CE573 Structural Dynamics [Fall 2008]

Transcription

1 CE573 Structural Dynaics [Fall 2008] 1) A rigid vehicle weighing 2000 lb, oving horizontally at a velocity of 12 ft/sec, is stopped by a barrier consisting of wire ropes stretched between two rigid anchors 100 ft apart. The wire ropes have a total cross-sectional area of 1.25 in 2 and a odulus of elasticity of 26,000 si, and are stretched to an initial tension of 1000 lb. The vehicle oves noral to the barrier and stries it at id-height. Find the axiu deflection of the barrier. Assue ideal conditions, that is, a rigid vehicle, weightless barrier, no friction, no daping, and perfectly elastic ropes. 50 ft 50 ft 2) A 60-in long 3 in x 3 in x 3/16 in tube cantilever structure supports a 2000-lb weight attached at the tip. The properties of the tube are as follows: cross-sectional area, A = 2.02 in 2 ; oent of inertia, I = 2.60 in 4 ; section odulus, S = 1.73 in 3 ; and, odulus of elasticity, E = 29,500 si. The syste is subjected to a sinusoidal force at the tip, acting horizontally in one of the planes of syetry. The force has an aplitude of 250 lb and oscillates at 3 cycles per second. Assuing that the syste is daped to 2 percent of critical daping, find the axiu steady-state tip displaceent and the axiu steady-state bending stress in the cantilever. Treat the attached weight as concentrated at the tip of the support structures and neglect the weight of the tube. Neglect P-Δ effects, that is, neglect the bending oent due to the eccentricity of the gravity force on the tip load with respect to the base of the cantilever. F(t) W 3 x3 x3/16 tube 60

2 3) Find the general expression for displaceent for t>τ, i.e. for the free-vibration stage, of an undaped single-degree-of-freedo structure forced with the finite duration load shown below. Do not assue that the loading is ipulsive. F(t) F 0 τ t 4) The 8-ft high two colun + slab storage syste shown below supports a collection of rigid crates bundled as a 10 ft x 10 ft x 10 ft cube and weighing 50 ips total. Each colun has a lateral stiffness of 5 ips/in. An explosion near the structure loads one side of the structure area over 0.01 sec with an average pressure of 10 psi. Initially the crate arrangeent is intact. However once the structure reaches its first axiu displaceent (i.e., the first axiu displaceent away fro the blast side), the straps holding the crate assebly breas and 1/4 of the crates fall fro the supporting slab. Find the aplitude of the next axiu displaceent (i.e., the first axiu displaceent in the direction of the blast location). Also state how long it taes for the structure to arrive at that displaceent state after the blast? Copared to the crates, coluns and the slab have negligible weight. Ignore the blast load on the frae of the structure and daping in the structure. Assue the slab to be rigid and neglect the axial and shear deforation in the coluns, and assue the structure behaves linear elastically. 10 ft Pressure (psi) 10 rigid slab 8 ft 0.01 Tie (sec) coluns

3 5) A pacage weighing 50 lb is suspended in a box, as shown below, by two springs with a stiffness of 250 lb/in each. The box is placed inside a truc that produces vertical haronic vibrations during transport of aplitude y(t) = 1.5 sin(4t) in. Deterine the axiu steady-state displaceent, velocity and acceleration experienced by the pacage. Ignore the transient response. Ignore daping. rigid W y(t)=1.5 sin(4t) truc bed 6) [BONUS] A paniced businessan weighing 200 lb wals on a tightrope between two structures to avoid the people on ain street. If the natural frequency of his vertical vibrations at the particular position shown below is 2 Hz, find the tension in the rope. 80 in 160 in

4 CE573 Structural Dynaics [Fall 2007] 1) The planar two-story frae structure setched below has rigid slabs. The slab at id-height is assless; the roof slab has ass M. All four coluns in the frae are identical: they have the sae odulus of elasticity E, oent of inertia I, and height H, and no ass. The colun-slab and colunground connections are rigid (no relative rotation between colun ends and slab/ground is possible). Assue that there is no daping in the structure and ignore the axial deforations in the coluns. a) What is the least nuber of coordinates, i.e. degrees of freedo, do you need to express the dynaic behavior of this structure properly? b) Find the natural period of the structure. c) During an earthquae, the ground oves. You are given the ground displaceent x ground (t). Write the differential equation of otion for the structure. (Do not solve the equation.) rigid slab, ass M E,I,H E,I,H E,I,H rigid slab, assless E,I,H x ground (t) 2) For each syste shown below, write the differential equation of otion and provide the expression for the natural period. Note that is ass, 1 and 2 are spring constants, f(t) is external force, and y(t) is support displaceent. f(t) f (t) 1 a) b) y(t) f (t) c) 1 2 Note the displaceent y(t) of the left-hand support.

5 3) On a nice weeend day, while you are riding your bicycle around in West Lafayette casually, you ae a wrong turn and end up riding on the infaously rough Lindberg Road by the Celery Bog. The Lindberg Road surface level varies haronically with ± 6 c undulations about the level surface elsewhere. The distance between consecutive peas of these undulations is easured to be 2. Assue that all of the suspension in your bie is thans to two springs under the seat of your bie. When you sit on the seat of your bie for your casual ride, the springs deflect 5 c. When you are seated, the daper under the seat provides an equivalent linear viscous daping of 10% of the critical daping. A siple representation of your ride on the never-ending rough road is shown below. a) If you are riding your bicycle at a horizontal speed v of 2.5 /sec, how uch buping up and down will you experience? In other words, what is the axiu vertical displaceent you will experience? b) On another Lindberg Road bie ride, you are carrying a bacpac which increases your on-seat weight by 20%. Assuing that you are still able to ride at v = 2.5 /sec, will this loaded ride be ore or less cofortable than your previous one, i.e. the ride without the bacpac? M c /2 /2 v 6 c 2

6 CE573 Structural Dynaics [Fall 2006] 1) A) The total wind drag acting on a flag of ass can be represented as f(t). The flag is hoisted to a assless flagpole which is rigidly connected at its base to a rigid assless girder. The girder is supported by two coluns. Colun connections, at both ends, are rigid. The coluns and the flagpole resist forces via bending. Their properties are shown on the figure. Assue that the flag can be luped as a point-ass at the tip of the flag-pole. Ignore gravity. a) Write the differential equation of otion for the syste. b) Write the expression for the natural frequency of the syste. wind H 2 EI 2 EI 1 EI 1 H 1 B) Two point asses of each are attached to a assless rigid rod of length L. The rod is hanging fro a frictionless pin and at ¾ L down, it is laterally supported by a spring with spring constant. Assue that the syste is in equilibriu when the rod is vertical. Do not ignore gravity. a) Write the differential equation of otion for sall angular oscillations about the equilibriu position. b) Write the expression for the natural frequency of the syste. L/2 L/4 L/4

7 2) At the Bowen Laboratory, a single-story test structure with heavy roof has been subjected to dynaic tests. The structure is 8-ft tall and is supported by four concrete prisatic coluns (E = 3,000 si). The coluns have a cross-section of 12-in by 12-in. An elevation view of the structure is shown below. In one of the tests, the structure was pulled statically with 40 ip force causing it to deflect 0.10 inches at its roof level. The load was then reoved instantaneously and the displaceent response of the structure was recorded. It was found that at the end of first swing, which too 0.5 seconds to coplete, the structure cae within 0.02 inches of the release location (i.e. it deflected 0.08 inches fro its undisturbed, vertical position). Ignore the ass of the coluns and the gravity. a) Find the daping ratio of the structure. b) Find the ass of the slab. c) Find the iniu nuber of full-cycles the structure needs to go through such that its deflections are reduced to no ore than ±0.01 inches about its equilibriu position. d(t) EI M 8-ft EI Plucing force 3) The ain entrance door to the CIVL building has a torsional spring and torsional viscous daper at its hinge. The rigid door is prisatic (1 eter wide and 3 eters tall) and has a ass of 100 g. Its centroidal ass oent of inertia about an axis parallel to its axis of rotation (and passing through its center of gravity) is 11.0 g 2. The torsional spring has a stiffness t of 25 N /rad. The door syste is critically daped. a) Write the differential equation of otion for the door s otion. b) What is the daping coefficient c t? c) The ailan ics the closed door to open it and get out with his hands full. What initial angular velocity ust his ic cause in the door so that the door opens to 75 o? θ t 3.0 c t 1.0

8 CE573 Structural Dynaics [Fall 2005] 1) A child weighing 100 lb is sitting at the free end of a assless plan. Her weight causes the plan s end to deflect 1.8 inches down. Assue that the plan is ade out wood (odulus of elasticity E = 1.495x10 6 lb/in 2 ) and has a cross-section 12 inches wide and 2 inches deep. What is the undaped natural period of the up-and-down oveent the child would experience as a light breeze passes by? pin support roller support 10 ft 5 ft 2) You are ased to review a space elevator. This elevator can be idealized as a chaber (ass ) supported fro above by a linear spring (stiffness ) and a linear viscous dashpot (daping constant c) which are both attached to a set of rocets (see illustration below). The rocets fly up vertically at a constant speed v o. Ignore the effect of gravity. a) Write the governing differential equation of otion for the elevator chaber. b) Solve the differential equation of otion to find the expression for the distance travelled by the elevator chaber, i.e. z(t), as the rocets lift off vertically with a speed v o =50 /sec. Assue that the chaber has a ass 1000 g and is initially at rest. Use = 4000 N/ and c = 2000 N sec/. v o v o c z(t) 3) An undaped single-degree-of-freedo syste is excited with a triangular forcing function as shown below. The syste weighs 483 lb, is initially at rest, and oves horizontally on a frictionless surface. Assuing that the linear spring stiffness constant = 60 lb/ft, F o = 4500 lb, and τ = 1 sec, find x(t), i.e. the displaceent of the ass in tie. Tae g = 32.2 ft/sec 2. x(t) f(t) f(t) F o τ t