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ISSN: 2165-784X
Journal of Civil & Environmental Engineering
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Prediction of the Design Load for High Strength Concrete Columns

Ayad Zeki Saber Agha* and Mereen Hassan Fahmi Rashid

Erbil Polytechnic University, Erbil Technical Engineering College, Civil Engineering Department, Erbil, Iraq

*Corresponding Author:
Ayad Zeki Saber Agha
Erbil Polytechnic University, Erbil Technical Engineering College
Civil Engineering Department, Erbil, Iraq
Tel: +18197620971
E-mail: [email protected]

Received Date: November 28, 2016; Accepted Date: December 05, 2016; Published Date: December 08, 2016

Citation: Agha AZS, Rashid MHF (2016) Prediction of the Design Load for High Strength Concrete Columns. J Civil Environ Eng 6:260. doi: 10.4172/2165- 784X.1000260

Copyright: © 2016 Agha AZS, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

This pap er presents a method to predict the strength of high strength concrete columns subjected to axial compressive load and bending moment in one direction (i.e. Uni-axial bending condition). General non-dimensional equations proposed by applying the least squares approximation and using the experimental data found in previous studies.

The results obtained from the proposed equations; showed good correlation with the tested result of square and rectangular high strength concrete columns. The proposed model is tested and applied on some columns found in previous studies and showed excellent agreement with the experimental data for high strength concrete columns.

Keywords

Column; High strength concrete; Uniaxial bending

Introduction

High strength concrete can be used to advantage in the design building since it leads to columns which can carry higher loads for the same size column cross section, or to smaller column cross section for the same size loads. Other reasons for the use of high strength concrete include its high elastic modulus, low creep deformations and improved ductility. The use of high strength concrete together with high yield strength steel reinforcement appears to be an attractive proposition for heavily loaded columns of building structures [1]. The principle reason for using high strength concrete is that it may offer that most costefficient solution for many structural advantageous in compression members. For this reason, the use of high-strength concrete in columns and core walls of buildings, among other applications, is increased [2].

Many studies [3-9] have demonstrated the economy of using high strength concrete in columns of high-rise buildings and low to mid rise buildings. In addition to reducing the column size, and producing a more durable material, the use of high strength concrete has been show to be advantageous with regard to lateral stiffness and axial shortening and reduction in cost of forms. There is no unique definition of high strength concrete. The Australian standard for concrete structures AS 3600-1994 [10] is limited to concrete compressive strength up to 50 Mpa, while Razvi and Soatcioglu [11], considered the strength of (41 Mpa) for normal weight concrete and (27 Mpa) for light weight concrete to be high strength concrete. This is found to be justifiable and since most of the ready-mix concrete supplied. There is no universal agreement on the applicability of ACI code requirement for calculating flexural strength of high strength concrete columns subjected to combined axial load and bending moment.

Columns are usually designed for combined for combined axial load and bending moment using the rectangular stress block shown Figure 1. This stress block was originally derived by Mattock et al. [12]. Based on the tests of un-reinforced concrete columns loaded with axial load and moments [13]. The concrete strength ranged up to 52.5 MPa parameters defined the stress block, the intensity of stress (α1) and stress block depth ratio of the neutral axis (β1), Mattock et al. [12] proposed:

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Figure 1: Modified stress block.

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Nedderman [14], proposed a lower limit on (β1) of 0.65 for concrete strength is excess of 55 MPa. New Zealand standard and ACI-Code recommended that the currently used parameters for the equivalent rectangular concrete compressive stress block shown in Figure 1 are applicable up to civil-environmental-engineering Forcivil-environmental-engineering it is recommended that β1 = 0.65 and 1(α ) reduced linearly with increase incivil-environmental-engineering to become a minimum of (0.75) at civil-environmental-engineering.

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Available test data indicate that typical stress-strain curves in compression for HSC are characterized by an ascending portion that is primarily linear, with maximum strength achieved at an axial strain between (0.0024 and 0.003). Therefore it may be more appropriate to use a triangular compression stress block shown in Figure 1 for HSC columns when civil-environmental-engineering exceeds 70 MPa intensity of compression stress equals (α1 = 0.63); rather than 0.85 civil-environmental-engineering or (α1 = 0.63); and the depth of the rectangular compression block is equal to α1 = 0.67c or β1 = 0.67.

The Canadian Code [15]; suggested the following modified rectangular stress block:

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Ibrahim et al. [16], compared the concrete component of the measured load and moment strength of (94) tests of eccentrically loaded columns with concrete strengths ranging up to 130 MPa and they conclude that the max. Concrete strain before spalling were greater than (0.003), and the HSC columns can be designed based on rectangular stress block with some modification of the parameters as below:

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Rangan and Lioyd [2]; presented a program on the behaviour and strength of HSC columns under eccentric compression, they developed a theory to predict the load-deflection behaviour and the failure load, based on a simplified stability analysis and a stress-strain relation of strength concrete in compression the average ratio of test failure load to predicted failure load was (1.13) with coefficient of variation of (10%).

Saatcioglu and Razvi [11], investigated the strength and deformability of confined high strength concrete columns based on available experimented data up to 250 columns, tested either under monotonically increasing concentric loading or reversed lateral loading were evaluated in terms of load, ductility and draft capacities. The results indicate that the confinement requirement for HSC columns significantly more stringent than those for NSC columns and it is possible to obtain ductile behaviour in HSC columns through proper confinement. The use of high strength confinement steel reduces the need to impose unrealistically high volumetric ratios to attain deformability usually expected of NSC columns. The results also indicate that the product of the volumetric ratio and strength of confinement steel, normalized with respect to concrete strength, can be used as a design parameter.

Setty and Rangan [17], presented a study on eccentrically loaded HSC columns, they proposed a modified method to predict the failure load of eccentrically loaded reinforced concrete columns using an equivalent rectangular stress block that applies to all grades of concrete. The calculated failure load correlates well with the failure load to predict of (143) test columns, the mean value of test failure load to predict failure load was (1.08) with coefficient of variation of (12%). Fafitis and Shah [18], proposed an analytical expressions for the stress strain curves of confined and unconfined high strength concrete. Also moment-curvature relationships were predicted for columns subjected to reversed lateral loading. The analysis and design based on the assumptions of the ACI Code [19] and principles of static and graphs of interaction diagram for short columns subjected to bending moments and axial compression load. The detailing of the design and analysis equations are found in many text books of reinforced concrete design [20-26]. In this paper, general equations containing non-dimensional terms in the form of second and forth degree polynomial are proposed. The equations have been used to analysis and design of high strength reinforced concrete short and tied columns subjected to uniaxial bending moment and axial load.

Theory and Analysis

Two general equations of 2nd and 4th degree were proposed to present the non-dimensional interaction surface in the following form:

civil-environmental-engineering (1)

and

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Where

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The coefficients of the equivalent rectangular stress block used in the analysis are shown below:

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The least squares approximations [27] were applied to minimize the error function (S) with respect to the coefficients:

civil-environmental-engineering (3)

Where Zc = Values given by the proposed equation.

Ze = Values from the test results.

S = Sum of errors.

N = No. of data.

Considering equation (1):

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Minimization with respect to the coefficients civil-environmental-engineering the following equation is obtained:

civil-environmental-engineering (5)

First equation (5) is solved by computer using experimental data shown in Tables 1 and 2 for normal and high strength concrete columns to show the accuracy of the proposed model.

  Ptest/Pcal
No. Ref. width
(in)
Depth (in) fc´(psi) fy´
(ksi)
Reinforcement  Eccen. ex(in)  Eccen. ey(in) pu (exp)kips EQ6 EQ7 EQ8 EQ9
1 [29,30] 4 4 5435 45.6 4 # 4 2.82 2.82 13.5 1.61 1.85 0.89 0.88
2   4 4 5435 45.6 4 # 4 2.82 2.82 14.3 2.01 2.15 0.88 0.89
3   6 8 3200 53.5 4 # 5 3 4 32 1.04 1.01 0.61 0.58
4   6 8 3700 53.5 4 # 5 6 8 17 3.43 3.06 1.61 1.53
5   6 8 3500 53.5 4 # 5 6 4 21 2.35 1.83 1.31 1.23
6   6 8 3800 53.5 4 # 5 3 8 24 1.69 2.43 1.08 1.38
7   4 4 3200 44.5 4 # 3 1 1.5 21 0.6 0.45 0.15 0.19
8   4 4 4095 44.5 4 # 3 1 1.5 24.8 0.08 0.05 0.03 0.04
9   4 4 3905 73 9 Φ 1/4 in 2.5 3.5 9.6 0.42 0.56 0.65 0.9
10   4 4 3806 73 9 Φ 1/4 in 3 3.5 8.7 0.52 0.65 0.79 0.91
11   4 4 3894 73 9 Φ 1/4 in 3.5 3.5 8        
12   4 4 3830 73 9 Φ 1/4 in 2 2 14.3        
13   4 4 3715 73 9 Φ 1/4 in 0.5 5.5 10.8        
14   4 4 3895 73 9 Φ 1/4 in 0.5 7 6.24 0.68 0.8 0.95 0.98
15   4.25 4.25 3545 44.5 8 # 3 3 2 13.9 1.02 0.95 1.05 0.94
16   4.25 4.25 3884 44.5 8 # 3 3.25 2.25 11.8 1.07 1.06 1.06 0.98
17   4.25 4.25 4227 44.5 8 # 3 2.5 3 13.6        
18   8 8 4230 46.8 8 # 5 0.8282 3.091 141.4 1.07 1.36 1.18 1.38
19   8 8 3735 46.8 8 # 5 0.764 1.848 173.5        
20   8 8 4860 46.8 8 # 5 2 3.464 120 0.68 0.76 0.98 1.21
21   8 8 4635 46.8 8 # 5 2.5 4.33 89 0.39 0.48 0.75 1.19
22   8 8 2805 46.8 8 # 5 1.414 1.414 134.5        
23   8 8 3998 46.8 8 # 5 2.546 2.546 112.5        
24   8 8 4275 47.6 8 # 5 2.828 2.828 116 0.8 0.8 1.09 1.07
25   8 8 4950 46.8 8 # 5 4 4 83.125 0.83 0.95 1.91 1.81
26   6 9 4590 46.8 8 # 5 0.9987 1.498 176.5        
27   6 9 3690 46.8 8 # 5 2.194 3.328 90 1.21 1.3 1.75 1.79
28   6 9 3548 46.8 8 # 5 2.992 4.493 70 0.29 0.33 1.2 0.64
29   6 9 3645 46.8 8 # 5 1.273 1.273 153 1.45 1.38 2.14 1.97
30   6 9 4482 46.8 8 # 5 3.182 3.182 85 0.4 0.37 0.88 0.52
31   6 9 3485 46.8 8 # 5 3.117 1.8 90        
32   6 12 3402 46.8 8 # 5 2.364 4.472 104.5 0.51 0.43 31.55 0.64
33   6 12 3105 46.8 8 # 5 3 6 70 0.56 0.65 1.36 1.21
34   6 12 4023 47.8 8 # 5 3.394 3.394 98 0.81 0.67 1.27 1.02
35   6 12 3800 71.6 8 # 5 2.598 1.5 122 0.46 0.37 0.54 0.8
36   5 5 4633 71.6 4 # 4 0.4087 0.9867 73 0.8 0.91 0.85 0.92
37   5 5 4633 71.6 4 # 4 0.4076 0.9839 77 0.9 1.06 0.97 1.08
38   5 5 4997 71.6 4 # 4 2.624 1.087 38 0.88 0.74 0.67 0.64
39   5 5 4997 71.6 4 # 4 2.624 1.087 35.8 0.62 0.53 0.48 0.44
40   5 5 4997 71.6 4 # 4 4.89 2.025 19.1 0.47 0.4 0.37 0.33
41   5 5 4997 71.6 4 # 4 5.028 2.083 17.6 0.52 0.44 0.42 0.37
42   5 5 4633 71.6 4 # 4 0.7623 0.7628 78.2 1.68 1.87 1.62 1.61
43   5 5 4633 71.6 4 # 4 0.7545 0.7545 75.5 1.46 1.59 1.41 1.38
44   5 5 5164 71.6 4 # 4 1.897 1.897 38.7 0.7 0.8 0.59 0.61
45   5 5 5164 71.6 4 # 4 1.947 1.947 37 0.57 0.66 0.48 0.5
46   5 5 5164 71.6 4 # 4 3.784 3.784 18.5 0.48 0.56 0.41 0.43
47   5 5 5164 71.6 4 # 4 3.725 3.725 18.9 0.46 0.54 0.4 0.41
48   5 5 3480 71.6 4 # 4 2.505 2.505 42.1 2.84 2.14 2.67 3.23
49   5 5 3480 71.6 4 # 4 4.889 2.025 18.5 1.59 1.73 2.11 1.97
50   5 5 3660 71.6 4 # 4 1.917 1.917 38.2 1.14 3.59 1.12 1.13
51   5 5 3660 71.6 4 # 4 3.712 3.712 18.2 0.8 0.77 0.69 0.63
52   10 10 5100 43.6 8 # 7 12.5 0 88 0.39 3.56 0.43 0.91
53   6 8 3700 53.5 4 # 5 6 0 24 0.38 0.4 0.42 0.43
54   6 8 3900 53.5 4 # 5 3 0 60 1.67 1.92 1.27 0.98
55   6 8 3700 53.5 4 # 5 0 4 70 0.91 2.27 0.82 0.72
56   6 8 4600 53.5 4 # 5 0 8 32 0.99 0.62 1.11 1.12
57   4 4 3426 44.5 4 # 3 5 0 6.445 1.09 0.71 1.18 1.43
58   4 4 3426 44.5 4 # 3 3 0 11.91 0.91 0.57 1.03 0.91
59   6 6 3000 40 8 # 4 2 0 72 1.04 0.65 1.2 1.18
60   6 6 3400 40 8 # 4 2 0 80 1.04 1.01 1.03 1.01
61   6 6 4200 40 8 # 4 2 0 100 1 1.01 0.97 0.97
62   6 6 4300 40 8 # 4 2 0 106 1.01 1.02 0.98 0.98
63   5 5 4688 71.6 4 # 4 1.031 0 78 0.99 0.99 0.97 0.97
64   5 5 4688 71.6 4 # 4 1.06 0 81.5 0.99 1.01 0.95 0.96
65   5 5 5376 71.6 4 # 4 2.8 0 42.3 0.96 0.98 0.93 0.94
66   5 5 5376 71.6 4 # 4 2.725 0 46 1.06 1.03 1.04 1.03
67   5 5 5376 71.6 4 # 4 5.25 0 23.6 1.05 1.02 1.04 1.02
68   5 5 5376 71.6 4 # 4 5.24 0 23.7 1.01 1.04 0.97 0.99
69   5 5 3666 71.6 4 # 4 2.73 0 42.6 1.56 0.97 1.37 1
70   5 5 3666 71.6 4 # 4 5.275 0 19.7 1.81 1.14 1.59 1.17
Average Ptest/Pcal =Variance 1.1 1.32 1.44 1.37
0.131 0.13 0.13 0.13

Table 1: Experimental data for Normal strength concrete columns.

  Ptest/Pcal =
No. Ref. width (mm) Depth (mm) fc
(M Pa)
fy
(M Pa)
Reinforcement  Eccen.
(e) mm
Pu
(exp)
kN
Pcal
(eq12) kN
Pcal
(eq13) kN
EQ6 EQ7 EQ8 EQ9 EQ12 EQ13
1 [17] 100 300 53.1 454 6 Φ12 10 1387 1506.43 1463.26 0.58 0.59 0.52 0.55 0.92 0.95
2   100 300 54.2 454 4 Φ12 10 1200 1432.35 1365.03 0.70 0.96 0.67 0.87 0.84 0.88
3   100 300 56.6 454 4 Φ12 10 1375 1475.09 1402.11 1.12 1.29 1.08 2.21 0.93 0.98
4   100 300 67.1 454 6 Φ12 10 1464 1732.09 1668.28 0.88 0.89 0.88 0.89 0.85 0.88
5 [2] 175 175 58 454 6 Φ12 15 1476 1467.59 1225.69 0.76 0.77 0.74 0.75 1.01 1.20
6   175 175 58 454 6 Φ12 50 830 948.49 584.02 0.90 0.91 0.88 0.88 0.88 1.42
7   175 175 58 454 6 Φ12 65 660 807.32 472.27 0.78 0.78 0.77 0.77 0.82 1.40
8   100 300 58 454 6 Φ12 10 1192 1596.56 1545.52 1.03 1.07 1.09 1.12 0.75 0.77
9   175 175 58 454 4 Φ12 15 1140 1336.46 1058.35 0.67 0.67 0.80 0.76 0.85 1.08
10   175 175 58 454 4 Φ12 50 723 792.69 467.29 0.47 0.47 0.57 0.53 0.91 1.55
11   175 175 58 454 4 Φ12 65 511     0.64 0.64 0.64 0.64    
12   100 300 58 454 4 Φ12 10 915 1497.26 1421.50 0.78 0.81 0.77 0.79 0.61 0.64
13   175 175 92 454 4 Φ12 15 1704 1883.83 1509.86 0.66 0.62 0.60 0.58 0.90 1.13
14   175 175 92 454 4 Φ12 50 1018 1138.76 677.29 0.44 0.44 0.74 0.73 0.89 1.50
15   175 175 92 454 4 Φ12 65 795 956.54 543.61 0.86 0.90 0.71 0.71 0.83 1.46
16   100 300 92 454 4 Φ12 10 1189 2088.53 1981.78 0.63 0.61 0.39 0.40 0.57 0.60
17   175 175 92 454 4 Φ12 15 1745 1726.97 1310.30 0.39 0.38 0.37 0.36 1.01 1.33
18   175 175 92 454 4 Φ12 50 908 959.68 549.79 0.38 0.38 0.72 0.69 0.95 1.65
19   175 175 92 454 4 Φ12 65 663 794.42 437.86 1.05 1.09 0.77 0.77 0.83 1.51
20   100 300 92 454 4 Φ12 10 1043 1980.99 1830.33 0.67 0.58 0.32 0.33 0.53 0.57
21   175 175 97 454 4 Φ12 15 1975 1791.96 1354.72 1.24 1.29 1.54 1.51 1.10 1.46
22   175 175 97 454 4 Φ12 50 1002 990.51 566.19 0.92 0.72 0.37 0.38 1.01 1.77
23   175 175 97 454 4 Φ12 65 746 819.24 450.72 0.40 0.33 0.27 0.26 0.91 1.66
24   100 300 97 454 4 Φ12 10 1610 2057.47 1892.52 0.67 0.68 0.64 0.64 0.78 0.85
25   175 175 97 454 4 Φ12 15 1932 1791.96 1354.72 1.19 1.23 1.07 1.10 1.08 1.43
26   175 175 97 454 4 Φ12 50 970 990.51 566.19 0.78 0.64 0.50 0.49 0.98 1.71
27   175 175 97 454 4 Φ12 65 747 819.24 450.72 0.40 0.33 0.26 0.25 0.91 1.66
28   100 300 97 454 4 Φ12 10 1650 2057.47 1892.52 0.70 0.71 0.67 0.67 0.80 0.87
29 [28] 200 200 103 576 8 Φ16 0 3983     1.07 1.04 1.05 1.04    
30 Series A 200 200 103 576 8 Φ16 5 3974 3578.90 3510.31 1.09 1.09 1.10 1.11 1.11 1.13
31   200 200 103 576 8 Φ16 10 3476 3471.93 3247.12 1.01 1.03 1.05 1.08 1.00 1.07
32   200 200 103 576 8 Φ16 20 3073 3193.02 2637.24 0.97 1.01 1.09 1.12 0.96 1.17
33   200 200 103 576 8 Φ16 20 3068 3193.02 2637.24 0.96 1.00 1.09 1.12 0.96 1.16
34   200 200 103 576 8 Φ16 0 4110     1.11 1.08 1.09 1.07    
35   200 200 103 576 8 Φ16 5 3861 3578.90 3510.31 1.08 1.08 1.10 1.11 1.08 1.10
36   200 200 103 576 8 Φ16 10 3176 3471.93 3247.12 0.90 0.91 0.93 0.95 0.91 0.98
37   200 200 103 576 8 Φ16 10 3990 3471.93 3247.12 1.20 1.23 1.27 1.30 1.15 1.23
38   200 200 103 576 8 Φ16 10 3606 3471.93 3247.12 1.05 1.08 1.11 1.13 1.04 1.11
39   200 200 103 576 8 Φ16 20 3019 3193.02 2637.24 0.94 0.98 1.06 1.09 0.95 1.14
40 [28] 200 200 101 576 8 Φ16 0 3438     0.92 0.89 0.91 0.89    
41 Series R 200 200 101 576 8 Φ16 10 2998 3423.04 3206.30 0.85 0.86 0.88 0.90 0.88 0.94
42   200 200 101 576 8 Φ16 20 2776 3152.86 2612.36 0.85 0.88 0.95 0.97 0.88 1.06
43   200 200 101 576 8 Φ16 30 2483 2859.05 2134.18 0.81 0.85 0.97 0.99 0.87 1.16
44   200 200 101 576 8 Φ16 50 1958 2339.31 1527.81 0.69 0.72 0.95 0.91 0.84 1.28
45   200 200 101 576 8 Φ16 0 3326     0.89 0.86 0.87 0.86    
46   200 200 101 576 8 Φ16 10 3002 3423.04 3206.30 0.85 0.86 0.88 0.90 0.88 0.94
47   200 200 101 576 8 Φ16 20 2643 3152.86 2612.36 0.79 0.82 0.87 0.90 0.84 1.01
48   200 200 101 576 8 Φ16 30 2456 2859.05 2134.18 0.91 0.95 1.11 1.12 0.86 1.15
49   200 200 101 576 8 Φ16 50 2111 2339.31 1527.81 0.81 0.83 1.19 1.09 0.90 1.38
50   200 200 101 576 8 Φ16 80 1434 1792.88 1059.02 0.52 0.52 0.84 0.72 0.80 1.35
51 [28] 200 200 92 560 8 Φ12 0 3383     0.84 0.82 0.83 0.82    
52 Series J 200 200 92 560 8 Φ12 10 2626 2731.06 2421.12 1.01 0.98 0.99 0.98 0.96 1.08
53   200 200 92 560 8 Φ12 20 2878 2386.26 1796.40 1.08 1.12 1.08 1.11 1.21 1.60
54   200 200 92 560 8 Φ12 0 2959     1.29 1.34 1.29 1.31    
55   200 200 92 560 8 Φ12 10 3091 2731.06 2421.12 1.30 1.30 1.28 1.27 1.13 1.28
56   200 200 92 560 8 Φ12 20 2784 2386.26 1796.40 1.19 1.20 1.18 1.18 1.17 1.55
57 [28] 200 200 87 560 8 Φ12 0 2541     1.02 0.99 1.00 0.98    
58 Series Z 200 200 87 560 8 Φ12 0 2946     1.11 1.15 1.11 1.14    
59   200 200 87 560 8 Φ12 10 2762 2635.03 2351.50 1.54 1.59 1.54 1.56 1.05 1.17
60   200 200 87 560 8 Φ12 20 2628 2316.65 1761.46 1.18 1.18 1.16 1.16 1.13 1.49
61   200 200 87 560 8 Φ12 30 2238 2012.10 1372.29 1.14 1.10 1.12 1.10 1.11 1.63
62   200 200 87 560 8 Φ12 30 2160 2012.10 1372.29 0.96 0.99 0.95 0.97 1.07 1.57
63   200 200 87 560 8 Φ12 0 2962     1.46 1.50 1.42 1.44    
64   200 200 87 560 8 Φ12 10 3000 2635.03 2351.50 0.97 0.94 0.95 0.94 1.14 1.28
65   200 200 87 560 8 Φ12 20 2886 2316.65 1761.46 1.21 1.26 1.20 1.23 1.25 1.64
66   200 200 87 560 8 Φ12 30 2145 2012.10 1372.29 1.37 1.41 1.33 1.36 1.07 1.56
67 [28] 200 200 87 560 8 Φ12 0 2586     0.86 0.84 0.85 0.83    
68 Series M 200 200 87 560 8 Φ12 0 3162     1.10 1.07 1.08 1.06    
69   200 200 87 560 8 Φ12 10 2998 2635.03 2351.50 1.22 1.26 1.22 1.25 1.14 1.27
70   200 200 87 560 8 Φ12 20 2793 2316.65 1761.46 1.45 1.49 1.44 1.47 1.21 1.59
71   200 200 87 560 8 Φ12 30 2176 2012.10 1372.29 1.22 1.22 1.20 1.19 1.08 1.59
72   200 200 87 560 8 Φ12 50 1887 1549.78 938.27 2.19 1.58 1.93 1.52 1.22 2.01
73   200 200 87 560 8 Φ12 0 3061     1.06 1.03 1.04 1.02    
74   200 200 87 560 8 Φ12 0 3758     1.34 1.31 1.32 1.30    
75   200 200 87 560 8 Φ12 5 3428 2765.45 2672.32 1.31 1.33 1.31 1.32 1.24 1.28
76   200 200 87 560 8 Φ12 10 3263 2635.03 2351.50 1.37 1.43 1.38 1.41 1.24 1.39
77   200 200 87 560 8 Φ12 20 2884 2316.65 1761.46 1.54 1.58 1.54 1.56 1.24 1.64
78   200 200 87 560 8 Φ12 20 2943 2316.65 1761.46 1.61 1.65 1.60 1.61 1.27 1.67
79   200 200 87 560 8 Φ12 0 3137     1.09 1.06 1.07 1.05    
80   200 200 87 560 8 Φ12 0 3927     1.41 1.37 1.39 1.37    
81   200 200 87 560 8 Φ12 0 3856     1.38 1.35 1.36 1.34    
82   200 200 87 560 8 Φ12 5 3484 2765.45 2672.32 1.34 1.36 1.33 1.35 1.26 1.30
83   200 200 87 560 8 Φ12 10 3243 2635.03 2351.50 1.36 1.42 1.36 1.40 1.23 1.38
84   200 200 87 560 8 Φ12 20 2890 2316.65 1761.46 1.55 1.59 1.54 1.56 1.25 1.64
Average Ptest/Pcal =Variance 1.1 1.315 1.44 1.374 0.985 1.282
0.131 0.131 0.131 0.131 0.029 0.093

Table 2: Experimental data for High strength concrete columns.

The nominal bending moment at balance strain condition and pure bending condition are calculated for each test about X and Y axces civil-environmental-engineering also the maximum axial strength (Pno) and nominal axial strength at balance strain condition (Pnb) are also determined for each test. Based on these data, the value of (X=Mnx/ Mnox or X=Mnx/Mnbx), (Y=Mny/Mnoy or Y=Mny/Mnby) and (Z=(Pn-Pnb)/(Pno-Pnb) or Z=Pn/Pno) are determined.

Substituting the values of X, Y and Z in equation (5) and solving this equation, the value of the coefficients civil-environmental-engineering are determined. Computers programs are prepared to perform all these calculations.

Pure bending moment condition

civil-environmental-engineering

civil-environmental-engineering (6)

and

civil-environmental-engineering (7)

Balance bending moment condition

civil-environmental-engineering

civil-environmental-engineering (8)

and

civil-environmental-engineering (9)

Equation (1) can be re-arranged in the following form for the purpose of solution:

civil-environmental-engineering (10)

Balance bending moment condition

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

Pure bending moment condition

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

Finally, equation (10) can be solved easily by computer or by hand calculations:

In case

civil-environmental-engineering

Also equation (2) re-arranged in the following form for the purpose of solution:

civil-environmental-engineering (11)

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

In case using balance strain condition; use the moments (Mnbx and Mnby) instead of (Mnox and Mnoy) respectively.

Because the 4th degree equation is more complicated for solution and results of equations (6-9) which are represent normal and high concrete strength columns subjected to uniaxial and biaxial bending conditions give results relatively far from the experimental data, so the study concentrate on the high strength concrete columns with uniaxial bending condition. And 2nd degree equation is selected for this purpose [27-30].

Pure bending moment condition

civil-environmental-engineering

civil-environmental-engineering (12)

Balance bending moment condition

civil-environmental-engineering

civil-environmental-engineering (13)

In general, the results obtained from equations (12) and (13) are found in good agreement with the experimental load for high strength concrete columns subjected to uni-axial bending condition, as shown in Figures 2 and 3.

civil-environmental-engineering-theoretical-load

Figure 2: Experimental load versus theoretical load form equation [12].

civil-environmental-engineering-Experimental-load

Figure 3: Experimental load versus the oretical load form equation [13].

The following examples explain the calculations and application of the proposed method.

Example1: (specimen IA/ref. [2]).

Experimental data:

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

Eccentricity (e) =15 mm

Pn exp. = 1476 kN

Solution:

civil-environmental-engineering

civil-environmental-engineering

Pure bending

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

Simplify to:

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

Application of equations 10 and 12:

civil-environmental-engineering

civil-environmental-engineering

A = -3.8 E-08

B = -7.17 E-04

C = 1.1339

civil-environmental-engineering

civil-environmental-engineering

Application of equations 10 and 13:

civil-environmental-engineering

civil-environmental-engineering

A = -6.586 E-08

B = -6.8 E-04

C = 1.1249

civil-environmental-engineering

civil-environmental-engineering

Example 2: Specimen/series J/Column #2/ref [28].

Experimental data:

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

Eccentricity (e) = 10 mm

civil-environmental-engineering

Solution: The following results are obtained using computer programs:

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

Application of equations 10 and 12:

civil-environmental-engineering

A = -6.688 E-09

B = -3.95 E-04

C = 1.12916

civil-environmental-engineering

civil-environmental-engineering

Application of equations 10 and 13:

civil-environmental-engineering

A = -7.931 E-09

B = -3.799 E-04

C = 1.12016

civil-environmental-engineering

civil-environmental-engineering

Example 3: Specimen/series A/Column #3/ref [28].

Experimental data:

civil-environmental-engineering

civil-environmental-engineering

Fy=576 MPa

Eccentricity (e) = 10 mm

civil-environmental-engineering

Solution: The following results are obtained using computer programs:

civil-environmental-engineering

civil-environmental-engineering

civil-environmental-engineering

Application of equations 10 and 12:

civil-environmental-engineering

Application of equations 10 and 13:

civil-environmental-engineering

Conclusions

1. An alternative method is proposed for analysis and design of normal and high strength concrete square and rectangular tied columns subjected to compression load with uni-axial and biaxial bending conditions.

2. General 2nd and 4th degree equations are proposed for this purpose. The coefficient values of these equations are determined using the experimental data of previous studies by applying the principles of least square method.

3. The method is applied on some specimens found in previous studies, and good agreement is found with the experimental results.

4. Computer programs are prepared to find the coefficient of the general equations, and performing all calculations and finding the theoretical load for normal and high strength concrete columns.

References

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