Comparison of various methods used in the analysis …€¦ · Comparison of various methods used...

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Comparison of various methods used in the analysis of silos without wall friction

M. Khouri Department of Civil Engineering, Lebanese University, Lebanon

Abstract

The use of circular silos with independent hopper structure in the process of storing grain and other granular material has become very extensive nowadays. The structural analysis of silos is of a complex nature and requires sophisticated mathematical techniques for proper calculation. Janssen, Airy and Reimbert suggested different material lateral pressure approaches that can be followed in the process of calculating silo wall forces. This paper compares the above three methods to the water behaviour approach and makes use of the finite element method to check the closed form techniques. Also, average values of the three approaches were determined for various silo dimensions and graphed for the use of engineers and designers.

1 Introduction and background

Different types of silos have been developed but the most used are circular ones. Recently, the use of a totally independent structure of hopper within the silo was found to be more economical due to the isolation and simplicity in the process of construction and design. Silos depend on many factors such as the stored material, the type of silo, the construction material properties, the height and diameter of the silo, wall thickness, the angle of the hopper, etc. Many investigators have studied circular silos. Timoshenko et al. [1,2] presented the differential equation that can be used in solving silo problems. The equation calculates the deformation shape of the wall for any pressure function by isolating a vertical strip in the wall and using symmetry. In 1895, Janssen [3] suggested a pressure application approach, while Airy [5] addressed his pressure application approach in 1897; Reimbert and Reimbert [4] in 1976 suggested a pressure application approach, which is similar in nature to the other two

Computational Methods and Experimental Measurements XII 425

methods. These material pressure application approaches can be used in Timoshenko's equation to calculate silo wall forces. Safarian and Harris [6] attempted to do an evaluation of the methods (Janssen, Airy, and Reimbert). Also Thonier [7] presented the strip analysis, which by itself, is a very interesting and useful approach. Many engineers including Safarian and Harris [6], Reimbert and Reimbert [4], and Fintel [8] have also worked on the analysis of silos, which led to the setting up of the ACI Committee 313, which is the “Committee on Construction of Concrete Bins, Silos, and Bunkers for Storing Granular Materials” [9]. Note that the angle of the hopper is not considered in this analysis and the flow of the material within the silo is assumed to be uniform (mass-flow) [10] where the outlet is considered to be sufficiently large to allow the flow without creating a stagnant solid in the area where the hopper meets the silo wall. In addition, the wall friction between the material and the silo wall is not taken into account in the differential equation, keeping in mind that the methods that are being compared consider friction in the determination of the lateral pressure. (Wall friction evaluation will be presented in a future paper following this). This study is divided into three parts, the first part of this work evaluates Janssen, Reimbert and Airy’s static lateral pressure application approaches, compares them to each other and to the water behavior pressure application approach. In the second part, the silo is modeled using the finite element method to check the mathematical closed form techniques. The third part however, deals with finding average values for Airy, Janssen, and Reimbert for various silos with various heights and diameters, and various wall thicknesses; the average values are then graphed to allow the designing engineer to simply use them. It is important to note that the following work is limited to static analysis and silos also need to be checked for wind and seismic loads.

2 Comparing the approaches

We begin by setting up the differential equation, perform the analysis for each approach, and compare the results for circular silos with both hopper and the wall structure being independent. (See Figure 1.)

Figure 1: Silo geometry.

426 Computational Methods and Experimental Measurements XII

The wall of the silo is divided into two parts: 1. The upper part where the lateral pressure “p” is applied between ho and (ho + h1). 2. The lower part where no lateral pressure is applied between the base and ho Timoshenko gives the differential equation to calculate the deformation shape of the wall by isolating a vertical strip in the wall and using symmetry.

The differential equation given by Timoshenko and Gere [1] is:

DEpy4λy 4(4) =+ (1)

in which y is the lateral displacement, λ is the elastic length given by,

epRν13λ −

= , and DE is the flexural rigidity given by ( )2

ν-112epDE

where ep is the wall thickness, R is the average radius, and ν is Poisson’s ratio. The solution of the differential equation (1) for the upper part h1 of the silo without the right term is:

[ ] [ ]λx λxhy e A1.cos(λ x) B1.sin(λ x) e A2.cos(λ x) B2.sin(λ x)−= + + + (2)

If the particular solution of equation (1) is yp, the general solution of the differential equation (1) for the upper part h1 of the silo is:

phg yyy += The general solution of the differential equation (1) for the lower part ho where there is no grain is:

[ ] [ ]λx λxhy e C1.cos(λ x) D1.sin(λ x) e C2.cos(λ x) D2.sin(λ x)−= + + + (3)

A1, A2, B1, B2, C1, C2, D1, and D2, are constants. Note that for the water behavior and Janssen approaches the method for finding the particular solution is direct, while in case of Airy and Reimbert approaches the use of the method of reduction of order is needed.

2.1 Boundary conditions

y1(x) is the general solution of the differential equation for the upper part of the silo h1, and y2(x) is the general solution of the differential equation for the lower part of the silo ho.

• Due to the existence of the cover at the top of the silo, it is assumed to be pinned or simply supported on its circumference: For x = h1 + ho y1(x) = 0 and y1’’(x) = 0

• At the bottom, the silo is fixed: For x = 0 y2(x) = 0 and y2’(x) = 0

• At the point where the upper part of the silo meet the lower part, the function and its derivatives are continuous and are represented as follows: For x=ho y1(x) = y2(x) y1’’(x) = y2’’(x) y1’(x) = y2’(x) y1’’’(x) = y2’’’(x)

The problem now is to solve the eight equations for eight unknowns given the above boundary conditions.

2.2 Water behavior approach

In this method, the distribution of the lateral pressure “p” applied to the wall is assumed to be linear (or triangular) with depth “x”, and can be represented as follows:

xhoh1pp o , in which x is taken from the bottom of the

silo, γ is the weight per unit volume, ko is the Rankine coefficient for active earth pressure, and po is the pressure of the stored material at depth h1 below surface of

stored material, h1γkp oo = , where ko is given by sinρ1sinρ1k o +

−= , where ρ is

the angle of internal friction for the stored material, and h1 is the height of the storage zone. The solution of the differential equation (1) for the upper part h1 of the silo without the right hand side term is obtained from equation (2), and the particular solution of equation (1) is:

fp1(x)h

xhoh1.KKp

0P ≡

−+= , in which .DE4.λ

RE.epKK 4

2 == , and

E is the elastic modulus. The general solution of the differential equation (1) for the lower part ho of the silo is obtained from equation (3). The resolution of the differential equation is made for function ff (presented later) which is the deflection multiplied by the elastic modulus; this allows the calculations independent of the silo material. The following analysis was done using MathCAD 2000 and can be summarized as follows: The boundary conditions give M * Constant vector = VV, where M is an 8x8 matrix corresponding to the application of the boundary condition, and the vector VV is:

fp1 ho h1+( )−

d2fp ho h1+( )−1

⋅⋅

fp1 ho( )−

d1f ho( )−1λ

d2fp ho( )−1

⋅⋅

d3fp ho( )−1

2 λ 3⋅⋅

where,

d1f x( )xfp1 x( )d

d2fp x( )

2xfp1 x( )d

d3fp x( )

3xfp1 x( )d

By solving this system of 8 equations, the values of the eight constants of integration AA, AA1, BB, BB1, CC, CC1, DD, and DD1 can be determined. Each function is divided into two intervals where u Є [0, ho] and x Є [ho, ho + h1]. The functions ff (x) and gg (u), which represent (Deflection * E) are:

The wall moment in the vertical direction is:

M(x) = DE . {d2fp(x) + (2- λ2) [ff(x)]}, and

M(u) = DE . (2- λ2) . [gg(u)], The wall shear in the vertical direction is:

V(x) = DE . {d3fp(x) + 2 . λ3 . [ff(x)]}

V(u) = DE . 2 . λ3 . [ff(x)] The force N in the circumferential direction is:

N(x) = ff(x) . [ep / R]

N(u) = gg(x) . [ep / R]

2.3 Janssen’s approach

Janssen’s method is based on equilibrium of a thin horizontal layer of stored materials and the equation for horizontal pressure is:

hR.Y.kµ

e1µγ.Rp in which, Y is the depth, ko is the Rankine

coefficient for active earth pressure, Rh is the Hydraulic radius, and µ1 is the angle of friction of the stored material against the wall. The procedure will be the same as for the water behavior approach. However the particular solution fp1(x) is:

fp1 x( )1

KKβ⋅ 1 4

α4 4 λ

⋅ e α− h1 ho+ x−( )⋅⋅−

with 2RR

Rkoγ.β

Rkoµ1.α h

Having the particular solution, the constants of integration and the functions that represent deflection*E, moment, shear and force N in the circumferential direction can be determined similar to the water behavior approach.

2.4 Airy’s approach

The Airy’s equation was derived by considering static equilibrium of wedge-shaped portions of the stored material above the plane of rupture. Airy’s equation for silos leads to lateral pressure at depth Y:

1 µ.µ1µµRY

µ11µµR..2p γ in which, µ1 is the angle

of friction of the stored material against the wall, µ is the internal friction of the material µ = tan (ρ), and ρ is the angle of internal friction for the stored material. The procedure is similar to the water behavior approach. However, the particular solution fp1(x) is calculated as a linear differential equation with constant coefficients using the method of reduction of order [11]. The differential equation can be written using the operator techniques in factored form as:

)().).().().(.( 43210 xRymLmLmLmLa =−−−− , (4) where m1, m2, m3, m4 are constants, L is the linear operator, and a0 is the coefficient of the highest derivative. This method yields the general solution if all arbitrary constants are kept, while if arbitrary constants are omitted it yields a particular solution [11]. In this case a0 = 1, m1 = -l(1+i), m2 = l(1-i), m3 = l(-1+i), and m4 = l(1+i). To solve the first order linear differential equation, use the general solution for the equation ⇒=+ q(x)p(x).y(x)(x)y'

∫ ∫+∫∫=−− pdxpdxpdx

c.edxQ.e.ey , where c is an arbitrary constant (take c = 0),

q x( ) δ1 1α

ε1 x⋅ β+

⋅:=, where,

δ1 γ1

µ1 µ+( )⋅2 R⋅DE

⋅:=α 1 µ

2+:= β 1 µ1 µ⋅−:= ε1 2

µ1 µ+( )2 R⋅

y1 x( ) em1 x⋅ xq x( ) e m1− x⋅⋅⌠⌡

d⋅:=

y2 x( ) em2 x⋅ xy1 x( ) e m2− x⋅⋅⌠⌡

d⋅:=

y3 x( ) em3 x⋅ xy2 x( ) e m3− x⋅⋅⌠⌡

d⋅:=

y4 x( ) em4 x⋅ xy3 x( ) e m4− x⋅⋅⌠⌡

d⋅:=

Let )Re(y4(x)z(x) = , (Re means the real part of) where x is the depth taken from the top to the base of the silo. After making the analytic transformation fp1 becomes, x).-h1z(hofp1(x) += Having the particular solution, one can calculate the constants of integration and determine deflections and forces as before [11].

2.5 Reimbert’s approach

Reimbert method for computing static pressure considers that at the bottom of the silo, the lateral pressure becomes asymptotic to the vertical axis. At depth ho, the lateral pressure reaches a maximum value equal to pmax. The procedure is similar to Airy’s approach where the lateral static pressure at depth Y given by the Reimbert equations is:

max 1CY1pp , in which

hmax µ

γ.Rp = with Rh being the Hydraulic

radius = R/2 for circular silo, and 01.k4.µ

DC= , with D being the diameter of silo.

2.6 Numerical comparison between the four approaches

In order to compare the four methods, a circular silo is taken with the following variables:

γ = 0.75 T/m3; ho = 10 m; h1 = 30 m; D =10 m; ep = 0.3 m. ν = 0.2; ρ = 30 o; µ1 = 0.3; Total Height = 40 m.

Figure 2: Pressure vs. height for the four methods.

Figure 2 shows the difference between the pressure application approaches for all four methods. The following Figures 3–6 represent the comparison of results (deflection, moment, shear and circumferential force) for the four methods. Note that, on the graphs, number 1 represents the water behavior approach, number 2 represents Janssen’s approach, number 3 represents Airy’s approach, and number 4 represents Reimbert’s approach.

Figure 3: Deflection*E vs. height for the four methods.

Figure 4: Moment vs. height for the four methods.

As can be seen in Figure 4, the moment is small as we move away from the top of the hopper upwards, while in the same region we have a deflection. This is due to the fact that the shell membrane forces compensate the deflection thereby decreasing the flexural forces in the wall of the silo. Note that to evaluate the deflection, values of the graphs (Deflection*E) should be divided by the elastic modulus E. It can be noticed from Figures 2–6 that Janssen, Reimbert, and Airy method give comparative results while the water behavior method gives larger values as compared to the above three methods. So, averages were done for the Reimbert,

Airy and Janssen methods only excluding the water behavior approach since it tends to exaggerate the values.

Figure 5: Shear vs. height for the four methods.

Figure 6: Force N in the circumferential direction vs. Height for the four methods.

In comparing the three methods (Janssen, Reimbert, Airy), Reimbert’s gives values little greater than the other two. In addition, it seems that Airy method does not give reasonable values at the top of the silo due to the fact that his pressure equation does not properly represent the behavior in that region, as can be seen in Figure 2.

3 Comparison with the finite element method

In order to verify the validity of the results obtained by solving the differential equation, analysis of the same silo studied above was done using the finite element method (Effel Program). See the finite element model in Figure 7. The lateral pressure considered was as defined by the water behavior approach and

the results of the deflection, moment, shear and circumferential force obtained by the finite element method were compared to the results obtained by solving the differential equation.

Figure 7: The finite element model.

Figure 8: Deflection vs. Height (with E = 2000000 T/m).

Figure 9: Moment vs. height.

The two methods are very close to each other as expected (See Figures 8–11). A little difference exists in the graphs due to the fact that the membrane forces in the vertical direction in the silo wall are assumed to be zero and are not included in the differential equation, while in the finite element method, these forces exist and tend to decrease the deflection and consequently the forces. Note that in Figures 8–11, number 1 represents results of the differential equation while number 2 represents results of the finite element method. Both methods are done for the water behavior approach.

Figure 10: Shear vs. height.

Figure 11: Force N in the circumferential direction vs. height.

4 Average results for different silo dimensions

After doing a comparison between all approaches, average values for Janssen, Airy and Reimbert’s are calculated for various silos, with various heights and diameters and wall thicknesses. Note that only silos (and not bunkers) are analyzed, and the ratio H/D is kept greater than 1.5 [6].

Table 1: Dimensions of silos used in the analyses.

D(m) ho(m) h1(m) Notation on graphs 5 10 10 1 5 10 20 2 5 10 30 3

10 10 20 4 10 10 30 5 10 10 40 6 10 10 50 7 15 10 25 8 15 10 30 9 15 10 40 10 15 10 50 11

Figure 12: Average deflection * E vs. height for silos with different dimensions. (ep = 30 cm).

Figure 13: Average moment vs. height for silos with different dimensions. (ep = 30 cm).

Figure 14: Average shear vs. height for silos with different dimensions. (ep = 30 cm).

Figure 15: Average force N in the circumferential direction vs. height for silos with different dimensions. (ep = 30cm).

The dimensions (presented in Table 1) of silos chosen in this analysis satisfy this condition and cover significant number of silo used. Other uncalculated dimensions can be interpolated as required by the designing engineer. Therefore, given a silo with specific dimensions, one can determine from the curves presented in Figures 12–19 the shear, moment, deflection and the membrane force N in the circumferential direction; this can be done directly or by interpolation between the curves. Note that in the above graphs, the material stored in the silo has a unit weight g = 0.75T/m3. To determine shear, moment, deflection or force N in the circumferential direction for a stored material with a different unit weight, the

values determined from the graphs need to be multiplied by (γ material/0.75) since values are proportional to the unit weight.

Figure 16: Average deflection * E vs. height for silos with different dimensions. (ep = 20 cm).

Figure 17: Average moment vs. height for silos with different dimensions. (ep = 20 cm).

5 Conclusion

This work is divided into three parts, the first deals with the comparison between all approaches (Janssen, Airy, Reimbert and water behavior), while the second part was done to check the closed form solution of the differential equation where finite element method was utilized to do this checking.

Figure 18: Average shear vs. height for silos with different dimensions. (ep = 20 cm).

Figure 19: Average force N in the circumferential direction vs. height for silos with different dimensions. (ep = 20cm).

In the third part, average values were found for the three methods (Janssen, Airy, and Reimbert). These average values were done for deflection, moment, shear, and the circumferential membrane force, and were graphed for silos with various heights, diameters, and wall thicknesses. From the above we can conclude the following: 1- Janssen, Airy and Reimbert lateral pressure application approaches

give comparative results, while the water behavior approach gives larger values than the above three methods especially in critical areas. Due to this reason, the water behavior method was excluded in finding average values.

2- As silo height increases, forces and deflection increase; as the silo diameter increases forces and deflection also increase.

3- The Finite Element Analysis is very close to whatever lateral pressure approach used, as expected.

4- Average values for deflection and forces can be used from the graphs for the purpose of design. Given a material with a unit weight g that needs to be stored in a certain silo volume, one can suggest a corresponding height,

diameter and wall thickness then determine the wall forces and perform the design. If the exact dimensions are not available in the graphs, one can interpolate and determine the corresponding values.

For further research, the author recommends that analysis be made for such silos with wall friction keeping in mind that including friction will somewhat complicate the differential equation. Also, additional work needs to be done in the area of dynamic analysis and design of silos.

Notation

γ : Weight per unit volume for the stored material. ho: Height from level 0 to the top of the hopper. h1: Height from top of the hopper to the top of the silo. E: Elastic modulus. DE: Flexural rigidity. ep: Silo wall thickness. ν: Poisson’s ratio. ρ : Angle of internal friction for stored material. ko: Rankine coefficient of active earth pressure – the ratio of the horizontal pressure to the vertical pressure. D: Diameter of the silo. R : Average radius. Rh : Hydraulic radius. λ : Elastic length. µ1: Angle of friction (stored material against wall). µ: Internal friction. y: Lateral displacement. po: Pressure of stored material at depth h1 below surface of stored material. (Water behavior approach). p: Lateral pressure. yg: General solution of the differential equation. yp (or fp1): Particular solution of the differential equation. yh: Solution of the differential equation without right term.

Acknowledgements

The author thanks the Lebanese University, Faculty of Engineering, Branch II for sponsoring part of this on going research. Also, the author thanks the engineers Bassam Mazloum, Georges Abi Saad and Issam Abou Antoun who helped in the set up of this paper.

References

[1] Timoshenko, S. and Gere, J., “Theory of Elastic Stability”, Second Edition, McGraw-Hill, Mechanical Engineering Series, 1963.

[2] Timoshenko, S. and Woinowsky-Krieger, S., “Theory of Plates and Shells”, Second Edition, McGraw-Hill, Engineering Mechanics Series, 1959.

[3] Janssen H.A., “Versuche Uber Getreidedruck in Silozellen”, VDI Zeitschrift, Dusseldorf, V.39, Aug. 31, 1885, pp 1045-1049.

[4] Reimbert M. and Reimbert A., “Silos-Theory and Practice”, Trans Tech Publication, 1st Edition, 1976, Claustal, Germany.

[5] Airy W., “The Pressure of Grain”, Minutes of Proceedings, Institute of Civil Engineers, London, V 131, 1897, pp 458-465.

[6] Safarian and Harris, “Design and Construction of Silos and Bunker”, Published by Van Nostrand Reinhold Company, New York, 1984.

[7] Thonier, H., “Conception et Calcul de Structure des Bâtiments”, Presse de l’Ecole Nationale des Ponts et Chaussées, 1993.

[8] Fintel, M., “Handbook of Concrete Engineering”, Published by Van Nostrand Reinhold Company, 2nd edition, New York, 1974.

[9] “ACI Manual of Concrete Practice”, Part 4,Committee313, American Concrete Institute, Redford Station, Detroit, Michigan, 1995.

[10] Jenike A.W., Elsey P.J., and Wooley R.M., “Flow Properties of Bulk Solids”, ASTM, Proceedings, V.60, 1960, pp. 1168-1181.

[11] Spiegel, M.R., “Advanced Mathematics for Engineers & Scientists”, Schaums Outline Series, McGraw-Hill Book Company, 1971.

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