Tenth U.S. National Conference on Earthquake EngineeringFrontiers of Earthquake Engineering July 21-25, 2014 Anchorage, Alaska 10NCEE

EVALUATION OF COLLECTOR DESIGN FOR CONCRETE DIAPHRAGMS

J. S. LeGrue1

ABSTRACT This paper provides a comparison of collector design methods for concrete diaphragms. The traditional method, most commonly associated with flexible plywood diaphragms, is generally considered overly conservative for the design of concrete diaphragms. To limit the amount of collector reinforcement required, practitioners use alternative methods that rely on the rigidity of the diaphragm to distribute inertial forces. Two such alternatives, one assuming ideally rigid diaphragm behavior and one assuming limited length collector elements, are considered. As is common in practice, collector designs are performed for example structures with varying diaphragm shear forces, diaphragm lengths, and configurations utilizing hand calculations. Finite element analysis is used to evaluate the effectiveness of the design methods. While all three methods result in acceptable performance, the flexible diaphragm method is conservative and the limited collector length method produces the most efficient design. However, the effect of building configuration on collector stresses is more significant than is assumed by the alternate methods. Consequentially, additional consideration must be given where walls are located only at one end of the line of resistance. However, in other conditions, the alternate methods provide comparable performance to the flexible diaphragm method at lower cost.

1Senior Project Engineer, Hohbach-Lewin, Inc., Structural Engineers, Eugene, OR 97401 LeGrue JS. Evaluation of Collector Design for Concrete Diaphragms. Proceedings of the 10th National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, 2014.

EVALUATION OF COLLECTOR DESIGN FOR CONCRETE

DIAPHRAGMS

J. S. LeGrue1

ABSTRACT This paper provides a comparison of collector design methods for concrete diaphragms. The

traditional method, most commonly associated with flexible plywood diaphragms, is generally considered overly conservative for the design of concrete diaphragms. To limit the amount of collector reinforcement required, practitioners use alternative methods that rely on the rigidity of the diaphragm to distribute inertial forces. Two such alternatives, one assuming ideally rigid diaphragm behavior and one assuming limited length collector elements, are considered. As is common in practice, collector designs are performed for example structures with varying diaphragm shear forces, diaphragm lengths, and configurations utilizing hand calculations. Finite element analysis is used to evaluate the effectiveness of the design methods. While all three methods result in acceptable performance, the flexible diaphragm method is conservative and the limited collector length method produces the most efficient design. However, the effect of building configuration on collector stresses is more significant than is assumed by the alternate methods. Consequentially, additional consideration must be given where walls are located only at one end of the line of resistance. However, in other conditions, the alternate methods provide comparable performance to the flexible diaphragm method at lower cost.

Introduction

Diaphragm classification is one of the first steps in the lateral analysis of a structure. Engineers traditionally classify a diaphragm as flexible or rigid based on its geometry, composition, and stiffness relative to the stiffness of the vertical elements supporting it. Diaphragm classification affects many aspects of the lateral analysis of the building including the distribution of lateral forces to vertical elements, how torsion is accounted for in the analytical building model, and determination of building deformations. However, one area of lateral design in which diaphragm classification is not consistently addressed is collector design. This is ironic as collectors are part of the diaphragm. Traditional collector design is consistent with the assumption of a flexible diaphragm. That is, it assumes that the diaphragm lacks stiffness and requires the addition of stiff collector elements to deliver diaphragm forces to vertical elements of the Lateral Force Resisting System (LFRS). This approach has its basis in the design of flexible plywood diaphragms and is presented in various introductory engineering texts such as [1]. It has obvious benefits for the diaphragm, which is designed for a low shear demand. However, the use of discrete collector elements loaded axially constitutes a load path with a less redundancy and less ductility than a

1Senior Project Engineer, Hohbach-Lewin, Inc., Structural Engineers, Eugene, OR 97401 LeGrue JS. Evaluation of Collector Design for Concrete Diaphragms. Proceedings of the 10th National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, 2014.

continuous diaphragm loaded in shear. Consequentially, building codes such as ASCE 7-10 require that most collector elements be designed to resist amplified seismic forces [2]. Alternatives to the traditional methodology are utilized in practice to eliminate “unnecessary” collectors and reduce collector demands where collectors are required. Two such alternatives are considered in this evaluation. The rigid diaphragm method relies on the stiffness of the diaphragm to redistribute diaphragm forces to the locations of vertical elements. This method does not utilize collector elements. Instead, the diaphragm is designed to have sufficient capacity to deliver the design force to the vertical elements within the length of the vertical elements. The limited collector length method is a hybrid approach that utilizes a collector of limited length determined so that diaphragm strengthening is unnecessary, however, collector forces do not reach the magnitude of a full-length collector. A series of example structures with varying diaphragm forces, collector lengths, and geometries is considered. Collector designs are performed using hand calculations for each of the three methods. These calculations provide a comparison of the material requirements and implied construction costs of the methods; however, they do not provide a basis for comparing the expected performance of the designs in a seismic event. To this end, finite element analyses are performed and steel and concrete stresses are evaluated.

Example Structures The example structures are single-story concrete shear wall buildings with concrete roofs as shown on Fig. 1.

Figure 1. Example structures (plan view).

Diaphragm length, wall location, and diaphragm design force, , have been varied as indicated on Table 1. Other design parameters are shown on Table 2.

Table 1. Example structure configuration.

Label Ldiaph

(ft) Wdiaph

(ft) Wall

Location Lwall

(ft) Fpx

(k) 1A 150 300 Center 25 750 1B 150 300 End 25 750 2A 150 300 Center 25 1200 2B 150 300 End 25 1200 3A 75 300 Center 25 750 3B 75 300 End 25 750 4A 75 300 Center 25 1200 4B 75 300 End 25 1200

Table 2. Diaphragm design parameters.

Diaphragm thickness, 10 in Diaphragm reinforcement ratio, 0.00194 (#5 @ 16 in o.c.) Specified concrete strength, 4000 psi

Concrete shear strength, 2 126 psi

Concrete tension strength, 7.5 474 psi

Concrete compressive strength, 0.85 3400 psi Reinforcing steel strength, 60000 psi

Although shear wall design is not part of this evaluation, it has been assumed that the structures are classified as special reinforced concrete shear wall buildings. Per ASCE 7-10 Table 12.2-1, the building response modification factor is R=5 and the overstrength factor is = 3.0. The walls on each side of the structure are of equal strength and stiffness. The diaphragm design force is uniformly distributed. Accidental eccentricity has been neglected. Therefore, the diaphragm force at each wall, , is

= . (1)

Diaphragm Shear Strength

The diaphragm and collector designs used as the basis of this evaluation have been performed in accordance with ACI 318-11 [3]. Diaphragm shear strength, , is given by ACI 318-11 Sec. 21.11.9 as = (2 + ). (2)

The resistance factor, , has been taken as 0.60 per ACI 318-11 Sec. 9.3.4 for shear-governed members designed to resist seismic forces. , the gross area of concrete parallel to

, is equal to the diaphragm length times its thickness. For normal-weight concrete, = 1.0. The reinforcement ratio, , is shown on Table 2. The diaphragm unit shear strength, , expressed on a per lineal foot basis is

= (0.60)(12in)(10in) 2√4000 + (0.00194)(60000psi) = 17.5k/ft. (3)

Collector Design

The following sections demonstrate collector design for the example structures in accordance with the three methods outlined above. Results have been summarized in Table 3 at the end of this section. Note that some methods do not result in collector designs for some of the example structures. Fig. 5 at the end of this section shows the reinforcement layout for all of the collector designs. Flexible Diaphragm The traditional approach to collector design, this method assumes that shear is uniformly distributed over the length of the diaphragm. Collector elements are provided at the portions of the diaphragm that do not abut the vertical LFRS members. Collector diagrams for the example structures are shown on Fig 2.

Figure 2. Flexible diaphragm collector diagrams. The maximum collector force, , is computed as follows.

= (4)

where and are the lengths of the collector and diaphragm, respectively. Eq. 5 contains the diaphragm unit shear expressed as

= . (5)

The required collector reinforcement, , is determined as follows.

= (6)

where = 0.90 per ACI 318-11 Sec. 9.3.2.1. The collectors have been detailed as slab collectors following an example presented by the Structural Engineers Association of California [4]. This method results in the same amount of collector reinforcement as a collector beam, but distributes the reinforcement over a larger area, potentially reducing rebar congestion and eliminating confinement reinforcement. A portion of the collector reinforcement, denoted for “concentrated”, has been placed directly in line with the shear wall and the remainder of the reinforcement, denoted for “distributed”, has been placed in the slab abutting the wall. A free-body diagram of the collector is shown on Fig. 3. The widths associated with the tension and compression forces are arbitrary; however, the designer is cautioned to be mindful of ACI 318-11 Sec. 21.11.7.5 when determining the compression width.

Figure 3. Slab collector free-body diagram.

The eccentric layout of the reinforcement with respect to the wall results in a moment, , which is resisted by the couple . = + = (7) where and are the collector compression and tension forces that are distributed in the slab and and are their respective eccentricities with respect to the wall centerline. Assuming that the compression force is uniformly distributed across the slab width used to resist compression, = . Tension is assumed to be proportional to the area of the collector reinforcement resulting in

= . (8)

Additional reinforcement at the ends of the wall, denoted , provides :

= . (9)

Rigid Diaphragm In contrast to the traditional method, this method relies on the strength and stiffness of the diaphragm to redistribute internal forces to the portion of the diaphragm abutting vertical members of the LRFS. Additional collector elements are not required if the diaphragm has adequate shear strength over the length of diaphragm abutting the vertical elements. Hence, diaphragm shear design is based on the length of the wall or frame instead of on the overall diaphragm length. One consequence of this method is that the result is independent of wall location and diaphragm length. Thus, for the example structures, only the two differing values of need to be considered. For = 750 kips, the diaphragm unit shear is

= kips( )( ft) = 15k/ft < = 17.5k/ft. (10)

Therefore, the typical diaphragm reinforcement is adequate and no additional reinforcement is provided. However, for = 1200 kips, = 24k/ft > , and additional diaphragm reinforcement is required. The amount of additional reinforcement to provide can be determined using Eq. 2. An additional #5 at 16 inches on center (i.e., #5 at 8 inches on center net) yields = 25.8 k/ft, which is adequate. Limited Collector Length This is a hybrid approach based on the work of Johnson and Welton [5]. It utilizes diaphragm rigidity to distribute forces internally similar to the rigid diaphragm method, but provides a collector element instead of additional diaphragm reinforcement. This method is especially applicable in situations where diaphragm strengthening is not practical, for example, due to rebar congestion or the diaphragm strength limits of ACI 318-11 Sec. 21.11.9.2. The collector element is provided over the minimum length, , required to achieve

= ≤ . (11)

That is,

= and (12)

= . (13)

The collector diagram is shown on Fig. 4. If < , no collector is required and the design is equivalent to Method 2.

Figure 4. Limited length collector diagrams.

Collector Design Summary Reinforcement requirements are shown for each of the example structures on Table 3. The weight of additional reinforcement (Wgt) is provided as a comparison of the methods because it accounts for the length of the reinforcement and is directly related to construction cost. Fig. 5 shows the reinforcement layout for each configuration.

Table 3. Collector design forces and reinforcement.

Flexible Diaphragm Rigid Diaphragm Limited Collector Length

Label Tu

(k) As

(in2) Ase

(in2) Wgt.(lbs)

Tu

(k) As

(in2) Wgt.(lbs)

Lmin

(ft) Tu

(k) As

(in2)

Wgt.(lbs)

1A 391 7.5 0.59 1941 N/A 0 0 21.4 N/A 0 0 1B 781 15.0 2.10 3929 N/A 0 0 21.4 N/A 0 0 2A 625 11.7 1.72 3055 N/A 5.8 2490 34.3 203 4.0 586 2B 1250 23.6 4.23 6297 N/A 5.8 2490 34.3 406 7.6 1014 3A 313 6.0 0.19 769 N/A 0 0 21.4 N/A 0 0 3B 625 11.9 0.85 1549 N/A 0 0 21.4 N/A 0 0 4A 500 9.5 0.80 1237 N/A 5.8 1245 34.3 203 4.0 586 4B 1000 18.8 1.82 2484 N/A 5.8 1245 34.3 406 7.6 1014

Figure 5. Reinforcement layout for example structures.

Finite Element Analysis Analytical models of select example structures were created and analyzed using the ETABs finite element program. The models consisted of concrete slabs (modeled with shell elements) and reinforcement (modeled with line elements). Diaphragm design forces, , were amplified by the building response modification factor, = 5, for special reinforced concrete shear walls. The design force was applied as a uniformly distributed surface load on the slab.

Iterative analysis was performed to account for cracking. Slab elements with tension stress greater than the concrete tension strength were removed from the model and the analysis was re-run until no such elements remained. Maximum stress ratios (i.e., the ratio of stress to yield stress) for the reinforcement in tension, concrete in shear, and concrete in compression are given on Table 4. Fig. 6 shows a comparison of the cracked regions (denoted by removed slab elements) for example structures 2A and 2B. Table 4. Maximum stress ratios for distributed reinforcement, (Fsd), concentrated reinforcement

(Fsc), concrete compression (Cc), and concrete shear (Vc).

Flexible Diaphragm Rigid Diaphragm Limited Collector Length

Label Fsd Fsc Cc Vc Fsd Fsc Cc Vc Fsd Fsc Cc Vc 1A 0.6 0.9 0.2 1.9 1.9 N/A 0.2 1.7 Same as rigid diaphragm 1B 1.2 1.9 N/A 2.7 5.4 N/A N/A 3.4 Same as rigid diaphragm 2A 0.7 1.2 0.4 2.1 2.0 N/A 0.3 2.5 2.2 0.9 0.4 2.4 2B 1.5 2.2 N/A 4.0 5.5 N/A N/A 5.1 5.0 2.9 N/A 3.8 3A 0.6 0.9 0.2 3.1 1.7 N/A 0.2 3.4 Same as rigid diaphragm 3B 1.1 1.8 N/A 2.8 5.0 N/A N/A 3.5 Same as rigid diaphragm 4A 0.8 1.2 0.3 5.0 1.7 N/A 0.3 5.4 1.1 1.6 0.3 4.9 4B 1.3 2.1 N/A 4.2 5.0 N/A N/A 5.3 5.1 2.3 N/A 4.4

Figure 6. Finite element models for structures 2A and 2B. Un-shaded elements have been

removed from the model due to tension stress greater than capacity.

Findings and Conclusions Overall, the three design methods resulted in stress ratios for concrete and steel that were generally less than or equal to the building response modification factor of = 5 indicating performance consistent with the intent of ASCE 7-10. Additionally, steel stress ratio was not significantly affected by diaphragm shear for any of the methods so the hand calculations appear to be sufficient, even for loads that produce non-linear responses such as extensive cracking. The steel stress ratio ranged from 0.6-1.9 for the flexible diaphragm method to 1.7-5.5 for the rigid and limited collector length methods, validating the hypothesis that that the flexible diaphragm method is overly conservative. Wall configuration also produced significant variation in steel stress for each of the design methods. Steel stress for end wall structures were 1.5-2.0 times as high as centered wall structures for the flexible diaphragm method and 2.7-3.0 times as high for the other methods. Variation in shear force and diaphragm length produced relatively minor (less than 30%) variations in steel stress ratio. In contrast, the concrete stress ratio was less sensitive to design method (25% maximum variation) and more sensitive to shear force (50% maximum variation). Concrete stress was more sensitive to wall configuration for the longer diaphragm structures 1 and 2 than for structures 3 and 4. In comparing the alternate methods, the limited collector length designs resulted in smaller steel and concrete stresses than the rigid diaphragm designs and did so with less additional reinforcement. Finally, wall location affected steel stress more than the other configuration parameters. This is notable because wall location is not considered in the rigid diaphragm and limited collector length methods. The end wall configuration used in the examples is an extreme case that eliminates force transfer via compression in the diaphragm. It should not be assumed that that the difference between the design methods would be as large if diaphragm compression was not completely eliminated. However, it is also not conservative to assume, as the alternate methods do, that wall location can be ignored completely. Based on these results, the limited collector length method produces the most efficient collector design. However, the designer is cautioned to consider the effect of wall placement on collector forces, even though that is not explicitly considered in the design method.

References 1. Breyer D, et al. Design of Wood Structures-ASD/LRFD. McGraw-Hill: New York, 2006.

2. American Society of Civil Engineers. Minimum Design Loads for Buildings and Other Structures, ASCE/SEI 7-10. ASCE: Reston, Virginia, 2010.

3. American Concrete Institute ACI 318 Committee. ACI 318-11: Building Code Requirements for Structural Concrete and Commentary. ACI: Farmington Hills, Michigan, 2011.

4. SEAOC Seismology Committee. Concrete slab collectors. The SEAOC Blue Book: Seismic Design Recommendations. Structural Engineers Association of California: Sacramento, 2008.

5. Johnson MW, Welton JC. Analysis of collector/drag elements in rigid diaphragms. Structural Engineers Association of Southern California: Fullerton CA, 2003.