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Created: 2024/05/28 19:19

Experiment no. 05

Experiment name: Buckling of slender column

Objective

The objective of this experiment is to,

1. Determine critical / buckling load of slender column (experimental).
2. Determine critical / buckling load of slender column (theoretical).
3. Compare the experimental and theoretical results.

Significance

Understanding buckling in columns is crucial in structural engineering because it directly impacts the structural integrity, safety and reliability of buildings, bridges, and other structures where columns are present. It is due to the fact that unlike tension members, for compressive members buckling can lead to catastrophic failure even if the material's ultimate strength is not exceeded.

Apparatus

1. Digital slide calipers
2. Column testing apparatus
3. Steel scale
4. Support systems

Data sheet

Calculation and graph

Theoretical critical stress

The column specimen's,

• Height, $h=19\text{[PARSE ERROR: Undefined("Command("space")")]}mm$
• Thickness, $b=3.25\text{[PARSE ERROR: Undefined("Command("space")")]}mm$
• Length, $L=650\text{[PARSE ERROR: Undefined("Command("space")")]}mm$
• Modulus of Elasticity, $E=210\times {10}^3\text{[PARSE ERROR: Undefined("Command("space")")]}Nm{m}^{-}2$

Hence, moment of inertia,

$$I_{min}=\frac{h{b}^3}{12}=\frac{19\times {3.25}^3}{12}=54.3529\text{[PARSE ERROR: Undefined("Command("space")")]}m{m}^4$$

Radius of gyration,

$$r=\sqrt{\frac{I_{min}}{A}}=\sqrt{\frac{54.3529}{19\times 3.25}}=0.9382\text{[PARSE ERROR: Undefined("Command("space")")]}mm$$

Therefore, slenderness ratio,

$$\frac{KL}{r}=\frac{650}{0.9382}\times K=692.8160K$$

Critical stress,

$${\sigma }_{cr}=\frac{{\pi }^{2}E}{(\frac{KL}{r}{)}^2}=\frac{4.3180}{K^2}\text{[PARSE ERROR: Undefined("Command("space")")]}MPa$$

Where, $K$ is the effective length factor.

For F-F condition

Effective length factor, $K=0.5$

Therefore, slenderness ratio,

$$\frac{KL}{r}=692.8160K=692.8160\times 0.5=346.41$$

Critical stress, $${\sigma }_{cr}=\frac{4.3180}{{0.5}^2}=17.27\text{[PARSE ERROR: Undefined("Command("space")")]}MPa$$

For H-F condition

Effective length factor, $K=0.707$

Therefore, slenderness ratio,

$$\frac{KL}{r}=692.8160K=692.8160\times 0.707=489.82$$

Critical stress, $${\sigma }_{cr}=\frac{4.3180}{{0.707}^2}=8.64\text{[PARSE ERROR: Undefined("Command("space")")]}MPa$$

For H-H condition

Effective length factor, $K=1$

Therefore, slenderness ratio,

$$\frac{KL}{r}=692.8160K=692.8160\times 1=692.82$$

Critical stress, $${\sigma }_{cr}=\frac{4.3180}{1^2}=4.32\text{[PARSE ERROR: Undefined("Command("space")")]}MPa$$

Experimental critical stress

The column specimen's,

• Height, $h=19\text{[PARSE ERROR: Undefined("Command("space")")]}mm$
• Thickness, $b=3.25\text{[PARSE ERROR: Undefined("Command("space")")]}mm$
• Area, $A=h\times b=61.75\text{[PARSE ERROR: Undefined("Command("space")")]}m{m}^2$

For H-F condition

Critical stress,

$${\sigma }_{cr}=\frac{P_{cr}}{A}=\frac{340\text{[PARSE ERROR: Undefined("Command("space")")]}}{61.75}\text{[PARSE ERROR: Undefined("Command("space")")]}Nm{m}^{-2}=5.51\text{[PARSE ERROR: Undefined("Command("space")")]}MPa$$

For H-H condition

Critical stress,

$${\sigma }_{cr}=\frac{P_{cr}}{A}=\frac{220\text{[PARSE ERROR: Undefined("Command("space")")]}}{61.75}\text{[PARSE ERROR: Undefined("Command("space")")]}Nm{m}^{-2}=3.56\text{[PARSE ERROR: Undefined("Command("space")")]}MPa$$

Result

The theoretical critical stress for the slender column in F-F, H-F, and H-H conditions was $17.27\text{[PARSE ERROR: Undefined("Command("space")")]}MPa$, $8.64\text{[PARSE ERROR: Undefined("Command("space")")]}MPa$, and $4.32\text{[PARSE ERROR: Undefined("Command("space")")]}MPa$ respectively while the experimental critical stress in H-F, and H-H conditions was $5.51\text{[PARSE ERROR: Undefined("Command("space")")]}MPa$ and $3.56\text{[PARSE ERROR: Undefined("Command("space")")]}MPa$ respectively.

Discussion

In this experiment, the buckling of a slender column was determined under three conditions - fixed-fixed, hinge-fixed, and hinge-hinge. For hinge-hinge condition, the critical stress was found to be $3.56\text{[PARSE ERROR: Undefined("Command("space")")]}MPa$ experimentally while the theoretical value was $4.32\text{[PARSE ERROR: Undefined("Command("space")")]}MPa$. Similarly, for hinge-fixed condition, the experimental and theoretical critical stress was found to be $5.51\text{[PARSE ERROR: Undefined("Command("space")")]}MPa$ and $8.64\text{[PARSE ERROR: Undefined("Command("space")")]}MPa$ respectively. The experimental critical stress of fixed-fixed condition was not determined in order to avoid permanent deformation. However, load of up-to $800\text{[PARSE ERROR: Undefined("Command("space")")]}N$ (or $12.96\text{[PARSE ERROR: Undefined("Command("space")")]}MPa$ stress) was applied without reaching critical limit. The theoretical critical stress under the same condition was found to be $17.27\text{[PARSE ERROR: Undefined("Command("space")")]}MPa$.

The column under hinge-hinge condition showed the least critical stress. This is because both ends of the columns were free to rotate, offering the least resistance to buckling.

The column under hinge-fixed condition had a lower buckling load compared to the fixed-fixed condition but higher than the hinge-hinge condition. The fixed end provided significant resistance to rotation, but the hinged end allowed some rotation, reducing the overall stability.

The column under fixed-fixed condition sustained the highest load without reaching the critical limit among these three conditions. This is because both ends of the column are restrained from rotating, providing the greatest resistance to buckling.

The experimental results are aligned well with the theoretical predictions. The column under fixed-fixed condition sustained the highest load, followed by the hinge-fixed, and then the hinge-hinge conditions before buckling.