Structural Analysis Laboratory. Michael Storaker, Sam Davey and Rhys Witt. JEE 332 Structural Analysis. 4 June 2012.

Transcription

1 Structural Analysis Laboratory Michael Storaker, Sam Davey and Rhys Witt JEE 332 Structural Analysis 4 June 2012 Lecturer/Tutor Shinsuke Matsuarbara 1

2 Contents Statically Indeterminate Structure Objective... 4 Theory... 4 Procedure... 4 Calibration... 4 Single Central Load... 5 Two Equidistant Loads... 5 Results... 6 Calibration... 6 Single Central Load... 7 Two Equidistant Loads... 9 Discussion Calibration Single Central Load Two Equidistant Loads Conclusions Column Buckling Objective Theory Euler s Method Southwell Plot Apparatus Procedure Results Southwell Plots Euler s Method Conclusion Shear Centre Introduction Aim Apparatus Theory Procedure

3 Results Experimental Beam 1: Beam 2: Beam 3: Theoretical Beam 1: Beam 2: Beam 3: Discussion Conclusion References Appendix A

4 Statically Indeterminate Structure Objective The objective of this experiment was to compare theoretical and experimental results from for the external reactions and deflections of a statically indeterminate beam. Theory The theoretical reactions were calculated using the three moment equation, presented here as Equation 1. ( ) (1) M A was calculated first using Equation 1, then the reaction forces were calculated using equilibrium equations (sum of forces and moments equals zero). Deflections were calculated using the following relationship, Equation 2. ( ) (2) Firstly the bending moment was found, as a function of x. The bending moment function is discontinuous, therefore the bending moment and hence deflections were found separately for two sections and three sections for the single central load and two equidistant loads respectively. Equation 2 was then integrated twice to obtain the deflection, y. Procedure Calibration The first step to be done was to calibrate the load cell. A simply supported beam was used for calibration, as in Figure 1. Figure 1 - Simply supported beam A simply supported beam is statically determinate therefore the load on the load cell can be known. This allows the reading from the load cell (or more importantly, the change in 4

5 reading) to be compared with the known load upon it, enabling the calculation of a calibration factor, as in Equation 3. 1 F CF LCR (3) Where CF = calibration factor F = force on load cell LCR = load cell reading 1 It should be noted that in this document is used in the following way: Value Value Value W 0 Several values for the calibration factor were found using different loads on the simply supported beam and the value used in subsequent calculations was the average of these. See Figure 1 for a graphical representation of this. This was done in order to reduce the effect of random error, due to friction and other effects within the load cell and dial gauges. Single Central Load For this part of the experiment the beam was setup such that it was fixed at one end and simply supported at the other end, as in Figure 2. Figure 2 Single Central Load A point load was applied using masses ranging from 0.5 kg to 3 kg in steps of 0.5 kg. For each applied load, the load cell reading at the simple support was taken, as well as the deflection at various points along the length of the beam using dial gauges. Two Equidistant Loads The same beam and supports were used in this part of the experiment as for the single central load part but the single central load was replaced with two equidistant loads and the dial gauges were positioned slightly differently. 5

6 CF [N] Figure 3 Two equidistant loads Other than the position of the loads and dial gauges, the procedure was the same as for single central load. Results Calibration CF Average CF C y [N] Figure 4 Calibration Factor 6

7 C y [N] A y [N] Single Central Load Theoretical 0.00 Experimental W [N] Figure 5 - Ay against W Theoretical 0.00 Experimental W [N] Figure 6 C y against W 7

8 Δy [mm] diff(%) M A [N] Theoretical 0.00 Experimental W [N] 10 Figure 7 = M A against W MA Cy -10 Ay W [N] Figure 8 Percentage difference of theoretical with respect to experimental reactions Experimental Theoretical x [m] Figure 9 - Deflection 8

9 D y [N] A y [N] Δy diff(%) kg kg x [m] Two Equidistant Loads Figure 10 Percentage difference of theoretical with respect to experimental Δy Theoretical 0.00 Experimental W [N] Figure 11 A y against W Theoretical 0.00 Experimental W [N] Figure 12 D y against W 9

10 Δy [mm] diff(%) M A [N] Theoretical 0.00 Experimental W [N] Figure 13 M A against W MA Dy Ay W [N] Figure 14 Percentage difference of theoretical with respect to experimental reactions x [m] Experimental Theoretical Figure 15 - Deflection 10

11 Δy diff(%) kg kg x [m] Figure 16 Percentage difference of theoretical with respect to experimental Δy Discussion Calibration Figure 1 shows how the potential error that could have been introduced, if only one value for the calibration factor had been calculated. The large spread meant that one particular data point cannot be trusted to give an accurate representation of the same data across a range, even if the data is theoretically constant. Single Central Load Figure 2 (and Figures 3, 4, 8, 9, and 10) show the linearity between the reactions and the applied load, which was expected. The percentage difference of the theoretical with respect to the experimental reactions is shown in Figure 5. This amount of error is quite reasonable. Some of this would be due to friction in the dial gauges and also the load cell and some would be due to the fact that the load cell doesn t obey Hook s law perfectly. Deflection is shown in Figure 6. Unfortunately, there is a very big difference between theoretical and experimental deflection, both for the single central load and two equidistant loads (percentage difference is shown for the single central load in Figure 7). Apart from the deflection magnitude, the shapes do look remarkably similar. An interesting graph to plot would have been the deflection, at a given point x, with respect to the applied load. This would provide a good check to see whether the bar obeyed Hooke s Law, or whether it showed slightly plastic behaviour. Two Equidistant Loads Both the single central load part and the two equidistant loads gave very similar results. They suggest that theory is very accurate for predicting the external reactions, but quite poor for predicting the shape. 11

12 It is interesting to see how the absolute error of the reactions in this part of the experiment are similar in shape to the same in the Single Central Load part, in that the error increases at higher values of W. The fact that the shape of the absolute error was repeated from one part to another suggests that significant bias error was involved. It was expected that, because materials often don t behave perfectly elastically, the experimental reaction forces might increase with respect to the theoretical data with increasing load. This would be because the rigidity decreases when leaving the elastic region and approaching the plastic region, and a less rigid beam would require a greater reaction at its simple support. However, this was not the case, the experimental reaction forces decreased for higher load. Conclusions The results from this experiment suggest that theory is very accurate for predicting the external reactions and the shape of a deflected beam but quite poor for predicting the magnitude of deflection. 12

13 Column Buckling Experiment Objective To investigate the buckling behaviour of a column subjected to pure compressive end loadings with differing end supports, and determine the critical buckling load. Theory A column is a long slender member that is subjected to axial compressive loads. When subjected to large loadings a column will endure to lateral deflection, known as buckling. The critical load (P CR ) is the maximum axial load that a column can support when it is on the verge of buckling. When loads greater than this critical load are applied, it will cause the beam to be under enormous amounts of stress, resulting in large deformations and a loss in its load carrying capacity. To determine this Critical Loading point a variety of analytical and experimental methods are available. These methods can be compared and assumptions made about their validity. A common analytical method for determination of the critical buckling load of a column is Euler s Method. A southwell plot will be completed from the experimental results to allow comparison for the critical loading points of the respective methods. Euler s Method Leonard Euler devised a formula that can predict the critical axial load that a column can take without buckling. Equation 4 displays this formula: Where E is the modulus of elasticity for the material ( ) 4 I is the moment of inertia for the columns cross sectional area K is the effective length factor L is the length of the column The effective length factors are determined from the following table (1): Table 1: Effective Length Factors End Connection K Pin - Pin 1 Pin - Fixed 0.7 Fixed - Fixed 0.5 Southwell Plot Southwell plots allow us to determine the experimental critical buckling load of a column. By plotting the deflection of the column against the variable u/n (deflection over load) and 13

14 developing a linear trendline we are able to determine the critical buckling load by taking the inverse of the slope of the added trendline. A typical southwell plot is shown below in figure 17. Figure 17: Example Typical Southwell Plot Apparatus Procedure 1. Release all tension from the spring balance using the hand wheel 2. Fit appropriate steel chucks onto the loading beam and machine base plate 3. Release the loading beam by loosening the release handle 4. Position the loading beam so it can accommodate the strut being tested 5. Fix the position of the beam by tightening the release handle 6. Balance the complete loading assembly, including spring loading wheel and link using the supplied weights 7. Place the strut in the apparatus taking care to set the loading beam horizontal. Fine tuning may be done using the fulcrum adjust capstan 8. Adjust the loading beam balance weight if required 9. Fit the cord pulley on the left of the vertical column in such a position that the loop on the end of the cord will be at the centre of the strut 14

15 10. Apply a 300g horizontal mass on the struts 11. Zero the dial gauge on the short vertical column and use the scale provided to contact the centre of the strut 12. Commence loading the column by turning the hand wheel. The loading beam must be brought into the horizontal using the fulcrum capstan each time before a reading of load and deflection can be recorded 13. Keep increasing the load until the dial gauge indicates large increases in deflection for a little increase in load 14. Carefully unload the strut & Tidy up! Reference 2012, JEE 332 Structural Analysis Laboratory Handout, Australian Maritime College. 15

16 Results Southwell Plots The results obtained from the experiment are shown in table 2 below: Table 2: Experimental Results Beam Connections Set Load (N) Actual Load (N) Deflection (mm) pin-pin pin-pin pin-pin pin-pin pin-fixed pin-fixed pin-fixed pin-fixed pin-fixed pin-fixed pin-fixed fixed-fixed fixed-fixed fixed-fixed fixed-fixed fixed-fixed fixed-fixed fixed-fixed fixed-fixed fixed-fixed fixed-fixed fixed-fixed fixed-fixed The results can then be plotted to show the load vs. deflection graphs. These plots are shown below in figures 18, 19 &

17 Load P (N) Load P (N) Load P (N) Deflection u (mm) Pin - Pin Figure 18: load vs. deflection, for pin pin connection Deflection u (mm) Pin - Fixed Figure 19: load vs. deflection, for pin fixed connection Fixed - Fixed Deflection u (mm) Figure 20: load vs. deflection, for fixed fixed connection 17

18 These graphs depict the deflection of a beam as a function of the actual load. They show a good representation of what is happening as a result of changing the connection type. As a result of changing to fixed connections over pined, the beam is able to absorb much more load (without deflecting). To complete the southwell plots a quick calculation for deflection over force had to be made to complete the plots. The following three table s 3, 4 & 5 show the data for the three different end supports Table 3: Pin Pin, Load Deflection Table for Southwell Plot Pin - Pin P u (m) u/p (mm/n) (N) E E E Table 4: Pin Fixed, Load Deflection Table for Southwell Plot Pin - Fixed P (N) u (mm) u/p (mm/n) E E E E E E-05 Table 5: Fixed Fixed, Load Deflection Table for Southwell Plot Fixed - Fixed P u (mm) u/p (mm/n) (N) E E E E E E E E E E-06 18

19 Using the data from these tables it is possible to create the southwell plots, as shown in figures 21, 22 & 23 below y = x + 2E µ/p (mm/n) Pin - Pin Linear (Pin - Pin) µ (mm) Figure 21: Southwell Plot; Pin Pin end connections y = x + 3E µ/p (mm/n) Pin - Fixed Linear (Pin - Fixed) µ (mm) Figure 22: Southwell Plot; Pin Fixed end connections 19

20 0.00 y = x + 6E µ/p (mm/n) Fixed - Fixed Linear (Fixed - Fixed) µ (mm) Figure 23: Southwell Plot; Fixed Fixed end connections Once the experimental data was plotted, a linear trendline was added which will allow us to determine the critical load. The equations for the added trendlines can be found and are shown below in equations 5, 6 & 7. Pin Pin 5 Pin Fixed 6 Fixed Fixed 7 These equations are then differentiated and P critical can be calculated for each different end loading using the equation (8) shown below: 8 The resultant critical loads obtained are shown below in table 6: Table 6: P CRITICAL determined from southwell plots End Connection P CRITICAL (N) Pin - Pin Pin - Fixed Fixed - Fixed The results shown in table # display the critical buckling load obtained using the southwell plot method. To determine accuracy these results can be compared to the results predicted by Euler s Method. Euler s Method As described in theory above the determination of the critical buckling load can be found using the equation (9) 20

21 ( ) 9 Young s Modulus of the steel beams is assumed to be 200x10^9 GPa The area moment of inertia can be determined using the equation 10 With values for b and h being m and m, respectively. Values for K vary for each strut, with L remaining constant. Using these assumptions, values for P CRITICAL using Euler s method are determined and shown below in table 7. Table 7: Euler s Method results End Connection P critical K L E I Pin - Pin E E-11 Pin - Fixed E E-11 Fixed - Fixed E E-11 To determine the accuracy and validity of these predicted results, a comparison against the values determined from the southwell plots can be made, as shown below in table 8. Table 8: Comparison of Experimental (southwell) and Analytical (euler) Results P Critical (N) Connections Euler Method Southwell Plot Pin - Pin Pin - Fixed Fixed - Fixed It can be seen from table 8 that the results predicted compared to those gathered and calculated experimentally are reasonably similar. Sources of error in the experiment could have led to this difference in results (Human error, etc) 10 Conclusion From the results obtained for P CRITICAL calculated by the southwell plots and upon comparison with Euler s results, I would have no problem predicting the critical buckling load of a column in compression using Euler s Method. An error of less than 2% is evident for all three end connection types, this could be improved with better equipment and more experiments though it is quite close. The results show that utilising fixed connections in a column beam increases the beams ability to absorb load. With a Pin Pin connection the beams, critical load value was very low about 415N. The next beam with a connection type Pin Fixed presented a higher ability to absorb the load, displayed by its P Critical value being about 834N. The Final beam 21

22 tested with end connections Fixed Fixed showed the greatest ability to absorb load with a P CRITICAL Load value of 1666N, almost double a Pin Fixed. 22

23 Shear Centre Experiment Introduction Steel beams are used for strengthening in many different types of structures including bridges, ships, buildings, sheds and many more. The beams are used to strengthen the structure and they are usually loaded in either tension or compression. When loads are applied to beams, they will twist. Different shapes of beams have different resistance to the amount of twist. In each beam there will be a point that when a force is applied there will be no twist of the beam and only shear. This point is called the shear centre. Aim The objective of this experiment is to determine the shear centre of 3 beams with different cross sections. The values obtained will then be compared to theoretical values to see whether the experiment was accurate. Apparatus Theory Shear Centre is defined by the McGraw-Hill Concise Encyclopedia of Engineering as a point on a line parallel to the axis of a beam through which any transverse force must be applied to avoid twisting of the section (2002). If a load is applied to the beam on any point off this shear centre line the beam will bend and twist at the same time. To experimentally find the shear centre of a beam loads are applied along the transverse axis of the beam until a point is found where there is no twist. On this longitudinal axis is where the shear centre of the beam. To find the shear centre theoretically a more complex approach is taken. These steps have been taken from Mechanics of Materials by Hibbeler (2011). 23

24 Step 1: Sum the moments of the shear-flow resultants about a point A and set this moment equal to the moment of V about A. Solve this equation to determine the moment-arm or eccentric distance e, which locates the line of action of V from A. Step 2: Find the axis of symmetry for the cross section. Step 3: If an axis of symmetry for the cross section exists, the shear centre lies at the point where this axis intersects the line of action of V. Procedure 1. Measure the dimensions of the beam. 2. Attach the specimen to the rig. 3. Position the dial gauges and tighten all necessary fasteners. 4. Zero dial gauges. 5. Apply the load at each one of the loading point and record the displacement shown on the gauges. 6. Remove the load and beam. 7. Repeat for 3 different beams 8. Plot results to find experimental shear centre of each beam 24

25 Results Experimental Below are results from the experiment. Figures 2, 4 and 6 represent the deflection of the beam in millimetres when a 2kg load is applied at different points along the transverse axis of the beam. The thickness of all the beams is 1.6mm. The raw data is supplied in Appendix A Beam 1: Beam 1 is the channel or C-beam. The left hand side and the dimensions of the beam are shown in Figure 1. The deflection of the LHS and the RHS have been graphed in Figure 2. Figure 24 - C-Beam Cross Section 25

26 Deflection (mm) y = x y = x Distance of Applied Mass from LHS (mm) Figure 25 - Deflections of the C-Beam over the width of the beam Equating these two lines will give and solving for x will give the shear centre distance from the LHS. Thus the experimental shear centre is millimetres from the LHS of the beam. 26

27 Deflection (mm) Beam 2: Beam 2 is the S-beam and the dimensions and the left hand side of the beam are shown in Figure 3. The deflection results of the LHS and the RHS have been graphed in Figure 4. Figure 26 - S-Beam Cross Section y = x y = x Distance of Applied Mass from LHS (mm) Figure 27 - Deflections of the S-Beam over the width of the beam Equating the two lines and solving for x the shear centre distance is obtained. 27

28 Deflection (mm) Thus the experimental shear centre for the z-beam is millimetres from the LHS of the beam. Beam 3: Beam 3 is the L-beam and the left hand side as well as the dimensions of the beam are shown in Figure 5. The deflections of the LHS and the RHS have been graphed in Figure 6. Figure 28 - L-Beam Cross Section y = x x y = x x Distance of Applied Mass from LHS (mm) Figure 29 - Deflections of the L-Beam over the width of the beam Equating the two lines and solving for x the shear centre distance can be obtained. 28

29 Thus the shear centre distance is millimetres from the LHS of the beam. Theoretical To find the theoretical shear centre the steps have been followed as listed above in the theory. For beams 2 and 3 the shear centres can be by just studying the shape of the beam. As beam 2 (s-beam) is symmetrical the shear centre will be located on the central axis in the vertical plan. Beam 1: The solving of the shear centre shown below is found with Figures 7 and 8. Figure 30 - Showing the arbitrary position 'x' with respect to the neutral axis (Hibbeler, 2011) Figure 31 - Forces acting in the c-beam (Hibbeler, 2011) Step 1: Find Inertia about the Neutral Axis assuming it is a thin wall channel. This means the t 2 terms and higher order t terms will be assumed to equal zero. [ ( ) ] (11) Simplifying Equation 11. ( ) (12) Step 2: Find the shear flow for an arbitrary position x from Figure

30 ( ) [ ] (13) (14) Substituting Q in the Equation 14. ( )[ ] ( ) (15) Simplifying Equation 15. ( ) [( ) ] (16) Step 3: Solve for F f Equation 17 can be found. (17) Integrating q between b and 0 to find Equation 18. [( ) ] (18) Step 4: To find the shear centre moments about point A must equal zero. Therefore Equation 19 must be true. [( ) ] (19) Solving for e distance of the shear centre from A equals. [( ) ] (20) Substituting values into Equation 20 we obtain the distance of the shear centre from point A. [( ) ( )] This value of e is the distance from the point A to where the shear centre of the beam is located. For this beam it is located millimetres from the LHS. Beam 2: Solving for the shear centre for the z-beam is much simpler than for the c-beam. This is beacuse, as Hibbeler states, the shear centre will always lie on an axis of symmetry (2011). As the shear flow will be symmetrical either side of the line of symmetry no twisting will 30

31 occur at the line of symmetry. Thus, from Figure 3 it can be seen that the shear centre will be positioned 110 millimetres from the LHS. Beam 3: The L-beam can be solved in the same manner as the C-beam. However, this is not necessary. As we are to take moments about the point A (which is located at the apex), the shear flow distributions produce zero moment about this point (Bauchau, 2010). This is also true if the shear force is applied at A. Therefore at point when a shear force is applied at point A there is no moment of twist applied in the beam. Thus we can assume that the shear centre for the L- beam is located at the apex. The apex is located 72.5 millimetres from the LHS of the beam as shown in Figure 5. Discussion As the results above show, for the C-beam the experimental shear centre is roughly 35 millimetres away from the theoretical result, with the experimental shear centre located at 56.07mm and the theoretical located at 90.15mm from the LHS. As the theoretical results are assumed to be correct the experiment must have been conducted incorrectly. An error that may have occurred during the experiment may have been that the dial gauges were not zeroed after each mass was applied. Another error that may have happened is the gauge was read incorrectly. Both these mistakes would have resulted in incorrect data being recorded. For the Z-beam the results from both the experimental and theoretical analysis show that the shear centre is located at 110 millimetres from the LHS. This also proves the point that the shear centre lies on the axis of symmetry. As the results were so similar (with only 0.65mm difference) it can be concluded that for this beam the results were very accurate. The results for the L-beam have a difference of 1.5 millimetres. Therefore this experiment was also conducted very accurately. The results prove that the shear centre will indeed lie on the vertical line crossing the apex for reasons mentioned in the results. The slight error may have come through incorrect reading of deflection or through bumping the beam as results were recorded. Conclusion Overall it can be seen that two of the three experiments were done extremely accurately. These were the Z-beam and the L-beam. The theoretical and experimental results are almost identical therefore validating that the shear centre locating has been found correctly. For the C-beam the results had a large error. Thus the experimental results cannot be considered valid and therefore the test for this beam should be redone. Errors in testing would have occurred when reading data from the deflection measurers or when loading the mass. In conclusion it can be perceived that this test is an accurate test to find the shear centre when conducted correctly. 31

32 References McGraw-Hill Concise Encyclopedia of Engineering. 2002, The McGraw-Hill Companies, Inc Accessed Date: 30/5/2012 Hibbeler. R. C., Mechanics of Materials, Eighth Edition in SI Units, 2011, Pearson Bauchau. C., 2010, mae.ucdavis.edu Accessed Date: 1/06/2012 Kopeliovich, Dr. D. (2012) SubsTech, viewed 3 June 2012, , JEE 332 Structural Analysis Laboratory Handout, Australian Maritime College. Hibbeler, R.C Mechanics of Materials, Eighth Edition in SI Units. Pearson Education, Singapore. Simitses, J. Hodges, D Fundamentals of Structural Stability. Butterworth-Heinemann, USA. 32

33 Appendix A Beam 1 C-beam Hole Def L Dial Def R Dial Beam 2 Z-beam Hole Def L Dial Def R Dial Beam 3 L-beam Hole Def L Dial Def R Dial