Analysis of irregular-shaped diaphragms

Address design issues to ensure complete load paths throughout structures.

By R. Terry Malone, P.E., S.E.

The structural configurations of many modern buildings require complex lateral load paths that incorporate diaphragms at different elevations, multiple re-entrant corners, multiple irregularities, and fewer vertical lateral force-resisting elements. It is important to address these design issues and irregularities to ensure complete load paths throughout the structure; however, this doesn’t have to be a daunting task.

Knowledge regarding the analysis of complex diaphragm layouts varies greatly within engineering and code enforcement communities. In some cases, it has become standard practice to treat all structures as if they were simple rectangular diaphragms, and the absence of continuous load paths, presence of discontinuities, and missing elements such as chords, collectors, and drag struts are commonly overlooked. This is largely due to the lack of concise information on how to design complex diaphragms.

The purpose of this article is to bridge that information gap by providing an overview of a method, based on simple statics, that can be used to analyze complex diaphragm structures, while guiding readers to more detailed information through the references.

Principles of effective diaphragm design

For simple and complex buildings, the load path for lateral loads typically includes:

• development of the seismic or wind loads in the roof or floor diaphragms;
• transfer of loads acting on the diaphragms to diaphragm boundary members and internal collectors;
• collection of loads along the length of boundary members to the shear walls;
• transfer of forces within the diaphragm across areas of discontinuity;
• transfer of loads through shear walls to the resisting elements; and
• resistance of loads at foundation to soil interface.

Integral parts of the load path for even simple rectangular buildings are chord elements, which resist the bending or moment action within a diaphragm, and struts and collectors, which collect and transfer the diaphragm shears to the shear walls or frames as shown in Figure 1. All edges of diaphragms must have boundary members consisting of drag struts, chords, collectors, or vertical lateral force-resisting elements. Boundary members include chords and drag struts at diaphragms and shear wall perimeters, interior openings, discontinuities, and re-entrant corners. Collector elements must be capable of transferring the seismic or wind forces originating in other portions of the structure to the elements providing resistance to those forces.

Following are two key definitions relative to establishing complete load paths and the analysis and design of complex diaphragms and shear walls:

Diaphragm boundary — In light-frame construction, a location where shear is transferred into or out of the diaphragm sheathing. Transfer is to either a boundary element or to another force-resisting element.

Drag strut; collector — A diaphragm or shear wall element parallel and in line with the applied load that collects and transfers diaphragm shear forces to the vertical elements of the lateral force-resisting system and/or distributes forces within the diaphragm.

All of these components of the load path, including connections, help to ensure effective performance during a significant wind or seismic loading event.

For the purpose of clarification on the use within this paper, drag struts and collectors function in the same manner and are therefore the same thing — collector elements. It is the author’s preference to designate a collector element that receives shears from one side as a “drag strut.” A collector element that receives shears from both sides is designated as a “collector.”

Discontinuities and irregularities

Discontinuities in diaphragms are often created when a portion of an exterior wall line is offset from the main wall line, causing a disruption in the diaphragm chord or strut. When this occurs, the disrupted chord or strut force must be transferred across the discontinuity through an alternate load path. It is important to remember that, at diaphragm discontinuities such as offsets, openings, or re-entrant corners, the design must assure that the dissipation or transfer of edge (chord) forces combined with other forces in the diaphragm is within the shear and tension capacity of the diaphragm. All irregularities and/or discontinuities within a system of diaphragms and shear walls must be addressed.

The term “combined with other forces” is frequently misunderstood. The main diaphragm is already under shear force from the applied loads. Additional shear force is applied at the transfer area from the discontinuous member force. These shears must be combined with the basic diaphragm shear to be in compliance with code.

The example diaphragm shown in Figure 2 contains many discontinuities and irregularities commonly seen in modern designs. When highly irregular diaphragms are viewed as a whole, a rational design of the lateral force-resisting paths may seem daunting; however, when approached one section at a time, keeping in mind the statics approach outlined below, a robust design can be developed.

Method of analysis

For practical purposes and ease of demonstrating the method, the simple diaphragm shown in Figure 3 will be reviewed. For those same reasons, the orientation of the main framing members must also be ignored. Diaphragm dimensions and loading are provided in the figure along with a complete set of calculations in the full article at www.woodworks.org/irregular_diaphragms. The diaphragm shown has a single horizontal (end) offset at the left support. The offset causes a discontinuity in the diaphragm chord.

To successfully transfer forces through areas of discontinuity, it is important to understand how shears are distributed into and out of a diaphragm. The discontinuous chord shown at grid line B/2 is typically extended into the main body of the diaphragm with the use of a continuous light-gauge steel strap and flat blocking between the framing members. The intent is to overlap and transfer the disrupted chord force into the main chord at grid line C. The strap is lapped onto the discontinuous chord and is then applied over the sheathing and blocking within the main diaphragm. The distance the strap is extended into the diaphragm will depend on the results of the calculations (discussed later in this paper), practicality, and engineering judgment.

When the chord along line C is under tension, the transfer area lying between grid lines B/2 and C/3, shown as a dashed rectangle in Figure 3, could rotate if there is no means to resolve the rotational movement from the chord forces. Framing members at grid lines 2 and 3 can be designed to oppose the rotating couple forces. Local over-stressing and deformation in this area of the diaphragm can occur if resolution of the rotational forces is omitted. The free-body diagram at the right of the figure illustrates a simple method that not only eliminates rotational problems but also provides a complete load path that complies with code requirements.

Briefly, the method utilizes a portion of the diaphragm to the right of the discontinuity as a sub-diaphragm, or transfer diaphragm (TD), which receives the disrupted chord force and distributes it out to the main diaphragm chords at grid lines A and C by beam action. The transfer diaphragm acts like a beam with a concentrated load applied as depicted by the inset diagram. This method of analyzing diaphragms with offsets and openings was developed in the early 1980s.

Figure 4 provides an overview of the method of analysis. Typical symbols for 1-foot by 1-foot pieces of sheathing called “sheathing element symbols” are shown on the lower left. The directions of the shears applied at each edge of the elements indicate whether the shears are positive or negative. The basic shear diagram, expressed in pounds per linear foot (plf), is plotted below the diaphragm. The diaphragm unit shears (plf) at grid lines 1, 2, and 3 are determined and designated as positive or negative by placing the appropriate sheathing element symbols on the basic shear diagram as shown in the figure.

The main diaphragm and transfer diaphragm area are already under shear from the uniform load, as calculated in the basic shear diagram. Additional transfer diaphragm shears are created by the disrupted chord force. The transfer diaphragm shears must be added to or subtracted from the basic diaphragm shears to accurately account for the combined localized effects within the transfer diaphragm, resulting in net shears occurring within the transfer diaphragm area. This is what is intended by the ASCE 7 requirement of “combined with other forces.” The transfer diaphragm is the only area affected by combined shears. The unit shears in the areas outside the transfer diaphragm remain unchanged.

The transfer diaphragm shears are determined by analyzing the transfer diaphragm as a simple span beam with a concentrated load. Since the disrupted chord force acts to the left, the reactions of the analogous beam will act to the right as shown on the right side of the figure. The unit shears in the transfer diaphragm caused by the disrupted chord force are equal to the calculated reactions divided by the depth of the transfer diaphragm (DTD). The key in determining if the shears in the transfer diaphragm are positive or negative is to understand that the unit shears just determined and the direction of the arrows are acting on the edge of the sheathing element and not on the outer diaphragm chords. Placing the sheathing element symbols next to the unit shears at the transfer diaphragm reaction area and completing the direction of the shears acting on the other edges of the sheathing element symbol will determine whether the shears are positive or negative.

The final transfer diaphragm shears are determined by adding or subtracting the transfer diaphragm shears from the basic diaphragm shears.

R. Terry Malone, P.E., S.E., is senior technical director, WoodWorks – Wood Products Council. Read the rest of this article at www.woodworks.org/irregular_diaphragms. WoodWorks offers free project support as well as education and resources related to the code-compliant design of commercial and multifamily wood buildings across the U.S. Visit www.woodworks.org to find the technical expert nearest you or email the WoodWorks Project Assistance Help Desk at help@woodworks.org.