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THEORY: STR4 Deflections of Beams and Cantilevers STR4 Practical 3 - Page 1 PRACTICAL 3 Deflections in Beams and Cantilevers

THEORY: STR4 Deflections of Beams and Cantilevers STR4 Practical 3 - Page 2 PRE-PRAC READING Aim: To explore the occurrence of deflection in a beam, and analyse features of a beam that characterize a deflection curve. Safety: Tool Operation: When using levers such as Allan Keys and/or Spanners to tighten or loosen fixtures always apply force to these levers directed away from your body. Equipment: Do not lean, hang or apply any additional form of loading on the beam other than the weights provided. Significant Principles Used : - Types of Beams, Loads and Reactions (Gere & Goodno, Ed.8, Chapter 4.2, page 366 - 372) - Deflections of Beams (Gere & Goodno, Ed.8, Chapter 9, page 754 - 762) What you have to do Before the lab 1. Read through the theory and then do the pre-lab calculations: calculate the defections you expect to obtain from the test and fill out the table. During the lab 1. Undertake the experiments and record the data as indicated. 2. Compare the observed deflections with those calculated in the pre-work.

THEORY: STR4 Deflections of Beams and Cantilevers STR4 Practical 3 - Page 3 Theory: We often think of structures failing due to shear, collapse or fatigue; however it is also quite common for structures to fail through not meeting performance levels of deflection. Any force when applied to a civil structure is associated with producing bending within the structure, regardless of how small a force this may be. In Structure design and analysis deflection is an important parameter used to describe the bending of a beam. The deflection of a beam at any point along its longitudinal axis is defined as the displacement normal to the origin of that point; hence measured in the vertical direction exampled in Figure 1 bellow. Figure 1 From Figure 1 we can also see that when a beam deflects the beam rotates producing an angle “ θ ” which is known as the Angle of Rotation, measured in radians. The deflection at any point on a beam can easily be found by using the relevant deflection equation. Due to the various forms of loading and types of reaction supports different deflection equations exist for different beams. During the lecture and tutorial work for this subject you will be learning how to derive deflection equations and also how to use tables of already derived deflection formulae. Refer to the relevant sections of the text if you want to work through the theory now.

THEORY: STR4 Deflections of Beams and Cantilevers STR4 Practical 3 - Page 4 General test procedure: A rig is available to you that will allow you to observe the deflection of particular beams. A picture of the rig and also a test procedure follows. Operation of Equipment Procedure: Figure 2 1) Ensure test frame is stable and level; if it is wobbly notify a lab assistant. 2) Turn on the digital dial indicator by pushing the green “on/off” button located on the dial face panel. 3) Position the roller support by sliding it to the left or right, use the distance scale printed on the frame to accurately position the support. 4) Ensuring all loads are removed from the beam, carefully Zero the digital dial indicator by sliding it to the zero position and pushing the “zero” button located on the dial face panel. 5) Carefully place hanger load on the metal load frames which are movable and clipped on the beam. Also ensure that the hanger is still and not swinging when taking readings from the digital dial indicator. Mass of empty hanger = 10g . 6) Slide the digital indicator to measure the deflections at the desired locations. 7) If you need to remove the clamp supports use an Allen key to unscrew beam from the clamp support and the knob on the backboard of the clamp support to remove clamp. 8) When finished turn off the digital dial indicator and ensure all hanging loads have been removed from the beam.

PRACTICAL: STR4 Deflections of Beams and Cantilevers STR4 Practical 3 - Page 5 PRE-PRAC CALCULATIONS Figure 3 shows an Aluminium beam loaded with a 500g mass located 300mm to the right of “A”. The beam is supported by a clamp at “A” and a roller support 500mm to the right of “A”. In this configuration the beam is acting as a ‘propped cantilever’. If the beam were free to rotate at A: i.e. a roller was put in place of the clamp at A, then the beam would be acting as a simply supported beam. If the roller support at C was removed, on the other hand, then the beam would be acting as a cantilever. You will notice in Appendix G of Gere and Goodno that there is no case provided for a propped cantilever. In addition you may observe that the propped cantilever is over-constrained or ‘statically indeterminate’. Read about this situation in Chapter 10 of the text. Often engineers try to avoid such situations given that loads applied to such structures depend on the geometry of the supports which will normally be dependent on the construction or manufacturing accuracy and positions of the supports which must vary from time to time. Assuming that the beam is initially straight and that there is no loads on the supports then the following deflection formulas apply: Note: the Force is pointing downwards and therefore Negative Between A and D: ? 𝐴? = ??𝑥 2 12?𝐼𝐿 3 × (3𝐿(𝑏 2 − 𝐿 2 ) + ?(3𝐿 2 − 𝑏 2 )) Between D and B: ? ?? = ??𝑥 2 12?𝐼𝐿 3 × (3𝐿(𝑏 2 − 𝐿 2 ) + ?(3𝐿 2 − 𝑏 2 )) − ?(𝑥−?) 3 6?𝐼 Calculate the theoretical deflection across various points on the beam as shown in Table 1. Assuming that the mass of the beam is zero. Material E I (GPa) (mm^4) Aluminium 69 27 Mass Load Load Position Gauge Positio n Theoretical Deflection (g) (N) (mm) (mm) (mm) 500 4.91 300 A 0 500 4.91 300 50 -0.246 500 4.91 300 100 -0.847 500 4.91 300 150 -1.647 500 4.91 300 200 -2.432 500 4.91 300 250 -3.020 500 4.91 300 300 -3.726 500 4.91 300 350 -2.918 500 4.91 300 400 -2.182 500 4.91 300 450 -1.162 500 4.91 300 500 0 500 4.91 300 C 0

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