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 318-11), 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.
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 (http://www.constructionknowledge.net/concrete/concrete_basics.php). 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.
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.
Presented below is the MATLAB final results output by the program described. *----------------------------------------------* Structural Concrete Design created by Joseph Harrington Original Version: 02.05.2013 Latest Update: 05.01.2013 *----------------------------------------------* Additional results displayed due to running as "Debug" version. Max Negative Moment for the beam design: 11229.6 k-in Max Positive Moment for the beam design: 5765.5 k-in Max Shear Force for the beam design: 152.4 k Max Axial Force for the column design: 2046.4 k Max Moment for the column design: 9740.5 k-in Results for Debugging Slab Design: cover = 1.000 modulus of rupture = 0.316 w = 0.190 Slab Spacing = 18.000 Mmax = 0.427 Phi for Tension = 0.650 Phi for Shear = 0.750 Mu = 1.644 effective depth (d) = 15.000 be = 24.500 w per Rib = 0.645 Effective Length = 32.833 Mu for the bottom = 521.631 Rebar for the bottom = 0.654 Shear Demand = 9.785 Shear Capacity = 10.175 Mu for the top = 758.736 For the negative moment, T section does not contribute, so be = 6.500 Rebar for the top = 1.033 Results for Debugging Flexural Design of Beams: be = 70.000 d = 18.500 Design for Positive Moment Current Phi = 0.900 a = 1.517 c = 1.785 A_c = 106.201 As_postiive (current) = 6.018035 Strain = 0.028094 New Phi = 0.900 Design for Negative Moment Phi = 0.900 a = 6.289 c = 7.399 Strain = 0.004501 As_negtaive (current) = 13.542851 New Phi = 0.857 First Minimum Check = 2.223 Second Minimum Check = 2.343
Rebar Required from Positive Moment = 6.018 Rebar Required from Negative Moment = 13.543 Results for Debugging Shear Design of Beams: d = 18.500 Vc = 88.923 L = 432.000 Vu = 139.347 0.5 * phi * Vc = 33.346 Vs = 96.873 S1 = 2.521 SMax_Check_1 = 9.250 SMax_Check_2 = 6.947 S2 = 6.947 Effective S1 = 2.500 Effective S2 = 6.500 Vs_min = 37.569 Vu_min = 94.869 Location of Vu_min = 81.539 Minimum Distance required for shear reinforcing = 168.738 Number of Stirrups in Section 1 = 27.000 Number of Stirrups in Section 2 = 14.000 Toal Rebar = 595.320 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 = 5.667 Min Rebar Required for rho of 0.02 (min) = 0.807 Max Rebar Required for rho of 0.03 (max) = 1.210 Rebar Number = 9 As = 1.000 Rho = 0.025 c = 35.020 Total number of rebar in side view = 4 Current distance from left edge = 2.500 Current strain = 0.001330 Current force = -212.800 L/2 = 11.000 DFL = 2.500 Current moment = 1808.800 Current distance from left edge = 8.167 Current strain = 0.001815 Current force = -106.400 L/2 = 11.000 DFL = 8.167 Current moment = 301.467 Current distance from left edge = 13.833 Current strain = 0.002300 Current force = -106.400 L/2 = 11.000 DFL = 13.833 Current moment = 301.467 Current distance from left edge = 19.500 Current strain = 0.002786 Current force = -212.800 L/2 = 11.000 DFL = 19.500 Current moment = 1808.800 Total tensile force from rebar = 0.000 Total compressive force from rebar = 638.400
Total moment from rebar = 4220.533 Axial Capacity = 3309.537 Axial Demand = 2046.400 Moment Capacity = 13984.938 Moment Demand = 9740.500 Total Rebar Area for Section = 12.000 Results for Debugging Footing Design: d = 43.500 Footing Area = 44064.000 Column Area = 484.000 SW = 0.006 Required Footing Area = 43718.123 Pu = 2046.400 q_ultimate = 0.046 Cx = 22.000 Cy = 22.000 L = 216.000 Cx = 22.000 d = 43.500 B = 204.000 q_ultimate = 0.046 Vu_L = 506.863 Vc_L = 729.073 Vu_B = 476.490 Vc_B = 771.960 b0 = 476.490 Betac = 1.059 Vu_2 = 1847.154 Vc_2 = 1872.718 Mu_L = 44570.781 RebarArea_L = 32.400 Mu_B = 50659.685 RebarArea_B = 30.600 r = 0.971 A1 = 484.000 X = 91.000 A2 = 41616.000 phi_compression = 0.650 N1 = 2139.280 N2 = 1604.460 N = 1604.460 As_Dowel = 11.332 The optimal footing dimensions (W x L x T) are: 204.0 in by 216.0 in by 48.0 in The optimal slab-joist dimensions (W x H x T) are: 6.5 in by 16.0 in by 2.0 in @ 18.0 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: 0.654 in^2 The optimal combination for rebar given the specific requirements is 2 number 6 bars, which results in an actual area of 0.880 in^2 The required rebar area for the top of the slab-joist system is: 1.033 in^2 The optimal combination for rebar given the specific requirements is 2 number 7 bars, which results in an actual area of 1.200 in^2
BEAMS The required rebar area due to the positive moment is: 6.018 in^2 The optimal combination for rebar given the specific requirements is 14 number 6 bars, which results in an actual area of 6.160 in^2 The required rebar area due to the negative moment is: 13.543 in^2 The optimal combination for rebar given the specific requirements is 18 number 8 bars, which results in an actual area of 14.220 in^2 Two sections shear reinforcing acceptable as follows: For the first section, 27 #3 bar stirrups @ 2.5" spacing from approximately 20.0 in to 81.539 in (for both sides of the beam span) For the second section, 14 #3 bar stirrups @ 6.5" spacing from approximately 81.5 in to 168.738 in (for both sides of the beam span) The total amount (volume) of rebar of 595.320 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: 12.000 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 32.760 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 31.200 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 12.000 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: 25894752 STEEL TOTAL: 289787 WEIGHT (in pounds) CONCRETE TOTAL: 2247808 STEEL TOTAL: 82173 COST (per frame) MATERIAL TOTAL: $86042.85
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.
The hand calculations which verified the MATLAB optimized slab and joist system design are revealed in the next two scanned images.
The hand calculations for the beam design are revealed in the next five scanned images.
The hand calculations which verified the MATLAB optimized square column are revealed in the next several scanned images. Reminder: f c=8000 psi.
Finally, the hand calculations which verified the MATLAB optimized isolated footing design are revealed in the following five scanned images.