Generalized Strain Energy Density Research Articles

Exploration of the behavior and evolution features of beam-column joints with T-stub using the structural strain energy method

Based on the structural strain energy method, the work behavior of four T-stub connection joints under cyclic loading is analyzed to visualize their working behavior and failure evolution. The experimental strain data are modeled as generalized strain energy density (GSED) to characterize the structural stress state and the corresponding characteristic parameters, and the Mann-Kendall (M-K) criterion is formulated to determine the performance characteristic points (P and Q) of the joint Mj-Ej curves and to derive the starting point of structural continuity damage. Then, by analyzing the parameters of the joint working state (strain, deformation, joint energy dissipation, connection stiffness, bolt force and pry force), the trends of the joint stressing state curves before and after the critical loading are reflected. Finally, the feature point data interpolation (EDI) method is introduced to effectively expand the joint test data, visualize the change process of the strain/stress field, and deeply reveal the mechanical response mechanism and evolutionary degradation law behind it. The contribution of the above work is to quantitatively identify the P and Q characteristic points in the changing trend of the joint working state, to reveal the mechanical mechanism and the failure degradation law from the perspective of energy evolution, and to visualize the mutation of the field data before and after the P/Q using the EDI method, which is a generalized method that can be applied to the other progressive failure structures.

Numerical study of multiple hydraulic fractures propagation in poroelastic media based on energy decomposition phase field methods

This paper proposes a novel phase-filed model for simulating hydraulic fracturing in poroelastic media under complex stress conditions. The main theoretical contribution lies in the fact that a generalized strain energy density decomposition is incorporated to hydraulic fracturing phase field model, which unifies hydrostatic-deviatoric and spectral decompositions and predicts material failure through classical Mohr-Coulomb failure criterion. Based on Biot’s poroelastic theory, an indicator function is used to distinguish between fluid flow in fractures and in pores. The fully coupled equations are solved by combining staggered schemes with Newton-Raphson methods. The proposed model is validated by comparing laboratory experimental results with numerical results. In order to demonstrate the influence of introducing hydrostatic-deviatoric and spectral energy decompositions, we simulated propagation of multiple hydraulic fractures and coalescence of hydraulic fractures with natural fractures. Finally, the effects of stress ratios (σx/σy) on multiple hydraulic fractures are analysed. It is found that when the stress ratio is 2, a more complex hydraulic fracture network is formed in the porous media.

Fracture analysis of Brazilian circular hole disk under mixed mode loading

The paper presents an investigation into a Brazilian disk specimen featuring a concentric circular hole, accompanied by two internal edge radial cracks. By introducing rotations of the cracks relative to the vertical axis, a combination of fracture modes I and II is induced. Employing the finite element method, we compute stress intensity factors related to fracture modes I and II, alongside T-stresses, while varying geometric parameters such as inner and outer diameters, crack length-to-ring width ratio, and crack inclination angle. Experimental analyses of ebonite's fracture toughness in the mixed mode are conducted, subjecting samples to static loading until complete failure. Each test provides the angle at which crack initiation occurs and the corresponding critical load. Three distinct fracture criteria - the generalized maximum tangential stress criterion (GMTS), extended maximum tangential strain criterion (EMTSN), and generalized strain energy density (GSED) criterion - are employed to predict fracture direction and critical load. The outcomes exhibit good correspondence between experimental critical loads and theoretical predictions.

The Stressing State Features of a Bottom Frame Structure Revealed from the Shaking Table Strain Data.

As a classic issue, structural seismic bearing capacity could not be accurately predicted since it was based on a structural ultimate state with inherent uncertainty. This result led to rare research efforts to discover structures' general and definite working laws from their experimental data. This study is to reveal the seismic working law of a bottom frame structure from its shaking table strain data by applying structural stressing state theory: (1) The tested strains are transformed into generalized strain energy density (GSED) values. (2) The method is proposed to express the stressing state mode and the corresponding characteristic parameter. (3) According to the natural law of quantitative and qualitative change, the Mann-Kendall criterion detects the mutation feature in the evolution of characteristic parameters versus seismic intensity. Moreover, it is verified that the stressing state mode also presents the corresponding mutation feature, which reveals the starting point in the seismic failure process of the bottom frame structure. (4) The Mann-Kendall criterion distinguishes the elastic-plastic branch (EPB) feature in the bottom frame structure's normal working process, which could be taken as the design reference. This study presents a new theoretical basis to determine the bottom frame structure's seismic working law and update the design code. Meanwhile, this study opens up the application of seismic strain data in structural analysis.

Failure evolution analysis of end-plate connection joint based on structural stressing state theory

Based on the structural stressing state theory, the structural behavior of four extended end-plate joints under cyclic loading was analyzed to visualize their operational performance and failure evolution. The experimental strain data are modeled as generalized strain energy density (GSED) to characterize the structural stressing state and the corresponding characteristic parameters, and the Mann-Kendall (M − K) criterion is coined to determine the critical loads (P and Q) of the Mj−Ej curves of the joints and derive the starting point of structural continuity damage. Then, the trend of the joint stressing state curves before and after the critical load is reflected by analyzing the parameters of the working state of the joint (hysteresis performance, strain distribution mode at key measurement points, bending and shear deformation development, bolt force and prying force). Finally, the sample point expansion (SPE) method is introduced to effectively extend the joint experimental data, visualize the change process of strain/stress field, and reveal the mechanical response mechanism and evolutionary degradation law behind it in depth. The results of the study provide a new perspective for conducting nodal analysis and a reference for failure analysis of end-plate connection structures.

Stressing state features of H-steel columns under cyclic biaxial bending action revealed from experimental residual strains

In this paper, the stressing state modeling analysis of the residual strain data of H-steel columns reveals three characteristic points that exist during the failure of H-steel columns. Also, the correctness of the stressing state analysis method, residual, and buckling characteristic pairs was verified. First, the experimental residual strains were transformed into generalized strain energy density (GSED) values as the state variables for establishing the stressing state mode and characteristic parameters (characteristic pairs). The Mann-Kendall (M-K) criterion is applied to the normalized GSED sum-j curves to reveal the characteristic points P, Q, and U of the evolving stressing state of the H-steel column. Characteristic point P is defined as the elastic-plastic branch point of the H-steel column, characteristic point Q is defined as the failure starting point, and characteristic point U is defined as the progressive failure point. Around the characteristic points of the H-steel columns, the directly modeled stressing state characteristic pairs, residual characteristic pairs, and buckling characteristic pairs produce significant mutation characteristics. This phenomenon verifies the correctness of the revealed H-steel column characteristic points and the rationality of this paper's stressing state modeling method. Then, it is proposed that the elastic-plastic branch point P can be directly used as the design reference point, and it is compared with the design point given by Code EN1993–1–5. In conclusion, this paper provides new ideas for analyzing steel structures and opens up the value of residual strain data in structural analysis.

Bending and tensile stressing state features of ductile iron pipe flange connections revealed from tested strain data

The existing theories and methods based on structural ultimate working states with empirical judgment could not estimate the load-bearing capacity of ductile iron pipe flange (DIPF) connections accurately. This paper tries to address the issue applying the novel structural stressing state theory and methods, leading to the revelation of the essential bending and tensile working features of DIPF connections. Firstly, the experimental strain data are transformed into the generalized strain energy density (GSED) values to build the stressing state modes and the characteristic parameters. Then, the Mann-Kendall criterion is adopted to detect the mutation points in the evolution curves of characteristic parameters. Correspondingly, it is verified that the evolutions of stressing state modes also present the mutation features. The mutation features are the embodiment of the natural law from quantitative change to qualitative change of a system, which defines the starting points of DIPF connections’ failure processes. Furthermore, the Mann-Kendall criterion distinguishes the elastoplastic branch (EPB) points in the stressing state evolutions of DIPF connections. Both failure starting points and EPB points provide the new foundation to accurately determine the working capacities of DIPF connections and to improve their design.

Analysis of The Working Performance of Large Curvature Prestressed Concrete Box Girder Bridges.

Based on numerical shape functions and the structural stressing state theory, the mechanical properties of the curved prestressed concrete box girder (CPCBG) bridge model under different loading cases are presented. First, the generalized strain energy density (GSED) obtained from the measured strain data is used to represent the stressing state pattern of the structure; then, the stressing state of the concrete section is analyzed by plotting the strain and stress fields of the bridge model. The stressing state pattern and strain fields of the CPCBG are shown to reveal its mechanical properties. In addition, the measured concrete strain data are interpolated by the non-sample point interpolation (NPI) method. The strain and stress fields of the bridge model have been plotted to analyze the stressing state of the concrete cross-section. The internal forces in the concrete sections are calculated by using interpolated strains. Finally, the torsional effects are simulated by measuring the displacements to show the torsional behavior of the cross-section. The analysis and comparison of the internal force and strain fields reveal the common and different mechanical properties of the bridge model. The results of the analysis of the curved bridge model provide a reference for the future rational design of bridge projects.

INVESTIGATION OF THE PERFORMANCE OF RC BEAMS REINFORCED WITH FRP AND ECC MATERIALS

This paper investigates the structural working behavior of reinforced concrete beams bonded with fiber reinforced polymer and engineered cementitious composite materials subjected to bending using structural stressing state theory. First, six reinforced concrete beams externally bonded with composite reinforcement layer and one control beam are tested to investigate the effects of the bond length, fiber reinforced polymer grid thickness and fiber content on the flexural behavior. Then, the finite strain data of RC beams are interpolated by the numerical shape function method. The generalized strain energy density model is established to characterize the stressing state of the structure. Through the MannKendall criterion, the characteristics load P and Q of the beams are detected, and the whole loading process is divided into three stage. Finally, the analysis of the strain and deformation on the beams reveals the effect of different parameters on different stage. The characteristic load P increases as the bond length increases, and the characteristic load Q increases as the thickness of the FRP and the fiber content increase. The vertical deformation of the strengthened beam for the characteristic load Q and ultimate load is significantly smaller than that of the unreinforced beam.

Bending working law of corroded ductile iron pipe flange connections revealed by structural stressing state theory

Applying structural stressing state theory, this study finds out the bending working law of corroded ductile iron pipe flange (CDIPF) connections from their tested strain data. Firstly, the strain data are transformed into the generalized strain energy density values to build the stressing state mode and the characteristic parameter. Then, the Mann-Kendall criterion is used to detect the mutation points in the evolution curves of the stressing state characteristic parameters with the load increase according to the natural law from quantitative change to qualitative change of a system. Meanwhile, it is verified that the evolutions of the stressing state modes also present the corresponding mutation features. The mutation points define the starting points of the bending CDIPF connections' failure processes. Furthermore, the Mann-Kendall criterion distinguishes the elastoplastic branch (EPB) points in the normal working processes of CDIPF connections. Both the failure starting point and the EPB point lay the new foundation to analyze the working performances of the CDIPF connections and to improve their design.

NSF-Based Analysis of the Structural Stressing State of Trussed Steel and a Concrete Box Girder.

This paper analyses the characteristics of the mechanical behavior of a trussed steel and concrete box beam under bending conditions based on the structural stressing state theory and the numerical shape function method. Firstly, the parametric generalized strain energy density was introduced to characterize the structural stressing state of trussed steel stud concrete box girders, and the strain energy density sum was plotted. Then the Mann-Kendall criterion was used to discriminate the leap point of the curve change and to redefine the structural failure load. By analyzing the strain and displacement, the existence of a sudden change in the structural response during the load-bearing process was again demonstrated. Afterwards, the numerical shape function method was used to extend the strain data, and further in-depth analyses of strain/stress fields and internal forces were carried out to show in detail the working characteristics of each under load. Through an in-depth analysis from different angles, the rationality of updating the failure load was verified. Finally, the effects of different structure parameters on the evolution of the structural stresses of the members were analyzed in a transversal comparison. The analysis results of the stress state of a steel-concrete truss structure reveal the working behavior characteristics of a steel-concrete truss structure from a new angle, which provides a reference for the design of a steel-concrete truss structure in the future.

Stressing State Analysis of Partially Prestressed Concrete Beams with High Strength Reinforcement Based on NSF Method.

This paper analyzes the flexural behavior of a partially prestressed steel high-strength reinforced concrete beams based on the structural stress state theory and the numerical shape function method. First, the generalized strain energy density is formed by the measured strain data of the test beam to reflect the structural stress state of the beams, and then the Mann–Kendall criterion is used to judge characteristic points of the generalized strain energy density curve. Two characteristic points, namely, post-elastic boundary load and failure load, are detected, so that the whole loading process is divided into three structural stressing state stages. Unlike the ultimate load, failure load is defined according to the general law from quantitative to qualitative change, which represents the starting point of the failure stage of the beam. Then, experimental strains and deflections, strain/stress fields interpolated by the numerical shape function method, and internal forces calculated by integration are respectively analyzed to obtain their changing characteristics and working behavior around the characteristic points, which can also verify the correction and effectiveness of the Mann–Kendall criterion. In addition, through the analysis above, it can be known that the failure loads of the test beams can be effectively improved by increasing the prestressed reinforcement ratio or concrete strength.

Three-Dimensional analysis of mixed mode Compact-Tension-Shear (CTS) Specimens: Stress intensity Factors, T-stresses and crack initiation angles

In order to characterize the crack tip stress field in three-dimensional fatigue and fracture problems, stress intensity factor and T-stress solutions are usually necessary. In this paper, extensive three-dimensional (3D) finite element models are constructed for Compact-Tension-Shear (CTS) specimen, which is one of the most widely used specimens for fatigue and fracture tests under mixed-mode I&II loading. In-plane and out-of-plane geometric dimensions including varying crack depths ratios (a/W = 0.3, 0.5 and 0.7) and thicknesses ratios (B/W = 0.1, 0.3 and 0.5) are considered under different loading angles (β = 0°, 15°, 30°, 45°, 60°, 75° and 90°). The results of stress intensity factors (KI,KII and KIII) and T-stresses (T11 and T33) along the crack front are computed and analyzed. The combined effects of geometric dimensions and loading angles on these fracture parameters have been illustrated. The solutions of the parameters are then described using empirical formulae by fitting numerical results with the least-squares method. Solutions obtained have been used to predict the crack initiation angles for CTS specimens based on mixed-mode fracture criteria including the maximum tangential stress (MTS) criterion, the generalized MTS (GMTS) criterion, the strain energy density (SED) criterion and the generalized SED (GSED) criterion. Solutions are useful for fracture and fatigue problems of mixed-mode I&II, such as analysis of the constraint effects on fracture toughness and propagation path of fatigue crack.

Stressing state analysis of multi-span continuous steel-concrete composite box girder

This paper uses the theory of structural stressing state to study the working characteristics of three multi-span continuous steel-concrete composite box girder (SCCBG) models with different prestress levels. Firstly, the generalized strain energy density (GSED) is constructed to reveal the working state of the structure based on the experimental strain data, the Mann-Kendall (M-K) criterion is applied to distinguish the leap characteristics load points (P and Q) of the structural stressing state and defined as the “failure load”. It is used to update the existing structural “destruction” load to more clearly reflect the entire structural failure process. Then, the working state parameters of the cross-section are described (strain distribution mode, midspan deflection, crack width, etc.) to analyze the working behavior of the box girder model. Finally, the non-sample point interpolation method (NPI) is used to reasonably expand the experimental strain data, to verify the rationality of the characteristic load points, and the stressing state characteristics of box girders with different prestressed levels are further discussed.

Hysteretic stressing state features of RCB shear walls revealed by structural stressing state theory

This study reveals the essential stressing state features of reinforced concrete block (RCB) shear walls in their hysteretic working process by applying structural stressing state theory and method. The experimental strains of the RCB shear wall are transferred into generalized strain energy density (GSED) values to build the stressing state mode and the parameter characterizing the mode. Then, the Mann-Kendall criterion is used to detect the mutation feature in the evolution of the stressing state characteristic parameter with the hysteretic load increase. Correspondingly, it is verified that the stressing state mode also presents the mutation feature. The stressing state mutation is the essential feature in the structural working process as structural failure starting point, which is the embodiment of the natural law from quantitative change to qualitative change of a system. Furthermore, the stressing state analysis reveals the elastic-plastic branch point in the working process of the RCB shear wall, which could be directly taken as the design point. In total, this study explores a new way to analyze the structural hysteretic working behavior and provides the rational reference to improve the existing design code.

Internal Forces Analysis of Prestressed Concrete Box Girder Bridge by Using Structural Stressing State Theory.

This paper analyzes the working behavior characteristics of a prestressed concrete transverse large cantilever continuous (PCTLCC) box girder bridge model based on structural stressing state theory and the numerical shape function (NSF) method. At first, the normalized generalized strain energy density (GSED) is established to model the stressing state of the bridge model. Subsequently, the Mann Kendall (M–K) criterion is applied to detect three characteristic loads, respectively, elastic–plastic branch load P (200 kN), failure load Q (300 kN), and progressive failure load H (340 kN), and the failure load Q is found to be the starting load of the damage process of the bridge model, rather than the ultimate load where the structure has been destroyed. Finally, the NSF method is adopted to interpolate the test data, and a detailed analysis for the variation characteristics of the working behavior of the bridge model under loads is performed based on the interpolation results. The characteristic load detection method and experimental data extension method for PCTLCC box girder bridge established in this study can provide valuable references for the design and analysis of such bridges.

The Hysteretic Failure Features of Reinforced Masonry Shear Walls Revealed by Modeling Experimental Residual Strain Data

ABSTRACT The hysteretic working capacity of reinforced masonry shear (RMS) walls has been estimated empirically and statistically referring to the structural ultimate state with considerable uncertainty. This investigation is to try to break this puzzlement of the uncertainty by revealing the hysteretic working features of RMS walls applying structural state-of-stress theory. The experimental residual strains are firstly modeled as generalized strain energy density (GSED) values (state variables) to describe structural state-of-stress mode and its characteristic parameter (GSED sum). Then, the slope increment criterion is proposed to detect the state-of-stress mutation point in the GSED sum-load curve. Accordingly, it is verified that the evolution of the state-of-stress mode presents the mutation feature. Thus, the starting point of the RMS wall’s failure process is revealed as the embodiment of the natural law from quantitative change to qualitative change of a system. And the hysteretic failure load of the RMS wall is defined at the state-of-stress mutation point with the attribute of certainty and generality. Furthermore, the GSED-based analysis reveals the working features between the state-of-stress submodes of structural components. This investigation explores a new way to analyze the hysteretic working behavior of RMS walls, which provides the rational reference to the accurate estimation of the failure loads of RMS walls for designs.

Stressing State Analysis of Reinforced Concrete Beam Strengthened by CFRP Sheet with Anchoring Device

This paper investigates the working performance of reinforcement concrete (RC) beams strengthened by Carbon-Fiber-Reinforced Plastic (CFRP) with different anchoring under bending moment, based on the structural stressing state theory. The measured strain values of concrete and Carbon-Fiber-Reinforced Plastic (CFRP) sheet are modeled as generalized strain energy density (GSED), to characterize the RC beams’ stressing state. Then the Mann–Kendall (M–K) criterion is applied to distinguish the characteristic loads of structural stressing state from the curve, updating the definition of structural failure load. In addition, for tested specimens with middle anchorage and end anchorage, the torsion applied on the anchoring device and the deformation width of anchoring device are respectively set parameters to analyze their effects on the reinforcement performance of CFRP sheet through comparing the strain distribution pattern of CFRP. Finally, in order to further explore the strain distribution of the cross-section and analyze the stressing-state characteristics of the RC beam, the numerical shape function (NSF) method is proposed to reasonably expand the limited strain data. The research results provide a new angle of view to conduct structural analysis and a reference to the improvement of reinforcement effect of CFRP.

Essential working characteristics of concrete-filled steel tubular arch supports discovered by modeling experimental strains

This study reveals the general working law of concrete-filled steel tubular (CFST) arch supports by exploring the experimental strain data based on structural stressing state theory. Firstly, the tested strains are transformed into the generalized strain energy density (GSED) values to model the stressing state mode and the characteristic parameters of CFST arch supports. Then, the Mann-Kendall criterion detects the mutation points in the evolving curves of characteristic parameters to the increasing load. Accordingly, it is verified that the stressing state modes also present the corresponding mutation features. The stressing state mutations reveal the failure starting point and the elastic–plastic branch (EPB) point of the CFST arch support’s working process, complying with the natural law from quantitative change to qualitative change of a system. Finally, the existing failure load and design load are redefined according to the revealed mutation features, which could update the working behavior analysis of CFST arch supports and promote the more rational design.

A generalized strain energy function using fractional powers: Application to isotropy, transverse isotropy, orthotropy, and residual stress symmetry

In this paper, we propose a generalized strain energy density function based on invariants of stretch tensor with arbitrary exponents. We employ polynomial, logarithmic and exponential functions of these invariants to develop the strain energy functions. We also study characteristics and applications of the proposed model for isotropy, transverse isotropy, orthotropy with a special focus on initial/residual stress symmetry. The proposed invariants generalize the existing strain energy potentials constructed with invariants of Cauchy stretch. We construct generalized strain energy functions for initial stress problems using initial stress symmetries. We obtain objectivity of strain energy for initially stressed transversely isotropic solids to derive the invariants and study resulting symmetry. By extending Dunford–Taylor integration based approach and tensor diagonalization approach, we obtain stress through differentiation of anisotropic scalar invariants with respect to a tensor. These approaches are usually applicable for derivative of isotropic tensor functions with respect to tensors. Using initial stress compatibility, we derive constraint equations for material parameters by evaluating the limits of Cauchy stress in the reference configuration. In order to apply the proposed model for initial stress problems, we further investigate bending and unbending of hyperelastic structures. We study bending of a rectangle to a cylinder and unbending of a cylinder to a rectangle in presence of initial/residual stress and observe both magnifying and moderating effects of initial stress for stress distribution and flexural characteristics. Using experimental data we substantiate the proposed model for seat foam, bovine pericardium and rabbit skin which represent compressible isotropic and incompressible orthotropic materials. For demonstration, we use both polynomial and exponential functions of these invariants. Furthermore, we develop a nonlinear finite element computational model for thermo-hyperelastic structure where thermal stress represents the initial stress. We corroborate this stress–strain data with the proposed model for initial stress symmetry and observe nice agreements.