ENGR 244 Lab 6

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Lab 6: Behaviour of Columns Under Axial Load Lab Section CI – X Winter 2022 Professor Ahmed Soliman Concordia University Montreal, QC, Canada Tuesday March 22 nd , 2022 Table of Contents Nomenclature pg 3 List of Tables pg 3 List of Figures pg 3

Objective pg 4 Introduction pg 4 Procedure pg 4 Results pg 4-5 Discussion pg 5-6 Conclusion pg 6 References pg 7 Appendix pg 8-13 Original Data pg 14 Nomenclature λ = slenderness ratio σ all = allowable stress σ crit experiment = experimental critical stress σ crit theo = theoretical critical stress A = cross sectional area D = diameter

d i = inner diameter d o = outer diameter E = modulus of elasticity I = polar moment of inertia K = effective length factor L = length L eff = effective length R = radius of gyration P crit experiment = experimental critical load P crit theo = theoretical critical load List of Tables Appendix A: Experimental and Theoretical Results for Hollow Specimens pg. 8 Appendix B: Experimental and Theoretical Results for Solid Specimens pg. 9 List of Figures Figure 1: Stresses vs Slenderness Ratio pg. 5 Appendix C: Fixed-Fixed Hollow pg. 9 Appendix D: Fixed-Fixed Solid pg. 10 Appendix E: Pin-Fixed Hollow pg. 10 Appendix F: Pin-Fixed Solid pg. 11 Appendix G: Pin-Pin Solid 74mm pg. 11 Appendix H: Pin-Pin Solid 125mm pg. 12 Appendix I: Pin-Pin Hollow 225mm pg. 12 Appendix J: Pin-Pin Solid 225mm pg. 13 Objective: The objective of this lab is to determine the experimental and theoretical buckling loads of columns with different lengths and different end conditions. The buckling loads will be found for both hollow and solid columns. Introduction: Deformation of a column can occur when an axial load is applied. As the load increases, the deformation shall also increase and will eventually reach a point called the critical buckling load. The critical buckling load will vary throughout columns based on the length, end conditions, cross sectional shape and the location of the load on the material [ CITATION Con \l 4105 ]. The critical stresses are also dependent on the stiffness, strength and ductility of the material used for the column [ CITATION RMA17 \l 4105 ]. Using the

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previous equation used for deflection, (d 2 y/dx 2 ) = M/EI, we can derive a new equation used to find the theoretical critical buckling load: P cr = (π 2 EI)/L 2 . This equation is called Euler’s formula which requires three conditions to be met: the load must be along the centroidal axis of the column, the ends of the column must be pinned and the column must behave elastically [ CITATION Con \l 4105 ]. In practice, these assumptions are not met as columns are expected to show inelastic behaviour to axial loads [ CITATION RMA17 \l 4105 ]. Since σ cr = P cr /A, we can also say that σ cr = (π 2 E)/(L/r) 2 where r = (I/A) 1/2 and (L/r) 2 = λ 2 . In this equation, r is the radius of gyration, A is the cross-sectional area of the column and λ is the slenderness ratio. Euler’s formula assumes that both ends of the column are pinned and so a new factor, K, comes in when different types of supports are used. Different support types give different “effective lengths” to the column. The effective length can be given by the following equation: L e = KL where K is the effective length ratio, which varies with each type of support. Although Euler’s formula gives the theoretical critical buckling loads, the value calculated tends to be significantly higher than the experimental values due to assumptions that are not met in real life [ CITATION Con \l 4105 ]. Procedure: Prepare the compression testing device and measure the inner diameter, outer diameter, and length of each specimen. For the 225 mm specimens, do tests using all of the support conditions provided (pin- pin, pin-fixed, fixed-fixed). When the specimen is well placed at the supports, start increasing the compressive load and watch for bending. Record the maximum load after the specimen bends. For the smaller specimen, 75mm and 125mm, repeat the same steps but only using the pin-pin supports. Results: Sample Calculations Formulas used: L eff = KL, r = (I/A) 1/2 , λ = L eff /r, P crit theo = (π 2 EI)/L eff 2 , σ crit experiment = P crit experiment /A, σ crit theo = (π 2 E)/λ 2 , σ all = (139 – 0.868λ) MPa, σ all = (351000/λ 2 ) MPa For hollow bar: A = (π/4)(d o 2 – d i 2 ), I = π(d o 4 – d i 4 )/64 For solid bar: A = (π/4)(d 2 ), π(d 4 )/64 Calculations for 225mm, pin-fixed, hollow tube L eff = (0.707)(225mm) = 159.075 Cross section: A = (π/4)(6.363 2 – 4.5825 2 )  A = 15.3062mm 2 I = π(6.363 4 – 4.5825 4 )/64  I = 58.8209mm 4 r = (58.8209/15.3062) 1/2 = 1.9603mm λ = 159.075/1.9603 = 81.1465 σ all = 351000/81.1465 2  σ all = 53.3049MPa P crit theo = (π 2 (70x10 9 )(58.8209))/0.159075  P crit theo = 1605.9262N σ crit theo = (π 2 )(70x10 9 )/81.1465 2 = 104.9200MPa

σ crit experiment = 1285/15.3062  σ crit experiment = 83.9529MPa 20 40 60 80 100 120 140 160 0 50 100 150 200 250 300 350 400 450 500 Stresses vs Slenderness Ratio Theoretical Critical Stress Exponential (Theo- retical Critical Stress) Experimental Critical Stress Exponential (Exper- imental Critical Stress) Allowable Stress Exponential (Al- lowable Stress) Slenderness Ratio Critical Stress (MPa) Figure 1: Stresses vs Slenderness Ratio Discussion: The results seen on tables 1 and 2 (page 8-9) are in line with our original assumptions. Due to having varying conditions, the experimental stresses tended to be lower than the theoretical stresses. Figure 1 above shows two cases where the experimental stress was higher than the theoretical stress, but this can be explained due to issues at the end points. Since both cases came from pin-pin end conditions, we can assume the higher experimental stress was due to a transfer of load onto the pins instead of the load being held by only the specimen. Transfer loads can be caused by the pins lacking mobility once connected to the columns and can therefore get jammed into unfavorable positions. In the opposite cases, where the experimental stress is much lower than the theoretical stress, it can be explained by unforeseen lateral forces. The equipment used did not have a reliable way to position the columns straight under the axial load, so it is very likely that the columns were at a slight angle which takes a portion of our axial load and turns it into a lateral load. Since columns are more susceptible to lateral loads [ CITATION RMA17 \l 4105 ], small angles can lead to much earlier buckling stresses. When comparing the experimental stresses to the allowable stresses, we can see in figure 1 that the curve is comparable, and the experimental values were higher. Since the allowable stress values were designed to include a factor of safety, it was expected that the critical stresses would be higher. Comparing the allowable stresses to the theoretical stresses, we can see that a factor of safety of approximately 2 was used. Although this factor of safety wasn’t reached with our experimental results, it can be explained by all the discrepancies mentioned between the theoretical and experimental stresses.

In construction, columns are commonly made of cement or concrete with possibilities of steel rod reinforcements[ CITATION فعج 15 \l 4105 ]. These types of columns are expected to be inelastic and therefore experiment permanent deformation if ever moved [ CITATION RMA17 \l 4105 ]. Since these types of columns are extremely resistant to buckling from axial loads. The most common sources of failure are due to unexpected lateral loads. These loads can range from strong winds, earthquakes, or high impacts [ CITATION فعج 15 \l 4105 ]. The steel rod reinforcements help to strengthen the columns against lateral loads, as steel has more buckling resistance than concrete when subjected to lateral loads [ CITATION فعج 15 \l 4105 ]. Though highly resistant to compressive loads, columns can still fail if the axial load exceeds the critical point [ CITATION فعج 15 \l 4105 ]. Changes in temperature can also compromise columns as they will expand or contract, but columns are designed to be able to withstand most temperature changes [ CITATION فعج 15 \l 4105 ]. It is also possible that the steel bars or the concrete get corroded or damaged due to water or acids [ CITATION فعج 15 \l 4105 ]. Comparing our results in tables 1 and 2 (page 7), we can see that the hollow specimens are significantly more resistant to buckling than their solid counter parts. Mathematically explained, it is simply because hollow columns have a greater moment of inertia compared to solid columns of the same cross-sectional area. It can also be explained by the lower slenderness ratio for higher moments of inertia. In other words, the columns are wider for the same cross-sectional area which provides more stability and strength. Hollow columns also lower the overall weight of the structure [ CITATION Ard19 \l 4105 ] which is a large factor in bigger buildings. Conclusion: Looking at our results on tables 1 and 2 (page 8-9) and figure 1 (page 5), we can say that the experiment was a success. The experimental values were where they were expected to be except for a few cases where equipment affected the outcomes. The specimen buckled where they were expected to buckle, and it was able to be concluded that the hollow specimen had higher buckling resistance than their solid counterparts. To get better and more conclusive results, more tests would need to be done while adhering to the requirements of Euler’s formula. References [1] C. U. E. a. C. Science, ENGR 244 Mechanics of Materials Lab Manual, Montreal: Concordia University. [2] R. A. e. A. Awazli, "Behaviour of Reinforced Concrete Columns Subjected to Axial Load and Cyclic Lateral Load," University of Baghdad Engineering Journal, vol. 23, pp. 21-40, 2017. [3] فاقسلا نيز يدنه رفعج .ا .ا .م ., .أ .م , "bayt Employer," 17 01 2015. [Online]. Available:

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https://specialties.bayt.com/en/specialties/q/148975/what-are-the-causes-for-buckling-and-yielding-of- reinforcement-in-concrete-structures/. [Accessed 25 03 2022]. [4] A. e. Margaret.Abraham, "Comparison of Seismic Behaviour of Solid and Hollow Conrete Members in R.C.C Framed Buildings using ETABS," International Journal of Applied Engineering Research, vol. 14, no. 0973- 4562, pp. 161-166, 2019. Appendix Appendix A: Experimental and Theoretical Results for Hollow Specimens Type of Column Hollow Hollow Hollow Hollow Hollow Column Length, L(mm) 75 125 225 225 225 End Condition Pin-Pin Pin-Pin Pin-Pin Pin-Fixed Fixed-Fixed Effective length factor, K 1 1 1 0.707 0.5 Effective length, 75 125 225 159.075 112.5

L eff (mm) Inner diameter, d i (mm) 4.6075 4.59 4.6 4.5825 4.6175 Outer diameter, d out (mm) 6.37 6.353 6.36 6.363 6.363 Cross sectional area, A(mm 2 ) 15.1958 15.1523 15.1500 15.3062 15.0533 Moment of inertia, I(mm 4 ) 58.6992 58.1741 58.3367 58.8209 58.1519 Radius of gyration, r(mm) 1.9654 1.9594 1.9623 1.9603 1.9655 Slenderness ratio, λ 38.1602 63.7950 114.6614 81.1465 57.2373 Allowable stress, σ all (MPa) 105.8769 83.6259 26.6976 53.3049 89.3180 Theoretical critical load, P crit theo (N) 7209.5381 2572.2160 796.1128 1605.9262 3174.3634 Theoretical critical stress, σ crit theo (MPa) 474.4428 169.7575 52.5487 104.9200 210.8749 Experimental critical load, P crit experiment (N) 3940 3542 2018 1285 2017 Experimental critical stress, σ crit experiment (MPa) 259.2823 233.7599 133.2013 83.9529 133.9906 Appendix B: Experimental and Theoretical Results for Solid Specimens Type of Column Solid Solid Solid Column Length, L(mm) 225 225 225 End Condition Pin-Pin Pin-Fixed Fixed-Fixed Effective length factor, K 1 0.707 0.5 Effective length, L eff (mm) 225 159.075 112.5 diameter, d(mm) 6.4 6.443 6.47 Cross sectional area, A(mm 2 ) 32.1699 32.6036 32.8775

Moment of inertia, I(mm 4 ) 82.3550 84.5907 86.0175 Radius of gyration, r(mm) 1.6000 1.6108 1.6175 Slenderness ratio, λ 140.6250 98.7553 69.5518 Allowable stress, σ all (MPa) 17.7493 35.9904 72.5588 Theoretical critical load, P crit theo (N) 1123.8872 2309.4924 4695.4753 Theoretical critical stress, σ crit theo (MPa) 34.9360 70.8355 142.8173 Experimental critical load, P crit experiment (N) 1752 2502 3131 Experimental critical stress, σ crit experiment (MPa) 54.4608 76.7400 95.2323 Appendix C: Fixed-Fixed Hollow

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