Figure 1. Types of Cross - Sections
Lateral-torsional buckling behavior of various types of steel trusses without and with intermediate lateral rigid supports is investigated by performing finite element analysis. Commonly used parallel chord trusses, such as Pratt, Howe, Warren (with verticals), K and Diamond trusses of span 8 m and rise 0.8 m are considered. Two different types of intermediate lateral rigid support systems are considered. Elastic critical buckling joint loads of the trusses are found by modeling and performing finite element analysis using ANSYS - commercial software. From the obtained results it can be understood that critical buckling load, the magnitude and shape out of plane deflection depends upon whether the intermediate lateral support is provided or not and if provided, the type of lateral supports and spacing between them. The pattern and inclination of the web lacing also affect the stability of the truss. It is found that the Diamond and K trusses resulted in better stability and hence suggested for use in place of large spans and heavy loading.
Steel parallel chord trusses are most commonly used in the industrial structures and bridges. Different types of parallel chord trusses, include Pratt, Howe, Warren (with and without verticals), K-type, and Diamond type. Various cross-sections that can be provided for truss members are shown in Figure 1. Open sections, such as single angles, double angles, I sections, built up I sections and closed hollow sections, such as Circular Hollow Section (CHS), Square Hollow Sections (SHS), Rectangular Hollow Sections (RHS) are commonly used in trusses. Bolted connections may be conveniently provided for open sections. But for closed hollow sections, bolting is not convenient and welding should be preferred. Generally, the joints are assumed to be pinned/ hinged. Consider a parallel chord tubular truss of Pratt type as shown in Figure 2, and the detailing is also depicted. The top members HI, IJ, JK, KL, LM, and MN are called top chords. They are subjected to compression under gravity loading. The bottom members AB, BC, CD, DE, EF, and FG are bottom chords and are subjected to tension under gravity loading. Members AH and GN are outer struts. Members BI, CJ, DK, EL, FM, and GN are intermediate struts. Members BH, CI, DJ, DL, EM, and FN are called inclined members or lacing. They may be subjected to tension/ compression. The inclined and intermediate members are also called as web members. The joints are assumed to be welded. Each parallel chord truss of a building / bridge / any metal structure is a plane truss as all members of the truss are in a plane and loading is within the plane of truss. Beams cannot be provided for larger spans as heavier sections need to be provided for preventing instability which results in uneconomical design. Though steel parallel chord trusses result in lesser weight, lateral-torsional buckling of the trusses (instability) may occur similar to that of the beams. Trusses of different types and configurations are affected by the instability. Hence, apart from simple analysis, the structural designer need to assess the stability of the trusses, especially spanning over long spans or subjected to heavy loading such as railway truss girder bridges. The truss when subjected to in-plane downward load (P), it initially bends within the plane. On load reaching a particular value '(P )', the compression chord (top chord) cr tends to buckle or deform out of plane of truss, whereas the tension chord (bottom chord) tends to remain stable resulting in lateral-torsional buckling of the entire truss (out of plane deformation of entire truss). The particular value of Load 'P ' is called critical buckling load.
Figure 1. Types of Cross - Sections
Figure 2. Pratt Truss with Detailing
The failure of several pony-truss bridges around nineteenth century grabbed the attention of the designers and researchers on the problem of compression chord buckling. Literature presented by Galambos (1998) states that many researchers conducted experimental tests on Pony truss bridge models to study the stability of the bridges. Timoshenko and Gere (1961) provided methodology for calculating the buckling load and reduced lengths of members of a simple truss. Cho and Chan (2005) performed second order analysis and designed the struts of a truss by following equivalent imperfection approach. Chan and Cho (2005) performed P-Δ-d analysis of trusses built with single angle sections considering semi-rigid connections and imperfections of members. Chan and Cho (2008) presented elastic analysis and design of angle trusses considering initial curvature or imperfections and residual stresses. Jankowska - Sandberg and Kolodziej (2013) conducted experimental and analytical tests on parallel chord Pratt truss (or N truss) for different lateral support positions.
From the review, it can be understood that less literature is available on stability of trusses considering the research till date. There are about five majorly used parallel chord trusses in industrial structures and railway steel bridges based on different types and configurations. Hence, in this study, stability of commonly used parallel chord trusses, namely Pratt, Howe, Warren (with verticals), K and Diamond trusses with and without rigid lateral supports is investigated.
For validating the results obtained by ANSYS, a previously analyzed truss model by Jankowska and Kolodziej (2013) is re-analyzed. The chords and supporting struts were modeled with 25 x 25 x 2 mm tube (E = 208.07 GPa). Web members (intermediate struts and diagonal lacings) are modeled with 20 x 20 x 2 mm tube (E = 210.83 GPa). The dimensions of the members are shown in Table 1. Load 'P' is taken as 1 kN. Meshing of members is done with BEAM 188 element. The results are presented in Table 1.
Table 1. Critical Buckling Load
Different types of parallel chord trusses considered with rigid lateral supports (•) at end chords are shown in Figures 3 to 7. For all the trusses, the chords and outer struts, a steel tubular section 25 x 25 x 2 mm is assigned. 20 x 20 x 2 mm steel tubular section is assigned for intermediate struts and inclined lacings. The bottom and top chords are of each 1 m length. The rise of the truss is 800 mm and total length of the truss is 8000 mm. Hinge and roller supports are provided for all trusses as shown in Figure 3. Linear elastic behavior of steel is considered. Modulus of elasticity of steel is taken as 210 GPa for all members. BEAM 188 element, which is a 2- noded beam element with six degrees of freedom (translations along x, y, and z axes and rotations about x, y and z axes) at each node is considered for meshing of members. It is based on Timoshenko's beam theory. Shear deformation effects are included. The element can analyze slender to moderately thick beam structures. The element is suitable for analyzing flexural, lateral, and torsional stability problems. Each member is descritized into five beam members. Point loads of magnitude 1000 N are applied at intermediate joints for all the types of trusses and cases. Block Lanczos method is followed for finding the critical buckling loads 'P '.
Figure 3. Pratt Truss
Figure 4. Howe Truss
Figure 5. Warren Truss with Verticals
Figure 6. K Truss
Figure 7. Diamond Truss with Verticals
Practically, the trusses are provided with intermediate rigid lateral supports perpendicular to the plane of truss. Two cases of rigid lateral supports have been considered. Figures 8 to 12 represent the case - I type rigid lateral supports assigned to the trusses. The image of Pratt truss modeled in ANSYS with case - I intermediate lateral rigid supports is depicted in Figure 13. Figures 14 to 18 represent the trusses with case - II type rigid lateral supports.
Figure 8. Pratt Truss
Figure 9. Howe Truss
Figure 10. Warren Truss with Verticals
Figure 11. K Truss
Figure 12. Diamond Truss
Figure 13. Pratt Truss with Rigid Intermediate Lateral Supports
Figure 14. Pratt Truss
Figure 15. Howe Truss
Figure 16. Warren Truss with Verticals
Figure 17. K Truss
Figure 18. Diamond Truss
The critical buckling loads for the considered types of trusses with and without intermediate lateral supports are presented in Table 2. The corresponding displacement vector sum values of trusses are presented in Table 3. The deflected shapes (buckled profile) of trusses without intermediate lateral rigid supports and with intermediate lateral rigid supports (Case I and Case II) are depicted in Figures 19 to 21. Figure 22 represents plot between critical buckling load P and truss type and Figure 23 depicts plot between the displacement and truss type. The following are observed from the results in Tables 2 and 3 and Figures 19 to 23.
Table 2. Critical Buckling Loads of Trusses with and without Intermediate Lateral Supports
Table 3. Displacement Vector Sum Values of Trusses with/ without Intermediate Lateral Supports
Figure 19. Deflected Shape of Truss without Intermediate Lateral Rigid Support (a) Isometric View, (b) Top View
Figure 20. Deflected Shape of Truss with Intermediate Lateral Rigid Support (Case – I) (a) Isometric View, (b) Top View
Figure 21. Deflected Shape of Truss with Intermediate Lateral Rigid Support (Case – II) (a) Isometric View, (b) Top View
Figure 22. Plot of Critical Buckling Load vs. Truss Type
Figure 23. Plot of Displacement vs. Truss Type
This paper presents the study on lateral-torsional buckling of various types of trusses subjected to in-plane joint loads. Trusses without intermediate lateral rigid supports and trusses with intermediate lateral rigid supports of Cases I and II are considered for performing finite element analysis by ANSYS. Totally, fifteen trusses are modeled and analyzed. From the results presented, the following conclusions may be drawn.