A beam grillage is a structure made of a planar network of connected beams loaded normally to their plane. The geometrical configuration and the mutual moment-resisting connection between elements allow flexural and torsional resisting mechanisms to develop within the grillage, with general small displacements and a great ability in load redistribution. For this reason, this type of structure is widely used in civil constructions, such as in bridges, and in marine structures [1]. Even in small grillage structures, the number of degrees of freedom is large enough that simple mathematical expressions for displacement and forces are difficult to formulate. Although the studies on the behavior of structures loaded with out-of-plane forces date back to the nineteenth century [2], in the sixties and the seventies the major improvements on the design of grillage structures are found. Holmes and Ray-Chaudhuri [3] propose a limit analysis for determining the ultimate load of grillages, while minimum weight and optimization studies were first proposed by Rozvany [4]. More recent studies focus on the dynamic behavior [5] or on the optimization of foundation grillages [6].
The so-called grillage analogy is a widely adopted strategy for solving structures with out-of-plane loads, such as in bridge superstructures analysis [7]. The validity of such approach has been validated by comparing numerical analyses and experimental data, such as in [8]. Grillage structural schemes have been also introduced for modeling elastic continua [9]. With reference to the numerical methods for solving the structures, before the introduction of FEM, matrix analysis was the usual mathematical tool for studying beam grillage [10]. As largely diffused in civil and mechanical engineering, beam grillage structures can be made of various materials: concrete, steel, timber or composite material and can be subjected to various types of loading, including moving forces [11] or impacts and blasts [12]. It is worth to be mentioned that, although made of a planar network of beams, load transfer in grillage of reciprocal beams is related only to the flexural mechanisms, as the elements are supported one each other [13].
Bridge Analysis Simplified By Bakht Jaeger Pdf 29
As already mentioned, grillage structures are widely adopted in civil engineering works, in particular in bridge decks. In particular cases, say in Gerber girders, the span is suspended over two cantilevers, which act as supports [17]. Half-joint beams are adopted not to modify the total beam depth. Relevant failures in the past, e.g., the collapse of Annone SS36 overpass bridge in October 2016 in Italy [18] due to an anomalous load configuration, construction and maintenance issues, suggest that the integrity of Gerber girder supports is a primary requirement for the robustness of the suspended span. To this aim, several studies concentrated on the failure mechanisms of the main beams at the support [19, 20], but disregarded the specific analysis of the failure of the support itself, which requires a detailed design and maintenance during the service life of the structure [21]. For concrete structures, the behavior of half-joints depends on the arrangement and amount of reinforcements, and the quality of the concrete [22, 23]. The forensic analysis of the failure of Annone overpass highlighted that cracking and corrosion on the reinforcement bars caused a reduction of the bearing capacity of one support that initially failed [18]. Meanwhile, the investigations showed that, in the decades preceding the failure, the reduction of reinforcement area caused a progressive settlement of the supporting half-joint. This result is in perfect agreement with the results of Desnerck and colleagues [20, 22], who pointed out an evidence of a reduction of the vertical stiffness of damaged supports, which are fragile components that have not to fail in a capacity design.
The analysis of beam grillage structures accounts both the flexural and the torsional characteristics of the elements. In the present analysis, a simple rectangular structure made of three main beams of length \(\ell \) and orthogonal transverse beams of length w at both ends is considered, as shown in Fig. 1 where the main and transverse beams are named as M and T, respectively. The grillage is subjected to a vertical uniform load q distributed along the main beams, only. The system is supported on vertical springs, to simulate the axial stiffness of the ground (if the grillage is part of a foundation system) or of a substructure (to simulate supported floors and bridge decks). The system is horizontally constrained at each node, as illustrated in the box of Fig. 1.
The structural behavior of T-frame bridges is particularly complicated and it is difficult using a general analytical method to directly acquire the internal forces in the structure. This paper presents a spatial grillage model for analysis of such bridges. The proposed model is validated by comparison with results obtained from field testing. It is shown that analysis of T-frame bridges may be conveniently performed using the spatial grillage model.
Typically, the design of highway bridges in China must conform to the General Code for Design of Highway Bridges and Culverts (JTG D60-2004) specifications. The analysis and design of any highway bridge must consider truck and lane loadings. However, the structural behavior of T-frame bridge is particularly complicated, and many rigorous methods for analysis of T-frame bridges are quite tedious and often difficult.
Pan et al. [1] carried out uncertainty analysis of creep and shrinkage effects in long-span continuous rigid frame of Sutong Bridge. Azizi et al. [4] used spectral element method for analyzing continuous beams and bridges subjected to a moving load. Wang et al. [5] analyzed dynamic behavior of slant-legged rigid-frame Highway Bridge. Dicleli [6] presented a computer-aided approach of integral-abutment bridges, and an analysis procedure and a simplified structure model were proposed for the design of integral-abutment bridges considering their actual behavior and load distribution among their various components [7, 8]. There were several approximate analysis methods for bridge decks, which include the grillage method and the orthotropic plate theory [9]. Yoshikawa et al. [10] investigated construction of Benten Viaduct, rigid-frame bridge with seismic isolators at the foot of piers. Kalantari and Amjadian [11] reported a 3DOFs analytical model an approximate hand method was presented for dynamic analysis of continuous rigid deck.
Mabsout et al. [12] reported the results of parametric investigation using the finite-element analysis of straight, single-span, simply supported reinforced concrete slab bridges. The study considered various span lengths and slab widths, number of lanes, and live loading conditions for bridges with and without shoulders. Longitudinal bending moments and deflections in the slab were evaluated and compared with procedures recommended by AASHTO [13].
This paper presents a spatial grillage model for the analysis of a T-frame bridge. Static and dynamic analysis results of spatial grillage model for the T-frame bridge are compared with results based on results obtained from field testing. The research results shown that analysis of T-frame bridges may be conveniently performed using the spatial grillage model.
Spatial grillage model is a convenient method for analysis of box-girder bridges. In the model, the box-girder slab is represented by an equivalent grid of beams whose longitudinal and transverse stiffnesses are approximately the same as the local plate stiffnesses of the box-girder slab.
In the spatial grillage model analysis, the orientation of the longitudinal members should be always parallel to the free edges while the orientation of transverse members can be either parallel to the supports or orthogonal to the longitudinal beams. According to the grillage model, the output internal force resultants can be used directly. The grillage model involves a plane grillage of discrete interconnected beams. The representation of a bridge as a grillage is ideally suited for carrying out the necessary calculations associated with analysis and design on a digital computer and it gives the designer an idea about the structural behavior of the bridge.
Determination of a suitable grillage mesh for a box-girder of rigid frame bridge is, as for a slab deck, best approached from a consideration of the structural behavior of the particular deck rather than from the application of a set of rules. Since the average longitudinal and transverse bending stiffness are comparable, the distribution of load is somewhat similar to that of a torsionally flexible slab, but with forces locally concentrated. The grillage simulates the prototype closely by having its members coincident with the centre lines of the prototypes beams. In addition, there is a diaphragm in the prototype such as over a support, and then a grillage member should be coincident. Based on section shape of the rigid frame bridge and support arrangements, a spatial grillage mesh should be represented by the above mentioned spatial grillage method. At the same time, according to the grillage equivalent theory, the following three important aspects have to be noted: according to mechanical behavior of a rigid frame bridge, one should place the grillage beams along the lines of designed strength; the longitudinal and transverse member spacing should be reasonably similar to permit sensible static distribution of loads; in addition, virtual longitudinal and transverse members are often employed for the sake of convenience in the analysis. The virtual members only offer stiffness, but its weight must be ignored.
Three-dimensional linear elastic finite element models of the spatial grillage model of the Quhai Bridge have been constructed using SAP2000 finite element analysis software. In the finite-element model, 3D beam4 elements were adopted to create the grillage model that will be used to determine the internal stress resultants, natural frequencies, and corresponding mode shapes. The spatial grillage model was shown in Figure 3, and the virtual beams only offer stiffness. In the finite-element model of the bridge, 3D 1104 elements (beam 4) and 817 nodes were used. The spatial grillage model of the bridge as a whole is shown in Figure 4. 2ff7e9595c
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