Third Avenue New York, NY. Lateral System Analysis and Confirmation Design November 13, Michelle L. Mentzer.

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1 Executive Summary The lateral system for Third Avenue is primarily a shear wall core surrounding the elevator shaft and stairwells. There are six slightly different configurations of shear walls in this building with each configuration spanning a number of floors. On many floors, sets of shear walls are connected by heavily reinforced link beams. These beams were neglected for simplicity in the stiffness calculations. Additionally, the behavior changes midway through one of the configurations due to a change in concrete strength between floors twenty and twenty-one. It is clear after distributing the lateral loads and performing a drift analysis, that the shear walls do not support the full lateral load alone. In the critical direction, the north/south direction, many of the columns are positioned in such a way that their strong axis will resist a fraction of the lateral loads. This was not taken into consideration in the drift calculations. Therefore, the drift calculated is greater than the actual drift that will occur. The calculated drift is approximately two inches greater than the generally accepted drift. However, once the effect of the columns is added, the overall drift and individual story drifts should fall within an acceptable range. Shear walls were spot checked to ensure that the reinforcement in each wall at a certain floor is sufficient to resist the shear forces from lateral loads distributed to that member. Some of the walls that appeared to be over-reinforced in the Structural Concepts/Structural Existing Conditions report were shown to be need that extra reinforcement for added loads from torsion. Some of the walls were still overreinforced for shear alone, but these walls occur at areas where extra stresses result from shifting gravity loads. The shifting loads cause lateral stresses that are transferred to the shear walls and must be resisted along with wind and seismic loads. The foundation of the building must be able to withstand the overturning moments at that accumulate at the building s base. Maximum overturning moments were taken from the Structural Concepts/Structural Existing Conditions report to spot check the foundation. Based on these moments, the total building weight, and the bearing capacity of the soil, the foundation should be capable of resisting the overturning moments in both directions. Page 1 of 8

2 Existing Lateral System The primary lateral system used in Third Avenue is a set of shear walls in the core of the building that house the elevators and stairs. Therefore, while the walls are relatively uniform all throughout the building, many floors have openings in the shear walls to provide doorways into the elevator and stair lobby. This creates complications in distributing the lateral forces to the different walls in the frame. The walls are connected above and below each of these door openings by link beams. The concrete strength in the shear walls is 8000 psi up to floor twenty and decreases to 5000 psi beginning with the walls supporting the twenty-first floor. In addition to the shear wall, concrete columns can be expected to resist a smaller fraction of the lateral loads. The effect of these columns will be ignored in the calculations for this assignment. Lateral Loads Lateral loads were determined based on the wind and seismic calculations from the Structural Concepts/Structural Existing Conditions report (A1-A3). Each floor was analyzed for the maximum lateral load at that level. This was either the wind load or seismic loads at that level. The attached design shear spreadsheet (A3) shows the maximum lateral force at each level and distinguishes whether wind or seismic loads govern at that floor. The loads on this spreadsheet are later divided for individual shear walls to resist. The slabs act as diaphragms for transferring lateral loads from the face of the building to the lateral resisting shear walls. The distribution of lateral loads by floor is shown on the following page. Page 2 of 8

3 Distribution of Forces All lateral forces were assumed to be resisted by the shear walls alone. Refer to the spreadsheet entitled Distribution of Forces (A9-A15) for a wall by wall distribution of the shear forces. The walls were labeled as shown on the typical shear wall plan for floors nine through twenty-five below. Lateral forces were distributed based on the relative stiffness of the resisting shear walls in the direction of each load. The spreadsheet titled Direct Stiffness by Floor (A4-A5) was used to determine the relative stiffness of each shear wall in a particular direction. Center of Rigidity (A6-A8) was used to incorporate the effects of eccentricity into the total forces. This spreadsheet calculates the center of rigidity as well as the eccentricity of the force on each floor. This information is referenced in the Distribution of Forces spreadsheet, where the torsional effects are calculated and added to the forces distributed through the direct stiffness method. Torsional effects were based on the Page 3 of 8

4 difference between the center of rigidity of the shear walls and the centroid of the lateral load. In the case were the center of rigidity occurred at the center of the wall, a minimum eccentricity of five percent of the building width was assumed. Additionally, the effects of torsion can only act against the walls. Torsion cannot be expected to help resist lateral loads, so in the cases where the force from torsion was negative, it was neglected. Link beams were ignored in the distribution of forces and the calculation of stiffness. This simplified the calculation of stiffness, allowing all walls to behave in a similar manner. Had the link beams been included in the calculations, they would increase the stiffness of the walls that they connect, drawing a larger percentage of these forces to these walls. It is likely that due to this simplification, the shear walls that are connected by these link beams will be designed to resist larger loads than those calculated for this report. As long as the walls that get draw more load due to this assumption are still strong enough in shear, the design should be acceptable. Typical Shear Wall Plan Floors 9-25 Page 4 of 8

5 Drift The changes in geometry, modulus of elasticity, and the link beams complicate drift calculations for Third Avenue. Several attempts at calculating the building and story drifts were made before any reasonable value for drift was found. While several different methods were used to attempt to find a reasonable drift, the conservative assumption that the shear walls are the only lateral resisting elements was likely the cause of higher than anticipated drifts. One approach was to calculate the drift per story based on the sum of stiffnesses ( k) for all of the shear walls on that floor alone. This gave the walls the illusion of being stiffer than they were by ignoring the compounding effects of flexural displacement. It also meant that many walls were assumed to behave as short walls even though they would be considered very tall if the total wall height were taken into consideration. In short walls, the majority of the deflection is due to shear, while tall walls have a greater flexural displacement. The result of these calculations is shown in the Story Drift spreadsheet (A16). A second approach to determining drift was to divide the building into sections according to the geometry and strength of the shear walls. Shear wall geometry changes on floors 3, 7, 9, 25, and the bulkhead. Concrete strength changes between floors 20 and 21. The same divisions used in the direct stiffness spreadsheet were used for this analysis. The overall drift in the north/south direction was inches. This is a high drift for the height of the building. The common rule of thumb for drift is that it should be limited to the overall height of the building divided by 400. This rule allows Third Avenue to experience a drift of up to inches. The east/west direction only experiences a drift of 1.60 inches according to this method, which seems suspiciously low. These calculations are shown on page A17 of the appendix. Since link beams were still ignored in this analysis, a third attempt at calculating drift was made by considering the walls connected by a link beam to perform as a single wall with a length equal to the sum of the lengths of the two walls being linked. This assumption combined with the segmental drift calculations used for the second approach gave a low drift of 4.78 inches in the north/south direction. This is a very low deflection for such a tall building, although it is still larger than the drift given by the original analysis method. In reality, the building link beams would increase the stiffness, but not to the point of the walls behaving as one wall with the sum of the two lengths. Therefore, the deflection would be expected to fall somewhere between the inches calculated above and the 4.78 inches calculated with this method. Therefore, the deflection will most likely be close to the h/400 value of inches. Page 5 of 8

6 A final method for drift analysis was performed using STAAD. Each floor was assigned a node, and the entire building was modeled as a single column with a stiffness equivalent to the stiffness of the entire building. Stiffness was assigned on a floor by floor basis. The Moments of Inertia for Entire Shear Wall Core spreadsheet (A18-A19) shows the calculations to get an equivalent moment of inertia. The moment of inertia was modified to account for the difference in modulus of elasticity between the walls and the assumed values in STAAD. Wind loads in each direction were applied to the set of shear walls. This analysis produced a maximum drift at the top of the building of 13.9 inches in the north/south direction. A 13.9 inch drift at the roof is equivalent to h/360. The typical rule of thumb is h/400, so this drift still seems a little high. Since some of the columns in the building are very large and many are aligned with the strong axis in the north/south direction they are most likely expected to provide some additional lateral resistance. The maximum overall drift in the east/west direction by this method is only inches. Once again, this is a much lower number than expected. Based on the STAAD analysis, the maximum story drift is 1 inch for a 21.5 foot high floor. This is an unusually high story height used for the top mechanical room. For this floor, h/400 is inches. Several other story drifts exceed h/400 as well, but when the effects of the columns are added, they should fall into a reasonable range. Refer to pages A21-A20 of the appendix for STAAD results. Spot Checks Shear reinforcement was checked in the shear walls at levels two, three, seven, nine, and twenty-six. The checks were based on the fraction of the factored load going to each wall that was calculated in the Distribution of Forces spreadsheet. After these checks, it is apparent that the shear walls that appeared to be overreinforced on level two in the Structural Concepts/Structural Existing Conditions report needed that additional shear reinforcement due to the extra shear forces caused by torsion. The shear walls at levels seven and nine still appeared to be over reinforced for shear. This could be related to the shift in the center of the force at level nine since there is a setback and cantilever occurring simultaneously at level eight. The west shear walls are more heavily reinforced than the east walls. This could be related to the fact that the centroid of the lateral forces is shifted west when the building shifts left as a result of the setback and cantilever combination. All shear walls between levels six and nine are more heavily reinforced than the average shear walls. This is the section of the building where a lot of column walks and shifting loads occur. Levels six and eight are the two levels where cantilevers begin, which cause shifts in Page 6 of 8

7 the loads and complicate stresses in all structural members. Shear wall spot checks are shown on pages A22-A25 of the appendix. Foundation Third Avenue rests on a mat foundation. The depth of the mat ranges from 60 inches on the west side to 42 inches on the east side. This foundation behaves as a fixed support to resist overturning moments transferred down through the shear wall lateral system. The maximum overturning moment to be resisted in the north/south direction is 172,141 ft-k. In the east/west direction, a moment of 125,257 ft-k needs to be resisted. These values, along with the overall building dead load were determined previously in the Structural Concepts/Structural Existing Conditions report. The dead load is sufficient to resist the uplift forces from the overturning moments, as shown in the foundation calculations in the appendix (A26). Data from the geotechnical report indicates that the bearing capacity of the soil below the foundation is two tons per square foot. Based on this value and the area of the mat, the total bearing capacity of the soil under the mat is great enough to resist the sum of forces from dead load and the downward forces from overturning moments. Conclusions Shear walls were proven not to be the only lateral resisting elements in Third Avenue. They still resist the majority of the lateral loads, however, loads are transferred from the cladding through the slabs to the shear walls and columns. Drift was calculated to be slightly above h/400 in the north/south direction and extremely small in the east/west direction. While the north/south direction has more shear walls to resist lateral loads, the north and south faces are also almost twice the width of the east and west faces. This makes a very large difference in the magnitude of wind loads in the two directions. It makes sense for the building to deflect more in the direction with a larger force, but it is surprising that the other direction would not deflect more than the calculations show. Link beams were ignored in the stiffness and drift calculations. The fact that the deflection is so high in the direction that most of the link beams span shows that they must provide a significant increase in stiffness to the walls they connect. Since link beams effect stiffness, they also would have altered the distribution of the loads. In the shear wall spot checks, one wall is capable of resisting a shear that is around 50 kips lower than the load distributed to that frame. Since other walls in that direction have link beams at that level, their stiffness would be increased, causing them to take some load away from the overloaded shear wall. This combined with the load taken away from the shear walls by the columns should be sufficient to reduce the force in that wall to an acceptable value. Page 7 of 8

8 The mat foundation was checked for its resistance to overturning moments from wind forces. Based on its surface area, the building self weight, and the bearing capacity of the soil, the foundation will work. It has plenty of capacity to resist the overturning moments in both directions. Page 8 of 8