Deflections in Beams and Cantilevers: Practical 3 Guide

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.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

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

PRACTICAL: STR4 Deflections of Beams and Cantilevers STR4 Practical 3 - Page 6 Table 1 EXPERIMENT 1 General method is to use the digital dial indicator to record the deflection at various points to plot a deflection curve. Figure 3 Question 1 i) Prior to applying the 500g load to the beam as shown in Figure 3 use the digital dial indicator to record the Datum Reading deflections for the Gauge Positions. Load the beam as shown in Figure 3 recording the Load Deflection at the Gauge Positions. Use values in Table 2 to calculate the Experimental Deflection. *Transfer your Theoretical Deflection calculations from the Pre – Lab Calculations to the theoretical deflection column in Table 2 . = Load Reading – Datum Reading Gauge Position Datum Reading Load Reading Experimental Deflection Theoretical Deflection (mm) (mm) (mm) (mm) (mm) A -0.31 -0.29 -0.05 0.00 50 -0.73 -0.32 -0.41 -0.246 100 -1.33 -0.21 -1.14 -0.847 150 -1.72 0.20 -1.92 -1.647 200 -1.81 0.88 -2.69 -2.432 250 -1.80 1.74 -3.34 -3.020 300 -1.64 2.11 -3.75 -3.726 350 -1.17 2.16 -3.33 -2.918 400 -0.71 1.62 -2.33 -2.182 450 -0.47 0.74 -1.21 -1.162 500 -0.28 -0.26 -0.02 0 C -0.29 -0.24 -0.05 0 D a b

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

PRACTICAL: STR4 Deflections of Beams and Cantilevers STR4 Practical 3 - Page 7 Table 2 ii) Using the data you recorded in Table 2 plot a graph of Deflection versus Gauge Position iii) Suggest 3 possible sources of experimental error and quantify the effects of each source? • Human error: errors made by my groupmates or me, either forgetting to zero the datum after each test or misreading a number. • Equipment defects: the beam we were given was partially bent, perhaps causing some marginal errors in the values given by the datum. • Calculation errors: due to the formulas being very long it was very easy to forget a 0.001 when putting new numbers into a calculator causing data to carry over from each test. iv) What is the importance of taking Datum Readings? Taking datum readings gives points of reference for the deflection, acting as a zeroing.

PRACTICAL: STR4 Deflections of Beams and Cantilevers STR4 Practical 3 - Page 8 Question 2 i) Load the beam as shown in Figure 3 and use the digital dial indicator to measure the deflection at “B”. Use Trigonometry to calculate the Angle of Rotation “θ” of the beam at “C” demonstrated in Figure 4. NOTE : Since there is no Bending Moment between B and C this part of the beam remains straight. Deflection “V” = 1.63 mm Angle of Rotation “ θ ”= 0.93 degrees = 0.016 radians ii) For Without measuring, what do you observe the angle of rotation “ θ ” to be on the other side of the loaded beam at the clamp support “A”?. I observe the angle of rotation at A should be 0 as A is a fixed clamp joint. Figure 4

PRACTICAL: STR4 Deflections of Beams and Cantilevers STR4 Practical 3 - Page 9 EXPERIMENT 2 Figure 5 shows an Aluminium beam with as yet un-specified supports at “A” and “B” 600mm apart. It is centrally loaded with a single mass: 300mm to the right of point ”A” three 500g masses located at 270mm, 300mm, and 330mm to the right of point “A”. Employ a combination of knife edge or clamp supports at “A” and “B” to support the loaded Aluminium beam. Make observations on the deflection curve produced for each different combination of supports used. Figure 5 Question 1 i) For each of the possible types of reaction supports used to support the aluminium beam at “A” and “B”, load the beam as show in Figure 5 and accurately sketching the deflection curve observed for each combination of reaction supports used. Use the symbols on the key bellow to indicate important points on your sketches. Use the Digital Dial Indicator to estimate the deflection under the middle load NOTE: For (b) ONLY place load at free end support and use a ruler to estimate the deflection at the free end.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

PRACTICAL: STR4 Deflections of Beams and Cantilevers STR4 ii) KEY Point of maximum deflection X Point associated with maximum rotation θ Points of contra-flexure if existent O Table 3

PRACTICAL: STR4 Deflections of Beams and Cantilevers STR4 iii) List Beams form part (i) in order of Maximum Deflection under load and hence Maximum Bending Moment? D, A, C, B in order of Maximum to minimum. iv) Identify the effect of the end moment reaction supplied by the clamp support? It provides stability to the bar as well as a tapering effect to the deflection as it approaches. v) Name 3 practical applications of a cantilever structure? • Bridges • Cranes • Balconies Practical 3 - Page 11

PRACTICAL: STR4 Deflections of Beams and Cantilevers STR4 Working space Practical 3 - Page 12

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

ogy418 lab 3

Related Documents