So, we're going to spend the next couple of hours talking about holocore diaphragm design. The general design criteria that we're following for this presentation is IBC 2015, which references ASCE 710. ASCE 710 references rigid versus flexible diaphragms and diaphragm design forces. We'll be following ACI 318.14 for design and detailing. And there are no exceptions to the design and detailing provisions of the document made in IBC or in ASCE 7, as happens with other parts of the exceptions that are taken by IBC to the code. ASCE 712.8.4, horizontal distribution of forces, recognizes that there are options of rigid diaphragms where the story shear is distributed to the vertical elements of the lateral force resisting system based on relative lateral stiffness, but also of flexible diaphragms where the story shear is distributed to the vertical elements based on tributary areas. There is a difference in how the system behaves, the figure of rigid versus flexible. In this case, with equal stiffness, the three walls would take the loads equally if the diaphragm is rigid, but the walls will take the lateral forces based on contributary area if the diaphragm is flexible. So there certainly is a difference in the demand on the vertical system and in the behavior of the diaphragm. So the fundamental question, is the diaphragm rigid or flexible? There are a couple of ways that ASCE 7 deals with the question. The first is a prescriptive approach, and the second is a calculation approach. The prescriptive approach contained in 12.3.1.1, the flexible diaphragm, diaphragms constructed of untopped steel decking or wood structural panels are permitted to be idealized as flexible if any of the following conditions exist. And A, in structures where vertical elements are steel or braced frames, steel and concrete composite braced frames, or concrete masonry steel or steel and concrete composite shear walls, with those floors, it's a flexible diaphragm and in one and two family dwellings. And C, in structures of light frame construction where all the following are met, topping of concrete or similar materials is not placed over wood structural panel diaphragms except for non-structural topping no greater than one and a half inches thick. And each line of the vertical elements in the seismic force resisting system complies with the allowable storage drift in table 12.2.1. Designation for rigid diaphragm. Diaphragms of concrete slabs or concrete-filled metal deck with span-to-deck ratios of three or less in structures that have no horizontal irregularities are permitted to be idealized as rigid. In other words, if you meet the layout requirement of flexible or meet the layout requirement of rigid, then you can claim those designations without doing further work. The flow chart that's shown on the screen is a question of starting on the left, is the diaphragm wood structural panels or un-topped steel decking, if yes, you go up, if no, you go down. And you follow this across and basically determine yes or no in order to determine that a diaphragm is rigid or a diaphragm is flexible, but you may end up in between. So diaphragms are permitted to be idealized as flexible where computed maximum in-plane deflection of the diaphragm under lateral load is more than two times the average storage drift of the adjoining vertical elements of the seismic force resisting system of the associated story under equivalent tributary lateral force. What that's saying is you can calculate if your drift of the vertical system is less than twice the drift of, let me say that correctly, if your in-plane deflection under lateral force is more than twice the average storage drift of the vertical elements, then you have a flexible diaphragm. So this is the calculation method. It's important to recognize, though, that this is a level by level calculation. It's story drift. And depending on the lateral force resisting system, although it may be unusual, a diaphragm may be rigid at one level, but flexible at another level based on the comparison of the diaphragm deformation or story drift associated with that level. So rigid versus flexible, the diagram is showing the maximum diaphragm deflection and the average drift of the vertical elements and making a comparison to assume that it's flexible or assume that it's vertical. Section 12.3.1 on diaphragm flexibility is that the diaphragm shall consider the relative stiffness of the diaphragms and the vertical elements of the seismic force resisting system. Unless a diaphragm can be idealized as either flexible or rigid in accordance with the section 12.3.1.1, 12.3.1.2, or 12.3.1.3, the structural analysis shall explicitly include consideration of stiffness of the diaphragm. In other words, you're going to have to do more work with a semi-rigid modeling assumption in your analysis. This usually lends towards computer analysis of the system. It may be appropriate to point out that the difficulty in making the calculation in more complicated real world configurations when we run into real buildings that aren't simple rectangles and simple layouts of walls. A diaphragm is rigid for the purpose of distribution of story shear and torsional moment when the lateral deformation of the diaphragm is less than or equal to two times the average story drift. What the code said before in the earlier version of ACI 318, ASCE 7, except where diaphragms are flexible or permitted to be analyzed as flexible. And the code 16.04 in IBC has changed to say where required by ASCE 7 provisions shall be made for the increased forces induced on resisting elements of the structural system resulting from torsion due to eccentricity between the center of application of the lateral forces and the center of rigidity of the lateral force resisting system. This is a change between IBC 2012 and 2015. And the eccentricity or the torsion may be real because the center of stiffness does not coincide with the center of mass. And there is also the requirement for accidental torsion, an extra 5% to be added to that, which needs to be considered in the diaphragm design as well. So ASCE 7.10 section 12.10 covers diaphragms, cords, and collectors. Section 12.10.1 deals with the diaphragm design forces. Section 12.10-2 addresses the collector elements. The lateral forces for the diaphragm design are not the same as the lateral forces that are used for the design of the vertical elements of the lateral force resisting system. So there's a different formula, which is 12.10-1. Now it's also important to recognize that W sub I in this formula, which is the weight of the tributary area of the level that we're investigating, may not be the same as the weight of the tributary of the weight tributary to the diaphragm at that level or at level X. And the difference here is you don't have to include the weight of the wall in the direction of the load in the diaphragm force because the diaphragm force is collecting loads to deliver them to the lateral force resisting system. So the actual weight of the lateral force resisting system that's accepting the load doesn't have to be included in the equation. Frequently it is done that way for convenience and calculation, but it's not essential. There are also threshold bounds for diaphragm design forces. Equation 12.10-2 gives us a minimum threshold of 0.2 S sub DS I times WPX. And 12.10-3 gives us a maximum threshold of 0.4 S sub DS I WPX. And when we're looking at the diaphragm at the level to be concerned, WI, as the figure shows, is all weights required to be included by 12.7.2 between the planes shown, halfway down to the next lower level, halfway up to the upper level, all contribute to the weight that's used in this calculation for the weight of the level. Comparing the diaphragm design force from the calculation for diaphragms with the lateral force that's applied to the lateral force resisting system, we can see that at the roof level, they're essentially the same number. Again, the blue line, which is the F sub PX, the diaphragm could be lower if there is significant weight of the lateral force resisting system in the direction under consideration at that level because the W sub PX is less than W sub I. The way that the formulas work, we continue to have a higher level of force by calculation until we reach the lower bound threshold in this chart, which shows that you can't get the diaphragm design force lower than 0.2 times S sub DS times the WPX. So, the same principle, the same force applies in this slide, it's essentially to show how you would go about getting the design force when you're doing a modal analysis. The square root of the sum of the squares from 12.9.3 in the code is the methodology used for the modal analysis. There is an inconsistency in applying the system response modification factor, the R factor in the equation to higher modes because generally, the higher modes are not going to result in yielding energy dissipation and therefore the reduction of the energy in the design forces, but it's in the equations and there is an inconsistency. There are provisions that recognize that. Now, step back and say, today we're talking about ASC 710 and the traditional methodology that's been used for decades in order to calculate diaphragm design. In the future, we will be talking about ASC 716 and the new methodology for diaphragm designs that have been developed specifically aimed at precast concrete in seismic design categories C and above that try to remove these inconsistencies and generally work to larger values of lateral force than we're discussing today. The importance of this, and this slide I've used before, provided by Dr. Gosu, helped in the preparation of the presentation slides for this webinar, is really stark in telling us something about where we are with ASC 710. The lower blue overlying section ASC 7 range provides the minimum to the maximum value with the idea that the maximum requirement for 0.4 times s of ds times i times w of px should be the maximum force that we have to design a diaphragm for. But all of those stars and squares and circles are all events above that where measured acceleration of the diaphragms were higher than the maximum required by ASCE 7 and it points to the issues and the problems that prompted the alternative method that is published in ASC 716. But right now recognize that we're dealing with this range that may be lower than what the building will actually experience. Now we recognized as we were developing the seismic design manual and also the recommendations that are in the PCI design handbook during the course of the DSDM research that was PCI sponsored on the diaphragm seismic design methodology that the forces that were being used for the design of diaphragms were likely to be lower than they needed to be and we didn't have the updated provisions so we made some recommendations in order to provide a safer level criteria for the design of the diaphragms. So for the recommendations that came from the seismic manual based on the preliminary results that was that for structures assigned for seismic design category of B and C you could take the the roof level force and use it for the entire structure. For structures assigned to design categories D, E, and F if the shear walls are part of the lateral force resisting that it was sufficient to supply a diaphragm load factor of two to the force at the uppermost level derived from ASCE 7 and then to design each floor for that force. Again it was an interim stop gap way to recognize that the levels of force were not sufficient to guarantee that we would not have a problem with diaphragm behavior. Still may be questionable and again the resulting changes in ASCE 7 are 16 or even higher but it was a step towards providing for an additional level of safety in the structures. There's also the question or issue of transfer forces. Transfer when what happens when a wall is not continuous to the base and the loads from that wall need to be transferred through the diaphragm to a different location or an offset location for the wall. Where the diaphragm is required to transfer design seismic force from the vertical resisting elements above the diaphragm to other vertical resisting elements below the diaphragm due to offsets or changes in the relative stiffness in the vertical elements the forces have to be added to the forces that you're using in the design from equation 12.10-1. In other words if you've got transfer forces they have to be added to the force that you calculated for the level. The redundancy factor rho applies to the design of diaphragms in structures assigned to seismic design category D, E, or F. For inertial forces calculated in accordance with 12.10-1 the redundancy factor is one. It's kind of an odd way that the code says well you have a redundancy factor but for the equation 12.10.1 the redundancy factor is one. For the transfer forces the redundancy factor rho is the same as that used for the structure. So if you're structure has a redundancy factor because of its configuration then that redundancy factor would be applied to the transfer forces but not directly to the level only diaphragm forces. For structures having horizontal or vertical structural irregularities the requirements of that section also apply which they're section 12.3.3.4 in section 12.3.3.4. Increases in the forces due to irregularities are applied for seismic design categories D through F. A horizontal irregularity of type four exists where there is a discontinuity in the lateral force resistance path such as out of plane offset or of at least one of the vertical elements as shown before pointing us to the table for horizontal irregularities. This irregularity points to section 12.3.3.3 for seismic design categories B, C, D, E, and F and requires structural elements supporting discontinuous walls shall be designed to resist the seismic loading effects including the over strength factor of section 12.4. Remind you that the over strength factor is omega sub zero in the lateral force resisting system. So when you have these irregularities the intent of the over strength factor is to design a load path that is not intended to yield for the for the strength of the system that it's being designed for. So this adds for shear walls for instance the omega sub zero is 2.5 and that additional factor has to be applied to the load combination in this case. The connections of such discontinuous walls or frames to the supporting members shall be adequate to transmit the forces for which the discontinuous walls or frames were required to be designed. That seems to be somewhat obvious. Transfer forces. This can happen when a bearing wall building with hollow core floors where the walls in the upper floors are eliminated to create large open space on the first level by bearing those walls on columns so that the diaphragm must transfer the in-plane forces to the walls at the ends of the space. It might occur in a hotel lobby so we have these large transfer forces. The question then becomes does this configuration with transfer diaphragm constitute a structural element supporting discontinuous walls horizontally and therefore require the application of the over strength factor to the transfer force applied to the diaphragm. Well the commentary says such offsets in this case impose vertical and lateral load effects on horizontal elements that are difficult to provide for adequately. The commentary to ASCE 7 section 12.3.3.3 states the purpose of requiring elements that support discontinuous walls or frames to be designed to resist seismic load effects including over strength is to protect the gravity load carrying system against possible overloads caused by over strength of the seismic torch resisting system. So the question that we're left with and it probably contradicts what we're thinking before about the design of the load path is that the section doesn't address loads from the diaphragm. The transfer forces become a collector force subject only to the over strength factor for seismic design categories C, D, E, and F. Now again collectors are subject to the over strength factor but the overall design of the diaphragm that is not a collector might be excluded from the requirement for the design for that additional over strength factor. So collector elements those are elements that are transferring the loads from the diaphragm into the lateral force resisting system. In our diagram here three walls with we have a diaphragm that's transmitting the loads from the diaphragm into the shear walls, those collectors in seismic design category C through F required design for the over-strength factor of the vertical element system. We can see the result of not providing over-strength in the load path of bringing loads to the vertical elements of the lateral force resisting system. Here, the diaphragm failed to get the load to the shear wall, which is the element that is supposed to have flexural yielding at the base, and therefore dissipation of energy and reduction of lateral forces on the building overall. Obviously, if you don't engage and enact the energy dissipating mechanism, and you don't design the diaphragm to have the sufficient strength in order to ensure that that happens, then you can have a failure that is fairly dramatic. So, section 12.10-.2.1, the design of collector elements goes through this idea of what are we designing for, and collector governing design force is the maximum of omega sub zero calculated using V from section 12.8 or 12.9, which is, again, our lateral force, not the diaphragm force, but the lateral force that the system is intended to be designed for. Or it could be omega sub zero Q sub E calculated for the diaphragm force F sub PX. Or, in some cases, it may be actually governed by the minimum force F sub PX, because there still is a minimum. If those other two forces with the omega factor are still greater than the minimum, then the minimum applies, and the minimum does not have the overstrength factor applied to it. But the force that you calculate need not exceed the maximum force, which was developed from equation 12.10-.3. And, in addition, transfer forces that we described earlier, they have to be considered. Concrete diaphragms are covered in ASCE 3.18.14. A reinforced concrete slab acting as a structural diaphragm must satisfy all applicable 3.18 requirements for a one-way or a two-way non-pre-stressed or pre-stressed slab, as well as applicable requirements for the new chapter 12 on diaphragms. If a diaphragm is part of a building assigned to seismic design category D, E, or F, it also has to satisfy the applicable requirements of section 18.12. Important to recognize here, in the reorganization of 3.18, a new chapter on diaphragms was developed, treated like a member chapter, and it provides basic provision or requirements for diaphragms for all structures. But when the code was reorganized, the decision was made to keep all of the diaphragm requirements for all of the requirements for seismic design, including diaphragms, are kept in chapter 18. So, although it would be convenient to look at chapter 12 and think that everything you need is there, it's still true that if seismic design requirements are part of the design of the structure, then you're going to have to go to the chapter on seismic design in order to pick up those additional requirements. So, for one-way slabs acting as diaphragms, we walk through the way that the code has been reorganized, provides almost a step-by-step, a task list organization in order to find the references that you need. We start in chapter 12, but the code is set up to point us to other chapters that pertain to particular issues or toolbox chapters. So, we start with one-way slabs acting as diaphragms, and first there is minimum thickness. If it's not pre-stressed, go to section 7.3.1. If it is pre-stressed, then it's section 12.2, where we calculate deflection within limits rather than work on prescriptive minimum thickness requirements. Then there is requirements for shrinkage and temperature reinforcement for these one-way slabs. Non-pre-stressed, we're pointed to 24.4.1 and 24.4.3.1. Normal to the flexural reinforcement for grade 60 reinforcement, we have a value of .0018 times the gross cross-sectional area. If it's pre-stressed, then the governing section is 12.4.4.1 or 7.6.4.2. We keep running through, but the flowchart provides us the direction and the structure of the component chapters in ACI 318 do this. We follow through. For flexural reinforcement, non-pre-stressed, we're given the reference to sections for A sub S minimum and spacing requirements. For pre-stressed, we're sent to the chapters that are the toolbox information for pre-stressed concrete flexural design. For crack control, we have requirements for non-pre-stressed one-way slabs and requirements for pre-stressed concrete slabs. Finally, though it's rare to occur in a pre-cast pre-stressed concrete building, there's a similar line of logic of paths to follow and references to sections for two-way slabs acting as diaphragms. Most two-way slabs are cast-in-place slabs, and this rarely will apply to pre-cast, but it's provided here as a review of diaphragm requirements in ACI 318 for completeness. We have, again, provisions for minimum thickness, provisions for shrinkage, and temperature reinforcement and the pointers to go for that. The flexural reinforcement for non-pre-stressed referenced to section 7.7.2.3. The section for pre-stress for 21.2.2, and then finally, again, the provisions for crack control in pre-stressed concrete members. Again, these are the references for two-way slabs likely to be cast-in-place post-tension when they're pre-stressed. Now, for diaphragms in buildings assigned to seismic design category D, E, or F, we're pointed to section 18.12, structural diaphragms and trusses. 18.12.1 provides the scope, 18.12.2 the design forces, and it refers to the design forces of a legally adopted general building code. It's essentially aiming at IBC and ASCE-7 in order to get the loads, in order to provide the loads for design rather than reproducing them in ACI. Section 18.12.3 describes and defines the seismic load path. Section 18.12.4 addresses cast-in-place composite topping diaphragms. Now, we're getting to topping on cast-in-place concrete on precast or other material, and then cast-in-place topping diaphragms, which may not be composite. So, 18.12 continues on to deal with minimum thickness of diaphragms. The minimum, and here again, we're relating the cast-in-place concrete diaphragms to topping on precast concrete construction. For, the limit for minimum thickness is two inches for concrete slabs and composite topping slabs serving as structural diaphragms. Two and a half inches is the minimum for topping slabs acting as structural diaphragms that are not relying on composite action. The important distinction, we get into a lot of discussion, we'll cover this a little bit more as we go down through it, but there really are two different ways of looking at this. This has become more important in the upcoming code to make the distinction between composite and non-composite diaphragms because of the detailing rules and the force rules, but not so much here. Here, it's not uncommon to consider the topping slab as non-composite and simply design it for that. Question about designing a composite topping slab usually has been, what do you do with the joints? When the composite topping is over double T's, they're distinct joints between the flanges of the double T's that might have mechanical connections that aren't recognized by ACI. But with hollow core slabs, the joints are actually fully grouted full height, and so we may consider in a composite topping condition with hollow core that the overall thickness of the diaphragm actually includes the thickness of the hollow core slab as an effective thickness through the location of the cores. Somewhat difficult to determine when the cores are round, but also because the joints are grouted solid, we're not losing thickness just because we have joints between the precast. We move to the reinforcement section 18.12.7, and they're requiring that the minimum reinforcement ratio comply with 24.4. Except for post-tension slabs, the reinforcement spacing in each way is less than or equal to 18 inches. But there are other requirements including important ones for collector elements that will apply. Then the flexural strength section does permit that we do not have to consider non-linear strain distribution of 22.2.1.2 when we're considering the diaphragm in the floor as a deep flat equivalent beam. We're not required to provide deep beam considerations when we're making the design in that manner. We go further in looking at the shear strength of the diaphragm, and now we're pointing in section 18, the shear strength of the structural diaphragm, and there's an upper limit on V sub n in structural diaphragms. V sub n above the joints between precast elements in non-composite and composite cast-in-place concrete topping diaphragm. Again, this is pointing to shear friction strength. We have an implicit assumption that the reinforcement across joints and precast concrete is transferring the load by shear friction. But shear friction carries with it limitations. Even though we're depending on steel area only, there are limitations in the area of concrete that are provided to ensure that we're not overstressing the concrete even as we're using the reinforcing steel to transmit the shear as a shear friction reinforcement. Section 18.12.10 on construction joints refers to 26.5.6 construction contraction and isolation joints and table 22.9.4.2 condition b requirements for clean interface pre-late of latents and intentionally roughened. So, 18.12 is a complete rewrite of ACI 3.18.08. What was ACI 3.18.11? 18.12 is going through the organization now identifies the seismic load path. It provides the flexural design generalized. It provides the shear strength of the joint generalized. It provides the shear strength of topping slab diaphragms and the shear reinforcement over joints in precast concrete elements. This is the new direction, the new organization of ACI 3.18.14. It provides in the scope floor and roof slabs acting as structural diaphragms, structures assigned to the seismic design category D, E, and F, and collector elements and trusses forming part of the seismic force resisting system. Again, the design forces obtained from legally adopted general building code for collector elements earthquake forces are amplified by omega sub zero as we've discussed before and the references back to the forces from 18.12.2 are pointing back at these load controlling documents. So, the seismic load path of 18.12.3.1, all diaphragms and their connections shall be proportioned in detail to provide a complete transfer of forces to the collector elements and to the vertical elements of the seismic force resisting system. Here we have our floor slab with its inertial forces transferring the loads into the walls or columns in order to resist the ground motion. The elements of a structural diaphragm are subjected to primarily to axial forces and used to transfer diaphragm shear or flexural forces around openings or other discontinuities shall comply with the requirements for collectors in 18.12.7.5 and 18.12.7.6. Careful about diaphragm openings, elements that are going to carry axial forces are pointed towards the direction of having reinforcement and detailing that meets these requirements. Cast-in-place composite topping diaphragms where the topping is reinforced and the surface is previously hardened concrete on which the topping is placed is clean, free of latents and intentionally roughened. Although the table 12.9.4.2 requires joint roughness for shear friction to be an amplitude of one quarter inch, and again this is going back to chapter 12, the amplitude of intentional roughness is not defined in 18.12.4.1 that governs cast-in-place composite topping slabs. It's been the bone of contention for decades of what is intentional roughness. Les Martin, who was chairman of 318G for many, many years, used to say that the intent was originally not intentionally made smooth. That is, you don't have to be very rough for the values that are permitted by code to be in effect for the shear values in a composite topping and there's been a lot of research, most recently a paper published in 2013 about the roughness of the surface and it argues that machine cast hollow core would provide sufficient roughness for composite behavior. Again, it's subject to a number of research studies. The lowest value for machine cast shear from a push-off test from the 2013 research was about twice, it was 158 PSI, where the limit in the code for the interface shear, whether it's intentionally roughened or not, when it has no reinforcement across the boundary, is 80 pounds per square inch. So in this regard, the code is very conservative. The seismic experts that want to limit this or want to argue about the roughness and the surface are concerned about the effect of vertical acceleration and the possibility that vertical acceleration could push up and loosen the interface between the cast-in-place concrete and the underlying substrata if the roughness is not sufficient to ensure proper bond. But again, it's been very difficult. No one seems to have exactly a way to measure or to determine how rough is rough and how rough does it need to be. Cast-in-place concrete topping slabs acting alone in proportion and detail to resist the earthquake forces are not uncommon. Because of the difficulties of ensuring the roughness or actually I should say prescribing and measuring the roughness of hollow core slab surfaces, sometimes it's just easier to say we're going to design it as non-composite for the purpose of diaphragm behavior and all of the detailing for reinforcement in edge strips and in the cast-in-place concrete topping is sufficient to transmit the diaphragm forces that are needed in a higher seismic application. There are 18.12.6 that provides minimum thickness requirements, concrete slabs and composite slabs serving as a structural diaphragm two inches of topping. This may not be hollow core, it may be on a precast concrete stay-in-place slab with the cast-in-place concrete topping on it as shown in the figure. Topping slabs placed over a precast floor or roof elements acting as structural diaphragms and not relying on the composite action again is two and a half inch thickness. Now the reinforcing steel requirements, we went through the pointing the locations we were pointed to. Now let's take a look at what those requirements are from the locations in the code where we're pointing. Shrinkage and temperature reinforcement is required in one-way slabs perpendicular to the direction of bending and is the minimum required also in the direction of the span of one-way slabs and in both directions for two-way slabs. The minimum reinforcement ratio is dependent on the grade of the reinforcement used. Usually we're going to be grade 60 reinforcement unless we're using welded wire reinforcement in which case the value may be lower. But again we have limitations for what we can do when we're dealing with shear reinforcement requirements. So these are not yet into the design but are the minimum steel requirements for shrinkage and temperature. So a little dense slide 7.6.4.2, 24.4.4.1, 18.12 reinforcing steel requirements. This is where we get down to when pre-stressing tendons are used. Minimum compressive stress in concrete will be 100 psi on gross section after losses. This is considered in the area of post-tension concrete. Usually we consider minimum pre-stressing for pre-tensioned as 225 but not for the purpose of the application of these provisions. When the spacing exceeds 54 inches additional shrinkage and temperature steel shall be provided between tendons at the slab edge. Bonded tendons used as reinforcement to resist collector forces and diaphragm shear forces shall not exceed 60 psi stress due to design earthquake forces. That part is for those who follow the practice of using un-pre-stressed pre-stressing strand as mild steel reinforcement for diaphragms. When that's done, even though the pre-stressing strand may have 270 psi yield strength, in its unstressed condition it's limited to 60 psi stress for the design of earthquake forces in the diaphragm. Type 2 splices are required when mechanical splices transfer diaphragm forces to vertical elements of the seismic force resisting system. This is a chapter 18 requirement. The idea is to get overstrength to make sure that the transfer is developing the yield strength of the material which may be greater than the minimum yield strength that's specified. Collector elements with compressive stresses exceeding 0.2 f sub c prime at any section shall have transverse reinforcing satisfying 18.10.6.4e shear wall boundary element transverse reinforcement over the length of the element. This is again in a seismic application to make sure that there's sufficient reinforcement to avoid for having a sudden compressive failure of the concrete that's acting in compression 18.12.7.1 where welded wire reinforcement is used as the distributed reinforcement to resist shear in topping slabs placed over precast floor and roof elements. The wires parallel to the span of the precast element shall be spaced not less than 10 inches on center. Reinforced provided for shear strength shall be continuous and shall be distributed uniformly across the shear plane. Here is a limit that was imposed in 3.18 in the seismic chapter in the 1990s, late 1990s, with a recognition that there were failures in the Northridge earthquake where joints opened up in the cast in place concrete topping and the welded wire reinforcement that was supposed to be holding the diaphragm together simply fractured. It fractured because there wasn't sufficient strain capacity to handle the opening of the joints. Some calculations were made and it was determined that in order to meet the strain requirements for the normal spacing of the joints in question, the 10 inches would provide sufficient spacing so that if the joint opened the wire that was crossing the joint would have room to stretch without having a fracture. Now there's a bit of a flaw with this today because the common practice when the code was changed was that double Ts that were probably subject to these questions were about 8 feet wide typically. And if we've now grown to 12 foot wide double Ts, although this is not a double T webinar, it's about hollow core, the question is, is the spacing sufficient to handle the strain if the joint spacing is wider? That's another question for another day but it's important to understand the background of where this 10 inch spacing came from. Again, we're on hollow core in our consideration today and with the ties at the end of the hollow core, our intent is to have the joints in the hollow core not open up because of sufficient proportioning of reinforcement. And although the requirement is still there, the actual effect on behavior may not be as significant. The minimum spacing requirement for welded wire fabric in topping slabs on precast floors is to avoid the fracture of the distributed reinforcement during an earthquake. Cracks in a topping slab do open immediately above the joints between the flanges and the wires are restrained by the transverse wires. So that's the explanation of where this requirement is coming from. The minimum spacing do not apply to diaphragms reinforced with individual bars because the strains are distributed over a longer length. The maximum spacing S of deformed reinforcement shall be the lesser of 3H and 18. And then the spacing of deformed shrinkage and temperature reinforcement shall not exceed the lesser of 5H and 18. If you have a shrinkage and temperature reinforcement that uses welded wire fabric and you have a thickness of topping that's 2 inches and the maximum spacing is 5H, then it would appear that the maximum spacing in one section is the minimum spacing in another and that you're constrained to a 2 inch spacing no matter what you do. But actually the 5H would apply for composite system and the overall effective thickness of the slab includes the precast in that case and therefore we don't have that seeming constraint to only be at 2 inches. If you're in a 2.5 inch thick slab that is non-composite, then the 5H at least gives us up to 12.5 inches as a possible maximum spacing in comparison to the minimum spacing of 10 inches with welded wire reinforcement. Reinforcing steel requirements are covered in table 24.3.2.1. There's a lot of, the trend was in ACI 318 in the reorganization to go and putting equations in tables, requirements in tables, so we now have the requirement for deformed bars or wires, bonded pre-stressed reinforcement and combined deformed bars or wires and bonded reinforcement, all provided with formulas that include the strength of the steel or the stress in the steel or the stress in the pre-stress. Again, following the reinforcing steel requirements, the longitudinal reinforcement for collector elements at splices and anchorage zone have either a minimum center spacing of three longitudinal bar diameters, but not less than an inch and a half and a minimum concrete clear cover of two and one half longitudinal bar diameters, but not less than two inches. This can actually cause the reinforcement cover to demand a thicker slab over the cord reinforcement in order to meet these cover requirements. Again, we may end up with bars resting directly on top of the hollow core slabs in order to provide for that two and a half inches, but if you start to get larger bar diameters for cord reinforcement, if it's in a cast-in-place concrete topping, the topping may actually have to get thicker in order to meet the cover requirement. Transverse reinforcements required by 9.6.3.3, except as required in Chapter 18, in 18.12.7.5. 18.12.7.6, longitudinal reinforcing detailers for collector elements at splices and anchorage zones have to satisfy A or B, center to center spacing of at least three longitudinal bar diameters, but not less than one and a half inches and concrete clear cover, same as, but not less than two inches. Area of transverse reinforcement providing A sub V of at least greater than these equations, which is a minimum area of reinforcement requirement. Again, except as modified by the other sections. So, now we look at the flexural strength of the diaphragm. Flexural design of diaphragms based on linear strain distribution. We don't have to worry about non-linear strain distribution. We can treat this as an equivalent beam. It's equivalent. The effect of openings have to be considered. For shear strength for cast-in-place slabs, the nominal shear strength shall not exceed a value that includes the strength of concrete and the ratio of the steel ratio times the yield strength of the reinforcing steel. The reinforcing has to be placed perpendicular to the span of the diaphragm for this requirement. For cast-in-place concrete docking diaphragms on precast floor or roof members, A sub CV is computed using the thickness of the topping slab for only for non-composite topping slab diaphragms and the combined thickness of the cast-in-place and precast elements for composite topping slabs. And again, we look at a hollow core slab and we're kind of saying, well, what's the effective thickness? There are guidance in the PCI design handbook and other cases that consider some of the thickness of the web as contributing to the effective thickness of the total flange. And there is some engineering judgment that is required when applying those conditions. And if you're considering this, the strength of the precast is greater than the strength of the cast-in-place concrete topping, then you need to consider that the strength is based on the smaller of the concrete strength values for this evaluation. Cast-in-place concrete topping slabs, both composite and non-composite, placed on precast concrete elements, the V sub N shall not exceed V sub N as A sub V, F sub Y mu. Again, this is our shear friction limitation. It neglects the contribution of the concrete due to construction practice that causes shrinking cracks under service loads over joints. In addition, the nominal shear strength can exceed eight times the area times the square root of F sub C prime. But in this case, the mu for this shear friction calculation is limited to 1.0 times lambda, the lightweight concrete modification factor. Even though the slab may be cast as monolithic cast-in-place, the mu value is limited as 1.0 in recognition of the possibility of developing that longitudinal crack along the joints between precast concrete elements. A sub V, S is the total area of shear friction reinforcement within a topping slab. It includes both the distributed and the boundary reinforcement that's oriented perpendicular to the joints in the precast system, and the coefficient of friction, again, mu is 1.0. Important provision here, at least one half of A sub V, S is required to be uniformly distributed along the length of the potential shear plane. The area of distributed reinforcement in topping slab needs to satisfy 7.6.1.1 in each direction. The importance of this provision is if one half must be distributed along the length of the potential shear plane, then one half of the reinforcement might be in the cord reinforcement. So, the shear strength of the cord reinforcement can be included in the total shear strength calculation. Not more than a half can be concentrated at the ends. So, how should shear strength be proportioned between the web and the cord? For free bodies cut along a joint, both the web and the cord reinforcement cross the plane. As a shear friction evaluation, all the demand in the steel is tension. The forces may be distributed to the reinforcement based on the total area of steel, but the cord area may be much larger because of the required cord tension. So, we are limited to how much we can attribute to the cord reinforcement. How much tension strength must be discounted because it's assigned to shear resistance? Is the limit based on the proportional area or is it based on the limit imposed that half must be carried by the web reinforcement? The contribution of cord reinforcement to shear strength is not limited by its primary cord demand. What's important here is to recognize that when you look at the shear friction provisions in ACI 318, it specifically says any net tension across a joint must be added to the shear friction reinforcement area. But the commentary further goes to say if the force from tension and reinforcement is resisting a moment, then there is a counteracting compression force that accompanies the flexure. And therefore, the flexural reinforcement is not additive to the shear friction reinforcement. So, if you're using the cord reinforcement for shear, you don't have to add the additional requirement for tension compression cord reinforcement to it. You simply have to find the one which governs the condition. Now, we have reached the point before I get into talking in more detail specifically about hollow core slabs. It's a two-hour long event here. So, what I'm going to do is at this point, I'm going to take a five-minute break. We'll be back in five minutes to pick up where we stopped. And we will finish going through the information that's provided in the PCI manual for the design of hollow core slabs on diaphragms and a design example in the second half of the webinar. So, I will look forward to seeing you back. Okay. For the second half of the webinar on hollow core diaphragms, we're actually going to get away from the deep weeds of code provisions and try to look at hollow core and the application of these provisions in real details and real calculations to design of hollow core diaphragms. The design issues, again, this is following the PCI manual for the design of hollow core slabs and walls, M&L 126, published by PCI. It was edited by S.K. Ghosh and prepared for the PCI hollow core slab producers committee. It has a lot of good information. We're covering one chapter in otherwise rather extensive reference to assist in the design of hollow core slab structures. And I can commend that to your reference library if you're doing work in hollow core. The design issues in a diaphragm comprised of hollow core slabs are the design of connections to get the loads into the diaphragm, although most of the earthquake forces originate in the diaphragm itself. The strength and the ductility of the system to transmit these loads to the lateral force resisting elements and the design of the connections required to direct the lateral forces from the diaphragm into the lateral force resisting elements. So we're dealing with these issues of lateral force resisting distribution, structural integrity requirements, elements of the diaphragm, longitudinal joints, transverse joints, boundary elements, and again the question of topped versus untopped diaphragm behavior. We talked in the first section about the difference between rigid and flexible diaphragms. Concrete diaphragms are normally considered to be rigid when compared with the lateral force resisting elements. Depending on the type and the magnitude of lateral forces applied, a hollow core slab diaphragm may need to be considered as a non-rigid diaphragm, but for most low and mid-rise structures in low seismic risk areas, an assumption of a rigid diaphragm would be reasonable. The lateral force resisting system distribution and the difference in behavior of flexible and rigid diaphragms we discussed before, but it's illustrated here in figure 4.3.1 from the design manual. A flexible diaphragm with a rigid support behaves as a continuous beam. A rigid diaphragm which distributes the loads based on relative stiffness is as if it has supports on the end with a reaction that is proportional to its distributed force in the center and we can see that the moment diagrams of the two different assumptions depending on the stiffness of the wall elements can be significantly different. Usually, because of some redistribution because of cracks that can occur along joints, the behavior that we see in application really is probably somewhere between the two. Structural integrity is one first part that we have to consider. We have minimum tie forces and minimum wall bracing forces. ACI 318 requires consideration of structural integrity for all precast concrete structures. While a proper detailing for lateral loads will satisfy the complete load path requirement for structural integrity, there are some minimum provisions in section 16.5 of ACI 318 that must be met. With specific regard to diaphragms, the provisions to be aware of are these tie requirements in bearing wall buildings. For buildings other than bearing wall buildings, the connection of the diaphragm to the members being laterally braced by the diaphragm is a minimum nominal tensile strength of 300 pounds per foot. For the large panel bearing wall systems, the summary of the tie forces is shown in the figure. T1, the tie force is 1,500 pounds per foot of floor or roof. The T2 is a 16,000 pound tie force. T3, transverse inside is 1,500 pounds per wall. So the elements of the diaphragms that we're going to consider is a boundary element, a cord, a collector or drag strut, longitudinal joints and transverse joints. These are the parts that we're going to be looking at. The boundary element is an edge member around the perimeter of the diaphragm or around the perimeter of an opening in a diaphragm which ties diaphragms together. The cord is a diaphragm boundary element perpendicular to the applied load that's assumed to take axial stress due to the diaphragm moment. A collector or drag strut is a diaphragm boundary element parallel to the applied load that collects and transfers diaphragm shear forces to the vertical elements. The longitudinal joint is oriented parallel to the slab span and the transverse joint is oriented perpendicular to the slab span. So when I talk about longitudinal and transverse, it's in order to give you an idea of the orientation that I mean. We're going to be looking at an example that includes this layout of a floor. It has shear walls at the ends that are 30 feet long, a shear wall in the center that's 20 feet long. It has side shear walls that are 30 feet long oriented towards the center of the structure. This orientation of walls near the center of stiffness, everything will provide rigidity focused on the center of stiffness, minimizes the volume change or temperature change forces that may have to be considered in this particular example. So I just point that out that this isn't going to have a lot of volume change restraint. So we'll mention it but we'll not be working on doing volume change calculations in order to try to predict it. To satisfy structural integrity, all diaphragms should have boundary elements of some type. The boundary elements ensure that the diaphragm will have the strength to transfer the lateral loads to the lateral force resisting system. Tension reinforcement is placed in the boundary element to enable it to act as a cord to allow the diaphragm to act as a deep horizontal beam or a tied arch. The reinforcement can also provide shear friction steel as we mentioned in the last segment. It can contribute to the shear friction force transfer. Collectors are required in all diaphragms to transfer shear forces from the diaphragm to the edges of the lateral force resisting system unless the entire edge of the diaphragm is supported continuously on shear walls or frames resisting lateral forces. This is illustrated in the figures in ASCE 7 that cover diaphragms. So for the longitudinal joints for hollow core, we have grouted shear keys that provide for vertical load transfer and for horizontal shear transfer. We have grouted reinforcement for ties that also can span across joints or particularly at the ends. Now in the U.S. practice, our grouted shear keys are generally a shape that is aimed at holding the vertical alignment of the hollow core. The amount of stress that is permitted in these is relatively low. There is European practice that involves waving joints and a little bit more this difference in the surface amplitude along the length of the joints that had been developed specifically for better performance in seismic design. Those things have been tested. They're good and it's valuable information. Just mention that it's not common practice in the United States that follows that to provide a waved edge of hollow core in order to promote higher horizontal shear transfer. The longitudinal joints, again, the grouted key ways between the slabs have a capacity to transfer longitudinal shear from one slab to the next. Using a shear stress of 80 psi, the usable design strength for longitudinal shear is this PV sub n is phi times 0.08 times the height times the length with phi being 0.75. When the grout strength is exceeded or ductile behaviors required, shear friction principles can be used to design reinforcement to be placed perpendicular to the longitudinal joints. The reinforcement is placed in the transverse joints at the slab ends rather than being distributed along the length of the joints. Because there's usually a machine fabricated product, it's not going to be easy to insert mechanical connectors. It's not going to be easy to have projecting reinforcement because it's all fabricated by slab. We're looking at providing details to meet these things. Again, it's important to recognize that the friction will transfer in the joints if the joints don't open. Even though we may have values that suggest that 80 psi is well above what we need for shear transfer in a grouted joint, we do want to provide some perimeter reinforcement to ensure that those joints don't open. Under load so that that force transfer is maintained. Longitudinal joints again may have some shear friction steel running across the butt joint. Transverse joints have reinforcement and transfer joints provide shear friction reinforcement per shear in the longitudinal joints. Again, shear friction is widely used as a method of calculating the reinforcement needed to ensure the calculation of the transfer of these forces. The transfer joint may also act as part of a drag strut with axial tension or compression to carry diaphragm loads to the lateral force resisting elements. A transverse joint may also be part of a cord member where the flexural tension is resisted. Transverse joints may be an interior joint that disrupts the web of a horizontal beam where the horizontal shear would have been transferred in order to maintain the full effective depth of the diaphragm. A lot of different functions. Cord tension is resisted by reinforcement that provides flexural strength to the diaphragm. The effective depth when we're considering the calculation, we usually think of the distance d minus a over two. Well, it's hard to calculate that in a floor diaphragm. More typically, we simply take 0.8 times the depth of the diaphragm as a value to use for D for calculating the effective depth for the cord reinforcement. Because diaphragms tend to act as tied arches rather than beams, the tension in the cord reinforcement doesn't go to zero at the ends of the diaphragm. The cord reinforcement must be anchored at the ends of the diaphragm where standard hooks at the ends of the cord will suffice. It's important to understand that tied arch behavior is not exactly flexure, and this does influence some design conditions and decisions that we'll talk about. Boundary joints for anchorage at transverse boundary elements. The bars may be grouted into keyways or into hollow core slab cores where the top of the core is cut away. Concrete is then used to fill the cores for the length of the bar embedment. It's not clear when the anchorage of the connector bars and keyways is sufficient when connector bars should be placed in hollow core slab cores. And again, how much can we rely on the keyway alone in developing the bar? There's a concern that as the boundary element in the keyway crack, the anchorage for connector bar in the keyway might be lost, so we can have to consider the details carefully. Deformation and reversible loading in a seismic event would suggest that anchoring connection bars and hollow core slab cores would be preferable in more intense seismic situations. In keeping with the code philosophy, it's suggested that bars be anchored in hollow core slab cores in structures assigned to seismic design category C or higher. So, boundary joints. The edge of a member, cord, collector. We can see that we have in one case a bond beam with reinforcement with ties that run into the core of hollow core in order to tie this together and engage the bond beam reinforcement that may be the cord reinforcement. Another option is a whole cast-in-place edge strip, probably more preferably used in a seismic application where we're trying to ensure continuity of the reinforcement and then the beam with the diaphragm floor. So, there's the consideration of tops versus tops. We covered some of the code requirements before about the thicknesses of two inches to two and a half inches depending on whether it's composite or not composite. The topping slab can be designed as non-composite without the consideration of the hollow core slabs and then it's treated more like a cast-in-place concrete slab. When the topping provides the strength and the stiffness for the diaphragm but the connections are made in the hollow core slabs, shear stresses are going to be present at the interface of the topping and the hollow core. The stresses will generally be well distributed through the interface but may be more highly localized near the connections. That's structural mechanics. The primary benefit with composite structural topping are to increase the stiffness and allow easier continuous ties and plans with irregular shapes or large openings. But the seismic areas, the additional topping weight increases the seismic design forces. Top diaphragms may be a necessity in buildings assigned to high seismic design categories with plan irregularities or large diaphragm span to depth ratios. Untopped hollow core diaphragms may be sufficient when the diaphragm floor system is straightforward and in-plane diaphragm deflections are acceptable. So, topped hollow core, again reflecting back to the first part of the presentation, is controlled by 18.12.4 and non-composite controlled by 18.12.5. Composite the topping diaphragm. Composite designed for gravity loads. Composite designed for gravity loads. We can assume or sometimes assume that, again, if the stress in the interface between the topping and the top of the hollow core is less than 80 psi then the topping may be effective in transferring gravity loads through the hollow core as a composite system. But remember the effects of camber. The topping thickness at the ends may be greater than the topping at mid-span if the floor is maintained level or the topping might follow the camber of the double T. Camber being normally a normal natural process that occurs when a member is pre-stressed. We have to consider that the topping may need to have control joints in order to avoid uncontrolled cracking. And we have to think about the bond of topping to the hollow core. This 80 psi limit, which is in table 16.4.2, is actually imposed for both intentionally roughened and not intentionally roughened in the table. So the design example, I showed that earlier, it's probably a little clearer now to show that we have 30-foot walls at the sides, 30-foot walls at the ends, and a 20-foot wall in the center of this rectangular shape that's 200 feet by 80 feet. The walls are taken as masonry walls for the purpose of the example. The building is six stories without a parapet, risk category two, 14-foot floor-to-floor height, and the dead weights of eight-inch hollow core, partitioned mechanical equipment, weight of concrete framing, weight of the exterior wall system, all to provide a way to do the calculations for loads. We have a basic wind speed that's relatively high of 130 miles per hour. The building is rigid in both directions. The hollow core handbook does a calculation of the building period in order to define it for wind as being a rigid structure. The roof height of this is 84 feet. We have a table in order to provide the wind pressures and the calculations for wind pressures on a level-by-level basis through the height of the structure. We note that the suction is uniform from base and the pressure increases from the base towards the roof. So we have the table of the tributary height of the floors, the pressures on the floors, the coefficients, the wind coefficients, the total design wind pressure, how that wind pressure breaks between suction and pressure, and calculation of the total wind forces that are applied to each level of the building. The wind forces are provided for the north-south and the east-west direction, and we're calling the top of the of the screen the north direction for the purpose of looking at the drawing so that east-west is side to side. The wind forces along the story height, they're lower at the roof because it's no parapet and the exposed height is less. We can see that the forces on the wide section of the building are greater than the forces on the end of the building. When we do a rigid diaphragm distribution to the walls, the end walls get 59.1 kips, the center wall gets 17.5. The location of the maximum moment is 87.1 kips feet from the left support. The maximum moment is 2577 foot kips. Notice that because this doesn't look like a continuous beam, it looks like a beam that has a little bump in it that reduces the moment in the center. That is the effect of having a wall with less stiffness and assuming a rigid diaphragm. We have a reduced moment, but the negative moment is not enough to counteract all of the positive moment that is coming because of the stiffness of the end walls. We want to use the perimeter beams as cords. Then we look at what is the tension force, the moment divided by phi d is 44.7 kips, where we took d as 0.8 times the depth of the diaphragm. We need to connect the beam. In this case, the cords are being developed in the perimeter beams. We connect the beams to the column for the force plus the forces due to volume change and gravity loads. We have not added anything for volume change in this case. This is not a moment resisting frame, so there's not additional forces that are related to gravity loads. We're looking at connection forces that are just from the behavior of the lateral force resisting system and the diaphragm as spanning between the wall elements. The cords have to continue through that center wall, so we have a calculation for the area of reinforcement as two number six bars. In this case, again, over here on the side, I've added this. There is a slide that will show all of these figures later, but I wanted to be able to at least reference the detail that I'm talking about as I went through this. We see a detail on the side where we have the two number six bars and a bond beam and the hollow core bearing on the bond beam with an additional keyway that is providing the number three bars for shear transfer that we'll look at later. The connection of the diaphragm web to the cords, and this cord tension is transferred to the web in shear. Now, this is not a calculation of VQ over I, which would be a shear flow calculation. It's based on the distribution of the cord force from a zero shear to a maximum shear through the alternate composite design method from ACI 318. That is, the total amount of shear that we need to carry from across the joint from zero shear to maximum shear has to equal the total tension force that we're trying to develop in the tie, which is why this looks like a cord calculation, but actually produces a value that we're going to divide over the length from zero moment to maximum moment in order to determine how much shear per foot are we carrying. Now, the negative wind pressure connection requires 381 pounds per foot. We saw from the wind load calculations, we have to make sure that our connection from the perimeter from the floor to the perimeter to collect that wind pressure into the diaphragm is also provided by the calculation. In this case, the shear distribution governs over the wind calculation. Shear friction for the shear with the bars placed in the keyway perpendicular to the transverse joint, if the keyways are three feet on center, three foot wide sections here, then we have a value for n sub u divided by phi f sub y plus v sub u. We are coming up with a requirement of 0.043 n squared per foot in the keyway in order to provide that we're showing number three bars every second keyway in order to transmit the forces. Maximum, the longitudinal shear, which is parallel to the longitudinal joints. Here we're talking about the longitudinal shear is the, this is the interior joint. The maximum longitudinal shear force is the first joint away from, the first joint away from the shear wall. The center bay connections are made directly to the shear walls, so the center bay joint length only, we again, we have a v sub u from the calculation and we're looking at 0.08 times the height times the, this is the limit for the shear transfer, which is equal to 86.4 kips, which is greater than the demand for that first joint to the carrying the load to the shear wall. The transverse shear reinforcement and the transverse joints at the ends, this is the center joint. Again, the calculation for v over phi f sub y mu is a shear friction calculation, shows 1.3 square inches per two transverse joints. So we're going to need two number eight bars in each transverse joint in order to carry the shear load that we're trying to pick up from the diaphragm for this, for this transverse section. And that's the little picture that we're down here showing the grouted area, the concrete area between the ends of the cores over the center beam. Shear friction to carry the connection to the 30-foot wall on the ends. We have shear friction reinforcement for the 30-foot wall is 1.3 square inches. Negative pressure is not concurrent with shear. So the structural integrity ties that will actually control the outer playing force, which is 0.03 times 20 is six kips for the bay. And we find that a sub s required is n sub u divided by phi divided by f sub y is 0.1 square inches. We end up using six number five bars for the shear requirement here at the top near the hollow core slab ends. And the detail that's shown on the side is showing how a wet joint is providing for the tie into the masonry wall with the perimeter reinforcement. But three number five bars are providing that tie over into a broken out hollow core slab core in order to develop that transfer from the floor to the shear wall. At the center wall, we have a similar condition. We have much lower load. V sub u divided by phi f sub y mu is only 0.19 inches. So we see in the center, again, we've got a wet joint. We have the reinforcement running as a collector through the joint into the masonry wall here. And we only need two number three bars located near the hollow core slab ends. Or we could use mechanical connections if there's a way to install them. You don't have to break out the whole core of a hollow core, you only need to break out enough to provide for the development of the tie reinforcement that you're using in your detail. Now, load applied perpendicular to the hollow core spans, to the hollow core spans, that is in the east-west direction, we can look at that's a fairly simple, the sidewalls, north and south edge walls are providing the lateral support. So it's a simple span moment with a simple shear diagram. The maximum load in the east-west direction occurs at the fifth floor level, 45.7 kips uniformly distributed. The shear distribution to the walls is 22.9 and we see the maximum moment shown here in the diagram on the side just for reference. The chord forces for this configuration is very small because of the, we're going for the length of the wall. We find that the number three bars across the transverse joints are adequate for this. This is the number three bar in the keyway that we have for other purposes. So we're not adding reinforcement to something in this direction that has already been provided for the transfer for shear friction in the other direction for the design in the other direction. So the longitudinal shear, parallel to the longitudinal joints and B sub u is equal to M sub u over D where D is taken as 0.8 times the diaphragm depth. We have this 45.7, the point got missed in the figure, sorry, but 2.9 kips will not control when compared to 59.1 kips that are applied in the north-south direction. Again, it's the same approach though that was used before. This is not based on VQ over I, but based on the total force from zero shear to maximum shear, average then distributed along the length of the joint. The shear connections to the walls on the sides, again using shear friction reinforcement, we see that the transfer of the load to the 30-foot sidewalls is 0.76 kips. With the bars in the keyways at three feet on center, we can use a number three bar at every other keyway and we're back to the looking at that's where the number three bar at the second, at the second, every second keyway came in the diagram. The diagram that we have on the edge, remember that we're on top of a bond beam with two number six bars that is the cord reinforcement for the other direction. For this case, the shear in the transverse joint is 5.77 kips. The shear reinforcement we need is 0.13 kips. We have a number three bar at every second keyway, adequate to transfer the shear in the longitudinal direction in this example. We go back and look, all of those little sections that I cut, that I pasted along with the calculation sheets are cut on the page that are shown here and collected on the slide that shows the letters and the details related to where they're cut in the example problem in order to be able to put that all together so that you can see in your handout from this sheet and from the detail sheet where those details were working. Now let's jump to a light seismic design, Seismic Design Category B. Keeping out of controversy of determining what we would do in a high seismic area because there are still some significant questions of the best and most efficacious design and detailing for high seismic with hollow core, we'll take a look at some of the differences for seismic design as opposed to wind design. The risk category for the example is two. The importance factor is one. The site class is D. So the seismic design category with the map spectral accelerations at the site based on the latitude and longitude or the postal address is corresponding to a 0.2 and a one second periods or S of S is 0.217, S of 1 is 0.069 G. We go through the seismic calculations and determine with the soil factors that we have from the code and S of DS value and S of D1 value in order to be able to calculate the lateral forces. The weight of the building attributed attributable to a typical floor is shown on the slide for the floors and for the roof it's just the accumulation from the criteria that we stated earlier. The calculations of C sub S are based on a building period for its height using the simplified equation in ASCE 7. We have a building period of 0.55 seconds and for this particular example we find that C sub S based on the one second return period actually governs the lateral force on the structure. So now we have a shear of 465 kips as the base shear for the weight of the building. We do the vertical distribution of forces for the purpose of the lateral force resisting system. In order to get the vertical distribution for the lateral forces which includes WX, W sub I, H to the HX to the 0.55 power. So we have the calculation for the lateral forces on the building. Well now we also have to do a separate calculation for the diaphragm design forces as we showed by code before. Now we have the example that shows that we have a maximum force in the diaphragm 181 kips. The lower bound diaphragm force is 90 kips. The design lateral force for the roof diaphragm is 128 kips. We can see from the table over here that we reach the minimum at the lower two levels of the building and when we compare the diaphragm forces to the vertical element forces we can see the same kind of pattern that we discussed in the first part of the session. We'll go through in order to to look at this. We're doing many of the same things that we did for the wind load design. I won't spend a whole lot of time going through the details of the calculations other than to point out that we're basically following the same procedures with a few different aspects in terms of the attention to detail. In the seventh edition of the PCI design handbook low and moderate seismic design category B and C were grouped together. There's no amplification for the code specified design force is considered necessary if the design force at the uppermost level is used for every floor diaphragm. The same will also apply to structures assigned to high seismic design category D, E, and F if the lateral force is resisting entirely by special moment frames. If the seismic design category D, E, and F structures where the shear walls are part of the lateral force resisting system, we would multiply the lateral force times two for the roof level diaphragm design and then use that amplified force constant down through the height of the building. For this particular case, the diaphragm force acting parallel to the hollow core slabs in the north-south direction is about 0.64 kips per foot. Using a rigid diaphragm, the shear is 0.7 kips on the 30-foot wall and 16.5 kips on the 20-foot wall in the center. We have a similar shear and moment diagram. They're comparable to the wind load case that we looked at. The core force is calculated in the same way. In this case, the perimeter or the boundary element is drawn a little different. Again, because of the seismic application and the intent of keeping everything consistent, we have used a pore strip on the edge that is in the depth of the hollow core and across the end of the hollow core in order to provide that cord reinforcement with floor number four bars to satisfy the requirement. But actually, we're using floor number six bars because the force in the east-west direction perpendicular to the span controls this condition, which we'll show later. The connection of the diaphragm to the web is a similar take using the similar approach of 37.9 kips distributed over the length from zero moment to maximum moment of 0.44 kips per foot. The connection also has to resist the outward force from the exterior wall system per ASCE 710. The design force for the wall anchorage would be greater of the following, which is 0.4 s of ds k sub a i w sub w or 0.2 w sub w, which is 0.098 kips per foot. The exterior force is governed still by the the connection of the diaphragm to the web per shear. At the transverse joint, the same shear parallel to the transverse joint as at the cord has to be transferred. However, the tension considers the inertial force. It should consider the inertial force from the weight of the exterior bay, which is the largest of the following values. And we're looking at a at a local condition of the bay force and ensuring that we have the force for transfer. 20% of w sub p, the 0.73 kips per foot controls. And we end up with number three at three feet on center in the key ways with two number fives continuous down the joint. Longitudinal shear parallel to the longitudinal joints, the maximum longitudinal shears at the first joint away from the 30 foot wall as it was for the wind load. We're providing shear friction reinforcement in the two transverse joints and the two boundary elements for shear resistance. We're conservatively considering a 5% minimum eccentricity being resisted only by the end walls. And so we get the demand of 1.38 square inches of the four joints and 0.35 square inches per joint in order to provide that shear resistance. In the seismic calculation, the shear reinforcement was distributed over four joints as opposed to two joints in the wind calculation. In the seismic detailing, a collector is provided so that the shear can be distributed over the full width of the building and the outside bays are available for the shear transfer. No collector was used in the wind calculation so the shear had to be resisted by the center bay only. The shear friction reinforcement is provided at the outside edges of the outside bay. The cord reinforcement is also located on the outside edges. It's been considered the practice to consider these effective additives since both cause tension in the reinforcement but that's conservative as I discussed earlier when we're looking at true shear friction provisions moment due to the moment that creates tension is not generally additive to the shear friction reinforcement area but we're running across three bays of a structure and we're considering the effect of the diaphragm as a tied arch and developing the tension at the ends of the tied arch. It is probably conservative but safe to consider them as additive because we're treating the tied arch bars as tension and not truly as flexural reinforcement. ACI 318.14 22.9.4.6 requires the area of reinforcement required to resist the net tension to be added to the tension to the reinforcement. Again that's what I've explained since the cord in the hollow core is more in tension we're using it that way. The longitudinal shear. The shear parallel to the longitudinal joints for the seismic case following the same approach that we used before. The moment is 164 kip feet. The A sub S requirement is 0.4 square inches in the transverse joint A sub S is 0.35 inches where the four number six bars that we have in the perimeter is okay for this calculation in the center it's where we have two number five bars to satisfy A sub S for 0.35. Shear parallel to the longitudinal joints is to transfer the shear to the wall and the collector element where 62.1 kips divided by 80 is 0.78 kip for feet that's 80 is the length of the collector. A sub V is a shear friction calculation with mu as mu equal to one we need 0.017 inch square per foot and we're doing that with number four bars at 18 on center and this is the edge condition again it's a fully grouted condition considering that we're concerned with seismic detailing. Seismic detailing the longitudinal shear parallel to the longitudinal joints this is the use of number four hairpins at eight feet on center in this in this case we have the collector element reinforcement of for 19.5 kips in seismic design category B the amplification factor of the collector design force is not required but it was included in the design example in the holocore manual to illustrate the procedure of applying the overstrength factor omega sub zero to the collector force to show the effect of increasing the reinforcing steel when a stronger collector force requirement is in fact in force again that would really only apply for seismic design category C and above but it was used in the holocore manual in order to illustrate the principle. The longitudinal shear parallel to the longitudinal joints at the shear wall connection this is the center connection down in in this section we're looking at the v sub u over the building width 0.14 kips per foot determine the ace of vf is actually very small using the number four bars at eight feet on center for the shear friction transfer into the center wall and there are two number four bars that are run in as the longitudinal reinforcement in the joint in the center wall. We have the longitudinal shear parallel to the longitudinal joints we've got the collector element reinforcement calculation shown with the detail that's there. The calculation of diaphragm deflection is also something to be checked to idealize the diaphragm section as a transfer section to solve for the neutral axis depth calculate an eye crack about the neutral axis and for rigid diaphragm the deflection can be calculated as a uniform load deflection between the end walls less the effect of the center wall as a concentrated beam. We end up with diaphragm design force acting perpendicular to the holocore slabs as a different calculation for the other direction the longitudinal the longitudinal shear doesn't control it as much as as it's much smaller than the longitudinal shear caused by the diaphragm the shear connection to the wall with five percent eccentricity provides a value that the that the number three bars at three feet is adequate the collector reinforcement is higher but the ace of s requirement was floor number fours the determination in the design example was to use the floor number six for this overall collector down the length of the building. There is finally a shear in the transverse joint calculation shown in the slides time is running short so I will move on to um to the to forge the conclusion part which shows again a layout that shows the location of the walls and the layout of the of the section cuts of where what came from the slides that you've seen for the calculation of the various levels of shear and then provided behind that the details that I distributed through the slides are collected together so that you can see the overall design for the example condition for seismic design category b and compare it back to the design for the wind load that was shown in the first part of the design example. Again these things all follow from the PCI holocore manual it's covered in a chapter that covers all about the diaphragm designs done under ASCE 710 and ACI 318 14 the the material is is there for reference it's sometimes difficult to follow all of the details when we try to put it all into a couple of hours but I hope that this provides sufficient introduction to see the detailing and the design requirements the code provisions that are applicable to holocore design for for the diaphragms and with that running to the end of my time I will say thank you and if if there are questions and we haven't addressed questions that you may have if you provide the questions to PCI I certainly will will gladly take them up and and try to make a response in time after this this presentation. Thank you. On behalf of PCI I'd like to thank Dr. Cleland for his presentation and all of our attendees for their participation. Unfortunately we don't have time for questions but they will all be forwarded to Dr. Cleland along with your contact information as well. As a reminder certificates of continuing education will appear in your account at www.rcep.net within 10 days. If there were multiple people at your location please send the completed attendance sign-in sheet to marketing at pci.org. A recording of this webinar will be uploaded within a week. If you have questions regarding the webinar content please email marketing at pci.org with the subject online holocore webinar. There will also be a pop-up survey that will automatically open once this webinar ends and the program closes. Thank you and have a great day.

holocore diaphragm design

rigid diaphragms

flexible diaphragms

determining diaphragm type

thickness requirements

reinforcement

seismic load path

cast-in-place composite topping

shear strength

flexural strength

ACI 716 methodologies

shear friction reinforcement

boundary elements

composite diaphragms

lateral forces