1.571 Structural Analysis and Control Prof. Connor Section 1: Straight Members with Planar Loading. b Displacements (u, vβ),

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1 1.571 Structural nalysis and Cntrl Prf. Cnnr Sectin 1: Straight Members with Planar ading Gverning Equatins fr inear ehavir 1.1 Ntatin Yv, a V M F Xu, a Defrmatin Displacement Relatins Internal Frces Y a β θ a v a a ssume β is small b Displacements (u, vβ), ngitudinal strain at lcatin y : Fr small β hen ε( y) uy ( ) uy ( ) u( 0) yβ vy ( ) v( 0) ε( y) u, yβ, ε a + ε b ε a u, stretching strain ε b yβ, bending strain Structural nalysis and Cntrl Sectin 1 Prf Cnnr Page 1 f 17

2 Shear Strain a b β a b θ γ decrease in angle between lines a and b γ θ β dv θ v, d γ v, β 1.3 Frce Defrmatin Relatins σ Eε stress strain relatins fr linear elastic material τ Gγ τ σ V M F X F σd M -yσd V τd Cnsider initial strain fr lngitudinal actins ε σ + ε ε + ε b a ε where ε σ ε strain due t stress initial strain hen ε ttal strain ε a + ε b ε σ ε ε σ E Structural nalysis and Cntrl Sectin 1 Prf Cnnr Page 2 f 17

3 ls σ E(ε + ε ) E(ε + ε b ε ) a Eu, yβ, ε ) ( F σd F E( u, yβ, ε )d F u, E β, -ye M M -yσd If ne lcates the X ais such that d + d + -ε Ed -ye( u, yβ, ε )d M u, -ye d + β, y 2 Ed + yε Ed ye d 0 the equatins uncuple t give: Define hen F u, Ed + -ε Ed M β, y 2 Ed + yε E d D F M Ed stretching rigidity 2 y Ed bending rigidity - ε Ed yε Ed F u, + F M D β, + M Cnsider n inital shear strain τ Gγ Gv, ( β) V Gγd G(v, β)d V (v, β) Gd Define D Gd transverse shear rigidity Structural nalysis and Cntrl Sectin 1 Prf Cnnr Page 3 f 17

4 then V D (v, β) 1.4 Frce Equilibrium Equatins M V + V M + M F m C F + F b V b y Cnsider the rate f change f the internal frce quantities ver an interval F -F+ F+ F+ b 0 F+ b 0 F y M c F b 0 -V+ V+ V+ by 0 V+ by 0 V b y 0 2 -M + M + M+ m b y V M m V b y 2 0 et 0 (i. e. d ) M m+ V b y F + b 0 V + by 0 M + V + m Structural nalysis and Cntrl Sectin 1 Prf Cnnr Page 4 f 17

5 1.5 Summary f Frmulatin Equatins uncuple int 2 sets f equatins; ne set fr aial lading and the ther set fr transverse lading. ial (Stretching) F, + b 0 F F + u, undary Cnditin F r u prescribed at each end ransverse (ending) V, + b y 0 M, + V+ m 0 M D β, + M V D (v, β) undary Cnditins M r β prescribed at each end and V r v prescribed at each end Nte: hese equatins uncuple fr tw reasns 1. he lcatin f the X ais was selected t eliminate the cupling term yed 2. he lngitudinal ais is straight and the rtatin f the crss sectins is cnsidered t be small. his simplificatin des nt apply when: i the X ais is curved (see Sectin 2) ii the rtatin, β, can nt be cnsidered small, creating gemetric nn linearity (see Sectin 4) Structural nalysis and Cntrl Sectin 1 Prf Cnnr Page 5 f 17

6 1.6 Fundamental Slutin Stretching Prblem Gverning Equatins: F + b 0 (i) undary Cnditins Frm (i) u F F + D S F F u u b F ( ) - b d + C 1 (ii) F ( ) -( b d) + C 1 F F hen which can be written as C 1 F + ( b d) F ( ) - b d + ( b d) + F F ( ) b d + F Nte: yu culd als btain this result by inspectin: b F ( ) F F ( ) b d + F Structural nalysis and Cntrl Sectin 1 Prf Cnnr Page 6 f 17

7 Frm (ii) F F u, F F u ( ) d + C 2 F F d + C 2 u DS 0 F F C 2 u d DS F F u ( ) d + u 0 F u ( ) u d + u p ( ) 0 F u ( ) u u p ( ) where u p ( ) particular slutin due t b and F. where F u u u, DS u, u p ( ) Structural nalysis and Cntrl Sectin 1 Prf Cnnr Page 7 f 17

8 1.7 Fundamental Slutin: ending Prblem V V, v M, β M V ( ) V, v M ( ) M, β Internal Frces V ( ) V M ( ) M + V ( ) Gverning Equatins fr Displacement M M M D β, + M β, D V V D (v, β) v, β D Integratin leads t: 2 M V ( ) β β β ( ) D D 2 M V 2 β( ) β β β 2, D D 2 3 M 2 V V v ( ) v + β v ( ) 2 D 2 6 D D M 2 V 3 V v ( ) v v + β v D, 2 D 3 D Structural nalysis and Cntrl Sectin 1 Prf Cnnr Page 8 f 17

9 1.8 Particular Slutins Set hen where β i, end rtatin at i due t span lad v i, end displacement at i due t span lad β i β i e + v i v i e +, β i,, v i, β i, e end rtatin at i due t end actins v i, e end displacement at i due t end actins Cncentrated Mment a b M* M* a β, D M* a 2 M* a v, ( a) 2 D 2 D Cncentrated Frce a P* b β, P* a D P* a P* a 3 P* a 2 v , ( a) D 3 D 2 D Structural nalysis and Cntrl Sectin 1 Prf Cnnr Page 9 f 17

10 Distributed ading bd d Replace P* with bd an d integrate frm 0 t 2 β, bd 0 2 D 3 2 v, bd bd bd ( ) 0 D 0 3 D 0 2 D fr b cnstan t (ie unifrmly distributed lading) β, 3 b D 2 4 b b v, D 8 D Structural nalysis and Cntrl Sectin 1 Prf Cnnr Page 10 f 17

11 1.9 Summary F, u u u DS M 2 V 3 V v v, v + β 2 D 3 D M V 2 β β, β D 2 D hese equatins can be written as D u v β u, v, β, F u v 3 D D 2 D V 001 β 2 M D D ls Rigid bdy transfrmatin frm t F F, F V V, V M M, M V F V M F, V, M, F V M Structural nalysis and Cntrl Sectin 1 Prf Cnnr Page 11 f 17

12 1.10 Matri Frmulatin Strai ght Members Define: u u v β Dis placement Matri F F V M End ctin Matri General Frce Displacemnt Relatin Epress displacement at as: u u, + f F + u u, : Due t applied lading f F : Due t frces at u : Effect f mtin at as ed n cantilever mdel Interpret f Member fleibility matri Ri gid bdy transfrmatin frm t Fr the prismatic case f D D 2 D D D Structural nalysis and Cntrl Sectin 1 Prf Cnnr Page 12 f 17

13 Frce Displacement Relatins 1 Define k f Member stiffness matri Start with u u, + f F + u Slve fr F Define hen f F u u u, F k u k u k u, F i, k u, F k u k u + F, i Net, determine F where F F, F k F ( )u + ( k )u + F, i F, i F, F, i Nte F i and, are the initial end actins with n end displacements, F i Finally, rewrite as F + F i k u + k u, F k u + k u + F i, Ntice that there are tw fundamental matrices: k and Structural nalysis and Cntrl Sectin 1 Prf Cnnr Page 13 f 17

14 Matrices fr Prismatic Case u v β u v β F V 0 12 D* ( 6)D* ( 12)D* ( 6)D* M 0 ( 6)D* (4 + a)d* 6 D* (2 + a)d* Ḇ F V 0 ( 12)D* 6 D* 12 D* 6 D* M 0 ( 6)D* (2 + a)d* 6 D* (4 + a)d* Ḇ D D a D* D (1 + a) Structural nalysis and Cntrl Sectin 1 Prf Cnnr Page 14 f 17

15 1.11 ransfrmatin Relatins Rigid dy Displacement ransfrmatin u u r ω ω ω ω u u + ω r u u v v + ω in tw dimensins ω ω u v ω u v ω u u Statically Equivalent Frce ransfrmatin ranslate frce system acting at t pint P P r m m P P m m + r P F F, F Structural nalysis and Cntrl Sectin 1 Prf Cnnr Page 15 f 17

16 Crdinate ransfrmatin y y z,z a a a y a ' a' a y' a z a' Ra a z' he inverse is ake cs θ sin θ 0 R sin θ cs θ cs θ cs ( θ) 1 sin θ sin( θ) R R 1 R R( θ) (, yz), Glbal frame (', y',z') cal frame l ( ) g F () R gl F ( ) R ( gl) R l Given k in lcal frame ( k () ), transfrm t glbal frame If hen l F () l () k () R gl u g k () u l l ( ) ( ) ( ) F g ( ) F () ( ) () gl g R lg k R u R lg l l ( ) ( ) ( ) F g ( ) g k g u ( ) ( ) k g ( ) () ( gl ) ( ) () gl k R R R lg l ( ) (R gl ) k l Structural nalysis and Cntrl Sectin 1 Prf Cnnr Page 16 f 17

17 1.12 Structural Stiffness Matri assembly g F g g g g g k u + k u + F i, g g g g g k F ( g ) u + k u + F i, F i g l R F i Use direct stiffness methd t generate the system equatins referred t the glbal frame. ake as the psitive end and as the negative end. n+ n- fr member n Write system equatin as P I + KU E Wrk with the partitined frm f system stiffness matri K. k in n+,n+ k in n,n k in n+,n with k F i in n,n+ in n+ f P I F, i, in n f P I Structural nalysis and Cntrl Sectin 1 Prf Cnnr Page 17 f 17