Iterative Optimization of a Typical Frame in a Multi-Story Concrete Building. Description of Program Created by Joseph Harrington

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1 Iterative Optimization of a Typical Frame in a Multi-Story Concrete Building Description of Program Created by Joseph Harrington Based on the given serviceable loads, a preliminary model is required for the building of interest. This can be completed in any general structural analysis software. A preliminary model is necessary in order to determine the maximum shear and bending moment demands imposed upon the structure. Incorporating the necessary design criteria established by the Building Code Requirements for Structural Concrete (ACI ), an iterative design process was developed through the MATLAB computational program to determine the dimensions of the slabs, joist system, columns, beams, and footings that optimally meet the design requirements. The maximum shear force and bending moment obtained from the aforementioned structural analysis results are input directly to the developed MATLAB program which then iteratively analyzes each structural component for acceptability. If the design is acceptable, the dimensions and volume of the component are computed. An estimated total cost of materials, based on the total volume required, is computed for each acceptable design iteration and compared against the current optimal design cost. The design with the lowest estimated cost is selected and this iterative process is continued and the design refined until the program determines the most cost effective, acceptable design. The final solution provided by the MATLAB program is to be checked for acceptability through hand calculations and selected as the final design upon confirmation. Some of the analysis completed in MATLAB required assumptions in order to complete the iterative process. While the following discussion is not an exhaustive list of the assumptions made, it outlines a general understanding of the types and the impacts upon the results. All assumptions align with the material taught in Dr. Fafitis Concrete Structures course at Arizona State University and all assumptions were determined acceptable through hand calculation verification (represented in the Appendix). One of the main assumptions made within the MATLAB is for the design of the columns. The design of this structural member proved to be the most difficult component of MATLAB program coding because of the typical use of interaction diagrams to determine possible reinforcement configurations and steel ratio required. With obvious difficulty of incorporating every individual interaction diagram into the iterative process, a few assumptions were made to analyze the columns. First, it was determined that the best method of coming up with the rebar configuration was to determine the minimum number of bars to satisfy the maximum 6 inch spacing requirement by the governing design code. Aligning with Dr. Fafitis recommendation, the steel ratio for the bars was constrained to fall within 2% and 3%. Once the program determined the appropriate rebar designation number to adhere to this constraint, the strain at each bar location was found with the assumption that the neutral axis extended beyond the edge of the column (forcing all of the rebar into compression). The ultimate capacities of the design were determined from finding the appropriate forces at each bar location from the strain discussed previously, and subsequently, the moments as well.

2 For the footings, it was assumed that the depth of embedment was equal to 2 feet and that this soil had a unit weight of 100 pounds per cubic foot. The cost analysis incorporated into the iterative process is computed on a per frame basis. Note that these figures are extremely approximate and just used for comparison purposes throughout the iterative process. The unit weight of each material, concrete and steel reinforcement, used was taken to be 150 pounds per cubic foot and 490 pounds per cubic foot respectively as described in the 2005 AISC Code of Standard Practice. After the weight of the frame was determined by applying these appropriate unit weights to the volume totals, the cost is determined by applying a cost factor for each material. For the rebar, 50 cents per pound was applied to the total weight, whereas 2 cents per pound was applied to the weight of the concrete ( An optimal design is selected based on the estimated cost. After the optimal design is determined from the iterative procedures completed by the program, specific rebar requirements are determined and additionally verified through hand calculations. Generally, the rebar configuration requirements are based on general construction considerations such as an allowable maximum number of reinforcing bars for a specific cross section, or the maximum and minimum reinforcing bar designation (all are input parameters in the program). When multiple possible rebar configurations given the constraining construction considerations are available, the program selects the reinforcing configuration that results in the least amount of excessive steel reinforcing area. The expectation of practicing structural engineers is that they typically have the ability to utilize a commercial software to carry out the calculations implemented into the program. However, it is good practice to know what specific calculations and code checks a commercial design software makes and what considerations/assumptions are appropriate for specific projects. Furthermore, this computer program would be extremely useful in providing personalization and a system of checks for a commercial software, or particularly, for structural engineers in a small firm that does not have a large commercial reinforced concrete program readily available for every engineer.

3 Appendix The information below contains an example solution for the problem statement for a project in the aforementioned Concrete Structures course at Arizona State University (Spring 2013), which is shown below.

4 Presented below is the MATLAB final results output by the program described. * * Structural Concrete Design created by Joseph Harrington Original Version: Latest Update: * * Additional results displayed due to running as "Debug" version. Max Negative Moment for the beam design: k-in Max Positive Moment for the beam design: k-in Max Shear Force for the beam design: k Max Axial Force for the column design: k Max Moment for the column design: k-in Results for Debugging Slab Design: cover = modulus of rupture = w = Slab Spacing = Mmax = Phi for Tension = Phi for Shear = Mu = effective depth (d) = be = w per Rib = Effective Length = Mu for the bottom = Rebar for the bottom = Shear Demand = Shear Capacity = Mu for the top = For the negative moment, T section does not contribute, so be = Rebar for the top = Results for Debugging Flexural Design of Beams: be = d = Design for Positive Moment Current Phi = a = c = A_c = As_postiive (current) = Strain = New Phi = Design for Negative Moment Phi = a = c = Strain = As_negtaive (current) = New Phi = First Minimum Check = Second Minimum Check = 2.343

5 Rebar Required from Positive Moment = Rebar Required from Negative Moment = Results for Debugging Shear Design of Beams: d = Vc = L = Vu = * phi * Vc = Vs = S1 = SMax_Check_1 = SMax_Check_2 = S2 = Effective S1 = Effective S2 = Vs_min = Vu_min = Location of Vu_min = Minimum Distance required for shear reinforcing = Number of Stirrups in Section 1 = Number of Stirrups in Section 2 = Toal Rebar = Results for Debugging Column Design: Number of Spaces = 3 Number of Bars Per Side (Additional to Corners) = 2 Total number of bars in column = 12 Effective Spacing = Min Rebar Required for rho of 0.02 (min) = Max Rebar Required for rho of 0.03 (max) = Rebar Number = 9 As = Rho = c = Total number of rebar in side view = 4 Current distance from left edge = Current strain = Current force = L/2 = DFL = Current moment = Current distance from left edge = Current strain = Current force = L/2 = DFL = Current moment = Current distance from left edge = Current strain = Current force = L/2 = DFL = Current moment = Current distance from left edge = Current strain = Current force = L/2 = DFL = Current moment = Total tensile force from rebar = Total compressive force from rebar =

6 Total moment from rebar = Axial Capacity = Axial Demand = Moment Capacity = Moment Demand = Total Rebar Area for Section = Results for Debugging Footing Design: d = Footing Area = Column Area = SW = Required Footing Area = Pu = q_ultimate = Cx = Cy = L = Cx = d = B = q_ultimate = Vu_L = Vc_L = Vu_B = Vc_B = b0 = Betac = Vu_2 = Vc_2 = Mu_L = RebarArea_L = Mu_B = RebarArea_B = r = A1 = X = A2 = phi_compression = N1 = N2 = N = As_Dowel = The optimal footing dimensions (W x L x T) are: in by in by 48.0 in The optimal slab-joist dimensions (W x H x T) are: 6.5 in by 16.0 in by in spacing The optimal beam dimensions (W x H) are: 38.0 in by 20.0 in The optimal column dimensions (W x H) are: 22.0 in by 22.0 in These cross-sectional dimensions resulted in the following rebar requirements: SLAB-JOISTS The required rebar area for the bottom of the slab-joist system is: in^2 The optimal combination for rebar given the specific requirements is 2 number 6 bars, which results in an actual area of in^2 The required rebar area for the top of the slab-joist system is: in^2 The optimal combination for rebar given the specific requirements is 2 number 7 bars, which results in an actual area of in^2

7 BEAMS The required rebar area due to the positive moment is: in^2 The optimal combination for rebar given the specific requirements is 14 number 6 bars, which results in an actual area of in^2 The required rebar area due to the negative moment is: in^2 The optimal combination for rebar given the specific requirements is 18 number 8 bars, which results in an actual area of in^2 Two sections shear reinforcing acceptable as follows: For the first section, 27 #3 bar 2.5" spacing from approximately 20.0 in to in (for both sides of the beam span) For the second section, 14 #3 bar 6.5" spacing from approximately 81.5 in to in (for both sides of the beam span) The total amount (volume) of rebar of in^3 for each beam span COLUMNS - Square (Evenly Distributed Reinforcing) Based on the design assumptions and results, the amount of rebar in the columns is: in^2 This was determined from the use of 12 number 9 bars FOOTING For the L span of the footing, the optimal combination for rebar given the specific requirements is 21 number 11 bars, which results in an actual area of in^2 For the B span of the footing, the optimal combination for rebar given the specific requirements is 20 number 11 bars, which results in an actual area of in^2 For the dowel bars, the optimal combination for rebar given the specific requirements is 12 number 9 bars, which results in an actual area of in^2 The dowels are required to extend 22 in into the column from the given specific requirements SUMMARY OF MATERIALS For the frame assigned, the approximate values were calculated: VOLUME (in cubic inches) CONCRETE TOTAL: STEEL TOTAL: WEIGHT (in pounds) CONCRETE TOTAL: STEEL TOTAL: COST (per frame) MATERIAL TOTAL: $

8 The hand calculated checks of the results produced by the MATLAB program are presented on the following pages. The hand checks follow the order in which the various elements were designed through the MATLAB optimization: slabs and the joist system, then beams, next columns, and finally footings.

9 The hand calculations which verified the MATLAB optimized slab and joist system design are revealed in the next two scanned images.

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11 The hand calculations for the beam design are revealed in the next five scanned images.

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16 The hand calculations which verified the MATLAB optimized square column are revealed in the next several scanned images. Reminder: f c=8000 psi.

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18 Finally, the hand calculations which verified the MATLAB optimized isolated footing design are revealed in the following five scanned images.

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