Group 4, Lab 9

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California State University, Northridge *

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Mechanical Engineering

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Feb 20, 2024

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Fall 2023 California State University, Northridge Experiment 9 Two Pinned Arch Mechanics Lab AM 317 Written By:

Abstract: Thrust force refers to the horizontal force exerted by a bridge or an arch on its supports. For engineers, it is critical to understand the stability and equilibrium states of arches and bridges because it determines the load that they can carry. One of the main things to consider when analyzing arches is the thrust line, the thrust line is a line along which the resultant thrust force acts withing the arch. It is not a physical line that is there but on that the forces in the arch act around. The thrust line should be within the middle third of the arch thickness to ensure stability and overturning. Another thing to consider is the stability of an arch. Arch stability depends on the distribution of forces and the construction of the arch. The thrust force must stay within the supports of the arch to prevent it from collapsing. In order to account for arch stability, the foundations of the arch are required to resist the horizontal component of the thrust force. The main consideration that we are going over in the lab is the direction and force of the thrust force. Thrust force on an arch typically pushes horizontally away from the center of the arch and pushes outwards against the supports. In a well built arch, the vertical component and horizontal components are resolved. The vertical components supports the weight of the arch itself, as well as the applied loads. The horizontal component pushes outwards against the supports of the arch. In the lab, we tested the thrust forces of a fully loaded arch and an arch where readings were taken at key points. What we found was a very similar outcome to what we hoped to see when we calculated the thrust force and when we measured the thrust force.

Introduction: Analyzing thrust forces in arches is fundamental in the field of engineering and structural engineering. In order to design a same and stable arch, engineers use a combination of analytical and measured methods to ensure that an arch can support its own weight and the loads without experiencing deformations or even failure. One of the main things to consider when analyzing arches is the thrust line, the thrust line is a line along which the resultant thrust force acts withing the arch. It is not a physical line that is there but on that the forces in the arch act around. The thrust line should be within the middle third of the arch thickness to ensure stability and overturning. Another thing to consider is the stability of an arch. Arch stability depends on the distribution of forces and the construction of the arch. The thrust force must stay within the supports of the arch to prevent it from collapsing. In order to account for arch stability, the foundations of the arch are required to resist the horizontal component of the thrust force. In this lab we hope to find the correlation with the calculated thrust force and the measured thrust force. We will be using the following equipment, the structures test frame, in order to hold the frame of all the equipment, the digital force display and the power supply in order to get a reading of the forces applied to the structures test frame, the aluminum arch itself, in order to actually apply the loads, the hangers and weights, in order to apply the loads onto the aluminum arch, and the scale, which will read out the forces and transfer its readings to the force display.

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Using these items and the formulas in the theory section, we hope to see a correlation between the measured values and the calculated values.

Theory: There are a handful of equations that we need to know in order to find the calculated thrust forces for this lab. Because this is a controlled experiment, the equation for finding the theoretical thrust force is constant, however, in the real world the equation may become more complex. The equation for finding the horizontal thrust force for a given load is as follows, where H B , the horizontal thrust reaction at B (N). is equal to 5 times P, the load (N), times x, the load location, distance from the left-hand side support (m), over 8 times r, the rise of the arch in meters, times L 3 , the span of the arch (m) cubed, all times L 3 again, plus x 3 minus 2 times L times x 2 . H B = 5 Px 8 r L 3 ( L 3 + x 3 − 2 L x 2 ) The equation above works for the first part of the lab, where the weights were applied one at a time and changed places depending on where the load was to be measured. That was done to simulate a single large load placed on different parts of the arch. The next equation is for when all of the hangers are loaded with weight and simulating an evenly distributed load. The equation for finding the horizontal thrust load when the load is equally distributed is as follows, H B , the horizontal thrust reaction at B (N), is equal to w, the intensity of uniform distributed loads (N/m), times L 2 , the span of the arch (m) squared over 8 times r, the rise of the arch (m). Using those two formulas along with the measurements of the arch, we hope to both measure and calculate the horizontal thrust load on the arch. We also expect to see a curved graph that should be similar between loads.

Procedures: The procedures in the lab were quite simple but tedious and time consuming. Overall though, we were able to complete the lab without much hassle, and where we did run into issues, we were able to resolve them quickly. To start the lab, we first had to plug in the power supply into the force display. We then measured the dimensions of the two-pinned arch. We took the measurements of its length and its height at its apex and recorded them in our table to later calculation. We then went on to load the hangers for the experiment. Initially, all the weights and the hangers were haphazardly placed back into the supply drawer, so it was a bit of an issue sorting that out. Once we had gotten the supplies sorted, we began loading the weight pan. The first part of the experiment had us loading the mid span hanger with a 500-gram preload. We loaded the weight pan and placed it on the mid span hanger. We then zeroed the digital force reader and prepared more weight pans, this time with 100 grams total. We then, with the 500-gram preload still on the midspan hanger, placed the 100-gram weight at each of the hangers, recording what the digital force reader read out. While recording we noticed that as we placed and removed the weight pans, the force of us moving the weight pans would affect the zero reading on the digital force scale. To mitigate this, we ended up zeroing the digital force scale each time a weight pan was removed. Doing this led us to having more consistent reading. Once we had finished this process, we repeated it for a 500-gram weight and recorded the results. For the next step, we removed the 500-gram preload and loaded all of the hangers at once with a 70-gram load. Doing this was tedious as we had to count and load all of the weight pans.

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We made sure to zero the digital force reader while there were no weights on an of the hangers. Once all the 70 grams weights were installed, we recorded the digital force reading and repeated the procedure with 140-gram weights. Once all the testing was done, we unplugged all the equipment and put everything back neatly into the supply drawer.

Results: Table 1: Arch Data Rise of Arch, r (m) 0.10 Length of arch, L (m) 0.51 Table 2: Weight Data Added mass (g) 100 500 Hanger mass (g) 15 15 Total mass 115 515 Total load (N) 1.12776475 5.05042475 Table 3: Data for a point load moving across a two-pinned arch. Load location x (m) Measured thrust force for 100 g load (N) Calculated thrust force for 100 g load (N) Measured thrust force for 500 g load (N) Calculated thrust force for 500 g load (N) 0 0.00 0 0.00 0 0.05 0.20 0.345983757 1.10 1.549405521 0.1 0.40 0.6559679448 1.70 2.937595579 0.15 0.60 0.9012591797 2.20 4.036073718 0.2 0.70 1.061134462 2.50 4.75203694 0.25 1.00 1.122841178 2.70 5.02837571 0.3 0.70 1.081597096 2.60 4.843673951 0.35 0.60 0.9405903696 2.00 4.212209046 0.4 0.40 0.7109795375 1.60 3.183951842 0.45 0.20 0.4118935219 1.10 1.844566642 0.5 0.00 0.07043162962 0.00 0.3154112109 Table 4: Data for a uniformly distributed load Total mass (g) Total mass of hangers (g) Uniform distributed load (N/m) Measured thrust force (N) Calculated thrust force (N) % Error 630 745 14.32540049 4.6 4.657545834 1.235539841 1260 1395 26.82407206 8.3 8.721176428 4.829353374

Graph 1: Measured Thrust Force (100g) 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.2 0.4 0.6 0.8 1 1.2 Measured Thrust Force (100g) Location Thrust Force Graph 2: Calculated Thrust Force (100g) 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.2 0.4 0.6 0.8 1 1.2 Calculated Thrust Force (100g) Location Thrust Force

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Graph 3: Measured Thrust Force (500g) 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.5 1 1.5 2 2.5 3 Measured Thrust Force (500g) Location Thrust Force Graph 4: Calculated Thrust Force (500g) 0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 2 3 4 5 6 Calculated Thrust Force (500g) Location Thrust Force

Discussion and Conclusions: For the most part, we saw in our graphs and tables the results that we expected to see from the lab. In Table 1, we recorded the values of the rise of the arch in meters as well as the length of the arch in meters. While the length of the arch was easy enough to measure, the rise of the arch was a little difficult to measure as we were measuring the top of the arch to the “bottom” which was empty space below the middle. To remedy this, we placed a ruler at either end of the supports and measured the vertical distance to from the top of the arch to the meter stick. In Table 2, we recorded the values that we got from the first part of the lab, using one hanger and multiple different points. There we also found the weight of the hanger and took that into account when doing calculations. In Table 3, we recorded our measured values as well as our calculated values. For the most part, our calculated values were higher than our measured values. They did, however, see a similar rise and fall with as the location on the two pinned arch changes. The graphs of the measured and calculated values are later in the report. In Table 4, we recorded the values of force when the arch was fully loaded. Our values for each were very close. The lighter loaded arch had a 1.24% error while the heavier loaded arch had a 4.83% error.

In the graphs based on Table 3, we saw a general curve on all the locations of the arch. In all the cases save for one in the measured thrust force for the 100-gram load, we saw a consistent curve.

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