Laboratory Analysis of Fixed-Arch - Assignment Example

ASSIGNMENT COVER SHEET Electronic or manual submission UNIT ENS6149 Structural Analysis NAME OF STUDENTS HAFSIA Mehdi NGUYEN Sy Tho TAJINDER Singh STUDENT ID NO. 10373208 10378888 10380149 NAME OF LECTURER Dr Alireza Mohyeddin DUE DATE 18/05/2015 Laboratory Report – Fixed Arch Group or tutorial (if applicable) Group 50 Course i59 CAMPUS JOONDALUP I certify that the attached assignment is my own work and that any material drawn from other sources has been acknowledged. This work has not previously been submitted for assessment in any other unit or course. Copyright in assignments remains my property. I grant permission to the University to make copies of assignments for assessment, review and/or record keeping purposes. I note that the University reserves the right to check my assignment for plagiarism. 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Agreement X Date 18/05/2015 FOR PROCEDURES AND PENALTIES ON LATE ASSIGNMENTS PLEASE refer to the University Admission, Enrolment and Academic Progress Rule 24, and the ECU Course and Unit Delivery and Assessment Policy Structural Analysis Laboratory Report Fixed Arch Group 50 Name Student # Signature Mehdi HAFSIA 10373208 Sy Tho NGUYEN 10378888 TAJINDER SINGH 10380149 TABLE OF CONTENTS 1 Abstract 1 2 Introduction 1 3 Material and Methodology 1 3.1 Material 1 3.2 Methodology 2 4 Results and discussion 3 5 Further investigations 5 5.1 Self-Stressing of the fixed arch 5 5.2 Impact of a horizontal displacement of one support 5 6 Conclusion 7 TABLE OF FIGURES Figure 1: Experimental apparatus 2 Figure 2: HB against the distance from end A to the point of application of the load 4 Figure 3: MA against the distance from end A to the point of application of the load 5 Figure 4: HB against the distance from end A of the applied load for span lengths of 500 mm (blue) and 505 mm (red) 6 Figure 5: MA against the distance from end A of the applied load for span lengths of 500 mm (blue) and 505 mm (red) 6 TABLE OF TABLES Table 1: Experimental and theoretical results for the reaction HB and fixing moment MA 3 1 Abstract The experiment entails loading a fixed parabolic aluminum arch by a 500 grams weight along its span. The fixing moment at end A (the left-hand side) and the horizontal reaction at end B (the right-hand side) supports are recorded for the individual positions of the weight. The experiment then entails comparing the results with those obtained from the derivation of the secant assumption formula. The report also discusses the discrepancies associated with the experiment. 2 Introduction The primary purpose of arches is to reduce the bending moments in long-pan structures. However, the rising of the arches creates a horizontal reaction at the supports even when loaded by a shear force. The case also studies a fixed arch. As a result, there is the creation of a fixing moment at the supports. The experiment involves the recording of the horizontal reaction (HB) at end B of the arch. The experiment also entails the measuring of the fixing moment (MA) at end A. The case then compares the reactions with the theoretical value obtained from secant assumption formula. The report also discusses the relevance of the theoretical formula. 3 Material and Methodology 3.1 Material The TecQuipment® device for the fixed arch study shown on Figure 1 has been used to conduct the experiment. The device has provisions for the recording of MA and HB for different positions of the hanging weight along the span. The characteristics of the arch used in the experiment as follows: Span: L = 0.500 mm. Pitch (distance between the different points of application of the load along the span): p = 50 mm. Hanging weight: W = 0.500*9.81 = 4.9 N Rise: r = 100 mm Figure 1: Experimental apparatus 3.2 Methodology The aim of the experiment is to study the relationship between the position of a load point W along the arch’s span from end A and the MA and HB values. Following the derivation of the secant assumption formula, the formula results in the following relationships in Equations 1 and 2, with: a: distance of the load from end A. b: distance of the load from end B. Equation 1: Secant Assumption Formula derivation for calculation of HB Equation 2: Secant Assumption Formula derivation for calculation of MA The experiment does not measure the fixing moment directly at end A. the moment force created by an arm on a force sensor is recorded. One can find the fixing moment at point A by multiplying the length (h) of the momentum arm by the moment force as shown in Equation 3. Equation 3: Calculation of experimental MA based on the recorded moment force The negative sign in front of h accounts for the sign of the moment. 4 Results and discussion Table 1 below shows the experimental results for MA (MA,exp) and HB (HB,exp) and the theoretical ones (MA,theo and HB,theo respectively) calculated using Equations 1 and 2. The table also includes the percentage deviation between theoretical and experimental results. Table 1: Experimental and theoretical results for the reaction HB and fixing moment MA Distance from end A (mm) Display moment force (N) Experimental fixing moment (MA,exp) (Nm) Theoretical fixing moment (MA,theo) (Nm) deviation between experimental and theoretical MA Experimental horizontal reaction at end B (HB,exp) (N) Theoretical horizontal reaction at end B (HB,theo) (N) deviation between experimental and theoretical HB 0 0 0 0 0.0% 0 0 0.0% 50 3.2 -0.16 -0.15 6.7% 0.8 0.74 8.1% 100 3.2 -0.16 -0.16 0% 2.5 2.35 6.4% 150 1.9 -0.095 -0.09 5.6% 4.3 4.05 6.2% 200 0.9 -0.009 0 0.0% 5.5 5.30 3.8% 250 -1.4 -1.55 0.08 2037.5% 5.9 5.75 2.6% 300 -2.0 -2.05 0.12 1808.3% 5.4 5.30 1.9% 350 -2.0 -2.05 0.12 1808.3% 4.2 4.05 3.7% 400 -1.5 -1.55 0.08 2037.5% 2.5 2.35 6.4% 450 -0.4 -0.45 0.03 1600% 0.1 0.74 86.5% 500 0 0 0 0.0% 0 0 0.0% The table shows a positive correlation between the calculated and experimental values of HB. The noted deviations are less than 9%. The placement of the load closest to the supports A and B results in the maximum deviation. This represents the case when HB is the smallest that turns out to be the normal case. A reaction of 0.8N results in an error of 0.1 that yields a deviation of 8.1%. The small error for HB equals to 4.2 N and it yields a deviation only 3.7%. However, the HB calculated values are smaller than the measured values. As a result, the use of the formula for design purposes may be problematic taking into account the fact. Overcoming the problem necessitates the use of an adequate safety factor. The conclusions are similar for the fixing moment MA. The maximum deviation is 6.7%. It occurs for a low MA value. However, a great number of the calculated fixing moments associated with MA are greater than the measured ones thereby exhibiting the conservative aspect of the formula. Figures 2 and 3 below represent the plots for MA and HB against the distance from A respectively; for both calculated and experimental results. Figure 2: HB against the distance from end A to the point of application of the load Figure 3: MA against the distance from end A to the point of application of the load The graphs confirm the observations made regarding the Table 1 results. The formula predicts that the application of a load at the arch’s crown yields the maximum value of HB. The experimental results in Figure 2 verify the prediction. Both the theoretical and experimental results shown in Figure 3 reveal that placing the load between 50 millimeters and 100 millimeters from end A produce the maximum MA. The results also show that the application of a load 200 millimeters (40% of the span length) from end A yields a zero MA. The interpolation performed between the experimental points exhibits the tendency of putting the maximum MA for the load at 60 millimeters from end A. On the other hand, the interpolation between the calculated results points yields a maximum MA for a load at 75 millimeters from end A. It is proper to conclude that the maximum fixing moment given by the formula occurs for a load at the same location as compared to the fixing moment measured during the experiment. Therefore, it is evident that the secant assumption formula provides an accurate prediction of the behavior of the arch. 5 Further investigations 5.1 Self-Stressing of the fixed arch It is proper to understand that the arch; being fixed at both ends, has the likelihood of undergoing self-stressing. Self-stressing refers to the condition in which a structure in a state of equilibrium, devoid of any applied external forces develops internal forces. The temperature gradient provides one common cause of self-stressing witnessed in metallic structures. An increase in temperature results into the expansion of the arch material. Since the arch is fixed, the support constrains the expansion. As a result, the arch undergoes a compressive stress emanating from the change in temperature in the absence of any external load applied to the metallic structure. 5.2 Impact of a horizontal displacement of one support Investigating the influence that a slight horizontal outward displacement of the support will have on MA and HB may necessitate the use of the secant assumption formula for an increase of the span length by 5 millimeters followed by the comparison to the theoretical results obtained in Table 1 for the original span. Figures 4 and 5 respectively represent plots for HB and MA for span lengths of 500 millimeters and 505 millimeters. Figure 4: HB against the distance from end A of the applied load for span lengths of 500 mm (blue) and 505 mm (red) Figure 5: MA against the distance from end A of the applied load for span lengths of 500 mm (blue) and 505 mm (red) The plots of both figures show a small deviation between the results for 500 millimeters and 505 millimeters. The deviations are less than 5%. Moreover, the maximum MA and HB deviation between the two cases is only 2%. The implication is that there is no significant impact created by a small outward displacement of the supports on the values of the reactions (fixing moment and horizontal reaction). However, the displacements reduce the arch’s rise. The decrease of the rise of 6mm and the horizontal displacement also make use of the same calculation. From the results, it is evident that there is an increase of approximately 7% for both the fixing moment and the horizontal reaction. The conclusions then remain as before. In actual practice, the installation of affixed arch requires the consideration of the soil onto which the arch is installed as the fixed support. As a result, it is proper to install a fixed arch on a hard soil such as rocks to avert support displacements. Nevertheless, as noted in the experiment, a small displacement on the support does not have a significant impact on the reactions. As a result, a fixed arch installed on a soft soil such as sand retains its integrity provided that the displacement of the support is small. 6 Conclusion The experiment has affirmed that there is a positive correlation between the calculated values of the fixing moment and horizontal reactions using the secant assumption formula at the supports of a fixed arch and the experimental reactions obtained from the experiment. Read More