L a) Calculate the maximum allowable midspan deflection (w o ) critical under which the beam will slide off its support.

Transcription

1 ecture 6 Mderately arge Deflectin Thery f Beams Prblem 6-1: Part A: The department f Highways and Public Wrks f the state f Califrnia is in the prcess f imprving the design f bridge verpasses t meet earthquake safety criteria. As a highly paid cnsultant t the prject, yu were asked t evaluate its sundness. Yu rush back t yur lecture ntes, and yu mdel the verpass as a simply supprted beam f span with an verhang =0.01. Assume that the distributed lad is a sinusidal functin. EI EA N a) Calculate the maximum allwable midspan deflectin (w ) critical under which the beam will slide ff its supprt. Part B: Assume that the abve design with an external axial frce N=0 and =0.01 has a safety factr f ne. The design f earthquake resistant structures requires a safety factr f five, meaning that (w ) critical must be increased by a factr f five withut the bridge cllapsing. Tw pssible design mdificatins were prpsed. In the first ne, the verhang is simply increased t new. In the secnd design, a tensile frce N is applied t the bridge t increase its transverse stiffness and thus reduce the central deflectin and the resulting mtin f the supprt. b) Fr the first prpsed mdificatin, what length new f the verhang will meet the requirement f a safety factr f five? Give yur result in terms f the riginal and ther parameters if needed. c) Fr the secnd design, what is the magnitude f the dimensinless tensile frce N/EA that will give a safety factr equal t five? d) Which design is better? Can yu think f a third alternative design slutin? 1

2 Prblem 6-1 Slutin: Recall: (a) Calculate the max deflectin w artical supprt Assume N 0 N 1 dw EA dx 0 dx at /under which the beam will slide ff its 1 dw dx 0 (1) dx Since the applied lad is a sin functin, we can assume the deflected shape will als be a sine functin x wx w sin x w' x w c s x w' x w c s Substitute the abve equatin int equatin (1), we have 1 x w d 0 cs x x 1 sin w x 0 1 w 0 1 w 1 w

3 Thus, the maximum allwable midspan deflectin is w w We re given 0.01 w w 5 w 5 (b) Case 1increase t meet safety factr f 5 5w new 1 5w new 1 w 5 1 w new 5 1 w Recall: new 5 (c) Case : apply a tensile frce N, s nw N 0 N 1 dw dx EA 0 dx N 1 w EA

4 We want t calculate N that will give us the equivalent effect f applying a safety factr f 5, in which case 1 w 5 In the case f /100 (d) Which design is better? N 5 EA N 0. EA 100 It is difficult t say which design is better. Each design has its advantage and disadvantages. Fr case 1, we will have a lng verhang which may nt be aesthetically pleasing. Fr case, it may be difficult t apply cnstantly a tensile frce. Other ptins include a stiffer simply supprted beam, r add cables t suspend the bridge.

5 Prblem 6-: A lng span aerial tramway steel cable f length =1km is laded by a hurricane wind with intensity q(x) sinusidally distributed between the end statins. The cable deflects by w=5m. E MPa y 00MPa D 60mm D Crss-sectin f cable x q ( x ) q 0 sin w 0 q 0 q ( x ) a) Calculate the resulting lad intensity q b) Calculate the tensin in the cable N. c) Calculate the tensile stress. d) Cmpare (c) with the yield stress, and determine the safety factr. Prblem 6- Slutin: Using the equatin f equilibrium EIw Nw'' q Hwever, a cable has n bending stiffness, s ur equatin becmes: Nw'' q q 1 x w'' q sin N N Integrate twice q x w' c s C1 N q x w sin C1x C N 5

6 Plug in the bundary cnditins t slve fr the cnstants 00 0 w w C 0 w 0 w0c 1 C 0C 1 0 w x q x sin N a) Calculate the lad intensity The cable deflects by w 5m at the middle pint x / q q w w sin N N q Nw b) Calculate the tensin n the cable(n) N 1 1 d w dx and q x Using wx sin EA 0 dx N w' q x cs N x w' x q x c N s EA q x N cs d N x 0 N x sin EA q x N 0 EA q 0 N EA q 6

7 1 q N EA Given w 5m, then q Nw 5N 5N EA q N EA 5EA N 5EA N N 666N c) Calculate the tensin stress n the cable N MPa A MPa d) Cmpare with the yield stress and determine the safety factr yield stress 00 safety factr wrking stress 1.95 safety factr.17 7

8 Prblem 6-: Plt the dimensinless deflectins (w /) versus the dimensinless line lad fr bth bending and membrane (cable) slutins ver a slender beam. At what dimensinless deflectins will the bending and membrane slutins be equal, assuming a length t thickness rati equal t 10? Prblem 6- Slutin: Recall bending and membrane slutins: Pure Bending Membrane P x P x wx sin wx n si EI N at x at x P P w w w w EI N P P w w EI N h 1 where I q 1 where N EA (Prblem 6-) P P 1 w 1 h Eh h q E s N Eh 1 w P 1 P P Eh h w 1 N q Eh P 1 Eq h where q P rearrange the abve expressin w 1 1 P h E 1 8

9 w P et s call y, x Eh We want t plt 1 Bending y x h 1 1 Membrane y x Use a length t thickness rati equal t 10: Bending y 1.x Membrane y x 10 h 9

10 At what dimensinless deflectins will the bending and membrane slutins be equal? w w bending membrane 1 1.x 0.5x x P S at Eh deflectins w , the bending and membrane slutins will be equal, where the dimensinless w

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