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ON CANCELLATION OF UNBALANCED FORCE VECTOR IN THE ELASTOPLASTIC INCREMENTAL ANALYSIS OF A FRAME骨組の弾塑性増分解析における不平衡力の解消について
Abstract
As a frame deforms under increasing external loads, relatively brittle member in the frame may fracture at an early point on the loading process. This causes sudden and considerable unbalanced force vector in the frame and then the released forces, which are axial force, bending moment, shear force and axial torsional moment of early fractured member, are redistributed into the remaining members of the frame. The elastoplastic incremental analysis to estimate the restoring force characteristics of a frame must be carried out with cancellation of above mentioned unbalanced force vector. However, it seems the dealing of those unbalanced force vector in the present most analysis codes is not necessarily clear and this reduces the reliability of the obtained values of horizontal load-carrying capacity.
In this paper a procedure to cancel a large unbalanced force vector in the elastoplastic incremental analysis of a frame is presented. The procedure is applicable only to the analysis method in which elastic and plastic components of deformations of each element can be separated explicitly from the largely deformed frame, e.g. the Fibered Plastic Hinge Method (FPHM)6), 7), 8). By using the procedure, redistribution of the released forces of early fractured relatively brittle member into the remaining members of the frame can be done, therefore, an accurate restoring force characteristics of the frame having both brittle and ductile members can be obtained. The validity and reliability of the procedure are demonstrated by the analysis of two-bay two-story steel frame.
The obtained findings are as follows:
1. Using the present procedure, a discontinuous restoring force characteristics of the frame having both brittle and ductile members can be obtained.
2. A final behavior of the restoring force characteristics of the present example frame approaches to that of the frame excluded initially the fractured members. This agrees qualitatively with the schematic diagram presented in Commentary on Structural Regulations of the Building Standard Law of Japan 2015 Edition¹⁾.
3. The reliability of the horizontal load-carrying capacity of a frame obtained by the analysis without unbalanced force cancellation may not be sufficient.
This paper presents a numerical method for three-dimensional (3D) seismic response analysis of a frame containing member failure. The base of proposed method is the Fibered Plastic Hinge Model (FPHM)9),10) in which elastic and plastic components of deformations of each element can be separated explicitly from a largely deformed frame. The analysis is performed with cancellation of a large unbalanced force vector caused by a sudden fracture of members in the dynamic elastoplastic incremental analysis of a frame. The FPHM program uses a gradient of existing elastic strain energy as an internal force vector, which is needed to evaluate unbalanced force vector, of a frame at each incremental step. A redistribution of member forces, which are axial force, biaxial bending moments, shear force and axial torsional moment, of early fractured low ductile member into the remaining members of the frame is done in each step, therefore, the dynamic response of a frame that contains both low ductile and ductile members can be obtained accurately. The validity of proposed method is verified through the numerical experiments on one-bay one-story braced steel frame having a low ductile tension brace.
Then, a possibility to use proposed method as a collapse analysis method for a 3D frame is examined by utilizing available shaking table test results on full-scale two-bay four-story steel building¹³⁾ assuming a simple fracture criterion for an element:
|ε|max = ηεy
where |ε|max is the maximum value of axial strain of a fiber due to varying axial force and biaxial bending moments at the element ends, εy is the initial yield strain of a fiber, and η is a reference value. Since the FPHM divides the element-end sections to fine fibers, |ε|max can be easily obtained in the numerical procedure.
Assuming η = 20, which was determined by trial and error, the obtained numerical results follow mostly the collapse behavior of the building, except that the deterioration behavior due to local buckling of the columns is different from the test result¹³⁾. In addition, the firstly and secondly fractured columns obtained by the present analysis are consistent with those observed in the test¹³⁾. Although a systematic way to estimate the value of η is unknown at the present time, η may be a parameter which relates a member which loses load carrying capacity by the local buckling to an element formulated according to the Bernoulli-Euler hypothesis.
A second-order spread-of-plasticity analysis program has been developed for analysing three-dimensional (3-D) steel frames. Material nonlinearity is modelled through the von Mises yield criterion in conjunction with the associated flow rule and the assumption of isotropic hardening. Gradual yielding is modelled through numerical integration of the material points on the cross-sections, which are located at the selected integration points along the member length. A mixed element formulation is proposed for analysing large-scale framework analysis. Established benchmark results are used to verify the proposed spread-of-plasticity analysis. The inelastic behaviour of a 3-D steel frame is investigated, and the computing efficiency and accuracy of using the mixed-element approach for advanced analysis of large-scale framework is illustrated.
The experimental study was made on the three dimensional elasto-plastic behavior of space frames consisted of two rigid frames in one direction and two tensile braced frames in the other direction which are general structural system of low steel structures. Test specimens were one story-one bay space frames constructed with four columns and four beams of H-shaped members and with four tensile braces of round steels, and were tested under horizontally loading combined with axial load. From these experimental results, the following difference of the elasto-plastic behavior of space frames from plane frames were clarified. 1. Additional axial force in columns caused by deformation of braced frames makes the difference to the restoring forces of two parallel rigid frames, and torsion of frame is accumulated with repetition of loading cycles. 2. The maximum restoring force and plastic deformation capacity in direction of strong axis of H-shaped columns decrease under the influence of the deformation of columns in direction of weak axis.
In this paper, the inelastic dynamic response of steel space frames with tensile bracing members is studied by the method presented in the first paper. The method assumes that a column segment consists of uniaxially stressed fibers along its length. The dynamic response is computed by assuming an elasto-plastic type hysteresis behavior considering strain hardening in stress-strain relationship for the columns and in tension only for the diagonal bracing members. The responses of single-story, single-bay braced space frames subjected to simultaneous action of two horizontal components of sinusoidal ground acceleration are studied using lumped mass, rigid-floor idealization. The results of the analyses include the responses of the space frames with eccentricity between centers of mass and elastic resistance. In addition, it is shown that the analytical solution predict well the general cyclic force-deformation behavior of the experimental results of a braced frame given in Ref.9. Some of the significant aspecst of the results are summarized in the following : 1) The maximum responses of the space frames are less than those of the plane frames in the direction of the braced frames and greater than those in the orthogonal direction. 2) The torsional responses of the space frames are affected by the ratios of elastic stiffness and maximum strength in each horizontal direction and increase in the case of a great difference between elastic stiffness and maximum strength in two horizontal directions compared with little difference. 3) The torsional responses of the space frames with eccentricity subjected to sinusoidal gronud acceleration with no phase difference between two horizontal components are greater than those with a phase difference. 4) The magnitude of the total input energy into the space frames with the same maximum strength in two horizontal directions subjected to the identical ground acceleration scarcely depends on the rations of elastic stiffness in two horizontal directions and eccentricity between centers of mass and elastic resistance. The maximum response of a column of a space frame is, however, unable to be estimated by only the total input energy into the space frame.
During past severe earthquakes, a number of R/C buildings designed by the old codes suffered heavy damages including collapse. To evaluate the seismic performance of old buildings for the ultimate limit state, it is important to study the nature of axial shortening for old columns subjected to earthquake loads. Half-scale model specimens with shear mode simulating old columns were tested until they came to be unable to sustain axial load. Using results of the tests, it was attempted to formulate the relations of vertical and lateral deformation based on the concept of failure surface contraction and theory of plastic flow.
This paper presents an analytical model for the simulation of steel braced frames in which both buckling and fracture of the braces are considered. The cross-section consists of many fiber segments, with each element taken to lose its resistance at the time of fracture. A sudden loss of resistance is converted to an immediate inertial force to maintain the overall equilibrium. The proposed model is applied to a three-story braced frame whose earthquake response behavior is examined for multiple near-fault ground motions. The effect of buckling on the maximum story drift is most notable with significant increases in the maximum story drift. The effect of brace fracture is found secondary particularly when the fracture occurs after buckling.
An accurate plastic hinge type beam elememt has been developed by the authors for three-dimensional (3-D) elastoplastic large deformation analysis of frames which contain all kinds of members: i.e. steel members, RC members, SRC members, CFT members, PC members, steel damper braces and tension braces. The element was originally proposed for pure steel frames by the first author (Shugyo 2003) and named Fibered Plastic Hinge Model (FPHM). The formulation procedure is a combination of the modified incremental stiffness method, the updated Lagrangian formulation, and numerical integration of fiber stiffnesses about the sections at the plastic hinges. The following assumptions are made to form the elastoplastic tangent stiffness matrix of the pure steel element: (1)members have thin walled closed or open sections, and cross sections remain plane and do not distort in the absence of cross-sectional warping, (2)deflection is large but elastic strain is small, (3)axial stress and the shear stress due to St. Venant torsion participate in yielding of fibers of members with closed sections, while only axial stress participates for members with open sections, (4)plastic deformation consists of only four components that correspond to axial force, biaxial bending moments, and torsional moment or bimoment, (5)there is no local buckling, (6)although an actual generalized plastic strain increments in a short element generally distribute nonlinearly, it is idealized as generalized plastic strain increments distribute linearly with the values at element nodes i and j, (7)incremental plastic deformations in the two half portions occur concentrically in the plastic hinge of zero length at element nodes i and j respectively. Because of the above mentioned assumption (6), the element requires at least four-element approximation for a frame member. This causes considerable increase of total degrees of freedom in the analysis of multi-bay multi-story frames.
In this paper a method to reduce the total degrees of freedom in frame analysis by the FPHM is presented. Introducing the plastic deformation reduction coefficient, the assumption (6) is modified as one-element approximation for a frame member has sufficient accuracy for practical use. The optimum value of the plastic deformation reduction coefficient is examined by using quasi-static analysis of four kinds of one-bay one-story plane portal frames, 3-D quasi-static analysis of two-bay four-story steel frame which contains composite beams and semirigid column bases, and 3-D quasi-static analysis of twenty-story eccentric steel frame with H-shaped steel columns.
This study investigates the influences of shear-mode mullion-type walls on the performance of RC rigid-frames. To express shear-axial coupled motion occurred with shear failure of walls, shear-axial softening behavior was modeled using the theory of plasticity and the concept of failure surface contraction. Pushover analysis was then conducted to RC frames that have various combinations of column over-design factors and wall breadths to simulate response to the seismic load and collapse mechanism. Some frames with low column over-design factor showed partial collapse after shear failure of walls, while others showed more ductile mechanism. The results revealed how shear-mode mullion-type walls can affect the performance of RC frames with different configurations.
A beam element is presented for analysis of the elastoplastic large deflection of three-dimensional (3D) frames that have steel members with semirigid joints. A plastic hinge type formulation was employed, combining the "modified incremental stiffness method," the updated Lagrangian formulation, and numerical integration about the end sections of the element. The end sections of the element are discretized into small areas to estimate the plastic deformations of the element. The elastic and plastic deformations of the element are treated separately. The behavior of a semirigid joint is modeled as the element-end compliance. The method can treat comprehensively the plastic deformations due to torsion and warping. Considering the assumptions of the method, a four-element approximation for a member gives excellent results for a 3D analysis of semirigid and pin--connected steel frames as well as for rigid frames. The adequacy of the method is verified by comparing the results with experimental ones obtained by the writer. Some examples are presented to demonstrate the accuracy and efficiency of the method.
Elastoplastic analysis of 3D steel frame with tension braces by the Fibered Plastic Hinge Model
- M Shugo