2. PROBLEM STATEMENT AND SOLUTION STRATEGIES. L q. Suppose that we have a structure with known geometry (b, h, and L) and material properties (EA).

Transcription

1 . PROBEM STATEMENT AND SOUTION STRATEGIES Problem statement P, Q h ρ ρ o EA, N b b Suppose that we have a structure wth known geometry (b, h, and ) and materal propertes (EA). Gven load (P), determne the dsplacements (), deformatons (ε ), nternal force (N) and resstng force (Q). Soluton strategy The formal soluton strategy s the followng: (A) Develop Q relaton can determne Q for a gven Determne ε from (Knematcs) Determne N from ε (Consttutve of materal) Determne Q from N (Eulbrum) (B) Gven P, determne by teratve algorthm (C) Determne N from ; thus havng N Can determne N for a gven P P relatonshp -

2 (A) Develop Q relatonshp Frst, determne ε from by nonlnear knematcs: ( ) b + h h ε + () From lnear materal consttuton: N EA ε () h N EA ε EA + (3) Then, by eulbrum n the deformed confguraton: Q Nsnρ N h ( ) b + h Q h h EA + b + h ( ) (4) (Exact) -

3 Compare to analyss wth small dsplacement assumpton From nonlnear knematcs E. () x x Taylor seres of + x x For small x, approxmate + x + Knematcs wth approxmaton becomes h h ε + (5) From eulbrum n the deformed confguraton E.(3), h h sn ρ + ( ) b h for small (6) Substtute (5) n () and together wth (6) n (3), Q h h EA EA 3 Q 3 ( 3h + h ) (7) - 3

4 Compare to the lnear analyss sn ρ near knematcs: o h ε (8) near materal consttuton: N EA ε () Eulbrum n undeformed confguraton: h Q Nsnρo N (9) Thus, h Q EA () 3 Q s a lnear functon of. Compare types of analyss for Q relaton TYPE Knematcs Materal Eulbrum. Nonlnear- arge dsplacement E. (4). Nonlnear- Second order E. (7) 3. near E. () Nonlnear E. () Approx. Nonlnear E. (5) near E. (8) near E. () near E. () near E. () Nonlnear (Deformed) E. (3) Nonlnear (Deformed) E. (6) near (Undeformed) E. (9) - 4

5 Compare types of analyss for Q relaton 3.5 h, b, EA arge dsp E.(4) Approx. E.(7) near analyss E. () Q() Q() from second-order analyss s ute accurate whle the euaton s much smpler. It can capture the snap of dsplacement when the load keeps ncreasng. Q() from lnear analyss, whch assumes nfntesmally small dsplacement and s based on eulbrum n the undeformed poston, s accurate only for very small dsplacement. When the dsplacement gets large, lnear analyss s not acceptable. - 5

6 (B) Iteratve algorthm From part (A), we now have the Q relaton n term of Q ( ) as a functon of. (B.) Newton-Raphson (NR) method For gven load P, the NR algorthm s dq ) Intalze, U P, and K d ) + Advance to next step 3) U Dsplacement ncrement K 4) + Dsplacement 5) Q Q( ) Resstng force 6) K dq Tangent stffness d d dq 7) U P Q Unbalance force 8) Check convergence U If < Tolerance ( -4, -6 ) then STOP. P Otherwse, repeat () to (8). Steps 5 and 6 are called State determnaton as they use the relaton developed n part (A). - 6

7 Example EA 3 From E.(7) Q 3 ( 3h + h ) dq EA 3 6 h + h 3 d ( ) P.35 Newton Raphson Method U U.5 Q() Q U/ K Q Q3 Q

8 (B.) Modfed Newton-Raphson (NR) method (Do not update stffness) For gven load P, the NR algorthm s dq ) Intalze, U P, and K d ) + Advance to next step 3) U Dsplacement ncrement K 4) + Dsplacement 5) Q Q( ) Resstng force 6) Do not update stffness; use ntal tangent stffness K. 7) U P Q Unbalance force 8) Check convergence U If < Tolerance ( -4, -6 ) then STOP. P Otherwse, repeat () to (8). Advantage: no update of stffness K Dsadvantage: (many) more teratons necessary - 8

9 (B.3) Secant method For gven load P, the NR algorthm s dq ) Intalze and K d ) + Advance to next step 3) P Dsplacement K 4) Q Q( ) Resstng force 5) K Q Update secant stffness 6) Check convergence If Q P, Repeat () to (6). Advantage: Need not know expresson for dq d. K can be calculated from a tral tral K ( ) Q tral tral. Dsadvantage: More teratons necessary. Iteraton converges slower than NR method. - 9

10 (C) Determne N from usng E. (3) from Part (A) h N EA ε EA + (3) Gven P, can be determned by methods n Part (B). Therefore, gven P, N can be determned usng E.(3) and results from Part (A) and (B). -

11 EXTENSION OF NEWTON-RAPHSON METHOD TO MUTI- DEGREE-OF-FREEDOM (MDOF) SYSTEMS Scalar uanttes: P, Q, U, and become vectors P, Q, U, Example and K becomes matrx K. Determne and due to load P and P. P, P, EA ρ α (deformed) k h Gven: h., b, EA, P.8, P.5 b k EA Governng euatons: Resstng force Q eual to external load P. (, ) (, ) Q Q P Q Q P Q Q dq d + d Q Q dq d + d -

12 In vector and matrx forms, where Q Q dq d dq Q Q d or dq K d Q Q Q(, ) Q ( ) K Q(, ) Q Q Need to set up relatonshp between Q and and stffness matrx K Eulbrum at node : Q Nsnα + S where N axal force S sprng force k Eulbrum at node : Q Ncosα Materal consttutve relaton: N AEε Knematcs: ε where (, ) ( b ) + ( h+ ) h+ b -

13 In eulbrum euatons, substtute N, ε, and snα h+ cosα b h+ Q (, ) EA + k Q (, ) EA ( h + ) + k (, ) b Q(, ) EA EA b (, ) Q Q EA + k + EA h + EA + k + h + ( ) ( h ) ( ) ( ) Q Q EA b + Q Q EA h+ ( ) ( ) b Q Q EA + Q ( ) - 3

14 Q EA b EA + b ( ) ( ) Then, perform Newton-Raphson Method., Iteraton : K ( ) ,.8 U P.5 ( ).46 K U Q ( ) K ( ) U P Q > Tolerance CONTINUE.8 Iteraton : ( ).6 K U Q ( ) K ( ) U P Q < Tolerance STOP.4 The soluton s.48 and.65 Ans - 4