Details about Lab 5

First, for a beam, we can find out the end support reaction by using the three equation of equilibrium:

1. Force in x direction $F_x$ = 0
2. Force in y direction $F_y$ = 0
3. Moment at a point $M_a$ = 0

Second, let’s look at the last week’s beam and their support conditions:

1. Cantilever : 1 fixed support and 3 reactions
2. Simply supported : 2 supports (hinge and roller) and 3 reactions

See something similar?

If the number of reactions = number of the equations of equilibrium, we have a statically determinant beam.

Now, lets look at the beams we used in this lab:

1. Beam A, Overhanged multi-spam beam : 3 supports (2 rollers and 1 pinned), 4 reactions
2. Beam B, Propped Cantilever : 2 supports (1 fixed and 1 roller), 4 reactions

As the number of reactions > number of equations of equilibrium, we have a statically indeterminate beam.

You will probably find statically indeterminate beams everywhere in the civil engineering structures. The extra unknown reactions add complexity to the analysis of these beams, and require special methods such as moment equilibrium, method of superposition, slope deflection etc etc.

One of the most important things is that any change in the elevation of the end support (rise or fall) causes a big change in the bending moments and the shear forces. This big change could cause a failure in a place where the failure was not expected.

Beam A: Overhanged Multispan

What did we do?

1. Loaded the beam in 3 different loading conditions: Load at A, Load at E, Load at A and E
2. Shifted the support D 3 mm below the horizontal and loaded the beam at A and E.

We measured:

1. End Reactions
2. Deflections at various points
3. Strain from the strain gages

What do I want in the report?

1. Sample Calculation

   1. Consider the final loading condition with load at A and E without change in support height, please provide me the equations and calculations for bending moment and shear forces for each subsection (AB,BD,DF).
   2. Bending stresses using the strain gage data for load at A and E, for both horizontal support and support shifted by 3 mm.

2. Results/Attachments

   1. Percentage difference in the bending stresses for horizontal support and support shifted by 3mm.
   2. Bending moment and shear force diagrams for all the loading conditions. Do not provide the rough diagram, I need the magnitudes at different points.

Beam B : Propped cantilever

What did we do?

Used principle of superposition to get the reaction at the roller end.

1. Loaded the beam to get a deflection on the free end (free loading).
2. Pushed the end support up so that the net deflection is zero and the beam comes back to its original horizontal position.

What do I want in the report?

1. Sample Calculation

   1. For free loading

      1. Calculate the theoretical free end deflection and moment at the support under the maximum load for the cantilever free condition. Use the equation provided in the previous lab.
      2. Calculate the moment at the support for this condition.
      3. Using the theoretical free end deflection as obtained in the first part, find the theoretical force required to counterbalance this deflection. Use the formula: \begin{equation} P_{theo} = \frac{\Delta 3 E I}{L^3} \end{equation}
      4. Calculate the moment just with the $P_{theo}$ at free end.
      5. Add the moments from point 2 and point 4 to get the theoretical moment $M_{theo}$ at support for propped cantilever .
         

   2. After pushing the end support

      1. Percentage difference between the $P_{theo}$ found above and the measured end reaction when the beam became horizontal.

2. Results/Attachment

   1. Bending moment and shear force for 3 conditions:
      1. Only the free loading with 15N
      2. Only the end support force
      3. Free loading + end support force
   2. Compare bending moment at the end support for the propped cantilever with free loading and end support force with the $M_{theo}$ found above.