Lab Report: Analysis of Unsymmetrical Portal Frame

The objectives of this experiment were:

• To determine the horizontal reaction component at a support of an unsymmetrical two-pinned portal frame through experimental means
• To determine the magnitude and direction of the sway caused by the applied vertical load
• To compare experimental results with theoretical calculations using the moment distribution method

• Unsymmetrical Portal Frame Apparatus (with columns of unequal height)
• Dial Gauges (for measuring horizontal reaction and sway)
• Weight Hangers
• Slotted Weights
• Stabilizing Loads

An unsymmetrical portal frame, in this case, one with columns of different heights, is a statically indeterminate structure. Unlike a symmetrical frame, it will sway horizontally even under a symmetrical vertical load. This sway occurs because the differing column lengths lead to unequal stiffness, causing an unbalanced distribution of forces.

To theoretically analyze such a frame, the Moment Distribution Method is employed, considering the effects of side-sway. The analysis is typically performed in two stages:

Non-Sway Analysis: The frame is artificially restrained from swaying, and the moments and reactions due to the external loads are calculated. This also yields the magnitude of the artificial restraining force required.

Sway Analysis: The external loads are removed, and an arbitrary sway is imposed on the frame. The moments and the sway force required to cause this arbitrary sway are calculated.

Finally, using a sway correction factor, the results from the sway analysis are scaled and superimposed onto the non-sway results to obtain the true final moments and reactions in the frame.

1. Initial stabilizing loads of 20 N and 2 N were applied to the designated hangers.
2. The experimental setup was gently tapped to minimize any internal friction.
3. A vertical load, starting at 10 N, was applied at the center of the beam.
4. To simulate a non-sway condition for measurement, an additional horizontal weight was added at support D until the dial gauge at that point returned to its initial reading. This added weight was recorded as the horizontal reaction.
5. The deflection reading from the dial gauge at point B, which measured the sway, was also recorded.
6. These steps were repeated for increasing vertical loads of 20 N, 30 N, and 40 N.

Experimental Observations:

The following table shows the measured horizontal reaction and the deflection at point B for various applied vertical loads.

Theoretical Calculation Summary (for 25 N Vertical Load):

The analysis was performed using the moment distribution method.

Non-Sway Analysis:

• Distribution factors and fixed-end moments were calculated.
• The moment distribution was carried out, and the restraining forces required to prevent sway were found to be Q_A = 1.702 N and Q_B = 5.09 N. The total restraining force was X = 3.388 N.

Sway Analysis:

• An arbitrary sway was assumed, and the corresponding fixed-end moments were calculated.
• The moment distribution for the sway case was performed, yielding a sway force of X’ = 34.848 N.

Sway Correction and Final Results:

• A sway correction factor, λ, was calculated using the formula X + λX’ = 0, which yielded λ = -0.0972.
• This factor was used to modify the moments from the sway analysis, which were then added to the moments from the non-sway analysis to get the final moments.
• From the final moments, the theoretical horizontal reaction at support D was calculated to be 2.439 N.

• From the experimental graph, for an applied vertical load of 25 N, the horizontal reaction was 2.0 N.
• From the experimental graph, for an applied vertical load of 25 N, the horizontal deflection (sway) was 0.32 mm.
• The theoretically calculated horizontal reaction for a 25 N load was 2.439 N.
• The percentage error between the theoretical and experimental horizontal reaction was calculated as:
  \[ \% \text{Error} = \left( \frac{2.439 – 2.0}{2.439} \right) \times 100\% = 18.0\% \]

The experiment was conducted to determine the horizontal reaction and sway in an unsymmetrical portal frame. The experimental value for the horizontal reaction was compared with the theoretical value obtained from the moment distribution method.

A difference of 18.0% was observed between the two values. This discrepancy can be attributed to several factors, including instrumental errors such as friction in the frame’s joints and supports, as well as potential personal errors in reading the dial gauges or applying the loads precisely. The results confirm that an unsymmetrical frame undergoes side-sway even when subjected to a symmetrical vertical load, a key characteristic that distinguishes it from a symmetrical frame.