In order to determine the dynamic behaviors of soils, are scarce, unrealistic
assumptions of existing studies about this subject like assuming the soils membranes
as rigid so disregard the interaction and assuming the earth movements as harmonic,
always exist. One of the numerical methods, finite element method is used at
the structural analysis which considers the mentioned interactions. This method
is applied to the interaction problem in the form of Euler and Lagrangian approaches
with Westergaards added mass approach (Karaca and
Durmus, 2000; Dogangun et al., 1996; Rammerstorfer
et al., 1990; Bathe, 1982; Adedeji
and Ige, 2011; Azizpour and Hosseini, 2009; Alsulayfani
and Saaed, 2009).
One of the new techniques in seismic design were based on changing the dynamic
characteristics of buildings by dissipating the energy with lower damage in
structural components of the system (Sharbatdar and Saatcioglu,
2009; Adedeji and Ige, 2011).
Regarding the properties of the soils, in this study, isoparametric element
which is used at the analytical solutions is assumed to be an elastic solid;
only the impulse pressure is taken into consideration at analytical methods.
And the silo-material interaction is examined according to the East-West component
of Erzincan in 1992 earthquake with the utilization of Lagrange formulation
by adapting the mentioned eight nodded three-dimensional isoparametric element
to the Structural Analysis Program (SAPIV) (Bathe et al.,
1973; Alsouki et al., 2008; Alfaaouri
et al., 2009). Fallah et al. (2009).
The application of modern control to diminish the effects of seismic loads on
structures offers an appealing alternative to traditional earthquake resistant
design approaches (Adedeji and Ige, 2011; Alsulayfani
and Saaed, 2009). Finally, the data obtained from the analysis of silo is
compared in-between them by many aspects and some justifications are reached
about the utility of Lagrange Approach at the cylindrical silos.
DETERMINATION OF DYNAMIC PRESSURE DISTRIBUTIONS USING SOME ANALYTICAL AND NUMERICAL METHODS
Some analytical methods: With the assumption that the slosh effects
at granular material could be neglected, besides Westergaard, Karman and Hoskins-Jacobsen
methods which are used at liquid containers and only consider the impulse effects,
The relations used at calculations are given at Table 1. At
these relations is the maximum acceleration of earth movement ρ is the
material unit mass, h is the height of the contained material and r is the radius
of the silo. Here it should be stated that these relations are appropriate for
rigid silos (Karaca and Durmus, 2000; Housner,
1957; Nasserasadi et al., 2008).
In adapted Veletsos method (Jardaneh, 2004), the impulse
pressure can be estimated from the following equation by obtaining the dimensions
qi(0) value from Fig. 1a according to h/r ratio
and the dimensionless qi(z) value from the chart of Fig.
|| Analytical methods used for determining dynamic pressure
|| qi (0) and qi (z) values for the estimation
of impulse pressure
Some numerical methods
Added mass approach: The principle of added mass approach is based on the
study made by Westergaard (1931) and Priestley
et al. (1986). In that study, Westergaard added a mass which creates
the dynamic pressure, to the structural mass at the interface of fluid-structure.
The value of added mass which has parabolic distribution from the material surface
to bottom can be obtained by the following expression:
where, h is totals material height, z is the depth of the material from the surface and ρ is the unit mass of the material.
In this study, the use of added mass approach with finite element method is made by adding an impulse mass determined using different methods for the materials to the mass of solid elements.
For this purpose, equation of motion given as:
can be written in the following form for the added mass approach:
The equation of the damped impulsive motion is known to be as following, where, M is mass matrix, C is damping matrix, K is rigidity matrix, u is displacement vector and a(t) is the acceleration of base motion.
In the added mass approach, this motion equation takes the following form, where, Ma is added mass matrix and M* (= M+Ma) is total mass matrix.
As it is seen from this relation, according to this approach it has been assumed that Ma mass vibrates simultaneously with the structure and because of the contained material, only the mass in the motion equation increases and the damping does not change.
This method which is not able to consider the slosh effects, can be practically
used at engineering structures like silos in which the impulsive effects dominates,
by adding the membrane to the finite elements model at the membrane-material
interface of the material mass (Celep and Kumbasar, 1992;
Hangai et al., 1983; Ching
et al., 2011).
Lagrangian approach: In this approach, material behaviour is expressed by the displacement term at the finite element node points and thus the equilibrium-appropriateness conditions are provided at the points of membrane-material interface automatically.
Assumptions made for this study are given below:
||Neglecting the slosh effects caused by base motion at the
granular material in the silo, only the impulsive effects are taken into
||The contained material is assumed to be compactable, behaves linearly
elastic, whose viscosity effects are negligible and at which the rotation
For the three dimensional model, the equation can be written as follows where,
εv is the unit volumetric strain, ux, uy
and uz are the material displacements along x, y and z axes, respectively,
p is pressure and Ev is the bulk modulus of material:
Rotations about x, y and z axes which are necessary in order to supply rotation constraints are, respectively expressed as:
From here, the rotation pressures, pxr, pyr and pzr can be as the following; where, Ψx, Ψy and Ψz shows constraint parameter coefficients for the axes orientations of x, y and z and E22 = ΨxEv, E33 = ΨyEv and E44 = ΨzEv.
Accordingly, the total potential (U) and kinetic energy (T) of the system is determined by the equations of:
where, E, ε and v shows elasticity matrix, strain and velocity vectors, respectively. Therefore, the Lagrange equation can be written as:
where, here ui and Fi show i numbered displacement component
and external load corresponding to this component, respectively and this equation
behaviors can be applied to the nonlinear systems as well as to the linear systems
(Karaca and Durmus, 2000; Celep and
Kumbasar, 1992; Hangai et al., 1983).
Utilization of the three dimensional isoparametric element by finite element modeling: In this study, the selected three dimensional isoparametric element with eight node points and general (x, y, z) and local (r, s, t) axes groups which are considered for this element, are given in Fig. 2.
|| Isoparametric element
Mass and stiffness matrices of this element should be known in order to determine the motion equations. Mass matrix of the element can be expressed as:
where, J is the Jacobian matrix, Q is the interpolation functions and ηi, ηj, ηk are weight coefficients.
In order to obtain the stiffness matrix, the elasticity matrix (E) whose diagonal elements are E11, E22, E33 and E44, respectively and the other elements are zero and the strain-displacement matrix (B) at the equation ε = B.u, where, εT = [εv, εxr, εyr, εzr], should be known. Thus, the element stiffness matrix is as:
After the mass and stiffness matrices of the selected element was determined
by the equations of (11, 12), potential
and kinetic energy expressions can be written as:
according to finite elements method. As it is seen from this equation, as only
the impulse effects are taken into consideration in granular material, no term
related with surface slosh takes place in potential energy expression. If the
mentioned energy expressions are replaced in the Lagrange equation of number
(10), the motion equation of the undamped system can be obtained
Silo application: Here, adapting the three dimensional isoparametric
element whose formulation was given before, to the structural analysis program
SAPIV (Bathe et al., 1973), such that the surface
slosh elements could not be used, computation of the silo whose characteristics
are given in Fig. 3, is done according to the East-West component
of the 1992 Erzincan Earthquake (Karaca and Durmus, 2000)
|| Silo plan and vertical section
|| Earthquake acceleration of the March 13, 1992 Erzincan earthquake
In this computation, the bulk modulus, Poisson ratio and the unit mass of
the contained wheat is taken as Ev = 4,167x107 N/m2,
v = 0.30 and ρ = 800 kg m-3, respectively and unit mass, Poisson
ratio and elasticity modulus of silo wall material is taken as ρ = 2500
kg m-3, v = 0.2 and E = 285x108 N m-2, respectively.
||Finite element meshes considered at rigid analysis of the
silo material dynamic pressure distributions obtained by analytical methods
and Finite Elements Method (FEM) by using Lagrangian approach is given in
Solution by assuming the silo to be rigid: In this solution, it is assumed that the silo base and walls are rigid and four models of the silo which are shown in Fig. 5 with unit width at diameter length in the direction perpendicular to base motion created by the earthquake, is considered in order to compare the obtained results with the ones obtained from analytical methods.
From this Fig. 6, it is seen that the difference between material dynamic pressure values estimated by finite elements method according to four different models does not exceed 12% at the base, pressures increase in case the finite element mesh is densed about the membrane considering a determined convergence, distribution obtained by the help of Model 1 practically coincides with the one obtained by.
Adapted Housner method, Westergaard method gives larger values at a maximum
of 19% from all of the models at the base and the distributions obtained by
finite elements method. This situation indicates that finite elements method
which uses the element selected by Lagrangian approach can be used effectively
as analytical methods at the rigid analysis of silos (Karaca
and Durmus, 2000).
Variation of the material dynamic pressure formed during earthquake at the
5 numbered element of Model 1 and Model 3 by finite elements method is given
in Fig. 7. As it is seen, variation of material dynamic pressure
due to time is the same of earthquake accelerogram (Karaca
and Durmus, 2000; Hangai et al., 1983; Durmus,
1997, 1993; Dogangun and Durmus,
|| Material dynamic pressure distributions estimated by analytical
and Lagrangian approach
|| Variation of material dynamic pressure at the 5 numbered
element of silo during earthquake
Flexible solution according to model considering the entirety of the silo
Lagrangian approach: In this solution, wall thickness (tw)
is taken to be 0.75 m. It is assumed that the walls have a prescribed flexibility
depending on material and geometric characteristics. The silo model for the
analysis by the finite element method by assuming walls to be flexible is given
in Fig. 8. In this model, it is assumed that the silo is fixed
to the base, so that all degrees of freedom at the silo base are zero. Truss
element is used for materials free vertical movement and equal horizontal
displacement with wall.
Considering the entirety of the silo, material dynamic pressure distribution, obtained from the analysis realized on the model seen in Fig. 8, is given in Fig. 9 with the pressure distribution obtained for silo model having a unit width with the same element dimensions and the variation of horizontal displacement with the silo membrane thickness is given in Fig. 10.
|| Silo model used for the entirety of the silo at the solution
by Lagrangian approach
|| Material dynamic pressures for the entirety and unit width
of the silo
These Fig. 9 and 10 show that material dynamic pressures obtained from the analysis of silo model with unit width, are smaller than the ones obtained from the analysis of the entirety of the silo, such kind of silos designed according to silo analysis with unit width might have remained at insecure side and as the membrane thickness increase, horizontal displacements decrease.
Added mass approach: Finite element mesh considered at added mass approach used in the comparison of displacement and stress values obtained from Lagrangian Approach is given in Fig. 11. The unit weight of the wall which is 25 kN m-3 with the use of this method is taken as 57.21 kN m-3. Horizontal displacement distributions obtained according to both of the two methods are given in Fig. 12.
|| Variation with membrane thickness of displacement
|| Finite element mesh used at added mass approach analysis
Normal stress distributions obtained from both of the two methods are given in Fig. 13. It is seen from Fig. 13 that horizontal displacement distributions obtained by Lagrange and Added mass approaches coincide up to half of the height from base, later Lagrangian approach gives 23% greater values at silo top end, stress distributions are similar to each other in form and the σz distributions obtained by both of the two methods give very close values to each other.
|| Horizontal displacement distributions
|| Normal stress (σx, σy, σz)
distributions obtained by Lagrange and added mass methods
Some conclusions and proposals which can be deduced from this study are summarized below:
Material dynamic pressure distributions, obtained from the rigid solution by Lagrangian Approach according to four different models of silo which are subjects to numerical applications, locate between the distributions which are determined by analytical methods. And this demonstrates that finite elements method which uses the element selected by Lagrangian Approach can be used effectively as the analytical methods at the rigid analysis of silos.
According to the results obtained from the flexible analysis of silo, on the contrary for the ones obtained from rigid analysis, material dynamic pressure reaches to maximum at the mid of the approximate depth, not at the silo base, and then the decrease in dynamic pressures.
Material dynamic pressure obtained from the flexible analysis with unit width of the silo are smaller than the ones obtained from the flexible analysis entirety of the silo and this demonstrates that such kind of silos which are designed according to silo analysis with unit width can remain at unsafe side.
It is seen that, stress distributions, obtained by Lagrange and Added Mass Approaches which are used at the flexible analysis of the entirety of silo, are similar to each other in form and σz distributions obtained by both of the two methods are very close to each other.
In summary, this study shows that isoparametric element selected by Lagrangian approach can be used efficiently in the earthquake estimation of reinforced concrete cylindrical silos by neglecting the slosh effects in compare with the analytical methods and added mass approach.
These conclusions are drawn from the numerical examples considered in this study. In order to generalize them, theoretical and experimental studies should be made on many models and the results of assemblage of models should be evaluated all together.
||Acceleration of base motion
||Maximum acceleration of earth movement
||Elasticity modulus of silo
||Bulk modulus of material
||Height of the contained material
||Total mass matrix
||Added mass matrix
|pxr, pyr, pzr
||General time varying load vector
||Radius of the silo
|ux, uy, uz
||Material displacements along x, y and z axes
||Material unit mass
|ηi, ηj, ηk
||Unit volumetric strain
|Ψx, Ψy, Ψz
||Constraint parameter coefficients for the axes orientations of x, y and