1.571 Structural Analysis and Control Prof. Connor Section 3: Analysis of Cable Supported Structures

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1 .57 Structural Analysis and Control Prof. Connor Section 3: Analysis of Cale Supported Structures Many high performance structures include cales and cale systems. Analysis and design of cale systems is a comple topic. Individual cales have a non-linear equivalent stiffness. In addition, the interaction etween multiple cales must e considered. In order to e ale to analyze and design cale supported structures, a numer of different topics must e covered. he topics to e covered are: 3. - Cale equations 3. - Modeling of eam with single cale using an equivalent spring Modeling of eam with multiple cale using a eam on equivalent elastic foundation Design procedures for cale/eam system Prof Connor Page of 8

2 3. Equations of a single cale 3.. Equilirium Equations Y w v ( ) X F cos + cos + ( cos) + 0 ( cos) + 0 F y sin + sin + ( sin) w 0 When 0 ; For self-weight ( sin) w 0 ( cos) 0 cos constant γ weight of cale / unit length w γ s wd γds d ds cos d w γ d cos γ cos Prof Connor Page of 8

3 So Knowing we get 3.. Eample γ ( sin) cos sin sin htan cos dv tan d dv γ d cos cos d - v d --- α --- γ --- dv d + dv d Y γ v ( ) X Boundary Conditions dv d 0 Solution so 0 v( 0) dv et d v p, - dp, and d d φ d dp φ + p d dp + p Integration y parts leads to φd ln( p + + p ) φ + C --- γ Prof Connor Page 3 of 8

4 for hen p + + p e φ C e C e φ dv d p 0 at 0 C p + p + e φ p + e φ p p + e φ pe φ + p + C e φ Integrating for Setting Finally (for convenience) at any point along the cale cos hen y at γ p e φ e φ e φ dv p ( e φ e φ ) d v C ( e φ + e φ ) + C φ φ γ C 3 0 v e γ γ e γ --- e φ v --- γ cosh--- caternary (chain) γ dv d Prof Connor Page 4 of 8

5 For w constant and 0 cos cos δ - v δ w then v C + C + --w Boundary Conditions v v v v 3..3 More Eamples Eample # ( sin) w 0 ( tan) w 0 v w 0 w h v then v 0 at v, 0 at 0 v C + C + --w v, C + w C 0 C w v ( ) w -- w v( 0) v ma --- h 8 Prof Connor Page 5 of 8

6 Eample # Yv, chord v X Point at Point at 0 v 0 at v v and C 0 C -- v --w v ( ) w --v } deflection from chord Prof Connor Page 6 of 8

7 3..4 Geometric Relations - Arc ength Y v ( ) X ds d + ( v, ) d ds d( + ( v, ) ) s s ds ( + ( v, ) ) d s Simplification for shallow curve So hen ( v, ) sin small wrt tan cos cos v, constant if there is only loading in the y-direction Consider cale in Eample # h v v ( ) h w w Prof Connor Page 7 of 8

8 Differentiate Approimate s w v, w v, w --- (for shallow curve) s ( + ( v, ) ) d + --( v, ) d s w d 0 s w s w s w w --- 8h w s hw w 3 --h h Note: s deformed length s o initial length he shallow assumption implies constant ε AE s εs o s AE o s s o + s s o + s AE o + AE s o w AE s --- o w AE Prof Connor Page 8 of 8

9 3..5 Equivalent angent Stiffness A o B A B u w v u B u AE o u -- v, d - Consider Cale AB with initial length o - Apply tension, which results in u - Apply transverse loading, w, which results in a negative end movement u. u is a function of. - he net movement is u u u B. o 0 Chord Shortening hen cos ( cos) for small and ( v, ) u -- v, d w v, --- u -- w --- d w o u B u u o w o --- AE 4 cos ---- v, Prof Connor Page 9 of 8

10 Perturation - Increment y an amount - Get corresponding change in u B u B + u B + o ---- w o AE 4 + For small wrt u B u B du B u Bd u B o ( w o) AE 4 3 du B o ( w o) AE 4 3 d f B tangent fleiility d k B k B --- du B k B du B f B tangent stiffness AE o AE --- wo + du -- B d AE Note: k B approaches as increases o A Write k B E eff o E eff effective modulus E AE --- wo + Prof Connor Page 0 of 8

11 Alternate forms (shallow cale) w h h w --- o Aσ* where σ* intial cale stress h o So E eff E E 8h σ* o E E h σ* o and w o 8 -- ( h o ) Aσ* A cale w --- o 8σ* ( h o ) Prof Connor Page of 8

12 3..6 Inclined Shallow Cale B, u w n h m φ A h cosφ Evaluation of w a ds w n k E eff A -- E eff E AE w n w n h m for self-weight ds φ w a ds φ w a unit weight per unit length of center line w a dscosφ φ hen w n w a cosφ w n ( w a cosφ) w a ( cosφ) w a h E E eff AE w a h Prof Connor Page of 8

13 3. Modeling of eam with single cale using an equivalent spring Eample #: Cantilever eam with single cale w d Reaction at A due to w d wd M A Deflection at B due to w d A B v wd v B 4 8EI Reaction at A due to M A sin A Beam and Spring Model B Deflection at B due to sin v 3 B -- 3EI k c B v B e c For small v B e c spring etension e c v B sin F c incremental force in cale F c k c e c k c sinv B Prof Connor Page 3 of 8

14 F c F c sin k c sin v B for pretensioned cale F c cos k c sincos v B k c E eff Non-inear Spring Model A c E eff c E AE w n k c, eq k c, eq k c sin k* his model is used to determine the incremental forces due to line loading. Illustration P M A A B k* Deflection at B P v B v Bp + v 3 F Bs --- s EI 3EI ( P F) 3 ( P k*v B ) 3 v 3EI B -- 3EI v 3EI B -- + k* P 3 Prof Connor Page 4 of 8

15 Moment Reaction at A M A ( P k*v B ) k*v B P P k* M A P --- 3EI -- + k* 3 M A P 3EI k* Note: Inclusion of cale reduces the negative moment at the support and also the deflection at the end point. his effect depends on the relative stiffness of the eam vs the cale. So, for M A P( α) α 3EI k* k* 0 α 0 and M A P k* α and M A 0 Prof Connor Page 5 of 8

16 Eample # Case a w w + V Positive Shear Shear V w w 8 M M + Positive Moment Moment Case F F F -- Shear - Moment F 4 Prof Connor Page 6 of 8

17 Case c w et hen w F -- M V F F F -- w F -- w + w α( w) w -- M w w ( α) -- V w ( α) + w w F -- M ma occurs at such that V( * ) 0 M min occurs at Optimum Case * -- ( α) M ma M* -- M min F w M* M ( ) ( α) α α ± For α < ( * must e positive) α w ( α) 8 w M peak --- ( 0.7) 8 * w --- ( α ) 8 Prof Connor Page 7 of 8

18 Case c (moment alanced) M* * M* Suppose F is a spring resultant w k hen Epress k in terms of δ deflection at ( +ve) δ δ w + δ s 5w δ 4 w EI F s 3 δ s EI δ δ kδ EI -- 5w 4 kδ EI 48EI 5w EI + k 3 F δk w αw 48EI k k α 48EI α α k Rigid Support For optimal design α 0 k 0 No Support α EI k opt EI Prof Connor Page 8 of 8

19 3.3 Modeling of eam with multiple cales using a eam on elastic foundation model A eam on many springs can e modelled as a eam on an elastic foundation. A simple analytic solution eists for constant foundation stiffness. his solution is useful for preliminary design Governing equations M V + V M + M o Y, v For small rotations and V F y V + V + V+ 0 V + 0 dv + 0 (i) d M o V + M + M M 0 M -- + V 0 dm - + V 0 (ii) d Xu, From eam theory γ v, β M β, D B V D Neglecting transverse shear deformation (ie γ 0) v, β and β, v, Prof Connor Page 9 of 8

20 hen From (ii) replacing into (i) M v, D B M D B v, dm V - M, d d M V, ---- d M, d M ---- d + 0 d - ( D d B v, ) + 0 Boundary conditions v or V prescried at each end and β or M prescried at each end 3.3. Winkler Formulation Model Winkler s hypothesis assumes the restraining force at point is in part a function of the displacement at. v ( ) ( ) k s v ( ) d his relation is the limiting form when there are many closely-spaced uncoupled springs supporting the eam k* s For k* and s constant k* k s s Prof Connor Page 0 of 8

21 hen where k s v + some prescried loading In this case the governing equation takes the form d - D d ( B v, ) + k s v Note: Boundary conditions have the same form and do not depend on the nature of the restraining foundation stiffness Solution with D B and foundation stiffness constant D B constant k s constant Define Solution has the form where d 4 v - d 4 4λ 4 k s + v D B k s D B D B v v part + e λ ( C sinλ + C cosλ) + e λ ( C 3 sin λ + C 4 cosλ) C, C, C 3, and C 4 are integration constants. Particular solution v part constant --- k s Characteristic length e λ decays with increasing For eample for λ 3 e e e λ 0 Define as the characteristic length Prof Connor Page of 8

22 D B λ k --- s 4 k s 4D B hen, for >, the e λ terms can e ignored. Also, if > Define hen C 3 and C 4 are evaluated using the B.C.s at Also At v v part ( ) + e λ ( C 3 sinλ + C 4 cosλ) sinλ φ cosλ φ cosλ + sinλ φ 3 cosλ sinλ φ 4 φ C 3 + φ C 4 φ C 3 + φ C 4 v ( ) v part ( ) ---- e λ v ( ) v part ( ) e λ v, v, ( part) + C 3 λ( cosλ + sinλ) + C 4 λ( cosλ sinλ) φ λc 3 + φ λc 4 v' ( ) v part '( ) -- e λ (i) (ii) From (i) and (ii) it can e deduced that C 3 and C 4 have a e λ factor. herefore, the C 3 and C 4 terms can e neglected for 0 herefore, the general solution can e approimated as 0 v v p + e λ ( C sin λ + C cosλ) v v p v v p + e λ ( C 3 sinλ + C 4 cosλ) Prof Connor Page of 8

23 End Zone Interior Zone End Zone 3 he solution then consists of end zone solutions and an interior zone solution when the memer length is greater than 3 λ DB k s Epansion of Solution Near 0 v v p + e λ ( C sin λ + C cosλ) v, β v p, + C { λe λ ( cosλ sinλ) } + C { λe λ ( cosλ sinλ) } v, β, v p, + C { λ e λ ( cos λ ) } + C { λ e λ ( sinλ) } M D B v, Set hen v, v p, + C { λ 3 e λ ( cosλ + sinλ) } + C { λ 3 e λ ( cosλ sin λ) } V D B v, ψ ψ ψ 3 ψ 4 e λ λ ( cos + sinλ) e λ sinλ e λ λ ( cos sinλ) e λ cosλ v v p + C ψ + C ψ 4 β v p, + C λψ 3 C ψ v, v p, + C { λ ψ 4 } + C { λ ψ } v, v p, + C { λ 3 ψ } + C { λ 3 ψ 3 } Prof Connor Page 3 of 8

24 3.3.5 Eamples of oadings and Boundary Conditions Case v p --- k s Boundary Conditions At 0 v 0 M 0 v, 0 hen C k 0 C s --- k s λ C 0 C 0 v M D B --- ( e λ cosλ) k s --- ( ψ 4 ) k s --- ( λ e λ sinλ) k s -----λ k ψ s Maimum value of M B at λ 0.8 ψ 0.3 ma M D B ma k ( 0.3) s 0.8 λ Maimum value of v at λ.4 ψ ma v ma --- ( + k s k s.4 λ Prof Connor Page 4 of 8

25 Case - Infinitely long eam with concentrated force at center P C Use symmetry at C to model as P Boundary Conditions P At 0 V -- D B v, β v, 0 So P -- v, C 0 C 0 C C and he solution is v, λ 3 ( C + C ) --- P P C 0 D C ---- B 8D B λ 3 Pλ v - ( ψ k + ψ 4 ) s M Pλ P ψ D B k 3 M ψ s 4λ 3 V D B C { λ 3 ψ } C λ 3 Pλ ( + { ψ 3 }) D B - k ( 4λ 3 ψ 4 ) s - Pλ k s Maimum value of M at 0 Pλ M 3 ma --- D k B s P λ Maimum value of V at 0 Pλ V ma - k s Prof Connor Page 5 of 8

26 3.4 Design procedures for cale/eam systems 3.4. Strength-Based Design cale spacing Set σ d allowale stress for dead weight σ d fσ u f 0.4to0.5 Determine area of cale w d A c σ d σ d sin Resulting Stiffness sin w d w d Equivalent vertical spring K v k c stiffness of cale where hen Define k v * K v K v k c k c sin AE* ---- E* E eff E* ( ) K v AE* ----sin w d - E* sin σ d distriuted stiffness k v * E*w - d sin σ d Prof Connor Page 6 of 8

27 arp Cale Arrangement - constant cos A c cos w d ---- sin constant σ d Note k v *( ) E*w - d cossin σ d - Stiffness decreases rapidly with distance from the tower - k v * at 0 is. herefore, need to modify arrangement in this region Fan Cale Arrangement - k v *( ) constant { + } sin --- cos k v *( ) -- tan E*w - d σ d --- E*w d σ d E*w - d E*w d --- σ d ( + ) σ d A c w d - σ d ower Geometry α ma α ma ---- where hen α α ( ma ) ( ma ) Prof Connor Page 7 of 8

28 3.4. Stiffness-Based Design Check for strength cale spacing k v * constant AE K* v k* v sin c A c ( k* v ) c - E*sin c e c v A c E* -- c sin v c et v movement due to live load and c Increment in cale tension due to live load c k c e c e c vsin k* v --- sin v total tension in cale c + c c w d ---- α sin w d k* sin α v v ---- { w sin dead + w live } arp Cale Arrangement - constant c cos A c ( k* v ) -- E* cossin Fan Cale Arrangement c sin constant ( k* A v ) c -- c 3 ( k* v ) -- ( + ) 3 E* E* A c ( k* v ) E* + 3 Prof Connor Page 8 of 8