Dynamics of structures

Fall 2019

University of Qom

By:

A. Shahiditabar

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CHAPTER 10

EVALUATION

OF STRUCTURAL

PROPERTY

MATRICES

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SELECTION OF THE DEGREES OF FREEDOM

In general, the dynamic response of a structure cannot be described adequately by a SDOF model; usually the

response includes time variations of the displacement shape as well as its amplitude. Such behavior can be

described only in terms of more than one displacement coordinate; that is, the motion must be represented by more

than one degree of freedom. As noted in Chapter 1, the degrees of freedom in a discrete parameter system may be

taken as the displacement amplitudes of certain selected points in the structure, or they may be generalized

coordinates representing the amplitudes of a specified set of displacement patterns. In the present discussion, the

former approach will be adopted; this includes both the finite element and the lumped mass type of idealization.

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Finite Element Stiffness

The flexibility or stiffness coefficients associated with

any prescribed set of nodal displacements can be

obtained by direct application of their definitions.

the structure is assumed to be divided into a system of

discrete elements which are interconnected only at a

finite number of nodal points.

The properties of the complete structure are then found

by evaluating the properties of the individual finite

elements and superposing them appropriately.

The two nodal points by which this type of element can be assembled

into a structure are located at its ends, and if only transverse plane

displacements are considered, it has two degrees of freedom at each

node, vertical translation and rotation.

The deected shapes

resulting from applying a unit displacement of each type at the

left end of the element

while constraining the other three nodal displacements are

shown in Fig. 104.

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Finite Element Stiffness

The deflected shapes resulting from applying a unit

displacement of each type at the left end of the element

while constraining the other three nodal displacements are

shown in Fig. 104.

These displacement functions could be taken as any

arbitrary shapes which satisfy nodal and internal

continuity requirements, but they generally are

assumed to be the shapes developed in a uniform

beam subjected to these nodal displacements.

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With these four interpolation functions, the deflected shape of the element can now be expressed in terms of its nodal

displacements:

The physical meaning of the stiffness : they represent the forces developed in the structure when a unit displacement

corresponding to one degree of freedom is introduced and no other nodal displacements are permitted

Stiffness

While the unit rotation is applied and equating the

work done by the external forces to the work done

on the internal forces:𝑊𝐸 = 𝑊𝐼 . In this case, the

external work is done only by the vertical force com-

ponent at a because the virtual displacements of all

other nodal components vanish; thus

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The internal virtual work is done by the internal moments associated with θ = 1 acting on the virtual curvatures, which are

However, the internal moments due to 𝜃𝑎 may be expressed as

Thus the internal work is given by

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For the special case of a uniform beam segment, the stiffness matrix resulting from Eq. (10-21) when the

interpolation functions of Eqs. (10-16) are used may be expressed by

As mentioned earlier, when the stiffness coefficients of all the finite elements in a structure have been evaluated, the stiffness

of the complete structure can be obtained by merely adding the element stiffness coefficients appropriately; this is called the

direct stiffness method. In effect, any stiffness coefficient 𝑘𝑖𝑗 of the complete structure can be obtained by adding together the

corresponding stiffness coefficients of the elements associated with those nodal points. Thus if elements 𝑚, 𝑛, and 𝑝 were all

attached to nodal point 𝑖 of the complete structure, the structure stiffness coefficient

for this point would be

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Example E101. The evaluation of the structural stiffness matrix is a basic operation of the matrix displacement method

of static structural analysis; although a general discussion of this subject is beyond the scope of this structural dynamics

text, it may be useful to apply the procedure to a simple frame structure in order to demonstrate how the element

stiffness coefficients of Eq. (10-22) may be used.

Consider the structure of Fig. E10-1a. If it is assumed that

the members do not distort axially, this frame has the three

joint degrees of freedom shown.

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The structure stiffness matrix finally obtained by assembling all these coefficients is

10-2 MASS PROPERTIES

The simplest procedure for defining the mass properties of any

structure is to assume that the entire mass is concentrated at the

points at which the translational displacements are defined. The

usual procedure for defining the point mass to be located at each

node is to assume that the structure is divided into segments, the

nodes serving as connection points. Figure 106 illustrates the

procedure for a beam type structure.

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The off-diagonal terms 𝑚𝑖𝑗 of this matrix vanish because an acceleration

of any mass point produces an inertial force at that point only. The

inertial force at 𝑖 due to a unit acceleration of point 𝑖 is obviously equal

to the mass concentrated at that point; thus the mass influence

coefficient 𝑚𝑖𝑖 = 𝑚𝑖 in a lumped mass system.

Consistent Mass Matrix

The degrees of freedom of the segment are the translation and rotation at each end, and it will be assumed that the

displacements within the span are defined by the same interpolation functions 𝜓𝑖 𝑥 used in deriving the element stiffness.

If the beam were subjected to a unit angular acceleration of the

left end ሷ𝑣3 = ሷ𝜃𝑎 = 1, accelerations would be developed along

its length, as follows:

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The inertial force resisting this acceleration is

the vertical force at the left end can be evaluated by introducing a vertical virtual displacement and equating the work done

by the external nodal force 𝑝𝑎 to the work done on the distributed inertial forces 𝑓𝑖(𝑥). Thus

Expressing the vertical virtual displacement in terms of the interpolation function and substituting Eq. (1026) lead nally to

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In the special case of a beam with

uniformly distributed mass the results are

Example E10-2. The structure of Example E10-1,

shown again in Fig. E10-2a, will be used to illustrate

the evaluation of the structural mass matrix

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10-3 DAMPING PROPERTIES

10-4 EXTERNAL LOADING

When a virtual displacement 𝛿𝑣1 is applied

Thus, the element generalized loads can be expressed in general as

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10-6 CHOICE OF PROPERTY FORMULATION

In the preceding discussion, two different levels of approximation have been considered for the evaluation of the mass,

elastic stiffness, geometric stiffness, and external load properties: (1) an elementary approach taking account only of the

translational degrees of freedom of the structure and (2) a .consistent. approach, which accounts for the rotational as well

as translational displacements.

The elementary lumped mass approach presents a special problem when the elastic stiffness matrix has been formulated

by the finite element approach or by any other procedure which includes the rotational degrees of freedom in the matrix.

If the evaluation of all the other properties has excluded these degrees of freedom, it necessary to exclude them also from

the stiffness matrix before the equations of motion can be written. The process of eliminating these unwanted degrees of

freedom from the stiffness matrix is called static condensation.

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Substituting this into the first of the submatrix equations of Eq. (1044) leads to

or

where

Example E103. To demonstrate the use of the static condensation procedure, the two rotational degrees of freedom will be

eliminated from the stiffness matrix evaluated in Example E101. The resulting condensed stiffness matrix will retain only the

translational degree of freedom of the frame and thus will be compatible with the lumped mass matrix derived in Example E10-2.

The stiffness submatrix associated with the rotational degrees of freedom of Example E10-1 is

and its inverse is

When this is used in Eq. (10-45), the

rotational degrees of freedom can be

expressed in terms of the translation:

The condensed stiffness given by Eq. (10-47) then is

Problems 10-1,2,3,8