DEFLECTION OF CURVE BARS AND DAVITS TITLE: DEFLECTION OF CURVE BARS AND DAVITS 1.0 OBJECTIVE To investigate the relationships between load, horizontal deflection and vertical deflection for a curved davit, an angled davit, a semicircle structure and quarter-circle structure. 2.0 INTRODUCTION Figure 1 shows the Curved Bars and Davits experiment. It consists of a back plate, a pair of dial indicators arranged at 90°, and one of four test structures. The two indicators are on a magnetic base. We can move the base to any position on the back plate. One of the indicators measures horizontal deflection, the other vertical deflection. The four structures are a quarter circles, a semicircle, a curved davit and an angled davit. Each structure has a boss fitted to the free end. This allows us to apply loads and measure deflections in both the horizontal and vertical directions. Printed on the back plate of the equipment is some useful information. Make a note of this - we will need it to analyze our results after we have completed the experiment. Figure 1 3.0 THEORY 1 MAKMAL PEPEJAL DEFLECTION OF CURVE BARS AND DAVITS We have seen that the elongation of a single truss member, when load is applied, is usually quite insignificant. However, when the accumulative effect of elongation in all members of a structure is taken into account, the displacements of some parts of the structure are found to be appreciable. When a beam is subjected to lateral loading, the deflection is usually much larger than for an axially loaded member. We have an interest in beam deflection because the design of floor beams is frequently controlled by the limit on deflection imposed by a code rather that by the allowable stress. We must also understand beam deflection in order to solve certain statically indeterminate problems that occur in beam design. In engineering mechanics, deflection is a term that is use to describe the degree to which a structural element is displaced under a load. The deflection of a member under a load is directly related to the slope of the deflected shape of the member under that load and can calculated by integrating the function that mathematically describes the slope of the member under that load. Deflection can be calculated by standard formula (will only give the deflection of common beam configurations and load cases at discrete locations), or by methods such as “virtual work”, “direct integration”, “Castigliano’s method”, “Macaulay’s method” or the “Matrix stiffness method” amongst others. An example of the use of deflection in this context is in building construction. Architects and Engineers select materials for various applications. The beams used for frame work are selected on the basis of deflection, amongst other factors. The elastic deflection f and angle of deflection φ (in radians) in the example image, a (weightless) cantilever beam, can be calculated (at the free end) using: fB FL3 3EI B FL2 2 EI Where, 2 MAKMAL PEPEJAL DEFLECTION OF CURVE BARS AND DAVITS F = force acting on the tip of he beam L = length of the beam (span) E = modulus of elasticity I = area moment of inertia The deflection at any point along the span can be calculated using the above-mentioned methods. From this formula it follows that the span L is the most determination factor, if the span doubles, the deflection increases. Building codes determine the maximum deflection, usually as a fraction of span e.g. 1/400 or 1/600. Either the strength limit state (allowable stress) or the serviceability limit state (deflection considerations amongst others) may govern the minimum dimensions of the member required. The deflection must be considered for the purpose of the structure. When designing a steel frame to hold a glazed panel, one allows only minimal deflection to prevent fracture of the glass. The deflective shape of a beam can be represented by the moment diagram, integrated. Deflection occurs when an object hits a plane surface. In physics deflection is the event where an object collides and bounces against a plane surface. Deflection is also a tactic used in battle that describes “leading the target” that is shooting ahead of a moving target so that the target and projectile will collide. This tactic is only necessary when using slow projectiles such as a crossbow bolt, or over long distances, such as in aerial dogfight. CAUTION: The most common mistake in computing deflection is caused by using "w" as load per foot instead of load per inch. The derivation of the deflection formulas uses the unit of inches for all the factors in the formula. Uniformly distributed beam loads "w" are normally described as load per foot which must be converted to the proper load per inch value before insertion into the deflection formula. If "W" is used for the total distributed load instead of the load per unit of length ("w") this conversion is not necessary. 3 MAKMAL PEPEJAL DEFLECTION OF CURVE BARS AND DAVITS 4.0 EQUIPMENT For student to investigate two common structures and two common davit structures Curve bar and davit equipment Curved davit deflections 4 MAKMAL PEPEJAL DEFLECTION OF CURVE BARS AND DAVITS 5.0 PROCEDURE 1. Referring to Figure 1, set up the equipment to test the semicircle first. Ensure you have mounted the semicircle with a plate each side of the end, and you have damped in securely. 2. Measure and record the breadth and, depth of the section checking several places on the structure and taking an average. 3. Clip the weight hanger onto the two lugs on the loading boss. Gently pull down on the weight hanger and note the direction the loading boss on structure moves. 4. Set the indicator positions so they contact the horizontally and vertically and have the maximum amount of travel in each direction. 5. Carefully zero the indicators 6. Apply a mass of 100g to the hanger; tap the test frame to reduce the effects of friction then take readings of both indicators. 7. Repeat with masses up to 500g in 100g increments tapping the test frame each time. 8. From the measurements of the section calculate the second moment of area ‘I’ Enter all of your results and values into Table 1. 9. Remove the semicircle and attaching it to the side of the frame rather than the bottom member, replace it with the quarter circle. 10. Ensure there are clamp plates each side of the structure and the indicator positions give the amount of travel needed for the maximum loading. 11. Repeat the experiment. Similarly repeat the experiment for the curved davit and the angle davit. 12. Enter all results into Table 1. Calculate the ‘I’ value for each structure as the manufacturing process may change the thickness and width of the material. 13. Plot graphs for each section, with load versus the horizontal and vertical deflection. Calculate the gradient of each line in mm/N. Compare this to values calculated form the standard formulae for each section or those calculated from first principles 5 MAKMAL PEPEJAL DEFLECTION OF CURVE BARS AND DAVITS 6.0 DATA Breadth Breadth Breadth 1 2 3 (mm) (mm) (mm) Semicircle Deflection Quarter circle Deflection Curved Davit Deflection Angle Davit Deflection Depth Depth Depth 1 2 3 (mm) (mm) (mm) Average depth (mm) 19.20 19.20 19.20 19.20 3.50 3.50 3.50 3.50 19.20 19.20 19.20 19.20 3.50 3.50 3.50 3.50 19.19 19.10 19.20 19.16 3.15 3.10 3.00 3.08 19.00 19.00 19.00 19.00 3.20 3.50 3.30 3.33 Mass(g) Load(N) 0 100 200 300 400 500 Average breadth (mm) 0 0.98 1.96 2.94 3.92 4.90 I value (m4) Semicircle Deflection (mm) V H 0 0.82 1.73 2.87 4.09 5.46 0 0.48 2.01 3.45 4.91 6.78 6.86x10-11 Quarter circle Deflection (mm) V H 0 0.63 1.37 2.21 2.45 3.38 0 0.49 1.03 1.60 1.76 2.40 6.86x10-11 Curved Davit Deflection (mm) V H 0 0.24 0.55 0.98 1.22 1.74 0 0.35 0.56 1.52 1.89 2.74 4.6652x10-11 Angle Davit Deflection (mm) V H 0 0.13 0.47 0.74 0.91 1.19 0 0.23 0.91 1.46 1.79 2.38 5.8466x10-11 NOTE: All the data shown above, get from calculation. 6 MAKMAL PEPEJAL DEFLECTION OF CURVE BARS AND DAVITS 7.0 CALCULATION The following information as well as that printed on the back plate is: type radius ∆H ∆V 2PR3 π PR3 R=150mm EI 2EI R=150mm PR3 π PR3 2EI 4EI PRL (2R + L) PR2 (4L+π R) 2EI 4EI PL1L2(0.707L1+L2) + PL23 PL22 (3L1+L2) Semicircle deflection Quarter circle deflection Curved davit deflections R=75mm L L=150mm Angel davit deflections 45o L2 R=105mm L1=150mm L1 L2=105mm 2EI 6EI 6EI For a rectangular section: I =bd 3 /12 Semicircle deflection 7 MAKMAL PEPEJAL DEFLECTION OF CURVE BARS AND DAVITS R = 150mm E 69GNm 2 b = 19.20mm d = 3.50mm I bd 3 12 19.20 x10 3 3.50 x10 3 3 12 6.86 x10 11 m 4 V P N PR 3 2 EI 2 PR 3 EI V 8 H MAKMAL PEPEJAL DEFLECTION OF CURVE BARS AND DAVITS 0.98 150 x10 3 2 69 x10 9 6.86 x10 11 3 0.98 1.0976 x10 3 1.3975 x10 m 1.96 150 x10 3 2 69 x10 9 6.86 x10 11 3 1.96 2.1952 x10 3 2.94 150 x10 3 2 69 x10 9 6.86 x10 11 3.2928 x10 3 4.90 3 3 3 3 3 3 m 2 2.94 150 x10 3 69 x10 9 6.86 x10 11 3 m m 3.92 150 x10 3 2 69 x10 9 6.86 x10 11 4.3904 x10 3 m 5.5900 x10 3 m 2 3.92 150 x10 3 69 x10 9 6.86 x10 11 4.90 150 x10 3 2 69 x10 9 6.86 x10 11 5.4880 x10 3 m 6.9876 x10 3 m 3 21.96 150 x10 3 69 x10 9 6.86 x10 11 4.1925 x10 m 3 3.92 3 2.7950 x10 m 3 2.94 2 0.98 150 x10 3 69 x10 9 6.86 x10 11 2 4.90 150 x10 3 69 x10 9 6.86 x10 11 Quarter circle 9 MAKMAL PEPEJAL DEFLECTION OF CURVE BARS AND DAVITS R=150mm b=18.20mm d = 3.20mm E 69x10 9 I bd 3 12 19.20 x10 3 3.50 x10 3 3 12 6.86 x10 11 m 4 V PR 3 4 EI PR 3 2 EI V 10 MAKMAL PEPEJAL DEFLECTION OF CURVE BARS AND DAVITS P N 0.98 150 x10 3 4 69 x10 9 6.86 x10 11 2 69 x10 9 6.86 x10 11 0.5488 x10 3 m 0.3494 x10 3 m 3 0.98 1.96 150 x10 3 4 69 x10 9 6.86 x10 11 3 1.96 1.0976 x10 3 1.6464 x10 3 3 2 69 x10 9 6.86 x10 11 2.1952 x10 3 m 1.3975 x10 3 m 3.92 150 x10 3 3 4.90 150 x10 3 4 69 x10 9 6.86 x10 11 2 69 x10 9 6.86 x10 11 2.7440 x10 3 m 1.7470 x10 3 m Curved Davit m 3.92 150 x10 3 4 69 x10 9 6.86 x10 11 m 2.94 150 x10 3 3 1.0481x10 m 3 4.90 2 69 x10 9 6.86 x10 11 3 3.92 1.96 150 x10 3 3 0.6988 x10 m 2.94 150 x10 3 4 69 x10 9 6.86 x10 11 3 2 69 x10 9 6.86 x10 11 3 2.94 0.98 150 x10 3 3 4.90 150 x10 3 3 Deflection 11 MAKMAL PEPEJAL DEFLECTION OF CURVE BARS AND DAVITS R = 75mm L = 150mm b=19.16mm d = 3.08mm E = 69 GNm-2 bd 3 12 I 19.16 x10 3 3.08 x10 3 12 4.6652 x10 V m 3 4 PR 2 4 L R 4 EI P N 11 PRL 2R L 2 EI V 0.98 12 MAKMAL PEPEJAL DEFLECTION OF CURVE BARS AND DAVITS 0.98 75 x10 3 2 4 69 x10 9 4.6652 x10 11 4150 x10 75 x10 3 3 0.2663 x10 3 m 1.96 75 x10 3 2 1.96 4 69 x10 9 4.6652 x10 11 4150 x10 75 x10 3 3 0.5326 x10 3 m 2.94 75 x10 3 2 2.94 4 69 x10 9 4.6652 x10 11 4150 x10 75 x10 3 3 0.7989 x10 3 m 3.92 75 x10 3 2 3.92 4 69 x10 9 4.6652 x10 11 4150 x10 75 x10 3 3 1.0652 x10 3 m 4.90 75 x10 3 2 4.90 4 69 x10 9 4.6652 x10 11 4150 x10 75 x10 3 3 1.3314 x10 3 m 0.98 75 x10 3 150 x10 3 2 69 x10 9 4.6652 x10 11 2 75 x10 150 x10 3 3 0.5137 x10 3 m 1.96 75 x10 3 150 x10 3 2 69 x10 9 4.6652 x10 11 2 75 x10 150 x10 3 3 1.0275 x10 3 m 2.94 75 x10 3 150 x10 3 2 69 x10 9 4.6652 x10 11 2 75 x10 150 x10 3 3 1.5412 x10 3 m 3.92 75 x10 3 150 x10 3 2 69 x10 9 4.6652 x10 11 2 75 x10 150 x10 3 3 2.0550 x10 3 m 4.90 75 x10 3 150 x10 3 2 69 x10 9 4.6652 x10 11 2 75 x10 150 x10 3 3 2.5687 x10 3 m Angle Davit Deflection 13 MAKMAL PEPEJAL DEFLECTION OF CURVE BARS AND DAVITS L1 = 150mm L2 = 105mm b = 19.00mm d = 3.33mm E = 69 GNm-2 bd 3 12 I 19.00 x10 3 3.33 x10 3 12 5.8466 x10 V P N 11 m 3 4 PL22 3L1 L2 6 EI PL1 L2 0.707 L1 L2 PL32 2 EI 6 EI V 0.98 14 MAKMAL PEPEJAL DEFLECTION OF CURVE BARS AND DAVITS 0.98 0.15 0.105 0.707 0.15 0.105 2 69 x10 9 5.8466.x10 11 6 69 x10 5.8466 x10 3x150 x10 3 105 x10 3 0.98 0.105 3 0.2477 x10 3 m 6 69 x10 9 5.8466 x10 11 0.98 105 x10 3 2 9 11 0.4506 x10 3 m 1.96 0.15 0.105 0.707 0.15 0.105 2 69 x10 9 5.8466.x10 11 6 69 x10 5.8466 x10 3x150 x10 3 105 x10 3 1.96 0.105 3 0.4955 x10 3 m 6 69 x10 9 5.8466 x10 11 1.96 105 x10 3 2 9 1.96 11 0.9013 x10 3 m 2.94 0.15 0.105 0.707 0.15 0.105 2 69 x10 9 5.8466.x10 11 6 69 x10 9 5.8466 x10 11 3x150 x10 3 105 x10 3 2.94 0.105 3 0.7432 x10 3 m 6 69 x10 9 5.8466 x10 11 2.94 105 x10 3 2 2.94 1.3519 x10 3 m 3.92 0.15 0.105 0.707 0.15 0.105 2 69 x10 9 5.8466.x10 11 6 69 x10 9 5.8466 x10 11 3x150 x10 3 105 x10 3 3.92 0.105 3 0.9910 x10 3 m 6 69 x10 9 5.8466 x10 11 3.92 105 x10 3 2 3.92 4.90 105 x10 3 2 4.90 6 69 x10 5.8466 x10 3x150 x10 105 x10 9 3 11 1.8026 x10 3 m 4.90 0.15 0.105 0.707 0.15 0.105 2 69 x10 9 5.8466.x10 11 3 4.90 0.105 3 6 69 x10 9 5.8466 x10 11 3 1.2387 x10 m 2.2532 x10 3 m Difference ratio Difference ratio = theory value – experiment value x 100% Theory value 15 MAKMAL PEPEJAL DEFLECTION OF CURVE BARS AND DAVITS Parts P= 4.90N V Semicircle deflection Quarter deflection Curved Davit deflection Angle Davit Deflection 5.4880 5.46 100% 5.4880 0.5% 2.7440 3.38 100% 2.7440 23.18% 1.3314 1.74 100% 1.3314 30.69% 1.2387 1.19 100% 1.2387 3.93% H 6.9876 6.78 100% 6.9876 2.97% 1.7470 2.40 100% 1.7470 37.38% 2.5687 2.74 100% 2.5687 6.67% 2.2532 2.38 100% 2.2532 5.63% 9.0 OBSERVATION 16 MAKMAL PEPEJAL DEFLECTION OF CURVE BARS AND DAVITS From the experiments that have been done, we found that there was a deflection at the horizontal and vertical axis. When the load had been applied to the structure; the indicator gave a value for both of the axis. The graph plotted is linear, when the more load applied to the structure using hangers and mass, there are more deflection happened. From collected data, found that there is a little bit different between theory and experiment. The most deflection was from semicircle structure on the horizontal and the lowest deflections value came from quarter circle structure. Experiment value has a different from calculation value. The differences percentage is 0.5% for the vertical axis and 2.97% for the horizontal axis for semicircle deflection. For quarter circle deflection is 23.18% is for vertical axis and 37.38% for the horizontal axis. Meanwhile for curved davit deflection for the vertical axis is 30.69% and for horizontal axis is 6.67%. For the angle davit deflection, the result for differential between calculation value and experiment value is 3.93% for vertical axis and 5.63% for horizontal axis. 10. RECOMMENDATION 17 MAKMAL PEPEJAL DEFLECTION OF CURVE BARS AND DAVITS The valued from the experiment and calculation have a bit differences. This is because of some error had happened when the experiment been done. This was happen because of some rectification factor. The factors are: i) ii) iii) It should be that the two digital deflection indicators is not really set at 90° to each other on the back-board The experiment equipment place on the Jared table, and this influence the reading of digital deflection indicators. The experiment equipment not complete. Solution: i. It should have the gadget to make sure the indicator is set in 90°. ii. Do the experiment on the floor or on the stable place to avoid it from jar. iii. There should be has a computer to transfer the data to get graph. 11. CONCLUSION 18 MAKMAL PEPEJAL DEFLECTION OF CURVE BARS AND DAVITS From the experiment, our objective which is to investigate the relationships between load, horizontal deflection and vertical deflection for a curved davit, an angled davit, a semicircle structure and quarter-circle structure are successful. There have the differences of deflection between for a curved davit, an angled davit, a semicircle structure and quarter-circle structure. The graph plotted is linear, when the more load applied to the structure using hangers and mass, there are more deflection happened. Even though there is a little bit difference between theoretical and experimental result. This is because there are some errors occur as we mention in recommendation. 12.0 REFERENCES 19 MAKMAL PEPEJAL DEFLECTION OF CURVE BARS AND DAVITS 1. Mechanic of material, sixth Edition SI unit, R.C Hibbeler. 2. Engineering mechanic, third revised edition, Tata McGraw-hill 3. Engineering Mechanic, static and dynamic, third edition McGraw-Hill book company, Singapore. 4. Engineering Mechanic( si unit), S.C Mathur, Fourth Edition, S.K. KATARIA & SON. 5. www.tqstructure.com 20 MAKMAL PEPEJAL