Part IB EXPERIMENTAL ENGINEERING MODEL STRUCTURES. 1. To compare the behaviour of various different linear-elastic structures with simple theory.

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1 Part IB EXPERIMENTAL ENGINEERING SUBJECT: INTEGRATED COURSEWORK LOCATION: STRUCTURES TEACHING LAB EXPERIMENT: A2 (SHORT) MODEL STRUCTURES OBJECTIVES 1. To compare the behaviour of various different linear-elastic structures with simple theory. 2. To give students practice in analysing discrepancies (if any) between theory and experiment, to discover the reasons for them (and so perhaps to improve the theory and/or their understanding of structures). 3. To give students opportunity to verify or disprove certain postulated theorems about linear-elastic structures. 4. To give students practice in oral reporting of their findings briefly but accurately to colleagues, so that the lessons learned may be more widely spread, without the necessity for every student to test every structure. 5. To show to students various aspects of the behaviour of structures that they can explore in more depth as part of an extended exercise. INTRODUCTION This laboratory is part of the Integrated Coursework on Earthquake-resistant structures. To understand the behaviour of a building to an earthquake requires knowledge of the structural mechanics of the building. In the first and second year lecture courses on Structural Mechanics, simplified Engineers theories are presented for the linear-elastic behaviour of structures such as trusses, beams or frameworks. These theories are based on certain simplifying assumptions to reduce the real structure to a theoretical model which can be analysed. For example, in beams each cross-section is assumed to remain plane and undistorted, and normal to the deflected beam centre-line. The question arises, whether these assumptions are always reasonable and if not, whether there are practical circumstances where incorrectness of the assumptions puts the theory into significant error. In this experiment, students compare the results of various tests against standard theory, and attempt to explain discrepancies (if any). A further question on linear-elastic structures is whether there are any general statements which may validly be made for example, on the magnitudes and directions of deflection caused by certain loads. Other groups of students will be given postulated statements or theorems of this type, and asked to verify or disprove them by testing simple models. 1

2 OUTLINE This short 2-hour test session is designed for 6 pairs of students together. Each pair does a different, but related, experiment: in the final 20 minutes, groups describe to others what they have discovered. Four experiments (A D) are essentially on beams. In each case you will be asked to predict the behaviour of a specimen using first - year Engineers Bending Theory. Dimensions and material data will be given to you when you arrive in the laboratory. You will then perform the test, see what actually happens, plot graphs of results, identify shortcomings of the theory (if any), and attempt to explain any discrepancies. Two experiments (E, F) are on bent cantilevers, one with cross-section similar to that of one of the beams (experiment C). Here there will be several statements or potential theorems about deflections. You are asked, by experimenting on the structures in your own way, to disprove or verify these statements (considering for example within what accuracy they might be true, or whether some rewording would be appropriate). In all 6 cases, the magnitude of the applied loads will be limited, so that the structure remains in its elastic range. WRITE-UP Do your preliminary calculations on the lined paper provided, and plot graphs on the squared paper. Follow up with a brief summary and discussion of your findings. Remark upon their implications for practical structures, and on the findings of other groups. DETAILS Further information about the six individual experiments is given below. Before coming to the laboratory, please read these details so as to get a general idea of what is to be done. However, do not attempt beforehand any of the calculations described you will not know until you arrive which of the six experiments is assigned to you. Experiment A: Tubular beam The tube is made from aluminium alloy (E 70 x 10 3 N/mm 2 ). The cross-section is approximately square, with side 50 mm and mean wall thickness 0.83 mm (you may wish to verify these measurements). Calculate I for the cross-section. Hence predict the elastic central deflection (δ = WL 3 /48EI) for each of three spans L (200, 400 and 650 mm) which you will test, for a simply-supported beam with central point load W = 100 N. Plot these on a single graph of W against δ (allowing for max. load 100 N and max. deflection five times that predicted for your longest beam). Draw straight lines to show the expected behaviour as W is increased gradually from zero. Now do the experiment with a truly central point load and four or five load increments. Take some readings as you unload, to see whether behaviour is elastic. There is no need to put down your readings in a table simply plot points on your W-δ graph as you proceed. Compare the theoretical and experimental results and consider the reason(s) for the discrepancy (if any) and how to modify the theory appropriately. Discuss with the demonstrator what to do next. 2

3 Experiment B: Sandwich beam The beam has aluminium alloy flanges glued to a foam infill. The aluminium sheets (E 70 x 103 N/mm2) are 51 mm wide and 0.75 mm thick. The foam has E 7 N/mm2, and the centre-centre separation of the flanges is 20mm. Calculate I for the composite section transformed to aluminium. Hence calculate the elastic central deflection (δ = WL 3 /48EI) for each of four spans L (range mm) which you intend to test, for a simply-supported beam with central load W = 50 N. Plot these on a single graph of W against δ (max. load 50 N, max. deflection twenty times that predicted for your longest beam). Draw straight lines to show the predicted behaviour as W is increased gradually from zero. Now do the experiment, with a few load increments up to the max. load of 50 N. Take some readings as you unload, to see whether behaviour is elastic. There is no need to put down your readings in a table simply plot points on your W-δ graph as you proceed. Compare the theoretical and experimental results, and consider the reason(s) for the discrepancy (if any) and how to modify the theory appropriately. Discuss with the demonstrator what to do next. Experiment C: Square tube with mitre bend L = 372 mm The specimens are made of steel (E 200 x 10 3 N/mm 2 ) and the cross-section is approx. 25 mm square with 1 mm wall thickness. First test the centrally-loaded simply-supported beam shown in Fig. 2 and plot F against v (max. load 100 N). Is it elastic? (Predicted deflection v = FL 3 /6EI.) Now sketch the bending moment diagram for the two figures, and consider the distorted geometry in the two cases (regarded as two connected cantilevers in each case). Hence predict a relation between the stiffnesses W/u and F/v. Hence, from the measured F/v graph, predict the expected graph of W against u. Plot this prediction on a separate graph, and then perform the test of Fig. 1 (max. load W = 50 N) plotting experimental points on the graph without writing them down in a Table. Consider possible reason(s) for the discrepancy (if any), and how to modify the theory for Fig. 1 appropriately. Discuss with the demonstrator what to do next. 3

4 Experiment D: Slim I-beam The overall dimensions of the cross-section are 12.5 x 29 mm, and pure-bending experiments show that EI = 34.6 Nm 2. Predict the elastic central deflection (δ = WL 3 /48EI) for a centrally-loaded simplysupported beam under 100 N load, for four values of span L you intend to test (range mm). Plot these on a single graph of W against δ (allowing for maximum load 100 N, max. deflection 6 mm). Draw lines to show expected behaviour as W increases from zero. Now do the experiment, starting with the shortest span. Use several load increments, and take some readings as you unload, to check whether behaviour is elastic. There is no need to put down your readings in a table simply plot points on your W-δ graph as you proceed. Increase W until its value reaches 100 N, or the sideways deflection reaches 10 mm, whichever occurs first. Use smaller weight increments in the vicinity of the buckling load, to obtain the buckling load as accurately as possible. Keep careful note of the sideways deflection at each load stage, so that each student can do at least one Southwell plot (see below). Consider how the buckling load (preferably for a perfect beam, from the Southwell plot) varies with span, and discuss with the demonstrator what to do next. Experiment E : Curved cantilevers In this experiment, a curved perspex cantilever with a central plane of symmetry is loaded in that plane by a tip force of constant magnitude but variable direction. The object of the experiment is for you, by making appropriate tests of your own devising, to verify or disprove the various hypotheses listed below. First set up the cantilever and paper sheet with the origin of axes for displacement directly below the hole next to the hole to which force is applied. Displacement of the cantilever under load can then be recorded by pricking through the displaced hole; and the corresponding force direction can be recorded from the cotton thread, again by pricking holes or otherwise. Test the cantilever with the same force (0.4 kgf plus scalepan) in various directions (all parallel to the drawing board) over as near as possible the full 360 range. Observe the tip displacement in each case; draw a curve through the displaced points; and indicate the corresponding force directions. Consider hypotheses (a), (b) and (d) in the light of these observations; note whether each hypotheses is true, false or badly worded, giving reason(s) why you reach this conclusion from the results. [Note for (d) that this is a 2- not 3-dimensional structure]. You have effectively measured the elastic flexibility of the cantilever (i.e. the displacement for unit force) for forces in varying directions. Now resolve each measured displacement (or flexibility) into directions parallel and perpendicular to the corresponding applied force, using an appropriate sign convention. On a separate piece of graph paper, to an appropriate scale, plot (displacement-in-the-force-direction) horizontally, and vertically the corresponding (displacement-perpendicular-to-the-applied-force). If you can, draw a curve through these points, and comment, in the light of hypothesis (d). Finally, by studying the observations already made, or by carrying out further special tests, consider hypothesis (c), especially version (i). How general can your conclusions be, and to what accuracy are various verified hypotheses true? 4

5 Experiment F : Right-angled cantilever tube The tube is of steel, with wall thickness 1 mm and a square section of side 25 mm. (A similar tube is being tested by another group of students). Your rig allows load (max. 50 N) to be applied in three directions at a point near the tip of the cantilever, and displacements to be measured at much the same point. Load can be applied in a wide range of directions in the cantilever plane; and load can be applied or deflections read at a second point, on the beam. The object of the experiment is for you, by making appropriate tests of your own devising on this bent cantilever, to verify or disprove the hypotheses listed below. First apply the three in-plane loads in turn. In each case, cycle the load once or twice between 0.2 and 4.2 kg in the scale pan : then take readings of the four dial gauges at 0.2, 4.2, 0.2, 4.2 kg etc. until you are satisfied with consistency. To eliminate the effect of friction on loads applied via pulleys, ensure that the load has been increased by at least 0.2 kgf just before you take each set of readings. Finally, take readings of the dial gauges with the out-of-plane load at 0.2, 2.2, 0.2 kgf etc. Estimate flexibilities or compliances in mm/n. If you have time later, take some intermediate readings to check on linearity. Then consider the hypotheses in the order given, and see whether your results prove or disprove them. If your experiments suggest that a hypothesis is true, or nearly so, devise a way of stating its accuracy. In some cases, rewording a hypothesis may be necessary. From some tests, you may obtain the 3 x 3 flexibility matrix F at a point, defined by the relation d = Fp between displacement d and applied load p. Calculate the eigenvalues and eigenvectors of F using Matlab. Then consider, consulting the demonstrator, what further experiments to do to verify the hypotheses. Note: For a right-angle bent cantilever with uniform members of length h, bending in plane only, with flexural modulus EI1 for the column and EL2 for the beam, the tip deflections according to simple Engineers Bending Theory are horiz. load H : Hh 3 /3EI 1 horiz. ; Hh 3 /2EI 1 vertical vert. load V : Vh 3 /2EI 1 horiz. ; V h vertical EI1 3EI2 Hypotheses (for experiments E and F) The following hypotheses are put forward as applying to all structures in the linear-elastic range of behaviour. (a) If a force is applied to the structure at any point A, the displacement of A is in the same direction as the force. (b) If a force is applied to the structure at any point A, the displacement of A in the force direction is greater than its displacement in any perpendicular direction. (c) A force is applied at point A on a structure in direction i and the displacement of point B in direction j is measured. This displacement is the same as that caused at A in direction i by the same force magnitude applied at B in direction j: (i) if A and B are the same point, or (ii) if A and B are any two points (the same, or distinct). (d) If a force is applied at any point A, and is to cause A to displace in the same direction as the force, the force must be applied in one of three definite mutually-perpendicular directions. One of these directions will show the maximum, and one the minimum, flexibility. 5

6 NOTE ON MATRICES In the elasticity experiment, you effectively measure the terms of the 3 x 3 matrix F relating displacement d to the applied load P at the same point, ie. D = FP. From Part IA Mathematics, if d is to be the same direction as P, ie d = λ P where λ is scalar, then [F - λ I] P = 0 (1) where I is the unit matrix. Obviously (1) is satisfied trivially when P = 0, but there may be some special cases when P is not zero but (1) is nevertheless satisfied. Then the determinant of [F - λ I] must be zero, giving a cubic equation for λ whose roots are the eigenvalues (here the flexibility of the structure in the special directions). To each eigenvalue there corresponds a special direction, the eigenvector. Each eigenvector can be obtained by substituting an eigenvalue into (1) and regarding P as unknown (but with one arbitrarily-chosen component set to unity). Eigenvalues and eigenvectors can be found using the command eig in Matlab (type help eig to learn more). NOTE ON SOUTHWELL PLOT The Southwell plot is a method of estimating the buckling load P cr for a perfect strut from the increase of sideways deflection of an imperfect specimen as the load P approaches P cr. All struts will have some initial imperfection in the eventual buckling mode, say e o at midspan. According to simple strut theory, this is magnified to w = e o (1 P/P cr ) as P P cr (leaving other shapes of imperfection insignificant). A gauge placed to measure sideways movement at midspan will record = w e o = e P o cr 1. A graph, the Southwell Plot of the ratio P /P against should then have slope 1/P cr. So P cr may be estimated, usually with good accuracy, from the slope of the best-fit line through experimental points. This applies strictly to pin-ended struts, and so will apply here if the top flange of the beam is regarded as similar to a pin-ended strut. If proper account is to be taken of the lateraltorsional behaviour here, with restraint of warping, the Southwell plot should be of /P n against giving slope 1/P cr n, where n is between 1 and 2 depending on the geometry of the beam. S.D. Guest F.A. McRobie September

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