Compressive Stress Arch Transmission - Lab Report Example

Abstract An arch is a structural member that transmits forces and stresses purely by compression [Cli11]. By the resolving the external forces it is subjected to into compressive stresses, and in turn eliminating tensile stresses, arches have found common application in structures requiring long spans, most notably bridges. This process of resolving forces to compressive stresses is referred to as arch action[Vai04]. When a load is applied at the top of the arch, it pushes outward at the base, referred to as thrust. As the load at the top increases, the arch tends to flatten out, resulting in more thrust at the bases. At this moment, all the fibres in a true arch are under compression. In order to ensure the arch does not flatten completely, or collapse, there is need for restraints. This can be done with the help of internal ties joining the ends of the arch or external bracings at the supports, usually in the form of abutments [Amb12]. There are various types of arches that have been used for a long time in the construction industry. The fixed arch is constructed using reinforced concrete for use in bridges and tunnels with small spans due to the buildup of internal stresses temperature differences. This makes the fixed arch statically indeterminate. The two- hinged arch has pinned connections at the base, allowing the structure to rotate freely to compensate for the internal thermal stresses. This allows two- hinged arch to support longer spans. The free movement of the structure could lead to development of internal stresses which makes the two- hinged arch statically indeterminate [Rey08]. The three- hinged arch has an additional hinge at its mid-span, which allows for movement in two directions to compensate for any internal stresses. This makes the arch statically determinate, able to support medium spanning structures, such as roofs for large buildings [Lue16]. This report focusses on the load distribution of a two- pinned arch, and a fixed arch subjected to a 500g load at incremental distances from the left support. Key words (arch, statically indeterminate, load distribution) Table of Contents Abstract 1 Table of Contents 3 List of Equations 3 List of Tables 3 List of Figures 4 1.Introduction 1 2. Two-Pinned Arch Experiment 2 2.1 Material 2 2.2 Methodology 2 2.3 Results and Discussion 3 2.4 Influence value 5 2.5 Inferences 6 3. Fixed Arch Experiment 11 3.1 Material 11 3.2 Methodology 11 3.3 Results and Discussion 13 3.4 Inferences 15 Conclusion 16 References 17 List of Equations Equation 2‑1: Theoretical Relationship between Position of Load and Reaction 3 Equation 2‑2: Influence value equation 5 Equation 3‑3: Theoretical Relationship between Position of Load and Reaction 12 Equation 3‑4: Theoretical equation for the fixing moment at A 12 Table 2‑1: Comparison between Experimental and Theoretical Values for Horizontal Reaction 3 Table 2‑2: Relationship between load position and influence value 5 Table 2‑3: Influence value for unit load on span 7 Table 2‑4: Bending Moment 8 Table 3‑5: Results table for fixed arch experiment 13 List of Figures Figure 2‑1: TecQuipment(R) two hinged arch apparatus 2 Figure 2‑2: Relationship between Load Position and Horizontal Reaction 4 Figure 2‑3: Influence value for the load at different positions 6 Figure 2‑4: Relationship between unit load position and influence value 7 Figure 2‑5: Bending Moment Diagram 9 Figure 2‑6: Bending Moment Diagram for Central Load on beam 9 Figure 3‑7: TecQuipment(R) fixed arch apparatus 11 Figure 3‑8: Relationship between Horizontal Reaction and Load Distance 14 Figure 3‑9: Relationship between fixing moment and load distance 15 1. Introduction In the experiment, a fixed load is applied at various positions across the arch and the effect this load has on the horizontal reaction investigated. In the cases of the two- hinged arch and the fixed arch, there are four unknown reactions but only three equations of static equilibrium, giving the structure an indeterminacy of one. This requires the use of the flexibility method. However, this experiment uses the secant formula which is a simplified formula which gives relatively good results for parabolic arch ribs. This gives the theoretical values for the reactions, which are compared with the experimental values collected during the experiment. Any discrepancies in the results obtained are noted and discussed. 2. Two-Pinned Arch Experiment 2.1 Material TecQuipment® two- hinged arch apparatus shown below is used to conduct the experiment. The apparatus allows for the recording of the thrust the arch places on its ends, which corresponds to the reaction forces at the supports. A 500g load is moved at intervals of 50 mm from the left end support. Figure 2‑1: TecQuipment(R) two hinged arch apparatus 2.2 Methodology When an arch is subjected to a load, it develops internal stresses which are transferred to its supports, generating reactions. The main aim of this experiment is to determine the relationship between the position of the load and the horizontal reactions developed at the support. The dial on the apparatus gives the experimental values for the reaction. The theoretical values for the horizontal reaction are calculated using the formula below, derived from the secant assumption; Equation 2‑1: Theoretical Relationship between Position of Load and Reaction Where is the horizontal reaction at B (N) W is the load (N) L is the span of the arch (m) x is the distance from the left hand side (m) r is the rise of the arch (m) 2.3 Results and Discussion The experimental results obtained for horizontal reaction at support B and the theoretical results obtained by calculation are as shown below; Table 2‑1: Comparison between Experimental and Theoretical Values for Horizontal Reaction Load Distance from Left Fraction of Span Displayed Horizontal Reaction Calculated Horizontal Reaction Variation g mm N N % 500 0 0 0 0 0 500 50 0.1 1.5 1.53 2.00 500 100 0.2 2.5 2.9 16.00 500 150 0.3 3.4 3.97 16.76 500 200 0.4 4.0 4.65 16.25 500 250 0.5 4.3 4.88 13.49 500 300 0.6 4.2 4.65 10.71 500 350 0.7 3.6 3.97 10.28 500 400 0.8 2.6 2.9 11.54 500 450 0.9 1.5 1.53 2.00 500 0 0 0 0 0 The results show that there is some correlation between the results obtained theoretically and those collected experimentally, with the greatest variation being just above 16%. This variation can be brought about by various factors, such as; Imperfections in the geometry of the experimental apparatus could result in the internal stresses not being distributed as is expected Lack of uniformity in the composition of the arch rib, which means the deflection is not uniform, resulting in uneven load distribution The secant formula is an approximation, and as such, does not produce perfect results for each load position- support reaction The horizontal reaction at support B varies, depending on the position of the load on the arch beam. The graph below shows the load position- reaction relationship for both the experimental and theoretical results; Figure 2‑2: Relationship between Load Position and Horizontal Reaction 2.4 Influence value In order to determine the position where the load will cause the largest horizontal reaction at the right support, an influence value is calculated using the formula; Equation 2‑2: Influence value equation The table below gives the experimental and theoretical influence value for the horizontal reaction for each load position; Table 2‑2: Relationship between load position and influence value Load Distance from Left Fraction of Span Experimental Horizontal Reaction Influence value Theoretical Horizontal Reaction Influence value Variation g mm N N % 500 0 0 0 0 0 500 50 0.1 0.30 0.31 2.00 500 100 0.2 0.50 0.58 16.00 500 150 0.3 0.68 0.79 16.76 500 200 0.4 0.80 0.93 16.25 500 250 0.5 0.86 0.98 13.49 500 300 0.6 0.84 0.93 10.71 500 350 0.7 0.72 0.79 10.28 500 400 0.8 0.52 0.58 11.54 500 450 0.9 0.30 0.31 2.00 500 0 0 0 0 0 There is a correlation between the experimental values collected and the calculated theoretical values, with very little variation. The results show that the load at 250mm has the greatest influence on the horizontal reaction, at 0.86. Figure 2‑3: Influence value for the load at different positions 2.5 Inferences The following can be inferred from the results of the experiment conducted; 1. The maximum horizontal reaction at the right support occurs when the load is at the mid- span of the arch (250mm from the left support) 2. The secant formula used is suitable as the deviation in the theoretical horizontal forces calculated and the experimental values recorded 3. The horizontal influence line value using a unit load being loaded at different points on the arch are as below; Table 2‑3: Influence value for unit load on span Unit Load Distance from Left Fraction of Span Horizontal Reaction Influence value mm N 1 50 0.1 0.32 0.31 1 100 0.2 0.60 0.58 1 150 0.3 0.82 0.79 1 200 0.4 0.94 0.93 1 250 0.5 0.97 0.98 1 300 0.6 0.93 0.93 1 350 0.7 0.80 0.79 1 400 0.8 0.59 0.58 1 450 0.9 0.28 0.31 Figure 2‑4: Relationship between unit load position and influence value 4. Find the horizontal reaction at A for a load of 4.9N (500g) at the crown (Ha = Hb with the load at the crown)? 5. The bending moment for the load on the arch span is as follows; Table 2‑4: Bending Moment Load Distance from Left Fraction of Span Horizontal Reaction Bending Moment g mm N Nm 500 0 0 0 0 500 50 0.1 1.5 0.15 500 100 0.2 2.9 0.25 500 150 0.3 4.0 0.34 500 200 0.4 4.6 0.4 500 250 0.5 4.7 0.43 500 300 0.6 4.6 0.42 500 350 0.7 3.9 0.36 500 400 0.8 2.9 0.26 500 450 0.9 1.4 0.15 500 500 0 0 0 Figure 2‑5: Bending Moment Diagram 6. Bending moment diagram to scale for beam loaded with 4.9 N at the centre Figure 2‑6: Bending Moment Diagram for Central Load on beam 3. Fixed Arch Experiment 3.1 Material TecQuipment® fixed arch apparatus shown below is used to conduct the experiment. The apparatus allows for the recording of the thrust the arch places on its ends, which corresponds to the reaction forces at the supports. A 500g load is moved at intervals of 50 mm from the left end support. Figure 3‑7: TecQuipment(R) fixed arch apparatus 3.2 Methodology When an arch is subjected to a load, it develops internal stresses which are transferred to its supports, generating reactions. The main aim of this experiment is to determine the relationship between the position of the load and the horizontal reactions developed at the support. The dial on the apparatus gives the experimental values for the reaction. The theoretical values for the horizontal reaction are calculated using the formula below, derived from the secant assumption; Equation 3‑3: Theoretical Relationship between Position of Load and Reaction Where is the horizontal reaction at B (N) W is the load (N) L is the span of the arch (m) a is the distance of the load from the left-hand side (m) b is the distance of the load from the right-hand side (m) r is the rise of the arch (m) The theoretical fixing moment is given by the formula below;- Equation 3‑4: Theoretical equation for the fixing moment at A Where is the fixing moment at A (N-m) 3.3 Results and Discussion The experimental results obtained for horizontal reaction at support B and the theoretical results obtained by calculation are as shown below; Table 3‑5: Results table for fixed arch experiment Distance from A Moment Force Fixing Moment Theoretical Fixing Moment Horizontal Reaction Theoretical Horizontal Reaction (mm) (N) (N-m) (N-m) (N) (N) 0 0 0 0 0 0 50 3 -0.15 -0.152 0.8 0.759 100 2.9 -0.145 -0.16 2.2 2.4 150 1.6 -0.08 -0.092 4 4.134 200 0 0 0 5.2 5.4 250 -1.6 0.08 0.078 5.5 5.859 300 -2.4 0.12 0.12 5.1 5.4 350 -2.3 0.115 0.118 4 4.134 400 -1.6 0.08 0.08 2.3 2.4 450 -0.5 0.025 0.028 0.6 0.759 500 0 0 0 0 0 Figure 3‑8: Relationship between Horizontal Reaction and Load Distance The plot of the horizontal reaction against the distance of the load from the left support shows that the horizontal reaction increases as the load moves away from the support, reaching it maximum value at the centre of the fixed arch (250mm from the left support). The plot also shows that there is a strong correlation between the values obtained during the experiment and the theoretical values calculated using the formula provided. Figure 3‑9: Relationship between fixing moment and load distance The fixing moment at support A starts out as negative and gradually reduces until the load reaches the middle point, where it changes to positive as the right support takes more of the load. The maximum fixing moment occurs when the load is at 100mm from the support. The plot of the fixing moment at support A shows that there is a close correlation between the experimental and calculated values, as is evidenced by the closeness of the two plots. 3.4 Inferences If the supports move outward out in proportion to the horizontal reaction, his has the effect of turning the support into a roller support. A roller support does not develop horizontal reaction, thus the value is reduced to zero. The roller support also does not allow moments to build up, causing the fixing moment value to reduce to zero as well. Imperfections in the geometry of the experimental apparatus could result in the internal stresses not being distributed as is expected, causing self-stressing. The fixed arch can be used in the construction of bridges with short spans. This is because the arch develops internal stressing, which could be catastrophic over long spans. Conclusion The experiment has proven that there is a good correlation between the experimentally measured and theoretically calculated values for both the two-pinned arch and the fixed arch. The horizontal reaction at the right support increases as the load nears the mid- span and decreases afterwards. The load was found to have the greatest influence value on the horizontal force when at the mid- span. The two-pinned arch does not have any fixing moment. The plot of the fixing moment on the fixed arch resembles a wave, increasing negatively when the load starts moving, then increasing to reach zero close to the middle, and eventually reducing again as the load nears the other support. References Cli11: , (Clive & Williams, 2011), Vai04: , (Vaidyanathan & Perumal, 2004), Amb12: , (Ambrose, 2012), Rey08: , (Reynolds, 2008), Lue16: , (Luebkeman, 1998), Read More