Engineering Mechanics Experiments - Deflection of a Simply Supported Beam - Research Paper Example

ENGINEERING MECHANICS EXPERIMENTS By Student’s name Course code and name Professor’s name University name City, State Date of submission Table of Contents Table of Contents 2 Experiment 1: Shear Force Variation with increasing load 3 Experiment 2: Shear force variation for various loading conditions 7 Experiment 3: Bending moment variation with increasing load 11 Experiment 4: Bending moment variation for various loading conditions 15 Experiment 6: Deflections of a Simply Supported Beam 23 Experiment 1: Shear Force Variation with increasing load Introduction Shear in beams is a common force that is experienced in structures such as road bridges. This prompts the designers to come up with practicable structure that are able to carry loads that are perceived to possess knife edge characteristic. This experiment aims at finding out the relationship between shear variation and increase in load. Aim The aim of this experiment was to determine the relationship between shear variation and increase in load. Method Figure1: Experiment 1 setup The digital display was set to zero and a point load of 100g applied between the two supports at 10cm from the left support. The digital shear force displayed was recorded while noting that the positive shear force meant downward force and vice versa for negative force. The load was increased by a 100g interval to the 400g mark. Theory Shear force moments are spread across a structural beam depending on the area of a cross section and the young modulus. When a beam is placed under a uniformly distributed load, the increase in shear force is directly proportional to the increase in load. The load applied on a beam shall be equal to the abrupt shear force changes due to change in shear force. Shear force usually remains constant when a concentrated load acts across its length. Also the intensity of the load can be used to indicate the shear force at a particular given state of a beam. The positive change in shear force represents positive change in bending moments. Results Mass(g) Load(N) Experimental Shear Force (N) Theoretical Shear Force(N) 100 0.9 0.2 0.22 200 1.96 0.4 0.44 300 2.94 0.6 0.66 400 3.92 1.0 0.89 Table1: Effect of loading on shear force Configuration W1(N) W2(N) F1(N) F2 (N) Experimental Shear Force(N) Theoretical Shear Force(N) 1 1.96 _ -0.6 2 0.98 1.96 1.6 3 0.98 1.96 0.4 Table2: Effect of location of load on shear force Analysis Graph1: Shear Force Versus Load Applied for both experimental and theoretical results. Discussion Structural wear and tear is blamed for the erroneous results that are usually given back on experimental setups. This experiment was mainly affected by zeroing error and mechanical wear and tear of the experimental setup due to usage over time. From the above graph, it is noted that the change in shear force is directly proportional to the shear force produced on a beam. F1 W1 W2 F2 Figure 2: Free body diagram for the experiment 1 Figure 3: bending moments diagram for experiment 1 Conclusion Shear force on a beam is directly proportional to the load applied to it assuming that all the other external forces are kept constant. Therefore, the intensity of the load can be used to calculate the shear force at a particular given state of a beam. Experiment 2: Shear force variation for various loading conditions Introduction Shear force and bending moments in a structural member vary with different loading conditions. For the sake of study, beams are assumed to be rigid and homogenous, therefore resistant to external forces. This experiment shall investigate the forces acting on an overhanging beam with double load acting upon it. Aim The main aim of this experiment was to investigate Shear force variation for various loading conditions. Method Figure 4: Setup for shear force experiment (single point load) Set the test apparatus to zero in readiness for applying of load. Apply a load as shown in the figure 1 above and record the shear deformation registered by the digital display at an interval of 100g. Apply point loads of 100 g and 200 g at 22 cm and 26 cm respectively from left support as shown in the figure 2 below. Vary the distance by 24cm and 40cm for 100g and 200g respectively and tabulate the results. Figure 5: Setup for shear force experiment (single point load) Theory When the end of a beam is extending beyond its supports, it is said to be overhanging. The forces affecting overhanging beam depends on the type of loading. Therefore the type bending moment diagram to be drawn from them entirely depends on it. Result Configuration W1 (N) W2 (N) F1(N) F2(N) Experimental Shear Force (N) Theoretical Shear Force (N) 1 1.96 0 2.66 -0.7 -0.6 -0.62 2 0.98 1.96 1.34 1.6 1.8 1.65 3 0.98 1.96 0.68 2.26 0.3 0.35 Table 3: Table showing experimental shear force and theoretical shear force Analysis In order to understand the forces acting on the setups in figure 1 and 2 above, the following free body diagrams express this. The x, y graph plotted indicates that the shear force generated is directly proportional to the external force under which it is subjected. W1 F1 F2 Figure 6: Free body diagram for experimental setup 1 W1 W2 F1 F2 Figure 7: Free body diagram for experimental setup 2 Figure 8: bending moment diagram for setup 2 Discussion The experiment above assume that the shear force is uniform across the width therefore homogeneity. Based on this fact, the shear force is not affected by the distribution of the bending moments. The forces acting across other planes such as the z-axis are not covered in this experiment as they are acting perpendicularly. The reactions F1 and F2 are successfully determined by the experiment. The comparison between the real figure and the theoretical figures show the fluctuation by up to 0.05N which is considerably low. This is due to the mechanical errors due forces such as the ones acting on other planes which are not covered by this experiment. Conclusion The Shear force and bending moments in a structural member vary with different loading conditions that it is imposed on. The loading conditions usually vary with the position of the load which is the main determinant of the reaction forces that the supports shall be put under. Experiment 3: Bending moment variation with increasing load Introduction The bending moment across beams varies with load and the configuration of the structural setup. Steady loading affects the shear forces and bending moments as the beam continues to fluctuate or buckle. It should however be remembered that the forces acting on point loads are more destructive than the cantilever and uniformly distributed loads. This experiment seeks to establish the bending moment variation with increase in load. Aim The main aim of this experiment is to investigate the bending moments with respect to steady change in load. Method The experimental setup was set to zero to ensure accuracy. A point load of 100g was applied 10cm from the left support and the displayed figure recorded in table 1 below. A positive figure on the display interface was noted to be the applied bending moments which was set to the right of the beam (Clockwise). A negative figure indicated a counterclockwise moment for the sake of this experiment. The same procedure was repeated for loads 200g, 300g and 400g and the results recorded. Theory The elastic theory of the bending moments assumes that the beam is made of homogenous material that has a constant modulus of elasticity throughout the span. The relationship between stress and strain is directly proportional for linearly elastic material. For analysis purpose, the material is not to be strained past the elastic limit. The neutral plane is assumed to be the length that remains unchanged on the beam during the loading experience. Results Mass(g) Load (N) Displayed force (N) Experimental bending moment (Nm) Theoretical bending moment (Nm) 100 0.98 0.2 0.025 0.028 200 1.96 0.4 0.050 0.056 300 2.94 0.7 0.088 0.084 400 3.92 0.9 0.113 0.111 Table 4: Results obtained from the experimental setup Analysis Graph 2: A graph of load against the experimental and theoretical bending moment Discussion The magnitude of error noted is up to 0.006 which is relatively low. Human error is blamed for this lose and also deteriorating device sensitivity which is caused by mechanical error due to age. Frictional forces were also to blame for these errors in the readings achieved in this assignment. Despite all these errors, the setup manages to prove that the bending moments vary proportionally with increase in load. Conclusion Bending moments vary with increase in load as per the elastic theory of moments. This experiment succeeds to illustrate this fact and also the human errors, mechanical errors and frictional forces which affect the experiment as a matter of calculation. The theoretical and experimental values differ negligibly thus the experimental setup is real if simulation for larger loads would be applied. Experiment 4: Bending moment variation for various loading conditions Introduction Bending moment is said to vary proportionally to the load applied to it. This also depends on which kind of configuration that is applied in the experiment. Several experiments have been setup to investigate the relationship between the bending moments and the loading conditions. For the sake of analysis the beam is subdivided into two sections which are said to be at balance in terms of clockwise and anticlockwise bending moments. It is with this note that this experiment seeks to establish the bending moment variation for various loading conditions. Aim This experiment seeks to investigate the bending moment variation for various loading conditions in a beam. Method The digital display was set to zero to ensure conformance of the data received according to the loads. A load of 200g was applied 14cm from the left support and the force data displayed recorded on the table below. The bending moment was calculated and recorded in the table after which two point loads of 100g and 200g were set at 22cm and 26cm respectively from the left support for analysis. The resulting or displayed force was recorded and the weights 100g and 200g moved to 24cm and 40cm respectively. The experimental bending moment at the cut section was calculated and recorded in the same table. Figure 7: Setup for configuration 2 to investigate bending moments Figure 8: Setup for configuration 3 to investigate bending moments Theory When a body is stationary, the following forces act on it for there to be balance: i. For vertical equilibrium, total forces acting upwards equal to those acting downwards, ii. For horizontal, the total forces acting on the right of the beam equals those on the left and iii. For moment equilibrium the total clockwise moments equal those acting anticlockwise. Results Configuration W1 (N) W2 (N) Displayed force (N) Experimental bending moment Theoretical bending moment 1 1.96 _ -1.2 0.150 0.167 2 0.98 1.96 1.0 0.125 0.161 3 0.98 1.96 1.0 0.125 0.133 Table 5: Showing the results obtained in the experiment conducted to establish the bending moment variation for various loading conditions. Analysis Figure 9: Showing the bending moment diagram for the experimental setup under study Discussion The experimental and theoretical values differ by up to a maximum variation of 0.036. It is established that all elements, that is, vertical, horizontal and moments must be at equilibrium for a beam to balance. Conclusion The bending moment variation for various loading conditions is successfully portrayed by the experimental configurations set up above. It is observed that the horizontal, vertical and moment equilibrium must be reached for a stationary beam to at equilibrium. These factors are however affected by the type of load whether point load or uniformly distributed load. Experiment 5: Bending Stress in a Beam Introduction The bending stress in a beam acts transversely due to loads thereby causing shear forces and bending moments. Beams bend due to lateral forces acting on them thus causing changes on the deformation axis or the deflection curve. All loads on a beam are assumed to be symmetrical for the purpose of analysis as they can be translated to the x and y axis. This experiment aims at investigating the bending stresses in beams. Stress analysis of beams is very important for the purpose of establishing mechanical properties Aim The main aim of this experiment was to investigate the bending stresses in a beam which is said to be homogenous all along its span. Method Figure 10: Bending stress in a beam experiment setup Theory In order to find the Young's modulus the following equation (i) is used: --------------------------------- (i) Where: E = Young's modulus of a beam (69GPa). = Stress = Strain The equation (ii) below is also used in cases where the Young's modulus is not provided in order to derive the bending moments of a beam and also to ascertain the second moment of a beam in cases where the latter is provided. Where: M = Bending moment (Nm) = Stress I = Second moment of inertia (m4) = Distance from the neutral axis. Results Gauge Number Load (N) 0 100 200 300 400 500 1 -16 -118 -242 -256 -460 -570 2 -15 -85 -171 -248 -321 -396 3 -14 -95 -180 -261 -332 -409 4 15 27 12 -1 -13 -26 5 7 -6 -20 -36 -47 -61 6 -45 -21 2 27 46 71 7 14 34 68 88 111 132 8 10 58 112 164 209 261 9 -15 31 93 140 193 243 Table 6: Result for Experiment 1 (uncorrected). Analysis Assuming that the beam contains homogeneous traits, the bending stress shall be distributed along equally. The bending stress in a beam shall be a balance between the compressive stresses and tensile stresses that originate internally. A simply supported beam shall hence have tensile stresses at the bottom and compressive forces at the top as a matter of fact. Discussion This experiment succeeds in ascertaining the bending stresses in a beam. The main constraint is that the mechanical losses and the zeroing error affects the results thus the results obtained have to be corrected. Conclusion Beams bend when lateral forces acting on them along the axis set as the datum. This experiment shows that Hooke’s law applies in the bending stress of beams as the proportionality is direct when a load is applied. Experiment 6: Deflections of a Simply Supported Beam Introduction A beam is said to be a structural element whose dimensions greatly differ from one which is considered to be the length. In addition a beam is usually supported at a few points along its span. When a beam is placed under a load the original dimensions change thereby leading to deformation. In order to design a beam, two parameters are taken into consideration mainly the strength and stiffness. This experiment seeks to investigate the maximum deflection of a simply supported beam when loaded. Aim This experiment was aimed at examining the deflection of a simply supported beam that was subjected to a point load by varying the beam length by changing the support distance. This was done in order to find the relationship between the deflection and the length of the beam. The maximum deflection is determined by the equation (i) below: Where: W = Load (N) L = Distance between supports. E = Young's Modulus. I = Second moment of inertia (m4). Method The digital display meter was set to zero. A load of 100g was placed on the hook at an interval. Figure: Configuration for experiment 6 Theory Deflection refers to the extent to which a structural element is displaced by a load from its original position. Maximum deflection is achieved when the beam is subjected to severe conditions of stress. The theory however follows the assumptions that the mechanical properties of the beam obey Hooke’s law, the curvature achieved is small and lastly that other forms of stress such as shear do not apply. Results Mass(g) Actual Deflection(mm) Theoretical Deflection 0 0 0 100 0.1 0.03 200 0.13 0.05 300 0.15 0.09 400 0.18 0.12 500 0.21 0.15 Table 7: Showing deflections of a simply supported beam Analysis Graph 3: Plotted for actual deflection against weight Discussion Bending moment on simply supported beams is linearly proportional to the mass being applied to it. Therefore, the actual deflection which is also a function of force is equally proportional to the bending moment. Hooke’s law applies up to the maximum elasticity limit during which stress is proportional to strain. The actual curve shown is real for lateral and transverse loads which are responsible for the highest deflection. Conclusion The deflection on a beam entirely depends on the strength or stiffness it possesses. This experiment was successful in showing the actual deflections in a beam which differed by up to 0.07. This experiment is therefore viable in showing the relationship between deflections and loads to which beams are imposed on in case of simulated loads. Read More