LECTURE+6+COE-3001-A

# Force ra axial load within the upper

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Unformatted text preview: NTINUED) ⌃ Fy = 0 N2= RA (reac>on force) RA + R D virtual cut (1) P1 = 0 ²་  NOTE: Ini>ally unsure about sign of RA (DIRECTION OF OTHER FORCE VECTORS KNOWN) ²་  We have two unknown forces so create a second equa>on ⌃ MB = 0 aRD = (a + b)P2 ✓ ◆ b R D = P2 1 + a ²་  Subs>tute for RD in eqn. (1) RD b R A + P 2 + P2 a N1 virtual cut N1 = P1 = 10kN Lower sec>on under axial tension 1/17/13 P2 M. Mello/Georgia Tech Aerospace P2 P1 = 0 b P2 a 12.3kN R A = P1 RA = Upper sec>on under axial compression ﬁnish up… 5 EXAMPLE 2- 3 (ﬁnish) ²་  FINALLY, COMPUTE THE TOTAL CHANGE IN LENGTH OF BAR ABC n X✓ L i ◆ ²་  RECALL, = N (2.6) i E i Ai i=1 2 X✓ L i ◆ L1 N1 L2 N2 + ²་  IN THIS PROBLEM: = N i OR, = E i Ai E 1 A1 E 2 A2 i=1 ²་  SUBSTITUTE: •  E1= E2 = E = 200 GPa = 200 x 109 N/m2 •  L1 = 500 mm ; A1 = 160 mm2 •  L2 = 750 mm ; A2 = 100 mm2 •  N1 = 10 kN •  N2 = RA = - 12.3 kN = 0.375mm 0.192mm = 0.183mm 1/17/13 M. Mello/Georgia Tech Aerospace 6 BARS WITH CONTINUOUSLY VARYING LOADS OR DIMENSIONS = RL 0 ( x) = RL N (x)dx 0 EA(x) (2.7) LIMITATIONS! ²་  LINEAR ELASTIC MATERIALS ONLY ²་  The normal stress formula only applies to prisma>c bars and not tapered bars. ²་  INTEGRAL FORMULA GIVES SATISFACTORY RESULTS ONLY IF THE “TAPER ANGLE” IS “SMALL”, SAY FOR ANGLES < 20° 1/17/13 M. Mello/Georgia Tech Aerospace 7 Example 2- 4: Axial displacement of a gently tapered bar GIVEN: ²་  A tapered bar of solid circular cross- sec>on of length L and elas>c modulus E ²་  Supported at end B (ﬁxed end) ²་  Subjected to tensile load P and free end A ²་  dA = diameter of bar at free end A ²་  dB = diameter of bar at ﬁxed end B Change in length of a tapered bar of solid circular cross sec>on PROBLEM STATEMENT: ²་  Elonga>on of the bar due to load P ²་  Assume angle of taper is small (prisma>c beam assump>on of normal stress deﬁni>on is technically violated but the error is small) SOLUTION STRATEGY: R RL L )dx ²་  Apply = 0 (x) = 0 N (x(x) EA ²་  NOTE THAT IN THIS CASE N(x) = P ²་  NEED TO DESCRIBE A(x):...
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