CE 320 Structures Laboratory 1 Flexure Fall 2006 General Note: All structures labs are to be conducted by teams of no more than four students. Teams are expected to meet to decide on an experimental design strategy and to carry out the required tests. However, all written materials that are turned in must be written individually by each student. Consider a linear elastic steel bar with solid rectangular cross-section that is to be tested in flexure. At its midpoint it has strain gages at the top and bottom that can be used to measure strains. You also will have available given weights that can be applied to the beam as loads, a measuring tape, displacement dial gages, micrometer, two tables that can be used to support the beam, and up to four C-clamps. You can use beam deflection formulas to interpret test results. Examples of beams with deflections and bending moments are given on the last page of this handout. Design two different experimental setups by which you can determine: a) The bending rigidity EI of the beam, based on your measurements only. Design your experiments to have measurements of deflections at more than one location in each experiment. b) Compute the moment of inertia I of the cross section about the axis of bending from your measurements of the cross-sectional dimensions. c) The maximum stress and strain that occurs in your experiment. If the location of the strain gages does not correspond to maximum stress and strain for your experimental design, use statics, i.e. proportionality of stress to the bending moment M, to help with the computation of maximum stress and strain values. d) Compare the stress results from (c) with those computed using the value of the moment of inertia I from (b) and the linear elastic bending stress formula σ = My/I where the bending moment M is calculated by statics using loads and measured distances along the beam. Explain any differences with stresses obtained in (c). For each of your two experimental design setups write the test procedure with appropriate figures to allow someone with no background in strength of materials and structural analysis to conduct the tests, obtain the necessary raw data, and carry out the required computations to determine the requested results (a), (b), (c), and (d). There is no need to write the details of how the strain gages will be read as the TA will help with the reading of the strains. After you submit your write-up it will be read by the TA and if approved you will make an appointment with the TA to carry out your experiments. A final report that includes your experimental procedures and explanation of the results in detail, will be due after you conduct the experiments. Both the experimental procedure write-up and the final report are to be written by each individual student. If you need any further clarification please consult the course instructor Prof. Levon Minnetyan or the teaching assistant Mr. Mashal Sheban. Due dates: Test procedure write-up due by Sept. 15, 2006 (Submit to TA in Camp Annex 198) Experiment due by September 29, 2006 (Make appointment with TA) Final report due by October 6, 2006 (Submit to TA in Camp Annex198)
Structures Laboratory 1-Appendix Measurement of Strain using Strain Gages The strain gage works on the principle of electrical resistance of metals discovered by Lord Kelvin in 1856. For a uniform conductor of length L, cross-sectional area A and specific resistance p, the resistance is given by R=pL/A Eq A-1 The gage is bonded to the metal bar by an adhesive strong enough so that the gage deforms the same amount as the metal bar subjected to load. Thus the change in the length and the area of the gage will change the resistance of the gage. The change in resistance of the gage can be measured by the Wheatstone bridge in the strain box. The resistance of the strain gage varies with the strain. The change in the resistance of the strain gage is related to the strain by the gage factor, F, defined as F=( R/R/( L/L)=( R/R)/ε Eq A-2 It can be shown that F depends on the change in specific resistance and changes in geometry due to Poisson's ratio of the strain gage material. The values of the gage factor, F, are determined by the manufacturer of the gages and printed on the package containing the gages. Most strain gages have F values of about 2, but F can range from -140 to +175. The foil gages used in this experiment are made from an alloy of copper and nickel formed into a foil 0.001 inch or less thick. The foil gage is made by etching or punching the gage shape from the strain sensitive alloy and mounting on a backing of polyester. Various configurations and sizes of gages can be easily manufactured. Stain gage costs vary from about $2 to over $100 per gage. The strain box consists of two Wheatstone bridges, one connected to the strain gages and other a standard bridge. The amount of unbalance caused by the strain gage in the active bridge can be balanced against the standard bridge. The change in resistance to balance the two bridges is calibrated so that the output of the strain box is directly proportional to the strain experienced by the gage bonded to the loaded metal bar. The basics of the simple four-arm Wheatstone bridge can be found in any college level physics book and is also detailed later in this appendix. PROCEDURE (TO BE DONE BY TA) Two electrical resistance foil strain gages will be bonded (glued) to a metal bar using a cyanoacrylate adhesive ("superglue"). One strain gage on each side of the bar and both will be parallel to the direction of stress. Cut 3 to 4 foot lengths of 22 or 24 gage twisted strand wire and solder them to the strain gage leads. Electrical insulation tape will be used to prevent contact between the soldered connection and the metal bar. Also the electrical tape should be wrapped around the wires from the strain gages and the test specimen to form a strain relief loop that will prevent.the accidental pulling of the wires that could break the strain gage. The strain gages will be checked for electrical continuity and that the gages are not short circuited to the metal bar. Using a YOM meter, the continuity Check should indicate a resistance of 120 ohms across each gage and several thousand ohms resistance between gage leads and he metal test specimen to insure that strain gage is electrically insulated from the test specimen. Stain gages are temperature compensated for material with a thermal coefficient of expansion α=13xi0-6 in/in/ F. If the strain gages are bonded to a material with the same coefficient α value, the strain caused by temperature variation (thermal strain) will not be added or subtracted from the mechanical strain due to the applied load acting on the specimen. For gages with α values the same as the test specimen, the strain gages can be connected to the strain box as a quarter arm bridge; i.e. one active resistor being the strain gage. If the α values of the gage and the test specimen are different, then a two arm or half arm bridge must be used. One resistor is the active strain gage and the adjacent resistor is a dummy or inactive gage. The
subsequent pages of this appendix detail the Wheatstone bridge circuit or the proper circuit connection to the strain box for strain measurements. Also see Figure A1a and A1b. Figure A1a. Two-arm bridge using a temperature compensating (dummy) gage Figure A1b. Quarter-arm bridge for strain gages with α values the same as the test specimen. The strain gage leads are to be connected to a strain box that measures the strain in microstrain units, i.e. 10-6 units. The output of the strain box for each gage will be recorded before and after loading the specimen. The difference in the strain box outputs will be the change in the strain due to the applied load. FIGURES Draw a dimensioned sketch showing the location of the gages on the test specimen for each of your experimental designs. Record the strain data for each preselected loading. Compute the stresses as required. DISCUSSION Laboratory report must contain a brief statement of the purpose of the lab experiment and a neat hand drawn sketch done in pencil of the test set up for each experiment. Write a discussion of the results that includes specific comments on the required items. Also comment upon any significant errors made in conducting the laboratory experiment
Deflections and Bending Moments of Example Prismatic Beams: Equations for the deflections and bending moments of these and similar beams can also be found in any structural analysis textbook