Mining Science and Technology, 6 (1988) 163-169
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
CREEP BENDING OF ROCK PLATES J.G. Singh and P.C. Upadhyay Departments of Mining and Mechanical Engineering, Institute of Technology, Banaras Hindu University, Varanasi 221 005 (India) (Received May 1, 1987; accepted August 6, 1987)
ABSTRACT Creep bending of a rock plate under constant loading, has been analysed Non-equal elastic and viscosity constants under compressive and tensile loading of rock material have been incorporated in the Burger's Model. Results are
1. INTRODUCTION An analysis of the constant load creep bending of rock beams, incorporating differing elasticity and viscosity constants under compressive and tensile loading, gave some interesting results and useful guidelines . Hence, in this paper, it has been attempted to examine and extend the analysis, for the rock plates. Further, because many rock plate structures obtained in underground mining configurations are thick, the effect of transverse shear deformation has also been included in the analysis. Results have been plotted in non-dimensional forms, and cover a wide range of rock constants obtained in the field. The problem is stated as follows: A thick rock plate undergoes creep bending, due to a normal traction q on one of its faces, parallel to its mid-plane. The rock 0167-9031/88/$03.50
presented in terms of non-dimensionalised parameters, whose influence has been studied over a wide range. The influence of some of these parameters, appears dominating and interesting.
material is represented by the Burger's Model (Fig. 1). The elasticity and viscosity moduli of elements in the model under tensile loading are very different from those under compressive loading. It is required to investigate the state of displacement and stress fields in the body of the plate, which is free of body forces. Ek
~ o.=Ern £'
~" o-=3JJm ~n
o-e=Ek Cm cry= 3 jLJk ~m 0~ =
Fig. 1. Burger's model representation for a visco-elastic material.
© 1988 Elsevier Science Publishers B.V.
The geometry of the rock plate and its coordinate axes, etc., are as shown in Fig. 2. Let the Neutral Surface (NS), whose position is at present unknown, divide the plate height h into unequal parts ho and h t. The cartesian coordinate system Oxyz has its axes 0x and 0y on the undeflected NS, and 0z directed in the direction of the applied traction q. During the elastic deflection of the rock plate, the stresses ax, Oy and ~'xy at the point P(x, y, z), are given as 
Ox=-Ecz/(1-P:)(W,xx+"cW,yy), =-Etz/(1--t'2t)(Wxxq-~'tW,yy), oy= - E c z / ( 1 - ~ ' : ) ( W , y y +
where, E is the Young's Modulus and i, the Poisson's Ratio (subscripts c and t indicate compressive and tensile loading, respectively), and w is the displacement of the point P in the z direction. Replacing Fo and F t for E c and E t , and assuming i,c = 11t = 0.5 , in eqn. (1), we obtain the expressions for the stresses ox, Oy and 'rxy during the constant load creep bending of the rock plate, viz.
Ox=-4Fcz/3(W,xx+O.5w,yy), = - 4 E t z / 3 ( W , x x + O . 5 w , y y ),
= - 4 F t z / 3 ( W yy +O.5w,xx),
"rxy= - 2Fj/3W,xy,
z < 0;
z Z 0
The terms F~ and F t have already been defined in ref.  with regard to the Burger's Model shown in Fig. 1, as
EmcEkctmc/(Ekc(tmc + t) +Emctmc(1-e-t/t~));
F t = EmcEkctmc/(
Ekc (qStmc +
+ Emctmo(l-e-*'/~'~°)B) /~ = E m c / E m t = E k c / E k t , -~" ~ mc/~ t mt = ~ k c / ~ kt"
/~/ / /
Here E is the Young's Modulus, /, is viscosity constant and t (with subscript) the relaxation time of the elements in the Burger's Model. Subscripts m and k indicate Maxwell and Kelvin elements, respectively; and the subscripts c and t, respectively, indicate compressive and tensile loading. Substituting the values of stresses from eqns. (2) into the eqns. (A5) (see Appendix), the moment intensities Mx, My and Mxy are obtained as
M x = -Dv(w,x x + 0.5W,yy), M y = -Ov(W,yy-.}-O.5w,xx) , and Fig. 2. Geometry of a rock plate.
Mxy = - 0.5Dvw,xy
165 where D~ is the flexural rigidity of the rock plate under creep conditions, and is given by Dv = 4(F~h 2 + Fth2)/9.
Using eqns. (5) into the equilibrium equations (A4), the transverse shear force intensities are
Qx=-Dv(V2W),x, Qy = -Do(vZw),y;
which when substituted into the third of the eqns. (A2) yield the desired deflection equation (7)
)t( Fh3/9)V4w= q,
where X = 4/(1 + 005) 2.
We, thus, notice that the deflection equation could well have been written straightaway by replacing E~, Et, E, v and 13 by F c, F t, F, 0.5 and 0, respectively, in the elastic deflection equation (B1). Effect of shear
Nx = 2 / 3 ( F ~ h ~ - Ftht2)(w,x x + 0.5W,yy),
With the correspondence of terms in elastic deflection and constant loading creep deflection, already established, we immediately note the deflection equation, incorporating the shear effect, by replacing Eo, Et, E, v and 13 by Fo, F t, F, 0.5 and 0, respectively, in the eqn. (B2). The resulting equation is
where, X is given by (10') and
which when plugged into the first of the eqns. (A2), yield
~/= (1.7(1 + 0 1 5 ) - 0 . 5 ( 1 +005))
OvV 4w = q.
But, the D~ given by the eqn. (5') has the terms h e and ht, which must be defined. The use of eqns. (2) into the eqns. (A3) gives the force intensities
2/3( Fch2c- Ft h2 )(V 2w ),x = 0.
(v2w),x*0. Hence, the eqn. (8) yields
rch-Fth2=O. This, when used with ho + h t = h, furnishes the relations
hc = h / ( l + O ° s ) ,
But, as Qx and Qy are finite, the first of eqns. (6) viewed with the eqn. (8), suggests
/(0.3(1 + 005)3).
3. EFFECT OF 13 AND
The bending and shear contributions to the maximum deflection of a rock plate is influenced by the rock properties 13 and ~, through the factors ~ and ('. These factors can be evaluated again by replacing Ec, Et, E, v and 13 by Fo, F t, F, 0.5 and 0, respectively, in the equations (B3'). The factors are = (1 + 005)2/4,
~ ' = (1.7(1 + 015) - 0.5(1 + e°'S)) which are identical to the corresponding relations for the rock beams , and define the position of the NS in the rock plate. Using the relations (9) and putting F~ = F, the deflection eqn. (7) assumes the usable
/(1.2(1 + 0°s)).
The rock properties 13 and qs, thus, determine the factors ~ and ~', through 0, which can be written in a non-dimensionalised form
166 as [11 0 = ((1 + 3 T ) + X(1 - e-8~'r))/3 / ( ( 1 + T) + X(1 - e - ; r ) )
where 8 = 4,///3,
= tmc//tkc ,
X = Emc//Ekc,
T = t//tmc.
The parameters X and f are, respectively, the ratios of stiffnesses and relaxation times of the Maxwell and Kelvin elements in the Burger's model. By suitably adjusting the values of X and f, the behaviour of the most of the rocks can be approximated. 4, has entered the expression for 0 through the parameter 8 (the ratio of 4, and /3); and instead of examining the effect o f / 3 and 4,, we shall examine the effect of/3 and 8.
4. R E S U L T S
Position of the Neutral Surface ( N S ) , at the particular instant, given by the relation, h t = 0 °'5 hc, (see eqn. 9), is the key to the plate bending analysis, under the conditions of constant load creep. It decides the deflection and stress distribution. The N S position is a function of 0, which itself is a function of the parameters /3, 8, X, ~, and T (eqn. 13). The effect of these parameters on 0 and thereby on the h t / h c ratio, is the same as already discussed in ref.  for the rock b e a m (because, the h t / h c ratio is equal to 0 0.5 there also). Initial position of the N S depends o n / 3 (elastic state), and the final position on 4' (viscous state). In this time span, the value of 4', obtaining at the particular instant, decides the N S position. Here /3 has a strong influence, 8 (the ratio of 4, and /3) has a dominating and interesting influence, X (the ratio of stiffnesses of Maxwell and Kelvin elements) has some influence, and f (the ratio of relaxation times of the two elements) has negligible influence.
Fig. 3. Effect of fl on the history of ~' for a rock plate.
Again, the expression for the factor ~(0), see eqn. (12), which governs the bending part of the m a x i m u m deflection, is identical to that in the case of the rock beam. Hence, the effect of parameters /3, 8, X, and ~" on the factor ~ is identical to that in the case of the rock beam (see ref. ). It corresponds to the influence of these parameters on the N S position. The shear part of the m a x i m u m deflection, is governed by the factor ~'(0). Effect of the ,o I--'--'- ....
~, 6 ~ . . . .
T X J62
Fig. 4. Effect of 8 on the history of ~' for a rock plate.
(NS) of the rock plate, corresponds to a
r x 16 2
Fig. 5. Effect of X on the history of ~' for a rock plate.
parameters fi, 8, X and ~ on ~' has yet to be examined. Figure 3 shows the effect o f / 3 on ~ ' - T variation, for fixed values 8 = 2, X = 100 and ~"= 15, a n d / 3 as parameter. For fi = 1 and 5, the total increases in the value of ~' are 83% and 117% over their initial values of 1.000 and 4.914, respectively. Thus, fi not only very significantly influences the elastic values of ~', as discussed in the ref. , it has a predominating influence on ~' under creep conditions ,as well. Again, ~' values practically stabilise as the time t approaches 1000 tmo. The effect of 8 on ~ ' ( T ) for fixed values /3 = 3, X = 100 and ~ = 15 has been presented in Fig. 4. For 8 = 1, ~' is fixed at 2.796 and does not vary with time. For 8 = 2 and 3, however, the total rises in ~' values as t ---, oo from the same initial value of 2.796, is 116 and 240%, respectively. Therefore, the effect of 8 on ~' is, again very significant. The influence of X on ~ ' ( T ) for fixed values of fl = 3, 8 = 2 and ~ = 15, has been shown in Fig. 5. 5. C O N C L U S I O N S
The main conclusions of this paper can be s u m m e d up as follows: (1) The initial position of the Neutral Surface
purely elastic state, and is governed by the value of /3 (the ratio of stiffnesses under compressive and tensile loading); while its final position corresponds to a purely viscous state, and is decided by the value of @ (the ratio of viscosity constants under compressive and tensile loading). During this time span, the NS position, governed by the value of 0(/3, 8, X, f, T), keeps shifting, before it stabilises at a position corresponding to a purely viscous state. (2) The factors ~ and ~' (that, respectively, decide the bending and shear components of m a x i m u m deflection), related to the NS position, also change their values before stabilising, as t ~ c~. Obviously, their values are influenced by the same parameters/3, 8, X and ~'. (3) The results are not equally sensitive to the parameters/3, 8, X and f. (4) Parameter fl has a significant influence on the values of the functions ~ and ~', respectively, especially on the latter. (5) Parameter 8 ( = ep//3) has a very dominant and interesting influence on the factors ~ and ~'. For 8 = 1 (ff = fl or ~c/~tt =Eo/Et), the position of NS and the f a c t o r s ~ and ~' are no longer time-dependent, i.e. they are static. For 6 > 1, however, they become time functions, and are significantly influenced by the value of 8. (6) Parameter X (the ratio of stiffnesses of Maxwell and Kelvin elements) has a relatively weak influence over these quantities, while the influence of ~" (the ratio of relaxation times of Maxwell and Kelvin elements) is virtually nil. Therefore, in the determination of the values of the components Em, ~m, E k and /~k of the Burger's model, some laxity in the fixation of the values Em/Ek, and more so, of tm/tk, would not affect the overall results, materially.
(7) The ratio of plate heights under tensile and compressive loading (h t / h c) and the functions ~ and ~' attain around 90% of the total rises in their values by the time t ~ 1000 tmc. That is, for all practical purposes, their values stabilise within the period 1000 tmc (tmc being the relaxation time of the Maxwell element, under compressive loading).
where (Nx, Ny) is the in-plane normal force, (N~y, Nyx) is the in-plane shear force, and (Q~, Qs), the transverse shear force; per unit length, along a line parallel to (0y, 0x). Similarly, multiplying the equilibrium equations (A1) by z, and then integrating through the height of the plate, the equilibrium equations
M~,x+Myx,y-Qx=O, and Mxy,x + My,y - Qy = O,
REFERENCES are obtained, in which, the moment intensity 1 J.G. Singh and P.C. Upadhyay, Creep bending of rock beams. Mining Science and Technology, 5(2): 163-169. 2 J.G. Singh, P.C. Upadhyay and S.S. Saluja, Bending of rock plates, Int. J. Rock Mech. Min. Sci., 17 (1980): 377-381. 3 A. Nadai, Theory of Flow and Fracture of Solids, Vol. 2. McGraw-Hill, 1963. 4 J.G. Singh and P.C. Upadhyay, Elastic bending of thick rock plates, Proc., Soc. Min. Eng., Amer. Inst. Min. Metall. Pet. Eng., New York, 1982, pp. 664-672.
( M x, My, M x,,) is defined as (Mx, My, Mxy)= fh, Z( Ox, oy, .rxy) dz, ,I _he
(A5) where, (Mx, My) is the bending moment, and (Mxy, Myx) the twisting moment; per unit length, along a line parallel to (0y, 0x). APPENDIX B
APPENDIX A Deflection
The differential equation governing the elastic deflection of a rock plate  is
The stress equilibrium equations
Ox,x + ry,,,y + rzx,z = O, rxy,x + Oy,y + 'rzy,z = O,
p 2 ) ) V 4 W = q,
'Txz,x + "ryz,y + Oz,z = 0
on integration through the plate height, give the equilibrium equations
X = 4 / ( 1 + f l ° ' 5 ) 2,
Nx,x + Nyx,y =0, Nxy,x+ Ny,y=O, and Qx,x + Qs,y + q = 0;
in which the force intensities Nx, Ny, Nxy, Qy and Qx are defined as (Nx, Ny, Nxy, Qy, Qx) ht = .'_fr'ho(Ox' Oy, "rxy , Tyz, 'rzx ) dz,
in which, E and E t are the Young's moduli under compressive and tensile loading, respectively, and v, the Poisson's ratio under compressive loading. However, when the effect of transverse shear is incorporated, the eqn. (B1) assumes the form 
169 where X is as given by the eqn. (BI') and = ((8 + 1,)(1+ ¢},.s)_ 5v(1 +/t°"5)) /((2-~)(1
stants ~ and ~' are given as = (1 + fi°s)2/4,
~ ' = ((8 + ~,)(1 + ill.s) _ 5u(1 + ri0.s)) / ( 4 ( 2 - ~ , ) ( 1 + B°s)).
Effect of 13
The maximum deflection of a thick plate is given as  ( W ) m a x = (~0/ q -
where a is the span of the plate; the constants a and a' depend on the plate geometry and the boundary conditions, while the con-
The first term in the eqn. (B3) represents the deflection due to bending, while the second term, that due to the transverse shear. Thus, the rock property p influences the bending and shear parts of the maximum deflection through the factors ~ and ~', respectively.