Linear Timoshenko Beam Element Research Articles

Study of the Discrete Shear Gap Technique in Timoshenko Beam Elements

A major difficulty in formulating a finite element for shear-deformable beams, plates, and shells is the shear locking phenomenon. A recently proposed general technique to overcome this difficulty is the discrete shear gap (DSG) technique. In this study, the DSG technique was applied to the linear, quadratic, and cubic Timoshenko beam elements. With this technique, the displacement-based shear strain field was replaced with a substitute shear strain field obtained from the derivative of the interpolated shear gap. A series of numerical tests were conducted to assess the elements performance. The results showed that the DSG technique works perfectly to eliminate the shear locking. The resulting deflection, rotation, bending moment, and shear force distributions were very accurate and converged optimally to the corresponding analytical solutions. Thus the beam elements with the DSG technique are better alternatives than those with the classical selective-reduced integration.

COLLAPSE RISK PREDICTION OF BUILDINGS ON FIRE USING KEY ELEMENT INDEX

It is difficult to estimate the redundant strength remaining in a building on fire, since the fire conditions such as their ranges and locations may be different in each case. A large-scale fire occurred on the New York World Trade Center 7 (WTC-7) after the 9.11 terrorist attacks in 2001. The building continued to burn for 7 hours, and ended in a total collapse. The official report on the collapse investigation of WTC-7, released by the National Institute of Standards and Technology (NIST) in 2005, had suggested that the total collapse was triggered by a severe damage of an important column of the building called the key element. On the other hand, there is also an example of a high-rise building on fire that avoided a total collapse, regardless of its main structure continuously heated by fire for a long time; the Windsor building of Madrid in 2005. From the viewpoint of preventing the fire-induced collapse of buildings, it is necessary to clarify the relationship between various structural parameters of buildings, fire locations, their ranges, and the scale of collapse. The purpose of this study is to predict collapse risks of buildings on fire using a key element index (KI). The KI is defined as the ratio of the ultimate yield strengths of the structure with one column eliminated and the initial, undamaged structure. It indicates the contribution of a structural column to the vertical capacity of the structure. We investigated the relationship between the sum of KI values in various cases of fire range and the sum of the height of remains after the collapse. We applied an adaptively shifted integration (ASI) - Gauss code utilizing linear Timoshenko beam elements to investigate the collapse behaviors of a ten-story steel framed building model with various fire patterns. Fracture contact, contact release and re-contact algorithms were implemented in the code. The reduction curves of elastic modulus and yield strength of steel related to elevated temperature, shown by NIST, were adopted to consider the heat effect of fire. Thermal expansion of materials was also considered. From the numerical results of the fire-induced collapse analyses, it is found that the scale of collapse highly depends on fire ranges and locations. The risk of total collapse tends to increase when a large range of fire occurs in the lower layer. In addition, the risk increases much higher in those cases when fire occurs at the peripheral area of the building, compared to those cases at the inner area. The sum of heights of remains seems to depend on the sum of KI values, and there is a specific threshold of the sum of KI values that relates to the initiation of a large scale collapse. Therefore, the sum of KI values may be used to predict and take measures to avoid the risk of collapse under various fire conditions.

Elastic-Plastic Non Linear Behaviors of Suddenly Loaded Structures

Problem statement: The adaptively shifted integration technique was applied to the elasticplastic analysis of framed structures under dynamic loading. Approach: This study used analysis of linear beam element, the reshifting of the of the integration points in the element is conducted in order to attain higher accuracy. Results: In nonlinear finite element analysis by the ASI technique, the highest computational efficiency has been achieved by shifting the numerical integration points for evaluation of stiffness matrices of linear Timoshenko beam element. A computer program has been Written by VBASIC for solve the problem of the steel frame, results of numerical examples demonstrate the validity of the computer program and study many parametric study, such as Time step and Mass of the structure. Conclusion/Recommendations: In the elastic analysis there is no distinction between the ASI technique and the conventional finite element method, but for elasticplastic nonlinear analysis under dynamic load Adaptive Shifted Integration (ASI) technique is capable of predicting with reasonable accuracy the behavior of steel frame structures.

Structural collapse analysis of framed structures under impact loads using ASI-Gauss finite element method

The main objective of this study is to devise a technique, which, when implemented into finite-element codes, is efficiently applicable to impact collapse analyses of framed structures. In this study, the formerly developed adaptively shifted integration (ASI) technique for the linear Timoshenko beam element is modified into the ASI-Gauss technique by placing the numerical integration points of the two consecutive elements forming an elastically deformed member in such a way that stresses and strains are evaluated at the Gaussian integration points of the two-element member. On comparison with the ASI technique, the ASI-Gauss technique proves its higher accuracy and efficiency in elastic range. Moreover, instead of applying impact loads in the form of nodal forces, we consider the impact phenomenon by means of contacts between the elements involved and the elemental contact algorithm is verified from the point of conservation of energy. Impact analyses considering member fracture with different sets of parameters are performed using a high-rise framed structure and a small aircraft. From the results obtained, we can observe propagation phenomena of impact loads and shock waves. Also, a proper difference in impact damage is obtained by different sets of parameters. The results also indicate that the mass of the aircraft has a stronger influence on impact damage than its velocity. Moreover, soon after impact, tensile stresses are observed in the columns that were compressed by dead loads before impact.

ASI-Gauss法による世界貿易センタービルの飛行機衝突解析

In this paper, the previously developed AST-Gauss technique is applied to an aircraft impact analysis of New York World Trade Center Tower 2 (WTC2), to evaluate the structural vulnerability and behavior of the aircraft, at the horrifying scene that occurred in 2001. The ASI-Gauss technique is a modified version of the formerly developed Adaptively Shifted Integration (ASI) technique for the linear Timoshenko beam element, which computes highly accurate elasto-plastic solutions even with the minimum number of elements per member. The ASI-Gauss technique gains still higher accuracy especially in elastic range, by placing the numerical integration points of the two consecutive elements forming an elastically deformed member in such a way that stresses and strains are evaluated at the Gaussian integration points of the two-element member. From the numerical results, propagation of the shock waves resulted from the impact and high-level stresses in the structural members were observed. We could also see how the main wings of the aircraft cut through the perimeter columns and spandrels, how the debris moved through the building and to where it caused damage. Moreover, the velocity reduction of the right engine and the location from where it moved out of the building were in good agreement with the observed data.

Finite element analysis of superelastic, large deformation behavior of shape memory alloy helical springs

Brinson's one-dimensional constitutive modeling for shape memory alloy (SMA) is extended to consider the asymmetric tensile and compressive behavior as well as the torsional behavior. The incremental finite element method using linear Timoshenko beam elements is formulated by the total Lagrangian approach for the superelastic, large deformation analysis of SMA helical springs. The NiTi helical springs are analyzed and the calculated results are compared with the experimental results to show the validity of the present computational procedure in actual design of SMA actuators.

Seismic collapse analysis of reinforced concrete framed structures using the finite element method

AbstractA new finite element code using the Adaptively Shifted Integration (ASI) technique with a linear Timoshenko beam element is applied to the seismic collapse analysis of reinforced concrete (RC) framed structures. This technique can express member fracture as a plastic hinge located at either end of an element with simultaneous release of the resultant forces in the element. Contact between members is also considered in order to obtain results that agree more closely with actual behavior, such as intermediate‐layer failure. By using the proposed code, sufficiently reliable solutions have been obtained, and the results reveal that this code can be used in the numerical estimation of the seismic design of RC framed structures. Copyright © 2003 John Wiley & Sons, Ltd.

SEISMIC DAMAGE ANALYSIS OF REINFORCED CONCRETE BUILDING CONSIDERING MEMBER FRACTURE

The Adaptively Shifted Integration (ASI) technique, which produces the highest computational efficiency in the finite element analyses of framed structures including static and dynamic collapse problems, is applied to the seismic damage analysis of a reinforced concrete building. By expressing member fracture by a plastic hinge located at the exact position with a simultaneous release of resultant forces in the element, discontinuous problem such as this kind can be easily analyzed even by the conventional finite element code with the displacemental form. By using the algorithms described in this paper, sufficiently reliable solutions for the practical use have been obtained in the seismic damage analysis of a five stories-five span reinforced concrete building. The present technique can be easily implemented with a minimum effort into the existing finite element codes utilizing the linear Timoshenko beam element.

Analysis of structurally discontinuous reinforced concrete building frames using the ASI technique

The Adaptively Shifted Integration (ASI) technique, which produces the highest computational efficiency in the finite element analyses of framed structures including static and dynamic collapse problems, is applied to structurally discontinuous problems of reinforced concrete building frames. A new numerical scheme based on the updated Lagrangian formulation (ULF) adaptation of the ASI technique is developed, by modeling the fracture of a section by a plastic hinge located at the exact position with a simultaneous release of resultant forces in the element. By using the algorithms described in this paper, the analyses became possible even by the conventional displacement-based finite element codes, and sufficiently reliable solutions for practical use have been obtained in the explosive demolition and seismic damage analyses of a five storied, five span RC building frame. The present technique can be easily implemented with minimum effort into the existing finite element codes utilizing the linear Timoshenko beam element.

Finite Element Crash Analysis of Framed Structures by the Adaptively Shifted Integration Technique.

In the present study, each member of a framed structure is subdivided with two linear Timoshenko beam elements at both ends and a cubic beam element based on Bernoulli-Euler hypothesis at the center. The adaptively shifted integration (ASI) techinique is used only in the linear Timoshenko beam elements. The proposed model is applied to the explicit finite element analysis of the crashing behaviors of impulsively loaded framed structures considering the effect of large deformation by the updated Largrangian formulation. Several numerical studies have been carried out in order to show the validity of the proposed numerical technique.

Finite Element Crash Analysis of Framed Structures by the Adaptively Shifted Integration Technique.

In the present study, each member of a framed structure is subdivided with two linear Timoshenko beam elements at both ends and a cubic beam element based on Bernoulli-Euler hypothesis at the center. The adaptively shifted integration (ASI) technique is used only in the linear Timoshenko beam elements, The proposed model is applied to the explicit finite element analysis of crushing behavlors of impulsively loaded framed structures considering the effect of large deformation by the updated Largranglan formulation. Several numerical studies have been carried out in order to show the validity of the proposed numerical technique.

Explicit Finite Element Collapse Analysis of Framed Steel Structures by the Adaptively Shifted Integration Technique.

The adaptively shifted integration (ASI) technique is applied to the explicit finite element analysis of dynamic collapse behaviors of impulsively loaded framed structures. In the present study, the linear Timoshenko beam element is employed, considering the effect of large deformation by the total Lagrangian approach. Numerical studies are conducted for several impact problems. The obtained solutions are compared with the conventional finite element solutions to discuss the validity of the ASI technique in the explicit large deformation analysis. In some of the examples, the folding behaviors of axially crushed members in the framed strucure are approximated by the rigid-plastic solutions for the mean crushing load to attain higher computational efficiency.

Debris Impact Analysis of Large-scale Space Structures by Using ASI-FEM.

In this paper, the Adaptively Shifted Integration (ASI) technique with the linear Timoshenko beam element, which can be easily implemented with a minimum effort into the existing finite element codes, is applied to the debris impact analysis of framed structures in space. In this technique, the numerical integration point in an elastically deformed element is firstly placed at the optimal point for linear analysis. Then the integration point is shifted immediately after the occurrence of a fully plastic section in the element, using the previously established relations between the location of numerical integration point and that of plastic hinge, to form a plastic hinge exactly at the position of that section. The technique prouduces higher computational accuracy with fewer elements than the conventional finite element code. By expressing member fracture by a plastic hinge located at the exact position with a simultaneous release of resultant forces in the element, discontinuous problem such as this kind can be easily analysed even by the finite element code with the displacemental form. Contact between members, and the distribution of element mass before and after member fracture has occurred, are also examined. By using the algorithms described in this paper, sufficiently reliable solutions have been obtained in the debris impact analyses of a space module unit and the International Space Station, which is due to be constructed from November. 1998.

A locking-free approximation of curved rods by straight beam elements

We consider an elastic model for a curved rod with arbitrary three-dimensional geometry, incorporating shear and membrane as well as bending and torsion effects. We define an approximation procedure based on a discretization by linear Timoshenko beam elements. Introducing an equivalent mixed problem, we establish optimal error estimates independent of the thickness, thereby proving that shear and membrane locking is avoided. The approximation scheme is tested on specific examples and the numerical results confirm the estimates obtained by theory.

Finite Element Collapse Analysis of Brittle Framed Structures by the Adaptively Shifted Integration Technique.

The adaptively shifted integration technique is applied to the finite-element collapse analysis of beams and framed structures composed of brittle materials. Linear Timoshenko beam elements are employed for the analysis. Plastic deformation and cracking on a cross section are treated using the layered approach. Points at which plastic deformation and cracking occur along the element length are accurately determined by the adaptive shifting of a numerically integrated section in each element. Numerical studies for beams and frames with and without ductile reinforcements are carried out to demonstrate the validity of the proposed numerical technique.

Finite Element Analysis of Dynamic Collapse Behaviors of Brittle Framed Structures by the Adaptively Shifted Integration Technique

The Adaptively Shifted Integration (ASI) technique, which produces the highest computational efficiency in the finite element analysis of framed structures including static and dynamic collapse problems, is applied to the explosive demolition analysis and seismic damage analysis of reinforced concrete buildings. By expressing an explosion or a fracture by a plastic hinge located at the exact position with a simultaneous release of resultant forces in the element, strong non-linear and discontin uous problems such as these can be easily analyzed even by the finite element code with the displacement form.By using the algorithms described in this paper, sufficiently reliable solutions for the practical use have been obtained in the explosive demolition and seismic damage analyses of a 5 stories-5 span reinforced concrete building.The present technique can be easily implemented with a minimum effort into the existing finite element codes utilizing the linear Timoshenko beam element.

Adaptively shifted integration technique for finite element collapse analysis of framed structures

AbstractThe present study is concerned with the improvement of the previously proposed ‘shifted integration technique’ for the plastic collapse analysis of framed structures using the linear Timoshenko beam element or the cubic beam element based on the Bernoulli‐Euler hypothesis. In the newly proposed ‘adaptively shifted integration technique’, the numerical integration points for the evaluation of the stiffness matrices are automatically shifted immediately after. the occurrence of plastic hinges according to the previously established relations between the locations of numerical integration points and those of plastic hinges. By using the adaptively shifted integration technique, sufficiently accurate solutions can be obtained in the non‐linear frame analysis by two‐linear‐element or only one‐cubic‐element idealization for each structural member. The present technique can easily be implemented in the existing finite element codes utilizing the linear or the cubic beam element.

Adaptively Shifted Integration Technique in the Finite Element Collapse Analysis of Framed Structures

The present study is concerned with the refinement of the previously proposed 'shifted integration technique' for the plastic collapse analysis of framed structures using the linear Timoshenko beam element or the cubic beam element based on the Bernoulli-Euler hypothesis. In the newly proposed 'adaptively shifted integration technique', numerical integration points are automatically shifted immediately after the occurrence of plastic hinges according to the previously established relations between the locations of numerical integration points and those of plastic hinges. By using the adaptively shifted integration technique, sufficiently accurate solutions can be obtained in the nonlinear frame analysis by two linear element or only one cubic element idealization for each structural member. The present technique can be easily implemented with a minimum effort into the existing finite element codes utilizing the linear and the cubic beam element.

Finite element crush analysis of framed structures

The present study is concerned with the development of a finite element algorithm for crush analysis of three-dimensional framed structures under quasi-static loading and its experimental verifications. Details of the incremental formulation by the updated Lagrangian approach is described, in which the linear Timoshenko beam element is employed with the shifted integration technique and with the elasto-plastic constitutive equation expressed in terms of resultant forces. The strain-hardening and the frictional contact are also taken into account in the present analysis. The results of finite element analyses are compared with the crush test results conducted for beams, columns and framed structures.

Simulation of Collapse of Framed Structures (Part 2)

The purpose of the present study is to develop the economical finite element analysis program for the collapse simulation of framed structures and to verify the validity of the developed code through the comparisons of the numerical results with the test results. In the first report, the details of the incremental formulations were described, in which the linear Timoshenko beam element was employed with the variable location technique of the numerical integration point and the elasto-plastic constitutive equation expressed in terms of resultant forces. In this second report, the finite element results by the updated Lagrangian formulation are compared with the crush test results conducted for beams, columns and space frames. The strain hardening and the frictional contact are taken into account in the present analysis. The obtained results can be summarized as follows : (1) From the results for beams and columns, it can be seen that the present analysis method gives sufficiently accurate solutions for the crush problem accompanied by large strains and large displacements, except for the cases that local deformations as shells are obvious(2) From the results for space frames, it can be seen that the developed code is sufficientlyeconomical and accurate from a practical point of view, however, some trial-and-error analyses are necessary for the crush behaviors in which friction and contact play important roles.