Because of their architectural versatility and energy

Transcription

1 Precast Concrete Corbels for Insulated Wall Panels Proposed system minimizes thermal bridging at wall connections by Mohamed Elkady, Maher K. Tadros, Mark Lafferty, George Morcous, and Doug Gremel Because of their architectural versatility and energy efficiency, precast concrete insulated wall panels are increasingly being specified on building projects. A common insulated wall system comprises 10 in. (250 mm) thick, composite panels. In this case, the designation indicates the thicknesses (in inches) of the exterior concrete wythe, insulation layer, and interior concrete wythe, respectively. Extruded polystyrene (XPS), expanded polystyrene (EPS), or polyisocyanurate (generally classified as polyurethane, or PU) boards are normally used as the insulation material. Energy-efficient panels are fabricated with fiber composite polymer connectors passing through the insulation layer. If the connectors also have high shear stiffness and capacity, an insulated panel will have the structural capacity of a 10 in. solid panel. Insulated panels can thus provide a weight savings of 40% over solid panels, and an insulation rating of nearly R-21 (thermal resistance of 21 h ft 2 F/Btu or 3.7 m 2 K/W). Because these panels are also air and moisture barriers and can provide both the exterior and interior finishes for a building all in one product they are very competitive with alternative cladding systems. Wall panels often require corbels to support roof and floor joists. Because corbels are very short cantilevers, they are subject to special loading and design criteria as given by Chapter 16 of the ACI Building Code 1 and Section of the seventh edition of the PCI Design Handbook. 2 The shear-flexure and shear-friction approaches are used to ensure that reinforcement crosses the nearly vertical cracks that are expected to develop at the interface between the corbel and the wall as the load approaches the corbel s ultimate capacity. For insulated wall panels, a common practice is to connect the two concrete wythes with a solid concrete block through the insulation at the corbel location (Fig. 1). As discussed in Section of the PCI Design Handbook, however, the resulting thermal bridge significantly reduces the energy Ci Design calculations are provided in an appendix, available with the online version of this article efficiency of the wall panel. For example, the handbook provides an example showing that the thermal resistance of a wall panel with a solid zone comprising 9% of the total panel surface area can be reduced by as much as 42%. An insulation penetration can also result in increased vapor transmission through the wall and/or condensation on the exposed surfaces. Depending on the climate and building use, the accumulation of moisture on the surfaces can lead to degradation of indoor air quality or the appearance of the panel. These effects are the main reasons for the research summarized in this article: an evaluation of the structural capacity of an insulated wall panel corbel system with minimal thermal bridging. This article describes the design and testing of a corbel that has been developed incrementally through a focused testing program. Design calculations are provided in an appendix, available with the online version of this article. Current Corbel Connections Figure 1 shows typical corbel connection details currently used with insulated concrete wall panels. For composite panels, which typically have relatively thin exterior and interior wythes, it is generally not possible in current practice to limit the steel reinforcement in the corbel to the interior wythe only. Thus, a solid block is created at the corbel by Interior wythe Corbel Corbel Typical wythe thickness (c) Corbel Fig. 1: Current practice for corbel connections: corbel with solid block; corbel with local thickening of interior wythe; and (c) corbel in noncomposite panel with thick interior wythe Ci October

2 cutting an opening (typically, 2 x 3 ft [0.6 x 0.9 m]) in the insulation layer (Fig. 1). To minimize thermal bridging, some designers opt to reduce the insulation thickness to allow extra interior wythe thickness to house the corbel reinforcement (Fig. 1); however, overcoming the associated detailing and fabrication issues can be time-consuming, and the detail will reduce the thermal efficiency of the panel. Noncomposite wall panels, which generally have load-bearing interior wythes that are at least 6 in. (150 mm) thick, can be constructed with no thermal bridging at corbels (Fig. 1(c)). However, this type of panel also requires significantly more materials and is significantly heavier than a composite insulated panel. Research Objective The objective of this research program was to develop corbel details for precast concrete insulated walls that: Minimize thermal bridging by eliminating the need for solid concrete penetrations or local reductions in thickness of insulation; and Ease panel fabrication by allowing producers to precast and pre-insert corbels in wall panel forms prior to placement of panel concrete. Loads and Materials The proposed corbel design is suitable for support of 12 ft (3.6 m) wide double tees spanning 60 ft (18 m), with selfweight of 85 lb/ft 2 (4.1 kn/m 2 ), superimposed dead load of 15 lb/ft 2 (0.7 kn/m 2 ), and live load of 50 lb/ft 2 (2.4 kn/m 2 ). 3 For creep analysis, 25% of the live load is assumed to be sustained. The resulting reactions are: Double-tee weight (D) = ( )/4/1000 = kip (68 kn); Superimposed dead load (SIDL) = ( )/4/1000 = 2.70 kip (12 kn); Live load (LL) = ( )/4/1000 = 9.00 kip (40 kn); Factored load = 1.2(D + SIDL) + 1.6L = kip (160 kn); Service load = D + SIDL + L = kip (120 kn); and Sustained load = D + SIDL L= kip (90 kn). The materials in the test corbels had the following properties and design parameters: Self-consolidating concrete, with specified strength of 8000 psi (55 MPa); Grade 60 bars per ASTM A615/A615M, with yield strength of 60,000 psi (414 MPa); Grade 60 bars per ASTM A706/A706M, with yield strength of 60,000 psi (414 MPa) for welded bars; High-density polyethylene (HDPE) plate, with tensile strength of 4600 psi (30 MPa) per ASTM D638 and Shore D hardness of 69 per ASTM D2240; and No. 3 (3/8 in. [10 mm] diameter) glass fiber-reinforced polymer (GFRP) bars with tensile strength of 110,000 psi (760 MPa) and modulus of elasticity of 6500 ksi (44,800 MPa). In conventional corbel design, the reinforcement quantity is determined using the strength limit state, while the reinforcement details are developed based on experimental results. However, due to the low stiffness of the GFRP bars and the HDPE plates, which are the primary tension and compression elements between the two wythes at the corbel, respectively, it is important to ensure that proposed corbel details have acceptable deformation under service loads. Therefore analyses are required for both service and strength limit states. The design must also include consideration of the effects of sustained loading and environmental effects on the long-term strength of GFRP bars, as well as the effect of the low ductility of GFRP bars on the safety of the assembly. Thus, ACI 440.1R 4 recommendations are used for the design of the GFRP bars in the corbel assembly, with the following factors applied to the GFRP bar manufacturer s published strength: A creep rupture factor of 0.2 is applied to account for sustained load effects; An environmental factor of 0.7 is applied to account for the effects of an exterior environment. Although this factor is appropriate for bars in severe environments such as in bridge decks, it is conservatively used here until further research justifies its removal; and A strength reduction factor, φ, of Structural Behavior and Analysis Methods Corbels are short cantilevers. If the span is very short, the shear-friction approach is appropriate. If the span is between 50 and 100% of the depth, the shear-flexure approach is appropriate. If the span is longer than 100% of the depth, diagonal tension cracking is expected to develop and conventional shallow beam design approaches can be used. Section of the PCI Design Handbook includes detailed examples for strength design of corbels, with both approaches checked and the larger amount of steel used. The Handbook also shows how the strut-and-tie method of strength analysis of disturbed regions can be used to determine the reinforcement required to meet strength demands. However, none of the three methods directly deals with deflection and crack control. They are indirectly incorporated through recommended details that have been subjected to experimental verification. Note that the main flexural reinforcement currently used is steel. It has a much higher modulus of elasticity than the GFRP bars proposed in this study. To develop corbel details incorporating GFRP tension ties, Elkady 5 built on information that has been used successfully for many years. In addition, he developed analysis methods for crack control under service loads as well as for evaluating creep of the GRFP bars under sustained loads. Also, he used finite element analysis (FEA) modeling to evaluate deformation and cracking at incrementally increasing loads. The following sections describe the corbel detailing that was developed after extensive analysis and numerous trials, which are not described here for brevity. More details are available in Elkady 5 and in Morcous and Tawardrous october 2015 Ci

3 An example analysis of the proposed corbel is provided in an appendix that is available with the online version of this article. Details of Proposed Corbel Figure 2 shows the dimensions of the corbel developed after a number of trials. The corbel was designed to be prefabricated and suspended while the top wythe concrete is placed around and beneath it. The design was also based on an assumption that each wythe of the wall panel has a thickness of at least 3 in. (75 mm). The ultimate production goal is to have the corbel, along with the extra reinforcement associated with it, pre-assembled ahead of the wall production and suspended from the forms before wall concrete is placed. In this way, valuable casting-bed time is preserved as much as possible. To ensure ductility, a relatively large corbel width of 18 in. (450 mm) was selected to allow meshing of the corbel reinforcement with added panel reinforcing in the vicinity of the corbel. The corbel height (10 in. [250 mm] plus a taper of 4 in. [100 mm]) was found to satisfy the maximum reinforcement limits of the ACI Code. To ensure interaction between the separately cast concrete components, the corbel was detailed to be embedded at a depth of 3/4 in. (20 mm) into the interior (top, as-cast) wythe. As would be required in any precast corbels, all test corbels were fabricated with the surface in contact with the concrete wythe deliberately roughened to a 1/4 in. (6 mm) amplitude before the concrete fully set. During the development of the corbel described in this article, a variety of details and options were analyzed and tested. One type, Type A, used GFRP NU-Tie bars (Fig. 3) as the main reinforcement connecting the corbel to the wall. A second type, Type B, used GFRP U-bars for main flexural reinforcement (Fig. 3). While tests verified that both types achieved the required performance goals (References 5 and 6), only the latest version of the Type B system, Type B2M, is covered in detail in this paper (Fig. 4 and 5). Flexure is transmitted by tension in the flexural reinforcement and compression block at the bottom of the corbel (Fig. 4). The insulation has a limited compression capacity and a very small stiffness. Thus, to ensure acceptable deformation and strength of the assembly, 1 ft 2 in. Roughen surface 10 in. 4 in. 8-3/4 in. 1 ft 2 in. 10 in. 4 in. 1 ft 6 in. Fig. 2: Precast corbel dimensions: side view (corbel is detailed to have 8 in. net projection from interior wythe of finished wall panel); and front view (Note: 1 ft = 0.3 m; 1 in. = 25 mm) 8 in. 3 ft 7-1/4 in. 1 ft 7 in. Fig. 3: Dimensions of GFRP bars used in test corbels: NU-Tie used in Type A corbels; and U-bar used in Type B corbels. NU-Ties were used as insulated panel connectors in all test panels. Note that the inside bend radius of the No. 3 (3/8 in. or 10 mm) GFRP U-bars used in this study was 1-3/4 in. (44 mm). While a larger radius would reduce the stress concentration at the bend, it would also occupy more space than available in the connection (Note: 1 ft = 0.3 m; 1 in. = 25 mm) 7-5/8 in. 1 ft 4 in. R 1-3/4 in. R-1 3/4 in. (1) No. 3 3 ft 10 in. (6) No. 3 GFRP U-Bars HDPE board (4) 1x4x18 in. 3 in. 4 in. 3 in. 3/4 in. 3/8 in. (2) No. 5 3 ft 5 in. a low-conductivity material of adequate strength and stiffness must be used to replace the insulation board in the anticipated compression block area. The product should have a modulus of elasticity comparable to that of the fiberglass bar material and absorption (3) No. 4 3 ft 10 in. 7-5/8 in. Embed PL 3/8x3x17 in. with 2 No. 4 bars (1) No. 3 closed stirrup 10 in. 9-3/4 in. 3/8 in. 1 ft 4 in. 1 ft 4 in. R 1/2 in. 1 ft 3-3/4 in. R 3/4 in. 1 ft 2 in. R 1-3/4 in. (2) No. 4 bars welded to plate 1/4 in. (1) No. 3 PL 3/8x3x17 in. Fig. 4: Proposed corbel Type B2M. The welded plate embed and the No. 3 closed stirrup tie the corbel to the interior wythe of the wall panel. The six No. 3 GFRP U-bars tie the corbel to the exterior wythe and provide tensile resistance for the corbel. Unless noted otherwise, steel bars are Grade 60 per ASTM A615/A615M. Welded bars to the 3/8 in. thick plate are Grade 60 per ASTM A706/A706M (Note: 1 ft = 0.3 m; 1 in. = 25 mm) Ci October

4 Fig. 5: Isometric views of corbel Type B2M: as assembled in a wall panel, with the GFRP and steel bars embedded in the precast concrete corbel; and an exploded view of the components and fire resistance properties comparable to those of the insulation. For the Type B2M detail, a 4 x 4 x 18 in. (102 x 102 x 457 mm) HDPE strip was used to transfer the compressive force at the base of the corbel. This material, commonly used by precasters for fabrication of formwork, is sold under many trade names (for example, CORRTEC-HITEC HDPE). In the Type B2M corbel, the applied corbel force is resisted by a combination of elements. Mechanical interlock is provided by roughening of the face of the corbel and embedding that face 3/4 in. (20 mm) into the interior concrete wythe. GFRP U-bars, which connect the corbel through the insulation to the exterior wythe, provide the tensile tie for the corbel. In addition, three steel stirrups are anchored in the interior wythe and contribute additional tensile resistance for shear friction and beam-shear crack control. The top two stirrups are welded to an anchorage plate located at the face of the insulation (Fig. 4). The plate anchor was implemented because a typical stirrup with 90 degree bends would not be practical within the 3 in. (75 mm) thickness of the interial wythe. The plate provides an additional advantage, as it rests on the insulation board during panel fabrication. Four horizontal No. 3 bars, each 46 in. (1170 mm) long, are also included in the corbel design. A No. 3 bar extending through the GFRP bar hooks in the exterior wythe helps anchor the GFRP U-bars. The three No. 3 bars in the interior wythe, together with two vertical No. 5 bars, each 41 in. (1040 mm) long, help increase the size of the concrete break-out zone and distribute cracking as the load approaches the ultimate capacity of the system. The efficacy of these bars was established with the aid of finite element analyses and experimental trials (per References 5 and 6). Although analyses showed that four No. 3 GFRP bars would be adequate for providing the tension tie for flexural strength purposes, an additional two No. 3 bars were used to Reaction wall Hydraulic ram and load cell Tie-back frame Precast corbel Test panel Loading frame 6 ft 0 in. Fig. 6: Schematic of test apparatus used to apply vertical load to corbels installed in insulated concrete test panels (Note: 1 ft = 0.3 m; 1 in. = 25 mm) allow the design to meet the requirements of ACI 440.1R for creep rupture. Having the vertical No. 5 bars tightly placed against the corners of the stirrups further enhances the stiffness of the connection in tension. Experimental Investigation Eleven full-scale tests were undertaken, including tests of Type A corbels, which incorporated GFRP NU-Ties. Type B corbels, described in detail herein, use GFRP U-bars to provide a tensile tie from the top of the corbel to the exterior wythe of the wall. The testing was done incrementally, with improvements introduced as information was gathered from preceding tests. Both connection types produced acceptable results, but in the interest of brevity, only the final three full-scale specimens, denoted Type B2M, are described in this article. 8 ft 0 in. 56 october 2015 Ci

5 Test specimens were 6 ft (1.8 m) wide, 8 ft (2.4 m) tall, insulated precast concrete panels with GFRP NU-ties connecting the two 3 in. (75 mm) concrete wythes through a 4 in. (100 mm) layer of XPS insulation. Each specimen was fabricated with a precast concrete Type B2M corbel installed at the panel centerline. After the concrete reached the design strength, a panel was erected against a reaction wall for lateral support (Fig. 6). A hydraulic ram was used to apply a vertical load on the corbel. The vertical force and overturning moment induced in the test panel were resisted by reactions at the base and a tie-back to the reaction wall (c) Fig. 7: Observed cracking in a Type B2M specimen: at a loading of 30 kip (133 kn); at a loading of 40 kip (178 kn); (c) after failure interior wythe; and (d) after failure exterior wythe (d) near the top of the panel. A potentiometer was attached to the edge of the corbel to monitor deflection. Cracking of Type B2M specimens was first observed at a load of about 30 kip (133 kn). The hairline crack was nearly horizontal at the top edge of the corbel (Fig. 7). As the load increased, cracks propagated in a 45-degree direction (Fig. 7). The three B2M specimens reached peak loads of 93.1 kip (414 kn), 96.6 kip (430 kn), and kip (448 kn), accompanied by concrete crushing in the interior wythe below the corbel (Fig. 7(c)). No sudden failure or separation of the corbel was observed in any of the three tests. Cracks developed near failure on both the front and back wythes of the wall, indicating engagement of both wythes in resisting corbel loading. Even near failure, cracking was gradual and controlled, and not excessive after the corbel reached the maximum load. Cracks were observed on both the front (Fig. 7(c)) and back (Fig. 7(d)) faces of the panel, indicating the efficiency of the proposed details in transmitting the corbel load to both concrete wythes. After failure, the corbel could not be removed from the panel, indicating that the bars were still securely anchored to the wall and that failure was due to excessive deformation followed by concrete crushing. Figure 8 shows the load-deflection behaviors for the three specimens as well as the design and service loads for the corbel. The plots indicate good consistency between all three Type B2M specimens and demonstrate that the required capacity was significantly lower than the observed capacity. Load, lb 100,000 90,000 80,000 70,000 60,000 50,000 40,000 30,000 20,000 10, Factored load = 36,000 lb Service load = 27,000 lb Sustained load =20,250 lb Corbel deflection, in. Fig. 8: Load-deflection relationships for the three Type B2M specimens (Note: 1 lbf = kn; 1 in. = 25 mm) Conclusions and Recommendations Through analysis, fabrication, and testing, the proposed insulated wall panel corbel details have been shown to: Eliminate thermal bridging that is normally caused by steel bars and concrete blocks interrupting the insulation; Be suitable for incorporation into a precast production process whereby the corbel and the associated reinforcement are prefabricated ahead of the panel production; and Provide structural performance satisfactory for supporting typical double tee floor and roof spans. Three methods of analysis were used to investigate the behavior of this innovative type of corbel: the traditional shear-friction, shear-flexure approach; the strut-and-tie method; and the FEA method. While the FEA method gave an indication of crack development with increasing load, it is admittedly difficult to use for daily design practice. While the strut-and-tie method was acceptable for determining the Ci October

6 corbel capacity, it did not provide adequate guidance relative to crack control detailing. It is therefore recommended that the traditional corbel design methods provided in the ACI 318 Code and the PCI Design Handbook be used in conjunction with additional calculations for creep rupture analysis as recommended by ACI 440.1R. The proposed stirrups showed satisfactory behavior. Despite the fact that much lower amounts of reinforcement are indicated by the referenced standards (refer to the appendix to this article available with the online version), the detailed shear reinforcement should be used until further research justifies smaller quantities. Acknowledgments and Disclaimer The experimental work reported in this paper was conducted at the University of Nebraska Structures Laboratory. M. Elkady conducted the majority of the research work as part of his MS degree under the supervision of M. Tadros and G. Morcous. Other graduate students, in particular R. Tawadrous, provided significant assistance during testing. Elkady received financial support for graduate school from e.construct, structural engineering consultancy, in Dubai, United Arab Emirates, and in Omaha, NE. The research was additionally funded by the partner companies of THiN-Wall: Concrete Industries, Lincoln, NE; Hughes Brothers, Seward, NE; and Tadros Associates, Omaha, NE. The opinions expressed in this article are those of the individual authors. The conclusions and recommendations are intended for corbels in composite insulated wall panels, regardless of the type of connector used to create composite action in the wall panel itself. References 1. ACI Committee 318, Building Code Requirements for Structural Concrete (ACI ) and Commentary (ACI 318R-14), American Concrete Institute, Farmington Hills, MI, 2014, 519 pp. 2. PCI Handbook Committee, PCI Design Handbook: Precast and Prestressed Concrete, seventh edition, Precast/Prestressed Concrete Institute, Chicago, IL, 2010, 804 pp. 3. Minimum Design Loads for Buildings and Other Structures (ASCE/SEI 7-10), American Society of Civil Engineers, Reston, VA, 2013, 636 pp. 4. ACI Committee 440, Guide for the Design and Construction of Structural Concrete Reinforced with Fiber-Reinforced Polymer (FRP) Bars (ACI 440.1R-15), American Concrete Institute, Farmington Hills, MI, 2015, 83 pp. 5. Elkady, M., Precast Concrete Insulated Wall Panel Corbels without Thermal Bridging, MS thesis, University of Nebraska-Lincoln, Lincoln, NE, Morcous G., and Tawadrous, R., Testing of THiN Wall Corbels, final report, University of Nebraska-Lincoln, Lincoln, NE, June 2015, 56 pp. Note: Additional information on the ASTM standards discussed in this article can be found at Mohamed Elkady is a Senior Engineer with e.construct.ae, Dubai, UAE. He received his BS in structural engineering from Ain Shams University, Cairo, Egypt, in 2002 and his master s degree in civil engineering from the University of Nebraska, Lincoln, NE, in He is a licensed professional engineer in Nebraska. Maher K. Tadros, FACI, is Principal at e.construct.usa, LLC, Omaha, NE. He is a member of Joint ACI-ASCE Committees 343, Concrete Bridge Design, and 423, Prestressed Concrete, as well as ACI Committee 546, Repair of Concrete. He received his PhD in structural engineering from the University of Calgary, Calgary, AB, Canada, in He is a licensed professional engineer in Nebraska and several other states. ACI member Mark Lafferty is General Manager, Concrete Industries, Inc., Lincoln, NE. He is a licensed professional engineer in Nebraska. ACI member George Morcous is an Associate Professor, University of Nebraska, Lincoln, NE. He is a member of ACI Committees 237, Self-Consolidating Concrete; 345, Concrete Bridge Construction, Maintenance, and Repair; and 347, Formwork for Concrete. He received his PhD in civil engineering from Concordia University, Montreal, QC, Canada. He is a licensed professional engineer in Nebraska. ACI member Doug Gremel is Manager, ASLAN Division, Hughes Brothers, Seward, NE. He is a member of ACI Committee 440, Fiber-Reinforced Polymer Reinforcement. He received his BS in engineering science, electrical engineering, and business administration from Colorado State University, Fort Collins, CO, in Received and reviewed under Institute publication policies. 58 october 2015 Ci

7 Appendix: Design Calculations As indicated in the PCI Design Handbook, a number of steps are required for a complete design of a corbel. The unique aspects related to design and detailing of the prototype corbels are emphasized in the following calculations. The main reinforcement comprises GFRP bars. It is recommended that these bars be designed to resist tension induced by the vertical and horizontal forces applied to the corbel. In conventional design, this reinforcement is also expected to comprise two-thirds of the shear friction reinforcement if that quantity exceeds the value from flexural analysis. Both checks will be made in this example. However, the flexural strength check will be performed using the recommendations of ACI 440.1R in order to acknowledge the special behavior of GFRP material, which does not exhibit a yield plateau. As such, ACI 440.1R recommends a relatively low strength reduction factor and also uses an assumption that the compression block is nearly triangular (rather that the equivalent rectangular stress block normally assumed with ductile reinforcement). With GFRP, another design criterion, checking against creep rupture of the reinforcement, may be critical. Fig. A1: Free body diagram of corbel Type B2. The main reinforcement, comprising 6 GFRP U- bars, is required to resist the tension induced by the vertical and horizontal loads applied to the corbel. Design steps correspond to the details recommended for implementation. They are consistent with those used in Type B2M. Figure A1 shows a diagram of the applied forces and the resistance forces at the interface between the corbel and the supporting wall. The applied load, Vu, at the strength limit state is assumed to be 36 kip, and Vsus, assumed as sustained load is kip. The horizontal force due to restraint at the bearing, Nu, is assumed to be zero, to match the experimental conditions. However, in most practical applications 10 to 20% of the vertical load is assumed for this value. The applied load eccentricity, a, is 5 in. The effective reinforcement depth is the corbel height, h, minus the cover depth and half the tension 1

8 bar diameter. Thus, the effective depth is (3/16) = in. The corbel width, b, is 18 in. The strength-reduction factor,, is 0.55, which is relatively low to reflect absence of a yield plateau in the GFRP bar stress-strain relationship. The shear span to depth ratio, a/d, is 5.00 / = This is less than Thus, corbel provisions apply. Minimum corbel size must be such that a check of maximum shear limit is met. According to the PCI Handbook, Table 5.3.1, the maximum limit is kip, which is greater than the required value of 36 kip. The minimum area of flexural reinforcement is intended to guard against sudden rupture without adequate warning. The PCI Handbook Eq is used below. However, the authors believe that this limit may not be applicable in this case, due to the special detailing recommended for this corbel and due to the fact that the reinforcement provided in the testing program resulted in experimental capacity far in excess of the required capacity. As min = 0.04 = = 0.69 in.2, Strength design: The required moment at the face of the support, Mu, can be obtained from handbook Eq. 5-86: Mu = = 180 kip-in. Flexural strength of the corbel cross section using the GFRP bar capacity, is calculated according to ACI 440.1R, Eq.8-6b: Mu Mn= CE Af ffu ( ) The neutral axis depth may be conservatively determined using the balanced depth formula given by ACI 440.1R, Eq.8-6c. The ultimate concrete compressive strain may be assumed and the ultimate reinforcement strain may be assumed , per the GFRP bar manufacturer s recommendations. The resulting value of cb is 1.91 in., which is a conservative value. More accurate analysis is not warranted, as flexural strength seldom covers corbel design. The ratio of the depth of equivalent rectangular stress block to the depth of the neutral axis, = 0.65, is at the low end of the range of. Experiments on which the ACI 440.1R formula is based indicate that the actual concrete compressive stress block near failure is triangular rather than rectangular. Mn = Af ( ) >Mu= 180 kip-in. Thus, Af > 0.33 in. 2. 2

9 The second limit on the area of the main reinforcement is determined as two thirds of that required by shear friction. According to PCI Handbook Eq. 5-33, the shear-friction coefficient µe= =.., = 3.85, but is not greater than 2.9. Use 2.9. The required area of reinforcement (PCI Handbook Eq. 5-87): Af = =. = The provided area of 0.66 in. 2 satisfies both requirements. The required area of horizontal stirrups is taken equal to one-third of that required for shearfriction, or one-half of the shear friction value of Af. Required horizontal stirrups, Ah = 0.5 (Af - An) = 0.5 x 0.33 = 0.17 in. 2 The provided steel stirrups are two legs of two No.4 bars in addition to one closed No.3 stirrup. Total area = 4 x x 0.11 = 1.02 in. 2. The authors recommend that the larger of this amount of stirrups or that calculated from above equation be used until further research justifies a smaller amount. Analysis for creep rupture of GFRP main flexural bars: Sustained vertical load was given to be equal to kip, acting at an eccentricity of 5.00 in. from the mid-thickness of the wall. The moment Msus is kip-in. The stress in the GFRP bars due to the sustained load moment, is calculated using cracked section analysis: Traditional working stress analysis, assuming cracked rectangular section may be used to determine the neutral axis depth. According to the PCA Notes on ACI (Table 10-2), the depth of the compression block of transformed cracked section (kd); 1 1, where Using the modular ratio n = Ef/Ec 1.19 and Af = 0.66 in2, the value of kd is 1.03 in. The stress in the reinforcement: Stress =Msus/(Af (d - kd/3)) = ksi According to ACI 440.1R, Table 8.3, the creep rupture capacity of GFRP bars is: f f,s =0.2 =0.2 x 0.7 x 110 = 15.4 ksi Thus, the provided six No. 3 GFRP bars are adequate for creep rupture resistance. 3