Civil Engineering Association

Approximate Torsional Analysis of Curved Box Girders by the M/R-Method

Curved bridges are becoming increasingly prevalent in highway construction because of improved geometric designs and construction techniques. The current trend in this type of structure is to shape the girders so that they follow the curvatures of the horizontal alignment, creating a continuous flow of the major structural elements. While the appearance and structural efficiency are often enhanced by the use of curved girders, the analysis and design of these members are likely to be more complicated, which, in some cases, may be the only major factor that prevents the adoption of such a system. It is desirable, therefore, to develop approximate methods which may help practicing engineers overcome this hindrance. The objective of this paper is to present to design engineers a simplified method for the torsional analysis of single-span or continuous curved box girders, which, by virtue of their excellent strength in resisting torsion, are generally recognized as ideal supporting elements for horizontally curved structures. The accuracy and limitations of the approximate method, as well as the effects of the various parameters inherent in the problem, are discussed herein. The results are then compared with those obtained in exact solutions based on the transfer matrix method.

Analysis of Curved Girder Bridges

Geometric, aesthetic, and economic considerations have led to the increased use of horizontally curved girders for highway bridges and interchange facilities which involve curved alignment. Despite the slightly higher fabrication costs associated with curving the girders, the net costs for such bridges, in certain cases, are lower than those for curved bridges with straight beams placed along the chords of the curved roadway. This overall economy is a result of the elimination of many substructure units (piers) and simplified form work necessary for the roadway slab. Since curved girders may be classified as linear elastic structures, all the various methods of linear structural analysis may be applied to the design of curved bridges. Although considerable analytical work has been done in this area, no attempt has been made to correlate the various methods of analysis or evaluate the degree of approximation inherent in several of the design procedures. The purpose of this paper is twofold. First, the fundamental differences between the behavior of individual straight girders and curved girders loaded normal to the plane of curvature will be established. Second, numerical results obtained from an approximate method of analysis widely used in the design of curved bridges3 will be compared with a rigorous analysis of curved grid systems.

---

Another Approach to Simplified Design of a Curved Steel Girder

In recent years several publications discussing an approximate analysis of a curved girder grid system were presented. In the first paper by Richardson, Gordon and Associates, a two-girder system connected with diaphragms was discussed. Further development for the multi-girder system connected by diaphragms was published by U. S. Steel Corporation. The presentation of W. T. Robertson at the ASCE meeting in October, 1965 and the article by James W. Gillespie4 also can be mentioned, among other publications concerned with the same subject. The principal assumption of analysis used in all the above mentioned publications is based on the well established fact that the change in torque between two points of a curved girder subjected to bending is equal to the area of bending moment diagram between these points, divided by the radius of curvature of the girder.

---

Analysis of Horizontally Curved Bridges

Many years ago, highway bridges were located by determining the most convenient crossing site, with little regard to the general alignment of the roadway. After the bridge location was established, the highway designer or surveyor laid out the highway to meet the bridge. During the last several decades, this situation has reversed and now bridges must fit the highway alignment that has been predetermined by many other considerations. The increasingly frequent occurrence of structures on curved alignment is presenting real challenges to engineers, especially in the design of urban freeways where multi-level interchanges must be built within tight geometric restrictions. The present-day emphasis on good appearance is also an important factor. Welding has helped to produce structures with smooth surfaces, interrupted by a minimum amount of detail. Outside transverse stiffeners are no longer used on many highway girders. The use of curved supporting beams or girders in a structure on curved alignment is a natural outgrowth of this trend toward aesthetic design.

---

Preliminary Design of Curved Bridges

In order to utilize several computer programs1 for the analysis of radial curved bridge systems of open cross-section, it is necessary to evaluate section properties. This paper presents and explains a simplified method for evaluating the internal forces in radial curved girder systems; the equations and factors presented permit the designer to determine required cross-sectional properties. A series of simplified equations are presented, which will permit evaluation of internal forces and deformation in a single, two- and three-span curved girder system. These forces can then be utilized to estimate initial section properties, which are necessary in utilizing various computer programs.

Effective Length of Columns with Semi-Rigid Connections

The effective length of a column depends on the boundary conditions at the ends of its unbraced length. In a framed structure, the boundary conditions depend on the stiffness of the beams framed to the column. Design guides available for determining the effective length are provided for cases where beams are rigidly framed to the column and have their far ends rigidly framed to another column, fixed, or hinged. This paper will present a solution for the case where beams are framed to columns using semi-rigid connections. Frequently, in the design of frames for tall buildings, it is found that beams must be selected based on their stiffness to resist frame drift, rather than on the strength needed to resist wind forces. In this case the actual bending moments required to be resisted at the ends of beams may be substantially smaller than the moment capacity of the beams. It is economically attractive to consider the use of semi-rigid connections adequate for the actual bending moment. However, the semi-rigid connection may have a lower stiffness than an equivalent portion of the beam, since its strength is weaker than the beam. This paper will present a set-up for a general solution for the stiffness of beams with semi-rigid connections. A solution will be made for a simplified case and available information for one type of semi-rigid connection will be used to illustrate the effect on the effective length of a column.

Unbraced Frames with Semi-Rigid Composite Connections

The benefits of semi-rigid connections are well known and much has been written about their use in braced frames and in Type 2 construction. One of the reasons these methods are not being used frequently by designers is that most semi-rigid connections are highly nonlinear, and analysis of the behavior of frames using them is difficult. One type of connection which may be able to overcome this difficulty is the semi-rigid composite connection. This is a connection made from a typical pinned end connection (e.g., seat angle and web clips or double web angles) and continuous slab reinforcement across the connection. Tests on the connection with seat angle (Fig. 1) by the authors8,9 have shown this connection type has a high degree of linearity in the service load region and that its ultimate capacity is easy to predict. This paper describes a design procedure for unbraced frames utilizing semi-rigid composite connections such as those described above. Two design examples, one for a four-story structure and one for a ten-story structure, and comparisons of their behavior with that of rigidly designed frames are included.

Design Aid of Semi-rigid Connections for Frame Analysis

In this paper, a useful design aid for determining the values of the initial connection stiffness Rki, the ultimate moment capacity M u, and the shape parameter n of a three-parameter power model describing the moment-rotation curve (M- qr) of semi-rigid connections with angles is prepared for its use in the practical design of flexibly jointed frames with angles. A set of nomographs allows the engineer to rapidly determine the M- qr curve for a given connection. Applying the design aid, numerical simulations on drift and column moment of a flexibly jointed frame with angles are illustrated.

Practical Advanced Analysis for Semi-rigid Frame Design

This paper presents three practical advanced analysis procedures for a two-dimensional semi-rigid steel frame design. Herein, the nonlinear behavior of beam-to-column connections is discussed, and practical modeling of these connections is introduced. The proposed methods can predict accurately the combined nonlinear effects of connection, geometry, and material on the behavior and strength of semi-rigid frames. The strengths predicted by these methods are compared well with those available experiments. Analysis and design procedures using the proposed methods are described in detail, and a case study is also given. The proposed procedures can be used for the LRFD design without tedious separate member capacity checks, including the calculations of K-factor. The procedures are suitable for adoption in practice.

Semi-Rigid Frame Design Methods for Practicing Engineers

Design of semi-rigid (PR) frames focuses on behavior characteristics of non-linear connections, including their substantially different loading and unloading characteristics. Moment-rotation connection representations, such as the three-parameter power model, facilitate the calculation of stiffness data required for frame analysis. In this paper, the connection characteristics are described in terms of linearized connection stiffnesses that are calculated on the basis of expected connection loads. This allows for the use of first-order analysis to determine structural stability, serviceability and member load effects. The design method detailed in this paper includes the concurrent selection of connection and member sizes. The LRFD approach of AISC is utilized, including the provisions that rely on amplification factors to account the provisions that rely on amplification factors to account for second-order effects. Member section checks are made with unbraced length K-factors determined from the alignment charts, using modified relative distribution factors to account for connection deformation.

Allowable Bending Stresses for Overhanging Monorails

In the design of monorails for trolley hoisting devices it is often necessary to extend the monorail out beyond the last available support (as in Fig. 1), so the lifting point is clear of obstructions. Such a monorail supported vertically at two locations can be analyzed as a simple beam with overhang so the maximum bending moment is easily obtained. However, the question of the allowable bending stress immediately presents itself. The intent of this paper is to address the problem and present a simple design solution.

Bracing Connections for Heavy Construction

The design of complex connections is not an exact science. Over the years, an intuitive approach to connection design has become widely accepted. This approach is based on the idea the structure (and parts thereof) will behave as the designer dictates, if he provides a path of adequate strength for the load (or loads) to follow. This "adequate strength path" is determined from the principles of statics and strength of materials. About 30 years ago, this intuitive load path method of design was put on a rigorous basis with the development of the Theorem of Limit Analysis—specifically the Lower Bound Theorem of Limit Analysis, which states: if a distribution of forces in the structure can be found which is in equilibrium with the applied loads, and if these forces everywhere within the structure are of such a magnitude that the yield stress (or yield criterion) is nowhere exceeded, then the applied loads are less than, or at most equal to, the loads required for collapse (unbounded yield deformations) to occur. Thus, if a load path is provided, the elements of which are in equilibrium with the applied loads, and if the stresses in these elements nowhere exceed the yield stress, a safe design will have been achieved. Also, the relative stiffness of the various connection elements should be considered in order to minimize the possibility of fracture.

Yield Line Analysis of Column Webs with Welded Beam Connection

In welded construction the connections, for the most part, are easily designed and fabricated. However, one of the more difficult connections occurs when a beam must be welded to a column web. This is so for two reasons: (1) analyzing the column web's capacity is arduous and (2) making the actual connection in accordance with the design is no simple task because of the space restrictions imposed on the welder by the column flanges. This paper will deal only with the first consideration—analyzing the column web strength—using Yield Line Analysis. The Yield Line approach has been utilized previously by Abolitz and Warner1 for brackets welded to column webs, and by Blodgett2 for welding rolled sections to box columns. The assumptions used here differ somewhat from those of the above authors. The final result is a series of curves covering a very high percentage of restrained beam to column web connections involving hot rolled shapes. The curves relate beam moments to column web thickness for a series of beam sizes and for W14 columns with yield strengths of 36 and 50 ksi. For special cases not covered by these curves, but for which the yield line pattern assumed is valid, direct solutions may be developed by use of the formulas used to construct these curves.

Torsional and Constrained-Axis Flexural-Torsional Buckling Tables for Steel W-Shapes in Compression

Torsional buckling (TB), an applicable limit state for W-shape members subject to axial compression, often controls when the torsional effective unbraced length exceeds the minor-axis flexural buckling effective unbraced length. Constrained-axis flexural-torsional buckling (CAFTB) is a potential limit state for W-shape members that are constrained to buckle with the center of twist at a location other than the centroidal axis, as is the case for a typical beam with one flange braced by a diaphragm and the other unbraced. Manual calculation of the TB or CAFTB available compressive strength is a somewhat lengthy process, especially when the section is slender for axial compression, and no design aid currently exists in the AISC Manual. This paper provides tables that facilitate the determination of TB and CAFTB available compressive strengths. Several example calculations are also provided.

Flange Bending in Single Curvature

Local bending of beam and column flanges is a common design consideration in steel structures. In most cases, the flange bends in double curvature due to the restraining effect of the connecting element. When a restraining force is not present, the flange will deform in single curvature. Common cases of single-curvature bending occur at the bottom flange of monorail beams and at hanger rod connections. In this paper, the equivalent-width method was explored in an effort to determine design procedures for elastic and plastic strength of flanges in single-curvature bending. This paper compares the available procedures for designing flanges bent in single curvature. New finite element models and yield line analyses are used to verify, expand and improve the existing design methods. Design recommendations are made for both elastic and ultimate strength approaches. Recommendations are also made for interaction of the local bending strength with longitudinal stresses in the flange. The effects of closely spaced loads and loads acting near the ends of members are also addressed.







==========================================================================================

ERRATA: Flange Bending in Single Curvature

-none-

Secondary Stresses in Trusses

Secondary stresses in steel trusses may be neglected in most cases. It is important, however, that secondary stresses be defined properly and the analysis and design be consistent with each other, as follows: If the truss members are designed for the axial forces that would occur if the members were pin-connected, then the flexural stresses indicated by a more refined analysis may be defined as secondary stresses and may be neglected, within reasonable limits. If the axial forces for member design are obtained from an analysis that includes flexural effects, flexural stresses cannot be dismissed as secondary stresses, since the presence of flexural effects in the analysis might have reduced the axial forces indicated by the analysis. In this case, the designer who wishes to neglect flexural stresses must first judge whether (and by how much) the axial forces indicated by the analysis were affected by flexural effects. And he must then make appropriate corrections in the axial forces to be used for design.

Cambering Steel Beams

Natural mill camber is the out-of-straightness remaining after the initial rolling, cooling, and straightening of the member at the mill. Tolerances for natural mill camber are listed in the AISC Manual of Steel Construction. Induced camber is that which is applied subsequent to the initial rolling and straightening process. Induced cambering can be done at either the rolling mill or the fabricating shop. Tolerances for induced camber are also listed in the AISC Manual of Steel Construction.





====================================================

Economical Use of Cambered Steel Beams

None