Dynamic Analysis Of Multi Degrees Of Freedom English Language Essay

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Earthquake excitation, in contrast with other dynamics excitations, is applied as support motion. The analysis of this support motion cannot be given by an exact solution, as it varies, depending to the seismic activity in the geographical area of the structure. This fact increases the difficulty in the prediction of the structural response to earthquakes. For these reasons, to accomplice a seismic analysis it is needed to take under consideration the previews incidents of the examined area that will provide the necessary data to prescribe the input seismic motion. [2] [3]

The calculation of corresponding stresses and deflection, for a single degree of freedom system model, could be achieved with an analytical solution. Because the majority of structures could be modeled as multiple degrees of freedom systems, the solution can be established with the use of numerical methods, or with Finite element methods.

In this paper a dynamic seismic analysis for framework buildings is presented, followed by employment of finite element method. The frameworks will be assumed as elastics.

Dynamic Analysis of multi degrees of freedom structures

Eigenvalues Analysis

Eigenvalue analysis helps to convert a multiple degrees of freedom system to a sum of single degree of freedom systems. By using in structural analysis the response spectra of each SDOF and the contribution of each mode, a very good approximation of floor's displacement and reactions could be achieved.

Eigenvalue analysis is essential to approach time depended problems of dynamic loading on structures, such as the non - periodic loading of an earthquake excitation. Because of the nature of dynamic problems, there is more than one answer to their solution. For each system the shape of the undamped free vibration could help describe the behavior and the response of the structure during a periodic or for a non - periodic loading. The solution of eigenvalue problem gives the natural frequencies ( λ = ω2 ) and modes of a system.

To determine the response of a building during an earthquake excitation two methods of analysis, based in eigenvalues theory, are generally used; response spectrum analysis and response history analysis. [4]

Mode superposition method

The Mode superposition method is the most effective and wide used analysis to evaluate and calculate the response of linear structures to an earthquake excitation. This method simplifies the large set of global equilibrium equations to a relative small number of uncoupled second order differential equations. This action ends up to an easier and simpler to handle system of equations. The main advantage of this method is that ends up solving the free vibrations mode shapes of uncoupled equations of motion. The new variables of the uncoupled equations are the modal coordinates. The solution derives after solving these modal equations independently. A superposition of modal coordinates gives solution of the original equations.

As an earthquake excitation seems to affect only the lower frequencies of a building, this method neglects the higher frequencies and mode shapes, without introducing any significant errors in the final solution [5]

Response Spectra Analysis

Response spectra analysis uses the principles of Mode Superposition Method, having as result to reduce the dynamic problem of earthquake excitation to a series of static analyses. By using a specific ground motion loading, a complete time history response of joint displacements and member forces are calculated. This procedure can be used only for linearly elastic analysis. The most important advantage of RSA is that only the maximum values of displacements and members forces of each mode are involved in the solution and this makes the method suitable for the computational analysis of the earthquake excitation problem. It is needed to be mentioned that response spectra is not accurate for the analysis of no-linear multi degrees of freedom structures. [6]

By using this approach the time depended variables are separated from others. So, displacement, velocity and acceleration can be expressed as:




Where Φ is a 'n x n' matrix having 'n' spatial vectors which are not variables of time, and Y ( t ), are the vectors containing variables of time.

To achieve a solution for this problem the space functions need to satisfy the following mass and stiffness orthogonality conditions:



where 'I' is a diagonal unit matrix and Ω2 is also a diagonal matrix, having as diagonal terms the. For simplicity it is assumed that vector Φn is always normalized, that leads the generalized mass to be equal to unity (.

The equation of motion is given as:


By using the above equations, after pre-multiplication by :

⇒ (1.7)

Where is the modal participation factor for load function j.

Even though for real structures the 'n x n' matric d is not diagonal, to uncouple the modal equations it is necessary to assume that there is no coupling between modes and d matrix is diagonal, having as diagonal terms:


For a three dimensional seismic problem, the typical modal equation is:


With mode participation factors:


The calculation of maximum displacements and peak forces must follow, for all directions of this 3-D problem. Then, it is needed to calculate the response of the system having the maximum responses of the three components present at the same time.

For one dimensional analysis Eq. (1.9) takes the form:


The difficulties in the use of response spectra analysis are concerning the large amount of output information produced which are needed for the complete design checking of the structure, as a function of time. Also, repeating this analysis for several different earthquake excitations is needed, to ensure that all significant modes are excited. [7]

Displacement response spectrum

With the assumption that and for a specific ground motion, equation (1.11) can be solved for different values of ω. Using these values, a curve of the maximum peak response can be plotted. This curve represents the displacement response spectrum.

Figure 1.1: Relative Displacement Spectrum - inches, for Loma Prieta earthquake [8]

Pseudo-velocity and pseudo-acceleration spectrum.

Using the same variables as before, plotting and give the pseudo-velocity and the pseudo-acceleration spectrum respectively.

Even though the importance of the displacement response spectrum, the pseudo-velocity and the pseudo-acceleration spectrum is limited, as for their physical values, there is a mathematical relationship that could lead us from the pseudo-acceleration spectrum to the total acceleration spectrum as follows:


For an undamped system, using this equation the total acceleration is equal to . To avoid that, the displacement response spectrum curve is not plotted as modal displacement versus ω, but it is presented in terms of S( ω ) versus a period T in seconds, having S( ω ) as:



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Figure 1.2: Pseudo-acceleration spectrum - percent of gravity, for Loma Prieta earthquake [9]

These response spectrum curves characterize an earthquake at a specific site, so they are not functions of the properties of the structure. Choosing the appropriate linear viscous damping properties of the structural system, a specific response spectrum curve is selected.

Solving (Eq.13) for , the maximum modal response for a typical mode n with Tn and corresponding spectrum response is:

and the maximum modal displacement response is:

Modal combination

For the calculation of the peak values of displacement or forces of a structure, under earthquakes' excitation loading, the modal combination is required. This can be achieved by the following methods:

Sum of the absolute values of modal response.

This is the most conservative method for the calculation of the peak values of displacement or force of a structure under earthquake excitation. In this method it is assumed that the maximum modal values, for all modes, occur at the same time.

Square Root of the Sum of the Squares (SRSS).

In SRSS method it is assumed all of the maximum modal values are statistically independent. For a tree dimensional structure this criteria is not valid, as a large number of frequencies are almost identical. [10]

Complete Quadratic Combination (CQC).

CQC method is based on random vibration theories and it is used in most of the modern computer seismic analysis programs. The peak value of a typical force is estimated from the maximum modal values by using the double summation equation that follows.

is the modal force for mode n.

This modal summation includes all modes. With a similar way they could be derived the peak values of displacements, relative displacements, base shear and overturning moments.

is the so-called cross-modal coefficient, and for CQC method with constant damping is given as:

Having as r:

The cross-modal coefficient array is symmetric and all terms are positive.

From these three types of modal combination, CQC is the one minimizes the introduction of errors to this problem's solution. [11] [12]

Response history analysis

Response history analysis is the analysis of a structure for a specific earthquake excitation, by numerical integration of the equation of motion. In order to perform response history analysis, it is necessary to have a digitized ground motion acceleration record. The main feature that makes this procedure useful is that allows the calculation of solutions of the deflected shape and force state of the structure at each instant time during the earthquake.

As the results of this type of analysis are only for the reaction of the structure during a specific earthquake excitation, to use the result of forces and displacement for design it is required the minimum of at least three earthquake records to be analyzed. Having the minimum of data required, only the maximum forces and displacement from each case will be used for the design analysis. If the existed records are above seven, then the mean forces and displacement could be used.

For the design analysis linear response history analysis is used rarely, as response spectrum analysis is more common. [13]

RHA procedure

For the response history analysis method the displacement u derives from the superposition of the modal contribution:

The force distribution can be calculated as the summation of modal inertia force distribution SR as:


So, the contribution of the nth mode to the excitation vector s = m ι is independent of the normalization of the mode and it is given from the following equation as:

The equation of motion is now given as:


The solution derives from the above equation, solving for a single degree of freedom problem, with the properties of the nth mode of the multiple degree of freedom system. By re- writing equation (26) and replacing u with Dn:



= (1.28)

Modal responses

The nth modes' contribution to the displacement is


By performing static analysis procedure, the equivalent static forces are calculated as:



There are two factors that determine the equivalent static force, the nth mode contribution to the spatial distribution mι of peff ( t ) and the pseudo-acceleration response of the nth mode SDF system to the ground acceleration.

Generalizing, the nth mode contribution to any response quantity is calculated by using static analysis of the structure, for external loading forces fn ( t ).

The equation for this contribution is:


The static displacement due to forces sn is:

By using eq. 2,

Finally, to find the total response of the structure, during an earthquake excitation, the summation of each of the response quantities for all modes must be calculated.

The modal displacements are:

Design Spectrum

The Design Spectrum should be representative of the ground motions that have been recorded during past earthquake excitations. If there are no past earthquake data for the site, design spectrum should be calculated based to ground motions of similar conditions sites.

Design spectrum is calculated as an average of many earthquakes response. A typical design spectrum is shown in the following figure:

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Figure 1.3: Typical Design Spectrum [14]

For major structures usually is developed a site-depended design spectrum, to include the effect of local soil conditions and distance to the nearest faults, as the Uniform Building Code has defined specific equations for each range of the spectrum for four different soil types.

Chapter 2

Elastic framework building under earthquake excitation

The system that will be examined is an n-story framework building with the same height for every story ( H ) and bay equal to L. It is assumed that the flexural rigidity of all beams is constant ( E lb ) and the column rigidity ( E lc ) does not vary with height. All floor are assumed to have the same mass ( m ). Finally, the damping ration of the n natural vibration modes is constant. Also the assumption of lumped mass for each floor has been made, to reduce the degrees of freedom of the building. The sketch of the pre-mentioned system follows:

Figure 2.1 :Model of a twenty story building with 2880 DOF and of framework lumped-mass building [15]

There are two key parameters that help in the understanding of the response of a linear framework building to an earthquake excitation, the beam-to-column stiffness ratio ( ρ ) and the fundamental natural vibration period ( T1 ). By defining these two parameters the system can be defined completely. [16]

Beam-to-column stiffness ratio ( ρ )

This parameter is based on the properties of the beams and columns in the story closest to the mid-height of the building. The value of ρ derives from the equation that follows:

Lb: length of beams

Lc: length of columns

This summation involves only the beams and the columns from the mid-height story of the structure.

Beam-to-column stiffness ration indicates how this system behaves during the excitation of an earthquake. In the case that ρ=0 the beams does not impose any restrain on joint rotation and the behavior of the frame is the same as of a flexural beam. If ρ=∞, there is no joint rotation, as it is not allowed any from the beams, and the structure behaves as a shear beam with double curvature bending of the columns in each floor. Finally, for an intermediate value of ρ beams and columns manage to undergo bending deformation with joint rotation. Normally, for an earthquake resistant structure, ρ should take a value that insures that the stiffness of the columns is greater than the stiffness of the beams.

The basic parameters of the frame that are related to the beam-to-column ratio are, the fundamental natural period, the relative closeness of the natural periods and the shape of the natural modes [17]

Beam-to-column stiffness ratio and fundamental natural period

A diagram that shows the variation of the fundamental period compared to the variation of ρ for a five-story framework building follows.

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Figure 2.2: Fundamental natural vibration period of uniform five-story frame [18]

For constant column stiffness and floor mass, the value of the fundamental natural period is rapidly decreasing, as ρ increases from 0 to ∞.

Beam-to-column stiffness ratio and natural vibration period ratios

Even though the ratios of the natural periods do not depend on the fundamental natural period, they strongly depend on ρ. This effect is even more noticeable on the higher mode periods. In the following diagram can be observed that the values of the natural periods of a building with small ρ are more separated from each other, in comparison with those with larger ρ.

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Diagram 2.3: Natrural vibration period ratios of a five-story building. [19]

Beam-to-column stiffness ratio and shapes of the natural vibration modes

The values of ρ have a crucial impact on the shape of the natural modes. In figure x are presented the shape of the natural vibration modes of a uniform five-story frame, for ρ=0, ρ=1/8 and r=.

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Figure 2.4: Natural vibration modes of a five-story building [20]

Fundamental natural vibration period ( T1 )

Fundamental natural vibration period and beam-to-column stiffness ratio and their influence on structure's response.

In the following diagrams are presented the normalized response quantities of a five-story building to the fundamental natural period T1, for different values of beam-to-column stiffness ratio. The normalized quantities are:

, with the displacement of the top floor (five-story building) and the peak ground displacement

, with the base shear and and is the effective modal mass for the first mode.

, with the base overturning moment and the effective modal height of the first mode

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Figure 2.5: Normalized response quantities [21]

From the above diagrams it is noticeable that the effect of ρ on the top floor displacement is almost absent. If the value of the period T1 is high, the top floor displacement is identical to the ground displacement, as the floor masses remain stationary, while the earthquake occurs.

Normalized shear diagram has the same form as the pseudo-acceleration spectrum. In the same way for small values of T1 the curve goes to 0.5g and for larger values tends to reach zero. There is a variation with ρ to the diagram of the normalized overturning moment, but only for the higher values of T1. Normalized base shear and overturning moment diagrams are not representing for the variation of base shear and overturning moments, as and are depending on ρ.

Design Spectrum of linearly Elastic Building

The design spectrum analysis properties have been introduced to the previews chapter. A figure of the design spectrum to ground motion of a frame with specified T1 and ρ, derived by using the response spectrum analysis follows.


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Figure 2.6: Design spectrum for ground motions [22]

Modal responses - Modal contribution factors

To calculate the peak value of the nth-mode contribution to a response quantity r the following equation is used:

An: ordinate of the pseudo-acceleration response or design spectrum of the nth-mode for natural period Tn and damping ratio ζn.

: Modal static response



Specifically, the modal static responses for the base shear, the top-story shear, the base overturning moment and for the top floor displacement are:




The modal contribution factors are dimensionless and it does not depend in the method of modes normalization. The summation the of all factor is equal to one.

Beam-to-column stiffness ratio and its influence on the higher mode response.

For the understanding of the influence of ρ and T1 on the higher mode response, they are plotted in the following diagram the base shear for all five modes, of a five story frame, and the base shear calculated only from the first mode, for three different values of ρ. Also, in the following diagrams one can observe that the importance of each of the response quantities of the higher modes is limited, in comparison with the significance of the first modes for ρ=∞ (frame behaves like a shear beam). This fact changes, as ρ is reduced and the response quantities of the higher modes have their greatest influence when ρ=0 (frame behaves like a flexural beam). The decreasing trend of ρ seems to affect to the opposite way the values of the higher modes contribution factors for the base shear and the top floor shear especially for the second mode. Also, the ratios of T1/Tn increases as ρ decreases. For that reason, the values of Tn are spread out over a wider period range of the design spectrum. Response quantities tend to have different reaction in the change of ρ on the higher mode response. For a decreasing ρ, the top floor's displacement higher modes contribution decreasing and the opposite reaction follow for the forces. It needs to be noticed that the small values of the displacements for the higher modes reduce their importance to this analysis.

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Figure2.7:Higher node response in Vb, V5, Mb and u5 for uniform five-story frame for three values of ρ.

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Figure 2.8: Normalized base shear in uniform five story frame for three values of ρ, using RSA , including one or five modes. [23]

Fundamental natural vibration period and its influence on the higher mode response.

From the above diagram (figure 2.8) derives the conclusion that the one-mode curves are independent of ρ and identical to the design spectrum. For the higher mode response of a frame and for the acceleration sensitive region of the spectrum, T1 the influence is not significant, but it is more effective, for higher values of T1 in the velocity and displacement sensitive region. This is a result of three factors. To begin with, the static value of remains the same in all modes. Also, for a fixed ρ the modal contribution factor does not depend on T1. Finally, the only factor that has a dependence on T1 is the pseudo-acceleration spectrum ordinate, but it is also depended on period ration Tn/T1, which for a certain value of ρ, it is independent from T1. For these reasons, the design spectrum defines the influence of T1 in higher-mode response Figure 2.9.

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Figure 2.9: Natural vibration periods and spectral ordinates [24]

If the values of T1 and ρ remain fixed, there is a difference between the values of the response contribution of the higher modes, depending to the response quantity. To be more specific, the higher-mode response is more significant for forces than for displacements (for shear frame utop and Mb are identical, as they are identical their modal contribution factors). Also, it is more important for base shear in comparison with base overturning moment. Finally, the higher-mode response is more significant for top-story than for base shear.

To conclude, one of the more important issues in an earthquake analysis is the number of mode one should include. If an exact analysis is to be made, all natural modes should be included. Otherwise, the first few modes are enough to derive an accurate result for the response of a structure during an earthquake excitation. The number of the modes that should be considered can be decided taking under consideration two important factors; The modal contribution factor and the spectral ordinate An.

If only J modes are included to an earthquake analysis, the error in static solution is given from the equation:

The number of included modes should be chosen carefully, so that the error is small.

Generally, it is needed to include more modes for an accurate analysis for flexural frames, than the modes needed for shear frames.

Chapter 3

Elastic framework building under earthquake excitation and finite element analysis

To derive the equation of motion for a frame structure using the finite element method, it is needed to proceed using the following steps:

Idealize the framework building:

A frame structure can be idealized as an assemblage of beams, columns and walls. The structure will be divided in to E finite element, interconnected only at the nodal points. The sizes of the elements are arbitrary; all the elements could have the same size, or different. For this idealization it is assumed that the beam and the columns of this structure are one dimensional.

Define the degrees of freedom at the nodes:

The displacements of the nodes are the degrees of freedom. Generally, in a planar two dimensional frame there are three degrees of freedom for each node; The axial deformation in the x direction ( u ), the deflection in the y direction ( v ) and the rotation in the x-y plane and with respect to the z-axis, θz. In the sum, each element with two nodes will have a total of six degrees of freedom. For the case of a three dimensional frame there are six degrees of freedom for each node, three translation in the ( x, y, z components ) and three rotation ( about the x, y and z axes ). If it is considered only a planar displacement, each node will have just two degrees of freedom, the transverse displacement and rotation.

The axial deformation of beams is usually neglected in the analysis of buildings. The axial deformation of columns is also ignored for low-rise buildings.

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Figure 3.1: DOF of a planar 2-D element: [25]

Elements Matrices and external force components

As first step for this procedure one has to calculate the stiffness ( ke )and the mass matrices ( me ) for each finite element and the applied force vector to this elements ( pe(t) ), with reference to the DOF for the element. The force vector is time depended. The procedure to evaluate the equations for the force - displacement and the inertia force - acceleration relations follows:

Generally, the external forces on the stiffness component of the structure are related to the resulting displacement uj. If the examined system is linear this relationship derives using the method of superposition and the concept of stiffness influence coefficients.

A unit displacement is applied along DOF j, while all other displacements are maintained at zero. To achieve that it is needed to apply forces along all degrees of freedom of the structure.

The stiffness influence coefficient kij is the force applied to DOF i due to unit displacement at DOF j.

Figure xx shows the stiffness coefficients for u1=1 and for u4=1 for a two story, two bay planar two dimensional frame structure.

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Figure 3.2 [26]

Using the method of superposition, the force fsi at DOF i and the relation with the displacements uj ( j=1 … N ) is given from the equation:


This equation can be written in a matrix formation:




For the finite element analysis, the stiffness component of the external force for each element is given as:


For a beam element the displacement is time depended and it is given as:

Where N is the degrees of freedom of the element, function is the interpolation function that defines the displacement of the element due to unit displacement while all other DOFs are zero. For planar displacement of a beam element and ( four DOF's ) needs to verify the boundary conditions that follows:

For i = 1 :

For i = 2 :


For i = 3 :

For i = 4 :

At Figure 3.3 are presented the degrees of freedom for a beam element and at figure 3.4 the shape of the interpolation functions

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Figure 3.3: DOFs of a beam element, considering planar displacement [27]

The shape of could be anything that would satisfy the boundary conditions. It is very difficult to derive the exact deflected shapes of a beam element if the flexural rigidity varies through the element's length. To avoid this difficulty the shear deformation could be neglected, so the equilibrium equation for a beam that is loaded only at its ends is:

From this equation derives a cubic polynomial that describes the displacement of the bar.

The constant variable could be calculated for each of the boundaries conditions and obtain the following equations for the shape functions:

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Figure 3.4: Interpolation functions of a beam element, considering planar displacement [28]

These equations for the interpolation are valuable for the formulation of the element matrices for no-uniform elements. This approach is suitable for one dimensional beam problems, but not for two or three dimensional finite elements. For that reason the finite element procedure is based on assumed relationships between the displacements at interior points of the element and the displacement at the nodes. This approach helps to the simplification of the analysis, but it introduces approximations in the solution.

For a beam element of length L that its flexural rigidity is E I (x), the stiffness influence coefficient kij is the force in DOF i due to unit displacement in DOF j. With the use of the principal of virtual displacement kij can be expressed as:

It is obvious that the symmetric form of this equation will lead to a diagonal symmetric stiffness matrix. In this equation flexural rigidity could be changed through the length of the beam. This does not happen to the interpolation function ψ, that it is exact only for uniform elements. This fact can introduce errors to the solution. These errors could be reduced at any desired level by using a finer finite element mesh. For the elastic framework building a uniform finite element is assumed ( ).

Damping forces

The model of the energy dissipation of a vibrating structure can be idealized by equivalent viscous damping. Using this assumption it is possible to relate the external forces ( fDj ) acting on the damping component of the structure to the velocities ( ). A unit velocity is assumed along DOF j, having all the other velocities in the other DOF equal to zero. These velocities will generate internal damping forces that resist the velocities. As the equilibrium of the forces should maintained, external forces are needed. The damping influence coefficient cij is the external force in DOF i due to unit velocity in DOF j. Using the method of superposition, the force fDi at DOF i and the relation with the velocities ( j=1 … N ) is given from the equation:


This equation can be written in a matrix formation:




For the finite element analysis, the damping component of the external force for each element is given as:


Generally, the coefficients cij are not calculated directly from the dimensions of the structure and the sizes of the structural elements, as it is a very difficult process. For this reason, the damping is specified by numerical values for the damping ratio, based on experimental data for similar structures.

Inertia Forces:

The inertia forces fIj, acting on the mass component of the structure, are related to the acceleration. Using the same procedure as previous, a unit acceleration is applied to the DOF j, and all accelerations of the other DOF are remaining zero. As a result of this action, the fictitious inertia forces oppose these accelerations ( D' Alembert's principle ). To keep the force equilibrium it is necessary to have external forces opposites to the inertia forces. The mass influence coefficient mij, is the external force that is applied in DOF I due to unit acceleration along DOF j. Using the method of superposition again, the force fIi at DOF i and the relation with the acceleration (j=1 … N ) is given from the equation:


This equation can be written in a matrix formation:




m is the mass matrix and it is symmetric ( = )

For the finite element analysis, the mass component of the external force for each element is given as:


The mass influence coefficient mij for a structure is the force in the i DOF due to unit acceleration in the j DOF. For a beam element using the principle of virtual displacement the following equation derives:


As previous with equation for kij, the symmetric form of this equation will lead to a symmetric mass matrix ( ). Having the assumption of a lumped mass element, where the mass is distributed as point masses along the translational DOF at ends and the two masses are calculated using static analysis of the beam, the lumped mass matrix takes the form:


All the off diagonal terms of the lumped mass matrix are zero,

The external force p ( t ) is now expressed to be distributed among the three components of the frame and can be calculated for each element as the summation of the stiffness component , the damping component and the mass component

Form the transformation matrix ( ae) that relates the element displacements ( ue) and forces ( pe ) to the displacement ( u ) and forces ( p ) for the finite element assemblage:

and (3.22)

In these equation represents a Boolean matrix consisting of zeros and ones. Its necessity is for locating the elements of and matrices at the proper locations in the global stiffness matrix, mass matrix and applied force vector respectively. This transformation matrix relives as from the obligation to carry out the transformations , and to transform the element stiffness and mass matrices and applied force vector to the nodal displacement for the assemblage.

Start with the composure of the element matrices to determine the stiffness and mass matrices and the applied force vector for the assemblage of finite elements:

(3.23) (3.24) (3.25)

The operator A denotes the "direct assembly" procedure for assembling according to matrix the elements of the stiffness and mass matrices and the elements of the force vector, for each element from e = 1 to e = N, where N is the total numbers of the elements of the structure and are assembled ,in to the global matrices.

Formulate the equation of motion for the prescribed finite element assemblage:


To conclude, finite element analysis of a framework elastic structure has the same solving procedure with the displacement method of analysis for the same structure, having as a difference the formulation of element stiffness and mass matrix.