Often referred to as grid methods, two-dimensional analysis methods consist at minimum of an interconnected series of beam elements that represent the major flexural members of a bridge superstructure and sometimes include the substructure. The deck slab can be either distributed and included with beam properties or explicitly modeled with shell elements. 2D analyses can be used simply to determine girder distribution factors with one-dimensional methods used to complete the design, or utilized to determine both dead load and live load envelopes for subsequent factoring and limit states checks.
Because 2D models explicitly account for live load distribution based on geometry and element stiffnesses, resulting designs are more efficient than those based on the approximate distribution factors in a 1D design. Skew factors also do not need to be applied, as skew effects are explicitly modeled. See Examples 1 and 2 in Section 7 for a comparison of 1D versus 2D
design.
distribute loads based on actual longitudinal and transverse member stiffnesses. The transverse member stiffnesses in the grid often require the use of an effective stiffness value, for instance when modeling steel cross-frames, since vertical geometry is not modeled.
Figure 2-16 – 2D grid model of three span continuous steel girder bridge for Example 2.
A grid requires the development of member properties that proportionally assign the material in the bridge to the mesh of line elements. By approximating the deck properties into line elements, the grid analysis does not fully model the membrane stiffness of the deck, but the outputs of shear, moment, and axial force are compatible with AASHTO design equations. For guidance on modeling basic grid analyses, see Section 3.6.1.
Even though basic grid analysis is a step up in rigor from a 1D analysis, it still suffers from many of the same limitations of the 1D analysis. In some cases a grid analysis may be acceptable for member design, but not for calculating camber and deflections at intermediate construction stages. Basic grid analyses are not appropriate in cases with:
• large second order effects, such as compression flange lateral bending stresses,
• geometrical sources of stiffness such as force couples from flange lateral bracing (pseudo box) or multiple bearings under box girders,
• significant lateral effects in multi-girder bridges, • significant torsional effects in open section girders,
• when other than no load fit is used with steel cross frames,
• when large shear membrane forces are present in the plane of analysis.
Improved grid analysis techniques have been developed to overcome many of these limitations.
DRAFT
2.4.3.2 Improved Grid Analysis
Over the years, both program enhancements and modeling techniques, not to mention increased computing power, have resulted in improvements to the basic grid analysis. The
plate with eccentric beam (PEB) model is now commonly used to model multi-girder bridges.
This model combines a line element girder and cross-frame/diaphragm grid with a shell element deck including the geometric offset of the deck from the girder centroids. A typical PEB model is illustrated in Figure 2-17.
Figure 2-17 – 2D PEB model of three span continuous steel girder bridge for Example 2, with internally offset girders and cross-frames bolded.
Improved grid models are still typically linear elastic and small deflection, but no longer require a single material transformed section. Models can use actual gross section material properties, and explicitly assign separate material properties to the girders and the deck. Varying section properties in the girders are still usually handled with stepped section properties in the models. Some programs now have beam elements with definable offsets such that the deck and girders can be modeled in a single plane, with the offsets providing the geometrical eccentricity, although rigid links are also effective when the offset needs to be modeled explicitly. Timoshenko beam elements that include shear deformations allow for improved cross-frame stiffness approximations, and in some programs, “exact” cross-frame stiffness properties can be programmed into user defined elements.
Techniques have been developed to calculate effective torsional constants in order to account for warping stiffness in thin walled open sections (see Section 3.6.4.1). Use of such techniques allows calculation of camber and deflections during construction of thin walled open section multi-girder steel bridges.
One of the biggest advantages of PEB models is that they can explicitly model the behavior of structures with large skew or curvature. A PEB analysis requires less compromise and fewer assumptions in defining the elements of the mesh, and does model the membrane stiffness of the deck. Modeling the deck as a continuum explicitly captures accurate transverse load distribution and torsional behavior more easily than a basic grid analysis, where the transverse load distribution is concentrated at the transverse grid members and the torsional behavior must be distributed among multiple grid members. Also eliminated is the requirement of assigning an
Although modeling the deck and girders separately makes extracting design forces a little more difficult, improved grid analysis is accurate enough to use for design of most multi-girder bridges. The improved analysis is not appropriate in cases with:
• large second order effects,
• geometrical sources of stiffness such as force couples from flange lateral bracing (pseudo box) or multiple bearings under box girders.
For guidance on modeling PEB analyses, see Section 3.6.2.