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.
Transcript
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.
