A Case Study Comparing Two Approaches for Applying Area Loads: Tributary Area Loads vs Shell Pressure Loads

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1 1 A Case Study Comparing Two Approaches for Applying Area Loads: Tributary Area Loads vs Shell Pressure Loads By Dr. Siriwut Sasibut (Application Engineer) S-FRAME Software Inc. # Commerce Parkway S-FRAME Software Inc. #282, 800 Village Walk S-FRAME Software (UK) Ltd. 21 Bloomsbury Square Richmond, BC V6V 2X7 Guilford, CT USA London UK, WC1A 2NS CANADA Phone: +44 (0) Phone: Phone: Please visit us at to request more information about this publication.

2 2 1. Introduction 2. A Case Study 2.1. Basic model details 2.2. Comparison and discussion of results 3. Summary 4. References Table of Contents

3 3 Two Approaches to Area load Distribution in S-FRAME 1. Introduction When a surface is subjected to an Area Load, the engineer usually has two approaches to model how this Area Load is transmitted to the surface s supporting members. One approach is based on the tributary area principle, where loads are distributed to the supporting members in proportion to their associated tributary areas which are defined by certain rules. Another method is to model the loaded surface using finite elements (FEs) and connect them to the supporting members. In this second method, each contributing structural element s relative stiffness will influence how the loads are distributed to the surface s supporting members. S-FRAME Analysis handles both of these approaches for distributing Area Loads to a surface s supporting members. This document illustrates, by example, the underlying implications of the two approaches to the distribution of area loads to the supporting members. 2. A Case Study A one-storey 3D steel frame is subjected to uniformly-distributed vertical loads applied on a 150mm-thick concrete slab at the top floor Basic model details The geometry of the frame and the sections of structural members are as shown in Figure 1. The applied area load is 15. Two primary models with some variations in the modeling technique are considered in this study. In the first model, as shown in Figure 2, the area load is distributed to the supporting beams on four sides using S-FRAME s Area load tool with the two-way span direction. The concrete slab, in this case, is not required by the engineer to be considered as part of the structural model and the area load is being distributed to the supporting beams based on the tributary area principle automatically by S-FRAME s Area Load tool. This approach is particularly appropriate when the supporting beams are relatively stiffer than the slabs where the slabs and beams can be almost independently analyzed and designed. The influence of this relative stiffness will be discussed in more detail in the next section.

4 4 Figure 1. One-Storey 3D Steel Frame Figure 2. Tributary Area Model Model 1 In the second model, as shown in Figure 3, the concrete slab is modeled by 6x8 shell elements which are in turn connected to the supporting beams on four sides. The loads are applied directly on the shells representing the concrete slab using the Uniform Pressure

5 5 tool in S-FRAME. With this approach, all contributing structural components are included in the model where their stiffness is represented by the elements used. How the area loads are being distributed to the supporting members will be directly influenced by the model set up. Note that this model is using Physical Member technology in S-FRAME where only 4 perimeter physical members are required. Internally, S-FRAME will break each physical member into its constituent finite element members and ensure consistency between these members and the interconnecting shell elements. However, the results reported are associated with physical members as shown in subsequent moment and shear force diagrams Comparison and discussion of results Figure 3. Finite Element Model Model 2 The two models are analyzed in S-FRAME using a linear static analysis. To investigate the effects of the two approaches on the distribution of area loads to the supporting members, bending moments (My) and shears (Fz) in the beams are considered, as shown in Figure 4-7. For the 8m beams, the maximum My are kn-m and knm and the maximum Fz are kn and kn for Model 1 and 2, respectively.

6 6 Figure 4. Moment Y (kn-m) in Beams Model 1 Figure 5. Moment Y (kn-m) in Beams Model 2

7 7 Figure 6. Shear Z (kn) in Beams Model 1 Figure 7. Shear Z (kn) in Beams Model 2

8 8 Figure 8 and 9 show the magnitudes of the vertical forces at each beam s end using a Free body forces tool in S-FRAME. These indicate the portions of the total area loads that are being taken by each beam. Note that the total of these forces are 720 kn and kn for Model 1 and 2, respectively. Clearly, for Model 1, the supporting beams are, as expected, taking all of the area loads of 720 kn before transferring them to the columns. In model 2, on the other hand, some portion of the total area loads (about 8.5%) is transferred directly from the shells to the columns. As mentioned earlier, this is mainly influenced by the relative stiffness of each contributing structural element. In the design codes, one way to address the above issue is by an introduction of a parameter which is used as a measure of the relative stiffness of the slabs and the supporting beams. For instance, in ACI 318 code, a parameter,, is used for this purpose. According to the ACI code, when is less than 1.0, the shear on the beam may be reduced by a linear interpolation method to account for the fact that some of the loads will be transferred directly from the slab to the columns. Interested readers may refer to a chapter on Two-way slab systems in the code for the definition of this parameter and related topics. However, for Model 2, provided that is equal to 1.51, one may assume, based on the pertinent code s requirement, that the supporting beams should have been taking all of the area loads. This is clearly not the case for this example, even when a much finer FE mesh is used to ensure that the analysis solution has converged which incidentally has already converged with the original model. To ensure that the supporting beams are properly loaded when using the second method, it is possible to achieve a similar behavior as the first method. A common practice is to reduce the shell thickness but doing so introduces inaccuracies in the automatic self weight calculations. Alternatively, one can define fictitious material properties with an increase in material density to account for the self weight loss due to a smaller thickness. Another approach is to reduce the material Young s Modulus (E) and assign the proper thickness. The latter approach is probably a preferred one since the model would then have the correct slab dimensions if it is ever required to be exported to a third party BIM product. To illustrate one of these modeling techniques, Model 2 is modified, namely, Model 2a, by reducing the Young s modulus (E) of the concrete slab to 25% of the original E. This stiffness reduction is based on an ACI code s recommendation where a reduced moment of inertia of flat plates and flat slabs of 0.25 is used in an elastic analysis for strength design. Table 1 summarizes the key results for the 8m beams obtained from these models. Given of 6.03, the maximum shear force and moment in the beams from Model 2a are very close

9 9 to those of Model 1. Evidently, the maximum shear force from Model 2a is even higher than that of Model 1. Another alternative finite element-based approach (the second method) that may be used to model the supporting beam as a steel-concrete composite beam will be explored. A Composite Beam Design capability in S-STEEL allows the engineer to consider and design, in accordance with AISC and CISC codes, a steel beam and concrete slab to act as a single unit in resisting flexure provided that the two parts are connected properly. Figure 10 and 11, respectively, show an example of the graphical representation and code details for the 8m composite beam which is designed to meet the pertinent requirements of Canadian CSA-S16-09 code. Upon returning to S-FRAME, the composite beam s strong-axis moment of inertia will be recalculated by S-FRAME taking into account a contribution from the concrete slab. Note that a reanalysis is required due to the changes in stiffness of the structure. The key results for the 8m beams obtained from this model, namely, Model 2b, are also summarized in Table 1. Based on the increased moment of inertia due to a composite beam action, is calculated as 12.9 and the maximum shear force and moment in the beams are increased by about 5% when compared to those of a noncomposite Model 2a. In this example, the consideration of a composite beam action in an FE model provides the results in terms of the maximum shear force and moment that are somewhat higher than those obtained from the tributary area approach. One of the pitfalls with the FE approach that may require an engineer s attention is when there are lateral floor bracings present within a slab. Any intermediate node (or seed node) falling within a floor bracing member, as shown in Figure 12, may prove to be difficult to be removed from the meshing algorithms unless meshing operations are allowed to be performed at design time rather than run time. Unlike some other software where meshing operations are only allowed to be performed at run time, meshing operations in S-FRAME are allowed to be performed at design time. This provides the user with an opportunity to first mesh the slab and subsequently add the floor bracings, thus allowing the floor bracings to be modeled as a part of the floor s lateral system while at the same time not receiving the vertical area loads from the slab. Figure 13 and 14 illustrate the differences between FE mesh with floor bracings when meshed at design time and run time, respectively. Note that the modeling of lateral floor bracings using S-FRAME s Area load tool (Model 1) can be also achieved easily by excluding the floor bracing members from an Area Load Members folder so that no area loads will be distributed to them.

10 10 Figure 8. Force Z in Beams Model 1 Figure 9. Force Z in Beams Model 2

11 11 Models Flexural Stiffness Ratio Max. Shear (kn) % Diff. Max. Moment (kn-m) % Diff. 1 N.A a b Table 1. 8m Beam s key results Figure 10. Composite Beam Model 2b

12 12 Figure 11. Composite Beam s Code Details Model 2b

13 13 Figure 12. Floor Bracings with a seed node Figure 13. FE and floor bracings with a seed node meshed at design time

14 14 Figure 14. FE and floor bracings with a seed node meshed at run time Distribution of area loads applied on a surface with complex geometry to the supporting members can be sometimes a daunting task for engineers. Using either a tributary area or FE approach, this can be easily achieved in S-FRAME. Figure 15 illustrates 2-way distributions of area loads applied on surfaces or panels with regular shapes and a more complex shape using the tributary area method. Figure 16 illustrates FE mesh of the same surfaces where S-FRAME s mesh generator is used to mesh the panels with appropriate user-defined mesh parameters.

15 15 Figure way area load distribution of complex geometry Figure 16. FE mesh of complex geometry

16 16 3. Summary While it is not the purpose of this document to recommend which approach one should use to distribute area loads, it is our opinion that both approaches should be made easily available to engineers so that they can decide which one to use depending on the requirements of their particular design problems. S-FRAME offers both modeling tools for engineers to use as needed. From our example, however, some conclusions may be drawn as follows: 1. When the supporting beams are relatively stiffer than the slabs, defined by, the Tributary Area approach may be a more appropriate way of distributing the area loads. 2. When the supporting beams are relatively softer than the slabs, defined by < 1.0, the Finite Element approach may be a more appropriate way of distributing the area loads. 3. Some modeling techniques used to adjust the stiffness of contributing structural elements may be adopted to manipulate the distribution of area loads. 4. References 1. S-FRAME R11 s Help System, S-FRAME Software Inc., CSA S Design of Steel Structures, ACI Building Code Requirements for Structural Concrete and Commentary, 2011

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