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Soil-structure interaction in the time domain
SOIL-STRUCTURE
INTERACTION IN THE TIME DOMAIN
M. N. VILADKAR,~ G. RANJAN? and R. P. SHARMA$ tCivi1 Engineering Department, University of Roorkee, Roorkee-247 667, India $Civil Engineering Department, Regional Institute of Technology, Jamshedpur-831014, (Rece&eff 3 seceder
India
1991)
Abstract-Saturated soils, which are two-phase media, display time-dependent behaviour at constant loading. The resulting phenomena of consolidation and creep causes deformations which are significant compared to instantaneous deformations or those which occur during the construction period of the structure. Redistribution of shear forces, bending and torsional moments in the structure occur due to differential settlements rather than the total settlements. Most of the interaction analyses presented so far are based upon the elastic behaviour of the whole system. An attempt has been made in this paper to present a thr~-dimensional v&co-elastic finite element formulation for studying the interactive behaviour of space frames, taking into account the stress/strain-time response of supporting soil medium. The paper also underlines the methodology for evaluating time-dependent visco-elastic constants for the soil mass and presents a comparison of the resulting structural behaviour with that when interaction is neglected.
lNTRODUCTION The mechanics of interaction between structure, foundation and subsoil must take into account the
effect of complex states of stress, strain and environment on the mechanical behaviour of different classes of materials. Calculation of mechanical behaviour under different states requires that different variables involved be related by means of fundamental equations including the equilibrium equations, the kinematic equations, the compatibility conditions, the constitutive equations and the boundary conditions. Soil behaves elastically under small magnitudes of stresses and the resulting strains are recoverable. If the stress is high, the irrecoverable inelastic strain results represent the plastic behaviour. Visco-elastic soils exhibit elastic behaviour upon loading followed by a slow and continuous increase of strain at a decreasing rate. A continuously decreasing strain upon stress removal follows an initial elastic recovery. Such soils are significantly influenced by the rate of strain or stress, the longer the time to reach the final stress level at constant rate of stress, the larger is the corresponding strain. The behaviour of v&o-elastic materials must therefore be expressed by constitutive equations which include time as an additional variable in addition to stress and strain. The general creep behaviour of any material includes a transient phase in which the strain rate which initially is very high and which sub~uen~y goes on decreasing until a steady-state is reached in which the strain rate is either zero or very small and deformations, quite large. This state is followed by a transient phase in which the strain rate is very high. This phase usually leads to failure. CAS 463-c
429
Time-dependent behaviour of saturated soils, which are primarily two-phase media, consists of two phenomena, namely, the primary consolidation in which deformations occur due to hydro-dynamic lag and creep or secondary consolidation which occurs due to structural breakdown of soil. The deformations due to these two phenomena are quite large compared to instantaneous deformations or those which occur during the construction period of the structure. PROBLEM ID~N~~CA~ON
Most of the interactive analyses of structures consider the elastic behaviour of the whole system. There is a major lack of knowledge as regards the understanding of the interactive behaviour of structnres fords on soil displaying time~~ndent constitutive refationships. An attempt is therefore made in the present work to give a fully three-dimensional visco-elastic finite element formulation for a structur+foundation-soil system. An approach for the evaluation of visco-elastic constants of soil is also discussed along with the interactive behaviour of space frames. EARLIER WORK 1. Interaction of structures with respect to consolidatioa behaviour of soil
Chamecki [l] used the expression for settlement from Terxaghi’s one-dimensional consolidation theory to compute the array of transfer coel%ients required for the interactive analysis of plane frames. Morris 121treated the problem of a continuous space
430
M. N.
VILADKAR er (II.
frame on soil media as a problem of frame with individual columns supported by a Kelvinian model in series with a Hookean spring to account for the time-dependent behaviour of soil. The coupling of the structure to the foundation mechanism was accomplished in the force-displacement relationship of the structure. The resulting visco-elastic equilibrium equations were solved to study the interactive behaviour of the structure. Heil[3] presented a method of settlement analysis in which the effect of time-settlement properties of subsoil, creep of concrete and procedure of structural construction were all accounted for. Larnach l4] computerized Chamechi’s [I] approach to allow sufficient iterations so as to ensure a fully converging solution. King and Chandrasekaran [5] used plane strain finite element modelling to study immediate and the long-term response of the structure which has been found to have been significantly affected by both the inhomogeneous and transversely isotropic nature of the clay stratum and the raft flexibility. Later, King and Chandrasekaran [6] presented a fully three-dimensional finite element formulation for studying both the immediate and the long-term behaviour of a space frame founded on a raft and supported by an overconsolidated clay stratum. The formulation makes use of the substructure technique, the undrained soil modulii for studying the immediate behaviour and the drained soil modulii for the long-term behaviour of the structure. Wood and Larnach [7] developed a method of assessing the combined behaviour but assuming that soil offers restraint only to the vertical settlement. King and Chandrasekaran [8] proposed a simplified method for evaluating raft support stiffness, and found on comparing the results with those obtained through FEM that almost identical results could be obtained. 2. Interaction of structures behaviour of soil
with respect to creep
Huang[9] idealized the problem of a raft on a linearly v&o-elastic foundation as a thin elastic plate connected to a complex rheological analogue consisting of a Maxwell model connected to a series of Kelvin models. Two procedures were used for the dete~ination of stresses and deflections in the raft, namely, the method of integral transforms which is a single-step procedure and a method based on time increments which can be adopted to consider the complicated time history of loading, such as the loading due to sequential construction. Brown [lo] examined the effect of linear soil creep with regard to differential settlements of structures. It was shown that for a flexible footing, the differential settlement tends to increase unless restrained by the structural stiffness. However, large values of relative stiffness of footing tend to reduce the differential settlement with time. Brown and Booker [ 1I] converted the problem of raft on visco-elastic foundation to an equivalent
elastic problem using the Laplace transforms. The solution was converted to an eigenvector expansion and then application of the inverse Laplace transform gave the solution in the time domain. The analysis neglects the effect of shear stresses, if any, developed along the base of the raft. SOIL AS A VISCO-ELASTIC
BODY
The time dependency of deformations in soil is usually attributed to two aspects, namely the primary consolidation and the secondary compression or creep of soil. Prediction of field behaviour usually assumes these as an uncoupled phenomenon despite the fact that Taylor and Merchant [ 121, Gibson and Lo [ 131, Schiffman et al. [141 and Barden (I 51 have discovered the coupled behaviour of soil under onedimensional loading. The time-dependent behaviour of clay soils changes depending upon drainage conditions. In undrained conditions, saturated soil behaves as an incompressible material, the volumetric strains as zero and the deviatoric strains are therefore the total strains. However, in a drained situation, both the volumetric and the deviatoric strain components will appear. One can thus refer to volumetric creep and deviatoric creep and therefore volumetric creep compliance (strain required to cause a unit change in hydrostatic stress) and deviatoric creep compliance (strain required to cause a unit change in deviatoric stress). VISCO-ELASTIC
APPROACH
(4 The structure may be either plane or space frame but all the joints in the structure act as rigid joints. (b) All the loads are supposed to act only at the joints. Cc) The foundation for the structure can be either individual column footings or the combined footing or a raft. (4 The superstructure and the foundation behave in a linearly elastic manner. The soil supporting the foundation is treated as W a linear visco-elastic half space and may be stratified in nature. In a finite element formulation, the soil is treated as homogeneous and isotropic within an element. The properties, especially in the clay strata, may differ from element to element. 07 Due to presence of clay soil, the problem is treated as quasi-static and isothermal. The shear strength of soil is never exceeded. Infinitesimal strain theory is valid. Constitutive law for visco -elastic body In the present analysis, saturated soil mass is considered in a drained situation and is idealized as a Kelvin body (Fig. 1) consisting of a spring and
r u
R
Comparison
of eqns (2) and (4) gives
P,=l,
Ql=(R.Cn$=(R’+n.D),
Pr’Sii(r) = Q2.4i(t),
t
Fig. 1. Kelvin model.
dashpot in parallel so that for constant applied stress, the strain experienced by the two components are the same whereas the stresses shared by the two are different. (i) Uni-axial stress situation. The time-dependent stress-strain relationship for the idealized soil model under uniaxial stress situation is given by Flugge [16] and Findley et al. [17] as
1
.D)G
f P:; n=l
where S, and d, are hydrostatic components of stress and strain and P2 and Qr are again time operators of the form Pz=Pi+
f P:; n=l
(64
Q,=Q;+
5 Q,;. n-l
t6b)
The stress-strain relationship for a Kelvin body under a hydrostatic stress situation, by analogy to the deviatoric case, can be expressed as
Comparison
of eqns (6) and ( 7) gives
P2= 1,
Q~=(R.+r”~)=(R.+9.D),
(34
Visco-elastic constants
The stress-strain relations for an elastic body for the deviatoric and hydrostatic cases are respectively S,=2G.d,
(9a)
aii=3K.cii.
W
and
By comparing eqn (3) with (9a) and (6) with (9b), the corresponding elastic constants are modified in terms of time operators as
G=!.%
(3b) If the material is isotropic and homogeneous, the stress-strain relation for a Kelvin body under a deviatoric stress situation can be written as
(8)
where R” and ?‘I are the spring and dashpot constants for Kelvin elements under hydrostatic stress conditions.
(2)
where S, and du are deviatoric components of stress and strain and P, and Q, are the operators of the form P,=P&+
(6)
(1)
where R and q are respectively the spring and the dashpot constants, cr, the applied constant stress, 6 and i are respectively the strain and strain rate and D = a/at is the differential operator. (ii) Multi-axial stress situation. The state of stress and strain, at any point within a loaded body can be decomposed into its hydrostatic and deviatoric components. (a) Deviatoric stress condition : The differential form of the constitutive law for deviatoric component is expressed as
f’, . s&) = Q, *d&),
(5)
where R’ and rl’ are the spring and dashpot constants for Kelvin elements under deviatoric stress condition. (b) Hydrostatic stress condition: The constitutive law for hydrostatic component, in differential form can also be written as
‘I
u
= (R +
431
Soil-structure interaction in the time domain
and
Kc!.@
3 P2’
2 p,
(10)
where G and K are shear and bulk modulii of the material. The modulus of elasticity, E, and Poisson’s ratio, v, are related to K and G as E=-
9KG 3K+G
and
v=-.
3K -2G 6K+2G
(11)
M. N. VILADKAR
432
Substituting the values of G and K in terms of time operators from eqns (10) into (1 l), visco-elastic material constants, namely, time-dependent Young’s modulus, E,(t) and time-dependent Poisson’s ratio, v,(t) can be obtained as E,(t) =
and Visco -elasticity
Equation (13) can therefore by rewritten as
{~I=
P, Q2- f’*Q, PzQ, + 2P, Qz’
Finite element formulation
(12)
~~i=([~l+~.t-‘[rTl)~~,~-~~-‘~rTl~~r-,~
matrix
=[R”,l{GI -h,lG-Ii>
For a visco-elastic material, the time-dependent stress-strain relationship can now be written for a general three-dimensional stress situation as (01 = [U {ci>
(13) {a} = [N]{he}
and matrix.
The visco-elasticity matrix can be obtained by substituting the values of E,,(t) and v,(t) from eqn (12) into a conventional elasticity matrix to get
[VI=;
(17)
where At = time increment, (6,) = strain vector at time t, {t,_ 1> = strain vector at time (t - l), [R,,] = ([a] + At -‘[ri]) and [n,] = At-‘[q]. Using
where
[V] = visco-elasticity
(16)
The stress/strain-time relationship for visco-elastic material given by eqn (16) can be expressed in finite difference form as
3Q1Q2 f’, Q, + 2P, Q,
v,,(t) =
et al.
and
{t} = [B]{6’},
(18)
where [N] and [B] are respectively the shape function and strain-displacement matrices, the governing equilibrium equations of an element can be obtained using the principle of virtual work. At any time instant, t, consider a single element acted upon by nodal forces, {Fe}, body forces, (b’}, and distributed traction forces, {q’}, which result in an equilibrating
2P,Q,+P,Q,
P,Q,--P,Q,
f',Qz - PzQ,
0
0
0
P,Q,-PIPI
2f-‘2Q,x f’,Q,
P,Qz-P,Q,
0
0
0
P,Q,-f’,Q,
P,Qz--f’zQ,
2f’, Q, + P, Qz
0
0
0
0
0
0
tP,Q,
0
0
0
0
0
0
;p,Q,
0
0
0
0
0
0
;P,Q,
(14)
This is the general form of the visco-elasticity matrix which can be expressed in terms of spring and dashpot constants of the Kelvin body by substituting expressions (5) and (8) into (14) to give 2R’+R” R”-R’
[VI=;
+;
R”-R’ 2R’-
R”-R’
R”
R” _ R’
RI’--R’
0
0
0
R”-R’
0
0
0
2R’-R”
0
0
0
0
0
0
;R’
0
0
0
0
0
0
;R’
0
0
0
0
0
0
5R’
2r7’+?1°
l”-rl’
q”--_11’
0
0
0
rl”-_?’
2q’+rl”
q”-q’
0
0
0
2q’+qU 0
&’ 0
0
0
q’-q’
= [RI + [tgD.
0
q“-q
0
0
0
0
0
iv’
0
0
0
0
0
0
$l’
(15)
433
Soil-structureinteractionin the time domain stress field, {a>. If this element is subjected to an arbitrary virtual nodal displacement pattern, {dP}, which results in a compatible internal displacement and strain distribution ds and de, then Internal work done over the volume of an element Wi =
s”
{de)r{a) do.
(1%
Substituting eqns (17) and ( 18) results in
{d~)TIWItt,lIWX. I 1dv. (20)
J
We = (d~~r{F”) +
J”
(d6}r[ NIT{b’}do
+ s G-WlWW~ ‘is. J
(21)
Equating the internal and external work done and rearranging the terms yields
= (f? + (5) + v;f +IcJ~~:-,~~
(22)
where
IK, I=
Discretization of the whole structun?foundation-soil system using suitable finite elements, Formation of the global load vector due either to the body forces or distributed loads on the structure Generation of the element stiffness matrices (i) Elastic stiffness for the finite elements in the structure, foundation and elastic soil domains according to the equation
”
External work done over the volume of an element
vq(W
conventional elastic &rite element formulation for structure and foundation can be integrated for the solution of interactive analysis in the following manner
J”
Pl’P$,l PI do
= first v&co-elastic stiffness matrix
(23)
= second v&o-elastic stiffness matrix
(24)
[Icy =
Jc
1NPl PI dv,
WI
where [D] is the elasticity matrix for the structure, foundation and elastic soil materials. (ii) Visco-elastic stiffness matrices as per eqns (23) and (24) for the clay soil. (iii) Assembly of the global stiffness matrix for the overall system leading to equation system of the type K*l
(6*}= {F*),
(28)
where [K*] is the assembly stiffness matrix, {S‘) is the nodal point displacement vector and (F*) is the global force vector, including the contribution from eqns (25) and (26) for structure and foundation elements. (iv) Solution of the equation system in the time domain: l For the first time increment, the term
and in eqn (22) would be zero as (6, _, > is treated as zero. Since the initial conditions are based on no settlement having yet taken place. The solution of eqn (28) yields displacements of the overall system. The stresses in the structure, foundation and elastic soil elements are computed as (&I = unknown displacements at the end of time interval, At, and (&_ , > = displacements at the beginning of the time interment At and can be found from initial conditions or a previous solution. Equations (23) and (24) are element stiffness matrices for viscoelastic material. These stiffness matrices can be evaluated using a Gaussian integration technique. Computational algorithm The v&co-elastic finite element formulation for saturated clay soils under drained conditions and
and stresses in clay soil are computed as per the eqn (17) in which the second term, i.e. (Iv,](e, _ t ) is zero. l For the subsequent time increments, the global force vector (F*j in eqn (28) is updated by adding tbe last term of eqn (22), where {a,_, } are the displacements at the end of the previous time increment in clay soil.
M. N. VILADKAR ei al
434
The modified global equation system is then resolved to get the displacement of the overalf system at the end of the current time increment. This means that the displacements obtained at the end of any time increment are the total displacements up to that time. The stresses at the end of any time increment are calculated according to eqn (29) in structure, foundation and elastic soil elements and according to eqn (17) in clay soil elements, where {c,_ ,) are the total strains which occurred in clay soil upto the end of the previous time increment. Step (iv) is repeated until the convergence is achieved indicating that the steady-state condition in clay soil is reached.
the two stress conditions are known at any time, T, and co~sponding to the steady-state. The hydrostatic (or volumetric) and deviatoric strains occur simultaneouly in a saturated soil mass subjected to a three-dimensional stress situation under fully drained conditions. In such a situation, if the vertical component of strain, de,, can be evaluated at any time during the process of deformation, then it can be separated into its hydrostatic and deviatoric components as
The displacements, strains and stresses at this stage in structure. foundation and soil would correspond to the final equilibrium condition of the system.
where
DETERMINA~ON
OF MATERIAL CONSOLIDATION
CONSTANTS DATA
T=
5 [1 -
e(R:V)T],
=’ R
R = a/c”.
The applied stress increment
ACT,= {(Acr, + Aa, + ACTS).
R is given by
can be expressed as
AC,.= 2
= mviAu, = 1.5.m;Aa,,
where mu, = coefficient of volume compressibility determined from triaxial isotropic consolidation test for a three-dimensional stress situation and m, = coefficient of volume compressibility for onedimensional consolidation (the relationship between the two coefficients is given by Head 1181).The vertical component of the volumetric strain, due to a hydrostatic stress condition, when steady-state is reached, is given by At~“=fA~~=f(l.S~m,Aa,)=fm,Aa,..
-RT
(35)
(31)
Taking the natural logarithm on both sides of eqn (1) and rearranging yields the dashpot viscosity coefficient $-,
(34)
(30)
T=CQ
and therefore the spring constant,
(33)
The volumetric strain, At,. due to hydrostatic stress
when the steady-state is reached theoretically at time, ErzCa
AC!= AC,-At-:.
USING
The solution of the governing differential equation [eqn (I)] subject to the initial condition, 6 T= 0 at time T = 0 is given by 6
Adc::=iA&,. and
Therefore, spring constant condition is given by
(32)
AS the stress in the proposed visco-elastic formulation has been split into its component deviatoric and hydrostatic stress situations, the spring and the dashpot constants (R’, q’ for the deviatoric and R”, 4” for the hydrostatic) can be determined for the two situations using eqns (31) and (32) if the strains under
(36)
R” under a hydrostatic
R” = Aa,/A@
(37a)
and the dashpot viscosity coefficient at any time, T, using eqn (32), is given by
?
0 -_
-R”T
CJb)
Soil-structure interaction in the time domain where AC:’ is the hydrostatic strain at any time, T, during consolidation and can be expressed as AC!’ = U, - A@,
rl
-R”T lo&(1 - V,)’
(39)
where time, T, required to reach a certain percentage of consolidation has the same definition as given by Terzaghi [19].
The deviatoric strain at infinite time when steadystate condition is reached is given by Acd” I 9 1 = Acp - Achm
VW
where AC:” is given by eqn (36) and the strain due to applied stress increment tensor [eqn (34)], AE~ is given by
ALr=w=
m;Au.H H
where A@ is the deviatoric strain at any time, T. Since Acf’and AefT [eqn (38)] correspond to the same time instant, T, the value of ud can be taken as that of U,. The degree of consolidation, U at any time, T during consolidation can be obtained directly using the expression given by Sivaram and Swamee [21] as
(4Tv/n)2.8]0.‘79
’
1
where As is the vertical compression of soil layer of thickness, H caused by increase in pore water pressure, AU given by Skempton and Bjerrum [20]. The spring constant R’ under deviatoric condition is therefore given by Ao, - Au, (41)
AC:”
and dashpot viscosity coefficient at any time, T, is given, using eqn (32), as -R’T rl’T= lO&(l- U,)’
(42)
(42W
where T,, is the time factor [19]. The dashpot viscosity coefficients, q’ [eqn (39)] and n ‘ [eqn (4211 are both functions of load and time and will vary accordingly. In the present analysis, average values of n” and n’ have been considered for the sake of simplicity by considering time factors corresponding to 33% and 67% degrees of consolidation under both stress conditions. Therefore
= m, [Aa, + A (Aa, - Au,)], (Mb)
R’=
(424
‘% = loo ’ [I +
2. Deviatoric stress condition
As
This is analogous to eqn (39) where u,, is the degree of deviatoric consolidation expressed as
(38)
where CJ,‘is the degree of hydrostatic consolidation. Therefore t‘T =
435
- R”T,,
“’ =j [ lo&(1 -0.33)+10&(1
- RRTs7 -0.67)
1
(42~)
for ?I, R” is replaced by R’. PROBLEM
The problem of a 2 x 2 bay, single-storey orthogonal space frame with individual footings (Fig. 2a) has been considered for analysis. The soil medium is stratified (Fig. 2b) with top and bottom layers of dense sand with 1.5 and 13.7m thicknesses and a clay layer, 2.3 m thick sandwiched between the sand layers. The frame rests on a clay layer. Details of the frame and the loads acting on it are given in Table l(a) and various sizes of footings in Table l(b).
Fig. 2(a). Sketch of single-storey space frame.
436
M. N. VILADKARer al. (b)
SoiC prof
iCe
Fig. Z(b). Single-storey space frame founded on stratified soil medium. Table i(a). Frame dimensions and loads on beams Transverse beams
Longitudinal beams
A,-Al-A, A,-Ad--A,
A,-AS-A, A-A, -A2
Item (I) Length (m) Cross-section (m x m) Load on beams (kN/m) Load on beams in terms of pressure (kN/m2)
ANALYSIS
From the geometry of the structure, it is clear that the structure is symmetrical. Therefore, the symmetry property of structure and loading are utilized and only one quarter of the structure-footing-soil system is considered for the analysis. The beams and columns are discretized into a number of threedimensional isoparametric paralinear elements with linear elements only at the beam-column junctions. The paralinear elements, with parabolic variation of displacements along the axial direction and linear variation in the cross-sectional plane of the member, represent the flexure in beams. The column footings and the soil medium are discretized into only isoparametric linear elements. Material properties
The frame and footings are considered to be made of concrete. The modulus of elasticity and Poisson’s ratio of the concrete used in the analysis Table l(b). Footing dimensions Footings (1) A, A2 -43
A4
Size (m x m) (2) 1.65 x 1.65 2.20 x 2.20 2.65 x 2.65 3.00 x 3.00
(2)
(3)
6.10 0.40 x 0.90 60.00
7.90 0.40 x 0.90 90.00 225.00
150.00
Columns (4) 4.30 0.40 x 0.40 -_
are 22.2 x lo6 kN/m2 and 0.15, respectively. The properties of sand and clay used for analysis are presented in Tables 2(a) and (b), respectively. Visco-elastic constant for clan
Due to the applied load on frame, the stresses in each element of the soil mass will be different. The stresses at the centre of each soil element were therefore evaluated from the elastic analysis of the frame-footing-soil system using the soil properties from Tables 2(a) and (b). From this analysis, the principal stresses and therefore the hydrostatic and deviatoric stress components acting at the centre of each soil element were evaluated. Equations (34)-(42~) were programmed to evaluate the viscoelastic contants R’, q’ and R”, II” for deviatoric and hydrostatic stress situations. Sample values of these constants for a typical element just below the footings A,-,$, are presented in Table 3. Total and differential settlements
Figure 3 shows the time-settlement curves for the footings ,4,--A, and Fig. 4 shows the differential settlement-time curves. Both the total and differential settlements increase initially at a faster rate and gradually become asymptotic as the steady-state is reached after the lapse of a total time period of 2.5 years corresponding to 95% consolidation. It can be seen from the frame and loading pattern that the footings AI-A4 carry loads in ascending order
Soil-structure interaction in the time domain Table 2(a). Properties of sand layer Drained modulus of elasticity Layer No. (I) First layer
(kzn’) 50,000.000
Second layer
65,OOO.OOO
431
179.60, 194.70 and 204.5Omm and those from interactive analysis are 149.40, 155.70, 167.13 and 173.35mm, respectively. This gives a percentage difference of 1.80, 15.35, 16.50 and 17.90, respectively, the settlements due to interactive analysis being on the lower side. Similarly, Table 5 shows the differential settlements between adjacent footings. The differential settlements between footings A2 and A,, A, and A,, A, and A, and A4 and A, obtained from the conventional method are 27.40, 42.50, 24.90 and 9.8Omm and those from interactive analysis are 6.30, 17.73, 17.65 and 6.20 mm, respectively. Due to the monolithic junction between the columns and their footings, the tendency of the individual columns of the structure to settle relative to each other is restricted by the combined stiffness of column-footing-soil system and also the stiffness of
Drained Poisson’s ratio (3) 0.250 0.250
of the subscripts. Naturally, it follows that the magnitude of settlement under these footings should also increase in the same order; this fact is also borne out in Fig. 3. Table 4 shows the comparison of total settlements of the footings obtained using both interactive analysis (settlement corresponding to about 95% consolidation period) and the conventional method [20]. The settlements of footings A,-& obtained from the conventional method are 152.20,
Table 2(b). Properties of clay layer Item (1)
Symbol (2)
Value (3)
Compression index Natural void ratio Coefficient of consolidation Unit weight of the soil Undrained modulus of elasticity Undrained Poisson’s ratio Pore pressure coefficient Allowable bearing capacity of soil
0.270 0.340 0.6221 m2/yr 17,000 kN/m’ 8000.000 kN/m* 0.480 0.600 150,000 kN/m2
Table 3. Sample values of visco-elastic constants Stress component Deviatoric
Element below the footings (1)
&y2
A,
rl’ (kN yrlm2 1 (3)
1
21.307 22.537 23.005 23.279
A2
A, A,
Hydrostatic R” ““4pl”
12.604 13.332 13.608 13.770
1” (kN yrlm2 1 (5)
137.740 147.940 152.920 159.580
81.478 87.514 90.456 94.400
‘ii
5 %
E
3 if
I20 -
8
Footings Al AZ Al A4
150 -
200
1
0
I 0.25
I
I
I
I
I
0.50
0.75
1.00
1.25
I.50
I l.7!5
I 2.00
I 2.25
I 2.50
Tlme (yeor)
Fig. 3. Timmttlement
curves for the footings of single-storey space frame.
I 2.75
438
M. N.
VILADKAR
et al.
Between
25 0
I
I
I
0.25
0.50
075
I
I
I
I
1.00
I 25
1.50
I.75
Time
I 2.00
footings
I 2.25
I 2.50
I 2 75
(year)
Fig. 4. Variation of differential settlement with time for single-storey space frame. Table 4. Comparison of settlements Settlement of the footings in mm
~~_ .~
Settlement obtained from
;12;
p3?
(Aoj
(“5;
149.40 152.20 1.80
155.70 179.60 15.35
167.13 194.70 16.50
173.35 204.50 17.90
(1) Interactive analysis Conventional method [20] Percentage difference between 72) and (1)
the interconnecting beams. This minimizes interaction analysis differential settlements.
reveals
the
that
total
the
and
Vertical stress in columns and in clay Figures 5(a, b) show the variation of vertical compressive stress with time in columns ,4-A, near column-beam and column-footing junctions, respectively. It is seen in the column of type A,, near the column-beam junction, that the vertical compressive stress increases initially up to a period of about 0.5 years corresponding to 55% consolidation beyond which there is a continuous release of stress with the passing of time. However, near the column-footing junction, it is seen that the stress increases continuously with time up to a period of 1.O year beyond which the increase is insignificant. In columns of type A,, steep increase in the vertical compressive stress is observed at both the locations for a period of about 0.5 years beyond which there is a continuous decrease of stress with time. A continous reduction of vertical stress has been observed from an initial maximum at the beginning
of the consolidation for columns of type A, near the column-beam junctions (Fig. 5a). However, near the column-footing junction, the stress is found to reduce for a period of 0.5 years beyond which the stress continuously increases and finally becomes asymptotic. In central columns of type A,, the stress decreases at both the locations at a high rate for a period of about 1.Oyear, beyond which the trend is asymptotic. Figure 5 reveals that the corner columns are heavily stressed and the central column is stressed to the minimum. Therefore, a certain part of the structure is stressed to the maximum, not at the end of the consolidation but at an intermediate period of consolidation. This may be due to the fact that the peak values of differential settlement may occur at any intermediate period of consolidation. According to the usual design concept, any structural member should be designed for the maximum stress acting on it and this study reveals that this maximum can occur at any intermediate time during consolidation. This, therefore, suggests that the time-independent interactive analysis of King and Chandrasekaran [6],
Table 5. Comparison of differential settlements Differential settlement between adjacent footing in mm
Differential settlement from (I) Interactive analysis
A,& (2) 6.30
Conventional method [20] Percentage difference
21.40 334.92
Al-A, (3)
A,-A, (4)
+A, (5)
11.13 42.50 139.70
17.65 24.90 41.07
6.20 9.80 58.06
439
Soil-structure interaction in the time domain 86
I14
CoCumn A,
c !
“O
F
lo6
5 0 g
102
is 96
-’
94
6E Time (year)
Time (year) loo-
Column A2
76
-
g ;
74
-
“E 2
72
-
column A,
5
92
I I
0
I 2
1 3
68
0
I
Time (year)
Time (year)
Fig. 5(a). Variation of vertical compressive stress, oY near column-beam single-storey space frame.
junctions
with time for
columns obtained from the independent analysis (assuming flxity of the base) and those from interactive analysis. The stress values for interactive analy sis correspond to 95% consolidation. In columns A,
which is a single-step elastic analysis, would not reveal this type of behaviour. Table 6 gives the value of the percentage difference of the maximum vertical compressive stress in
80
68 r 67 0
I 3
2
r
67 e
64
70
“E 5 76 -Y
Column A,
Column A3
t i?! .+ 74
56
v) t!
1
I
0
I
52
I 2
1 3
Time (year)
I
I
2
J 3
Time (year)
-
66
I
76
r
Column A2
Column A4
66-
-
64
62:/y 60
]
0
2
Time (year)
3
Time (year)
Fig. 5(b). Variation of vertical compressive stress, zaneeu column-footing junctions for single-storey space
M. N. VILADK~Ret al.
440
Table 6. Comparison of maximum vertical compressive stress ov in columnst (stress in kN/m2 x 102) Column A, Location (1)
Ind. (2)
Int. (3)
Near columnbeam junction
86.88
95.76
Near columnfooting junction
14.32
63.11
Column A, % diff. (4)
Ind. (5)
Int. (6)
+ 9.27
90.55
92.36
+77.30
25.34
63.51
Column A, % diff. (7)
Ind. (8)
+ 1.96 81.72
f60.10
70.56
Column A,
bit. (9)
% diff. (10)
Ind. (11)
bit. (12)
% diff.. (13)
66.04
-23.75
84.91
68.58
-23.81
79.56
f11.31
81.90
69.10
-18.52
t Obtained on basis of: Ind., independent analysis (with fixed base); Int., interactive analysis; and diff., difference.
and A,, interactive analysis gives higher values of vertical stress than those in the independent analysis. The trend is the opposite in cases of columns A, and A,. Figure 6 shows the variation of vertical compressive stress with time in the clay element just below the footings. The magnitude of this stress continuously increases below the footings A, and A, and decreases below the footings A, and A,. The rate of increase or decrease of stress in columns and in clay below the footings, in general, is high in the early period of consolidation and later it gradually becomes asymptotic, the reason for which can be attributed to the differential settlement. Figure 7 shows the final settlement profile at about 95% consolidation of the footings in transverse and longitudinal directions. The profile is concave upward. This type of settlement profile is valid for all the times during consolidation and can be confirmed
from time-settlement plots (Fig. 3). The concave settlement profile results in the transfer of loads from heavily loaded interior columns to relatively lightly loaded exterior columns. The magnitude of this transfer generally follows the magnitude of differential settlements. Bending moments in beams Figure 8 shows the variation of maximum bending moments in all the transverse (A,-AZ-A, and A,-Ad-A,) and longitudinal (A,-A,-A, and A,-A,-A,) beams with time. The rate of increase of bending moment is also initially high which gradually decreases with lapse of time. The reason for the continuous increase of the bending moment is the increase of differential settlements (Fig. 4) with time. Figures 9(a, b) show the final bending moment diagrams for transverse and longitudinal beams
Footing A,
Footlng A,
Time (yeor)
Tlme (yeor)
44.6 r Fbotlng A4
c
43.0
0
I
I
I
I
2
3
5Ooi
Time (year)
Time (yeor)
Fig. 6. Variation of vertical compressive stress,
Soil--structure in~r~tion Transverse
ii
441
in the time domain drrectlon
IS0 -
E
s E
A2
160 -
A4
I80
6.lm
6.lm
cl
Longitudinal direction
I-
7.9m
I_
7.9m
J
Fig. 7. Final settlement profile in transverseand lon~tu~nal directionsof footingsof single-storeyspace
frame. obtained from interactive and independent analyses, respectively. The bending moment diagram for interactive analysis corresponds to the end of the ~nsotidation period (i.e. 95% co~li~tion). The values of maximum bending moment in all the beams obtained via interactive analysis are on the higher side whereas independent analysis underestimates the values of maximum bending moment. It is also observed that the bending moment obtained by the interactive analysis changes its sign at the middle support (column A2 or A.& This observation is common for beams both in longitudinal and transverse directions.
CONCLUSIONS
1. The constitutive law for a visco-elastic material under a multiaxial stress sytem is derived taking into account its behaviour under hydrostatic and deviatoric stress conditions. 2. A threedimensional v&o-elastic finite element formulation has been presented in an explicit form. 3. The approach for determination of visco-elastic soil constants under hy~os~tic and deviator& stress conditions is outlined. 4. The total and differential settlements and also the bending moments vary with time due to the Beams /
3’ 600
A,-A,-A,
c) 1 400 .% 200
Time (year)
Fig. 8. Variation of bending moment in beams with time for single-storey space.frame.
442
M. N. VILADKARet ul. 400 6. Severe stresses are obtained in outer columns of the frame during and not at the end of the consolidation period, an observation not predicted by time-independent analysis.
200 0
REFERENCES S. Chamecki, Structural rigidity in calculating settlements J. Srruci. Mech. Div., ASCE 82, l-9 (1956). D. Morris, Interaction of continuous frames with soil media. J. Struct. Div., ASCE 92, 13-44 (1966). H. Heil, Studies on the structural rigidity of reinforced concrete building frames on clay. Proc. 7th Int. Conf. on Soil Mech. and Found. Engng, Vol. II, p, 197. Montreal, Canada (1969). 4. W. J. Larnach, Computation of settlements of building frames. Civ. Engng Pub. Works Rev. 65, 1040-1044
200 :
I 2 400
Central
c 5 E ? .g 400 : d
beam A,-A4-A,
I - --
Independent
-
Interactive
analysis analysis
/
200
0
/
/’
/
/\
(1970).
I\ / /
‘\
\
I
\ 200 End beam A,-A,-A,
400 6.lm
6.lm
2
Fig. 9(a). Final bending moment diagram for transverse beam of single-storey space frame.
Centml beam E
,:
g e 400 -
---
Independent
-
Interactive
05
// 200
m
analysis analysis A ‘1,
/
/
’
\ \
5. G. J. W. King and V. S. Chandrasekaran, An assessment of the effects of interaction between a structure and its foundation. Proc. British Geotech. Sot. Conf on Settlement of Structures, pp. 368-383. Cambridge (1974). 6. G. J. W. King and V. S. Chandrasekaran, Interactive analysis of a rafted multistorey space frame resting on an inhomogeneous clay stratum. Proc. fnt. Conf. on Finite Element Melhod in Engng, pp. 4933509. University of New South Wales (1974). 7. L. A. Wood and W. J. Larnach, The interactive behaviour of a soil-structure system and its effect on settlements. Symp. on Recent Developments in the Analysis of Soil Behaviour and their Application to Geotechnical Structures. pp. 75-87. University of New
South Wales (1975). 8. G. J. W. King and V. S. Chandrasekaran, Interactive analysis using a simplified soil model. Proc. Inr. Symp. on Soil-Structure Interaction, Vol. I, pp. 93-100. Roorkee (1977). 9. Y. H. Huang, FE analyses of rafts on visco-elastic foundations. Inr. Co& on Numer. Methods. in Geomech.. Vol. 1, pp. 4633474. Virginia (1976). 10. P. T. Brown, Structure-foundation interaction and soil creep. Proc. Ninth Inr. Conf on Soil Mech. and Found. Engng, Vol. 1, pp. 4399442. Tokyo (1977). Il. P. T. Brown and J. R. Booker, Numerical analysis of rafts on visco-elastic media using eigenvector expansions. Int. J. Numer. Anal. Meth. Geomech. 3, 63-78 (1979). 12. D. W. Taylor and W. Merchant, A theory of clay consolidation accounting for secondary compression. J. Math. Phys. 9, 167-185 (1940). 13. R. E. Gibson and K. Y. Lo, Theory of consolidation for soils exhibiting secondary compression. Norwegian Geotech. Inst., Publication No. 41. 14. R. L. Schiffman, C. C. Ladd and A. T. F. Chen, The secondary consolidation of clay. Symp. on Rheology and Soil Mech., International Union of Theor. and Appl. Mech., Grenoble (1964). 15. L. Barden, Consolidation of clay with non-linear viscosity. Geotechnique 15, 345 (1965). 16. W. Flugge, Visco-elasticity. Blaisdell (1967). 17. N. W. Findley, J. S. Lai and E. Onaran, Creep and Relaxation
Fig. 9(b). Final bending moment diagram for longitudinal beams of single-storey space frame.
visco-elastic nature of clay until a steady-state condition is reached. 5. The rate of variation is initially very high and then becomes asymptotic.
of
Non-linear
Visco-elastic
Materials.
North-Holland, Amsterdam (1976). 18. K. H. Head, Manual of Soil Laboratory Testing, Vol. 3, pp. 1028-1031. Pentech Press, London (1986). 19. K. Terzaghi, Theoretical Soil Melchanics. John Wiley, New York (1943). 20. A. W. Skempton and L. Bjerrum, A contribution to settlement analysis of foundation in clay. Geotechnique 7, 168-178 (1957).
21. B. Sivaram and P. K. Swamee, A computational coefficient. J. Soils method for consolidation Foundations
17, 48-52 (1977).
INTERACTION IN THE TIME DOMAIN
M. N. VILADKAR,~ G. RANJAN? and R. P. SHARMA$ tCivi1 Engineering Department, University of Roorkee, Roorkee-247 667, India $Civil Engineering Department, Regional Institute of Technology, Jamshedpur-831014, (Rece&eff 3 seceder
India
1991)
Abstract-Saturated soils, which are two-phase media, display time-dependent behaviour at constant loading. The resulting phenomena of consolidation and creep causes deformations which are significant compared to instantaneous deformations or those which occur during the construction period of the structure. Redistribution of shear forces, bending and torsional moments in the structure occur due to differential settlements rather than the total settlements. Most of the interaction analyses presented so far are based upon the elastic behaviour of the whole system. An attempt has been made in this paper to present a thr~-dimensional v&co-elastic finite element formulation for studying the interactive behaviour of space frames, taking into account the stress/strain-time response of supporting soil medium. The paper also underlines the methodology for evaluating time-dependent visco-elastic constants for the soil mass and presents a comparison of the resulting structural behaviour with that when interaction is neglected.
lNTRODUCTION The mechanics of interaction between structure, foundation and subsoil must take into account the
effect of complex states of stress, strain and environment on the mechanical behaviour of different classes of materials. Calculation of mechanical behaviour under different states requires that different variables involved be related by means of fundamental equations including the equilibrium equations, the kinematic equations, the compatibility conditions, the constitutive equations and the boundary conditions. Soil behaves elastically under small magnitudes of stresses and the resulting strains are recoverable. If the stress is high, the irrecoverable inelastic strain results represent the plastic behaviour. Visco-elastic soils exhibit elastic behaviour upon loading followed by a slow and continuous increase of strain at a decreasing rate. A continuously decreasing strain upon stress removal follows an initial elastic recovery. Such soils are significantly influenced by the rate of strain or stress, the longer the time to reach the final stress level at constant rate of stress, the larger is the corresponding strain. The behaviour of v&o-elastic materials must therefore be expressed by constitutive equations which include time as an additional variable in addition to stress and strain. The general creep behaviour of any material includes a transient phase in which the strain rate which initially is very high and which sub~uen~y goes on decreasing until a steady-state is reached in which the strain rate is either zero or very small and deformations, quite large. This state is followed by a transient phase in which the strain rate is very high. This phase usually leads to failure. CAS 463-c
429
Time-dependent behaviour of saturated soils, which are primarily two-phase media, consists of two phenomena, namely, the primary consolidation in which deformations occur due to hydro-dynamic lag and creep or secondary consolidation which occurs due to structural breakdown of soil. The deformations due to these two phenomena are quite large compared to instantaneous deformations or those which occur during the construction period of the structure. PROBLEM ID~N~~CA~ON
Most of the interactive analyses of structures consider the elastic behaviour of the whole system. There is a major lack of knowledge as regards the understanding of the interactive behaviour of structnres fords on soil displaying time~~ndent constitutive refationships. An attempt is therefore made in the present work to give a fully three-dimensional visco-elastic finite element formulation for a structur+foundation-soil system. An approach for the evaluation of visco-elastic constants of soil is also discussed along with the interactive behaviour of space frames. EARLIER WORK 1. Interaction of structures with respect to consolidatioa behaviour of soil
Chamecki [l] used the expression for settlement from Terxaghi’s one-dimensional consolidation theory to compute the array of transfer coel%ients required for the interactive analysis of plane frames. Morris 121treated the problem of a continuous space
430
M. N.
VILADKAR er (II.
frame on soil media as a problem of frame with individual columns supported by a Kelvinian model in series with a Hookean spring to account for the time-dependent behaviour of soil. The coupling of the structure to the foundation mechanism was accomplished in the force-displacement relationship of the structure. The resulting visco-elastic equilibrium equations were solved to study the interactive behaviour of the structure. Heil[3] presented a method of settlement analysis in which the effect of time-settlement properties of subsoil, creep of concrete and procedure of structural construction were all accounted for. Larnach l4] computerized Chamechi’s [I] approach to allow sufficient iterations so as to ensure a fully converging solution. King and Chandrasekaran [5] used plane strain finite element modelling to study immediate and the long-term response of the structure which has been found to have been significantly affected by both the inhomogeneous and transversely isotropic nature of the clay stratum and the raft flexibility. Later, King and Chandrasekaran [6] presented a fully three-dimensional finite element formulation for studying both the immediate and the long-term behaviour of a space frame founded on a raft and supported by an overconsolidated clay stratum. The formulation makes use of the substructure technique, the undrained soil modulii for studying the immediate behaviour and the drained soil modulii for the long-term behaviour of the structure. Wood and Larnach [7] developed a method of assessing the combined behaviour but assuming that soil offers restraint only to the vertical settlement. King and Chandrasekaran [8] proposed a simplified method for evaluating raft support stiffness, and found on comparing the results with those obtained through FEM that almost identical results could be obtained. 2. Interaction of structures behaviour of soil
with respect to creep
Huang[9] idealized the problem of a raft on a linearly v&o-elastic foundation as a thin elastic plate connected to a complex rheological analogue consisting of a Maxwell model connected to a series of Kelvin models. Two procedures were used for the dete~ination of stresses and deflections in the raft, namely, the method of integral transforms which is a single-step procedure and a method based on time increments which can be adopted to consider the complicated time history of loading, such as the loading due to sequential construction. Brown [lo] examined the effect of linear soil creep with regard to differential settlements of structures. It was shown that for a flexible footing, the differential settlement tends to increase unless restrained by the structural stiffness. However, large values of relative stiffness of footing tend to reduce the differential settlement with time. Brown and Booker [ 1I] converted the problem of raft on visco-elastic foundation to an equivalent
elastic problem using the Laplace transforms. The solution was converted to an eigenvector expansion and then application of the inverse Laplace transform gave the solution in the time domain. The analysis neglects the effect of shear stresses, if any, developed along the base of the raft. SOIL AS A VISCO-ELASTIC
BODY
The time dependency of deformations in soil is usually attributed to two aspects, namely the primary consolidation and the secondary compression or creep of soil. Prediction of field behaviour usually assumes these as an uncoupled phenomenon despite the fact that Taylor and Merchant [ 121, Gibson and Lo [ 131, Schiffman et al. [141 and Barden (I 51 have discovered the coupled behaviour of soil under onedimensional loading. The time-dependent behaviour of clay soils changes depending upon drainage conditions. In undrained conditions, saturated soil behaves as an incompressible material, the volumetric strains as zero and the deviatoric strains are therefore the total strains. However, in a drained situation, both the volumetric and the deviatoric strain components will appear. One can thus refer to volumetric creep and deviatoric creep and therefore volumetric creep compliance (strain required to cause a unit change in hydrostatic stress) and deviatoric creep compliance (strain required to cause a unit change in deviatoric stress). VISCO-ELASTIC
APPROACH
(4 The structure may be either plane or space frame but all the joints in the structure act as rigid joints. (b) All the loads are supposed to act only at the joints. Cc) The foundation for the structure can be either individual column footings or the combined footing or a raft. (4 The superstructure and the foundation behave in a linearly elastic manner. The soil supporting the foundation is treated as W a linear visco-elastic half space and may be stratified in nature. In a finite element formulation, the soil is treated as homogeneous and isotropic within an element. The properties, especially in the clay strata, may differ from element to element. 07 Due to presence of clay soil, the problem is treated as quasi-static and isothermal. The shear strength of soil is never exceeded. Infinitesimal strain theory is valid. Constitutive law for visco -elastic body In the present analysis, saturated soil mass is considered in a drained situation and is idealized as a Kelvin body (Fig. 1) consisting of a spring and
r u
R
Comparison
of eqns (2) and (4) gives
P,=l,
Ql=(R.Cn$=(R’+n.D),
Pr’Sii(r) = Q2.4i(t),
t
Fig. 1. Kelvin model.
dashpot in parallel so that for constant applied stress, the strain experienced by the two components are the same whereas the stresses shared by the two are different. (i) Uni-axial stress situation. The time-dependent stress-strain relationship for the idealized soil model under uniaxial stress situation is given by Flugge [16] and Findley et al. [17] as
1
.D)G
f P:; n=l
where S, and d, are hydrostatic components of stress and strain and P2 and Qr are again time operators of the form Pz=Pi+
f P:; n=l
(64
Q,=Q;+
5 Q,;. n-l
t6b)
The stress-strain relationship for a Kelvin body under a hydrostatic stress situation, by analogy to the deviatoric case, can be expressed as
Comparison
of eqns (6) and ( 7) gives
P2= 1,
Q~=(R.+r”~)=(R.+9.D),
(34
Visco-elastic constants
The stress-strain relations for an elastic body for the deviatoric and hydrostatic cases are respectively S,=2G.d,
(9a)
aii=3K.cii.
W
and
By comparing eqn (3) with (9a) and (6) with (9b), the corresponding elastic constants are modified in terms of time operators as
G=!.%
(3b) If the material is isotropic and homogeneous, the stress-strain relation for a Kelvin body under a deviatoric stress situation can be written as
(8)
where R” and ?‘I are the spring and dashpot constants for Kelvin elements under hydrostatic stress conditions.
(2)
where S, and du are deviatoric components of stress and strain and P, and Q, are the operators of the form P,=P&+
(6)
(1)
where R and q are respectively the spring and the dashpot constants, cr, the applied constant stress, 6 and i are respectively the strain and strain rate and D = a/at is the differential operator. (ii) Multi-axial stress situation. The state of stress and strain, at any point within a loaded body can be decomposed into its hydrostatic and deviatoric components. (a) Deviatoric stress condition : The differential form of the constitutive law for deviatoric component is expressed as
f’, . s&) = Q, *d&),
(5)
where R’ and rl’ are the spring and dashpot constants for Kelvin elements under deviatoric stress condition. (b) Hydrostatic stress condition: The constitutive law for hydrostatic component, in differential form can also be written as
‘I
u
= (R +
431
Soil-structure interaction in the time domain
and
Kc!.@
3 P2’
2 p,
(10)
where G and K are shear and bulk modulii of the material. The modulus of elasticity, E, and Poisson’s ratio, v, are related to K and G as E=-
9KG 3K+G
and
v=-.
3K -2G 6K+2G
(11)
M. N. VILADKAR
432
Substituting the values of G and K in terms of time operators from eqns (10) into (1 l), visco-elastic material constants, namely, time-dependent Young’s modulus, E,(t) and time-dependent Poisson’s ratio, v,(t) can be obtained as E,(t) =
and Visco -elasticity
Equation (13) can therefore by rewritten as
{~I=
P, Q2- f’*Q, PzQ, + 2P, Qz’
Finite element formulation
(12)
~~i=([~l+~.t-‘[rTl)~~,~-~~-‘~rTl~~r-,~
matrix
=[R”,l{GI -h,lG-Ii>
For a visco-elastic material, the time-dependent stress-strain relationship can now be written for a general three-dimensional stress situation as (01 = [U {ci>
(13) {a} = [N]{he}
and matrix.
The visco-elasticity matrix can be obtained by substituting the values of E,,(t) and v,(t) from eqn (12) into a conventional elasticity matrix to get
[VI=;
(17)
where At = time increment, (6,) = strain vector at time t, {t,_ 1> = strain vector at time (t - l), [R,,] = ([a] + At -‘[ri]) and [n,] = At-‘[q]. Using
where
[V] = visco-elasticity
(16)
The stress/strain-time relationship for visco-elastic material given by eqn (16) can be expressed in finite difference form as
3Q1Q2 f’, Q, + 2P, Q,
v,,(t) =
et al.
and
{t} = [B]{6’},
(18)
where [N] and [B] are respectively the shape function and strain-displacement matrices, the governing equilibrium equations of an element can be obtained using the principle of virtual work. At any time instant, t, consider a single element acted upon by nodal forces, {Fe}, body forces, (b’}, and distributed traction forces, {q’}, which result in an equilibrating
2P,Q,+P,Q,
P,Q,--P,Q,
f',Qz - PzQ,
0
0
0
P,Q,-PIPI
2f-‘2Q,x f’,Q,
P,Qz-P,Q,
0
0
0
P,Q,-f’,Q,
P,Qz--f’zQ,
2f’, Q, + P, Qz
0
0
0
0
0
0
tP,Q,
0
0
0
0
0
0
;p,Q,
0
0
0
0
0
0
;P,Q,
(14)
This is the general form of the visco-elasticity matrix which can be expressed in terms of spring and dashpot constants of the Kelvin body by substituting expressions (5) and (8) into (14) to give 2R’+R” R”-R’
[VI=;
+;
R”-R’ 2R’-
R”-R’
R”
R” _ R’
RI’--R’
0
0
0
R”-R’
0
0
0
2R’-R”
0
0
0
0
0
0
;R’
0
0
0
0
0
0
;R’
0
0
0
0
0
0
5R’
2r7’+?1°
l”-rl’
q”--_11’
0
0
0
rl”-_?’
2q’+rl”
q”-q’
0
0
0
2q’+qU 0
&’ 0
0
0
q’-q’
= [RI + [tgD.
0
q“-q
0
0
0
0
0
iv’
0
0
0
0
0
0
$l’
(15)
433
Soil-structureinteractionin the time domain stress field, {a>. If this element is subjected to an arbitrary virtual nodal displacement pattern, {dP}, which results in a compatible internal displacement and strain distribution ds and de, then Internal work done over the volume of an element Wi =
s”
{de)r{a) do.
(1%
Substituting eqns (17) and ( 18) results in
{d~)TIWItt,lIWX. I 1dv. (20)
J
We = (d~~r{F”) +
J”
(d6}r[ NIT{b’}do
+ s G-WlWW~ ‘is. J
(21)
Equating the internal and external work done and rearranging the terms yields
= (f? + (5) + v;f +IcJ~~:-,~~
(22)
where
IK, I=
Discretization of the whole structun?foundation-soil system using suitable finite elements, Formation of the global load vector due either to the body forces or distributed loads on the structure Generation of the element stiffness matrices (i) Elastic stiffness for the finite elements in the structure, foundation and elastic soil domains according to the equation
”
External work done over the volume of an element
vq(W
conventional elastic &rite element formulation for structure and foundation can be integrated for the solution of interactive analysis in the following manner
J”
Pl’P$,l PI do
= first v&co-elastic stiffness matrix
(23)
= second v&o-elastic stiffness matrix
(24)
[Icy =
Jc
1NPl PI dv,
WI
where [D] is the elasticity matrix for the structure, foundation and elastic soil materials. (ii) Visco-elastic stiffness matrices as per eqns (23) and (24) for the clay soil. (iii) Assembly of the global stiffness matrix for the overall system leading to equation system of the type K*l
(6*}= {F*),
(28)
where [K*] is the assembly stiffness matrix, {S‘) is the nodal point displacement vector and (F*) is the global force vector, including the contribution from eqns (25) and (26) for structure and foundation elements. (iv) Solution of the equation system in the time domain: l For the first time increment, the term
and in eqn (22) would be zero as (6, _, > is treated as zero. Since the initial conditions are based on no settlement having yet taken place. The solution of eqn (28) yields displacements of the overall system. The stresses in the structure, foundation and elastic soil elements are computed as (&I = unknown displacements at the end of time interval, At, and (&_ , > = displacements at the beginning of the time interment At and can be found from initial conditions or a previous solution. Equations (23) and (24) are element stiffness matrices for viscoelastic material. These stiffness matrices can be evaluated using a Gaussian integration technique. Computational algorithm The v&co-elastic finite element formulation for saturated clay soils under drained conditions and
and stresses in clay soil are computed as per the eqn (17) in which the second term, i.e. (Iv,](e, _ t ) is zero. l For the subsequent time increments, the global force vector (F*j in eqn (28) is updated by adding tbe last term of eqn (22), where {a,_, } are the displacements at the end of the previous time increment in clay soil.
M. N. VILADKAR ei al
434
The modified global equation system is then resolved to get the displacement of the overalf system at the end of the current time increment. This means that the displacements obtained at the end of any time increment are the total displacements up to that time. The stresses at the end of any time increment are calculated according to eqn (29) in structure, foundation and elastic soil elements and according to eqn (17) in clay soil elements, where {c,_ ,) are the total strains which occurred in clay soil upto the end of the previous time increment. Step (iv) is repeated until the convergence is achieved indicating that the steady-state condition in clay soil is reached.
the two stress conditions are known at any time, T, and co~sponding to the steady-state. The hydrostatic (or volumetric) and deviatoric strains occur simultaneouly in a saturated soil mass subjected to a three-dimensional stress situation under fully drained conditions. In such a situation, if the vertical component of strain, de,, can be evaluated at any time during the process of deformation, then it can be separated into its hydrostatic and deviatoric components as
The displacements, strains and stresses at this stage in structure. foundation and soil would correspond to the final equilibrium condition of the system.
where
DETERMINA~ON
OF MATERIAL CONSOLIDATION
CONSTANTS DATA
T=
5 [1 -
e(R:V)T],
=’ R
R = a/c”.
The applied stress increment
ACT,= {(Acr, + Aa, + ACTS).
R is given by
can be expressed as
AC,.= 2
= mviAu, = 1.5.m;Aa,,
where mu, = coefficient of volume compressibility determined from triaxial isotropic consolidation test for a three-dimensional stress situation and m, = coefficient of volume compressibility for onedimensional consolidation (the relationship between the two coefficients is given by Head 1181).The vertical component of the volumetric strain, due to a hydrostatic stress condition, when steady-state is reached, is given by At~“=fA~~=f(l.S~m,Aa,)=fm,Aa,..
-RT
(35)
(31)
Taking the natural logarithm on both sides of eqn (1) and rearranging yields the dashpot viscosity coefficient $-,
(34)
(30)
T=CQ
and therefore the spring constant,
(33)
The volumetric strain, At,. due to hydrostatic stress
when the steady-state is reached theoretically at time, ErzCa
AC!= AC,-At-:.
USING
The solution of the governing differential equation [eqn (I)] subject to the initial condition, 6 T= 0 at time T = 0 is given by 6
Adc::=iA&,. and
Therefore, spring constant condition is given by
(32)
AS the stress in the proposed visco-elastic formulation has been split into its component deviatoric and hydrostatic stress situations, the spring and the dashpot constants (R’, q’ for the deviatoric and R”, 4” for the hydrostatic) can be determined for the two situations using eqns (31) and (32) if the strains under
(36)
R” under a hydrostatic
R” = Aa,/A@
(37a)
and the dashpot viscosity coefficient at any time, T, using eqn (32), is given by
?
0 -_
-R”T
CJb)
Soil-structure interaction in the time domain where AC:’ is the hydrostatic strain at any time, T, during consolidation and can be expressed as AC!’ = U, - A@,
rl
-R”T lo&(1 - V,)’
(39)
where time, T, required to reach a certain percentage of consolidation has the same definition as given by Terzaghi [19].
The deviatoric strain at infinite time when steadystate condition is reached is given by Acd” I 9 1 = Acp - Achm
VW
where AC:” is given by eqn (36) and the strain due to applied stress increment tensor [eqn (34)], AE~ is given by
ALr=w=
m;Au.H H
where A@ is the deviatoric strain at any time, T. Since Acf’and AefT [eqn (38)] correspond to the same time instant, T, the value of ud can be taken as that of U,. The degree of consolidation, U at any time, T during consolidation can be obtained directly using the expression given by Sivaram and Swamee [21] as
(4Tv/n)2.8]0.‘79
’
1
where As is the vertical compression of soil layer of thickness, H caused by increase in pore water pressure, AU given by Skempton and Bjerrum [20]. The spring constant R’ under deviatoric condition is therefore given by Ao, - Au, (41)
AC:”
and dashpot viscosity coefficient at any time, T, is given, using eqn (32), as -R’T rl’T= lO&(l- U,)’
(42)
(42W
where T,, is the time factor [19]. The dashpot viscosity coefficients, q’ [eqn (39)] and n ‘ [eqn (4211 are both functions of load and time and will vary accordingly. In the present analysis, average values of n” and n’ have been considered for the sake of simplicity by considering time factors corresponding to 33% and 67% degrees of consolidation under both stress conditions. Therefore
= m, [Aa, + A (Aa, - Au,)], (Mb)
R’=
(424
‘% = loo ’ [I +
2. Deviatoric stress condition
As
This is analogous to eqn (39) where u,, is the degree of deviatoric consolidation expressed as
(38)
where CJ,‘is the degree of hydrostatic consolidation. Therefore t‘T =
435
- R”T,,
“’ =j [ lo&(1 -0.33)+10&(1
- RRTs7 -0.67)
1
(42~)
for ?I, R” is replaced by R’. PROBLEM
The problem of a 2 x 2 bay, single-storey orthogonal space frame with individual footings (Fig. 2a) has been considered for analysis. The soil medium is stratified (Fig. 2b) with top and bottom layers of dense sand with 1.5 and 13.7m thicknesses and a clay layer, 2.3 m thick sandwiched between the sand layers. The frame rests on a clay layer. Details of the frame and the loads acting on it are given in Table l(a) and various sizes of footings in Table l(b).
Fig. 2(a). Sketch of single-storey space frame.
436
M. N. VILADKARer al. (b)
SoiC prof
iCe
Fig. Z(b). Single-storey space frame founded on stratified soil medium. Table i(a). Frame dimensions and loads on beams Transverse beams
Longitudinal beams
A,-Al-A, A,-Ad--A,
A,-AS-A, A-A, -A2
Item (I) Length (m) Cross-section (m x m) Load on beams (kN/m) Load on beams in terms of pressure (kN/m2)
ANALYSIS
From the geometry of the structure, it is clear that the structure is symmetrical. Therefore, the symmetry property of structure and loading are utilized and only one quarter of the structure-footing-soil system is considered for the analysis. The beams and columns are discretized into a number of threedimensional isoparametric paralinear elements with linear elements only at the beam-column junctions. The paralinear elements, with parabolic variation of displacements along the axial direction and linear variation in the cross-sectional plane of the member, represent the flexure in beams. The column footings and the soil medium are discretized into only isoparametric linear elements. Material properties
The frame and footings are considered to be made of concrete. The modulus of elasticity and Poisson’s ratio of the concrete used in the analysis Table l(b). Footing dimensions Footings (1) A, A2 -43
A4
Size (m x m) (2) 1.65 x 1.65 2.20 x 2.20 2.65 x 2.65 3.00 x 3.00
(2)
(3)
6.10 0.40 x 0.90 60.00
7.90 0.40 x 0.90 90.00 225.00
150.00
Columns (4) 4.30 0.40 x 0.40 -_
are 22.2 x lo6 kN/m2 and 0.15, respectively. The properties of sand and clay used for analysis are presented in Tables 2(a) and (b), respectively. Visco-elastic constant for clan
Due to the applied load on frame, the stresses in each element of the soil mass will be different. The stresses at the centre of each soil element were therefore evaluated from the elastic analysis of the frame-footing-soil system using the soil properties from Tables 2(a) and (b). From this analysis, the principal stresses and therefore the hydrostatic and deviatoric stress components acting at the centre of each soil element were evaluated. Equations (34)-(42~) were programmed to evaluate the viscoelastic contants R’, q’ and R”, II” for deviatoric and hydrostatic stress situations. Sample values of these constants for a typical element just below the footings A,-,$, are presented in Table 3. Total and differential settlements
Figure 3 shows the time-settlement curves for the footings ,4,--A, and Fig. 4 shows the differential settlement-time curves. Both the total and differential settlements increase initially at a faster rate and gradually become asymptotic as the steady-state is reached after the lapse of a total time period of 2.5 years corresponding to 95% consolidation. It can be seen from the frame and loading pattern that the footings AI-A4 carry loads in ascending order
Soil-structure interaction in the time domain Table 2(a). Properties of sand layer Drained modulus of elasticity Layer No. (I) First layer
(kzn’) 50,000.000
Second layer
65,OOO.OOO
431
179.60, 194.70 and 204.5Omm and those from interactive analysis are 149.40, 155.70, 167.13 and 173.35mm, respectively. This gives a percentage difference of 1.80, 15.35, 16.50 and 17.90, respectively, the settlements due to interactive analysis being on the lower side. Similarly, Table 5 shows the differential settlements between adjacent footings. The differential settlements between footings A2 and A,, A, and A,, A, and A, and A4 and A, obtained from the conventional method are 27.40, 42.50, 24.90 and 9.8Omm and those from interactive analysis are 6.30, 17.73, 17.65 and 6.20 mm, respectively. Due to the monolithic junction between the columns and their footings, the tendency of the individual columns of the structure to settle relative to each other is restricted by the combined stiffness of column-footing-soil system and also the stiffness of
Drained Poisson’s ratio (3) 0.250 0.250
of the subscripts. Naturally, it follows that the magnitude of settlement under these footings should also increase in the same order; this fact is also borne out in Fig. 3. Table 4 shows the comparison of total settlements of the footings obtained using both interactive analysis (settlement corresponding to about 95% consolidation period) and the conventional method [20]. The settlements of footings A,-& obtained from the conventional method are 152.20,
Table 2(b). Properties of clay layer Item (1)
Symbol (2)
Value (3)
Compression index Natural void ratio Coefficient of consolidation Unit weight of the soil Undrained modulus of elasticity Undrained Poisson’s ratio Pore pressure coefficient Allowable bearing capacity of soil
0.270 0.340 0.6221 m2/yr 17,000 kN/m’ 8000.000 kN/m* 0.480 0.600 150,000 kN/m2
Table 3. Sample values of visco-elastic constants Stress component Deviatoric
Element below the footings (1)
&y2
A,
rl’ (kN yrlm2 1 (3)
1
21.307 22.537 23.005 23.279
A2
A, A,
Hydrostatic R” ““4pl”
12.604 13.332 13.608 13.770
1” (kN yrlm2 1 (5)
137.740 147.940 152.920 159.580
81.478 87.514 90.456 94.400
‘ii
5 %
E
3 if
I20 -
8
Footings Al AZ Al A4
150 -
200
1
0
I 0.25
I
I
I
I
I
0.50
0.75
1.00
1.25
I.50
I l.7!5
I 2.00
I 2.25
I 2.50
Tlme (yeor)
Fig. 3. Timmttlement
curves for the footings of single-storey space frame.
I 2.75
438
M. N.
VILADKAR
et al.
Between
25 0
I
I
I
0.25
0.50
075
I
I
I
I
1.00
I 25
1.50
I.75
Time
I 2.00
footings
I 2.25
I 2.50
I 2 75
(year)
Fig. 4. Variation of differential settlement with time for single-storey space frame. Table 4. Comparison of settlements Settlement of the footings in mm
~~_ .~
Settlement obtained from
;12;
p3?
(Aoj
(“5;
149.40 152.20 1.80
155.70 179.60 15.35
167.13 194.70 16.50
173.35 204.50 17.90
(1) Interactive analysis Conventional method [20] Percentage difference between 72) and (1)
the interconnecting beams. This minimizes interaction analysis differential settlements.
reveals
the
that
total
the
and
Vertical stress in columns and in clay Figures 5(a, b) show the variation of vertical compressive stress with time in columns ,4-A, near column-beam and column-footing junctions, respectively. It is seen in the column of type A,, near the column-beam junction, that the vertical compressive stress increases initially up to a period of about 0.5 years corresponding to 55% consolidation beyond which there is a continuous release of stress with the passing of time. However, near the column-footing junction, it is seen that the stress increases continuously with time up to a period of 1.O year beyond which the increase is insignificant. In columns of type A,, steep increase in the vertical compressive stress is observed at both the locations for a period of about 0.5 years beyond which there is a continuous decrease of stress with time. A continous reduction of vertical stress has been observed from an initial maximum at the beginning
of the consolidation for columns of type A, near the column-beam junctions (Fig. 5a). However, near the column-footing junction, the stress is found to reduce for a period of 0.5 years beyond which the stress continuously increases and finally becomes asymptotic. In central columns of type A,, the stress decreases at both the locations at a high rate for a period of about 1.Oyear, beyond which the trend is asymptotic. Figure 5 reveals that the corner columns are heavily stressed and the central column is stressed to the minimum. Therefore, a certain part of the structure is stressed to the maximum, not at the end of the consolidation but at an intermediate period of consolidation. This may be due to the fact that the peak values of differential settlement may occur at any intermediate period of consolidation. According to the usual design concept, any structural member should be designed for the maximum stress acting on it and this study reveals that this maximum can occur at any intermediate time during consolidation. This, therefore, suggests that the time-independent interactive analysis of King and Chandrasekaran [6],
Table 5. Comparison of differential settlements Differential settlement between adjacent footing in mm
Differential settlement from (I) Interactive analysis
A,& (2) 6.30
Conventional method [20] Percentage difference
21.40 334.92
Al-A, (3)
A,-A, (4)
+A, (5)
11.13 42.50 139.70
17.65 24.90 41.07
6.20 9.80 58.06
439
Soil-structure interaction in the time domain 86
I14
CoCumn A,
c !
“O
F
lo6
5 0 g
102
is 96
-’
94
6E Time (year)
Time (year) loo-
Column A2
76
-
g ;
74
-
“E 2
72
-
column A,
5
92
I I
0
I 2
1 3
68
0
I
Time (year)
Time (year)
Fig. 5(a). Variation of vertical compressive stress, oY near column-beam single-storey space frame.
junctions
with time for
columns obtained from the independent analysis (assuming flxity of the base) and those from interactive analysis. The stress values for interactive analy sis correspond to 95% consolidation. In columns A,
which is a single-step elastic analysis, would not reveal this type of behaviour. Table 6 gives the value of the percentage difference of the maximum vertical compressive stress in
80
68 r 67 0
I 3
2
r
67 e
64
70
“E 5 76 -Y
Column A,
Column A3
t i?! .+ 74
56
v) t!
1
I
0
I
52
I 2
1 3
Time (year)
I
I
2
J 3
Time (year)
-
66
I
76
r
Column A2
Column A4
66-
-
64
62:/y 60
]
0
2
Time (year)
3
Time (year)
Fig. 5(b). Variation of vertical compressive stress, zaneeu column-footing junctions for single-storey space
M. N. VILADK~Ret al.
440
Table 6. Comparison of maximum vertical compressive stress ov in columnst (stress in kN/m2 x 102) Column A, Location (1)
Ind. (2)
Int. (3)
Near columnbeam junction
86.88
95.76
Near columnfooting junction
14.32
63.11
Column A, % diff. (4)
Ind. (5)
Int. (6)
+ 9.27
90.55
92.36
+77.30
25.34
63.51
Column A, % diff. (7)
Ind. (8)
+ 1.96 81.72
f60.10
70.56
Column A,
bit. (9)
% diff. (10)
Ind. (11)
bit. (12)
% diff.. (13)
66.04
-23.75
84.91
68.58
-23.81
79.56
f11.31
81.90
69.10
-18.52
t Obtained on basis of: Ind., independent analysis (with fixed base); Int., interactive analysis; and diff., difference.
and A,, interactive analysis gives higher values of vertical stress than those in the independent analysis. The trend is the opposite in cases of columns A, and A,. Figure 6 shows the variation of vertical compressive stress with time in the clay element just below the footings. The magnitude of this stress continuously increases below the footings A, and A, and decreases below the footings A, and A,. The rate of increase or decrease of stress in columns and in clay below the footings, in general, is high in the early period of consolidation and later it gradually becomes asymptotic, the reason for which can be attributed to the differential settlement. Figure 7 shows the final settlement profile at about 95% consolidation of the footings in transverse and longitudinal directions. The profile is concave upward. This type of settlement profile is valid for all the times during consolidation and can be confirmed
from time-settlement plots (Fig. 3). The concave settlement profile results in the transfer of loads from heavily loaded interior columns to relatively lightly loaded exterior columns. The magnitude of this transfer generally follows the magnitude of differential settlements. Bending moments in beams Figure 8 shows the variation of maximum bending moments in all the transverse (A,-AZ-A, and A,-Ad-A,) and longitudinal (A,-A,-A, and A,-A,-A,) beams with time. The rate of increase of bending moment is also initially high which gradually decreases with lapse of time. The reason for the continuous increase of the bending moment is the increase of differential settlements (Fig. 4) with time. Figures 9(a, b) show the final bending moment diagrams for transverse and longitudinal beams
Footing A,
Footlng A,
Time (yeor)
Tlme (yeor)
44.6 r Fbotlng A4
c
43.0
0
I
I
I
I
2
3
5Ooi
Time (year)
Time (yeor)
Fig. 6. Variation of vertical compressive stress,
Soil--structure in~r~tion Transverse
ii
441
in the time domain drrectlon
IS0 -
E
s E
A2
160 -
A4
I80
6.lm
6.lm
cl
Longitudinal direction
I-
7.9m
I_
7.9m
J
Fig. 7. Final settlement profile in transverseand lon~tu~nal directionsof footingsof single-storeyspace
frame. obtained from interactive and independent analyses, respectively. The bending moment diagram for interactive analysis corresponds to the end of the ~nsotidation period (i.e. 95% co~li~tion). The values of maximum bending moment in all the beams obtained via interactive analysis are on the higher side whereas independent analysis underestimates the values of maximum bending moment. It is also observed that the bending moment obtained by the interactive analysis changes its sign at the middle support (column A2 or A.& This observation is common for beams both in longitudinal and transverse directions.
CONCLUSIONS
1. The constitutive law for a visco-elastic material under a multiaxial stress sytem is derived taking into account its behaviour under hydrostatic and deviatoric stress conditions. 2. A threedimensional v&o-elastic finite element formulation has been presented in an explicit form. 3. The approach for determination of visco-elastic soil constants under hy~os~tic and deviator& stress conditions is outlined. 4. The total and differential settlements and also the bending moments vary with time due to the Beams /
3’ 600
A,-A,-A,
c) 1 400 .% 200
Time (year)
Fig. 8. Variation of bending moment in beams with time for single-storey space.frame.
442
M. N. VILADKARet ul. 400 6. Severe stresses are obtained in outer columns of the frame during and not at the end of the consolidation period, an observation not predicted by time-independent analysis.
200 0
REFERENCES S. Chamecki, Structural rigidity in calculating settlements J. Srruci. Mech. Div., ASCE 82, l-9 (1956). D. Morris, Interaction of continuous frames with soil media. J. Struct. Div., ASCE 92, 13-44 (1966). H. Heil, Studies on the structural rigidity of reinforced concrete building frames on clay. Proc. 7th Int. Conf. on Soil Mech. and Found. Engng, Vol. II, p, 197. Montreal, Canada (1969). 4. W. J. Larnach, Computation of settlements of building frames. Civ. Engng Pub. Works Rev. 65, 1040-1044
200 :
I 2 400
Central
c 5 E ? .g 400 : d
beam A,-A4-A,
I - --
Independent
-
Interactive
analysis analysis
/
200
0
/
/’
/
/\
(1970).
I\ / /
‘\
\
I
\ 200 End beam A,-A,-A,
400 6.lm
6.lm
2
Fig. 9(a). Final bending moment diagram for transverse beam of single-storey space frame.
Centml beam E
,:
g e 400 -
---
Independent
-
Interactive
05
// 200
m
analysis analysis A ‘1,
/
/
’
\ \
5. G. J. W. King and V. S. Chandrasekaran, An assessment of the effects of interaction between a structure and its foundation. Proc. British Geotech. Sot. Conf on Settlement of Structures, pp. 368-383. Cambridge (1974). 6. G. J. W. King and V. S. Chandrasekaran, Interactive analysis of a rafted multistorey space frame resting on an inhomogeneous clay stratum. Proc. fnt. Conf. on Finite Element Melhod in Engng, pp. 4933509. University of New South Wales (1974). 7. L. A. Wood and W. J. Larnach, The interactive behaviour of a soil-structure system and its effect on settlements. Symp. on Recent Developments in the Analysis of Soil Behaviour and their Application to Geotechnical Structures. pp. 75-87. University of New
South Wales (1975). 8. G. J. W. King and V. S. Chandrasekaran, Interactive analysis using a simplified soil model. Proc. Inr. Symp. on Soil-Structure Interaction, Vol. I, pp. 93-100. Roorkee (1977). 9. Y. H. Huang, FE analyses of rafts on visco-elastic foundations. Inr. Co& on Numer. Methods. in Geomech.. Vol. 1, pp. 4633474. Virginia (1976). 10. P. T. Brown, Structure-foundation interaction and soil creep. Proc. Ninth Inr. Conf on Soil Mech. and Found. Engng, Vol. 1, pp. 4399442. Tokyo (1977). Il. P. T. Brown and J. R. Booker, Numerical analysis of rafts on visco-elastic media using eigenvector expansions. Int. J. Numer. Anal. Meth. Geomech. 3, 63-78 (1979). 12. D. W. Taylor and W. Merchant, A theory of clay consolidation accounting for secondary compression. J. Math. Phys. 9, 167-185 (1940). 13. R. E. Gibson and K. Y. Lo, Theory of consolidation for soils exhibiting secondary compression. Norwegian Geotech. Inst., Publication No. 41. 14. R. L. Schiffman, C. C. Ladd and A. T. F. Chen, The secondary consolidation of clay. Symp. on Rheology and Soil Mech., International Union of Theor. and Appl. Mech., Grenoble (1964). 15. L. Barden, Consolidation of clay with non-linear viscosity. Geotechnique 15, 345 (1965). 16. W. Flugge, Visco-elasticity. Blaisdell (1967). 17. N. W. Findley, J. S. Lai and E. Onaran, Creep and Relaxation
Fig. 9(b). Final bending moment diagram for longitudinal beams of single-storey space frame.
visco-elastic nature of clay until a steady-state condition is reached. 5. The rate of variation is initially very high and then becomes asymptotic.
of
Non-linear
Visco-elastic
Materials.
North-Holland, Amsterdam (1976). 18. K. H. Head, Manual of Soil Laboratory Testing, Vol. 3, pp. 1028-1031. Pentech Press, London (1986). 19. K. Terzaghi, Theoretical Soil Melchanics. John Wiley, New York (1943). 20. A. W. Skempton and L. Bjerrum, A contribution to settlement analysis of foundation in clay. Geotechnique 7, 168-178 (1957).
21. B. Sivaram and P. K. Swamee, A computational coefficient. J. Soils method for consolidation Foundations
17, 48-52 (1977).