OverviewThis module performs dynamic analysis of a Block Foundation resting on ground and

reports frequencies and amplitudes corresponding to different modes of vibration.

The analysis is based on the method proposed by Barkan [1]. Block Foundation (as shown in Fig. 1) is idealized as a system with six degrees of freedom - three translations along three mutually perpendicular axes x, y and z and three rotations about the same axes. The modes corresponding to translation along the vertical axis (z) and the rotation about the same axis are treated as uncoupled. This is based on the assumption that the common centre of gravity of machine and foundation and the centroid of the base area lie on the same vertical line or within a tolerable limit. On the other hand, the sliding and rocking motions in each of the two vertical planes (xz and yz) passing through the common centre of gravity of machine and foundation are separately coupled.  

 

The effect of damping has been ignored and the supporting soil is modelled by linear weightless springs. Some soil parameters have been introduced to yield the spring stiffness of soil in various modes. These parameters are Coefficient of elastic uniform compression (Cz), Coefficient of elastic non-uniform compression (C), Coefficient of elastic uniform shear (C), and Coefficient of elastic non-uniform shear (C).

The spring stiffnesses of soil (K) for the various modes of vibration are calculated as

For vertical motion   Kz= Cz Af

For horizontal (or sliding) motion K = C  Af

For rocking motion K = C Ix  (or y)

For torsional motion (rotation about vertical axis)  Iz

K = C

                                     

where,   

Af = area of horizontal contact surface between foundation and soilIx (or  y) = second moment of contact area about the horizontal axis (x or y)

passing through the centroid of the base and normal to the plane of rocking

Iz = second moment of contact area about the vertical axis passing through the centroid of  base area    

Vertical Translation (along z axis)

The equation of motion for the undamped, single degree of freedom system is expressed as:

where,

z = displacements along z-axism = mass of the foundationPz(t) = time varying exciting force

along z-axis

 

The circular natural frequency (z) is

The vertical amplitude (az) under the action of an exicting force Pzsinmt is

Where m is the circular operating frequency of the machine.

[Analysis for the torsional (rotation about z-axis) motion i.e. yawning as shown in Fig.1 is not carried out]

Sliding and Rocking Motion  (in xz plane)

The equations of motion for the undamped, two degree of freedom system are expressed as:

For horizontal motion (along x-axis)

For rocking motion (about y-axis)

where,

x = displacement along x-axis y =  rotation about y-axis W = weight of the foundationPx(t) = time varying exciting force along x-axis My(t) = time varying exciting moment about y-axisS = height of the centre of gravity of the foundation

from the base

 

The circular natural frequencies for coupled modes are obtained from the following equation

where, y is the ratio of the mass moment of inertia (y) about the y-axis passing through the centre of gravity to the mass moment of inertia (0y) about a parallel axis through the centre of elasticity of the base support.

The limiting frequencies of the coupled motion are expressed as

and

x and y correspond to the pure sliding along x-axis and pure rocking about y-axis respectively.

The horizontal amplitude (ax) and rotational amplitude (ay) under the action of an exciting force Pxsinwmt and exciting moment Mysinmt are expressed as

and

where,

So the net displacement along the x-axis at top of the foundation is

Sliding and Rocking Motion  (in yz plane)

The natural frequencies and the amplitudes are calculated using equations similar to those derived for motion in xz plane with the suffixes x and y interchanged in it.

The output reports the natural frequencies and amplitudes corresponding to the modes as discussed above.

References:

1. Barkan D. D. Dynamics of Bases and Foundations, McGraw-Hill Book Co. Inc., New York 1962

2. Major A. Dynamics in Civil Engineering Analysis and Design, Akade/miai Kiado/ , Budapest 1980

3. Srinivasulu.P and Vaidyanathan.C.V. Handbook of Machine Foundation, Tata McGraw-Hill Publishing Company Ltd., New Delhi 1996

4. Arya C.S., O’Neill W.M. and Pincus G. Design of Structures and Foundations for Vibrating Machines, Gulf Publishing Company, Book Division, Houston 1979.

Machine Data

Input Page 1 of 4 represents Machine Data :

Operating speed of engine(r.p.m)

The operating speed of the engine in r.p.m.The programwill convert it in specific unit system as per requirement

Weight of Machine The weight of machineX Co-ord of CG of machine

X Co-ordinate of Center of Gravity of the Machine

Y Co-ord of CG of machine

Y Co-ordinate of Center of Gravity of the Machine

Z Co-ord of CG of machine

Z Co-ordinate of Center of Gravity of the Machine

Horizontal exciting force along X-axis

Horizontal exciting force along X-axis

Horizontal exciting force along Y-axis

Horizontal exciting force along Y-axis

Exciting Moment about Y-axis

Exciting moment about Y-axis

Exciting Moment about X-axis

Exciting moment about X-axis

Height of foundation Height of foundationHeight of acting point of horz. exciting force from the top of foundation

Height of acting point of horizontal unbalanced force from the top of foundation

Vertical exciting force (along Z-axis)

Vertical Exciting force along Z-axis

Permissible percentage limit for eccentricities  in either direction

Permissible limit for checking the eccentricities in either direction   

Maximum allowable horizontal  amplitude

Maximum allowable horizontal amplitude

Maximum allowable vertical  amplitude

Maximum allowable vertical amplitude

 

Geometry Data

Input Page 2 of 4 represents Block Foundation Plan :

Block No The No. of Block(s)X-Coordinates of CG The X-Coordinate of CG of the

BlockY-Coordinates of CG The Y-Coordinate of CG of the

BlockZ-Coordinates of CG  The Z-Coordinate of CG of the

BlockUnit The unit of X,Y &Z Coordinate of

CG of the BlockX-Dim (Block Dimensions)

The Length of the Block in X-direction

Y-Dim (Block Dimensions)

The Length of the Block in Y-direction

Z-Dim (Block Dimensions)

The Length of the Block in Z-direction

Unit The unit of Length of the Block in X,Y&Z direction

 Soil Data

Input Page 3 of 4 represents Soil Data :

Unit weight of Soil The unit weight of SoilBearing capacity of Soil The allowable Bearing Capacity of SoilCo-efficient of elastic uniform compression

The Co-efficient of elastic uniform compressionof Soil

Co-efficient of elastic non- The Co-efficient of elastic non uniform

uniform compression compression for rocking mode of SoilCo-efficient of elastic uniform shear

The Co-efficient of elastic uniform shear of Soil

Co-efficient of elastic non-uniform shear

The Co-efficient of elastic non uniform shear ofSoil

 

Material Properties

Input Page 4 of 4 represents Material Data :

Unit Weight of Concrete

The unit weight of concrete

 

Output Data

The Output consists of the Analysis Results The results for all the design modules are presented in a grid or spreadsheet format which is compatible with Microsoft ExcelÓ . section.

Analysis Results

Analysis Results reports the natural frequencies and the amplitudes corresponding to the modes as discussed in Overview.