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日期：2024-08-27 04:13

ES196 Statics and Structures

Beam Bending Laboratory Briefing

Bring to the laboratory:

Laboratory Note Book

This hand-out

PPE Equipment (safety shoes mandatory, lab coat recommended)

Before the laboratory:

Complete the ‘Lab Preparation Online Course’ on Moodle and the associated quiz.

Read the Lab Briefing document (containing instructions and background material) on the ES196 web pages.

Read the Safety sheets for the lab on the ES196 web pages

Sanitise your hand before and after the lab exercise

Summary

By the end of the laboratory assignment the student should:

1. Understand the basic principles of bending moments

2. See how these principles can be demonstrated experimentally.

3. Have studied a determinant overhanging beam, at which concentrated loads are applied through its length.

4. Compare experimental results to theoretical calculations

1. Introduction

Structural members, including beams, are subjected to a series of external forces, which generate internal axial force, shear force and bending moment. Bending, in particular, refers to the behaviour of a structural member exposed to external loads acting perpendicularly to its longitudinal axis. Therefore, the magnitude and the direction of bending are dependent on the magnitude, type and direction of the loads.

2. Main purpose of the experiment

Performing the experiment of calculating the bending moment of a beam, the visual display and the experimental proof of the basic theory of the bending moments is achieved. In particular, the main goal of this experiment is the comprehension of the function of a determinant overhanging beam, at which different loads are applied through its length. Moreover, the computation of the bending moments acting on sections along this beam is performed by experimental and theoretical means. For this reason, the experimental set up shown in Fig. 1 is used.

This document aims to provide a description of how to set up and perform. bending moment in Beam experiments.

Figure 1. Experimental set up for the computation of the bending moments in a beam [1].

3. Theoretical background

Members that are slender and support loadings that are applied perpendicular to their longitudinal axis are called beams. In general, beams are long, straight bars having a constant cross-sectional area and often, they are classified based on their supports. Thus, a simply supported beam is pinned at one end and roller supported at the other, a cantilevered beam is fixed at one end and free at the other, and an overhanging beam has one or both of its ends freely extended over the supports (Fig. 2).

When a rigid body is subjected to a system of forces, which all lie in the x-y plane, three equations are necessary and sufficient for its equilibrium. These equations are the following:

ΣFx = 0

ΣFy = 0                         (1)

ΣMO = 0

Here ΣFx and ΣFy represent, respectively, the algebraic sums of the x and y components of all forces acting on the body, and ΣMO represents the algebraic sum of the couple moments and the moments of all the force components about the z axis, which is perpendicular to the x-y plane and passes through the arbitrary point O. The set of equations (1) is used for the resolution any determinant beam which yields to the calculation of its support reactions.

Figure 2. Different types of beams [2].

Because of the applied loadings, beams develop an internal shear force and bending moment that, in general, vary from point to point along the axis of the beam. In order to properly design a beam it therefore becomes necessary to determine the maximum shear and moment in the beam. One way to do this is to express shear force V and bending moment M as functions of their arbitrary position x along the beam’s axis. These shear and moment functions can then be plotted and represented by graphs called shear and moment diagrams. The maximum values of V and M can then been obtained from these graphs.

In order to formulate V and M in terms of x we must choose the origin and the positive direction for x. Although the choice is arbitrary, most often the origin is located at the left end of the beam and the positive direction is to the right.

In general, the internal shear and moment functions of x will be discontinuous, or their slope will be discontinuous, at points where a distributed load changes or where concentrated forces or couple moments are applied. Because of this, the shear and moment functions must be determined for each region of the beam between any two discontinuities of loading. For example, coordinates x1, x2 and x3 will have to be used to describe the variation of V and M throughout the length of the beam in Fig. 3, where imaginary cuts will be placed to determine the functions of V and M. These coordinates will be valid only within the regions from A to B for x1, from B to C for x2 and from C to D for x3.

Before presenting a method for determining the shear and moment as functions of x and later plotting these functions, it is first necessary to establish a sign convention so as to define “positive” and “negative” values for V and M. Although the choice of sign convention is arbitrary, here we will use the one often used in engineering practice and shown in Fig. 4.

Figure 3. Coordinates x1, x2 and x3 used to describe the variation of V and M [2].

Figure 4. Beam sign convention [2].

The shear and moment diagrams for a beam can be constructed using the following procedure:

• Determine all the reaction forces and moments acting on the beam, and resolve all the forces into components acting perpendicular and parallel to the beam’s axis.

• Specify separate coordinates x having an origin at the beam’s left end and extending to regions of the beam between concentrated forces and/or couple moments, or where there is no discontinuity of distributed loading.

• Section the beam at each distance x, and draw the free-body diagram of one of the segments. Be sure V and M are shown acting in their positive sense, in accordance with the sign convention given in Fig. 4.

• The shear is obtained by summing forces perpendicular to the beam’s axis.

• To eliminate V he moment is obtained directly by summing moments about the sectioned end of the segment.

• Plot the shear and the moment diagrams. If numerical values of the functions describing M are positive, the values are plotted below the x axis, whereas negative values are plotted above the axis.

Following the above procedure the shear and moment diagrams of the beam shown in Fig. 5(a) are constructed. The free-body diagram of this beam is shown in Fig. 5(b). The beam is resolved and the reaction forces are calculated:

Using Eqs 2 and 3 we calculate the reaction forces:

The shear and bending diagrams of this beam are derived by using sections 1 to 3 (Fig. 5(b)) and applying the equations of equilibrium for the free-body diagrams of Figs 5(c)-(e):

Figure 5. Derivation of shear and moment diagrams.

For section 1, 0≤x≤L1/2 (Fig. 5(c)):

For section 2, L1/2≤x≤ L1 (Fig. 5(d)):

For section 3, L1≤x≤L1+L2 (Fig. 5(e)):

The shear and moment diagrams shown in Fig. 5(f) are obtained by plotting Eqs. 6-11 for L1=4 m, L2=2 m, w=5 kN/m and P=10 kN.

4. Experimental set up and procedure

4.1. Health and safety

Even if it looks like a very “safe” and straightforward experiment to deal with, you need to be fully aware of the possible risks involved and how you can mitigate them. Take few minutes to read this briefing document, familiarise with the exercise and think about the possible risks and mitigations related to it:

Table 1. Health and Safety

Hazard                                                                     Mitigation / Precaution

4.2. Equipment set up and check

Before setting up the equipment, you must

- visually inspect all parts such as electrical cables and components for damage

- check that all connections of all electrical components are correctly secured

- check that each part of the equipment is not loose or fixed wrongly onto the frame.

- make sure that the surface where the frame. is set up is clear, dry and on a levelled surface

- make sure the four securing nuts in the top member of the beam are tightened

- make sure the Digital Force Display is turned on and displays correctly

- gently zero the force measure rotating the black dial visible on the upper left portion of the equipment; after that apply (gently) a load with your finger on the beam and then release

To perform. the experiment, the experimental set up of Fig. 1 is used, which shows a beam having two supports. In the left support, there is a roller pivot, whereas in the right support there is a pin. The beam is installed in an aluminium frame. as shown in Fig. 1.

The main point of the experiment is the existence of a real cut perpendicular to the longitudinal axis of the beam, which is used for the determination of the bending moment that is developed in the beam. In order to avoid collapse of the beam, in the point where the cut is located, there is a moment arm that joints the cut with the rest of the beam (Fig. 6). On this arm a force sensor is installed to measure force developed at the edge of the arm. Knowing the length of the arm, 125 mm, one can compute the moment developed at the cut position.

Figure 6. Beam used for the experiment [1].

The beam is supplied with specific grooved hangers every 20 mm for placing weights as concentrated loads. The details of the hanging set up are displayed in Fig. 7, and each small disc has a weight of 10 g. When the beam is hung on the upper part of the beam, the weights are hung on the grooved hangers and the beam remains horizontal relative to the weights. A special digital force display apparatus is installed on the frame. to give the measurements of the reaction force at the moment arm during the experiment.

Figure 7. Detail of the hanging set up [1].

4.3. Experiment 1 – Bending moment change at point of loading

In the beginning of the first experiment the weights are hung at the position of the cut as shown in Fig. 8, and the variation of moment of the beam at the point of the application of the load is examined. The free-body diagram of the beam in this case is depicted in Fig. 9, where l and a equal to 440 and 300 mm, respectively, while the distance between the left end of the beam and the pivot roller on the left is equal to 140 mm.

Figure 8. Experimental set up when the weights are placed below the cut [1].

Figure 9. Free-body diagram of the beam used in the experiment [1].

Firstly, a load of mass 100 g is hung at the cut position and the force that is displayed on the digital force display apparatus is reported in Table 2. Accordingly, the same procedure is repeated using the following mass values: 200 g, 300 g, 400 g and 500 g. The mass of the first column of Table 2 is converted to load (N) in the second column of the Table by multiplying it by the acceleration of gravity g, equal to 9.81 m/s2 . Then, the force that is developed at the moment arm is converted to the experimental moment (Nm) at the cut position by simply multiplying the force by 0.125 m.

Finally, the theoretical value is computed using the procedure described above and summarised in Fig. 5 for a generic case. Of course, the free body diagram to consider for the theoretical calculations is the one shown in Fig. 9. The theoretical calculations need to be carried for all 5 cases and results have to be included in the last column of Table 2.

At the end of this experiment, bending moment diagram for the case of 500g has to be included in the relevant section of the Beam Bending Lab Sheet.

In your lab report you should be able to compare the experimental and the theoretical results (using graphs and plots) and highlight the differences between them. What can you conclude from these results? How does the bending moment vary at the point of the cut? How far are the experimental results from the theoretical ones?

Table 2. Results of experiment with the weights in the cut position

4.4. Experiment 2 – Bending moment change far from point of loading

The scope of this second experiment is to show how bending moment changes at the cut position of the beam when different weights are placed at different positions of the beam.

1. For the first case, a weight of 400 g is placed on the left edge of the beam (Fig. 10). The experimental force is measured, converted to moment as previously, and the theoretical value is computed. The first row of Table 3 is then filled in. Please also draw the bending moment diagram in the relevant space of the Beam Bending Lab Sheet.

2. For the second case, two weights of 400 g and 200 g are placed in the positions shown in Fig. 11. Again, the experimental force is measured, converted to moment as previously, and the theoretical value is computed. The second row of Table 3 is then filled in. Please also draw the bending moment diagram in the relevant space of the Beam Bending Lab Sheet.

3. For the last case, two weights of 400 g and 500 g are placed in the positions shown in Fig. 12. Again, the experimental force is measured, converted to moment as previously, and the theoretical value is computed. The third row of Table 3 is then filled in. Please also draw the bending moment diagram in the relevant space of the Beam Bending Lab Sheet.

In your lab report again compare the results obtained theoretically and experimentally. Is there any relation between different forces applied at different points of the beam? Is there any significant difference to highlight in the beam bending graphs for each load case?

Figure 10. Experimental set up when one weight is placed at the left edge of the beam [1].

Figure 11. Experimental set up when two weights are placed at the centre of the beam [1].

Figure 12. Experimental set up when the weights are placed at the centre and right [1].

Table 3. Results of experiment with the weights in different locations

For your lab report, please comment on the following points:

- how does the bending moment change in shape when different point loads are applied in different locations?

- Why does the value in the digital force display for the configuration of Figure 10 show a negative value?

- How does the slope of the BM diagram vary as a function of the loads applied?

4.5. Experiment 3 (optional) – Bending moment change with a distributed load

For the last experiment, the case of “distributed load” applied to the beam is analysed. Three different weights of 200 g each are applied as shown in Fig. 13 (the length over which the distributed load is applied can be deducted using the previous images). Similarly to what has been done previously, the experimental force is measured, converted to moment as in the previous experiments, and the theoretical value of bending moment at cut is computed. The first row of Table 4 is then filled in. Please also draw the bending moment diagram in the relevant space of the Beam Bending Lab Sheet.

Finally, keeping the distributed load as it is, apply a point load placing a weight of 300g at the left end of the beam, as shown in Figure 14. Repeat the same steps followed for the previous exercises: the experimental force is measured, converted to moment as in the previous experiments, and the theoretical value of bending moment at cut is computed. The second row of Table 4 is then filled in. Please also draw the bending moment diagram in the relevant space of the Beam Bending Lab Sheet.

Figure 13. Experimental set up when the distributed load is applied [1].

Figure 14. Experimental set up when the weights are placed at the left edge of the beam in combination with the distributed load [1].

Table 4. Results of experiment with the distributed load