Study on the Sensitivity of Mesh Size and Properties in ...

2020 SEAOC CONVENTION PRCEEDINGS

Study on the Sensitivity of Mesh Size and Properties in the Analysis of Semirigid Diaphragms

Allen Adams, P.E., S.E., Chief Structural Engineer Bentley Systems, Inc.

Carlsbad, CA Abstract The behavior of a building structure in response to an earthquake or wind, including the distribution of the lateral forces to the lateral force resisting elements such as moment frames, braced frames or shear walls, is heavily dependent upon the nature, extent and stiffness of the floor and roof diaphragms. It is important that the building model used in the analysis and design of a building accurately – or adequately – captures the effects of these diaphragms. A study to determine the sensitivity of mesh size and diaphragm properties on the analysis results was undertaken, involving several analysis models. These models were investigated using a wide range of diaphragm mesh sizes and material properties, notably the diaphragm effective E. The models were also analyzed employing the rigid diaphragm assumption. The resulting frame story shears, diaphragm forces, and drifts from each of these scenarios were compared, to determine the sensitivity of the analysis to these diaphragm parameters, to identify the degree to which the results change for a given change in properties. The results of this study are presented, with useful insights and practical recommendations based on the results. Introduction ASCE 7 has extensive requirements related to diaphragms, but confidently determining the proper way of performing the necessary analyses that satisfy those requirements can be challenging. Increasingly, structures are required to be analyzed with the diaphragms modeled as semirigid diaphragms. Determination of the properties to assign to these diaphragms in the analysis can be difficult, tedious, and time-consuming, with low confidence in their suitability. Historically diaphragms have been considered as either Flexible or as Rigid because they accommodated hand calculations and simple analysis in computers with limited resources. Even when the shortcomings of those

assumptions were recognized for particular building configurations, computers lacked the speed to perform more robust diaphragm analyses. That is no longer the case, and as a result the Building Codes have taken a more stringent approach to what is permitted in regard to diaphragms, especially for seismic analysis. Section 12.3.1 of ASCE 7-16 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., semirigid modeling assumption).”

ASCE 7-16 Section 12.3.1.1 permits a diaphragm to be considered Flexible when the diaphragm consists of untopped steel decking or wood structural panels in structures where the lateral force resisting system is steel or composite steel braced frames, or concrete, masonry, steel or composite steel shear wall. These braced frame and shear wall systems are explicitly identified because they are very stiff relative to an untopped steel deck or wood panel floor system. Diaphragms in one- and two-family dwellings, and structures of light-frame construction – with some restrictions – are also permitted to be considered Flexible. Section 12.3.1.3 further permits any diaphragm to be considered Flexible if the deflection of the diaphragm is greater than two times the drift of the adjacent frames. See Figure 1.

Figure 1

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Conversely, in Section 12.3.1.2 the only diaphragm condition that is permitted to be considered Rigid is diaphragms of concrete slabs or concrete-filled metal deck with diaphragm span-to-depth ratios of 3 or less in structures that have no horizontal irregularities. All other diaphragms, according to Section 12.3.1, are required to be considered as Semirigid. This represents a significant portion of structures. Although historically the influence of diaphragms was given lesser consideration, and simplified approximated methods were allowed due to the need to use more simple hand and analytical techniques, increasingly the structures designed and built today need more robust analysis. Incorporation of semirigid diaphragms in the analysis is part of that. Determination of the properties to assign to these diaphragms in the analysis can be difficult, tedious, and time-consuming, with low confidence in their accuracy and suitability. Several case studies were undertaken to determine the sensitivity of the diaphragm meshing and properties to the analysis results, and the suitability of simplifications and approximate values. This would give an indication of the degree of effort warranted to get adequate design results. Influence of Mesh Size Two case studies were performed to measure the influence that the diaphragm mesh size has on the analytical results. The first case study consisted of a four-story structure with an asymmetric layout of steel moment frames. The diaphragm was a very common 3-1/4” light-weight concrete fill on 3” metal deck. The second case study was identical except that the frames were braced frames rather than moment frames, providing a comparison of the impact that the frame stiffness has on the diaphragm behavior. Case Study 1: The model used for Case Study 1 is shown in Figure 2.

Figure 2

The configuration was chosen to exaggerate the role that the diaphragm plays in the lateral analysis; the asymmetric frame layout and the narrow diaphragm through the mid-section forces the diaphragm to act as a ‘bridge’ between frames. The frames are laid out as shown in Figure 3, with the frames parallel to the Y-axis labeled Frame 1 through Frame 3 and the frames parallel to the X-axis labeled Frame 4 and Frame 5.

Figure 3

The diaphragm at each level has the configuration shown in Figure 4.

Figure 4

The diaphragms were assigned to be Semirigid diaphragms, and the model was analyzed using five different mesh sizes: 1’, 2’, 4’, 8’, and 15’ meshes. For comparison, the diaphragm was also analyzed as Rigid. The analyses were performed using the RAM Structural System, but any software capable of analyzing semirigid diaphragms could have been used. The 1’ mesh is shown in Figure 5.

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Figure 5

The 15’ mesh is shown in Figure 6. Note that the bays are 30’, so the 15’ mesh is half of the bay size.

Figure 6

As a measure of the influence of the mesh size on the distribution of the applied lateral story forces, the total shear at each story was determined and tabulated for each frame. The results for Frame #1 from the model analyzed with the 1’ mesh are shown in Figure 7.

Figure 7

The results for each of the mesh sizes for each frame are shown in Table 2. It is generally assumed that the finer the mesh, the more accurate the results. That is not necessarily true, but a discussion of that is beyond the intent of this paper. For purposes of this paper it is assumed that the finer mesh (in this case the 1’ mesh) is giving the most accurate results; shown in parentheses is the percent difference between the given value and the corresponding value from the 1’ mesh. It is remarkable to note that the degree of error between the 1’ mesh and the worst mesh, the 15’ mesh, is less than 1%. For example, the greatest difference is at the 5th floor of Frame 3: the story shear for the 15’ mesh varies from that of the 1’ mesh by 0.7%. That percentage is merely differences at the first decimal place (86.79 kip vs. 87.40 kip). This indicates that for this structure, the mesh size has virtually no impact on the accuracy of the analysis. In contrast, the frame story shears vary by as much as 11% between the rigid diaphragm analysis and the semirigid diaphragm analysis with the 1’ mesh. This supports the more-restrictive prohibition of use of the Rigid diaphragm analysis by more recent versions of ASCE 7. As a practical consideration it is instructive to note the large differences in the analysis times between the various mesh sizes, shown Table 1.

Table 1 – Analysis Time

Mesh Size Analysis Time 1’ 5:41 min 2’ 49 sec 4’ 13 sec 8’ 4 sec 15’ 2 sec

Rigid 1 sec This can be significant during the design process when models are repeatedly analyzed and refined. A significant amount of time can be saved by using a larger mesh without significantly sacrificing accuracy.

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Table 2 – Frame Story Shear – Case Study 1

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Case Study 2: It was recognized that the behavior of the diaphragm is dependent upon the stiffness of the frames with which it is interacting. To measure the magnitude of that impact, the model in Case Study 1 was modified; in this case study the frames are braced frames rather than moment frames. The frames are shown in Figure 8.

Figure 8

In every other respect the model used in Case Study 2 is identical to the model used in Case Study 1. As in Case Study 1, the frame story shear was tabulated for each frame for each mesh size. Those results are shown in Table 3. While for Case Study 1 with moment frames the largest variation between the 1’ mesh and the 15’ mesh was 0.07%, for Case Study 2 with braced frames the largest variation between the 1’ mesh and 15’ mesh was 1.5%. While this is still a relatively insignificant difference, it does indicate that the mesh size has a bigger impact when the frames are the stiffer braced frames than when the frames are the more flexible moment frames. The conclusion that can be drawn here is that the greater the difference between the stiffness of the diaphragms and the stiffness of the frames, the greater the sensitivity of the mesh size in the analysis. The error between the Rigid diaphragm and the Semirigid diaphragm is significant, nearly 60%. This shows that for some configurations the simplifying Rigid diaphragm assumption is a poor assumption, resulting in unacceptable degrees of error. Continuing with the model used in Case Study 2, the impact of diaphragm mesh size on story drift was investigated. Drift was determined and tabulated for the center of the layout as well as the extreme corners as shown in Figure 9.

Figure 9

The drifts for each level at these points were tabulated for each mesh size. This was done for the drifts in both the global X and Y directions. See Table 4. Again, the value in parentheses indicates the percent difference for that mesh size versus the 1’ mesh. The error in the story drift values for the large meshes are a few percentage points. This is likely due in part to the fact that the points at which drift is being measured are in the diaphragm, so the drift values have a component based on the deformation of the diaphragms independent of the displacement of the frames; the larger mesh is stiffer and thus results in smaller drifts. But at approximately 6%, these differences even with the coarsest mesh are probably not significant. Again, the Rigid diaphragm is a poor predictor of the story drifts. Finally, a representative section through the diaphragm at the top story was chosen, and the shear, moment, and axial force across that section were tabulated for each of the mesh sizes. The section cut was taken at the location shown in green in Figure 10.

Figure 10

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Table 4 – Drift – Case Study 2

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Table 5 – Diaphragm Forces – Case Study 2

The forces through that section cut are tabulated in Table 5, for the seismic load cases in both the X (E1) and Y (E2) directions. The 15’ mesh is showing some fairly large discrepancies, although on small force values. Otherwise, the variation in mesh size has little impact on the diaphragm forces at that location. Influence of Diaphragm Properties The determination of diaphragm properties for concrete slabs and concrete on metal deck is straightforward. In an analysis of semirigid diaphragms it is necessary to specify the diaphragm thickness, modulus of elasticity E, and poisson’s ratio. Generally, for concrete slabs the values are based on those of the concrete; cracked factors are often applied. For concrete on metal deck it is customary to use the properties of the concrete fill, with the diaphragm thickness defined as the thickness of the concrete fill plus half of the rib depth, or even as merely the thickness of the concrete fill. The contribution of the metal deck is generally ignored. However, for diaphragms of metal deck without fill, such as at a roof, or of wood panels, the determination of an effective modulus of elasticity can be tedious and difficult. The stiffness of a diaphragm of metal deck, for example, is a function of several things: fastener type, size, and spacing; deck profile; deck thickness; and deck span. Shown in Figure 11 is a page from the Vulcraft Roof Deck catalog indicating the calculations required. Furthermore, different areas of the roof plan are likely to have different deck configurations due to difference in deck spans or connection requirements, with the corresponding differences in the diaphragm stiffnesses. The final case study was conducted to determine the sensitivity to the

analysis of the diaphragm properties. If the analysis is not sensitive to the properties, simplifications such as assigning a single representative set of diaphragm properties to the entire roof may be acceptable; precise complicated diaphragm stiffness calculations may be avoided.

Figure 11

Case Study 3: For this case study a single-story structure was selected, similar in shape and configuration to the previous case study models, with braced frames, see Figure 12. The diaphragm was modeled as a Semirigid diaphragm with a 22-gage roof deck (thickness = 0.0299”).

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Figure 12

The braced frames were arranged in the configuration shown in Figure 13, with the frames parallel to the Y-axis labeled Frame 1 through Frame 3 and the frames parallel to the X-axis labeled Frame 4 through Frame 7.

Figure 13

The diaphragm is as shown in Figure 14.

Figure 14

The model was analyzed using seven widely different values of the diaphragm effective modulus of elasticity, E: 500 ksi, 1000 ksi, 2000 ksi, 4000 ksi, 8000 ksi, 12000 ksi, and 24000 ksi. For comparison, the diaphragm was also analyzed as Rigid. Depending on the spans, deck profile, and deck connection, there is a “correct” value of E, but that wasn’t calculated as part of this study; rather the focus is on these theoretical values and a comparison of resulting differences in the analysis. 500 ksi was arbitrarily chosen as the baseline, but not because it is the “correct” value (it is almost certainly not). A 4’ mesh was used for this study. For each frame, the Frame Story Shear was tabulated for each value of Effective E. The values are shown in Table 6, the values in parentheses indicates the percent deviation from the story shears resulting from the value of E of 500 ksi. Again, remarkably even with a variation in Effective E of 500 ksi to 24000 ksi (a factor of 48), the frame story shears vary by less than 4%. This indicates that any value between 500 ksi and 24000 ksi could be used in this analysis in this model and the results would be off by less than 4%.


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The results from the Rigid diaphragm analysis vary significantly from the Semirigid analyses and indicate why it is rarely acceptable to analyze a roof as Rigid. A representative section through the diaphragm was chosen, and the shear, moment, and axial force across that section were tabulated for each of the mesh sizes. The section cut location is shown in green in Figure 15. The forces through that section cut are tabulated in Table 7 for the seismic load cases in both the X (Load Case E1) and Y (Load Case E2) directions.

Figure 15

Between the smallest E and largest E the diaphragm moment varies 23%. Considering the magnitude of the difference between the smallest and largest E (a variation of 48X), this is still a relatively small discrepancy, indicating that a large error in the Effective E value will have a small impact on the diaphragm forces. A more useful comparison is between, say the results for the values of 4000 and 8000: based on the respective moments of 505 k-ft and 524 k-ft, if the value of E used in the analysis was in error by 100%, the resulting diaphragm forces would only be off by less than 4%. Based on this, reasonably accurate values of E, even if not exact, will yield adequately accurate diaphragm force values. Story drifts were also investigated and the results tabulated. First, the drifts at two points on the diaphragm that coincide with points directly on the braced frames were selected, as shown in Figure 16. These values are an indicator of the drifts of the frames themselves. The drift values are tabulated in Table 8. Again, the large variation in Effective E has a negligible impact on the drift of the frames.

Figure 16

Then the drifts at points on the diaphragm away from frame members were also investigated, shown in Figure 17.

Figure 17

For these drifts, the value of Effective E used had a significant impact on the results, shown in Table 9. For comparison purposes, the values in parentheses are a fraction rather than a percent. For example, doubling the value of E between 500 ksi and 1000 ksi resulted in drifts that were ½ as large. An E value of 4000 is 8 times as large as 500, and the resulting drifts are approximately 1/8 as much. This indicates that at points on the diaphragm away from the frames the drift is very nearly proportional to the value of E used. However, keep in mind that the analysis did not include the stiffening effects of drags, chords, and gravity framing. Because of that, these results are dubious and probably aren’t indicative of the true drifts nor of the true impact of the Effective E on those drifts at those extreme points.

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Table 7 – Diaphragm Forces – Case Study 3

Table 8 – Drift at Frames – Case Study 3

Table 9 – Drift at Diaphragm Points – Case Study 3

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Table 10 – Frame Story Shears – Case Study 4 – Baseline = 500

Table 11 – Frame Story Shears – Case Study 4 – Baseline = 12000

Case Study 4 Case Study 4 used the same model as Case Study 3, except a response spectra analysis was performed rather than equivalent static forces. The Frame Story Shears for the response spectra analysis are listed in Table 10. Response Spectra analyses tend to exaggerate structural flaws and irregular conditions. In this model the frames in the Y-axis direction are eccentrically placed; as a result the structure is highly torsional and the diaphragm plays a large role in distributing the response of the masses to the frames. These results indicate that for response spectra analysis the diaphragm properties have a much larger impact than for static wind and seismic load case analyses.

For consistency with the presentations in the other studies the values are shown in Table 10 with the percentages indicating the difference between that value and the corresponding value for E = 500 ksi. However, for this case study it is instructive to use one of the other, more likely values as the baseline. For example, assume that the “correct” value of Effective E is 12000 ksi. Using that as the baseline, the degree of error in the frame story shear can be determined if other values of Effective E are assigned. Table 11 shows a portion of Table 10, with 12000 ksi as the baseline, rather than 500 ksi For this model, if a response spectra analysis is being performed, if the correct value of Effective E is 12000 ksi but a value of 24000 ksi is erroneously specified, the results

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would be in error less than 4%; if a value of 8000 ksi is erroneously specified, the results would be in error 13%. This confirms that for response spectra analysis, more accurate diaphragm properties are required but indicates that there is still great latitude on the precision of the values. Conclusions Historically the influence of the diaphragm on the distribution of the lateral forces to the frames was performed manually. Software was unavailable or computers lacked the capacity to perform more rigorous analyses. For simplicity diaphragms were treated as either flexible or rigid, they were rarely rigorously investigated as semirigid. As analysis and design techniques have improved, as there is a tendency to push structures to their capacity limits in order to achieve more economical structures, and with structures becoming more architecturally challenging, it has become increasingly obvious that those old assumptions are in many cases inadequate and unsafe. As a result, the codes have become more reluctant to permit the use of the rigid diaphragm assumption, allowing it only under a narrow set of conditions. Fortunately, software has become more accommodating and computers have become more capable of handling the large analytical models created by the inclusion of semirigid diaphragm elements. These studies indicate to what degree the modeling of the diaphragms impact the analysis results, and provide useful guidance in the modeling effort. Correctly categorize each diaphragm: Flexible, Semirigid, or Rigid. In particular, improperly implementing the Rigid diaphragm model in the analysis model can result in significantly erroneous results. The analysis results are not particularly sensitive to the size of the mesh, but analysis time is. When modeling a diaphragm as Semirigid, use as large a mesh as reasonable: 4 ft or even 8 ft. The analysis time is substantially reduced, with little impact on the accuracy of the results. For very large structures it may be advantageous to use a very large mesh during the preliminary design phase where the model needs to be analyzed and reanalyzed frequently, and then use a more refined mesh for the final analysis and design. Braced frames and shear wall structures are slightly more sensitive to the mesh size than are the less-stiff moment frame structures. When modeling a diaphragm as Semirigid, use reasonably accurate material property values, but don’t waste time in an effort to have a high degree of precision. Be practical. As was shown by the case studies, the analysis results aren’t particularly sensitive to the diaphragm properties. If the

values that are used are reasonably close, that is good enough. Response spectra analysis requires a higher degree of accuracy in the modeling of diaphragm properties, yet even that allows for considerable discrepancies without significantly impacting the accuracy of the results. Don’t be careless and sloppy. Be practical. The results given in this paper are based on a select few models. The most valuable recommendation that can be given is to use the analysis tools to experiment and investigate. Create various models and see how changes in the types of diaphragms and changes in the diaphragm properties impact the analysis results. For the various configurations in the model, look at and compare displaced shapes, building periods, frame story shears, and drift, among others. Doing so will give confidence in the analytical results even when the properties are not known exactly; it will enable the engineer to work more practically and become more productive, while providing better designs.

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