Dynamic Analysis of Frames

Two-dimensional Frames The matrices formed for flexural members and already used for a cantilever beam can be used for the

dynamic analysis of two-dimensional frames. The elements of the i th row and j column of the mass and stiffness matrices are given by Eq. (13.12) in integral form and can be evaluated once the shape functions

i and j are known or assumed [as shown in Eq. (13.11)]. However, the axial displacements of joints (u 1A , u 1B ) are also considered for frames in addition to the transverse displacements (u 2A , u 2B ) and

rotations ( 3A , 3B ) about the out-of-plane axis considered in forming the matrices for beams, so that the size of the matrices is (6 6) instead of the (4 4) matrices shown for beams.

If shape functions of Eq. (13.11) are assumed for frame members of uniform cross-section, the member mass and stiffness matrices take the following forms in the local axes system

0 0 -S x

0 156 22L 0 54 -13L

2 0 S S 3 2 0 -S S 4 ……..(15.1)

M m = (mL/420) K m =

x -S

0 54 13L 0 156 -22L

1 -S 2 0 -S

1 0 S -S 2

2 0 -13L -3L 2 0 -22L 4L

0 -S 2 S 3

3 where S 2

x = EA/L, S 1 = 12EI/L ,S 2 = 6EI/L ,S 3 = 4EI/L, S 4 = 2EI/L

The member matrices formed in the local axes system by Eq. (15.1) can be transformed into the global axes system by considering the angles they make with the horizontal. A transformation matrix, T m [shown in Eq. (15.2)] is used for to represent the relation between the local vectors (e.g., displacement, velocity, acceleration, force) and global vectors (Fig. 15.1). The transformation matrix is also (6 6) instead of the (4 4) matrix for truss members.

Fig. 15.1: Local and global joint displacements of a frame member

-S

0 0 1 0 0 0 T m = [where C = cos , S = sin ] ..……....(15.2)

0 0 0 -S

This matrix can be used with further mathematical manipulations to obtain the global mass matrix and global stiffness matrix as was the case with truss members [Given by Eqs. (14.3)~(14.5)]. If the structural member is uniform (i.e., same area A and unit mass m) throughout its length, the calculations of Eqs. (14.3) and (14.4) can again be carried out explicitly. However, the explicit expressions are not shown here because they are not as convenient to write as were the matrices for truss members.

The mass and stiffness matrices and load vector of the whole structure can be assembled from the

G G member matrices and vector (M G m ,K m and f m ) and obtained in their final forms (along with matrix C) after applying appropriate boundary conditions.

Three-dimensional Frames The formulation of mass and stiffness matrices for the dynamic analysis of three-dimensional frames follows the same procedure as the formulation for two-dimensional frames discussed before. Since the

joint displacements and rotations in the x, y and z-axis (u 1A ,u 2A ,u 3A , 1A , 2A , 3A ,u 1B ,u 2B ,u 3B , 1B , 2B , 3B ) are considered in forming the matrices, the size of the matrices is (12 12).

A new feature of the displacements of three-dimensional frame element is the presence of torsional rotations 1A and 1B . In addition to the biaxial transverse displacements (u 2A , u 3A , u 2B and u 3B ) and rotations ( 2A , 3A , 2B and 3B ), they add to the complications in solving the three-dimensional frame problem. The member mass and stiffness matrices in the local axes system are shown in Table 15.1 and Table 15.2 respectively. The elements in Table 15.1 have a multiplying factor (mL/420) with them.

Table 15.1: Elements of member Mass Matrix in the local axis system

54 -13L

22L

54 -13L 140r 2 70r 2

-3L

(mL/420 ) 22L

-22L 70r 2 140r 2

4L 2 -13L

-13L

-3L 2 -22L

4L 2

[where r is the polar radius of gyration of the cross-section]

Table 15.2: Elements of member Stiffness Matrix in the local axis system

[where S = EA/L, S = 12EI /L 3 ,S = 6EI /L x 2 1z z 2z z ,S 3z = 4EI z /L, S 4z = 2EI z /L, T x = GJ/L, S 1y = 12EI y /L 3 ,S 2y = 6EI y /L 2 ,S 3y = 4EI y /L, S 4z = 2EI y /L]

The transformation matrix for 3D frames is quite complicated and is not shown here. It can be derived by three-dimensional vector algebra or by applying the axes rotations of the global axis system one by one.

After making the transformation from local to global axes system with an appropriate transformation matrix T m and using Eqs. (14.3)~(14.5), the global mass matrix and global stiffness matrix are obtained for a three-dimensional frame member.

The subsequent matrix assembly, setting boundary conditions, forming C matrix and carrying out the numerical integration follow the usual procedures.

Example 15.1 For the 2-dimensional concrete frame structure shown below, modulus of elasticity E = 450000 ksf, cross-

2 2 2 sectional area A = 1 ft 2 , moment of inertia I = 0.08 ft , mass per unit length m = 0.045 k-sec /ft for all the members. Formulate the mass matrix and stiffness matrix of the frame.

Solution u 1 ~u 3 u 4 ~u 6 The frame has 5 joints, therefore 15 DOF. A B u 7 ~u 9

The displacements u 7 ~u 15 are restrained, so that only six DOF (u 1 ~u 6 ) can possibly be non-zero.

There are four members in the frame, all with 10

the same cross-sectional properties and lengths.

u 10 ~u 12 D u 13 ~u 15 The member mass and stiffness matrices are E

0 156 220 0 54 -130 0 432 2160 0 -432 2160

AB = 1.071 10

G -4

0 220 400 0 130 -300 K AB = 0 2160 14400 0 -2160 7200

0 54 130 0 156 -220 0 -432 -2160 0 432 -2160

0 -130 -300 0 -220 400 0 2160 7200 0 -2160 14400 DOF [1 2 3 4 5 6]

DOF [1 2 3 4 5 6]

The same matrices for BC, with DOF u 4 ~u 9

0 220 400 0 -130 -300 K AD = 0 2160 14400 0 -2160 7200

54 0 -130 156 0 -220

-432 0 -2160 432 0 -2160

0 -45000 0 0 45000 0

0 2160 7200 0 -2160 14400 DOF [1 2 3 10 11 12]

0 -130 -300 0 -220 400

DOF [1 2 3 10 11 12]

The same matrices for BE, with DOF u 4 ~u 15 .

The structural mass and stiffness matrices (15 15) can be assembled from the member matrices by locating the elements at appropriate rows and columns. If only six DOF (u 1 ~u 6 ) are active, the final mass and stiffness matrices become

296 0 220 70 0 0 45432 0 2160 -45000 0 0

0 296 220 0 54 -130 0 45432 2160 0 -432 2160

M -4 = 1.071 10 220 220 800 0 130 -300 K = 2160 2160 28800 0 -2160 7200

Dynamic Analysis of Frames