## Stiffness Method- Using 3 Degree of Freedom Beam

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**Abstract**

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The analysis of an unbraced frame, portal frame was performed to asses the structural behaviour of a simple, unbraced statically indeterminate rigidly jointed plane frame. This is evaluated by a series of tests on a portal frame and adding weights at different critical points of the structure. Three test were taken to asses the critical points, the first test had a vertical load at the middle of the beam, second was a horizontal load attached to the corner of the beam and the column and finally the last test was a combination of the two loads at the respective locations. Data was acquired during the lab session and were analysed using graphs to see the various bending moments in each of the three situations. In order to check the suitability of the experiment the values were then compared with values acquired using the Simple Beam Theory and the Stiffness Method.

**Theory**

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Statically Indeterminate Structures:

A statically indeterminate structure is a system whereby reactions and forces cannot be analysed with the equations of static alone. The indeterminacy of the structure could most likely be external, internal and in some cases external and internal.

These statically indeterminate structures have some components such as slabs, beams, columns and foundations being assembled together.

Simple Beam Theory:

The simple beam theory is used to measure the bending moment and its location on the strain gauges. The simple beam theory is used to derive the forces from the experimental data.

**Equation 1.**

Where;

M= The bending Moment (Nm)

I= 2^{nd} moment of area about the neutral axis (m⁴)

σ= Stress acting on a particular point on the beam (Pa)

y= Distance from the neutral axis (m)

E= Young’s Modulus (Pa)

R= Radius of curvature at any point along the beam (m)

This equation is used to measure not only the bending moment but also serves as a guide to solve any of the parameters depending on the values provided. The equation is then rearranged to give an equation for the bending moment.

**Equation 2**

In order to find the bending moment, we need to find the 2^{nd} moment of area about the neutral axis, I and the stress. These are found by;

**Equation 3, ** **Equation 4.**

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Where , is the various strains measured in the lab with the strain gauge.

Stiffness Method- Using 3 Degree of Freedom Beam

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**Procedure.**

**1. **The portal frame is set up with sixteen (16) strain gauges at various points around it.
**2. **The values of each gauge was recorded with no weight on the frame. **(zero load strain)**
**3. **A load of 400g was added to the midpoint of the beam, this was applied as a vertical load. The values were then recorded. **(uncorrected load strain)**
**4. **The 400g load was replaced by another 400g load on the third node of the portal frame, this was applied to act as a horizontal load. The values of the strain were recorded.
**5. **A 400g load was then added back to the midpoint of the beam, with the horizontal load of 400g still in place. The strain values were then recorded.
**6. **The strain is measured for point 1, 4, 8, 9, 13 and 16, using weights of 100g, 200g and 300g. The values are then recorded.

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**EVALUATION.**

a)

LOADCASE 1.

During the experimental process we found out that the maximum bending moment of the first load case was -0.25Nm, where as the theoretical value of the maximum bending moment was -0.3Nm. There were some errors when comparing the two valuea. Some of these reasons could be the swaying of mass on the portal frame causing fluctuations in values when read on the gauge. From figure 2, the maximum moment at A, B, C and D for the experiment was 0.11Nm, 0.21Nm, 0.25Nm and 0.11Nm. Whereas the theoretical value of the same points was, 0.10Nm, 0.20Nm, 0.20Nm and 0.1Nm respectively. It can be seen that some had approximately the same bending moment whereas some had some amount of difference between them. There were errors could be brought from faulty equipment as well as human errors.

LOADCASE 2

Load case 2, also using figure 2, had its experimental values at point A, B, C and D, to be 0.18Nm, 0.08Nm, 0.14Nm and 0.2Nm respectively. This was concluded as opposed to the theoretical values of 0.18Nm, 0.18Nm, 0.18Nm and 0.306Nm respectively. These massive errors could be brought about from various reason such as temperature on the frame and the improper calibration of the strain gauge. The errors could have mainly come from the incorrect position of the weight at point C of the frame, this could bring about a huge effect on the reading of the values.

LOADCASE 3.

The experimental data from load case 3 which was a combination of loads in between point B-C and point C in figure 2. This is combination of the two load cases, 1 and 2. Using an online plotter, it was seen that the maximum bending moment was 0.29Nm. It was found out during the experiment that the experimental maximum bending moment approximately 0.28Nm. At points A to D, the experimental bending moments were, 0.5Nm, 0.57Nm, 0.35Nm and 0.29Nm. The theoretical values were 0.21Nm, 0.012Nm,0.38Nm and 0.41Nm. The errors between the two values are seen to be high in some areas and low in others. These errors could have come from human errors as well as material imperfections.

Despite the errors presented when comparing the experimental and theoretical values, the experiment was a success because the experimental values validates that of the theory.

b)

The principle of superposition in the case of the above load cases states that bending moment of when the two loads are put on the frame, load case 3 should be equal to the sum of the load case with the weight in the middle and the load case with the weight at end of the frame.

From the experimental values of the sum of load cases 1&2 are 0.29, 0.29, 0.39 and 0.31. Load case for load case 3 is 0.5, 0.57, 0.35 and 0.29. There are some errors in this calculations which maybe as a result of cumulative errors or human errors when reading the values. The extent of the errors was minimal and therefore be takes as negligible. This means that the experimental values and the theoretical values validate the principle of superposition.

**HEALTH AND SAFETY**

- Weights could drop on the foot of people participating in the experiments. The way to mitigate this is to wear safety boots which will prevent foot injuries.
- Electricals used in the laboratory could injure any person using the equipment at any point in time. All equipment must be checked to ensure no wires are exposed.
- Sharp edged of hooks holding the weights could injure a person doing the labs. These hooks should be placed in their containers when they are not being used.

**CONCLUSION**

It can be concluded from the laboratory experiment that even though there was not a significant amount of errors we can state that the principle of superposition is valid in the system. The values of the experimental data also validated that of the theoretical data, this shows that the experiment was a success.

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