Truss and Vertical Bracing Efficient Geometry - Maxwell's Load path theorem

Introduction

In this article, the Optimization of Vertical bracing and truss bridge is performed using Maxwell's theorem of load path theorem (1864). In a truss, structure load travels from the point of application to support through tension and compression in individual members.

In a truss, members can be grouped into two groups, members under tension (pull) and members under compression (push). For each group, a sum of the product of member force and member length is calculated. As per Maxwell's theorem, in an optimized truss geometry the sum of the above two quantities is the least ( shorted load path is one where the absolute sum of the product of force and length is least)

Validation using Work Done

The Work done is a product of force and displacement, it is further divided by two as the force application is gradual (static) and sudden (dynamic)

Bracing Geometry

Vertical bracing optimization was performed for, braced bay width 4 m and floor height of 4m, with a horizontal force of 200 kN at each floor level. Six different vertical bracing systems as shown in Figure 2. The first three alternatives are N-type, Inverted V-type and X-type bracing. In the next three alternatives, the internal angle between short-length bracing members was kept at 126.89 degrees, 141.9 degrees and 114.3 degrees.

The internal angle between short side bracing 126.89 is arrived using load path theorem on equivalent truss shown in Figure 3. For given span (L) and depth (D) the point load at point M1, the only variable is internal angle alpha. Maxwell's load path theorem is utilized to arrive at the alpha value. Table 1 provides a summary of the sum of the product of force and length for the range of alpha value, The alpha value 126.89 degrees gave the least value of the sum 550.0 kN*m. Coincidentally, the ratio of length L2 over L1 is close to the golden ratio 0.618 for bay width to floor height ratio 1. It establishes a connection between elegance and efficiency.

The sum of the product of force and length is calculated and summarized in Table 2

The bracing system fourth, with an internal angle between short sides 126.89 degrees provides the least load path.

Truss Optimization

Maxwell's load path theorem is applied to find the optimum geometry of a 60 m span through a truss. The total depth and total load (900 kN) were kept constant in all three alternatives.

Greiner and Pratt truss provides optimum geometry.

Conclusion

Maxwell's load path theorem application is presented through two example problems ( Vertical bracing and Bridge truss). It provides a first step in efficient geometry generation for truss structures.

References

1. Maxwell, J. C. (1864). On reciprocal figures, frames, and diagrams of forces. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 27 (182), 250–261.
2. William F. Baker, Lauren L. Beghini, Arkadiusz Mazurek, Juan Carrion and Alessandro Beghini (2015). "Structural Innovation: Combining Classic Theories with New Technologies," Engineering Journal, American Institute of Steel Construction, Vol. 52, pp. 203‐217.
3. DTUdk YouTube Channel - William F. Baker: "On the Harmony of Theory and Practice in the Design of Tall Buildings"