代写EMS410U: Beam Deflection Experiment 2025-26调试R语言

EMS410U: Beam Deflection Experiment 2025-26

Please refer to the detailed schedule and group lists on QMplus for the times and locations of the activities below.

Note: Although you experiment with the group, you submit your work individually. Thus, every aspect of this lab is essential for every student in the group.

Objectives

The aims of this experiment are:

•   To evaluate the Young’s modulus of a material using a beam under two different load and support configurations.

•   To explore the experimental concept of reading errors of measured quantities and their propagation during the calculation of derived quantities.

Introduction

In previous EMS410U (Experimental Design and Practice 1) experiments, you conducted investigations on "Ideal Gas" and "Viscosity," where you gained hands-on experience in data collection, analysis, and processing. These experiments introduced you to calculating uncertainties and errors in measurements and presenting your findings in a clear, informative manner.

In EMS412U (Mathematical and Computational Modelling I) lectures, you learned how to calculate the second moment of inertia and developed the skills to calculate and plot a bending moment diagram for a beam. This allows you to identify the location and magnitude of the maximum bending moment in the beam, as well as calculate the maximum bending stress and deflection.

This experiment will evaluate how effectively you can integrate the theoretical knowledge from EMS412U with the practical skills developed in EMS410U.

This experiment consists of three parts:

• Pre-experiment phase:

Conducted in your EMS410U Group Management Meeting as a group activity before you experiment in the lab.

In this first part, you will use the information given on the following pages to select the parameters of the experiment for measuring the deflection of a beam for two different load and support configurations. You are also required to write two short experimental procedures.

• Experimental phase:

Conducted in the Lab (G27 in weeks 9, 10 and 11 as a group activity)

In the second part, you will conduct the actual experiment for two different load and support configurations.

• Post-experiment phase/writing up:

Conducted individually after you have completed the lab as a group activity. The submission Deadline will be 2 weeks after you do the Beam Bending Lab. It should be submitted to the  submission point on the EMS410U QMPlus site.

In this section, you will analyse potential reading errors in the beam's dimensions, applied load, and measured deflection. You will then calculate the resulting uncertainty in the derived Young's modulus for the beam material. Following your analysis, you will prepare a comprehensive report detailing your methodology, calculations, and findings. Guidance on report writing will be provided in the Week 9 lecture.

Apparatus

Before the experimental phase of this laboratory, examine the apparatus (Figure 1) and match each of the underlined components in the description below with the labels in Figure 1. Note in Figure 1 the beam is configured as “simply supported” . We will also do experiments in a “cantilever” configuration (see Figure 3)

The apparatus is mounted onto a frame, which consists of two end supports, on which a beam may be simply supported or rigidly fixed (see Figure 1). Both fixed and non-fixed supports do not permit vertical displacements. The non-fixed end support is mounted on a trolley, which permits horizontal travel. The fixed end support does not permit horizontal movement.

The load is applied by using the loading mechanism. The resulting deflection is measured by a digital dial gauge. Both the dial gauge and the loading mechanism are connected by cables to the HDA200  Interface. This interface will allow you to read the applied force and the resulting deflection.

Figure 1 – Main components of the experimental apparatus

(configured as a simply supported beam).

Pre-Experiment Phase: Design of the parameters of the experiment

Work in the Groups assigned to you for EMS410U

This phase should be carried out as a group activity in your Group Management Meeting before you do the experiment (see full schedule on QMplus). In this pre-experiment phase, your group will calculate the parameters of the experiment using the information given below and then write your own experimental procedure for:

1.    A simply supported beam with load applied at the mid-span, see Figures 1 and 2.

2.    A cantilever beam with a load applied at the free end of the beam, see Figure 3.

Thus, you are actually conducting two experiments on beam deflection. This means you will have to perform. two sets of calculations to get the parameters of each experiment and write two experimental procedures. In the event of a problem, please consult the lab facilitators.

First, perform. the calculations necessary using the information below and show your results to the lab facilitators at the start of the Lab session. Your results should summarise all the necessary parameters in each experiment.

Then, write concise experimental procedures for both support and load configurations. Pay particular attention to the initial set-up of the experimental apparatus and zeroing, the set of data you are going to get from the experiment and any repeatability checks you intend to make. Discuss your procedures with the lab facilitators, and if approved, you will be allowed to conduct the experiment.

For your facilitation, five videos have been provided to you on QM+. Please watch these videos before designing your experiment.

1.    Introduction to Stress and Strain

2.    Understanding Young's Modulus

3.    Understanding the Area Moment of Inertia (Second Moment of Area)

4.    Understanding the Deflection of a Beam

5.    How to write an experimental procedure.

Information needed for calculations:

This section is based on material you covered in EMS412U – Mathematical and Computational Modelling I. For more information, go back to the notes/webinars for that module.

The yield strength of steel is in the range of 250-400 MPa. Apply a safety factor of 2.0 during your calculations. You must select experimental parameters (load) such that the yield strength of the beam is not exceeded.

The normal stress in a beam due to bending moment can be calculated as follows.

(1)

where M is the bending moment of the beam, I is a second moment of area of the beam’s cross- section. For (for rectangular cross-section), see Figures 2 and 3.

Figure 2 – Simply supported beam with load applied at the mid-span.

Figure 3 – Cantilever beam with load applied at the free end.

The maximum deflection of a simply supported beam (Figure 2), resulting from a load applied at the mid-span, occurs at the point where the load is applied (i.e. mid-span) and is calculated as follows.

(2)

where F is the applied load, L is the distance between supports, E is Young’s modulus (modulus of elasticity) and I is a second moment of area of the beam’s cross-section.

Similarly, the maximum deflection of a cantilever beam (Figure 3), resulting from a load applied at the free end, occurs at the point where the load is applied (i.e. the free end) and is calculated as follows.

(3)

Hint: Your experimental data will be used to calculate the Young’s modulus (modulus of elasticity) of the beam material. How does this information influence the design of the parameters of the experiment?

Experimental Phase

Work in the Groups assigned to you.

In this part of the lab, you will conduct two experiments using the lab procedures you have written. Useful information and tips:

•    Figure 4 shows the position of the dial gauge.

•    Load can be applied by turning the hand wheel clockwise.

•    The travel of the gauge is 12.7 mm. Discuss with the lab facilitators what to do if the deflection is greater than 12.7 mm.

•    Make sure to check that the results are repeatable.

•    Make a note of the equipment’s accuracy.

Figure 4 – Position of the dial gauge.

Post-Experiment Phase

In this final stage of the lab session, you will explore the propagation of reading errors. To do this, measure the beam’s dimensions with the calliper and a tape measure. Note the reading error for both instruments and calculate the value of the second moment of area I, including the overall error of I. Discuss your results with the lab facilitators.

Data Analysis

Calculate Young’s modulus E from Eq. (2) and Eq. (3) using the corresponding experimental values. Also, using the reading errors of quantities measured by the dial gauge, tape measure and calliper, evaluate the overall error of E by applying the propagation of errors method as discussed in the lecture.

Compare the calculated modulus of elasticity with the values cited in the literature. Assume the beam’s material is mild steel. You need to use a reliable source and reference it in your final report, e.g., a source found on Knovel.

For more information on how to analyse your data, see the synchronous live sessions and additional asynchronous material on the module QMplus site.

Writing up This Laboratory

This is an individual part of the assessment and should be submitted on the QMplus submission point.

More information and guidance will be given in Week 10.