- Part 1 Deflection Of Simply Supported Beam With Point Load Consider The Simply Supported Beam Subjected To A Concentra 1 (68.91 KiB) Viewed 61 times
- Part 1 Deflection Of Simply Supported Beam With Point Load Consider The Simply Supported Beam Subjected To A Concentra 2 (101.49 KiB) Viewed 61 times
- Part 1 Deflection Of Simply Supported Beam With Point Load Consider The Simply Supported Beam Subjected To A Concentra 3 (79.45 KiB) Viewed 61 times
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Part 1 - Deflection of Simply Supported Beam with Point Load Consider the simply supported beam subjected to a concentrated load P as shown. Let E and I denote Young's modulus and the second moment of area of the beam, respectively. Р A B 11 L Using the double integration method, the equations of the elastic deflection curve y(x) can be derived as follows: x <a Pb y.(x) = b x 2 a Pa y2(x) = [-x3 + 3Lx2 - (a? + 2L2)x+ a²L] 6 EIL 6 EIT [x3 – (L? – 62)x] - x L Based on the above equations, find the expressions of 0.(x) and 02(x). 1.b. Verify that: Pb(L2 - 62) Pa(L? - a) 0A = and @g = + 6 EIL 6 EIL It is known that: • Ifa > b, the maximum deflection will occur at Xm <a • If a <b, the maximum deflection will occur at xm > a Accordingly: 1.C. Show that for a > b, the maximum deflection occurs at: Xm = 1²b² 3 1.d. Show that for a <b, the maximum deflection occurs at: Xm = L - 1² - a² 3
Part 2- Shaft Assessment Based on Deflection Considerations Shown is a shaft mounted in bearings at A and D and having pulleys at B and C. The forces shown on the pulley surfaces represent the belt tensions. Shaft diameter=1.45 in 6-in D 300 lbf 50 lbf 27 lbf 360 lbf D C 6 in 8-in D 8 in Static force analysis yields the following results for the bearing reactions in xy and xz planes. uy plane xz plane Fb = 127 lbf 387 lbf E = 223 lbf B 8" D А. 5 D B C 106 lbf 350 lbf Fo, - 281 lbf 2.a. Based on the static analysis along with the results of Part 1, find Xm in xy-plane and calculate the absolute maximum deflection Ymar. Take E = 29700 ksi. Also, find the slopes of the deflection curve y(x) at A and D. 2.b. Similarly, find xm in xz-plane and calculate the absolute maximum deflection Zmax- Also, find the slopes of the deflection curve z(x) at A and D. 2.c. Find ye. Yc, Zg, and zc. The total deflection and total slope can be calculated as follows: Total Deflection 8 = y2 + z2 Total Slope @y)? + (0,) A=
C1 Design Constraint Bearings at A and D being spherical ball bearings and referring to Table 7-2 from Shigley's Mechanical Engineering Design Book, the slope magnitude at the bearings should not exceed 0.052 radian. Design Constraint The magnitude of transverse deflection at a pulley's location on the C2 shaft should not exceed 0.02 in. 2.d. Check whether constraints C1 and C2 are satisfied. If constraints C1 and C2 are not both satisfied, then you will be required to find a new adequate diameter for the shaft as explained below. • If the allowable deflection is exceeded, since I is proportional to dt, a new diameter can be found from: Bold dnew = data • If the allowable slope is exceeded, a new diameter can be found from: Saral dnew = dold In case several corrections are applicable, take the diameter with the largest drew/dold ratio. Sartowabte allowable 2.e. If applicable, determine the new shaft diameter and recalculate the total slopes at the bearings and the total deflections at the pulleys accordingly.