Engineering Mechanical Laboratory DEFLECTION OF BEAM

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1 14391 Engineering Mechanical Laboratory DEFLECTION OF BEAM Abstract If a beam is supported at two points, and a load is applied anywhere on the beam, the resulting deformation can be mathematically estimated. Due to improper experimental setup, the actual results experienced varied substantially when compared against the theoretical values. The following procedure explains how the theoretical and actual values were determined, as well as suggestions for improving upon the experiment. The percent error remained relatively small, around 10%, for locations close to supports. As much as 30% error was experienced when analyzing positions closer to the center of the beam. Objectives 1. To measure deflections in a simply supported steel and brass beam. To use measured deflections and theory to evaluate the Young s modulus of the materials 3. To compare the analytical and experimental values of deflection in the simply supported and cantilever beam 4. To note the source of error in a typical simply supported and cantilever beams experiment 1. Introduction & Background 1.1 General Background If a beam is supported at two points, and a load is applied anywhere on the beam, deformation will occur. When these loads are applied either longitudinally outside or inside of the supports, this elastic bending can be mathematically predicted based on material properties and geometry. 1. Determination of Curvature Curvature at any point on the beam is calculated from the moment of loading (M), the stiffness of the material (E), and the moment of inertia (I). The following expression defines the curvature in these parameters as 1/ρ, where ρ is the radius of curvature.

2 1 M (1.1) EI Equation (1.1) does not account for shearing stresses. Curvature can also be found using calculus. Defining y as the deflection and x as the position along the longitudinal axis, the expression becomes d y 1 dx dy 1 dx d y dx 3 (1.) 1.3 Central Loading (Simply supported beam) Central loading on a beam can be thought of as a simple beam with two supports as shown below. Fig. 1 Simply supported beam Applying equilibrium to the free body equivalent of Fig. 1, several expressions can be derived to mathematically explain central loading. Fx 0 Ax 0 L P M A 0, NCLP 0 NC (1.3) P F 0 y Ay NC P0 Ay Fig. act as free body diagrams for the section between AB and BC respectively. (a) 0 x L (b) L x L Fig. Free Body Diagram

3 Solving the reactions between AB and BC, equation (1.1) can be expressed as d y Px L EI 0 x dx (1.4) d y Px PL L EI + x L dx Integrating twice, Equation (4) becomes 3 Px L EIy C1x C 0 x 1 (1.5) 3 Px PLx L EIy C3x C4 x L 1 4 To determine the constants, conditions at certain positions on the beam can be applied. Knowing the deflection at each of the supports, as well as the slope at the top of the curve is zero; the constants can be derived to 3 PL 3PL PL C1 C 0 C3 C 4 (1.6) Combining Equations (1.5) with (1.6), the expressions for deflection can be expressed as 3 Px PL L EIy x 0 x Px PLx 3PL PL L EIy x x L (1.7) 1.4 Bending Stress The bending stress at any location of a beam section is determined by the flexure formula: My (1.8) I where: M - moment at the section y - distance from the neutral axis to the point of interest I - moment of inertia The largest stress at the same section follows from this relation (Equation (8)) by taking y at an extreme fiber at distance c, which leads to: Mc (1.9) I

4 . Experiments.1 APPARATUS.1.1 Simply-supported and Cantilever rectangular beams.1. Weights.1.3 Micrometer.1.4 Ruler.1.5 Dial Gauges. PROCEDURE PART A - SIMPLY SUPPORTED BEAM (1) Record the beam dimensions and calculate the moment of inertia (I) using I=bh 3 /1 () Calculate the maximum permissible loads for mid-span, where maximum allowable stress is..mpa. IMPORTANT - Check these calculated maximum loads with instructor before proceeding with the experiment. (3) Load the beam at the mid-span. Record the deflection at the point of loading at each incremental load. Small divisions on the dial gage are.mm. One full revolution of the dial is.m (. small divisions). (4) Load the beam at the mid-span in. kg. increments. Record the deflection at the points along the beam. PART B- CANTILEVER BEAM (1) Record the beam dimensions and calculate the area moment of inertia (I). () Calculate the maximum permissible loads for cantilever beam, where maximum allowable stress is..mpa. (3) Load the beam at the free end in. kg. increments. Record the deflection at the points along the beam. PART C - THE PRINCIPLE OF SUPERPOSITION (1) Place a single concentrated load at end point and measure the resulting deflection (δ 1 ) at the reference point. () Remove the first load, and place a second load at same or another point on the beam

5 and measure the resulting deflection (δ ) at the reference point. (3) Apply both loads simultaneously and measure the resulting deflection (δ 3 ) at the reference point. The sum of the single deflection (δ 3 ) should closely approximate the total deflection (δ 1 + δ )..3 REPORT.3.1 Plot the curve of load versus deflection for mid-span loading configurations of the simply supported beam. Show loads as ordinates and deflections as abscissas..3. Determine the values of modulus of elasticity for mid-span loading conditions. Create a table of the results you obtained for the modulus of elasticity from the simply supported beam for steel and aluminum beams..3.3 Plot position versus deflection for the simply supported beam and cantilever beam at many loads..3.4 Show the graph or table of results for the study of principal of superposition. 3. Example of results and discussions 3.1 Results Fig. 3 Example of results

6 3. Discussions The theoretical results were not as expected or experienced. There was significant error between the actual results and theoretical value, especially as the distance studied approached the midpoint of the beam. Though the difference in inches was small, the percent error could be as high as 30%. The main source of error within this experiment occurs due to the improper testing procedure. As seen in Fig. 4, the theory used within this exercise is based upon a beam with one fixed support allowing one degree of freedom, a second support allowing two degrees of freedom, and a central load. This produces dramatically different results when compared against the actual setup. When using two knife supports, the setup contains two supports allowing two degrees of freedom and a central load. This is pictured in Fig. 4. Since both ends are under-constrained, the analysis for the experiment with the above theory is not accurate. (a) Theory (b) Experiment Fig. 4 Discussion the supports Another cause of error in the theoretical is the effect of gravity on the beam. With no applied load, the equations above would return a zero result. This is inaccurate for beams that are not specifically supported such that gravitational factors are overcome. 3.3 Conclusions When a load is applied to a beam, either centrally over at another point, the deflection can be mathematically estimated. Due to the error that occurred in this exercise, it is clear that margins in safety factors, as well as thorough testing, is needed when utilizing beam design. It is also important to ensure the scope of the testing closely models real-world practicality.

7 4. Raw data WORK SHEET FOR BEAM DEFLECTION CENTRALLY LOADED BEAM CRITICAL DIMENSIONS: L =..m (test length) W =..m (width) t =..m (thickness) POSITIONS OF DIAL GAUGES: X1 =.m X6=..m X =.m X7 =.m X3 =.m X8 =.m X4 =.m X9 =.m X5 =.m X10 =..m LOAD (kg) MATERIAL. DEFLECTION DATA X1 X X3 X4 X5 X6 X7 X8 X9 X10

8 WORK SHEET FOR BEAM DEFLECTION CANTILEVER BEAM CRITICAL DIMENSIONS: L =..m (test length) W =..m (width) t =..m (thickness) POSITIONS OF DIAL GAUGES: X1 =.m X6=..m X =.m X7 =.m X3 =.m X8 =.m X4 =.m X9 =.m X5 =.m X10 =..m LOAD (kg) MATERIAL. DEFLECTION DATA X1 X X3 X4 X5 X6 X7 X8 X9 X10

9 APPENDIX: Deflection of beam