INTRODUCTION and THEORY BEAM LOADING AND REACTIONS POSITIONS W= APPUED LOAD Figure I Beam setup and schematic As well as the information given on the unit you will need the following formula: E=εσ Where: σ= Stress (Nm−2)ε= Strain E= Young’s modulus for the beam material (Nm−2) (Typically 69×109Nm−2 or 69GPa ) and IM=yσ M= Bending moment (Nm) I= Second moment of area of the section (m4) σ= Stress (Nm−2) y= Distance from the neutral axis (m) Figure 2 Beam nominal dimensions and strain gauge positions Figure 3 Bending sfress in a beam experiment in the structures frame Ensure the beam and Load Cell are properly aligned. Turn the thumbwheel on the Load Cell to apply a positive (downward) preload to the beam of about 100 N. Zero the Load cell using the control. Take the nine zero strain readings by choosing the number with the selector switch. Fill in Table 1 with the zero force values. Increase the load to 100 N and note all nine of the strain readings. Repeat the procedure in 100 N increments to 500 N. Finally; gradually release the load and preload. Correct the strain reading values for zero (be careful with your signs!) and convert the load to a bending moment then fill in Table 2. Table I Strain (pe) results for experiment 5. Table 2 Strain ( με ) readings for experiment 5 (corrected) Table 3 Averaged strain (με) readings for experiment 5 Results and Discussions 1. From your results, plot a stress distribution over the cross section of beam. 2. Calculate the second moment of area by means of theory and experimental datas, after compare the both results.
Certainly! The images you provided describe a beam bending experiment where the relationship between the bending moment and the strain in a beam is investigated. The experiment is set up to measure strain at various points on the beam using strain gauges as the load is applied.
Let’s go through the steps to calculate the stress distribution and the second moment of area based on the given experimental data.
Step 1: Plot a Stress Distribution Over the Cross Section of the Beam
To plot the stress distribution, we first need to calculate the stress at different points using the formula:
- M is the bending moment
- I is the second moment of area of the beam cross-section
-y is the distance from the neutral axis (for the topmost fiber, y will be negative, and for the bottommost fiber, y will be positive)
First, calculate the second moment of area I for the given cross-section. For a rectangular section, I is calculated using:
- b is the width of the beam
- h is the height of the beam
Given the cross-section dimensions in Figure 2, calculate I for the beam. However, the exact dimensions are not fully visible in the image. Assuming a symmetric I-beam, we would need the dimensions of each segment to calculate I accurately.
Next, use the bending moment values from Table 2 to calculate the stress at each gauge’s position. Since the values of y are the distances from the neutral axis, you can use them directly in the stress formula.
Step 2: Calculate the Second Moment of Area
The second moment of area, or the area moment of inertia, is a geometrical property of a section which reflects how its cross-sectional area is distributed about a certain axis (neutral axis for bending). It’s crucial for determining the bending resistance of a beam.
To calculate I theoretically for the given cross-section, we need the dimensions of the beam. For complex shapes like an I-beam, I can be calculated by breaking down the cross-section into simpler shapes, calculating the I for each shape, and then applying the parallel axis theorem as necessary.
For experimental determination of I, we would use the strain readings and the flexure formula to back-calculate the value of I. This involves measuring the actual strain at various points on the beam for known moments, and using the relationship between moment, strain, and distance from the neutral axis to find I.
Once I is calculated both theoretically and experimentally, you can compare the results to evaluate the experiment’s accuracy.
In the images provided, the detailed dimensions for the cross-section are not completely visible, and the strain readings are also not filled in. Therefore, I cannot perform exact calculations. However, the method I’ve described above is how you would proceed with the given data.
To complete the calculations, you would
- Measure or obtain the full dimensions of the beam’s cross-section.
- Fill in the strain readings for each load condition in the experiment.
- Use the given formulas to calculate the stresses and the second moment of
Once you have the actual dimensions and strain readings, you can follow these steps to perform the necessary calculations.