HW5soln - CE 573: Structural Dynamics HW#5 Solutions Given:...

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CE 573: Structural Dynamics HW#5 Solutions Given : Another common model used for buildings is the portal frame model, which tries to account for flexibility of both horizontal and vertical members by allowing for rotations at connections. In this problem, we seek to obtain the dynamic properties of the simple portal frame shown above. Assume that it has the three degrees of freedom shown (notice that 1 x is the same at both connections), a uniform weight per unit length kip ft 1.2 w = for the horizontal member (including dead load), negligible weight for the vertical members, a length parameter ft, 18 L = 29000 E = ksi, and . 4 430 in I = Assumptions : a) Mass matrix: for translational degrees of freedom, the mass is simply the total mass of the member that is translating. If there is more than one translational degree of freedom for a given member, the total mass is split among all DOF according to their tributary lengths. For rotational degrees of freedom, the “mass” is actually the mass moment of inertia for the member 2 1 12 rotation I = m L for rotation about the center of mass and 2 1 3 rotation I = m L for rotation about the end of a member. If one member has more than one rotational degree of freedom associated with it, the member is split into tributary lengths and the mass moment of inertia is computed using the shorter tributary lengths. (b) Stiffness matrix: A convenient way to compute the stiffness matrix is to use the force method – for each DOF in turn, give it a unit “displacement” (translation or rotation) while requiring the other DOF to have no displacement. Compute the forces and moments associated with keeping the structure in this deformed shape. If DOF #i is given a unit displacement (with all other DOF=0) and DOF #j needs a “force” ij F to maintain this, then the stiffness coefficient ij ij kF = . (A chart is attached that shown the required forces and moments for typical cases in the application of this method.) Note that when multiple members are associated with a given DOF, the stiffness coefficient is computed by summing all “forces” created by each member. Required : a) Show that the mass and stiffness matrices for this portal frame are: [] [] 33 3 3 23.65 1.6036 10 1.4255 10 0.08385 366.8 ; 1.6036 10 384.9 10 76.98 10 366.8 1.4255 10 76.98 10 384.9 10 MK −×−× ⎡⎤ ⎢⎥ == × × × −× × × ⎣⎦ where the basic units used are kips, inches, and seconds.
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b) Compute the natural frequencies and mode shapes for this system. (Express the mode shapes as vectors of unit length.) Sketch the mode shapes, and show that they are orthogonal to both the mass and stiffness matrices. Calculate the modal mass and modal stiffness for each mode. c) To better understand what happens when we make the shear frame model approximation, suppose that the moment of inertia for the horizontal member only is increased by a factor of 100.
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This note was uploaded on 05/09/2008 for the course CE 573 taught by Professor Whalen during the Fall '05 term at Purdue University-West Lafayette.

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HW5soln - CE 573: Structural Dynamics HW#5 Solutions Given:...

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