Measure and record the cross-section dimensions
Introduction
The purpose of this lab is to study the behavior of axially loaded compression members. The experiment focuses on two parameters affecting the load capacity of slender members in compression: length and support conditions. We will be using the 100kN Instron 5582 Universal Testing Machine to test eight steel bars with constant cross-section and different lengths. The specimens will be tested to failure. Support conditions will vary from pinned to fixed ends and combinations of these.
Procedure
Measure and record the cross-section dimensions and the length for all specimens. Calculate the expected theoretical buckling load using Euler’s formula. Test the specimens and record the experimental buckling load.
Formulas:
σcr = Critical stress
A = area of the column
I = moment of inertia
E = Modulus of elasticity = 200 GPA
r = radius of gyration
P = Applied Load
Results
Given data
| Column Dimensions | |
| b | h |
| 12.555 | 6.4 |
| 12.55 | 6.4 |
| 12.55 | 6.5 |
| 12.6 | 6.35 |
| 12.6 | 6.35 |
Average | 12.571 | 6.4 |
Table 2 Effective lengths of the columns
Columns | Support Conditions | L'(mm) |
1 | Pinned-Pinned | 118 |
2 | Pinned-Pinned | 143 |
3 | Pinned-Pinned | 170 |
4 | Pinned-Pinned | 195 |
5 | Pinned-Pinned | 250 |
6 | Pinned-Pinned | 300 |
7 | Pinned-Pinned | 350 |
8 | Pinned-Pinned | 225 |
9 | Fixed-Pinned | 220 |
10 | Fixed-Fixed | 180 |
Table 3 Calculated values of stress and load for theoretical and experimental data
Spec | Area | Moment of inertia (I) | L’ | r | L’/r | (L’/r)2 | Theoretical | Experimental | ||
P’cr | σ’cr | Pcr | σcr | |||||||
mm2 | mm4 | mm | mm | mm | mm2 | kN | MPa | kN | MPa | |
1.00 | 80.45 | 274.62 | 118.00 | 1.85 | 63.87 | 4079.30 | 4.2 | 52.20 | 4.37 | 54.29 |
2.00 | 80.45 | 274.62 | 143.00 | 1.85 | 77.40 | 5990.92 | 5.59 | 69.48 | 5.61 | 69.68 |
3.00 | 80.45 | 274.62 | 170.00 | 1.85 | 92.02 | 8466.80 | 8.9 | 110.62 | 9.09 | 112.97 |
4.00 | 80.45 | 274.62 | 195.00 | 1.85 | 105.55 | 11140.14 | 13.7 | 170.28 | 13.93 | 173.18 |
5.00 | 80.45 | 274.62 | 250.00 | 1.85 | 135.32 | 18310.55 | 17.23 | 214.16 | 17.27 | 214.67 |
6.00 | 80.45 | 274.62 | 300.00 | 1.85 | 162.38 | 26367.19 | 7.45 | 92.60 | 24.85 | 308.86 |
7.00 | 80.45 | 274.62 | 350.00 | 1.85 | 189.44 | 35888.67 | 5.4 | 67.12 | 37.03 | 460.22 |
8.00 | 80.45 | 274.62 | 225.00 | 1.85 | 121.78 | 14831.54 | 11.2 | 97.45 | 11.51 | 100.12 |
9.00 | 80.45 | 274.62 | 220.00 | 1.85 | 119.08 | 14179.69 | 21.01 | 182.76 | 21.10 | 183.55 |
10.00 | 80.45 | 274.62 | 180.00 | 1.85 | 97.43 | 9492.19 | 29.08 | 253.01 | 29.00 | 252.33 |
Table 4 the experimental values of k
Spec | Area | Moment of inertia (I) | Pcr | K | ||
mm2 | mm4 | kN | Exp | Theo | ||
8.00 | 80.45 | 274.62 | 11.51 | 0.96 | 0.70 | |
9.00 | 80.45 | 274.62 | 21.10 | 0.73 | 0.70 | |
10.00 | 80.45 | 274.62 | 29.00 | 0.76 | 0.70 | |
Conclusion
The difference between the theoretical and experiential values is 0.2 to 0.8 kN. This may be due to the loading placements or machine calibration tolerances. The error in the reading is less than 5% so it’s acceptable. Also from the plot between stress and L’/r, it’s a straight line for the pinned support.
References
- Timoshenko, S. P.; Gere, J. M. (1961). Theory of Elastic Stability (2nd ed.). McGraw-Hill.
- Nenezich, M. (2004). “Thermoplastic Continuum Mechanics”. Journal of Aerospace Structures. 4.
- Koiter, W. T. (1945). The Stability of Elastic Equilibrium (PDF) (PhD Thesis).
- Rajesh, Dhakal; Maekawa, Koichi (2002). “Reinforcement Stability and Fracture of Cover Concrete in Reinforced Concrete Members”. Journal of Structural Engineering.