PLANE_FRAME or BEAM ANALYSIS static, dynamic, natural frequency or buckling ! APPLICATION: PLANE FRAME ANALYSIS, N_SPACE = 2, ! NOD_PER_EL = 2, N_G_DOF = 3, N_LP_FLO = 3, N_EL_FRE = 6 ! REAL ELEMENT PROPERTIES: ! 1 = CROSS_SECTIONAL AREA 1 (e) 2 ! 2 = MOMENT OF INERTIA P_1 *------ A, E, I --------* P_2 ! 3 = YOUNG'S MODULUS OF ELASTICITY ! 4 = DISTRIBUTED LOAD, NODE 1 (TO RIGHT FROM NODE 1 TO 2) ! 5 = DISTRIBUTED LOAD, NODE 2 (TO RIGHT FROM NODE 1 TO 2) ! Optional mass density: 6 = MASS DENSITY ! Optional compression load: 7 = AXIAL_LOAD if BUCKLING or GEOMETRIC_K DATA_SET 01 Weaver plane frame example X----------\ F_y=-32 K, M_z=-1050 in K E=10000ksi, A=10 in sq 2 (1) 1 \ at node 1 I=1000 in^4, L_1=100 in \ (2) L_2x=100 in, L_2y=-75 in \ Node 1 disp: -0.0202608 in, -0.0993600 in, X [no uniform loads] and -0.00179756 radians 3 DATA_SET 02 Plane Frame, Smith-Griffiths, Prog. FEM, 2-ed, p115 Unknowns: X-disp, Y-disp, Z-rot Load | | Load E=1e6 kN/m^2, p=20 kN/m, Load=60 kN | | Horizontal El :A=5e3 m^2, I=6e-2 m^4 [Pin] p p V p V p p Other El :A=1e3 m^2, I=2e-2 m^4 1------2---7---8---4---6 Node 2 results:3.65e-8 -8.32e-7 -9.50e-4 (1) |(2) (7) (8)|(3) Reactions, node 1:-3.04e1 4.03e1 0.0 | / | Reactions, node 3: 3.89e1 2.34e2 3.42e-1 |(4) / |(6) Reactions, node 5:-8.51e0 1.26e2 1.42e1 | / | PARAMETER MAXIMUM, NODE MINIMUM, NODE | (5) | DOF_1, 6.4355E-08, 4 0.0000E+00, 1 | / | DOF_2, 2.8798E-03, 6 -4.2215E-03, 8 | / | DOF_3, 1.7735E-03, 4 -1.7022E-03, 7 | / | 3 [Fixed] | [Fixed] 5 DATA_SET 03 Plane frame, Smith-Griffiths, Prog FEM, 2-ed, p116-117 Frame with an internal hinge (pin) connection. 4 o [Pin] Unknowns: X-disp, Y-disp, Z-rot / Results, node 2: Pressure --> /(4) 6.5455E-08 -2.9951E-08 -8.4029E-04 / Reactions, node 1: [Pin] 3 o 5 1.8906E+01 3.7439E+01 -2.5207E+01 /(3) Reactions, node 4: Load --->6 -3.2350E+01 -1.0550E+01 0.0 /(2) I = 6e-2 m^4, 2 | A = 5e3 m^2, | (1) E = 1e6 kN/m^2, | Pin at node 3 = 5 | 1 [Fixed] DATA_SET 04 Single member, uniform load, one inclined roller. E=3e4 ksi, A = 1 in sq. 1 pressure | 2 / I = 0.08333 in^4, *-----------(1)----------|----o/ [Roller P1 = P2 = 0.005 K/in, [Pin] V / slope of L=100 in / 3/4 DATA_SET 05 Beam, Smith-Griffiths, 2-ed, p102, settlement, triangle pressure Beam foundation settlement. Unknowns: X-disp, Y-disp, Z-rot E=1e6 kN/m^2, p=4 kN/m on (2) | Load 20 drops to 0 on (3) | First bay : A=0.0 m^2, [ang=-.001] V p p p 0 I=4e-2 m^4 [v=0] 1-------5------2-------3-------4 Other bays: A=0.0 m^2, [u=0] (1) (4) (2) (3) I=2e-2 m^4 [v_2 = -.005] [v_3 = 0] DATA_SET 06 Cook, Bar multipoint constraint test, p274 Unknowns: X_displacement (Y-disp & Z-rot are zero) [fixed] E=1, A=1, I=1, p=0, k=AE/L=1 1 2 3 4 F=2, Constraint u_3==u_4, u_1=0 *----(1)----*---(2)--*--(3)---* Result: u_2=3F/k=6, u_3=5F/k=10 | | | Reaction, node 1: F_x=-3P=-6 F ---> F ---> F ---> DATA_SET 07 Bathe, Full frame version, p 143, first ed. The full frame to the right: F-->8 |(14) F k=AE/L=1 10-(15)-6-(16)-4 : Node_1 fixed |(12) |(13) |(3) V (n) Denotes element n 13-(10)-12-(9)-1-(1)--2-(2)--3 No line loads ^ |(11) |(5) |(4) E=1, A=1, I=1 : 11-(8)--7-(7)--5 F=2 F |(6) Cyclic constraints: u5=v4,v5=-u4,a5=a4 9<--F DATA_SET 08 BATHE, 1/4 frame with cyclic bc via mpc, p 143, first ed. Unknowns: X-disp, Y-disp, Z-rot E=1, A=1, I=1, p=0, k=AE/L=1 Rightmost 1/4 of the frame is used. F-->8 F |(14) F 4 : 10-(15)-6-(16)-4 : |(3) V |(12) |(13) |(3) V 1-(1)--2-(2)--3 Full: 13-(10)-12-(9)-1-(1)--2-(2)--3 |(4) ^ |(11) |(5) |(4) 5 : 11-(8)--7-(7)--5 F |(6) Cyclic constraints: u5=v4,v5=-u4,a5=a4 9<--F F=2, node_1 fixed. DATA_SET 09 "Frame, Shih Prob 7.2 " 1'=12" o 4 Inclined 4:3 Roller slope (3)/p E = 10e6 psi 3 @ center 1-4 /p p=50 ppf = 4.1667 lb/in 3 *p A = 0.75" * 1.5" = 1.125 in^2 p=800 | lb (2)/p I=b h^3 / 12 = 0.2109 in^4 1 v 2 /p Shih's Problem 7.2 frame o-(1)---*----(4)--* (Result is "large deflection") Pin 5 Al 6061 T6 1-4 = 6' horiz, 4-2 = 3' vert, 4-2 = 4' horiz (ref) DATA_SET 10 "Shih plane frame example 7.1" ! Unknowns: X-disp, Y-disp, Z-rotation ! Pin 4 ! Shih plane frame 7.1 o---*---------* ! E=2e11 N/m^2, 2 (3) (1) 1 \ p_13=50 kN/m ! I=6.51e-5 m^4 \ (2) ! A=1.25e-2 m^2 \ ! Node 1 Y disp: inclined o roller ! Node 3 rotation: XXXX 3 DATA_SET 11 "S_G, 2ed Beam-Column vibration, p302" ! Freq = 3.516/L^2 sqrt(E*I/Rho*A)= 0.063 3 L2 elements Fixed: 1----2----3----4 Free DATA_SET 12 "S_G, 3ed Beam-Column vibration, p415" ! Freq = 3.516/L^2 sqrt(E*I/Rho*A)= 0.063 6 L2 elements Fixed: 1----2----3----4----5----6 Free DATA_SET 13 "frequency of cantilever with equal end mass" ! Freq = 1.559 sqrt(E*I/M*L^3)= 2.81e-2 Point mass 6 L2 elements Fixed: 1----2----3----4----5----*6 DATA_SET 14 "Thomson portal Ex 10.5-4, p 278" ! Equal length portal 2 3 ! Freq = c * sqrt (EI/mL^4) *----(2)----* ! 1st anti-symmetric, c=3.21 | L | ! 1st symmetric, c=15.14 (1)L L(3) ! 2nd anti-symmetric, c=32.68 | | ! (Neglecting axial effects that 1 * Fixed 4* Fixed ! are included here) Here EI/mL^4 = 8.33333e-6