Arch Appl Mech (2013) 83:549–558DOI 10.1007/s00419-012-0703-8

ORIGINAL

Korhan Ozgan · Ayse T. Daloglu · A. Ihsan Karakas

A parametric study for thick plates resting on elasticfoundation with variable soil depth

Received: 22 December 2011 / Accepted: 21 September 2012 / Published online: 30 September 2012© Springer-Verlag 2012

Abstract The soil depth is generally considered to be constant for the analysis of plates resting on elasticfoundation in the literature. However, it is most reasonable to have a variable subsoil depth as the plate dimen-sions get larger. In present study, linearly varying subsoil depth is considered as well as constant, linear andquadratic variation of modulus of elasticity with subsoil depth. Also, a parametric study is performed to dem-onstrate the behavior of thick plates on elastic foundations with variable soil depth. Modified Vlasov Model isused for the analysis of the plate foundation system, and 8-noded Mindlin plate element incorporating shearstrain throughout plate thickness is used for the finite element model. Numerical examples are obtained fromthe literature to compare results and to show the influence of variable soil stratum depth on the behavior ofplates. Displacements, bending moments, and shear forces are presented in tabular and graphical formats. Asfar as results are compared, it can be concluded that variable soil depth significantly affects the variation of thedisplacements and therefore the internal forces of the plate while keeping it constant ends up with unrealisticresults.

Keywords Thick plate · Elastic foundations · Variable soil depth · Modified Vlasov model ·Finite element model

1 Introduction

Structures on flexible rafts such as foundations of heavy buildings, highways, airports’ concrete, and industrialground floors are usually considered as the plates freely resting on elastic foundations. Winkler Model calledas one-parameter model and Vlasov Model called as two-parameter model are widely used for the analysis ofsuch kinds of problems [1–7]. If two-parameter model is used, it is necessary to have a specific subsoil depthfor the analysis. However, especially in problems of plates resting on large foundation areas such as airportsand highways, it may not be reasonable to consider a constant subsoil depth. The depth of rigid base underthe subsoil is supposed to be variable. For this reason, especially in the analysis of plates resting on such largefoundation areas, the effect of variable subsoil depth on behavior of plates becomes significant.

K. Ozgan (B) · A. T. Daloglu · A. I. KarakasDepartment of Civil Engineering, Karadeniz Technical University, 61080 Trabzon, TurkeyE-mail: korhanozgan@yahoo.com; kozgan@ktu.edu.tr

A. T. DalogluE-mail: aysed@ktu.edu.tr

A. I. KarakasE-mail: aliihsanka@yahoo.com

550 K. Ozgan et al.

On the other hand, one of the theories commonly used for the analysis of plates resting on elastic foun-dations is classical thin plate theory in which the effect of shear strain throughout plate thickness is ignored.However, as the plate thickness increases, consideration of this effect becomes inevitable [8–10].

In this study, behavior of thick plates resting on elastic foundations with variable subsoil depth is investi-gated using Modified Vlasov Model for various plate thicknesses and aspect ratios. In addition, the alterationof subsoil modulus of elasticity with depth is considered. For this purpose, an 8-noded Mindlin plate elementwith 3 degrees of freedom (dof) per node is used to take the shear strain through plate thickness into consid-eration. Selectively reduced integration technique is applied to prevent shear locking problem arising as theplate thickness decreases.

2 Finite element model

2.1 Mathematical formulation of foundation model

Plates on elastic foundations represent a complex soil–structure interaction problem. The most importantconcept here is to include all the energy in the plate and soil below and surrounding the plate. In the study, min-imum potential energy principle is used to derive the field equations of this complex soil–structure interactionproblem.

Total potential energy of the plate-subsoil system (Fig. 1) is

∏= 1

2

∫

�

(∂2w

∂x2 ,∂2w

∂y2 , 2∂2w

∂x∂y

)[D]

(∂2w

∂x2 ,∂2w

∂y2 , 2∂2w

∂x∂y

)T

dxdy

+1

2

H∫

0

+∞∫

−∞

+∞∫

−∞

(σxεx + σyεy + σzεz + τxyγxy + τyzγyz + τxzγxz

)dxdydz (1)

−∫

�

qwdxdy

where w, D, q , and H are vertical deflection of the plate, flexural rigidity of the plate, uniformly distributedload, and subsoil depth, respectively.

It is customary to assume a displacement pattern for developing the stiffness matrices of the plate and soilsystem. The above elastic foundation problem is a three-dimensional one, and the displacement of the soil inthe x, y, and z directions are defined as u, v, and w, respectively. From the practical point of view, the lateraldisplacements in the soil are negligible compared with the vertical displacements, and hence, to simplify themodel, following assumptions are made.

u (x, y, z) = 0v (x, y, z) = 0w (x, y, z) = w (x, y) · φ(z)

(2)

Subsoil

Plate

Loadφ(0)=1

φ(z)

φ(H)=0Rigid Base

Fig. 1 Plates on elastic foundation

A parametric study for thick plates resting 551

φ(z) is mode shape that gives the variation of the deflection in the z direction (see Fig. 1). The value of φ(z)equals to one at the top and zero at the bottom of the subsoil. Then, the displacement of the surface of the soilequals the displacement of the middle surface of the plate.

Substituting Eq. (2) in Eq. (1), the total potential energy of the plate soil system takes the following form.

∏= 1

2

∫

�

(∂2w

∂x2 ,∂2w

∂y2 , 2∂2w

∂x∂y

)[D]

(∂2w

∂x2 ,∂2w

∂y2 , 2∂2w

∂x∂y

)T

dxdy

+1

2

+∞∫

−∞

+∞∫

−∞

{kw2 + 2t (∇w)2}dxdy −

∫

�

qwdxdy (3)

Outside the domain of the plate at z=0, the field equation that controls the surface displacement of the soilstratum is

− 2t∇2w + kw = 0 (4)

where k and 2t are called soil parameters and calculated as

k =H∫

0

Es (1 − νs)

(1 + νs) (1 − 2νs)

(∂φ

∂z

)2

dz 2t =H∫

0

Es

2 (1 + νs)ϕ2dz (5)

Here, Es and νs are modulus of elasticity and Poisson ratio of the subsoil, respectively.The field equation representing the deformation characteristic of the soil continuum is

− m∂2φ (z)

∂z2 + nφ (z) = 0 (6)

where

m =+∞∫

−∞

+∞∫

−∞

Es (1 − νs)

(1 + νs) (1 − 2νs)w2dxdy n =

+∞∫

−∞

+∞∫

−∞

Es

2 (1 + νs)(∇w)2 dxdy (7)

As mentioned before, φ (z) is a function that gives the variation of vertical deformation with subsoil depthsuch that φ (0) = 1 and φ (H) = 0. The solution of differential equation (6) applying the boundary conditionsyields,

φ (z) = sinh(γ (1 − z

H ))

sinh γ(8)

and the third parameter of the subsoil, γ , is

( γ

H

)2 = (1 − 2νs)

2 (1 − νs)

∫ +∞−∞

∫ +∞−∞ (∇w)2 dxdy

∫ +∞−∞

∫ +∞−∞ w2dxdy

(9)

As can be seen from above equations, soil parameters k and 2t depend on material properties (Es and νs anddepth (H) of subsoil, and function φ (z) which is a function of subsoil depth (H) and γ parameter in turn. Thevalue of γ parameter varies with displacement of the plate, w, Poisson ratio of the subsoil, νs , and the depthof subsoil, H . So, the solution of this complex soil–structure interaction problem can be performed using aniterative technique.

The process includes the effect of the surrounding soil domain on the plate behavior. Equation (4) has to besolved in a domain outside the plate boundaries with z = 0. Vlasov and Leont’ev [11] assumed an approximatesolution for the displacement function w(x, y). The domain outside the plate is divided into eight sub-domainsas shown in Fig. 2. For a rectangular plate with dimensions of 2a in x direction and 2b in y direction, theassumed functions are,

552 K. Ozgan et al.

x

y

plate

a a

b

b

x=aw=Wa

w=0x=∞

w(x)=Wae-λ(x-a)

w(y)=Wbe-λ(y-b)

w=Wb

y=b

y=∞w=0

w(x)=Wce-λ(x-a) e-λ(y-b)

I IIVI

VII VIII IX

V IV III

Fig. 2 Plate-soil surface divided into regions

a < x < ∞ and − b < y < b w (x, y) = Wae−λ(x−a)

b < y < ∞ and − a < x < a w (x, y) = Wbe−λ(y−b)

a < x < ∞ and b < y < ∞ w (x, y) = Wce−λ(x−a)e−λ(y−b)

(10)

where Wa, Wb, and Wc are vertical displacement of a discrete point at the boundary of the plate at x = a, y = band the corner of the plate, respectively, and the boundary conditions outside the domain of the plate are givenas at x = ±∞ and y = ±∞. Further detail is available in [7,12] and [13].

2.2 Evaluation of the stiffness matrix

In the study, 8-noded (3 degrees of freedom per node) Mindlin plate element (PBQ8) is used for the finiteelement model of this soil–structure interaction problem (Fig. 3).

15

2

68

347

η

ξ

Fig. 3 8-node 24 degrees of freedom element

A parametric study for thick plates resting 553

Displacements at each node are

u = zϕy = z8∑

i=1Niϕyi

v = −zϕx = −z8∑

i=1Niϕxi

w =8∑

i=1Niwi

(11)

and displacement function is

w = [N1 N2 . . . N8

] {we} (12)

[N ] is a matrix containing the shape functions [14] and {we} is the displacement vector containing all 24components of the type shown in Eq. (11). Substituting Eq. (12) into Eq. (3), strain energy stored in the platesoil system can be written as

U = 1

2{we}t ([

kp] + [kw] + [k2t ]

) {we} (13)

Here, [kp], [kw], and [k2t ] are stiffness matrix of the plate, stiffness matrix of the Winkler foundation, andstiffness matrix of the Vlasov foundation, respectively [12,13].

The stiffness matrix for the axial strain effect in the soil, [kw], is obtained by minimizing the total energywith respect to each component of displacement vector and may be written as

[kw] = kab

+1∫

−1

+1∫

−1

[N ]T [N ] dξdη (14)

in which a and b are shown in Fig. 3 and are natural coordinates. The stiffness matrix of the Vlasov foundationfor the shear strain within the soil is evaluated as

[k2t ] = 2tab

+1∫

−1

+1∫

−1

(1

a2

[∂ N

∂ξ

]T [∂ N

∂ξ

]+ 1

b2

[∂ N

∂η

]T [∂ N

∂η

])dξdη (15)

Finally, the system of equation in global coordinates is as follows

[K ] {W } = {F} (16)

in which, [K ], {W }, and {F} are the global stiffness matrix of plate-subsoil system, global displacement, andglobal load vector, respectively.

3 Numerical examples

3.1 Rectangular plate subjected to uniformly distributed load

A rectangular plate on elastic foundation taken from the literature [3,6] and [7] is analyzed to show the accuracyof the approach. Comparisons of the results are made in Table 1. Later, the effect of variable subsoil depth onthe behavior of the plate is investigated.

The modulus elasticity of the plate is 20,685,000 kN/m2 and Poisson ratio of the plate is 0.20. The thick-ness of the plate and plate dimensions are initially considered as 0.1524 m and 9.144 × 12.192 m, respectively.Then, the same example is resolved for h/ lx = 0.05 and 0.01 where h is the thickness and lx is the length ofthe short side of the plate. For each value of h/ lx ratio, the analysis is repeated for ly/ lx = 1.33, 2.00 and3.00 keeping lx constant as 9.144m. The modulus of elasticity of the subsoil is 68,950 kN/m2 and Poissonratio of the subsoil is 0.25. The problem is solved for constant, linearly varying modulus of elasticity and also

554 K. Ozgan et al.

Table 1 Comparison of maximum displacements and bending moments of plate on elastic foundation

Ref H1 = H2 3.048 m H1 = H2 6.096 m H1 = H2 9.144 m H1 = H2 15.24 m

w (cm) Mx (kNm) w (cm) Mx (kNm) w (cm) Mx (kNm) w (cm) Mx (kNm)P. study 0.0873 0.0490 0.1532 0.2794 0.1908 0.3917 0.2238 0.4311[3] 0.0871 – 0.1530 – 0.1896 – 0.2205 –[6] 0.0853 0.0445 0.1526 0.2880 0.1893 0.4109 0.2212 0.4671[7] 0.0872 0.0529 0.1524 0.3113 0.1890 0.4224 0.2070 0.4892

q=23.94kN/m2

Plate

Elastic Foundation

H1 H2

Rigid Base

ly

α

E1 E1 E1

E2 E2=10.E1E2=10.E1

Constant Linear Quadratic

Fig. 4 Plate on elastic foundation with variable subsoil depth

Fig. 5 Maximum displacement, bending moment, and shear force for various values of α

considering quadratic variation for modulus of elasticity of the subsoil (Fig. 4). The uniformly distributed loadis taken as 23.94 kN/m2.

From the comparisons of Table 1, it is observed that the results are in good agreement with each other.After the presented model is validated, a parametric study performed to supply information on the behavior

of the thick plates on elastic foundation having linearly variable subsoil depth. As seen from Fig. 4, the subsoildepth is considered to be constant and varying linearly along the long span of the plate. Accordingly, keepingH1 constant as 3.048 m, the depth of H2 is calculated for α = 0◦, 15◦, 30◦ and 45◦ separately. The results areshown in Figs. 5, 6, and 7.

Figure 5 shows maximum displacement, bending moment, and shear force as a function of α for a plate hav-ing various ratios of h/ lx and ly/ lx and constant modulus of elasticity of the subsoil. As seen from the figures,maximum displacements and shear forces increase as the value of α increases but there is not a steady increasein the bending moments. It can be said that variable subsoil depth has a significant effect on the behavior ofthe plate. This is understandable since a plate on an elastic foundation with a larger subsoil depth becomesmore flexible and has larger displacements. Moreover, lines move away from each other as α increases. This

A parametric study for thick plates resting 555

Fig. 6 Maximum displacement, bending moment, and shear force for various values of α (h/ lx = 0.016)

Fig. 7 Variation of displacement, bending moment, and shear force of the pate along y axis

556 K. Ozgan et al.

Table 2 Maximum displacements, bending moments, and shear forces of the plate for h/ lx = 0.05 and ly/ lx = 2.00

H1 = 3.048 mH2 (m) Constant Linear Parabolic

w (cm) Mx (kNm) Vx (kN) w (cm) Mx (kNm) Vx (kN) w (cm) Mx (kNm) Vx (kN)3.048 0.089 7.26 46.56 0.017 3.21 25.66 0.024 3.69 24.857.948 0.154 10.12 170.72 0.035 3.88 100.34 0.051 5.48 91.3213.607 0.194 9.73 357.63 0.049 3.66 216.87 0.075 5.62 205.3621.336 0.223 8.24 602.74 0.063 3.10 378.28 0.100 5.01 379.02

means that the effects of aspect ratio of the plate (ly/ lx ) and ratio of plate thickness to short span (h/ lx ) onthe results increase as α increases.

Figure 6 shows the effect of changes of the modulus of elasticity of the subsoil through the depth onthe behavior of the plate for h/ lx = 0.016. As expected, the results show that the maximum displacements,bending moments, and shear forces decrease for the linear or quadratic variation of modulus of elasticity ofthe subsoil compare to a constant value for modulus of elasticity of the subsoil. This is understandable in thatthe foundation becomes more rigid because the modulus of elasticity at the bottom of the subsoil is 10 timesof the modulus of elasticity at the top of the subsoil for the linear or quadratic variation. Clearly, aspect ratioof plate has considerable effects on the results for high values of α for all cases; constant, linear, and quadraticvariation for modulus of elasticity of subsoil.

The location of maximum displacement, bending moment, and shear force of the plate can vary withincreasing H2. The variation of displacement, bending moment, and shear force along y axis for various valuesof α is presented in Fig. 7 for h/ lx = 0.10 and ly/ lx = 1.33.

Results are listed in Table 2 for h/ lx = 0.05 and ly/ lx = 2.00 for comparison purposes.Maximum displacement, bending moment, and shear force of the plate for a constant subsoil depth (H1 =

H2 = 3, 048 m) are calculated as 0.089 cm, 7.26 kNm, and 46.56 kN, respectively. Then, the subsoil depth isconsidered to vary as linear and as quadratic fashion. The angle (α) of the rigid base with horizontal axis is takenas 15◦(H2 = 7.948 m), 30◦(H2 = 13.607 m), and 45◦(H2 = 21.336 m), respectively. When the maximumvalues are compared to the minimum values for all H2, it is seen that values of displacement, bending moment,and shear force increase approximately 2.50, 1.39, and 12.90 times, respectively. If the same comparisons aremade for the variable elasticity modulus of subsoil through the depth, values of displacement, bending moment,and shear force increase approximately 3.70, 1.25, and 14.74 times, respectively, for linearly varying modulusof elasticity of the subsoil and approximately 4.17, 1.52, and 15.25 times for quadratic variation of modulusof elasticity of the subsoil. Displacement and shear force increased significantly with increasing H2. It canbe said that changes in shear force is more than that of displacement and bending moment, and the effects ofvariable subsoil depth on the results increase in case of quadratic variation of the modulus of elasticity of thesubsoil. Therefore, assuming a constant subsoil depth may end up with unrealistic results. For this reason, thecomplete description of status of the plate-subsoil system and modeling it correctly is of great importance forsuch soil–structure interaction problems.

3.2 Raft foundation

A raft foundation solved before by Çelik and Saygun [6] is considered for second numerical example (Fig. 8).As in the previous numerical example, the effects of linearly varying subsoil depth to the rigid base in additionto variable subsoil modulus of elasticity with the depth by assuming constant, linear, and quadratic variation onthe behavior of the plate are investigated. The modulus elasticity of the plate is 200,000,000 kN/m2 and Poissonratio of the plate is 0.16. The thickness of the plate equals to 0.6 m. Plate dimensions are initially considered as11.6m for each direction. The analysis is repeated for ly/ lx = 2.00 keeping lx constant as 11.6m. The modulusof elasticity of the subsoil is 80,000 kN/m2 and Poisson ratio of the subsoil is 0.25. The problem is solved forconstant, linearly varying modulus of elasticity and also considering quadratic variation for modulus of elas-ticity of the subsoil. E2/E1 ratio is taken as 10 for linear and quadratic variation of modulus of elasticity. Here,E1 represents the modulus of elasticity of the subsoil at the top and E2 represents the modulus of elasticity ofthe subsoil at the bottom. Accordingly, keeping E1 constant, E2 is calculated for E2/E1 ratio.

As seen from Fig. 8, the subsoil depth is considered to be constant and to vary linearly through the longside of the plate. Accordingly, keeping H1 constant as 5 m, the depth of H2 is calculated for α = 0◦, 15◦, 30◦and 45◦ m separately. The results are shown in Figs. 9 and 10.

A parametric study for thick plates resting 557

P3 P3P2

P2 P2P1

P3 P3P2

P1=3200kNP2=2000kNP3=1200kN

E1

E1

E1

E2

E2

E2Es, νs

Rigid base

H1

H2

H2

ly

lx

α

Fig. 8 Raft foundation with variable subsoil depth

Fig. 9 Maximum displacement, bending moment, and shear force of raft foundation for various values of α

The effect of the modulus of elasticity of the subsoil through the depth on the behavior of the raft foundationis shown in Fig. 9. As expected, the maximum displacements, bending moments, and shear forces decrease asthe subsoil is more rigid. The results show that the effects of aspect ratio on the behavior of raft foundationare greater for higher values of α.

Displacements and shear forces increase with increasing the angle (α) of the rigid base with horizontalaxis in other words increasing H2 while the values of bending moment decrease. In this example, changes inthe results are around twenty percent because the modulus of elasticity of the soil for the raft foundation ishigher than that of the plate in the previous example.

Maximum displacement and bending moment with increasing H2 may occur at different location of theraft foundation. The variation of displacement and bending moment of the raft foundation along y axis forvarious values of α for ly/ lx = 1.0 is presented in Fig. 10 to show this situation more clearly. Especially, linesmove away from each other on the side of higher subsoil depth.

4 Conclusions

The effects of variable subsoil depth on the response of thick plates on elastic foundations are investigated.Modified Vlasov Model is used for the analysis of plate foundation system as well as Mindlin plate theory.And, 8-noded (3 degrees of freedom per node) rectangular plate bending element including shear strain effectthrough the thickness of the plate is used for the finite element model of this soil–structure interaction problem.A computer program is coded in Fortran 6.6 for the purpose. The analysis results indicate that the incorporation

558 K. Ozgan et al.

Fig. 10 Variation of displacement and bending moment of the raft foundation along y axis

of the variable subsoil depth in the modeling of plates on elastic foundation significantly affects the variation ofthe displacement, bending moment, and shear force of the plate, whereas keeping the subsoil depth as constantmay produce unrealistic solutions.

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