Grillage Method Research Articles

GRILLAGE ANALYSIS OF CELLULAR PLATE STRUCTURESOF VARYING DEPTH WITH INCL

A static analysis of cellular plate structures of linearly varying depth is presented by using thesimplified grillage method. The cellular plate structure in this study consists of top and bottomflange plates (non-parallel) separated by longitudinal webs having varying depth. A variationalformulation of thin-walled beam element with linearly varying depth has been carried out for thederivation of the stiffness matrix of the grillage members for various warping restraint effects. Theeffect of warping restraint in stiffness is included by two methods, firstly; the implicit method(through the torsional constant) and secondly; by the explicit method (as independent warpingdegree of freedom). Two differer,t approaches are used to derive the torsional stiffness matrix,namely:by the assumed displacement field and by the differential equation of equilibrium. Theeffect of secondary shear stresses from the warping restraint is also considercd.The varying cross section properties of the eguivalent grillage members are derived from theproperties of the actual cellular plate structure.Several numerical examples and applications are presented . The results are compared with thoseobtained from the flat plate / shell elements of MSCA.{ASTRAN program. Acceptable agreement isobtained.

Bridge deck analysis of transversely post-tensioned concrete box girder bridges

For rural bridges in Australia, a common design practice is pouring in-situ concrete on top of beams in order to tie all the beams together and distribute load. However, pouring concrete on-site creates more risk and contractors prefer to avoid it. Another method is using transverse post tensioning to tie beams. This article investigated the behaviour of transverse post-tensioning bars in providing load distribution between beams and ultimately comment on their effectiveness compared to in-situ poured decks. Currently, the industry has not completely investigated this matter in order to design post-tensioning accurately. Conservative estimates are currently used in industry today. Current practice is 50% of the design load on the beam where the load is applied in their design assumptions which is quite high. The team modelled concrete box girder bridges with transverse post-tensioning using grillage method. Several factors were investigated including bridge length and width, bridge skew and beam type. From the models, the team concluded that increasing the bridge span increases the load distribution, the load distribution difference is negligible for skew between 0 and 20 degrees and larger shear actions are observed with increased skew and width. It was determined that the worst-case total load on the beam where the load as applied was found to be 40.5%, 9.5% less than current practice. It is recommended that a similar investigation is conducted using a finite element method to gain a deeper understanding.

Analysis of Slab Culvert Bridges using Conventional Method and Grillage Method

IRC 112:2011, the Indian standard pertaining to concrete bridge design is based on the limit state design principle. It suggests the use of effective width method for the analysis of the structure. For many years, the analysis part of concrete bridge structures design has been based on two-dimensional (2D) analysis and the results obtained from longitudinal analysis were supposed to be valid over the whole bridge width. In this paper, the grillage method has been used to model and analyse the super-structure to get the three-dimensional effect, and a comparison of results between effective width method and grillage method is presented.

Research on the stiffness of Grillage Method in Virtual Cross Beam based on Fuzzy recognition

The stiffness of virtual cross beam is an important parameter which affects the calculation results when using beam grillage method to analyze the stress of bridge structure. In order to obtain a reasonable value of the stiffness about the virtual cross beam, it is necessary to make comparative analysis. On the basis of the comprehensive comparative analysis of the stress and deflection of the control section in hollow slab bridge and simply supported beam bridge with twin-box single cross section, the value range of the virtual cross beam stiffness is obtained. Based on the theory of fuzzy mathematics, a fuzzy recognition model for the stiffness of the virtual beam was proposed. By calculating the relative membership degree vector of the stiffness in each virtual beam, the length and width of the cross section of the virtual beam is obtained to be 0.7 times height of the beam. In order to further verify the reasonable of the value, the stress and deflection values of the control section have been compared. These values are taken from the single-box and double-chamber plexiglass box girder model and highway single-box three-chamber box beam. The compared results show that the solution of the grillage model with the stiffness of the virtual beam in this paper is in good agreement with shell solutions and solid solutions.

Three-dimensional modelling of shear keys in concrete gravity dams using an advanced grillage method

Contraction joint shear keys are resilient features of gravity dams that can be considered to increase the sliding safety factors or minimise seismic residual sliding displacements, allowing costly remedial actions to be avoided. This paper presents a novel, robust, and computationally efficient three-dimensional (3D) modelling and simulation strategy of gravity dams, using a series of adjacent cantilever beam elements to represent individual monoliths. These monoliths are interconnected in the longitudinal direction by 3D no-tension link elements representing the lumped shear key stiffness contributions at a particular elevation. The objective is to assess the shear key internal force demands, including the axial force, shear, and moment demands. Shear key demand-capacity ratios can then be assessed with related multi-axial failure envelopes. The 3D link element stiffness coefficients were derived from a series of 3D finite element (FE) solid models with a detailed representation of geometrical features of multiple shear keys. The results from the proposed method based on advanced grillage analysis show strong agreement with reference solutions from 3D FE solid models, demonstrating high accuracy and performance of the proposed method. The application of the proposed advanced grillage method to a dam model with two monoliths clearly shows the advantage of the proposed method, in comparison to the classical approach used in practise.

Girder Lattice Model and Single Beam Model In Load Test of Continuous Box Bridge Computational Comparison and Analysis

The static load test of a three-span continuous box girder in Huzhou City, Zhejiang Province is analyzed and calculated by using the grillage method and the single main girder method. The rationality of the two calculation models is compared with the field load test data. The calculation results show that the results of grillage method are more consistent with the experimental results than that of single girder method, and the safety of the calculation results is higher. Therefore, in the load test of multi-compartment box girder bridge, the grillage method is more reasonable and can better reflect the mechanical characteristics of the bridge structure.

Refined Vehicle‐Bridge Interaction Analysis Using Incompatible Solid Finite Element for Evaluating Stresses and Impact Factors

The vehicle‐bridge interaction can induce bridge vibration and consequently fatigue, durability deterioration, local damage, and even collapse of bridge structure. In this paper, a solid vehicle‐bridge interaction (VBI) analysis method is developed to provide refined analysis on the bridge responses including displacement and local stress under vehicle loads. The incompatible solid finite element (FE) is introduced to model the bridge, where the element shear locking is alleviated by incompatible displacement modes without sacrificing the computational efficiency. Benchmark example shows the incompatible solid element has superior computational efficiency compared to the conventional solid element. By virtue of the mass‐spring‐damper vehicle model, the interaction between vehicle and bridge is simulated with point‐to‐point contact assumption and the coupled dynamic equations are solved via nonlinear iteration. A case study on a simply supported T‐girder bridge is conducted to validate the developed solid VBI analysis method and then the dynamic impact factor (DIF) of the bridge is evaluated based on the computed stress results and compared to code values. Results show that the solid VBI analysis method yields more accurate time‐history bridge responses including displacement and stress under moving vehicles than the grillage method despite higher computational cost. Particularly, it can simulate realistic stress distribution and concentration along any concerned sections as well as in local components, which can provide detail information on the bridge behavior under dynamic loads. On the other hand, the DIF based on the computed stress result generally agrees well with the code values except for heavy vehicles where the stress‐based DIF is slightly higher than the value in Chinese code while lower than that of AASHTO, suggesting the value specified by Chinese code may underestimate the DIF of heavy vehicles in certain circumstances to which more attention should be paid.

New extended grillage methods for the practical and precise modeling of concrete box-girder bridges

The Hambly plane grillage method has been regarded as one of the classic numerical methods in the design field for modeling wide box-girder bridge structures. However, when it comes to the in-depth design applications, its strict division rules and insufficient mechanical explanations often make engineers inconvenient or puzzled at bridge modeling. This article investigates whether this method may be extended to become more adaptable for the design of current concrete box-girder bridge structures in consideration of both convenience and precision. To this end, the defects of Hambly plane grillage method are recognized, and new extended grillage methods, including the single-layer folding surface grillage and spatial grillage, are proposed respectively, to deal with different bridge design objects completely in a beam-oriented environment. The former allows freer cross-sectional division by breaking the basic rule of Hambly plane grillage method to include longitudinal separate-type beam components for a better exhibition of shear lag effects. The latter emphasizes a complete consideration of spatial behavior, including the easily missed in-plane effects of the top and bottom plates. The effectiveness of the methods are demonstrated by comparison case studies in some benchmark models and by a discussion of their applications.

Differences between hollow slab Grillage model and single beam model in design and analysis

Hollow slab girder bridge has the advantages of factory prefabrication, fast construction speed, small girder height and low cost. It is one of the main types of small and medium span bridges in China. In this paper, a hollow slab bridge with an effective span of 10.22m is taken as the research object. The Grillage method and single girder method of MIDAS/Civil finite element software are used to model and calculate the bridge, and comparing the difference between the results of two models in design analysis. The calculated value of single beam model is larger than that of Grillage model, and more insecure. The difference of stress calculation between the two models can be up to 1.2 MPa. For safety reasons, two kinds of modeling methods can be used for analogy in design, but when the structure is more complex, the model of the Grillage model should be closer to the actual situation.

Study on Transverse Load Distribution of Hinged Hollow Beam

The bridge is a kind of space structure, when the car load on a part of the bridge, the impact of its load will be transmitted to the surrounding. In this paper, the hinge plate method is used to calculate and analyze the simply supported hollow slab of a certain arch bridge. Considering the hinge plate mounting method is suitable for pouring concrete bridge connecting the longitudinal tongue and groove joints, horizontal beams fabricated separate beam only in the middle between the free flaps or reinforced with steel connection. Therefore, the transverse analysis and calculation of the superstructure of box girder are carried out by using hinge plate method. And mechanical analysis of the transverse beam with finite element software MIDAS Civil grillage method.

Rigidity and moment distribution of steel-concrete composite waffle floor systems considering the spatial effect

A steel-concrete composite waffle floor system (SCCWFS) that consists of orthogonal steel beams and a flat RC slab has potential applications in long-span building floors because of its large loading capacity, low depth-to-span ratio, and excellent ductility, as demonstrated by a previous experimental program. According to the test results, the spatial composite effect between the orthogonal steel beams and concrete slab significantly influences the performance of the SCCWFS. However, the complexity of the spatial composite effect in the SCCWFS makes it much more difficult to determine the rigidity and the internal force distribution in a routine design practice than the one-way steel-concrete composite floor or the RC waffle slab. To address this problem, a parameterized grillage method is developed in which intrinsic factors are defined to describe the critical properties of the deformation pattern, the reaction force and moment distribution of the SCCWFS; relation factors are defined to relate these properties of the SCCWFS to that of its corresponding steel grillage. Based on the parameterized grillage method, parametric analyses are conducted using a beam-shell mixed finite-element (FE) model and the influence of various parameters on intrinsic and relation factors is investigated, in which the beam height, slab thickness, length-to-width ratio, and so on are shown to be of importance. The data of a total of 5190 numerical models associated with seven parameters covering almost all the practical cases are then obtained, and a step-shaving procedure is proposed to derive formulas that are influenced by multiple variables. Based on the parameterized grillage method and the step-shaving procedure, formulas to predict the vertical displacements, reaction force and moment distribution of the SCCWFS are derived and verified. Finally, design procedures, recommendations and simplified forms of the formulas are proposed, in which the formulas are proved to have an error of approximately 10%.

基于梁格法的小半径钢筋混凝土弯箱梁桥空间受力特性分析

弯箱梁桥造型优美，能很好地适应和改善道路线型，近年来被广泛地应用于城市桥梁中。但弯箱梁受力复杂，需采用空间分析方法进行分析。本文采用MIDAS/Civil建立全桥梁格模型，分析了小半径弯箱梁结构的空间受力特性，研究了其在荷载作用下的位移以及支座反力特性。通过分析空间模型的计算结果，小半径弯箱梁结构存在明显的弯扭耦合效应，并可引起弯箱梁截面的扭转以及支座反力的不均匀等现象，通过设置支座预偏心可有效消减支座反力不均匀现象。 Curved box girder bridge is beautiful, can adapt to and improve the road linearity. In recent years, it has been widely used in urban bridges. But the force of curved box girder bridge is complex, it needs to use the spatial analysis method for analysis. In this paper, the full-bridge grid model was established by MIDAS/Civil based on grillage method, and the spatial force characteristics of the small radius curved box girder were analyzed, and the displacement and bearing reaction characteristics under the load were studied. By analyzing the calculation results of the space model, it can be seen that there are obvious bending and torsional coupling effects of the small radius curved box girder structure, which can cause the torsion of the cross section of the bending box and the nonuniformity of the bearing reaction. However, there is an effective reduction of the nonuniformity of the bearing reaction by bearing eccentricity.

Grillage Method Applied to the Planning of Ship Docking

Grillage Method Applied to the Planning of Ship Docking

Seismic analysis of half-through steel truss arch bridge considering superstructure

This paper takes a half-through steel truss arch bridge as an example. A seismic analysis is conducted with nonlinear finite element method. Contrast models are established to discuss the effect of simplified method for main girder on the accuracy of the result. The influence of seismic wave direction and wave-passage on seismic behaviors are analysed as well as the superstructure and arch ring interaction which is mostly related with the supported bearings and wind resistant springs. In the end, the application of cable-sliding aseismic devices is discussed to put forward a layout principle. The main conclusions include: (1) The seismic response isn't too distinctive with the simplified method of main girder. Generally speaking, the grillage method is recommended. (2) Under seismic input from different directions, arch foot is usually the mostly dangerous section. (3) Vertical wave input and horizontal wave-passage greatly influence the seismic responses of arch ring, significantly increasing that of midspan. (4) The superstructure interaction has an obvious impact on the seismic performance. Half-through arch bridges with long spandrel columns fixed has a less response than those with short ones fixed. And a large stiffness of wind resistant spring makes the the seismic responses of arch ring larger. (5) A good isolation effectiveness for half-through arch bridge can be achieved by a reasonable arrangement of CSFABs.

INFLUENCE OF SPAN LENGTH AND CROSSBEAM ON LOAD DISTRIBUTION FACTOR FOR GIRDER BRIDGES

Load distribution factor at concrete girder bridges and steel girder bridges are analyzed with finite element method to see effect of span length and cross beam to load distribution factor. Span lengths of analyzed bridge models are 30m, 40m, 50m and 60m. The number of intermediate cross beam is increased from one to until distance between cross beams becomes 5m. The finite element analysis results show that concrete girder and steel girder can use same load distribution factor and span length doesn’t affect to load distribution factor. Even though load distribution factor in interior girders is not influenced by cross beam, in exterior girders it is influenced by cross beam. Effect of cross beam in exterior girder is influenced by the number of lanes and distance from exterior girder to curb. Since design code introduces conservative load distribution factor, economically improved load distribution factor is proposed. The proposed load distribution factor includes cross beam effect with the number of lanes and distance from exterior girder to curb. The proposed equation is compared with AASHTO code and grillage method which is well-known method to calculate load distribution. The comparison results showed that the proposed equation is more efficient and useful than AASHTO and safer than the grillage method.

APPLICATION OF A SOFTWARE PROGRAMME LANGUAGE CALLED PYTHON IN DETERMINING THE LOADS AND LOAD DISTRIBUTION IN THE DECKS OF A LONG SPAN REINFORCED BRIDGE DECK

This research was to create software that can analyze, design and determine the cost of a beam/ deck for a simple supported bridge, the name of the software is Ivy. The method employed for the analysis is a grillage method, using finite element (beam element) and comparing the results with a manual approach (moment distribution and influence line). A 15m span was considered with a width of 7.5m, using a grillage distance of 1.5m, a knife edge load of 120KN and a uniformly distributed load of 30KN/m. The BS5400 and BS 8110 were used for the analysis and design. The result shows that values for the maximum bending moment locating the knife edge at 3m from the support is Mmax = 412KNm (manual approach), Mmax= 506.25KNm (Ivy) to give a percentage difference of 18.6%, at 6m, Mmax= 653KNm (manual approach), and Mmax= 566.25KNm with a percentage difference of -15.3%, at mid span 7.5m, it is Mmax= 789KNm (manual approach), Mmax= 768.75KNm(ivy) indicating a percentage difference of -2.63%. Similarly, the shear force at support locating the knife edge load at 3m is Vmax= 24.81KN (manual approach), and Vmax= 33.75KN (ivy) to give a percentage difference of 26.4%, at 6m, Vmax = 57.58KN (manual approach), and Vmax= 67.50KN (ivy) to give a percentage difference of 14.6% while at mid span 7.5m,Vmax = 38.98KN (manual approach), and Vmax= 67.50KN (ivy) to give a percentage difference of 42.2%. The main reinforcement area of steel locating the knife edge load at 3m is As = 5419mm 2 (manual approach), and As = 5271.72mm 2 (ivy) with a percentage difference of -2.7%; at 6m, As = 6145mm 2 (manual approach), and As = 5910.52mm 2 (ivy) with a percentage difference of -3.9%, while at mid span 7.5m, As = 8617mm 2 (manual approach), and As = 8088.6mm 2 (ivy), giving a percentage difference of -6.5%

Analysis of corrugated steel web beam bridges using spatial grid modelling

Up to now, Japan has more than 200 corrugated steel web composite beam bridges which are under construction and have been constructed, and China has more than 30 corrugated steel web composite beam bridges. The bridge type includes the simply supported beam, continuous beam, continuous rigid frame and cable stayed bridge etc. The section form has developed to the single box and multi-cell box girder from the original single box and single chamber. From the stress performance and cost saving, the span range of 50~150 m is the most competitive. At present, the design mostly adopts the computational analytical method combining the spatial bar system model, plane beam grillage model and solid model. However, the spatial bar system model is short of the refinement analysis on the space effect, such as the shear lag effect, effective distribution width problem, and eccentric load factor problem etc. Due to the similarity of the plane beam grillage method in the equivalence principle, it cannot accurately reflect the shearing stress distribution and local stress of the top and bottom plates of the box type composite beam. The solid model is very difficult to combine with the overall calculation. Moreover, the spatial grid model can achieve the refinement analysis, with the integrity of the analysis and the comprehensiveness of the stress checking calculation, and can make up the deficiency of the analytical method currently. Through the example verification of the solid model and spatial grid model, it can be seen that the calculation results for the stress and the displacement of two models are almost consistent, indicating the applicability and precision of the spatial grid model.

The Meticulous Analysis of Skew Beam Based on the Space Skew Beam Element Method

When a skew bridge is analyzed, it is difficult to treat skew boundary conditions with general space beam element in single beam calculation of skew beam bridge. The stiffness of skew bridge cannot be simulated efficiently. In order to improve the accuracy of skew bridge, the stiffness matrix of the space skew beam element is derived based on the principle of minimum potential in this paper, this element is suitable for single beam of skew bridge and grillage method. A corresponding program is completed by Visual C++, the calculated results of different finite element models are compared, so the validity of the proposed model in this paper is validated. This paper can provide reference for plane and space calculation of the skew bridge design.

The Application of Reliability Based Optimization of Tophat Stiffened Composite Panels Subject to Bi-Directional Buckling Loads

Stiffened composite panels are used within many applications, from aerospace to marine applications. Stiffened panels are utilized for their high strength to weight ratio and flexibility of layups while counteracting the low stiffness exhibited by composites. Complications arise when attempting to utilize the full variability of layups in conjunction with reliability constraints creating a complex design problem when constrained by both buckling and material strength. To aid the process of optimizing the design of composite structures and layups, while ensuring a low mass, this paper presents a bi-level optimization scheme for minimization of the weight of tophat stiffened composite panels with probabilistic deflection constraints. To improve the computational efficiency, an energy based grillage method is formulated and applied for the investigation of buckling problems under bi-directional in-plane loads. The method is validated by comparing the results obtained from FE model calculations. The variables that have a large impact on the structural safety have been identified by both safety index and COV based reliability analysis. A parametric study of plate dimensions and loading ratios is conducted to investigate the coupling effects on critical buckling load. The method presented in this paper, makes it possible for engineers to improve their designs, at an early stage, with an integrated consideration between product performance and design parameters.

Application on Grillage Method in Box Girder Bridges

Grillage method was an effective method for analyzing bridge structures. In order to validate the grillage method met the requirements of engineering precision in calculation about box girder bridges, the grillage model and the plate element model of the bridge were built by using the finite element software Midas/Civil; under the premise of the same loading location and the load value, the deflection values and the stress values calculated from the two models were compared, and the corresponding error ranges were obtained. The results show that the differences between them meet the engineering accuracy and using the grillage method in box girder bridges is reasonable.