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Displacements under linearly distributed pressures by extended Mindlin’s equations
Computers and Geotechnics 50 (2013) 143–149
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Displacements under linearly distributed pressures by extended Mindlin’s equations H.S. Sun a, G.H. Lei a,⇑, C.W.W. Ng a,b, Q. Zheng a a b
Key Laboratory of Geomechanics and Embankment Engineering of the Ministry of Education, Geotechnical Research Institute, Hohai University, 1 Xikang Road, Nanjing 210098, China Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Special Administrative Region
a r t i c l e
i n f o
Article history: Received 27 November 2012 Received in revised form 11 January 2013 Accepted 30 January 2013 Available online 27 February 2013 Keywords: Deformation Elasticity Footings/foundations Mindlin’s solution
a b s t r a c t Analytical elasticity solutions provide an efficient means of performing a first approximate analysis in foundation engineering. One of the well-known solutions is Mindlin’s solution to the stress and displacement induced by a point load at an embedment depth in a half-space. This solution is more superior but less widely used than Boussinesq’s solution. To promote this situation, Mindlin’s displacement equations are integrated to obtain a complete set of explicit formulae for calculating the displacements at an arbitrary point. The displacements are induced by uniformly and triangularly distributed horizontal or vertical pressures, which are exerted over a horizontal or vertical rectangular area in the interior of a homogeneous, isotropic, elastic half-space. These formulae facilitate the future development of computer programs for the analysis of related practical problems in foundation engineering. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Analytical elasticity solutions play an important role in preliminary predictions and in calibrating any sophisticated numerical schemes that are ultimately used for solving more complex practical problems in geotechnical engineering. Among many available solutions for stress and displacement in the fields of soil and rock mechanics, the solutions for loading over a rectangular area in a homogeneous, isotropic, elastic half-space are the most frequently applied ones in foundation engineering analysis [1–9]. These solutions are commonly expressed in the form of graphs and tables. They are, however, derived only for the computing points at or below the corners of the loaded rectangle. The method of superposition has to be used to calculate other points. Vaziri et al. [10] proposed the displacement solutions for uniform vertical or horizontal pressure over a vertical or horizontal rectangular area embedded in the interior of the half-space. The displacement solutions are derived for an arbitrary point within a predefined coordinate system. They have been readily coded into Oasys’s Frew [11], a software specifically developed for the analysis of flexible retaining walls [12]. In this paper, the uniform loading case studied by Vaziri et al. [10] is expanded into a combined uniform and triangular loading
⇑ Corresponding author. Tel.: +86 13851922201/25 83787216. E-mail addresses: [email protected] (H.S. Sun), [email protected] (G.H. Lei), [email protected] (C.W.W. Ng), [email protected] (Q. Zheng). 0266-352X/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compgeo.2013.01.009
case. The displacement formulae are derived by double integration of Mindlin’s equations and expressed in an explicit form for practical engineering applications such as calculations of ground displacement under anchor plates, rectangular piles or barrettes, eccentrically loaded buried rectangular footings, earth-retaining structures and diaphragm wall panel excavations. 2. The coordinate system and notations The coordinate system for an elastic half-space defined in Fig. 1 is the same as that of Vaziri et al. [10]. The origin of the coordinate system is located on the boundary surface of the half-space. A horizontal point load H or a vertical point load V is applied at an arbitrary point M (u, v, w) away from the origin. The displacements at any point N (x, y, z) due to the point load can be obtained by using Mindlin’s equations [13]. Nevertheless, it should be noted that the coordinate system adopted here is slightly different from that of Mindlin [13], where the point load H or V is applied at a point (u = 0, v = 0, w = c) under the origin. Similarly to the work of Vaziri et al. [10], the following notations are assumed for the purpose of presenting the derived displacement solutions in a compact form.
X ¼xu
ð1Þ
Y ¼yv
ð2Þ
Z1 ¼ z w
ð3Þ
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Nomenclature E G p, q x, y, z u, v, w
Young’s modulus shear modulus pressures normal and tangential to the loaded rectangle Cartesian coordinate axes coordinates of points in the loaded rectangle
Z2 ¼ z þ w Ri ¼
ð4Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 2 þ Y 2 þ Z 2i ði ¼ 1 or 2Þ
T ai ¼ arctan
m Poisson’s ratio dx, dy, dz displacements in the x, y and z directions H, V horizontal and vertical point loads
XYZ i a2 Ri
ða ¼ X; Y;
ð5Þ
or Z i ;
i ¼ 1 or 2Þ
ð6Þ
e.g.,
T X1 ¼ arctan
YZ 1 XR1
ð7Þ
T Z1 ¼ arctan
XY Z 1 R1
ð8Þ
B ðA ¼ X; Y; A ¼ 1 or 2Þ
T AB ¼ arctan
or Z i ;
B ¼ X; Y;
or Z i ;
i ð9Þ Fig. 1. The coordinate system (OABC lies on the surface of the half-space).
e.g.,
T XY ¼ arctan
Y X
ð10Þ
Z1 Y
ð11Þ
T YZ 1 ¼ arctan b ¼ 8pE
1m ¼ 16pGð1 mÞ 1þm
ð12Þ
where E is Young’s modulus, G is the shear modulus, and m is Poisson’s ratio. 3. Derivation procedures Following the coordinate system in Fig. 1 and the assumed notations, Mindlin’s original equations [13] for obtaining the displacements at any point N (x, y, z) can be readily rewritten as follows by a shift of origin. For a horizontal point load, the displacements in the x, y and z directions are given by
dx ¼
( ! ! H 1 X2 1 X 2 2wz 3X 2 ð3 4mÞ þ 3 þ þ 3 þ 3 1 2 b R1 R2 R2 R1 R2 R2 " #) 2 4ð1 mÞð1 2mÞ X 1 þ R2 þ Z 2 R2 ðR2 þ Z 2 Þ "
#
ð13Þ
VX Z 1 Z 1 6wzZ 2 4ð1 mÞð1 2mÞ dx ¼ ð3 4mÞ 3 þ 3 þ b R2 ðR2 þ Z 2 Þ R52 R2 R1
ð14Þ
" # VY Z 1 Z 1 6wzZ 2 4ð1 mÞð1 2mÞ dy ¼ ð3 4mÞ 3 þ 3 þ b R2 ðR2 þ Z 2 Þ R52 R2 R1
#
HXY 1 1 6wz 4ð1 mÞð1 2mÞ ð3 4mÞ 3 þ 3 5 b R2 R2 R1 R2 ðR2 þ Z 2 Þ2 " # HX ð3 4mÞZ 1 Z 1 6wzZ 2 4ð1 mÞð1 2mÞ dz ¼ þ 3 þ b R2 ðR2 þ Z 2 Þ R52 R32 R1 dy ¼
Fig. 2. Horizontal pressures over a vertical rectangle.
"
ð15Þ
For a vertical point load, the displacements in the x, y and z directions are given by
ð16Þ
ð17Þ
" ! # V 1 Z 22 2wz Z 21 6wzZ 22 8ð1 mÞ2 ð3 4mÞ ð3 4mÞ dz ¼ þ þ 3 þ 3þ b R1 R32 R2 R52 R2 R1 ð18Þ
The displacements (dx, dy, dz) at point N (x, y, z) due to horizontal or vertical pressure over a vertical rectangle (u = constant, v = v1 to v2,
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Xðp2 p1 Þ ½fð3 4mÞ½R2 z lnðZ 2 þ R2 Þ bðw2 w1 Þ " # Z1 z2 Z 2 2 2z þ þ lnðZ 2 þ R2 Þ þ ð1 mÞð1 2mÞ 3 R2 ðX 2 þ Y 2 ÞR2 Z 2 ðz 2wÞ 3z lnðZ 2 þ R2 Þ 2R2 þ þ R1 R2 þ Z 2 iv ¼v 2 Xðw p w p Þ w¼w 2 1 1 2 z lnðZ 1 þ R1 Þgw¼w21 þ v ¼v 1 bðw2 w1 Þ "( 2z 2z2 Z 2 ð3 4mÞ lnðZ 2 þ R2 Þ lnðZ 1 þ R1 Þ þ þ 2 R2 ðX þ Y 2 ÞR2 w¼w2 #v ¼v 2 Z2 þ2ð1 mÞð1 2mÞ þ lnðZ 2 þ R2 Þ ð21Þ R2 þ Z 2 w¼w1
dy ¼
v ¼v 1
Xðp2 p1 Þ bðw2 w1 Þ
dz ¼
Fig. 3. Horizontal pressures over a horizontal rectangle.
w = w1 to w2; Figs. 2 and 4) can be derived by double integration of Eqs. (13)–(18) with respect to v and w. Similarly, the displacements due to horizontal or vertical pressure over a horizontal rectangle (u = u1 to u2, v = v1 to v2, w = constant; Figs. 3 and 5) can be derived by double integration of Eqs. (13)–(18) with respect to u and v. As an example, consider the horizontal pressure over a vertical rectangle in Fig. 2. The point load on a rectangular element at the point (u, v, w) can be readily derived as follows:
H¼
1 ½ðw2 p1 w1 p2 Þ þ ðp2 p1 Þwdv dw ðw2 w1 Þ
ð19Þ
"
ð3 4mÞ
X 2 2z2 T X2 þ 3z lnðY þ R2 Þ X
þY lnðZ 2 þ R2 Þ þ XT X1 Y lnðZ 1 þ R1 Þ z lnðY þ R1 Þ " # Yw2 2zT X2 þ 2z þ 2lnðY þ R2 Þ 4ð1 mÞð1 2mÞ X ðX 2 þ Z 22 ÞR2 " ðX 2 Z 1 Z 2 Þ ðT XY T X2 Þ þ z lnðY þ R2 Þ 2X w¼w2 #v ¼v 2 Y Xðw2 p1 w1 p2 Þ þ lnðZ 2 þ R2 Þ þ ½fð3 4mÞ 2 bðw2 w1 Þ w¼w1 v ¼v 1 2z 2zwY 2zT X2 þ T X2 lnðY þ R2 Þ lnðY þ R1 Þ X X X 2 þ Z 2 R2 2
w¼w2 #v ¼v 2 Z2 þ4ð1 mÞð1 2mÞ lnðY þ R2 Þ þ ðT X2 T XY Þ X w¼w1
ð22Þ
v ¼v 1
where p1 and p2 are the pressures at the points (u, v1, w1) and (u, v1, w2), respectively. Substituting this point load into Eqs. (13)–(15) and taking double integration with respect to v and w give the required displacement solutions.
For the case of combined uniform and triangular horizontal pressures over a horizontal rectangular area as shown in Fig. 3, the displacements are given by
R1 R2 Y x½Z 1 T Z1 þ Z 2 T Z2 2 ! X 2 þ Z 21 Y lnðX þ R1 Þ X lnðY þ R1 Þ Y lnðX þ R2 Þ 2x #
Z2 R2 þ 2 lnðY þ R2 Þ þ R1 Y x 2 " ! X 2 þ Z 22 x Z 1 T Z1 þ Z 2 T Z2 Y lnðX þ R1 Þ X 2x # 2 Z lnðY þ R2 Þ Y lnðX þ R2 Þþ 1 lnðY þ R1 Þ x 2 3 uXY 4 þ lnðY þ R2 Þ5 þ 2ð1 mÞð1 2mÞ 2wz X 2 þ Z 22 R2 h YR2 þ2xZ 2 ðT X2 T XY Þ þ 2xX X 2 Z 22 lnðY þ R2 Þ iv ¼v 2 u2 q1 u1 q2 u¼u 2YZ 2 lnðZ 2 þ R2 Þgu¼u21 þ ½fð3 4mÞ v ¼v 1 bðu2 u1 Þ
dx ¼ 4. Displacements due to horizontal pressures For the case of combined uniform and triangular horizontal pressures over a vertical rectangular area as shown in Fig. 2, the displacements are given by dx ¼
"(
p2 p1 bðw2 w1 Þ 1 1 2 X þ Z 21 2zZ 1 ð3 4mÞ YR1 þ zY lnðZ 1 þ R1 Þ 2 2
lnðY þ R1 Þ þ zXðT X2 T X1 Þ þ X 2 lnðY þ R2 Þ þ z½3XT X2 þ Z 2 lnðY þ R2 Þ 2 3 YðX 2 Z 1 þ z2 Z 2 Þ Yz2 Z 2 5 Y lnðZ 2 þ R2 Þ þ 2z4 þ 2 2z lnðY þ R Þ 2 ðX þ Y 2 ÞR2 X 2 þ Z 22 R2 YZ 2 ð2w zÞ YR2 2zXT X2 ðX 2 Z 1 Z 2 Þ lnðY þ R2 Þ þ 2ð1 mÞð1 2mÞ 3ðR2 þ Z 2 Þ 3 1 1 2 þ 6zY lnðZ 2 þ R2 Þ YR2 X þ Z 22 lnðY þ R2 Þ þ zXT X1 2 2 ow¼w2 v ¼v 2 w p w p 2 1 1 2 þ X 2 lnðY þ R1 Þ þ ½fð3 4mÞ½Y lnðZ 1 þ R1 Þ w¼w1 v ¼v bðw2 w1 Þ 1 þ Z 1 lnðY þ R1 Þ XðT X1 þ T X2 Þ Y lnðZ 2 þ R2 Þ þ Z 1 lnðY þ R2 Þ ! 2zY wX 2 zZ 2 þ XðT X1 þ T X2 Þ þ z þ 2 2 2 þ4ð1 mÞð1 2mÞ 2 2 Z 2 R2 X þ Z 2 X þY w¼w2 #v ¼v 2 Y YZ 2 ð20Þ þ XT X2 lnðZ 2 þ R2 Þ Z 2 lnðY þ R2 Þ þ 2 2ðZ 2 þ R2 Þ w¼w1 v ¼v 1
q2 q1 bðu2 u1 Þ
ð3 4mÞ
½Y lnðX þ R1 Þ þ X lnðY þ R1 Þ Z 1 T Z1 þ Y lnðX þ R2 Þ Z 2 T Z2 þ Y lnðX þ R2 Þ þ X lnðY þ R2 Þ Z 2 T Z2 þ Y lnðX þ R1 Þ 2wzXY þ 4ð1 mÞð1 2mÞ Z 1 T Z1 þ R2 X 2 þ Z 22 iv ¼v 2 u¼u ½X lnðY þ R2 Þ þ Z 2 ðT X2 T XY Þgu¼u21 v ¼v 1
ð23Þ
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i q q1 3 4m h dy ¼ 2 ð2x þ XÞR2 Y 2 þ Z 22 lnðX þ R2 Þ bðu2 u1 Þ 2 i 1h þ ð2x þ XÞR1 Y 2 þ Z 21 lnðX þ R1 Þ 2 u 2wz þ lnðX þ R2 Þ ð1 mÞð1 2mÞ R2
dx ¼
þ XY lnðZ 2 þ R2 Þ X½XT X1 þ z lnðY þ R1 Þ þ Y lnðZ 1 þ R1 Þ 2 3 2 w XY þ 2zT X2 þ 2X lnðY þ R2 Þ52ð1 mÞ þ 2z4 X 2 þ Z 22 R2
½ð2X 4xÞR2 þ 4YZ 2 ðT YX T Y2 Þ þ 2xZ 2 lnðX 2 þ Y 2 Þ 2 16 X 2 þ Z 22 þ XR2 þ ðYZ 2 Þ2 2Y 2 lnðX þ R2 ÞþZ 22 ln X 4 Z 42 Y 2 þ Z 22 39u¼u2 3v ¼v 2 2 > 4 X 2 þ Z 22 þ Z 2 R2 þ ðXYÞ2 7> = 7 7 7 þ 2xZ 2 ln 7 5 2 6 2 2 > 5 X Z 2 ðX þ Y Þ > ; u¼u1
ð1 2mÞ½ðX 2 Z 1 Z 2 ÞðT X2 T XY Þ X½2z lnðY þ R2 Þ i w2 q1 w1 q2 v ¼v w¼w2 þ ½fð3 4mÞ½2zT X2 þ Y lnðZ 2 þ R2 Þgv ¼v 21 w¼w1 bðw2 w1 Þ 2 3 wXY 4 5 X lnðY þ R2 Þ X lnðY þ R1 Þ þ 2z T X2 þ X 2 þ Z 22 R2 2 þ 4ð1 mÞð1 2mÞ½Z 2 ðT XY T X2 Þ X lnðY þ R2 Þgvv ¼¼vv 21 w¼w w¼w1 ð26Þ
v ¼v
1 2wz 4ð1 mÞ ð3 4mÞR2 R1 R2 iv ¼v 2 u¼u ð1 2mÞ½R2 þ Z 2 lnðZ 2 þ R2 Þgu¼u21
dy ¼
u2 q1 u1 q2 þ bðu2 u1 Þ
v ¼v 1
dz ¼
ð24Þ
q2 q1 ½fð3 4mÞZ 1 ½Z 2 T Z2 þ Y lnðX þ R2 Þ bðu2 u1 Þ
þx lnðY þ R2 Þ Z 1 ½Z 1 T Z1 þ Y lnðX þ R1 Þ þ xlnðY þ R1 Þ 2 3 uYZ 2 þ T Z2 5 þ 2ð1 mÞð1 2mÞ þ2wz4 X 2 þ Z 22 R2
v ¼v 1
þ 2YZ 2 lnðX þ R2 Þ þ 2xZ 2 lnðY þ R2 Þ iv ¼v 2 u q u q u¼u 2 1 1 2 þ2xY lnðZ 2 þ R2 Þgu¼u21 þ ½fð3 4mÞZ 1 v ¼v 1 bðu2 u1 Þ 2wzZ 2 Y þ4ð1 mÞ lnðY þ R2 Þ Z 1 lnðY þ R1 Þ þ X 2 þ Z 22 R2 u¼u
q2 q1 bðw2 w1 Þ 3 4m ½ðw 5zÞR2 ðX 2 þ Y 2 4z2 Þ lnðZ 2 þ R2 Þ 2 1 þ ½Z 2 R1 ðX 2 þ Y 2 Þ lnðZ 1 þ R1 Þ þ 2z 2 w2 2z lnðZ 2 þ R2 Þ ð1 mÞð1 2mÞ½ð4z þ Z 2 ÞR2 2R2 R2
v ¼v 2 #w¼w2 2 2 X þ Y 2Z 1 Z 2 lnðZ 2 þ R2 Þ w¼w1
w2 q1 w1 q2 þ ½fð3 4mÞ½R1 þ 2z lnðZ 2 þ R2 Þ R1 bðw2 w1 Þ w þ2z þ lnðZ 2 þ R2 Þ 4ð1 mÞð1 2mÞ½R2 R2 i v ¼v w¼w2 þ Z 2 lnðZ 2 þ R2 Þgv ¼v 21
½XY þ ðT YX T Y2 ÞY 2 þ ð2xX X 2 ÞðT XY T X2 Þ Z 22 T Z2
ð1 2mÞ½Y lnðZ 2 þ R2 Þ þ Z 2 lnðY þ R2 Þ þ XðT XY T X2 Þgu¼u21
q 2 q 1 hn ð3 4mÞ½ð2z2 X 2 ÞT X2 þ 3zX lnðY þ R2 Þ bðw2 w1 Þ
w¼w1
ð27Þ
q2 q1 1 ð3 4mÞ YR1 þ zX lnðZ 1 þ R1 Þ 2 bðw2 w1 Þ 1 2 2 X þ Z 1 2zZ 1 lnðY þ R1 Þ zXT X1 YR2 zXT X2 2 i 2z X 2 lnðY þ R2 Þ þ Yz lnðZ 2 þ R2 Þ þ ½ðX 2 z2 ÞT X2 X 2zX lnðY þ R2 Þ XY lnðZ 2 þ R2 Þ YR1 zXT X1 2 Yw2 Z 2 2 X lnðY þ R1 Þ þ zY lnðZ 1 þ R1 Þ2z4 2 X þ Z 22 R2 # 3X 2 z2 T X2 þ 4z lnðY þ R2 Þ þ 3Y lnðZ 2 þ R2 Þ þ ½8ð1 mÞ2 X 1 ð3 4mÞ YR2 zXT X2 þ zY lnðZ 2 þ R2 Þ 2 v ¼v 2 #w¼w2 1 2 2 X þ Z 2 2zZ 2 lnðY þ R2 Þ 2 v ¼v 1
dz ¼ iv ¼v 2 v ¼v 1
ð25Þ
5. Displacements due to vertical pressures For the case of combined uniform and triangular vertical pressures over a vertical rectangular area as shown in Fig. 4, the displacements are given by
w¼w1
w2 q1 w1 q2 þ ½fð3 4mÞ½Y lnðZ 1 þ R1 Þ þ Z 1 lnðY þ R1 Þ bðw2 w1 Þ þ Z 2 lnðY þ R2 Þ XT X1 þ Y lnðZ 1 þ R1 Þ XT X1 þ 2z lnðY þ R2 Þ 2zwYZ 2 8ð1 mÞ2 ½Y lnðZ 2 þ R2 Þ þ Z 2 lnðY þ R2 Þ þ X 2 þ Z 22 R2 2 XT X2 gvv ¼¼vv 21 w¼w w¼w1
Fig. 4. Vertical pressures over a vertical rectangle.
ð28Þ
For the case of combined uniform and triangular vertical pressures over a horizontal rectangular area as shown in Fig. 5, the displacements are given by
H.S. Sun et al. / Computers and Geotechnics 50 (2013) 143–149
dx ¼
147
p2 p1 ½fð3 4mÞZ 1 ½Z 2 T Z2 Y lnðX þ R2 Þ x lnðY þ R2 Þ bðu2 u1 Þ þ Z 1 ½Z 1 T Z1 Y lnðX þ R1 Þ x lnðY þ R1 Þ 2 3 h uYZ 2 T Z2 5 2ð1 mÞð1 2mÞ XY þ Y 2 T YX þ 2wz4 2 2 X þ Z 2 R2 þ ð2x XÞXT XY Z 22 T Z2 þ ðX 2 2xXÞT X2 Y 2 T Y2
þ2YZ 2 lnðx þ R2 Þ þ 2xZ 2 lnðY þ R2 Þ iv ¼ v 2 u p u p u¼u 2 1 1 2 ½fð3 4mÞZ 1 þ þ2xY lnðZ 2 þ R2 Þgu¼u21 v ¼v 1 bðu2 u1 Þ 2wzYZ 2 4ð1 mÞð1 2mÞ lnðY þ R2 Þ Z 1 lnðY þ R1 Þ R2 X 2 þ Z 22 iv ¼v 2 u¼u ð29Þ ½Y lnðZ 2 þ R2 Þ þ Z 2 lnðY þ R2 Þ XT X2 gu¼u21 v ¼v 1
dy ¼
p2 p1 ½fð3 4mÞZ 1 ½R2 x lnðX þ R2 Þ bðu2 u1 Þ 2 3 xX 1 þ 5 þ Z 1 ½R1 x lnðX þ R1 Þ 2wzZ 2 4 Y 2 þ Z 22 R2 R2 1 1 4ð1 mÞð1 2mÞ xX þ R22 Z 2 R2 þxYðT YX T Y2 Þ 4 2 u¼u2 #v ¼v 2 1 2 2 þ xZ 2 lnðX þ R2 Þ þ ð2xX X Y Þ lnðZ 2 þ R2 Þ 2 u¼u1 v ¼v 1
u2 p1 u1 p2 þ ½fð3 4mÞZ 1 lnðX þ R2 Þ Z 1 lnðX þ R1 Þ bðu2 u1 Þ 2wzXZ 2 4ð1 mÞð1 2mÞ½YðT YX T Y2 Þ þ Z 2 lnðX þ R2 Þ Y 2 þ Z 22 R2 iv ¼v 2 u¼u ð30Þ þ X lnðZ 2 þ R2 Þgu¼u21 v ¼v 1
dz ¼
p2 p1 1 ð3 4mÞ YR1 xZ 1 T Z1 þ xY lnðX þ R1 Þ 2 bðu2 u1 Þ 1 2xX X 2 Z 21 lnðY þ R1 ÞþZ 2 ½xT Z2 þ Z 2 lnðY þ R2 Þ þ 2 x T Z2 þ lnðY þ R2 Þ þ Z 1 ½xT Z1 þ Z 1 lnðY þ R1 Þ 2wz Z2 2 3 2 ðZ þ xXÞY xXY x 2 þ þ T Z2 5 þ ½8ð1 mÞ2 þ 2wz4 X 2 þ Z 22 R2 Y 2 þ Z 22 R2 Z 2 1 ð3 4mÞ YR2 xZ 2 T Z2 þ xY lnðX þ R2 Þ 2 u¼u2 #v ¼v 2 1 þ 2xX X 2 Z 22 lnðY þ R2 Þ 2 u¼u1
Fig. 5. Vertical pressures over a horizontal rectangle.
6.1. Vertical pressure on an embedded horizontal area Vertical displacements at the corner K of an embedded horizontal rectangular area (shown in Fig. 6) due to uniform vertical pressure on the area have been given by Groth and Chapman [3,5]. It should be noted that for computing displacements at the boundary including at corners of the rectangular area, the integrated displacement solutions proposed in the current paper are singular. This is because some values of X, Y, Z1 and R1 defined in Eqs. (1), (2), (3) and (5) would be zero and their corresponding natural logarithm terms and denominator terms in Eqs. (20)–(31) are also singular. In order to obtain the displacements at the corner but avoiding the singularity problem, a substituting point, which is very close to the corner, is adopted. For example, suppose the coordinates of a corner are (1, 1, 1), the coordinates of the substituted point used are assumed to be (1.0001, 1.0001, 1.0001). Double-precision arithmetic calculation is used to obtain the displacements. The calculated results from this approximated approach are compared with the results from Groth and Chapman [3,5]. For comparison purpose, the parameters are selected as a = 1 m, b = 1, 5, 10 m, h/b = 0, 0.5, 1.0, 1.5, 2.0, p = 1 kPa, E = 1 MPa, m = 0.5, where a and b are the lengths of the shorter and longer sides of the loaded area, respectively; h is the embedded depth of the loaded area; and p is the vertical pressure. Fig. 6 shows the results calculated by Groth and Chapman [3,5] and by this study using Eq. (31) and applying p1 = p2 = p = 1 kPa. It can be seen that good agreement is obtained.
v ¼v 1
u2 p1 u1 p2 þ ½fð3 4mÞ½Y lnðX þ R1 Þ þ X lnðY þ R1 Þ Z 1 T Z1 bðu2 u1 Þ ! 2wzXY 1 1 þ½8ð1 mÞ2 þ þ Z 2 T Z2 þ Z 1 T Z1 þ R2 X 2 þ Z 22 Y 2 þ Z 22 iv ¼ v 2 u¼u ð3 4mÞ½Y lnðX þ R2 Þ þ X lnðY þ R2 Þ Z 2 T Z2 gu¼u21 ð31Þ v ¼v 1
6. Verification Derivation of the above equations is complex. Therefore, the accuracy of these equations should be verified by the available displacement solutions for some special loading cases. For this purpose, two cases are selected and the verification processes are as follows.
6.2. Vertical pressure on an embedded vertical area Vertical displacements at the corner M of an embedded vertical rectangular area (shown in Fig. 7) due to uniform vertical pressure on the area have been given by Groth and Chapman [3,5]. For comparison purpose, the parameters are selected as a = 1 m, b = 0.1, 0.5, 1, 5, 10 m, h/b = 0, 0.5, 1.0, 1.5, 2.0, q = 1 kPa, E = 1 MPa, m = 0.5, where a and b are the lengths of the shorter and longer sides of the loaded area, respectively; h is the embedded depth of the loaded area; and q is the vertical pressure. Fig. 7 shows the results calculated by Groth and Chapman [3,5] and by this study using Eq. (28) and applying q1 = q2 = q = 1 kPa. It can be seen that good agreement is also obtained.
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Fig. 6. Calculated vertical displacements for the corner K of an embedded horizontal area subjected to uniform vertical pressure.
Fig. 7. Calculated vertical displacements for the corner M of an embedded vertical area subjected to uniform vertical pressure.
7. Conclusions The displacements at an arbitrary computing point caused by a linearly-distributed normal or tangential pressure applied at an embedded vertical or horizontal rectangular area have been formulated algebraically. Good agreement has been obtained between the calculated results from the formulae proposed in this study and from the available solutions of some special loading cases in the literature. This suggests that the formulae are valid for the more general cases of load intensity. The formulae provide an efficient tool for understanding the displacement fields around embedded structures as encountered in most practical situations. Acknowledgments This study was sponsored by the National Natural Science Foundation of China (Grant No. 51278171), the Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT1125), the 111 Project (Grant No. B13024), the Fundamental Research Funds for the Central Universities of China (Grant Nos.
2011B02814 and 2010B28114), the Qing Lan Project of Jiangsu Province of China, the Chang Jiang Scholars Program of the Ministry of Education of China and the Research Grants Council of the Hong Kong Special Administrative Region (Grant No. 617608).
References [1] Douglas DJ, Davis EH. The movements of buried footings due to moment and horizontal load and the movement of anchor plates. Géotechnique 1964;14(2):115–32. [2] Giroud JP. Settlement of a linearly loaded rectangular area. J Soil Mech Found Div ASCE 1968;94(4):813–32. [3] Groth NN, Chapman CR. Computer evaluation of deformations due to subsurface loads in a semi-infinite elastic medium. Bachelor thesis, University of Sydney, Sydney, Australia; 1969. [4] Giroud JP. Stress under linearly loaded rectangular area. J Soil Mech Found Div ASCE 1970;96(1):263–8. [5] Poulos HG, Davis EH. Elastic solutions for soil and rock mechanics. New York: John Wiley & Sons; 1974. [6] Milovic´ D. Stresses and displacements for shallow foundations. Amsterdam: Elsevier Science Publishers B.V.; 1992. [7] Algin HM. Stresses from linearly distributed pressures over rectangular areas. Int J Numer Anal Meth Geomech 2000;24(8):681–92.
H.S. Sun et al. / Computers and Geotechnics 50 (2013) 143–149 [8] Algin HM. Vertical stress formula for pressure over rectangular areas. Géotechnique 2001;51(8):719–22. [9] Algin HM. Elastic settlement under eccentrically loaded rectangular surface footings on sand deposits. J Geotech Geoenviron Eng ASCE 2009;135(10):1499–508. [10] Vaziri H, Simpson B, Pappin JW, Simpson L. Integrated forms of Mindlin’s equations. Géotechnique 1982;32(3):275–8.
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[11] Oasys Ltd. Frew Oasys GEO suite for windows, version 19.0. London: Oasys Ltd.; 2011. [12] Vaziri HH, Troughton VM. An efficient three-dimensional soil-structure interaction model for analysis of earth retaining structures. Can Geotech J 1992;29(4):529–38. [13] Mindlin RD. Force at a point in the interior of a semi-infinite solid. Physics 1936;7(5):195–202.
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Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo
Displacements under linearly distributed pressures by extended Mindlin’s equations H.S. Sun a, G.H. Lei a,⇑, C.W.W. Ng a,b, Q. Zheng a a b
Key Laboratory of Geomechanics and Embankment Engineering of the Ministry of Education, Geotechnical Research Institute, Hohai University, 1 Xikang Road, Nanjing 210098, China Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Special Administrative Region
a r t i c l e
i n f o
Article history: Received 27 November 2012 Received in revised form 11 January 2013 Accepted 30 January 2013 Available online 27 February 2013 Keywords: Deformation Elasticity Footings/foundations Mindlin’s solution
a b s t r a c t Analytical elasticity solutions provide an efficient means of performing a first approximate analysis in foundation engineering. One of the well-known solutions is Mindlin’s solution to the stress and displacement induced by a point load at an embedment depth in a half-space. This solution is more superior but less widely used than Boussinesq’s solution. To promote this situation, Mindlin’s displacement equations are integrated to obtain a complete set of explicit formulae for calculating the displacements at an arbitrary point. The displacements are induced by uniformly and triangularly distributed horizontal or vertical pressures, which are exerted over a horizontal or vertical rectangular area in the interior of a homogeneous, isotropic, elastic half-space. These formulae facilitate the future development of computer programs for the analysis of related practical problems in foundation engineering. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Analytical elasticity solutions play an important role in preliminary predictions and in calibrating any sophisticated numerical schemes that are ultimately used for solving more complex practical problems in geotechnical engineering. Among many available solutions for stress and displacement in the fields of soil and rock mechanics, the solutions for loading over a rectangular area in a homogeneous, isotropic, elastic half-space are the most frequently applied ones in foundation engineering analysis [1–9]. These solutions are commonly expressed in the form of graphs and tables. They are, however, derived only for the computing points at or below the corners of the loaded rectangle. The method of superposition has to be used to calculate other points. Vaziri et al. [10] proposed the displacement solutions for uniform vertical or horizontal pressure over a vertical or horizontal rectangular area embedded in the interior of the half-space. The displacement solutions are derived for an arbitrary point within a predefined coordinate system. They have been readily coded into Oasys’s Frew [11], a software specifically developed for the analysis of flexible retaining walls [12]. In this paper, the uniform loading case studied by Vaziri et al. [10] is expanded into a combined uniform and triangular loading
⇑ Corresponding author. Tel.: +86 13851922201/25 83787216. E-mail addresses: [email protected] (H.S. Sun), [email protected] (G.H. Lei), [email protected] (C.W.W. Ng), [email protected] (Q. Zheng). 0266-352X/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compgeo.2013.01.009
case. The displacement formulae are derived by double integration of Mindlin’s equations and expressed in an explicit form for practical engineering applications such as calculations of ground displacement under anchor plates, rectangular piles or barrettes, eccentrically loaded buried rectangular footings, earth-retaining structures and diaphragm wall panel excavations. 2. The coordinate system and notations The coordinate system for an elastic half-space defined in Fig. 1 is the same as that of Vaziri et al. [10]. The origin of the coordinate system is located on the boundary surface of the half-space. A horizontal point load H or a vertical point load V is applied at an arbitrary point M (u, v, w) away from the origin. The displacements at any point N (x, y, z) due to the point load can be obtained by using Mindlin’s equations [13]. Nevertheless, it should be noted that the coordinate system adopted here is slightly different from that of Mindlin [13], where the point load H or V is applied at a point (u = 0, v = 0, w = c) under the origin. Similarly to the work of Vaziri et al. [10], the following notations are assumed for the purpose of presenting the derived displacement solutions in a compact form.
X ¼xu
ð1Þ
Y ¼yv
ð2Þ
Z1 ¼ z w
ð3Þ
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Nomenclature E G p, q x, y, z u, v, w
Young’s modulus shear modulus pressures normal and tangential to the loaded rectangle Cartesian coordinate axes coordinates of points in the loaded rectangle
Z2 ¼ z þ w Ri ¼
ð4Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 2 þ Y 2 þ Z 2i ði ¼ 1 or 2Þ
T ai ¼ arctan
m Poisson’s ratio dx, dy, dz displacements in the x, y and z directions H, V horizontal and vertical point loads
XYZ i a2 Ri
ða ¼ X; Y;
ð5Þ
or Z i ;
i ¼ 1 or 2Þ
ð6Þ
e.g.,
T X1 ¼ arctan
YZ 1 XR1
ð7Þ
T Z1 ¼ arctan
XY Z 1 R1
ð8Þ
B ðA ¼ X; Y; A ¼ 1 or 2Þ
T AB ¼ arctan
or Z i ;
B ¼ X; Y;
or Z i ;
i ð9Þ Fig. 1. The coordinate system (OABC lies on the surface of the half-space).
e.g.,
T XY ¼ arctan
Y X
ð10Þ
Z1 Y
ð11Þ
T YZ 1 ¼ arctan b ¼ 8pE
1m ¼ 16pGð1 mÞ 1þm
ð12Þ
where E is Young’s modulus, G is the shear modulus, and m is Poisson’s ratio. 3. Derivation procedures Following the coordinate system in Fig. 1 and the assumed notations, Mindlin’s original equations [13] for obtaining the displacements at any point N (x, y, z) can be readily rewritten as follows by a shift of origin. For a horizontal point load, the displacements in the x, y and z directions are given by
dx ¼
( ! ! H 1 X2 1 X 2 2wz 3X 2 ð3 4mÞ þ 3 þ þ 3 þ 3 1 2 b R1 R2 R2 R1 R2 R2 " #) 2 4ð1 mÞð1 2mÞ X 1 þ R2 þ Z 2 R2 ðR2 þ Z 2 Þ "
#
ð13Þ
VX Z 1 Z 1 6wzZ 2 4ð1 mÞð1 2mÞ dx ¼ ð3 4mÞ 3 þ 3 þ b R2 ðR2 þ Z 2 Þ R52 R2 R1
ð14Þ
" # VY Z 1 Z 1 6wzZ 2 4ð1 mÞð1 2mÞ dy ¼ ð3 4mÞ 3 þ 3 þ b R2 ðR2 þ Z 2 Þ R52 R2 R1
#
HXY 1 1 6wz 4ð1 mÞð1 2mÞ ð3 4mÞ 3 þ 3 5 b R2 R2 R1 R2 ðR2 þ Z 2 Þ2 " # HX ð3 4mÞZ 1 Z 1 6wzZ 2 4ð1 mÞð1 2mÞ dz ¼ þ 3 þ b R2 ðR2 þ Z 2 Þ R52 R32 R1 dy ¼
Fig. 2. Horizontal pressures over a vertical rectangle.
"
ð15Þ
For a vertical point load, the displacements in the x, y and z directions are given by
ð16Þ
ð17Þ
" ! # V 1 Z 22 2wz Z 21 6wzZ 22 8ð1 mÞ2 ð3 4mÞ ð3 4mÞ dz ¼ þ þ 3 þ 3þ b R1 R32 R2 R52 R2 R1 ð18Þ
The displacements (dx, dy, dz) at point N (x, y, z) due to horizontal or vertical pressure over a vertical rectangle (u = constant, v = v1 to v2,
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Xðp2 p1 Þ ½fð3 4mÞ½R2 z lnðZ 2 þ R2 Þ bðw2 w1 Þ " # Z1 z2 Z 2 2 2z þ þ lnðZ 2 þ R2 Þ þ ð1 mÞð1 2mÞ 3 R2 ðX 2 þ Y 2 ÞR2 Z 2 ðz 2wÞ 3z lnðZ 2 þ R2 Þ 2R2 þ þ R1 R2 þ Z 2 iv ¼v 2 Xðw p w p Þ w¼w 2 1 1 2 z lnðZ 1 þ R1 Þgw¼w21 þ v ¼v 1 bðw2 w1 Þ "( 2z 2z2 Z 2 ð3 4mÞ lnðZ 2 þ R2 Þ lnðZ 1 þ R1 Þ þ þ 2 R2 ðX þ Y 2 ÞR2 w¼w2 #v ¼v 2 Z2 þ2ð1 mÞð1 2mÞ þ lnðZ 2 þ R2 Þ ð21Þ R2 þ Z 2 w¼w1
dy ¼
v ¼v 1
Xðp2 p1 Þ bðw2 w1 Þ
dz ¼
Fig. 3. Horizontal pressures over a horizontal rectangle.
w = w1 to w2; Figs. 2 and 4) can be derived by double integration of Eqs. (13)–(18) with respect to v and w. Similarly, the displacements due to horizontal or vertical pressure over a horizontal rectangle (u = u1 to u2, v = v1 to v2, w = constant; Figs. 3 and 5) can be derived by double integration of Eqs. (13)–(18) with respect to u and v. As an example, consider the horizontal pressure over a vertical rectangle in Fig. 2. The point load on a rectangular element at the point (u, v, w) can be readily derived as follows:
H¼
1 ½ðw2 p1 w1 p2 Þ þ ðp2 p1 Þwdv dw ðw2 w1 Þ
ð19Þ
"
ð3 4mÞ
X 2 2z2 T X2 þ 3z lnðY þ R2 Þ X
þY lnðZ 2 þ R2 Þ þ XT X1 Y lnðZ 1 þ R1 Þ z lnðY þ R1 Þ " # Yw2 2zT X2 þ 2z þ 2lnðY þ R2 Þ 4ð1 mÞð1 2mÞ X ðX 2 þ Z 22 ÞR2 " ðX 2 Z 1 Z 2 Þ ðT XY T X2 Þ þ z lnðY þ R2 Þ 2X w¼w2 #v ¼v 2 Y Xðw2 p1 w1 p2 Þ þ lnðZ 2 þ R2 Þ þ ½fð3 4mÞ 2 bðw2 w1 Þ w¼w1 v ¼v 1 2z 2zwY 2zT X2 þ T X2 lnðY þ R2 Þ lnðY þ R1 Þ X X X 2 þ Z 2 R2 2
w¼w2 #v ¼v 2 Z2 þ4ð1 mÞð1 2mÞ lnðY þ R2 Þ þ ðT X2 T XY Þ X w¼w1
ð22Þ
v ¼v 1
where p1 and p2 are the pressures at the points (u, v1, w1) and (u, v1, w2), respectively. Substituting this point load into Eqs. (13)–(15) and taking double integration with respect to v and w give the required displacement solutions.
For the case of combined uniform and triangular horizontal pressures over a horizontal rectangular area as shown in Fig. 3, the displacements are given by
R1 R2 Y x½Z 1 T Z1 þ Z 2 T Z2 2 ! X 2 þ Z 21 Y lnðX þ R1 Þ X lnðY þ R1 Þ Y lnðX þ R2 Þ 2x #
Z2 R2 þ 2 lnðY þ R2 Þ þ R1 Y x 2 " ! X 2 þ Z 22 x Z 1 T Z1 þ Z 2 T Z2 Y lnðX þ R1 Þ X 2x # 2 Z lnðY þ R2 Þ Y lnðX þ R2 Þþ 1 lnðY þ R1 Þ x 2 3 uXY 4 þ lnðY þ R2 Þ5 þ 2ð1 mÞð1 2mÞ 2wz X 2 þ Z 22 R2 h YR2 þ2xZ 2 ðT X2 T XY Þ þ 2xX X 2 Z 22 lnðY þ R2 Þ iv ¼v 2 u2 q1 u1 q2 u¼u 2YZ 2 lnðZ 2 þ R2 Þgu¼u21 þ ½fð3 4mÞ v ¼v 1 bðu2 u1 Þ
dx ¼ 4. Displacements due to horizontal pressures For the case of combined uniform and triangular horizontal pressures over a vertical rectangular area as shown in Fig. 2, the displacements are given by dx ¼
"(
p2 p1 bðw2 w1 Þ 1 1 2 X þ Z 21 2zZ 1 ð3 4mÞ YR1 þ zY lnðZ 1 þ R1 Þ 2 2
lnðY þ R1 Þ þ zXðT X2 T X1 Þ þ X 2 lnðY þ R2 Þ þ z½3XT X2 þ Z 2 lnðY þ R2 Þ 2 3 YðX 2 Z 1 þ z2 Z 2 Þ Yz2 Z 2 5 Y lnðZ 2 þ R2 Þ þ 2z4 þ 2 2z lnðY þ R Þ 2 ðX þ Y 2 ÞR2 X 2 þ Z 22 R2 YZ 2 ð2w zÞ YR2 2zXT X2 ðX 2 Z 1 Z 2 Þ lnðY þ R2 Þ þ 2ð1 mÞð1 2mÞ 3ðR2 þ Z 2 Þ 3 1 1 2 þ 6zY lnðZ 2 þ R2 Þ YR2 X þ Z 22 lnðY þ R2 Þ þ zXT X1 2 2 ow¼w2 v ¼v 2 w p w p 2 1 1 2 þ X 2 lnðY þ R1 Þ þ ½fð3 4mÞ½Y lnðZ 1 þ R1 Þ w¼w1 v ¼v bðw2 w1 Þ 1 þ Z 1 lnðY þ R1 Þ XðT X1 þ T X2 Þ Y lnðZ 2 þ R2 Þ þ Z 1 lnðY þ R2 Þ ! 2zY wX 2 zZ 2 þ XðT X1 þ T X2 Þ þ z þ 2 2 2 þ4ð1 mÞð1 2mÞ 2 2 Z 2 R2 X þ Z 2 X þY w¼w2 #v ¼v 2 Y YZ 2 ð20Þ þ XT X2 lnðZ 2 þ R2 Þ Z 2 lnðY þ R2 Þ þ 2 2ðZ 2 þ R2 Þ w¼w1 v ¼v 1
q2 q1 bðu2 u1 Þ
ð3 4mÞ
½Y lnðX þ R1 Þ þ X lnðY þ R1 Þ Z 1 T Z1 þ Y lnðX þ R2 Þ Z 2 T Z2 þ Y lnðX þ R2 Þ þ X lnðY þ R2 Þ Z 2 T Z2 þ Y lnðX þ R1 Þ 2wzXY þ 4ð1 mÞð1 2mÞ Z 1 T Z1 þ R2 X 2 þ Z 22 iv ¼v 2 u¼u ½X lnðY þ R2 Þ þ Z 2 ðT X2 T XY Þgu¼u21 v ¼v 1
ð23Þ
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H.S. Sun et al. / Computers and Geotechnics 50 (2013) 143–149
i q q1 3 4m h dy ¼ 2 ð2x þ XÞR2 Y 2 þ Z 22 lnðX þ R2 Þ bðu2 u1 Þ 2 i 1h þ ð2x þ XÞR1 Y 2 þ Z 21 lnðX þ R1 Þ 2 u 2wz þ lnðX þ R2 Þ ð1 mÞð1 2mÞ R2
dx ¼
þ XY lnðZ 2 þ R2 Þ X½XT X1 þ z lnðY þ R1 Þ þ Y lnðZ 1 þ R1 Þ 2 3 2 w XY þ 2zT X2 þ 2X lnðY þ R2 Þ52ð1 mÞ þ 2z4 X 2 þ Z 22 R2
½ð2X 4xÞR2 þ 4YZ 2 ðT YX T Y2 Þ þ 2xZ 2 lnðX 2 þ Y 2 Þ 2 16 X 2 þ Z 22 þ XR2 þ ðYZ 2 Þ2 2Y 2 lnðX þ R2 ÞþZ 22 ln X 4 Z 42 Y 2 þ Z 22 39u¼u2 3v ¼v 2 2 > 4 X 2 þ Z 22 þ Z 2 R2 þ ðXYÞ2 7> = 7 7 7 þ 2xZ 2 ln 7 5 2 6 2 2 > 5 X Z 2 ðX þ Y Þ > ; u¼u1
ð1 2mÞ½ðX 2 Z 1 Z 2 ÞðT X2 T XY Þ X½2z lnðY þ R2 Þ i w2 q1 w1 q2 v ¼v w¼w2 þ ½fð3 4mÞ½2zT X2 þ Y lnðZ 2 þ R2 Þgv ¼v 21 w¼w1 bðw2 w1 Þ 2 3 wXY 4 5 X lnðY þ R2 Þ X lnðY þ R1 Þ þ 2z T X2 þ X 2 þ Z 22 R2 2 þ 4ð1 mÞð1 2mÞ½Z 2 ðT XY T X2 Þ X lnðY þ R2 Þgvv ¼¼vv 21 w¼w w¼w1 ð26Þ
v ¼v
1 2wz 4ð1 mÞ ð3 4mÞR2 R1 R2 iv ¼v 2 u¼u ð1 2mÞ½R2 þ Z 2 lnðZ 2 þ R2 Þgu¼u21
dy ¼
u2 q1 u1 q2 þ bðu2 u1 Þ
v ¼v 1
dz ¼
ð24Þ
q2 q1 ½fð3 4mÞZ 1 ½Z 2 T Z2 þ Y lnðX þ R2 Þ bðu2 u1 Þ
þx lnðY þ R2 Þ Z 1 ½Z 1 T Z1 þ Y lnðX þ R1 Þ þ xlnðY þ R1 Þ 2 3 uYZ 2 þ T Z2 5 þ 2ð1 mÞð1 2mÞ þ2wz4 X 2 þ Z 22 R2
v ¼v 1
þ 2YZ 2 lnðX þ R2 Þ þ 2xZ 2 lnðY þ R2 Þ iv ¼v 2 u q u q u¼u 2 1 1 2 þ2xY lnðZ 2 þ R2 Þgu¼u21 þ ½fð3 4mÞZ 1 v ¼v 1 bðu2 u1 Þ 2wzZ 2 Y þ4ð1 mÞ lnðY þ R2 Þ Z 1 lnðY þ R1 Þ þ X 2 þ Z 22 R2 u¼u
q2 q1 bðw2 w1 Þ 3 4m ½ðw 5zÞR2 ðX 2 þ Y 2 4z2 Þ lnðZ 2 þ R2 Þ 2 1 þ ½Z 2 R1 ðX 2 þ Y 2 Þ lnðZ 1 þ R1 Þ þ 2z 2 w2 2z lnðZ 2 þ R2 Þ ð1 mÞð1 2mÞ½ð4z þ Z 2 ÞR2 2R2 R2
v ¼v 2 #w¼w2 2 2 X þ Y 2Z 1 Z 2 lnðZ 2 þ R2 Þ w¼w1
w2 q1 w1 q2 þ ½fð3 4mÞ½R1 þ 2z lnðZ 2 þ R2 Þ R1 bðw2 w1 Þ w þ2z þ lnðZ 2 þ R2 Þ 4ð1 mÞð1 2mÞ½R2 R2 i v ¼v w¼w2 þ Z 2 lnðZ 2 þ R2 Þgv ¼v 21
½XY þ ðT YX T Y2 ÞY 2 þ ð2xX X 2 ÞðT XY T X2 Þ Z 22 T Z2
ð1 2mÞ½Y lnðZ 2 þ R2 Þ þ Z 2 lnðY þ R2 Þ þ XðT XY T X2 Þgu¼u21
q 2 q 1 hn ð3 4mÞ½ð2z2 X 2 ÞT X2 þ 3zX lnðY þ R2 Þ bðw2 w1 Þ
w¼w1
ð27Þ
q2 q1 1 ð3 4mÞ YR1 þ zX lnðZ 1 þ R1 Þ 2 bðw2 w1 Þ 1 2 2 X þ Z 1 2zZ 1 lnðY þ R1 Þ zXT X1 YR2 zXT X2 2 i 2z X 2 lnðY þ R2 Þ þ Yz lnðZ 2 þ R2 Þ þ ½ðX 2 z2 ÞT X2 X 2zX lnðY þ R2 Þ XY lnðZ 2 þ R2 Þ YR1 zXT X1 2 Yw2 Z 2 2 X lnðY þ R1 Þ þ zY lnðZ 1 þ R1 Þ2z4 2 X þ Z 22 R2 # 3X 2 z2 T X2 þ 4z lnðY þ R2 Þ þ 3Y lnðZ 2 þ R2 Þ þ ½8ð1 mÞ2 X 1 ð3 4mÞ YR2 zXT X2 þ zY lnðZ 2 þ R2 Þ 2 v ¼v 2 #w¼w2 1 2 2 X þ Z 2 2zZ 2 lnðY þ R2 Þ 2 v ¼v 1
dz ¼ iv ¼v 2 v ¼v 1
ð25Þ
5. Displacements due to vertical pressures For the case of combined uniform and triangular vertical pressures over a vertical rectangular area as shown in Fig. 4, the displacements are given by
w¼w1
w2 q1 w1 q2 þ ½fð3 4mÞ½Y lnðZ 1 þ R1 Þ þ Z 1 lnðY þ R1 Þ bðw2 w1 Þ þ Z 2 lnðY þ R2 Þ XT X1 þ Y lnðZ 1 þ R1 Þ XT X1 þ 2z lnðY þ R2 Þ 2zwYZ 2 8ð1 mÞ2 ½Y lnðZ 2 þ R2 Þ þ Z 2 lnðY þ R2 Þ þ X 2 þ Z 22 R2 2 XT X2 gvv ¼¼vv 21 w¼w w¼w1
Fig. 4. Vertical pressures over a vertical rectangle.
ð28Þ
For the case of combined uniform and triangular vertical pressures over a horizontal rectangular area as shown in Fig. 5, the displacements are given by
H.S. Sun et al. / Computers and Geotechnics 50 (2013) 143–149
dx ¼
147
p2 p1 ½fð3 4mÞZ 1 ½Z 2 T Z2 Y lnðX þ R2 Þ x lnðY þ R2 Þ bðu2 u1 Þ þ Z 1 ½Z 1 T Z1 Y lnðX þ R1 Þ x lnðY þ R1 Þ 2 3 h uYZ 2 T Z2 5 2ð1 mÞð1 2mÞ XY þ Y 2 T YX þ 2wz4 2 2 X þ Z 2 R2 þ ð2x XÞXT XY Z 22 T Z2 þ ðX 2 2xXÞT X2 Y 2 T Y2
þ2YZ 2 lnðx þ R2 Þ þ 2xZ 2 lnðY þ R2 Þ iv ¼ v 2 u p u p u¼u 2 1 1 2 ½fð3 4mÞZ 1 þ þ2xY lnðZ 2 þ R2 Þgu¼u21 v ¼v 1 bðu2 u1 Þ 2wzYZ 2 4ð1 mÞð1 2mÞ lnðY þ R2 Þ Z 1 lnðY þ R1 Þ R2 X 2 þ Z 22 iv ¼v 2 u¼u ð29Þ ½Y lnðZ 2 þ R2 Þ þ Z 2 lnðY þ R2 Þ XT X2 gu¼u21 v ¼v 1
dy ¼
p2 p1 ½fð3 4mÞZ 1 ½R2 x lnðX þ R2 Þ bðu2 u1 Þ 2 3 xX 1 þ 5 þ Z 1 ½R1 x lnðX þ R1 Þ 2wzZ 2 4 Y 2 þ Z 22 R2 R2 1 1 4ð1 mÞð1 2mÞ xX þ R22 Z 2 R2 þxYðT YX T Y2 Þ 4 2 u¼u2 #v ¼v 2 1 2 2 þ xZ 2 lnðX þ R2 Þ þ ð2xX X Y Þ lnðZ 2 þ R2 Þ 2 u¼u1 v ¼v 1
u2 p1 u1 p2 þ ½fð3 4mÞZ 1 lnðX þ R2 Þ Z 1 lnðX þ R1 Þ bðu2 u1 Þ 2wzXZ 2 4ð1 mÞð1 2mÞ½YðT YX T Y2 Þ þ Z 2 lnðX þ R2 Þ Y 2 þ Z 22 R2 iv ¼v 2 u¼u ð30Þ þ X lnðZ 2 þ R2 Þgu¼u21 v ¼v 1
dz ¼
p2 p1 1 ð3 4mÞ YR1 xZ 1 T Z1 þ xY lnðX þ R1 Þ 2 bðu2 u1 Þ 1 2xX X 2 Z 21 lnðY þ R1 ÞþZ 2 ½xT Z2 þ Z 2 lnðY þ R2 Þ þ 2 x T Z2 þ lnðY þ R2 Þ þ Z 1 ½xT Z1 þ Z 1 lnðY þ R1 Þ 2wz Z2 2 3 2 ðZ þ xXÞY xXY x 2 þ þ T Z2 5 þ ½8ð1 mÞ2 þ 2wz4 X 2 þ Z 22 R2 Y 2 þ Z 22 R2 Z 2 1 ð3 4mÞ YR2 xZ 2 T Z2 þ xY lnðX þ R2 Þ 2 u¼u2 #v ¼v 2 1 þ 2xX X 2 Z 22 lnðY þ R2 Þ 2 u¼u1
Fig. 5. Vertical pressures over a horizontal rectangle.
6.1. Vertical pressure on an embedded horizontal area Vertical displacements at the corner K of an embedded horizontal rectangular area (shown in Fig. 6) due to uniform vertical pressure on the area have been given by Groth and Chapman [3,5]. It should be noted that for computing displacements at the boundary including at corners of the rectangular area, the integrated displacement solutions proposed in the current paper are singular. This is because some values of X, Y, Z1 and R1 defined in Eqs. (1), (2), (3) and (5) would be zero and their corresponding natural logarithm terms and denominator terms in Eqs. (20)–(31) are also singular. In order to obtain the displacements at the corner but avoiding the singularity problem, a substituting point, which is very close to the corner, is adopted. For example, suppose the coordinates of a corner are (1, 1, 1), the coordinates of the substituted point used are assumed to be (1.0001, 1.0001, 1.0001). Double-precision arithmetic calculation is used to obtain the displacements. The calculated results from this approximated approach are compared with the results from Groth and Chapman [3,5]. For comparison purpose, the parameters are selected as a = 1 m, b = 1, 5, 10 m, h/b = 0, 0.5, 1.0, 1.5, 2.0, p = 1 kPa, E = 1 MPa, m = 0.5, where a and b are the lengths of the shorter and longer sides of the loaded area, respectively; h is the embedded depth of the loaded area; and p is the vertical pressure. Fig. 6 shows the results calculated by Groth and Chapman [3,5] and by this study using Eq. (31) and applying p1 = p2 = p = 1 kPa. It can be seen that good agreement is obtained.
v ¼v 1
u2 p1 u1 p2 þ ½fð3 4mÞ½Y lnðX þ R1 Þ þ X lnðY þ R1 Þ Z 1 T Z1 bðu2 u1 Þ ! 2wzXY 1 1 þ½8ð1 mÞ2 þ þ Z 2 T Z2 þ Z 1 T Z1 þ R2 X 2 þ Z 22 Y 2 þ Z 22 iv ¼ v 2 u¼u ð3 4mÞ½Y lnðX þ R2 Þ þ X lnðY þ R2 Þ Z 2 T Z2 gu¼u21 ð31Þ v ¼v 1
6. Verification Derivation of the above equations is complex. Therefore, the accuracy of these equations should be verified by the available displacement solutions for some special loading cases. For this purpose, two cases are selected and the verification processes are as follows.
6.2. Vertical pressure on an embedded vertical area Vertical displacements at the corner M of an embedded vertical rectangular area (shown in Fig. 7) due to uniform vertical pressure on the area have been given by Groth and Chapman [3,5]. For comparison purpose, the parameters are selected as a = 1 m, b = 0.1, 0.5, 1, 5, 10 m, h/b = 0, 0.5, 1.0, 1.5, 2.0, q = 1 kPa, E = 1 MPa, m = 0.5, where a and b are the lengths of the shorter and longer sides of the loaded area, respectively; h is the embedded depth of the loaded area; and q is the vertical pressure. Fig. 7 shows the results calculated by Groth and Chapman [3,5] and by this study using Eq. (28) and applying q1 = q2 = q = 1 kPa. It can be seen that good agreement is also obtained.
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Fig. 6. Calculated vertical displacements for the corner K of an embedded horizontal area subjected to uniform vertical pressure.
Fig. 7. Calculated vertical displacements for the corner M of an embedded vertical area subjected to uniform vertical pressure.
7. Conclusions The displacements at an arbitrary computing point caused by a linearly-distributed normal or tangential pressure applied at an embedded vertical or horizontal rectangular area have been formulated algebraically. Good agreement has been obtained between the calculated results from the formulae proposed in this study and from the available solutions of some special loading cases in the literature. This suggests that the formulae are valid for the more general cases of load intensity. The formulae provide an efficient tool for understanding the displacement fields around embedded structures as encountered in most practical situations. Acknowledgments This study was sponsored by the National Natural Science Foundation of China (Grant No. 51278171), the Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT1125), the 111 Project (Grant No. B13024), the Fundamental Research Funds for the Central Universities of China (Grant Nos.
2011B02814 and 2010B28114), the Qing Lan Project of Jiangsu Province of China, the Chang Jiang Scholars Program of the Ministry of Education of China and the Research Grants Council of the Hong Kong Special Administrative Region (Grant No. 617608).
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