In three previous articles the authors began the discussion of the use of various diaphragms in analysis software, including: 1) No diaphragm/flexible diaphragm; 2) Rigid diaphragm; 3) Semi-rigid diaphragm; and 4) Pseudo-flexible diaphragm with simplified methods. This article will focus on semi-rigid diaphragms, including a discussion of when the use of rigid diaphragms would not apply and, therefore, a semi-rigid diaphragm approach would be more appropriate. Semi-rigid diaphragms represent the most “complicated” analysis method for diaphragms because of the use of plate/shell elements in the analysis.

The rigid diaphragm is a convenient analytical technique for distributing the lateral forces to the frames and walls; forces are distributed to those elements as a function of their relative stiffnesses and position. Analysis using the rigid diaphragm assumption is generally adequate when the diaphragm in-plane stiffness is high relative to that of the frames. There are some circumstances, however, where the rigid diaphragm assumption may not be appropriate: floors with numerous openings, roof diaphragms of metal decking without concrete fill or of plywood sheathing, etc. Long, narrow diaphragms may be considered rigid in one direction but not in the other. For structures with multiple wings, such as L- or C-shaped buildings where the ends of the wings can drift independently of each other, the rigid diaphragm analysis may not be appropriate since it would lock the ends of the wings together, constraining them to move in unison. In these cases it may be necessary or required to analyze the structure modeled with semi-rigid diaphragms. It is often appropriate to analyze some stories using the rigid diaphragm assumption and other stories using the semi-rigid assumption.

Building codes often have prescriptive requirements that indicate when it is necessary to analyze a structure using the semi-rigid diaphragm assumption. See for example Section 12.3.1 of ASCE 7-10, which states: “Unless a diaphragm can be idealized as either flexible or rigid in accordance with 12.3.1.1, 12.3.1.2, or 12.3.1.3, the structural analysis shall explicitly include consideration of the stiffness of the diaphragm (i.e., semi-rigid modeling assumption).” The document indicates in part that a diaphragm can be considered “flexible” if it is constructed of untopped steel decking or wood structural panels and the structural system is steel, concrete or masonry frames or walls, or when the maximum in-plane deflection of the diaphragm under lateral load is more than two times the average drift of the adjacent lateral frames. In contrast, it specifies that a diaphragm can be considered “rigid” if it is a concrete slab or concrete-filled metal deck, with some limitations. Otherwise, the diaphragm must be modeled and analyzed as semi-rigid.

Figure 1: Diaphragm model.

It is interesting to note that while IBC 2012 categorizes flexible diaphragms the same way that ASCE 7-10 does (in Section 202 of IBC 2012 the definition of “Diaphragm, Flexible” states that a diaphragm is flexible where so indicated in Section 12.3.1 of ASCE 7-10), it categorizes rigid diaphragms very differently than ASCE 7-10. In Section 202 of IBC 2012 the definition given for “Diaphragm, Rigid” states: “A diaphragm is rigid… when the lateral deformation of the diaphragm is less than or equal to two times the average story drift.”

Hence, per the IBC, all diaphragms can be considered as either flexible or rigid, with the definition of flexible (through ASCE 7-10, Section 12.3.1.3) being any diaphragm for which the in-plane deflection is greater than two, and the definition of rigid being any diaphragm for which the in-plane deflection is less than or equal to two. Simply put, the IBC requires that you categorize the diaphragm by comparing diaphragm deflection to frame deflections: if that ratio is more than two the diaphragm can be analyzed as flexible; if that ratio is less than or equal to two the diaphragm can be analyzed as rigid. Hence, no diaphragms are required by the IBC to be analyzed as semi-rigid, although it may be good engineering judgment to do so for the reasons already mentioned. It may also be more practical to analyze the diaphragms as semi-rigid and avoid the need to perform the deflection calculations necessary to classify the diaphragms otherwise. It is permissible in any case to model and analyze the diaphragm as semi-rigid.

When modeling with a semi-rigid diaphragm, it is necessary to indicate the appropriate in-plane properties, including thickness, modulus of elasticity and Poisson’s ratio. Determination of these properties is fairly simple for diaphragms consisting of concrete slabs or concrete fill on metal deck by using the concrete properties and slab thickness, to which some engineers also apply a crack factor. Determination of these properties for diaphragms of other materials, such as roof diaphragms of metal decking without concrete fill or of plywood sheathing is not so straightforward, and can be rather laborious. The diaphragm must also be meshed into finite elements. Note that a rather coarse mesh is generally adequate since the purpose of the analysis is to get the structural behavior or the diaphragm forces along a plane, rather than the highest localized diaphragm stresses. The structure shown in Figure 1 was analyzed in Bentley Systems’ RAM Frame, the lateral analysis module of the RAM Structural System. This example structure was selected because of the asymmetry of the frames and the narrow diaphragms through the center of the floor plan.

The structure was analyzed several times, using a range of mesh sizes. Figures 2 and 3 show the diaphragm meshes at the roof for the 1-foot mesh and the 30-ft. mesh, respectively. Note that meshes of these dimensions are two extremes and are not generally recommended.

 Maximum Mesh Size Level 30′ 16′ 8′ 4′ 2′ 1′ Roof 33.55 [-0.6%] 32.63 [-0.3%] 32.80 [+0.2%] 32.77 [+0.1%] 32.73 [-0.0%] 32.74 [0.0%] 4th 63.19 [-2.7%] 63.97 [-1.5%] 64.33 [-1.0%] 64.70 [-0.4%] 64.85 [-0.2%] 64.96 [0.0%] 3rd 85.74 [-3.0%] 87.02 [-1.6%] 87.64 [-0.9%] 88.09 [-0.4%] 88.27 [-0.1%] 88.40 [0.0%] 2nd 103.43 [-1.4%] 104.33 [-0.5%] 104.38 [-0.5%] 104.68 [-0.2%] 104.77 [-0.1%] 104.87 [0.0%]
Table 1
Frame story shear for Frame 1.

Figure 2: 1-foot mesh at roof.

Table 1 shows the story shears (kips) at each story of Frame 1 (the frame on the right side of the structure as viewed in Figure 1, on the left side as viewed in Figures 2 and 3) for several mesh sizes ranging from a maximum of 1 ft. to a maximum of 30 ft.

Figure 3: 30-ft. mesh at roof.

Using the 1-ft. mesh as a baseline, the values shown in brackets are the percent difference between frame story shears for the 1-ft. maximum mesh size and the listed maximum mesh size. For this model, it is shown that the size of the mesh used in the analysis has very little impact on the distribution of the lateral forces; the analysis is not sensitive to the mesh size. It is instructive to point out large differences in analysis times: 8 seconds for the 30-ft. mesh, 17 seconds for the 8-ft. mesh, and over 25 minutes for the 1-ft. mesh. From this exercise, it is seen that a small refined mesh isn’t necessary, and that a reasonably large mesh is acceptable with the benefit of substantially faster analysis times. For comparison, note that the same analysis using the rigid diaphragm assumption took less than 2 seconds.

In building structures with well-distributed frames with adequate stiffness and stability, the analysis is not particularly sensitive to the diaphragm properties used in the analysis. Using the model with an 8-ft. mesh as the baseline, four analyses were performed using varying diaphragm stiffness properties. The resulting frame story shears are shown in Table 2.

 Diaphragm Stiffness Level 0.5 x E’ E’ 2.0 * E’ Rigid Roof 32.55 [-0.2%] 32.63 [0.0%] 32.80 [0.5%] 32.77 [0.4%] 4th 63.19 [-1.2%] 63.97 [0.0%] 64.33 [0.6%] 64.70 [1.1%] 3rd 85.74 [-1.5%] 87.02 [0.0%] 87.64 [0.7%] 88.09 [1.2%] 2nd 103.43 [-0.9%] 104.33 [0.0%] 104.38 [0.0%] 104.68 [0.3%]
Table 2
Frame story shear for Frame 1.

In this table, E’ is the effective modulus of elasticity representing the stiffness of the baseline model. The model was modified and analyzed using a stiffness of one-half of that of the baseline model for the diaphragms at each level, and then modified and analyzed again using a stiffness of twice that of the baseline model for the diaphragms at each level. Note that despite the large variation in the stiffness values, the resulting frame story shears vary very little from those of the model using the actual stiffness. This indicates that the results are not sensitive to the values used to define the diaphragm stiffness, and even a large error will have little impact on the resulting designs.

For comparison, the model was analyzed again with the diaphragms defined as rigid. Even for this model with narrow diaphragms, the differences in the frame story shears between the semi-rigid diaphragm and the rigid diaphragm are not more than 1.2 percent.

It is important to stress that the conclusions drawn here regarding mesh size and diaphragm properties are valid for the example model used, but may not be valid for any particular model. It is recommended that the engineer initially experiment and evaluate using different meshes and properties in order to be satisfied of the suitability of the values used. Generally, it is recommended to use a larger mesh size in order to reduce analysis time as long as it doesn’t compromise the design of the structure.

In some cases, an analysis using a semi-rigid diaphragm is necessary because of code requirements or slab configurations and properties. Computing tools are now capable of handling these larger and more complex models, and with the proper settings and values these analyses can provide robust structural designs. It is important to remember, however, that this level of sophistication and complication is not always necessary, and adequate designs can often be obtained more quickly, with simpler output, and with less effort using the assumptions of flexible diaphragms or of rigid diaphragms, when those assumptions are acceptable.

Allen Adams, P.E., S.E., is chief structural engineer at Bentley Systems, Inc. He can be reached at Allen.Adams@Bentley.com. Brian Quinn, P.E., is president and founder of SE Solutions, LLC, and Lisa Willard, P.E., is vice president at SE Solutions, LLC. They can be reached at Brian.Quinn@LearnWithSEU.com and Lisa.Willard@LearnWithSEU.com, respectively. Visit their SE University website, www.LearnWithSEU.com