Details about Lab 4

Let’s start with a quick difference between columns and beams:

1. Columns

   1. Axis - Vertical
   2. Loading - Vertical (for gravity loads) and horizontal (for earthquake and wind loads)
   3. Failure mode - Buckling (for long columns) and material failure (for short columns)

2. Beams

   1. Axis - Horizontal
   2. Loading - Vertical (for gravity loads) and horizontal (for earthquake and wind loads)
   3. Failure mode - Bending/Flexure (for long beams) and shear (for short beam)

A beam is a structural element that primarily resists loads applied perpendicular (vertical) to the beam’s axis (horizontal). The application of load does 3 things:

1. Cause deflections in the beam.
2. Cause internal bending, shear and axial stresses.
3. Cause changes in the support reactions.

Hopefully, you can appreciate what our beams go through with the small in class activity.

Beams have to be supported on supports (columns or footings) and similar to columns, beams also have various end support conditions. We did two different end conditions in our lab:

1. Cantilever: 1 fixed support at one end with 3 reactions (2 Forces and 1 Moment)
2. Simply supported: 2 pinned supports at two ends with 3 reactions (3 Forces)

Let’s go over what to put in the report:

Cantilever beam

For the cantilever, we:

1. Calculated theoretical deflection at A,B,C,D,E,F,G,H with 15N load
2. Measured experimental deflections with 15N load
3. Measured extreme fiber strain for 15N load

What do I need for lab report?

1. Sample Calculation

1. Pick one point from A,B,C,D,E,F,G,H and calculate the theoretical deflection.
2. Calculate the theoretical extreme fiber bending stress.
3. Calculate the experimental extreme fiber bending stress from the measured strain.

Bending stresses

We used the thickest steel beam for cantilever so use the section 6x20x1000 for these calculations.

1. Experimental:
   Multiply the Young’s modulus of steel with the strain to get stress. For strain, use (final - initial) value.

2. Theoretical:
   The bending stress for a beam is given by:

\begin{equation} \sigma = \frac{M y}{I_{zz}} \end{equation}

where M is the bending moment, y is the distance of the extreme fiber from the neutral plane and $I_{zz}$ is the moment of inertia.

Use:

1. M = Bending moment at the end of cantilever that we calculated in the lab.
2. y = 3mm , as the beam is symmetrical the neutral plane will be at the center, so the extreme fiber is at 3mm from neutral plane.
3. $I_{zz} = \frac{b h^3}{12}$, where h is 6mm and b is 20mm.

2. Results/Attachments

1. Report the theoretical bending stress, experimental stress and percentage difference between them.
2. Plot the theoretical deflections and experimental deflections with distance.
3. Draw the shear force diagram (SFD) and bending moment diagram (BMD) for your beam. You can draw it on a paper and paste the picture in your report, but it should be clear. Please talk to me after the next lab about SFDs and BMD.

Simply supported beam

For the simply supported beam, for Beam 2,3 and 4, we:

1. Put load in the middle and got the experimental mid point deflection.
2. Put the load in the middle and got the end reactions.
3. Changed the support to 10mm below its original position and got the end reactions.

What do I need for lab report?

1. Sample Calculations

1. Pick one beam (from Beam 2,3,4) and calculate the mid point deflection, the formula is in the report.
   

2. Results/Attachments

1. Put a table with end reactions for all the beams.
2. Report the end reactions when the support was shifter 10mm below the original level.
3. Plot the theoretical deflections and experimental deflections with load.