Most structural engineers learn about live load reduction (LLR) for the first time in school. If you are lucky, you may even be presented with the basic equation for LLR from ASCE 7 (ASCE 7-16: Minimum Design Loads and Associated Criteria for Buildings and Other Structures) and given a simple problem to work through, where the reduced design live load can be determined as follows:
L=L0 (0.25+0.15√(KLLAT))
Eq. 1
where L0 is the unreduced live load, KLL is the element factor, and AT is the tributary area. In the author’s case, he did not learn about LLR until his first job. He was asked to perform LLR on a 12-story boomerang-shaped building with irregular grids, curved slab edges, elevator core walls, and many transfer girders. To make things worse, the firm was not using analytical software that could compute live load reduction. So it had to be done manually. And, of course, it had to be done fast and efficiently because the project was already over budget and needed to make up time in the schedule.
LLR is important for column designs but less critical for walls as they are typically part of the gravity and lateral load resisting systems, and the lateral forces generally govern their designs. It is, however, important to consider the walls as they relate to determining the tributary areas to the columns. Beams can be reduced per ASCE 7; however, the tributary areas of such elements are generally low enough that it is common not to reduce the live load for them unless there are many repetitive beams where large material savings can be had.
Some software packages provide accurate results for live load reduction for certain structural geometries using a stiffness-based approach. However, while this is an acceptable method, it does not correlate well with how the building code defines tributary areas and the associated reduction calculation. Additionally, many software packages do not adequately address transfer elements and walls. And finally, having a hand method for back-checking software is always an important tool for structural engineers to ensure model accuracy.
This article provides a systematic approach for determining the tributary areas of columns for any common structural configuration to perform the calculations for LLR as presented in ASCE 7. The method presented can be done graphically by hand or using computer-aided drafting (CAD) software. The method begins with picking appropriate supports. Then the minimum length lines are drawn between those supports. Next, the support lines are bisected, creating the borders for the tributary areas. Finally, the areas are drawn, and the square footage is calculated, which gives the tributary areas for each particular support at any level. From here, the method in ASCE 7 is followed to determine the live load reduction.
The Method
Defining some basic symbols and drawing standards is important to begin an overview. Figure 1 shows symbols for supports, support lines, perpendicular bisectors, overlap regions, and tributary areas. This set of symbols is used throughout this article.
The method is based upon the Perpendicular Bisector Theorem of a Triangle taken from geometry, which states that the perpendicular bisector lines for each side of a triangle always intersect at a common point, called the circumcenter (Euclid & Heath, 1956). Furthermore, this point is exactly equidistant from the triangle’s three vertices (A, B, and C) (Figure 2).
It also can be shown that every point along a given bisector is equidistant from the two vertices of the side of the triangle being bisected. Finally, it should be noted that for right triangles, the circumcenter always falls on the midpoint of the hypotonus of the triangle. This situation often arises when using this theorem for LLR.
Step 1: Picking Supports
The principle of live load reduction is to reduce the design load for the supports that carry the load vertically from one floor down to the next. Supports in building structures are usually columns and walls. For columns, a point support is assumed at the center of each column. Graphically, we can use the support symbol (Figure 1) and draw a support at each column and assign it a unique number S1,S2,…Sn (Figure 3a and 3b).
Walls are more complicated. Often, it is assumed that walls are a continuous line support along the entire length of the wall. This is generally a good assumption for most structural analyses; however, in this method, this assumption makes the problem more difficult. The key is discretizing the line support into a series of point supports (Figure 4). The question arises: how many supports does one need to include for the analysis to be accurate? Typical walls should assume a minimum of three supports between column grids. For long walls, more supports should be used that are equally spaced along the length of the wall.
Step 2: Generating Minimum Support Lines
After the supports have been determined, support lines are constructed between every combination of two supports La,b, where a and b are the support numbers (Figure 3c). The method relies on the fact that no lines intersect except at the support points (or endpoints of the support lines). If an intersection occurs in the construction of the lines, the smaller distance line is selected, and the other is discarded. This ensures that the correct load path for any point on a slab is to the nearest support. In the case of Figure 3c, L2,3<L1,4 and therefore L2,3 is kept and L1,4 is discarded.
Step 3: Constructing Perpendicular Bisectors and Tracing the Tributary Areas
After generating the minimum length support lines, the perpendicular bisectors can be constructed, and the circumcenter determined for each triangular support region (Figure 3e). For perimeter support lines, no triangle exists. In these cases, the perpendicular bisectors are extended until they hit the edge of the slab/floor under consideration. After constructing the bisectors, tributary areas can be traced along the bisectors and slab/floor edges to form the tributary area polygon for each support (Figure 3f). The areas of these regions are reported in most CAD software packages. They can also be calculated using various geometry and calculus techniques; however, this is beyond the scope of this article.
Dealing with Rectangular Support Layouts
In common rectilinear grid systems, the lengths of the diagonal lines (hypotenuses) equal each other (Figures 5a and 5b). To deal with this, the engineer simply chooses the line they want to keep and removes the other. The solution does not change and yields the same tributary areas (Figures 5a and 5b).
Dealing with Overlap
For rare column layouts, selecting the smallest length support line does not produce the correct tributary area. This can be demonstrated in Figures 6a and 6d, where L1,4<L2,3. The perpendicular bisectors are constructed; however, overlap occurs on the perpendicular bisector, which produces overlap in the tributary areas (Figure 6b). This phenomenon can be explained using Delaunay triangulation, which is beyond the scope of this graphical method (Delaunay, 1934). To address this, the engineer has two options, redraw the larger support line with length L2,3 and delete the shorter length line L1,4, and then proceed as described above (Figure 6d, e, and f), or construct a modified perimeter line that connects the vertices that are adjacent to the overlap vertices within the overlap polygon region (Figures 6a, b, and c). Figures 6c and 6f show that both approaches produce the exact same tributary area.
Dealing with Transfers
Transfer elements are becoming more common as architects push the boundaries of open floor plans with fewer columns. However, given the criticalness of transfer elements, the engineer may choose not to use live load reduction to design supporting columns. If one does choose to use LLR, this method provides a simple way to determine tributary areas.
Figure 7 shows a building of n levels with a typical transfer element at some level k, where a column line at grid B-3 shifts laterally to a different location at grid B-2 within the structure. When this occurs, the tributary load from above is now split between two columns (B-2 and B-4). In computing the amount of tributary area to each respective column line supporting the transfer girder, one simply needs to use the ratio of the location of the supported (upper) column with respect to each supporting (lower) column. In mathematical terms, the total tributary areas AT,column(grid),j for each respective column supporting the transfer element at some lower level j below the transfer level k can be calculated as follows:
where At,column(grid),i is the tributary area to a column at the ith level. The tributary areas can be constructed for each level using the same techniques as presented above.
Conclusions
Live load reduction is an important technique for reducing column sizes in mid to high-rise buildings. ASCE 7 provides a method using the tributary areas for columns to determine the appropriate reduction. However, determining the tributary areas is difficult for complex structural layouts. This article presents a systematic approach for determining the tributary areas for buildings with irregular geometries (slab shapes and column layouts), walls, and transfer levels for determining the live load reduction for columns. The method can be done graphically and is easily implemented within a structural engineer’s design workflow.
References
ASCE 7-16: Minimum Design Loads and Associated Criteria for Buildings and Other Structures. (2017). American Society of Civil Engineers.
Delaunay, B. (1934). Sur la sphère vide. Bulletin de l’Académie des Sciences de l’URSS, Classe des Sciences Mathématiques et Naturelles(6), 793–800.
Euclid, & Heath, T. L. (1956). The thirteen books of Euclid’s Elements. New York: Dover Publications.
RAM Concept User Manual. (2021). Bentley