Two Pinned Arch Experiment - Lab Report Example

Running Head: LABORATORY REPORT Laboratory Report Name Institution Experiment one Table of Contents Abstract 3 1.0 Introduction 3 2.0 Test Data Analysis 4 2.1 Two –Pinned Arch Experiment Overview 4 2.2 Two-pinned Arch Experiment Results 6 2.3 Fixed arch experiment overview 9 2.4 Fixed Arch Experiment Results 10 3.0 Conclusion 14 Abstract 17 1.0Introduction 17 2.0Test Data Analysis 18 2.1Test Overview 18 2.2Statically Determinate Results 19 2.3Statistically indeterminate Truss Results 23 Conclusion 26 Abstract Arch beam got analyzed in two different states. The first state was a two-pinned arch used in determining the horizontal reaction forces at point B. The second state case was that of a fixed arch experiments and got used in determining horizontal reaction forces at one end and fixing moment at the other end. There was an acceptable margin between the theoretical and experimental findings from the experiment. The force was minimal (Zero) at both ends and maximum at the middle. The variation of moments from one end of the beam to the other was similar to that of the force. 1.0 Introduction In the experiment under consideration, an arch beam was analyzed in two different states. The first state consisted of a beam constrained in a two-pinned arch. The beam got subjected to a loading of 0.5kg at various parts of the beam. Theoretical and experimental values of horizontal reactions at point B then obtained. The second state got carried out when both ends of the arch were fixed. Fixing both ends of the arch limited the motion of the beam such that the beam could neither undergo translationally nor rotational motion. Theoretical and experimental values of both horizontal reactions at one end and fixing moment at the other end got obtained. The values were achieved by recording the experimental results from the gauges and using the appropriate formulas for the theoretical values. 2.0 Test Data Analysis 2.1 Two –Pinned Arch Experiment Overview Point A can only move in two directions. Point B can undergo vertical sliding, however, when forced down by the hanging weight, there is a resultant horizontal reaction force. The reaction force is as a result of point B pushing the barrier in the horizontal direction. The dimension of the arch were as follows; Height of the arch was 0.1 of a meter while the mid-span was found to be 0.5 of a meter wide. The setup of the two-pinned arch analysis summary is as shown in the diagram shown below. Fig 1: A two-pinned arch with displaced suspending weights along its length. A mass of 0.5 kg suspended in the arch was moved gently from the left-hand side to the right-hand side of the beam at intervals of 50 millimeters. At each location, an analysis of horizontal reactions was carried out by means of measuring and calculation. The horizontal reaction load is also calculated by dividing the horizontal reaction obtained with weight of the suspended mass. 2.2 Two-pinned Arch Experiment Results Below are the results from the experiment. Location Horizontal Reaction   Distance from left(mm) From left(m) Displayed horizontal reaction (N) Calculated (N) Influence Value Fraction Span 0 0.000 0 0.00 0.00 0.00 50 0.050 1.2 1.50 0.31 0.11 100 0.100 2.8 2.84 0.58 0.22 150 0.150 4.5 3.89 0.79 0.33 200 0.200 5.8 4.56 0.93 0.44 250 0.250 6 4.79 0.98 0.56 300 0.300 5.8 4.56 0.93 0.67 350 0.350 4.6 3.89 0.79 0.78 400 0.400 2.9 2.84 0.58 0.89 450 0.450 1.2 1.50 0.31 1.00 Table 1: Table showing the measured and calculated horizontal reactions at various locations The graphs below therefore is a graphical representation of the data given in the table. Fig 3: Graph of Distance from left against horizontal reaction Fig 4: Graph of fraction span against horizontal reaction Graph 3 and 4 shows that there is a slight difference between theoretical and experimental values. In both graphs, horizontal reaction at point B is minimum at both ends and maximum at the middle. Figure 4 gets referred to as an influence curve for horizontal reaction forces at points B. 2.3 Fixed arch experiment overview The fixed arch analysis gets performed in a state where both left-hand side and right-hand side can neither slide nor rotate. The same as that applied in the previous section was used, and the procedure performed similarly to that presented in the previous section. In this case, both the reaction at point B as well as the fixing moment at point A is determined. The experiment got carried out as shown in the diagram below. Fig 5: Fixed arch experimental set up 2.4 Fixed Arch Experiment Results Tabulated values of measurements as well as force and moment curves for the experiment are as shown below. Horizontal Reaction Fixing moment at A From A(mm) a Moment (N) From B (mm) b Experimental Hb (N) Theoretical Hb (N) Experimental MA (Nm) Theoretical MA (Nm) 0 0 0 450 0.45 0 0.00 0.00 0.00 50 0.05 3 400 0.4 1.2 0.81 -0.15 -0.14 100 0.1 3.1 350 0.35 2.8 2.47 -0.16 -0.13 150 0.15 1.8 300 0.3 4.5 4.09 -0.09 -0.05 200 0.2 0.2 250 0.25 5.8 5.05 0.01 0.03 250 0.25 -1.4 200 0.2 6 5.05 0.07 0.09 300 0.3 -2.2 150 0.15 5.8 4.09 0.11 0.11 350 0.35 -2.2 100 0.1 4.6 2.47 0.11 0.08 400 0.4 -1.5 50 0.05 2.9 0.81 0.08 0.03 450 0.45 -0.5 0 0 1.2 0.00 0.03 0.00 Table 2: Measured and theoretical reactions force as well as fixing moments for fixed arch at different loadings. Fig 6: Graph of distance from x against horizontal reaction Fig 7: Graph of fixing moment at A against Distance from x There is an acceptable variation between the experimental and theoretical values as shown in figure 6 and figure 7. Like in the other cases, the horizontal reaction will rise from zero at the far left of its maximum value at the middle and fall back to a minimum value of zero at the far right end. Fixing moments would also be zero at both ends since there are no external forces at these ends. The experimental values obtained were almost similar to the theoretical values although there was a slight difference. 3.0 Conclusion There was a significant relationship between the experimental and theoretical values for all the parameters obtained in the experiment. The slight variation gets attributed to systematic and other types of errors accrued during the trial. There is a general trend in the change of horizontal reaction from point A to point B of the arch. In all the cases, the ends experience the lowest reactions while the midspan experiences the highest reactions. Experiment Two Table of Contents Abstract 3 1.0 Introduction 3 2.0 Test Data Analysis 4 2.1 Two –Pinned Arch Experiment Overview 4 2.2 Two-pinned Arch Experiment Results 6 2.3 Fixed arch experiment overview 9 2.4 Fixed Arch Experiment Results 10 3.0 Conclusion 14 Abstract 17 1.0Introduction 17 2.0Test Data Analysis 18 2.1Test Overview 18 2.2Statically Determinate Results 19 2.3Statistically indeterminate Truss Results 23 Conclusion 26 Abstract The experiment was carried out in order to analyze a truss in two states. The first analysis consisted of truss with the redundant member while their other one had no redundant member. For all the states, the values of pure strain, deflection as well as forces were measured and recorded. Graphical representations of the parameters obtained got plotted. The graphs included the load against deflection curves as well as load against strain curves. Theoretical weight values were calculated by use of appropriate formulae and compared with the experimental pressure values. 1.0 Introduction Trusses are support structures that consist of a maximum of two force members. If the members are not well fixed or constrained, then the members would move relative to each other. In such a case, the truss assembly will be referred to as a mechanism. In situations where a truss contains excess constrain members, and the truss is in equilibrium, the truss is said to have a statically indeterminate loading. Force analysis of a statically indeterminate truss poses a lot of complexities and is difficult. The experiment, therefore, aims at analyzing trusses under different kinds of loading and also enable comparison between the findings and the theoretical values of the parameters under consideration. 2.0 Test Data Analysis 2.1 Test Overview For the experiment under consideration, the device consists of seven members for statically determinate situations and eight members for statically indeterminate cases as shown in the figure below. Various points of the truss can get subjected to varied loadings and the resultant measurements of strains, forces and deformations from different members recorded. The experiment demands a procedural approach is paying attention to all the details in the test manual. The loadings got varied at intervals of 50N from 0N to 250N and applied to the right-hand side of the truss. Fig 1: Truss analysis experimental setup 2.2 Statically Determinate Results For statically determinate truss, member six was removed and the following data and graphs were obtained from the experiment. Load Displacement strain 1 strain 2 strain 3 strain 4 strain 5 strain 6 strain 7 strain 8 Rod Diameter Material N mm me me me me me me me me mm GPa 0 0 0.2 -0.2 -0.3 0.3 -0.3 --- 0.1 -0.5 6 210 49 0.014 9.9 -9.5 -8.9 -18.1 0.4 --- 12.3 12.7 6 210 99 0.016 21.1 -19.6 -18.5 -39.4 0.3 --- 26.2 26.3 6 210 151 0.041 31.7 -30.1 -28.9 -59.9 0.9 --- 39.8 40.5 6 210 200 0.052 40.3 -39.2 -37.4 -78.2 1 --- 51.8 52.5 6 210 249 0.056 49 -47.6 -45.7 -95.6 1.4 --- 63.2 63.7 6 210 Table 1: strain readings in statically determinate trusses. Load True strain 1 True strain 2 strain 3 strain 4 strain 5 strain 6 strain 7 strain 8 N me me me me me me me me 0 0 0 0 0 0 0 0 0 49 9.7 -9.3 -8.6 -18.4 0.7 0 12.2 13.2 99 20.9 -19.4 -18.2 -39.7 0.6 0 26.1 26.8 151 31.5 -29.9 -28.6 -60.2 1.2 0 39.7 41 200 40.1 -39 -37.1 -78.5 1.3 0 51.7 53 249 48.8 -47.4 -45.4 -95.9 1.7 0 63.1 64.2 Table 2: True strain readings in statically determinate trusses Fig 2: Graph of loading against strain and true strain values for member 5 Fig 3: Graph of loading against true strain for member 7 For figure 9, the strain values are positive thus, it is true to conclude that member 5 is in tension. However, the variation between the member and the force is not linear. Figure 10 shows the member having a negative slope. It is thus, an indication that member 7 undergoes compression. A graphical representation of variation between deflection and load is as shown in the graph below. Fig 4: Graph of deflection against loading A comparison between the theoretical and experimental forces in the various members should be analyzed in the next stage using the maximum 250N force. Theoretical force is given by the expression shown below. Thus, calculating, the theoretical forces for each member strain, the results are tabulated in the table below. Member Member strain Experimental member force(N) Theoretical Member force (N) 1 48.8 292.6 271.69 2 -47.8 -283.6 -291.98 3 -45.9 -271.6 -286.76 4 -95.8 -567.5 -503.21 5 1.2 9.6 5.4 6 0.0 0 0 7 63.0 374.6 397.67 8 63.5 379.7 453.45 Table 3: Measured and theoretical member forces According to the results shown in the table, members 2, 3 and four are all under compression due to the negative forces. The other members are under tensile forces due to the active forces. 2.3 Statistically indeterminate Truss Results For statically indeterminate results, member 6 was added on the test machine. The following table shows the data obtained during the experiment. Load Displacement strain 1 strain 2 strain 3 strain 4 strain 5 strain 6 strain 7 strain 8 0 0 -0.6 -0.6 0 -0.7 -0.2 0.3 0.1 0.9 1 0 -0.6 -0.8 0 -0.5 -0.5 0.5 0.6 1.2 49 0.017 11.3 -7.7 -8.8 -16.5 2.3 -2.5 12.3 9.7 99 0.032 25 -14.6 -18.5 -33.9 5.6 -6 25.3 19.6 149 0.04 39.8 -22.7 -29.7 -53 9 -9.7 40.6 30.1 201 0.048 50.7 -27.7 -36.9 -65.8 11.5 -12.8 50.9 36.4 250 0.054 63.9 -33.3 -45.9 -80.8 15.6 -17.4 63.7 43.8 Table 9: Strain reading for statically indeterminate stresses Load Displacement strain 1 strain 2 strain 3 strain 4 strain 5 strain 6 strain 7 strain 8 0 0 0 0 0 0 0 0 0 0 1 0 0 -0.2 0 0.2 -0.3 0.2 0.5 0.3 49 0.017 11.9 -7.1 -8.8 -15.8 2.5 -2.8 12.2 8.8 99 0.032 25.6 -14 -18.5 -33.2 5.8 -6.3 25.2 18.7 149 0.04 40.4 -22.1 -29.7 -52.3 9.2 -10 40.5 29.2 201 0.048 51.3 -27.1 -36.9 -65.1 11.7 -13.1 50.8 35.5 250 0.054 64.5 -32.7 -45.9 -80.1 15.8 -17.7 63.6 42.9 Table 10: True strain for statically indeterminate truss Fig 5: Graph of strain against loading for member 7 Fig 6: Graph of strain against loading for member 8 For the exerted load of 250N, the theoretical and experimental member forces are obtained as shown in the table below. Member Member Strain Experimental member force Theoretical force 1 64.5 379.4 386.78 2 -32.7 -198 -187.67 3 -45.9 -272.5 -269.56 4 -80.1 -479.6 -465.56 5 15.8 92.3 101.34 6 -17.7 -103.3 -112.56 7 63.6 378 380.23 8 42.9 260.3 257.56 Table 6: Measured and theoretical member forces for statically indeterminate trusses From the above table, members 2, 3, four and 6 are subjected to compression while the rest are under tensional forces. The difference between theoretical and experimental values is slight as compared to those indeterminate truss loading. Conclusion The margin between the theoretical and experimental values is slightly smaller for statically indeterminate truss loading as compared to the statically determinate truss loading. It is, therefore, the difference between the pure strain and strain readings gets attributed to the errors experienced during the experiment. The errors can get avoided is proper procedures gets followed. However, some errors are unavoidable and in such cases, use of error factors can be applied. Bibliography Megson, T. H. G. (2005). Structural and stress analysis. Amsterdam: Elsevier Butterworth-Heineman. Read More

The experiment got carried out as shown in the diagram below. Fig 5: Fixed arch experimental set up 2.4 Fixed Arch Experiment Results Tabulated values of measurements as well as force and moment curves for the experiment are as shown below. Horizontal Reaction Fixing moment at A From A(mm) a Moment (N) From B (mm) b Experimental Hb (N) Theoretical Hb (N) Experimental MA (Nm) Theoretical MA (Nm) 0 0 0 450 0.45 0 0.00 0.00 0.00 50 0.05 3 400 0.4 1.2 0.81 -0.15 -0.14 100 0.1 3.1 350 0.35 2.8 2.47 -0.16 -0.13 150 0.15 1.8 300 0.3 4.5 4.09 -0.09 -0.05 200 0.2 0.2 250 0.25 5.8 5.05 0.01 0.03 250 0.25 -1.4 200 0.2 6 5.05 0.07 0.09 300 0.3 -2.2 150 0.15 5.8 4.09 0.11 0.11 350 0.35 -2.2 100 0.1 4.6 2.47 0.11 0.08 400 0.4 -1.5 50 0.05 2.9 0.81 0.08 0.03 450 0.45 -0.5 0 0 1.2 0.00 0.03 0.00 Table 2: Measured and theoretical reactions force as well as fixing moments for fixed arch at different loadings.

Fig 6: Graph of distance from x against horizontal reaction Fig 7: Graph of fixing moment at A against Distance from x There is an acceptable variation between the experimental and theoretical values as shown in figure 6 and figure 7. Like in the other cases, the horizontal reaction will rise from zero at the far left of its maximum value at the middle and fall back to a minimum value of zero at the far right end. Fixing moments would also be zero at both ends since there are no external forces at these ends.

The experimental values obtained were almost similar to the theoretical values although there was a slight difference. 3.0 Conclusion There was a significant relationship between the experimental and theoretical values for all the parameters obtained in the experiment. The slight variation gets attributed to systematic and other types of errors accrued during the trial. There is a general trend in the change of horizontal reaction from point A to point B of the arch. In all the cases, the ends experience the lowest reactions while the midspan experiences the highest reactions.

Experiment Two Table of Contents Abstract 3 1.0 Introduction 3 2.0 Test Data Analysis 4 2.1 Two –Pinned Arch Experiment Overview 4 2.2 Two-pinned Arch Experiment Results 6 2.3 Fixed arch experiment overview 9 2.4 Fixed Arch Experiment Results 10 3.0 Conclusion 14 Abstract 17 1.0Introduction 17 2.0Test Data Analysis 18 2.1Test Overview 18 2.2Statically Determinate Results 19 2.3Statistically indeterminate Truss Results 23 Conclusion 26 Abstract The experiment was carried out in order to analyze a truss in two states.

The first analysis consisted of truss with the redundant member while their other one had no redundant member. For all the states, the values of pure strain, deflection as well as forces were measured and recorded. Graphical representations of the parameters obtained got plotted. The graphs included the load against deflection curves as well as load against strain curves. Theoretical weight values were calculated by use of appropriate formulae and compared with the experimental pressure values. 1.0 Introduction Trusses are support structures that consist of a maximum of two force members.

If the members are not well fixed or constrained, then the members would move relative to each other. In such a case, the truss assembly will be referred to as a mechanism. In situations where a truss contains excess constrain members, and the truss is in equilibrium, the truss is said to have a statically indeterminate loading. Force analysis of a statically indeterminate truss poses a lot of complexities and is difficult. The experiment, therefore, aims at analyzing trusses under different kinds of loading and also enable comparison between the findings and the theoretical values of the parameters under consideration. 2.0 Test Data Analysis 2.

1 Test Overview For the experiment under consideration, the device consists of seven members for statically determinate situations and eight members for statically indeterminate cases as shown in the figure below.

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