Leon Kempner Jr. & Thien T. Do, Bonneville Power Administration, & Wendelin Mueller III, Portland State University
The method of analysis for transmission line towers has advanced since the use of graphically procedures. Before the advent of the computer, the transmission engineer used graphical analysis developed to a professional art for determining member forces in three dimensional space truss towers. With the introduction of the computer, analysis calculations using structural matrix techniques were developed. BPA's Tower computer program was one of the original programs developed specifically for transmission line lattice structures. Now with the power of the personal computer (PC) the availability of tower analysis computer programs are numerous, such programs include BPA's Tower, PLS Tower, GT Tower, etc.
The standard method of computer analysis for a lattice steel transmission line tower assumes three-dimensional truss behavior. The structural computer model is based on the tension and compression behavior of the individual members. These tower analysis programs are based on linear elastic structural performance, whereby pinned connected members are assumed to be axially loaded. The member forces determined from the computer model are compared to the allowable design capacity.
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With the maturity of the transmission tower engineering profession and the advances in PC calculation power, the transmission engineer can now push the margin of analysis beyond the simple linear truss model. The simple linear truss model is still the main analysis tool of the transmission engineer, but advanced tools for evaluating the performance of space truss towers, including effects other then simple truss action, are now being utilized. This article presents one such program that includes the non-linear effect of the after failure "post-buckling" load resistance of the members.
LIMIT
The Bonneville Power Administration (BPA) and Portland State University (PSU) developed a computer program called LIMIT. LIMIT performs a first-order non-linear analysis accounting for post-buckling performance of individual members in a lattice steel transmission tower. A direct iteration, Secant Method, procedure is used to account for this non-linear member behavior. The results are used to determine the "collapse" mechanism, and the load flow through the members as the tower approaches failure.
The LIMIT analysis uses individual member post-buckling performance curves to track the member's load when it exceeds its design compression capacity. Two options are used to model the post-buckling member performance: bilinear and normalized (empirical) curves. Normalized member performance curves were developed from actual tests of angle steel members. The LIMIT program has 29 normalized curves that are used to represent the post-buckling member performance of steel angles. These curves represent the performance of equal and unequal leg angles, and double angles. Both the bilinear and normalized curves assume linear behavior until the member reaches its design compression capacity. The figure shows examples of the bilinear and normalized curves and a curve that is representative of the actual performance of a member.
Member performance curves
The LIMIT program has three post-buckling analyses options: Deterministic (referred to as LIMIT), Probability-Based-Analysis (PBA), and Capacity-Variation-Analysis (CVA). The deterministic analysis, LIMIT, calculates the collapse load of the tower using individual member capacities and post-buckling performance. PBA analysis varies member capacities based on selected yield strength distributions. CVA analysis varies member capacities based on a user selected +/- percentage variation about a calculated "base" member capacity. Both PBA and CVA provide a simplified design tool that accounts for variation in parameters which can affect member capacities, such as yield strength, connection eccentricity, engineer judgment, etc.
PBA and CVA use the Monte Carlo technique to randomly vary the capacity of each member in the tower. A first order non-linear post-buckling analysis, LIMIT, is repeated using the random member capacity to obtain the tower collapse load distribution. The results of two analyses are presented: 230 kV single circuit transmission tower and a microwave tower. The analyses are compared to the results of full-scale tests.
The 230 kV tower is 75.1 ft (22.9m) tall, the distance between the footing supports is 17.1 ft (5.2m), and the tower bridge spans 78.1 ft (23.8m). The test load was a simple transverse load across the lower chord of the tower bridge. The failure load was 25.4 kips (113 kN) with a transverse deflection of 6 in. (15.24 cm). The failure occurred in the upper tower body above the waist. An elastic analysis using minimum yield strengths and calculate member capacities resulted in a failure load of 16.31 kips (72.5 kN) and a deflection of 2.14 in. (5.44 cm). A LIMIT analysis using the same capacities calculated a failure load of 18.0 kips (80.1 kN) and 2.53 in. (6.43 cm).
Member capacities were adjusted based on actual member yield strengths obtained from coupon tests. Both an elastic and LIMIT analysis were performed with the new capacities. The elastic analysis failure load increased to 20.6 kips (91.6 kN) with a deflection of 2.7 in. (6.86 cm), and the LIMIT results increased to 24.3 kips (108.1 kN) and 4.22 in. (10.72 cm). This LIMIT analysis predicted the actual tower test failure mechanism. A more refined LIMIT analysis was performed where the failed member capacities were adjusted based on strain gage test data. This analysis gave a failure load of 25.2 kips (112.1 kN) and a deflection of 4.22 in. (10.72 cm). The failure mechanism remained the same.
The microwave tower is 102.16 ft (31.14m) tall with a base width of 19.48 ft (5.94m). The test load was applied at the bridge in the direction normal to the axis of the bridge. The failure load divided by the load at first yield, Failure Load Factor (FLF), was 1.54. A FLF of 1.31, 1.38 and 1.39 was predicted by a LIMIT, PBA and CVA analysis respectively.
These two examples demonstrate the application of post-buckling performance and capacity variation for better defining the failure limits beyond first yield of lattice steel transmission towers. This advanced tool was developed to help design engineers structurally evaluate transmission towers. This has been found useful in full-scale testing, evaluation of existing designs that were done using graphical analysis, and for failure investigation.
Leon Kempner, Jr., PE, PhD, is a structural engineer for the Bonneville Power Administration. Leon has 28 years experience in electrical transmission tower analysis, design, and research. Contact can be made at 360-619-6556 and lkempnerjr@bpa.gov.
Thien T. Do, PE, is a structural engineer for the Bonneville Power Administration. Thien has 13 years experience in electrical transmission tower analysis, design, and research. Contact can be made at 360-619-6552 and ttdo@bpa.gov.
Wendelin Mueller III, PE, is a Professor of Civil and Environmental Engineering at Portland State University. His research has a focus on the analysis and testing of full-scale structures, and/or their components. Mueller's research is done in the Seismic Testing and Applied Research (STAR) laboratory that houses a Seismic Shake table. Information about his research can be found at http://www.ce.pdx.edu/star/. Contact can be made at: 503-725-4257 and wendell@cecs.pdx.edu.
