for seismic response analysis and optimization is initially demonstrated using an existing planar frame case. Subsequently, to validate the suitability of the surrogate model for Nonlinear Time History Analysis (NTHA) tasks, the proposed approach is applied to optimize a 3D steel frame equipped with nonlinear viscous dampers. Herein, a three-objective optimization problem is formulated, employing the Non-Dominated Sorting Genetic Algorithm (NSGA-II), driven by the trained rcGAN, to identify the Pareto front. The optimum design is subsequently selected from this front utilizing a multi-criteria decision-making technique. The outcomes from three optimization tests indicate that our approach effectively enhances the seismic performance of the frame while achieving substantial economic benefits, ultimately reducing the construction cost of the benchmark structure by up to 31.1 %.

Quantification and parametric analysis of large overhang RC planar frames subjected to horizontal and combined horizontal–vertical earthquake loading

Quantification and parametric analysis of large overhang RC planar frames subjected to horizontal and combined horizontal–vertical earthquake loading

Improved PBPD method for dual steel system considering the global effect of space frames

Performance-based plastic design (PBPD) method has been proposed to design single bay planar frames for twenty years. For a dual steel system, such as an eccentrically braced frame (EBF) structure, there will be both EBF bays and moment resisting frame (MRF) bays. So, the global effect between different frame bays cannot be considered when designing the global structure using the traditional PBPD method. In this paper, a ten-story high-strength steel composite K-shaped eccentrically braced frame (HSS-K-EBF) structure was firstly used for performance design and analysis based on a single bay EBF. The results show that the global structural model obtained by direct performance design of the single bay model cannot achieve the desired performance objectives. To this end, an improved PBPD method for the dual steel system was proposed considering the global effects of space frames. The elastic and elastic–plastic shear distribution relations were calculated for different bays of the frames including EBFs and MRFs. The performance objectives of the global structure were designed and evaluated more accurately by considering the over-strength contribution of shear links and the plastic development rate index ξ of energy dissipation members. The improved PBPD method was used to design the ten-story global HSS-K-EBF structure. The results of nonlinear dynamic time history analysis show that the improved PBPD method can ensure that the plastic damage distribution of energy dissipation members of different bays is more uniform and the plastic damage degree is more consistent with expectations.

Improved State-Space Approach Based on Lumped Mass Matrix for Transient Analysis of Large-Scale Locally Nonlinear Structures

Due to the assumption of acceleration variation in traditional step-by-step integration methods such as Newmark, sufficiently small time steps are required to ensure numerical stability and accuracy in dynamic systems. In contrast, the state-space approach, based on piecewise interpolation of discrete load functions, does not rely on predetermined acceleration assumptions and has demonstrated high efficiency in terms of stability and accuracy. The original state-space method requires the calculation of the inverse of the structural mass in the transition matrix. However, when a lumped mass matrix is used, this computation renders the entire mass matrix singular, resulting in an invalid solution expression. To address this issue, this study proposes an improved state-space approach for the transient analysis of large-scale structural systems with local nonlinearities. In this approach, a nonlinear force corrector is introduced as an external force term applied to the linear elastic system to account for the nonlinear behavior of locally yielding components. Consequently, the original nonlinear dynamic system can be transformed into an equivalent linear elastic transient system. Furthermore, based on the lumped mass matrix, a first-order ordinary differential state-space equation for such an equivalent linear elastic transient system is derived. Simulation results from three transient system examples show that the state-space approach outperforms the Newmark method in terms of accuracy and stability for dynamic systems characterized by high frequency and low damping. The prediction results show that the state-space approach appears to be insignificantly affected by the choice of the consistent or lumped mass matrix. The numerical results show that the root-mean-square errors between the consistent and lumped matrices in the top displacement time histories of a 15-storey plane frame under various seismic intensities are all less than 1%, and in the base reaction time histories responses the discrepancies are only about 0.5%, indicating that the use of lumped mass matrices is quite reliable. When many nodes or degrees of freedom have no assigned mass, the dimensionality of the state-space equation can be significantly reduced using the lumped mass approach. Therefore, the simulation of large-scale systems can be simplified by employing the improved state-space approach with lumped mass matrices, yielding results nearly identical to those obtained using traditional methods. In conclusion, the improved state-space approach has great potential for the simulation of transient behavior in large-scale systems with local nonlinearities.

IDA-Based Seismic Fragility Analysis of a Concrete-Filled Square Tubular Frame

Based on the incremental dynamic analysis (IDA) method, this paper conducts seismic fragility analysis of a CFST plane frame, a CFST spatial frame under 1D (one-dimensional) ground motions, and a CFST spatial frame under 2D (two-dimensional) ground motions, with different attacking angles. Firstly, nine-story, three-span CFST frame structures (including the plane frame and spatial frame) were modeled in OpenSees, based on the accurate simulation of the hysteresis performance of the test CFST frames. Then, twenty-five groups of ground motions were employed to analyze the seismic response. Lastly, the IDA curve clusters, probabilistic demand models, and seismic fragility curves of frame structures were researched, respectively. The analytical results showed that the exceeding probability of the spatial frame under 2D ground motions was successively greater than that under 1D ground motions, and greater than the plane frame, and the maximum difference at each performance level was up to 6% and 16%, respectively. The fragility analysis result of the spatial frame was sensitive to the attacking angle of ground motion, and the exceeding probability of the 135°, 150°, and 165° fragility curves was larger than that of the 0° (original attacking angle) fragility curve at each performance level. The research results provide a reference for seismic fragility analysis of CFST frame structures employing the IDA method.

A novel method to evaluate combined global seismic damage index using recorded floor-displacement data for RC plane frames

A novel method to evaluate combined global seismic damage index using recorded floor-displacement data for RC plane frames

The α-generalized implicit method associated with Potra–Pták iteration for solving non-linear dynamic problems

Generally, direct time integration procedures are used for solving the equations of motion in transient analysis of structures with large displacements. In this context, we propose an algorithm that combines the α-Generalized implicit integration method with the Potra–Pták two-step iterative scheme. The free Scilab program develops a computer code for the non-linear dynamic analysis of plane frames with large displacements and rotations. The FEM corotational formulation discretizes the structures considering the Euler-Bernoulli beam theory. The null-length connection element described by the axial, translational and rotational stiffnesses simulate the behavior of the beam-column connection. Jacobi’s method and the Scilab’s spec function determine the natural frequencies. The developed program is used for modal and transient dynamic analyses of frame problems available in the literature. The numerical results show that the Potra-Pták scheme obtains approximate solutions with fewer cumulative iterations until convergence and shorter processing time compared to the standard Newton-Raphson scheme. Further-more, the results show that the type of beam-column connection affects the vibratory behavior of the structure as well as the values of its natural frequencies.

Study on interior beam–column joint sub-assemblage under bi-directional loading using finite element analysis

Purpose The purpose of this paper is to study the shearing performance under bi-directional loading of an interior beam–column joint (BCJ) sub-assemblage using the finite element analysis (FEA) tool (midas fea), validated in this research. Design/methodology/approach The BCJ can be defined as an essential part of the column that transfers the forces at the ends of the members connected to it. The members of the rigid jointed plane frame resist external forces by developing twisting moment, bending moment, axial force and shear force in the frame members. On the type of joints, the response to the action of lateral loads depends on reinforced concrete (RC) framed structures. The joint is considered rigid if the angle between the members remains unchanged during the structural deformation. This work examined the shear deformation, load displacement and strength of a non-seismically detailed internal concentric RC joint using non-linear FEA. The bi-directional loading imposes the oblique compression zone on one joint corner. This joint core’s oblique compression strut mechanism differs significantly from that under unidirectional loading. The numerical results are compared with experimental results in this study, with the data published in the literature. Findings Numerical analysis results show that, in the comparative study of numerical and experimental values, the FEA tool predicts the behaviour of the RC BCJ well. The discrepancy between the experimental and numerical results amounts to 6 to 12% end displacement of the beam, 7% resultant joint shear force, 4.23% column bar strain and 0.70% hoop strain. Originality/value The current code of practice describes the joint sub-assemblage behaviour along the single axis individually. In the non-orthogonal system, the superposition of the two axes for joint space results in overlapping the stresses and, hence, the formation of the oblique strut. This may result in a reduction in the joint capacity under bi-directional loading. The behaviour must be explored in depth, and an attempt is made for further exploration.

Detecting localized damage in cantilevered structures under nonstationary ambient excitations via Gabor spectral mode transmissibility functions

A method based on Gabor spectral mode transmissibility functions (GSMTFs) is proposed to detect local damage in a cantilevered structure under nonstationary ambient excitations. Gabor transformation and singular value decomposition are used to reduce the influences of other vibration modes on Gabor spectral mode transmissibility functions and process nonstationary structural responses, respectively. A new state characteristic based on the fundamental structure frequency is formulated on the basis of the GSMTFs, eventually leading to the development of a new damage indicator. The probability density functions of the damage indicator for healthy and damaged states can be estimated from the measured data, and the receiver operating characteristic (ROC) curve derived from these probability distributions and the corresponding area under the ROC curve (AUC) are used to determine the damage location. A six-degree-of-freedom system and a typical transmission tower are numerically studied, and the results show that the proposed method can estimate the structural damage location under nonstationary random loads. The proposed method is further validated with a planar frame in the laboratory, which exhibits multiple damage elements via random force hammer excitations. The results show that the AUC values computed for certain parts of the structure containing the damaged elements are greater than those for other parts of the structure, indicating the effectiveness of the proposed method. Moreover, the proposed method is compared with the dot product difference (DPD) index, and the results from the laboratory planar frame demonstrate that the proposed method can better identify damage.

A Novel Weak-Form Plane Frame Quadrature Element and its Application to Bridge Analysis

A novel weak-form plane frame quadrature element (WPFQE) is developed. A complex structure can be divided into several non-homogeneous (or homogeneous) elements with large size. The Chebyshev–Lobatto differential quadrature dealing with differentiation and the Gauss–Lobatto quadrature dealing with integration have the same integration points including the ends of each element. Based on the energy variational principle, the diagonal mass matrix and positive definite real symmetric stiffness matrix for the element can be obtained. The unknown quantities at the internal quadrature points of the element can be represented by those at the endpoints of the element based on the principle of static equal effect anterior to the development of global matrices, which significantly reduces the sizes of modified element matrices while maintaining sufficient accuracy, thereby obtaining the small-sized global matrices. Assembly of elements can be performed efficiently just as that in the finite element (FE) method. In the case study, the static and dynamic performances of a three-span high-pier rigid-frame bridge with variable cross-section are studied by applying the proposed method. The results indicate that the orders of structural matrices and calculation time obtained by the WPFQE method are much smaller than those of the FE method.

Fire performance of steel reinforced concrete column to reinforced concrete beam planar frames: Experiment

This paper investigates the fire performance of large-scale steel reinforced concrete (SRC) column to reinforced concrete (RC) beam planar frames by a series of fire resistance tests. The testing parameters included beam load ratio, column steel ratio and column dimension. The test results indicated that, in fire condition, the SRC column to RC beam frame exhibited two primary failure modes: beam shear failure and column compression failure. In the case of beam shear failure, the fire resistance of frame specimens increased as the beam load decreased. In the scenario of column compression failure, out-of-plane failure occurred due to the weak axis of the SRC column being located outside the frame plane. Concurrently, the RC beam displayed significant shear deformation at the ends and notable bending deformation in the middle. The test results indicated that the column cross-sectional size notably influenced the fire resistance of SRC column to RC beam frame, however, the influence of column steel ratio was minimal.

Damage detection of frame structure using a novel time-domain regression method

Shear structure model is the most frequently used to model for the damage detection of frame building structures. However, due to the existence of modelling error, using a shear structure model to perform damage detection of a complex frame structure often results in inaccurate detection results. In this paper, a novel reduced model for the frame is proposed, which converts a multi-story multi-bay plane frame into a beam-like model, having one translational and two rotational degrees-of-freedom for each floor. Based on the new model, a novel time-domain regression method (TDRM) was established using the spectral density function between the horizontal acceleration of the frame floor and the reference response to identify the equivalent layer stiffness and damping parameters. Finally, a five-story two-bay frame structure is used to demonstrate the efficacy of the proposed time-domain regression method of estimating structural parameters and identifying structural damage.The results show that this method can identify, locate, and quantify the structural stiffness changes accurately.

A novel family of strain-based finite elements for the analysis of the material softening of planar frames

The article presents a novel strain-based finite element family for the analysis of material softening of planar frame structures. In our case, the softening zone is described by a discrete crack, which is considered as an ‘excluded’ finite element point, i.e., the deformation quantities in the crack are considered separately from the deformation quantities of the element. They are connected to the element only through kinematic quantities, used to describe the crack opening. The criterion for crack initiation is defined as the limit axial-bending resistance of the cross-section. The advantage of the presented model is that it is not necessary to define cracks or softening zones in advance and that the solution is mesh-independent in the sense that no further densification of the mesh is needed purely on account of capturing material softening. The accuracy and efficiency of the presented finite element family is illustrated by the example of a clamped–simply supported concrete beam and a portal concrete frame. The examples demonstrate that even with a minimum number of finite elements of suitable accuracy, sufficiently accurate results are obtained for normal engineering practice.

Cross-domain structural damage identification using transfer learning strategy

In damage identification of civil structures, training deep neural networks (DNNs) often requires a large volume of annotated training data. However, artificial damage to real-world structures is always forbidden, resulting in insufficient data for training. Transfer learning provides a solution that allows structural damage information in a data-rich domain to be transferred and shared as the prior knowledge to a data-scarce domain, thereby indirectly augmenting available training data in the latter domain. To this end, a multi-task transfer learning strategy has been adopted for the purpose of achieving cross-domain damage identification between a plane frame in the source domain and a three-dimensional frame in the target domain. The strategy can share the learning knowledge from the domain with sufficient training data to the target domain with insufficient data. Thereby, the DNN (named DNN#2) in the target domain can be trained on a small amount of training data. Owing to their ability in data feature extraction, stacked auto-encoders (SAEs) are used to construct the desirable DNN#1 and DNN#2 corresponding to different damage identification tasks, named Task#1 and Task#2, in the two domains. The two DNNs share the hidden layers in order to share damage feature information between the source and target domains. The training datasets of the two domains are first used to jointly pre-train the auto-encoders’ parameters in an unsupervised learning way. Afterwards, a supervised fine-tuning step is carried out to retraining the entire SAEs for better performance. By these means, Task#2 receives some prior knowledge from Task#1, and thus, it is accomplished on limited training data when Task#1 is synchronously achieved. The analysis results demonstrate that the proposed multi-task learning strategy requires only a single training session to simultaneously realize the cross-domain damage identification of two different steel frames.

On optimization of lightweight planar frame structures: an evolving ground structure approach

On optimization of lightweight planar frame structures: an evolving ground structure approach

A numerical assembly technique for free vibration of plane frames using a shifted and deflated Newton method

A semi-analytic method, namely the Numerical Assembly Technique (NAT) is extended to compute natural frequencies of arbitrary planar frame structures. The frame structure is systematically defined by a set of nodes, beams, bearings, springs, and external loads and the corresponding boundary and interface conditions are formulated. We consider the Timoshenko-Ehrenfest beam theory and take axial, bending and shear deformations as well as linear and rotational inertia into account. Since we use the exact solutions of the governing differential equations in the NAT, the accuracy of the results is independent from the spatial discretization. However, the resulting eigenvalue problem is non-linear and requires a numerical algorithm to solve. We consider three different representations of exact homogeneous solutions and study the numerical stability of the resulting system matrix. To solve for the eigenvalues we propose a Newton method enhanced with a shifted deflation algorithm. The stability and the efficiency of this approach is illustrated in numerical examples.

Damage and Nonlinearity Effects on Stress Wave Propagation in Planar Frame Structures: A Machine Learning Classification Approach Based on Stress Wave Amplitude Solution

Damage and Nonlinearity Effects on Stress Wave Propagation in Planar Frame Structures: A Machine Learning Classification Approach Based on Stress Wave Amplitude Solution

Closed-form solutions for axially non-uniform Timoshenko beams and frames under static loading

This paper presents the Green’s Functions Stiffness Method (GFSM) for solving linear elastic static problems in arbitrary axially non-uniform Timoshenko beams and frames subjected to general external loads and bending moments. The GFSM is a mesh reduction method that seamlessly integrates elements from the Stiffness Method (SM), Finite Element Method (FEM), and Green’s Functions (GFs), resulting in a highly versatile methodology for structural analysis. It incorporates fundamental concepts such as stiffness matrices, shape functions, and fixed-end forces, in line with SM and FEM frameworks. Leveraging the capabilities of GFs, the method facilitates the derivation of closed-form solutions, addressing a gap in existing methods for analyzing non-uniform reticular structures which are typically limited to simple cases like single-span beams with specific axial variations and loading scenarios. The effectiveness of the GFSM is demonstrated through three practical examples, showcasing its applicability in analyzing non-uniform beams and plane frames, thereby broadening the scope of closed-form solutions for axially non-uniform Timoshenko structures.

Experimental investigation of non-prismatic hybrid plane frame containing glass fiber reactive powder concrete and conventional concrete under static load

Experimental investigation of non-prismatic hybrid plane frame containing glass fiber reactive powder concrete and conventional concrete under static load

Static analysis of functionally graded porous beam-column frames by the complementary functions method

In this paper, an efficient and accurate numerical approach is implemented to examine the static behavior of Functionally Graded Porous (FGP) planar frames for the first time. The symmetric material constitutive relationship (SMCR), monotonic material constitutive relationships (MMCR), and uniform material constitutive relationship (UMCR) models are used to define the gradation of porosity along the thickness direction. The effects of porosity coefficient and material porosity distribution type on the static behavior of the FGP frames are investigated in detail. A direct relation between porosity coefficient and nodal deformations is observed.The effect of shear deformation is considered in the governing equations based on the Timoshenko Beam Theory (TBT). Within the scope of the presented procedure, fifth-order Runge-Kutta (RK5) algorithm is applied in the solution of the initial value problems. Imposing the rigidity matrix-based Complementary Functions Method (CFM) on the static analysis of the FGP planar frames is the novelty of this research. The results of the suggested approach are verified with the those of the finite element method (FEM), and those of the available literature. It has been carried out that type of the porosity distribution and porosity gradient index have an important impact on the response of FGP planar frames. By increasing the porosity coefficient, the displacement values of the frames made of UMCR and MMCR porosity distribution are affected more than those of the UMCR distributions. It can be noted that the SMCR can be preferred for designing the FGP beam-column structure as the minimum deformation values are obtained.