Flexural Test of a Reinforced Concrete Beam
Jeffrey Fain CAE 307 Section 3 Group 1 (Jason Shattuck, Shanli Wu, Songtao Cui)
Concrete Laboratory Illinois Institute of Technology Chicago, Illinois
April 16, 2012
Abstract: The purpose of this experiment was to determine the ultimate flexural strength of a reinforced concrete beam and to verify the analytical methods set forth in the ACI code for the design of similar reinforced concrete beams. The specified concrete mix was prepared and slump tested in accordance with relevant ACI code provisions. The reinforcing steel was fabricated and placed within preassembled formwork, after which the concrete was placed in the formwork and vibrated. Lastly, the beam was positioned within a custom made testing apparatus and loaded at its third points until failure. Data collected during the test included; load data, deflection data, and strain data. These data were evaluated on the basis of an analytical assessment of the same beam using the ACI code formulae for nominal flexural strength, ultimate flexural strength, shear reinforcement requirements and spacing, deflection limitation, and strain behavior. The results of the test data and the ACI based evaluation of the beam were in good agreement with 1.9% error in the assessment of maximum flexural strength and a 6% margin of error between the calculated and measured location of the neutral axis. The ACI code method of calculating deflections did not prove accurate, but it was consistent in providing a conservative guide to limiting deflections.
ii
Table of Contents Title Page
i
Abstract
ii
Introduction
1
Test Procedure Phase 1: Mix Design and Slump Testing
1
Phase 2: Placement and Attachment of Reinforcing Bars
3
Phase 3: Placement and Vibration of Concrete
5
Phase 4: Testing of the Beam and Concrete Cylinders
6
Results Concrete Compressive Strength
7
Relevant ACI Formulae
8
Flexural Strength Analysis
9
Shear Strength Analysis
10
Shear Reinforcement Spacing
11
Serviceability Analysis
12
Strain Behavior
16
Conclusions Ultimate Compressive Strength
18
Nominal/Ultimate Flexural Strength of the Beam
19
Deflections at Various Loads
20
Strain Behavior of the Beam
20
Error
20
Appendices Appendix A: Apparatus
21
Appendix B: Shear, Moment, and Deflection Diagrams
22
References
23 iii
Introduction: The purpose of this experiment is to determine the ultimate flexural strength of a reinforced concrete beam and to verify the analytical methods set forth in the ACI code for the design of reinforced concrete beams. The first phase of this investigation includes the specification of the mix and slump testing of the freshly mixed concrete. The second phase involves the arrangement and attachment of the reinforcing bars within the beam. The third phase involves the placement and vibration of the concrete. The fourth and final phase involves the loading and strain analysis of the beam and cylinders until failure. The discussion of testing procedures will take place within the sections related to each phase of investigation. The results and conclusions sections will draw together an aggregate analysis of all of the experimental data from each phase of the investigation and explain any error or deviation of the experimental results from analytical results based on the ACI code.
Test Procedure: Phase 1: Mix Design and Slump Testing The mix design for this investigation is shown in Table 1. Note that the only difference between the mixes was the quantity of water added to each. The aggregate mixes were obtained from an aggregate storage and batching unit manufactured by the Winslow Scale Company. The nominal maximum acceptable coarse aggregate size was specified as 0.75 inches. Portland cement was provided in 50 lb. bags and weighed separately on a 100 lb. capacity Toledo scale. The required quantity of water was weighed separately on the same Toledo scale. The cement and aggregate were placed in the hopper of a revolving drum concrete mixer manufactured by Acme. The dry contents of the hopper were emptied into the revolving mixing drum and the water was slowly added to the drum through the outlet chute. The concrete was allowed to mix for 15 to 20 minutes before removing a sample for slump testing. The assumed ultimate compressive strength of the concrete thus mixed was 3000 psi. Table 1 Mix Design
Mix Design Used Material
Batch 1
Batch 2
Batch 3
Type
Weight (lbs)
Weight (lbs)
Weight (lbs)
Cement
56
56
56
Water
36
34
32
Coarse Aggregate
207
207
207
Fine Aggregate
142
142
142
ÎŁ
441
439
437
Final Slump (in.) 2.5 2.5 3 *Wet Basis (Assumed): 3% H2O - Fine Aggregate, 0.5% H2O on Coarse Aggregate
1
Slump testing was carried out in accordance with ASTM C 143 / C 143M - 10a and the sampling procedures specified in ASTM C172 / C172M - 10.
Figure 1 Batching, Mixing, and Slump Testing
Cylinder samples were taken from each batch and prepared/cured in accordance with ASTM C39 / C39M - 12.
Figure 2 Casting Compression Cylinders
2
Phase 2: Placement and Attachment of Reinforcing Bars Figure 1 below illustrates the placement and attachment of the reinforcing bars within the pre-constructed steel formwork. The main tensile reinforcement consisted of two No. 5 steel reinforcement bars (db = 0.625 in. Ab = 0.31 in2). The stirrups were fabricated from No. 2 (db = 0.25 in. Ab = 0.1 in2) undeformed steel bars, as were the stirrup alignment bars at the top of the beam. The reinforcement cage was fastened with 16.5 gauge tie wire and placed on spacers to ensure a concrete cover depth of 0.75 inches. The assumed ultimate tensile strength of all reinforcing steel was 60 ksi (ASTM-615 grade 60). All dimensions relevant to the arrangement of reinforcing steel within the beam are given in Figures 4 and 5 below.
Figure 3 Tying Reinforcing Bars
Before the reinforcement cage was placed inside the formwork, a release agent was applied to its interior surfaces to prevent any bonding of the concrete to the formwork. Additionally, the rebar cage was thoroughly cleaned with mineral spirits in order to remove any dirt and oil which could discourage bonding between the concrete and the reinforcing steel. Finally, the reinforcing steel cage was carefully lowered into the formwork and centered from end to end and side to side.
3
Figure 4 Transverse and Longitudinal Beam Sections
Figure 5 Overall Transverse Beam Section
4
Phase 3: Placement and Vibration of Concrete The beam was cast in three lifts corresponding to each of the 3 cubic foot batches. The first and second batches were used to pour approximately 4.5 inch deep lifts, and the final batch was used to fill the remainder of the beam. Between each lift, a reciprocating concrete vibrator was placed in the concrete at regular intervals in order to ensure complete paste coverage of all aggregate and even distribution of the aggregate throughout the beam cross section. After the final lift was placed and vibrated, the top surface of the beam was troweled smooth and hoist anchors were placed above the support points in the top surface of the beam. The beam and cylinders were covered with plastic sheets in order to prevent moisture loss due to evaporation and left to cure for 33 days.
Figure 6 Placing and Vibrating Concrete
5
Phase 4: Testing of the Beam and Concrete Cylinders Testing of the beam was carried out using a custom built support and loading apparatus designed to load the beam at its third points, and thus place the section of the beam between the two load points in pure bending. A hydraulic ram provided the force required to fail the beam, and the line pressure of the ram was calibrated to indicate the value of one load P in kips. (A sketch of the shear, moment, and deflection diagrams is included in the Appendix of this report.) Figure 7 Failed Beam
Initially, the load was increased gradually up to about 5.7 kips and then decreased to 0.0 kips. This initial loading of the beam cracked the section below the neutral axis. Finally, the load was incrementally increased to failure at 10.4 kips. During both of these procedures, the strain and deflection of the beam at the mid-span were carefully monitored and recorded. In order to measure the strain a Whittemore Gauge was used to measure the distance between a series of dots placed on the beam as shown below.
Figure 8 Whittemore Button Locations
6
The Whittemore Gauge was equipped with a dial display accurate to ±0.001 in. Strain measurements were recorded in a table along with load and level data. Deflection measurements were made on either side of the beam at mid-span throughout the entire test. Deflection measurements were taken at regular intervals of loading and recorded along with the loads. The dial displays for both deflection gauges were accurate to ±0.001 in. The concrete cylinders were each tested in a Tinius Olsen compression testing machine in compliance with ASTM Standard C39 / C39M – 12.
Figure 9 Beam Apparatus
Results: Before the results were compiled a thorough analysis of the beam, as designed, was conducted according to the ACI code. Before any calculations were undertaken, the experimental strength of the batch cylinders was assessed.
Concrete Compressive Stress Batch 1 Batch 2 Batch 3
Max Load (kips) Cross-section Area (in2) 135.5 28.27433388 108.8 28.27433388 134.4 28.27433388 Ultimate Compressive Strength (ksi)
f'c (ksi) 4.79 3.85 4.75 4.46
Table 2 Compressive Strength
The measured average ultimate compressive stress of 4460 psi along with the geometrical data and steel specifications were used to compute the expected design flexural strength (Mn), ultimate flexural strength (Mu), ultimate shear strength (Vu), required stirrup spacing, and expected deflection at the loads imposed in the test. The nominal flexural strength (Mn = 30.6 ft-kips) was assumed to be the nearest estimate of the actual moment capacity at failure due to an expected load of Pn = 10.2 kips.
7
Relevant ACI formulae are listed below: √
ACI 8.5.1
(
)
(
ACI 9.3.2.2
)(
)
√
ACI 11.2.1.1
√
ACI 9.5.2.3
(
)
[
(
) ]
ACI 11.4.7
√
8
Table 3 Flexural Strength Analysis
b (in) 6
Dimensions h (in) 12
f' c (psi) 4465
d (in) 10.6875
Ec (psi) 3808602.9 Steel Specifications fy (psi) 60000
Diameter (in) 0 0.00
Compression Reinforcement Bar Area (in2)
Concrete Specifications fr (psi) 501.1
Es (psi) 29000000
# of Bars 0
Bar Area (in2) 0.31
Tensile Reinforcement Diameter (in) 0.625
Length (in) 108
Tensile Steel yields
Flexural Strength Analysis d' (in) 0
β1 0.827
φ 0.75 ρmin 0.00334 OK ρb 0.03096
Tension Reinforced Analysis R (psi) Mn (in-lb) 535.64 367096
ρcy 0.00000
n, Es/Ec 7.6
A' s (in2) 0.00
Av (in2) 0.10 ρmax 0.02242 OK
f' s* 60000.0
c (in) 1.98
As (in2) 0.62
ρ 0.00967
ρb 0.03096
a (in) 1.63
# of Bars 2
ρ' 0.00000
φ 0.9
Shear Reinforcement
εt 0.0132
Compression Steel yields
Mn (ft-kip) 30.59
Mu (in-lb) 330386
Mu (ft-kip) 27.53
9
The measured load at failure was 10.4 kips giving a moment at failure of 31.2 ft-kips. Thus, the ACI design calculations were somewhat conservative with an actual moment capacity at failure which was 1.9% larger than the calculated nominal flexural strength (Mu = 30.6 ft-kips), and an actual moment capacity at failure which was 13.3% larger than the calculated ultimate strength of the beam (Mu = 27.5 ft-kips). Table 4 Shear Strength Analysis
Shear Strength Analysis X (ft)
X (in) 0 0.5 0.89 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9
0 6 10.69 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108
V(x) (lb) 10197.10 10197.10 10197.10 10197.10 10197.10 10197.10 10197.10 10197.10 0.00 0.00 0.00 0.00 0.00 -10197.10 -10197.10 -10197.10 -10197.10 -10197.10 -10197.10 0.00
Vc (lb) 8569.36 8569.36 8569.36 8569.36 8569.36 8569.36 8569.36 8569.36 8569.36 8569.36 8569.36 8569.36 8569.36 8569.36 8569.36 8569.36 8569.36 8569.36 8569.36 8569.36
Vc Limit (lb) 14996.37 14996.37 14996.37 14996.37 14996.37 14996.37 14996.37 14996.37 14996.37 14996.37 14996.37 14996.37 14996.37 14996.37 14996.37 14996.37 14996.37 14996.37 14996.37 14996.37
φVc (lb) 6427.02 6427.02 6427.02 6427.02 6427.02 6427.02 6427.02 6427.02 6427.02 6427.02 6427.02 6427.02 6427.02 6427.02 6427.02 6427.02 6427.02 6427.02 6427.02 6427.02
φVc / 2 (lb) 3213.51 3213.51 3213.51 3213.51 3213.51 3213.51 3213.51 3213.51 3213.51 3213.51 3213.51 3213.51 3213.51 3213.51 3213.51 3213.51 3213.51 3213.51 3213.51 3213.51
10
The shear strength analysis of the beam shows that stirrups will be required in the outer thirds of the beam based on a maximum shear calculated from the nominal flexural design strength (Vmax = 10.2 kips). The spacing design is verified in the following graph. Table 5 Shear Reinforcement Spacing 1 S1
2 S2 12.76
4 S4 -7.48
5 S5 -7.48
Shear Reinforcement Spacing Analysis 3 S3
6 S6 -2.89
7 S7
-2.89
8 S8
-2.89
9 S9
-7.48
0
5.34
12.76
0
5.34
12.76
0
5.34
12.76
0
5.34
0 S0
0
5.34
S Stheoretical
0
2.5 5 5 5 5 5 5 5 5 5
3.75
5.34
ÎŁS (ft)
0
ÎŁS (in) 45
#xS
S (in)
5.34
9x 5
X(in)
0
Series1
Actual Spacing
S min
S max
5.34
10
0
8
5.34
6
0
4
5.34
2
X(ft) 0.208333333 0.625 1.041666667 1.458333333 1.875 2.291666667 2.708333333 3.125 3.541666667 3.958333333 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5
Smin
-10.00
-5.00
0.00
5.00
10.00
15.00
0
X (ft)
Smax
Spacing (in)
11
As can be seen from the spacing diagram, a stirrup spacing of 5 inches for a distance of 4 feet from each end is adequate according to the relevant ACI code provisions governing shear reinforcement. Table 6 Serviceability
yA (in3) yt (in) 5.69 3.39
Jd (in)
Serviceability Analysis Kd (in)
9.56
Mn (ft-kip) 30.59
A (in2)
Mn (in-lb) 367096
y (in)
Mcr (ft-kip) 6.52
(in)
Calculated Deflection
432 1.28 433.28
Mcr (in-lb) 78220
Ie
72 4.10 76.10
Ie (in4) 334.73
(in4)
6 0.3125 ÎŁ
Icr (in4) 329.32
(in)
Measured Deflection
Calculated Deflection at Failure Iut (in4) 888.69
(in-lb)
Max Moment
0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0324 0.0573 0.1019 0.1446 0.1845 0.1537 0.1230 0.0922 0.0615 0.0307 0.0061 0.0000 0.0307 0.0615 0.0922 0.1230 0.1845 0.2224 0.2591 0.2948 0.3301 0.3649
(ft-kip)
888.686 888.686 888.686 888.686 888.686 888.686 821.155 581.140 435.557 383.713 360.798 360.798 360.798 360.798 360.798 360.798 360.798 360.798 360.798 360.798 360.798 360.798 360.798 349.143 342.600 338.647 336.120 334.429
Max Moment
0.0000 0.0000 0.0000 0.0045 0.0085 0.0125 0.0180 0.0280 0.0640 0.1065 0.1445 0.1360 0.1175 0.0980 0.0760 0.0530 0.0360 0.0000 0.0065 0.0195 0.0410 0.0635 0.1050 0.1425 0.1740 0.2120 0.2500 0.2920
(lb)
0 13608 27216 40824 54432 68040 81648 102060 136080 170100 204120 170100 136080 102060 68040 34020 6804 0 34020 68040 102060 136080 204120 238140 272160 306180 340200 374220
Live Load, P 0.00 1.13 2.27 3.40 4.54 5.67 6.80 8.51 11.34 14.18 17.01 14.18 11.34 8.51 5.67 2.84 0.57 0.00 2.84 5.67 8.51 11.34 17.01 19.85 22.68 25.52 28.35 31.19
(kips) 0 378 756 1134 1512 1890 2268 2835 3780 4725 5670 4725 3780 2835 1890 945 189 0 945 1890 2835 3780 5670 6615 7560 8505 9450 10395
Live Load, P 0.000 0.378 0.756 1.134 1.512 1.890 2.268 2.835 3.780 4.725 5.670 4.725 3.780 2.835 1.890 0.945 0.189 0.000 0.945 1.890 2.835 3.780 5.670 6.615 7.560 8.505 9.450 10.395
P (lb) 10197.1
Δ (in) 0.358
12
Serviceability analysis shows that deflections are conservatively calculated using the ACI code conventions. The following deflections vs. load graphs illustrate the divergence between the measured deflection/load relationship and the calculated deflection/load relationship. At the failure load the calculated deflection (Δcalc = 0.365 in.) is 25% larger than the measured deflection. (Δmea = 0.292 in.) Since the service load of this member is not specified, we cannot compare the measured and calculated deflections for this loading. Figure 10 Deflection vs. Live Load Graphs
13
The raw deflection data without calculated deflections are presented below.
Slump Original Height (in) 12 12 12
Table 7 Deflection Data
Batch 1 Batch 2 Batch 3
1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.89
Calibration Factor
Final Height (in) 9.5 9.5 9
Live Load, P (kips) 0.00 0.38 0.76 1.13 1.51 1.89 2.27 2.84 3.78 4.73 5.67 4.73 3.78 2.84 1.89 0.95 0.19 0.00 0.95 1.89 2.84 3.78 5.67 6.62 7.56 8.51 9.45 10.40
Slump 2.5 2.5 3
North (in)
South (in) 0.000 0.000 0.000 0.005 0.009 0.013 0.019 0.029 0.066 0.109 0.147 0.137 0.118 0.098 0.076 0.053 0.035 0.000 0.007 0.020 0.042 0.065 0.107 0.145 0.177 0.216 0.254 0.296
Batch 1 Batch 2 Batch 3
0.000 0.000 0.000 0.005 0.009 0.013 0.018 0.028 0.064 0.107 0.145 0.136 0.118 0.098 0.076 0.053 0.036 0.000 0.007 0.020 0.041 0.064 0.105 0.143 0.174 0.212 0.250 0.292
Average Deflection, Δ (in)
Concrete Mix Data
0.000 0.000 0.000 0.004 0.008 0.012 0.017 0.027 0.062 0.104 0.142 0.135 0.117 0.098 0.076 0.053 0.037 0.000 0.006 0.019 0.040 0.062 0.103 0.140 0.171 0.208 0.246 0.288
Deflection
Deflection Test Data
Mix Design Used Material Batch 1 Batch 2 Batch 3 Type Weight (lbs) Weight (lbs) Weight (lbs) Cement 56 56 56 Water 36 34 32 Coarse Aggregate 207 207 207 Fine Aggregate 142 142 142 Σ 441 439 437 Final Slump (in.) 2.5 2.5 3 *Wet Basis (Assumed): 3% H2O - Fine Aggregate, 0.5% H2O on Coarse Aggregate
Line Pressure, p (ksi) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.5 2.0 2.5 3.0 2.5 2.0 1.5 1.0 0.5 0.1 0.0 0.5 1.0 1.5 2.0 3.0 3.5 4.0 4.5 5.0 5.5
Concrete Compressive Stress Max Load (kips) Cylinder Cross-section (in2) 135.5 28.27433388 108.8 28.27433388 134.4 28.27433388 Average Compressive Stress
Max Moment (ft-kip) 0.00 1.13 2.27 3.40 4.54 5.67 6.80 8.51 11.34 14.18 17.01 14.18 11.34 8.51 5.67 2.84 0.57 0.00 2.84 5.67 8.51 11.34 17.01 19.85 22.68 25.52 28.35 31.19
f'c (ksi) 4.79 3.85 4.75 4.46
14
Figure 11 Raw Deflection vs. Live Load Graphs
15
The strain test data presented below were gathered with a Whittemore strain gauge using Whittemore buttons as specified above and recorded as follows:
Strain Test Data Line Pressure (ksi) Calibration Factor Live Load (kips) 12 11 10.25 9.5 8.75 8 Level 7.25 6.5 5.75 5 4.25 3.5
0 1.89 0 --491 350 965 479 456 669 629 734 938 725 764
0.2 1.89 0.378 --490 350 965 478 455 669 630 733 938 726 764
0.4 1.89 0.756 --490 348 965 480 457 669 630 731 940 727 766
0.6 1.89 1.134 --489 347 964 479 455 669 630 731 940 727 766
0.8 1.89 1.512 --488 346 963 478 455 669 629 735 940 728 767
1 1.89 1.89 --486 345 962 477 454 668 630 726 941 730 769
1.2 1.89 2.268 --485 344 961 476 454 668 630 724 942 731 771
1.5 1.89 2.835 --482 342 959 475 454 668 631 725 944 733 774
2 1.89 3.78 --472 335 957 475 457 674 642 746 964 760 803
2.5 1.89 4.725 --465 340 955 479 464 686 658 763 988 785 832
3 1.89 5.67 --459 328 955 480 470 694 671 776 1006 805 855
3.5 1.89 6.615 --450 323 952 481 473 702 681 789 1008 825 877
4 1.89 7.56 --442 319 950 480 477 708 688 812 1010 840 894
Figure 12 Strain Test Data
16
Figure 13 Level vs. Strain Data Line Pressure (ksi) Calibration Factor Live Load (kips) 12 11 10.25 9.5 8.75 8 Level 7.25 6.5 5.75 5 4.25 3.5
0 1.89 0 --0 0 0 0 0 0 0 0 0 0 0
0.2 1.89 0.378 ---1 0 0 -1 -1 0 1 -1 0 1 0
0.4 1.89 0.756 ---1 -2 0 1 1 0 1 -3 2 2 2
0.6 1.89 1.134 ---2 -3 -1 0 -1 0 1 -3 2 2 2
0.8 1.89 1.512 ---3 -4 -2 -1 -1 0 0 1 2 3 3
1 1.89 1.89 ---5 -5 -3 -2 -2 -1 1 -8 3 5 5
1.2 1.89 2.268 ---6 -6 -4 -3 -2 -1 1 -10 4 6 7
1.5 1.89 2.835 ---9 -8 -6 -4 -2 -1 2 -9 6 8 10
2 1.89 3.78 ---19 -15 -8 -4 1 5 13 12 26 35 39
2.5 1.89 4.725 ---26 -10 -10 0 8 17 29 29 50 60 68
3 1.89 5.67 ---32 -22 -10 1 14 25 42 42 68 80 91
3.5 1.89 6.615 ---41 -27 -13 2 17 33 52 55 70 100 113
4 1.89 7.56 ---49 -31 -15 1 21 39 59 78 72 115 130
17
The Level vs. Strain plot in Figure 13 nicely illustrates the existence of a neutral axis 3.3 inches from the top of the beam at the final loading of 7.6 kips corresponding to a moment of 22.68 ft-kips. The steel stress at this loading can be shown to be:
The concrete stress at this loading can be shown to be:
This means that the steel is still within its elastic range, and the concrete is a bit outside its elastic range. However, the best approximation available for c, the neutral axis, at this loading is to assume that the behavior of both materials is elastic and the section is cracked. In this scenario “kd� gives the location of the neutral axis.
[√(
)
]
The calculated location of the neutral axis for this loading (kd = 3.4 in.) is in agreement with the test data indicating the neutral axis at a distance of 3.4 inches from the top surface of the beam.
Conclusions: The primary criteria for evaluating our results are the design values calculated based on the ACI code. The critical design values calculated and tested within this experiment were the ultimate compressive strength of concrete, the nominal flexural strength of the beam, the ultimate flexural strength of the beam, the magnitude of deflection at various loads, and the strain behavior of the beam. These topics will be discussed sequentially. Ultimate Compressive Strength: The ultimate compressive strength of the concrete used in the beam is obviously critical to its strength characteristics; however, the order of placement of the batches into the formwork and the respective strength of each batch plays an important role in determining the ultimate strength of the member. The concrete above the neutral axis of the beam after cracking is the only part of the concrete in compression; therefore, a stronger or weaker concrete batch at the top of the beam can affect the overall strength of the beam markedly. In this test, the batch placed at the top of the beam had a compressive strength of 4750 psi. The average compressive strength across all three batches was lower at 4460 psi. This difference in compressive strength of the concrete could easily account for the slight over-strength of the beam when compared to the ACI calculated nominal flexural strength. (Mn,test = 18
31.19 ft-kips; Mn,calc = 30.6 ft-kips) The lower water content of the later batches probably accounts for the higher compressive strengths of the cylinders.
Nominal/Ultimate Flexural Strength of the Beam: As mentioned above, the measured flexural strength of the beam at failure exceeded the calculated nominal flexural strength of the beam by 1.9%. Thus, the ACI design formulae provided a slightly conservative, but fairly accurate estimate of the member strength. The flexural strength of the beam at failure was 13.3% larger than the ultimate (factored) flexural strength of the beam according to the ACI code. This 13.3% excess capacity is meant to account for uncertainties in the design and construction of the member in the field. Deflections at Various Loads: The deflection equations within the ACI code are meant to give a general, conservative estimate of the maximum deflections for beams under service loading. The equations are not intended to predict deflections accurately. In this test, the ACI deflection equations performed as expected, consistently mirroring the actual deflections of the beam under load. When calculating the deflections for a beam which has been previously loaded and cracked it is important to keep in mind that the beam cannot be “uncracked�. The moment of inertia of the beam at its highest load after cracking will be the moment of inertia of the beam until a higher load induces more cracking, thus lowering the moment of inertia further. This must be accounted for when reloading a previously loaded beam. Strain Behavior of the Beam: The strain behavior of the beam was most effectively illustrated by the Level vs. Strain diagram provided in the results section of the report. The linear variation of strain with height clearly shows the neutral axis (plane of zero strain) at approximately 3.3 inches below the top surface of the beam. This data agrees very well with the calculated position of the neutral axis for elastic/cracked crosssections at 3.4 inches. This result confirms the basis of the ACI flexural equations, which assume that strain varies linearly with increasing load and plane sections remain plane under bending forces. Error: There were multiple sources of potential error in this multi-phase test, but the most significant source of error was likely human error in measurement and recording of data. The data showed several anomalies which were likely caused by the misreading of instruments and/or the miscalibration of test equipment. However, the test data set was large enough to yield acceptable results independent of a few minor errors. Another potential source of error in this test procedure was the control of the curing conditions and time of the beam. The beam was kept under a vapor barrier for the entire curing period; however, the beam was allowed to cure for 33 days instead of the standards 28 days. This additional curing time could account for the slightly increased strength of the beam. 19
In summation, the ACI equations performed exceptionally well yielding very good estimates of the actual flexural strength and strain behavior of the beam. The deflection equations were less accurate, but performed their intended function of limiting deflections in the interests of serviceability.
20
Appendix A: Apparatus Tinius Olsen compression testing machine Winslow Scale Company Batching Unit Acme revolving drum concrete mixer Toledo 100 lb. scale Electric Gantry Hoist Pre-constructed formwork Gilson concrete vibrator Custom designed concrete beam testing machine 2-3 Five gallon plastic buckets 2 sawhorses Tie wire tools 1 wheelbarrow 3 labeled concrete cylinder forms Water hose Threaded hoist connectors Slump testing cone and rod Measuring Tape Whittemore Gauge and Buttons Deflection gauges 2-3 Shovels Plastic sheeting Miscellaneous concrete finishing tools
21
Appendix B: Shear, Moment, and Deflection Diagrams
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References 1. ASTM C143/ C143M – 10a, “Standard Test Method for Slump of HydraulicCement Concrete," Vol 04.02, 2010. 2. ASTM C172 / C172M - 10, “Standard Practice for Sampling Freshly Mixed Concrete," Vol 04.02, 2010. 3. ASTM C39 / C39M – 12, “Standard Test Method for Compressive Strength of Cylindrical Concrete Specimens,” Vol 04.02, 2012. 4. ASTM C192 / C192M – 07, “Standard Practice for Making and Curing Concrete Test Specimens in the Laboratory,” Vol 04.02, 2007. 5. ACI 318, 2011, Building Code Requirements for Reinforced Concrete, American Concrete Institute, Detroit, Michigan. 6. Guralnick, Sidney, “CAE 307 Lab Manual”, Illinois Institute of Technology, Spring 2008. 7. Nilson, Arthur, et al; ”Design of Concrete Structures”; McGraw Hill, New York, NY; 13th Edition.
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