Technical Paper by JP Giroud, KL Soderman, T. Pelte and JF Beech

Technical Paper by J.P. Giroud, K.L. Soderman, 

T. Pelte and J.F. Beech 

DESIGN METHOD TO PREVENT GEOMEMBRANE 

FAILURE IN TANK CORNERS 

ABSTRACT: How close to the corner of a tank should a geomembrane be installed 

to prevent bursting of the geomembrane when the tank is being filled? This paper provides 

a method to answer this question based on the following parameters: the allowable 

tension and strain in the geomembrane, the maximum pressure exerted by the liquid on 

the geomembrane, the interface friction between the geomembrane and the walls of the 

tank, and the angle of the corner. Alternatively, when a geomembrane is installed at a 

certain distance from the tank corner, the method presented in the paper makes it possible 

to determine (with a factor of safety, if required) the maximum height of liquid 

that the geomembrane can withstand. The method also makes it possible to design a 

rounded corner or a chamfer to prevent rupture of the geomembrane in cases where the 

geomembrane cannot be installed close enough to the corner. As the use of the method 

is complicated, graphical solutions are provided to help design engineers, and examples 

are presented to illustrate the method. A comparison based on the proposed method 

shows that chamfers are less effective than rounded corners. 

KEYWORDS: Geomembrane, Tank corner, Failure, Chamfer design, Unsupported 

geomembrane. 

AUTHORS: J.P. Giroud, Senior Principal, K.L. Soderman, Project Engineer, and T. 

Pelte, Staff Engineer, GeoSyntec Consultants, 621 N.W. 53rd Street, Suite 650, Boca 

Raton, Florida 33487, USA, Telephone: 1/407-995-0900, Telefax: 1/407-995-0925; 

and J.F. Beech, Principal, GeoSyntec Consultants, 1100 Lake Hearn Drive N.E., Suite 

200, Atlanta, Georgia 30342, USA, Telephone: 1/404-705-9500, Telefax: 

1/404-705-9400. 

PUBLICATION: Geosynthetics International is published by the Industrial Fabrics 

Association International, 345 Cedar St., Suite 800, St. Paul, Minnesota 55101, USA, 

Telephone: 1/612-222-2508, Telefax: 1/612-222-8215. Geosynthetics International is 

registered under ISSN 1072-6349. 

DATES: Original manuscript received 28 February 1994, revised manuscript received 

26 May 1995 and accepted 1 July 1995. Discussion open until 1 July 1996. 

REFERENCE: Giroud, J.P., Soderman, K.L., Pelte, T. and Beech, J.F., 1995, “Design 

Method to Prevent Geomembrane Failure in Tank Corners”, Geosynthetics 

International, Vol. 2, No. 6, pp. 971-1018. 

GEOSYNTHETICS INTERNATIONAL S 1995, VOL. 2, NO. 6 

971

GIROUD, SODERMAN, PELTE AND BEECH D Geomembrane Failure in Tank Corners 

1 INTRODUCTION 

On a number of occasions, geomembranes lining concrete tanks have failed when the 

tank was filled, because they could not withstand the liquid pressure while being locally 

unsupported in the corners of the tanks (Figure 1a). To prevent this type of failure, it 

is typically recommended to install the geomembrane as close as possible to the corners 

of the tank and/or to chamfer the corners (Figure 1b). Both recommendations are sound, 

but they are only qualitative. The method presented in this paper makes it possible to 

provide quantitative recommendations. The method allows the design engineer to select 

an appropriate distance between the installed geomembrane and the corner to prevent 

geomembrane failure. If the geomembrane cannot be installed close enough to the 

corner, the method allows the design engineer to select a chamfer that would prevent 

geomembrane failure. 

2 DESCRIPTION OF THE MECHANISM 

2.1 Pressure Distribution 

As a tank is being filled, the pressure applied by the liquid on a given point of the geomembrane 

increases proportionally to the height of liquid above the considered point. 

This is expressed by the following equation: 

p = ρ gh (1) 

where: p = pressure exerted by the liquid on the considered point of the geomembrane; 

ρ = density of the liquid; g = acceleration of gravity; and h = height of liquid above the 

considered point of the geomembrane. 

At a given time during the filling of the tank, the pressure applied by the liquid on 

the geomembrane is proportional to the height of liquid above the considered point of 

(a) 

(b) 

Figure 1. 

Geomembrane in the corner of a tank: (a) failure; (b) chamfer. 

972 GEOSYNTHETICS INTERNATIONAL S 1995, VOL. 2, NO. 6


the geomembrane (Equation 1). The tensions in the geomembrane due to liquid pressures 

are therefore greater in the lower part of the tank than in the upper part. As a result, 

the Poisson’s ratio effect tends to cause the geomembrane to move downward as it 

moves toward the corners. Therefore, the problem is three-dimensional. However, because 

of friction between the geomembrane and the tank walls, downward movements 

of the geomembrane are limited and the description of the mechanism that follows only 

considers the horizontal elongation of the geomembrane and its resulting displacement 

toward the corners. Accordingly, the theoretical analysis considers a two-dimensional 

problem in a horizontal plane. 

The two-dimensional approach is believed to overestimate strains in the geomembrane, 

because it does not take into account the downward movement of the geomembrane 

that tends to release the strain in the horizontal direction. Therefore, the method 

presented in this paper is believed to be conservative. However, no attempt has been 

made to evaluate the degree of conservativeness. 

2.2 Evolution of Geomembrane Deformation 

In accordance with the fact that a two-dimensional approach is used, the behavior of 

the geomembrane is analyzed at a given elevation. As soon as the liquid level reaches 

the considered elevation, the portion of the geomembrane that is unsupported in the 

tank corner takes a circular shape and is subjected to a uniform tension. As the tank is 

being filled, the pressure on the geomembrane at the considered elevation increases; 

as a result, the tension in the geomembrane increases and the geomembrane elongates. 

Therefore, a portion of the geomembrane that was previously unsupported comes in 

contact with the wall (Figure 2). During the process, the portion of the geomembrane 

that was already in contact with the wall does not undergo any large sliding movement 

with respect to the wall, because as the liquid pressure increases the interface shear 

strength between the geomembrane and the wall increases more than required to balance 

the increased tension in the geomembrane. (This is an important result from the 

theoretical analysis presented hereinafter.) However, when the liquid pressure increases, 

a fraction, or all, of the portion of the geomembrane in contact with the wall 

elongates as the required interface shear strength is progressively mobilized. As a result, 

there is a small relative movement between the wall and a fraction, or all, of the 

portion of the geomembrane in contact with the wall. At any given time, the tensile 

strain in the geomembrane is distributed as shown in the example given in Figure 3. If 

the liquid pressure continues to increase, the geomembrane moves closer to the corner 

and the strain in the unsupported portion of the geomembrane increases, as shown in 

Figure 2. 

2.3 Modes of Failure 

As the liquid pressure increases when the tank is being filled, the strain in the geomembrane 

increases and two cases are possible: 

S the strain in the unsupported portion of the geomembrane reaches a value that causes 

failure of the geomembrane; or 


973


Unsupported under pressure p =0 

Unsupported under pressure p 1 

p = p 3 , ε 3 > ε 2 

p = p 2 , ε 2 > ε 1 

p = p 1 , ε 1 > ε 0 

p =0,ε =0 

Figure 2. 

increases. 

Evolution of the geomembrane position and strain as the liquid pressure 

Unsupported under pressure p =0 

Unsupported under pressure p 

0% 2% 4% 6% 8% 

8% 

Configuration and strain 

distribution under pressure p 

8% 

(Not to scale) 

6% 

4% 

Figure 3. Distribution of strain in the geomembraneat a given pressure(example when the 

strain is 8% in the unsupported portion of the geomembrane). 

(Note: It is assumed that the liquid pressure has increased continuously between p =0andp.) 



S when the tank is full, a portion of the geomembrane is still unsupported, but the geomembrane 

does not fail. 

One may wonder if a third case may exist: the geomembrane would reach the corner 

before the tank is full and, consequently, would not fail regardless of the pressure applied 

subsequently. It will be shown in this paper (see Equation 72) that this case cannot 

exist because, regardless of its tensile characteristics, a geomembrane can reach a corner 

only if it undergoes an infinite elongation, which indicates that the geomembrane 

would have failed before. 

Three modes of failure can be considered depending on the shape of the geomembrane 

tension-strain curve (Figure 4): 

S If the geomembrane tension-strain curve does not exhibit a yield peak or plateau, 

which is the case of polyvinyl chloride (PVC) geomembranes (Curve a, Figure 4), 

failure occurs when tension and strain in the geomembrane reach the values of tension 

and strain at the break point (Point B). 

S If the geomembrane tension-strain curve exhibits a yield peak, which is the case of 

high density polyethylene (HDPE) geomembranes (Curve b, Figure 4), the geomembrane 

material yields when the tension and strain in the geomembrane reach the values 

of tension and strain at the yield peak (Point Y). At this point, even if the liquid 

pressure ceases to increase, the geomembrane tends to elongate, which releases the 

tension according to the subsequent theoretical analysis (see Equation 18). The ten- 

B 

Pi 

Figure 4. Typical geomembrane tension-strain curves: (a) continuously increasing curve; 

(b) curve with a yield peak; (c) curve with a plateau. 


975


sion in the geomembrane can only decrease to the level of plateau P (Curve b, Figure 

4), at which point the geomembrane does not actually break, but it exhibits a very 

large elongation. It will take an increase of liquid pressure to actually break the geomembrane, 

but from a design and performance standpoint, the geomembrane is so 

mechanically deteriorated by the very large elongation that it must be considered that 

failure occurs when the yield peak, Y, is reached. Also, if there is a scratch in the unsupported 

portion of the geomembrane, rupture of the geomembrane occurs at a strain 

close to the yield strain, as shown by Giroud et al. (1994). This confirms that geomembranes 

with a yield peak should be considered to have failed when the yield peak has 

been reached. 

S If the geomembrane tension-strain curve exhibits a plateau or a quasi-plateau (defined 

by a very slight increase of tension for a large increase in strain, as shown by 

Curve c in Figure 4), which is the case of very low density polyethylene (VLDPE) 

geomembranes, then the large elongation that occurs after the beginning of the plateau 

has been reached (at Point Pi) tends to stabilize the geomembrane, albeit with 

large strain, as explained above. A pressure increase will be necessary for the geomembrane 

to actually break. However, it is clear that, from a design and performance 

standpoint, failure should be considered to occur when the plateau or quasi-plateau 

is reached (Point Pi). 

In conclusion, the maximum allowable tension and strain are: the tension and strain 

at break for geomembranes that do not exhibit a yield peak, a plateau, or a quasi-plateau; 

and, the tension and strain at the yield peak, or the beginning of the plateau or quasi-plateau 

for the other geomembranes. It is also important to note that the allowable tension 

and strain should be measured in a test with a plane-strain biaxial state of stress to be 

consistent with the two-dimensional approach used in the theoretical analysis. Finally, 

geomembrane creep (i.e. strain increase with time under a constant tension) is not considered. 

3 PRELIMINARY STEPS OF THE THEORETICAL ANALYSIS 

3.1 Assumptions 

As indicated above, vertical movements of the geomembrane are not considered. 

Therefore, a two-dimensional analysis can be conducted in a horizontal plane at a given 

elevation. A corner with an exterior angle θ (as defined in Figure 5) is considered. The 

liquid applies on the geomembrane a uniform pressure, p, which is proportional to the 

height of liquid above the considered horizontal plane, as shown by Equation 1. When 

the pressure is p, the distance between the corner and the point where the geomembrane 

is tangent to the wall is a (Figure 5). The initial value of a at the considered elevation 

is a install (i.e. a install is the value of a for p =0). 

As the pressure, p, increases, the geomembrane moves toward the corners. As a result, 

interface shear stresses develop between the geomembrane and the wall. In the most 

general case, shear stresses depend on interface friction and adhesion between the geomembrane 

and the wall. It is assumed that there is a friction angle, δ, and no adhesion 

between the geomembrane and the wall. 



a 

θ 

Ci 

Figure 5. Geometry of the geomembrane in the vicinity of a corner. 

(Note: Arc AB is referred to as “the unsupported portion of the geomembrane”.) 

3.2 Geometry of the Geomembrane 

Since it is subjected to a uniform pressure, the geomembrane takes the shape of a circular 

arc (arc AB in Figure 5). The length of arc AB is: 

L = R θ 

(2) 

where: R = radius of curvature of the geomembrane; and θ = exterior angle of the tank 

corner, defined in Figure 5. 

The distance, a, between the corner and the point, A or B, where the geomembrane 

is in contact with the wall of the tank is: 

a = R tan(θ∕2) 

(3) 

The distance, d, between the geomembrane and the corner (Figure 5) is: 

hence: 

d = R 

1 

− 1 (4) 

cos(θ∕2) 


977


d = 1 − cos(θ∕2) 

sin(θ∕2) 

a 

(5) 

In the case of a right-angle corner (Figure 1a), θ = π/2, and Equations 2, 3, 4 and 5 

become: 

L = πR∕2 

(6) 

a = R (7) 

d = R 2 − 1 = R∕( 2 + 1) = a 2 − 1 = a∕ 2 + 1 (8) 

In the case of a chamfered right-angle corner (Figure 1b), θ = π/4, and: 

L = πR∕4 

(9) 

a = 2 − 1 R = R tan π∕8 

(10) 

d =22 − 2 1∕2 − 1 R = 0.0824R 

(11) 

d = R[1∕ cos(π∕8)] − 1 = 0.0824R 

(12) 

d = 

a 

2 − 2 + 2 1∕2 = 0.199a 

2 − 2 1∕2 

(13) 

d = a[1 − cos(π∕8)]∕ sin(π∕8) = 0.199a 

(14) 

3.3 Force Equilibrium 

The next step consists of balancing the forces. According to a classical demonstration, 

the action of a uniform pressure, p, on arc AB is equal to its action on chord AB 

(Figure 5). The length of chord AB is: 

L AB = 2a sin(π − θ)∕2 = 2a cos(θ∕2) 

(15) 



Therefore, the force per unit height exerted by the pressure p on arc AB is: 

F AB = 2ap cos(θ∕2) 

(16) 

This force per unit height is perpendicular to chord AB. It must be balanced by the 

projection of the tension, T, in the unsupported portion of the geomembrane at both ends 

of arc AB. The projection on the direction CCi (perpendicular to chord AB) of the force 

equilibrium is expressed by: 

2ap cos(θ∕2) = 2T sin(θ∕2) 

(17) 

hence: 

T = 

pa 

tan(θ∕2) 

(18) 

Equation 18 shows that, if a geomembrane elongates without an increase of pressure 

(i.e. if the geomembrane yields), the geomembrane tension, T, decreases because the 

distance, a, between the corner and the point where the geomembrane is tangent to the 

wall decreases while the pressure, p, is constant. Therefore, if the geomembrane yields, 

some stabilization of the geomembrane deformation mechanism takes place as discussed 

earlier in Section 2.3. However, this stabilization occurs too late, because the 

geomembrane has already undergone major irreversible strain and, from a design standpoint, 

it should be considered to have failed even if it does not break immediately. 

Combining Equations 5 and 18 gives: 

T = pd 

cos(θ∕2) 

1 − cos(θ∕2) 

(19) 

In the case of a right-angle corner (Figure 1a), θ = π/2, and, according to Equations 

18 and 19, the tension in the unsupported portion of the geomembrane is: 

T = pa= pd(1 + 2 ) 

(20) 

It should be noted that, in the case of a right-angle corner, a is also the radius of curvature 

of the geomembrane. 

In the case of a right-angle corner with a 45_ chamfer (Figure 1b), θ = π/4, and, according 

to Equation 18, the tension in the unsupported portion of the geomembrane is: 

T = 1 + 2 pa 

(21) 

It should not be concluded from a comparison of Equations 20 and 21 that the presence 

of a chamfer increases the tension, T, in the unsupported portion of the geomem- 


979


brane. In fact, the chamfer greatly reduces the tension because, as seen in Figure 1b, 

it greatly reduces the value of a (defined in Figure 5). 

3.4 Geomembrane-Wall Interface Shear Stresses 

The tension in the unsupported portion of the geomembrane is balanced, totally or 

partly, by the interface shear stresses between the geomembrane and the wall of the tank 

(Figure 6). The interface shear stresses, τ, between the geomembrane and the wall of 

the tank are given by: 

τ = p tan δ 

(22) 

where δ is the interface friction angle between the geomembrane and the wall. 

In order to totally balance the tension, T, in the unsupported portion of the geomembrane, 

the shear stresses, τ, must be mobilized over a length L F (Figure 6a) given by: 

L F = T∕τ 

(23) 


L F = T∕(p tan δ) 

(24) 


L F = 

a 

tan(θ∕2) tan δ 

(25) 


L F = 

d cos(θ∕2) 

[1 − cos(θ∕2)] tan δ 

(26) 

The tension in the unsupported portion of the geomembrane is totally balanced by the 

interface shear stress if the tank is large enough that the required value of L F exists. As 

shown by Equation 25, L F is maximum when a is maximum, which occurs at the beginning 

of the filling of the tank, i.e. when a = a install . Hence, with a = a install in Equation 

25: 

L Fmax = 

a install 


(27) 

In the case of a right-angle corner, θ = π/2 and the value of L Fmax is: 



(a) 

D 

D/2 

(1) 

M 

L F 

τ 

a 

(1) 

Geomembrane tension 

T X 

L F 

T 

p 

M 

O 

(D/2) --- a 

X 

(b) 

D 

D/2 

a 

(1) 

M 

τ 

(1) 

Geomembrane tension 

p 

T X 

T 

(D/2) --- a 

T M 

O 

X 

Figure 6. Geomembrane-wall interface: (a) case where L F < (D/2) - a and, consequently, 

the geomembrane tension is zero over a certain length around the midpoint, M, between two 

identical corners; (b) case where the condition L F < (D/2) - a would not be met and, therefore, 

the geomembrane tension is nowhere equal to zero. 

(Note: (1) Unsupported portion of the geomembrane.) 


981


L Fmax = a install 

tan δ 

(28) 

According to Figure 6a, there is enough space for L F to develop if: 

D∕2 > L Fmax + a install 

(29) 

Combining Equations 27 and 29 gives the following condition: 

a install < 

D 

= 

21 + 

1 

tan(θ∕2)tanδ 

a crit 

(30) 

In the case of a right-angle corner, θ = π/2 and Equation 30 becomes: 

a install < 

D 

21 + 1 

tanδ = a crit 

(31) 

Values of D/a crit are presented in Table 1. The two corner angles used in Table 1 (i.e. 

the exterior angle and the interior angle) are defined in Figure 7. It should be noted that 

the following relationship exists: 

D min ∕a install = D∕a crit 

(32) 

where D min is the minimum distance between two identical tank corners to provide 

enough space for L F to exist. 

Table 1. Values of D min /a install = D/a crit . 

Angle of the corner 

θ 

(_) 

45 

60 

90 

120 

135 

ω 

(_) 

135 

120 

90 

60 

45 

Interface friction angle between geomembrane and wall 

δ (_) 

0 5 10 15 20 25 30 45 90 

∞ 

∞ 

∞ 

∞ 

∞ 

57.2 

41.6 

24.9 

15.2 

11.5 

29.4 

21.6 

13.3 

8.5 

6.7 

20.0 

14.9 

9.5 

6.3 

5.1 

15.3 

11.5 

7.5 

5.2 

4.3 

12.4 

9.4 

6.3 

4.5 

3.8 

10.4 

8.0 

5.5 

4.0 

3.4 

6.8 

5.5 

4.0 

3.2 

2.8 

2.0 

2.0 

2.0 

2.0 

2.0 

Notes: The above table was established using Equations 30, 31 and 32. The relationship between the exterior 

angle, θ, and the interior angle, ω, is illustrated in Figure 7. 



ω 

θ 

Tank 

Figure 7. Relationship between the exterior angle, θ, and the interior angle, ω. 

For example, Table 1 shows that, to provide enough space for L F to exist in the case 

δ =20_, the distance between two adjacent tank corners with a right angle must be equal 

to or greater than 7.5 times the value of a install . Thus, if the geomembrane is installed 

0.3 m from the tank corners (i.e. d install = 0.3 m), the value of a install given by Equation 

8 is 0.72 m. Therefore, according to Equations 31 and 32, the value of D min is 5.4 m. If 

the distance between tank corners is less than the value of D min , there is not enough space 

for L F to develop and the geomembrane tension is balanced otherwise, as discussed 

hereafter. 

If the condition expressed by Equation 30 or 31 is satisfied, it appears from the above 

discussion that the tension in the unsupported portion of the geomembrane is totally balanced 

by the shear stresses that are mobilized over the length L F (Figure 6a). As a result, 

the tension in the geomembrane is zero in a certain zone around the midpoint, M, between 

two tank corners. 

If the condition expressed by Equation 30 or 31 is not satisfied, the tension in the unsupported 

portion of the geomembrane is not totally balanced by the shear stresses and, 

consequently, the tension in the geomembrane at the midpoint, M, between two tank 

corners is not zero (Figure 6b). The geomembrane cannot have large movements because 

its tension on one side of M is balanced by its tension on the other side of M. 

Therefore, the only movements of the geomembrane are the small movements that result 

from strains in the geomembrane. 

As discussed in Sections 3.5 and 4.1, the fact that the geomembrane cannot have large 

movements is true not only in the case where the condition expressed by Equation 30 

or 31 is not satisfied, but also when it is satisfied. 

3.5 Required Friction 

Equation 26 shows that, as the distance, d, between the geomembrane and the tank 

corner decreases (which occurs when the pressure, p, increases), the length, L F , over 

which the shear stresses are mobilized decreases. This is a remarkable result: it means 

that the required interface friction between the geomembrane and the wall decreases 

as the pressure increases. As a result, if there is no general slippage of the geomembrane 

when the first increment of pressure is applied, there cannot be general slippage as the 

pressure increases, assuming that the interface friction angle, δ, does not decrease with 

time and/or as the pressure increases. (However, there is a small relative movement between 

the geomembrane and the wall to mobilize the interface shear stresses as discussed 

in the next section.) 


983


The remarkable result presented above deserves some explanation to help understand 

the mechanisms involved. As the pressure, p, increases, the distance, d, between the 

geomembrane and the corner decreases, as mentioned above, and the geomembrane 

tension, T, increases, which is illustrated in Figure 2 and will be demonstrated later in 

this paper (see Example 3 in Section 4.4). Equation 19 shows that the increase of T is 

less than proportional to the increase of p, because of the decrease of d. (In other words, 

successive pressure increments of equal magnitude cause tension increments of decreasing 

magnitude.) At the same time, the shear stresses, τ, increase proportionally to 

p, as shown by Equation 22. Therefore, as the pressure, p, increases, the available interface 

shear strength per unit length of wall increases faster than the interface shear 

strength required to balance the slowly increasing geomembrane tension, T. Therefore, 

the required interface friction angle, δ, decreases, or, for a given δ, the length, L F , over 

which shear stresses are mobilized decreases. In other words, a significant increase of 

p causes only a moderate increase of T, which requires a moderate increase of the interface 

shear strength, p tanδL F ; the moderate increase of p tanδL F results from the significant 

increase of p and a decrease of tanδL F ; and the decrease of tanδL F results in a decrease 

of L F or a decrease of the required value of δ. 

4 THEORETICAL ANALYSIS 

4.1 Geomembrane Tension and Elongation 

From the above discussion, it appears that in general there is no general slippage of 

the geomembrane. However, it should be recognized that the portion of the geomembrane 

that is in contact with the wall elongates as part of the process that mobilizes the 

interface shear stresses. This elongation of the geomembrane can be evaluated by first 

evaluating the geomembrane tension. At abscissa x on the portion of geomembrane 

along which interface shear stresses are mobilized (Figure 6), the following relationship 

exists between the geomembrane tension and the shear stresses: 

dT x = τ dx 

(33) 

where T x is the geomembrane tension at abscissa x. 

Two cases should be considered for the integration of Equation 33. 

First Case. If the condition expressed by Equation 30 or 31 is satisfied, the interface 

shear stresses are mobilized over the length L F (Figure 6a). In this case, if the origin of 

abscissae is located at the extremity of L F where T x = 0 (Figure 6a), integration of Equation 

33 gives: 

T x = τx 

(34) 

Therefore, the tension in the geomembrane varies linearly from T x = 0 at abscissa 

x =0toT x = T at x = L F , according to Equation 23 (T being the tension in the unsupported 



portion of the geomembrane). The strain in the geomembrane varies from ε x = 0 at abscissa 

x =0toε x = ε at abscissa x = L F (ε being the strain in the unsupported portion of 

the geomembrane). The resulting elongation of the geomembrane is: 

e = L F 

0 

ε x dx 

(35) 

If the tension-strain curve of the geomembrane is linear, the strain is proportional to 

the tension: 

ε x = T x ∕ J 

(36) 

where J is the geomembrane tensile stiffness. In this case, the strain varies linearly as 

a function of x, just like the tension. 

Combining Equations 34, 35 and 36 gives: 

e = τ F 

J L x dx 

0 

(37) 

hence: 

e = (1∕2)τL 2 F∕J 

(38) 

According to Equation 36: 

ε = T∕J 

(39) 

where: ε = strain in the unsupported portion of the geomembrane; and T =tensionin 

the unsupported portion of the geomembrane. 

Combining Equations 23, 25, 38 and 39 gives: 

e = 

a ε 

2tan(θ∕2) tan δ 

(40) 

In the case of a right-angle corner, the angle θ is equal to π/2 and Equation 40 becomes: 

e = 

a ε 

2tanδ 

(41) 


985


Second Case. If the condition expressed by Equations 30 or 31 is not satisfied (Figure 

6b), the geomembrane tension is nowhere equal to zero. Its minimum value is T M at the 

midpoint, M, between two identical corners. If the origin of abscissae is M, integration 

of Equation 33 gives: 

T x = T M + τx 

(42) 

Therefore, the tension in the geomembrane varies linearly from T x = T M at abscissa 

x = 0 (i.e. at point M) to the value T x = T for x =(D/2) - a. Therefore: 

T = T M + τ[(D∕2) − a] 

(43) 


τ = 

T tan(θ∕2) tan δ 

a 

(44) 


T M = T1 − (D∕2) − a 

a tan(θ∕2) tan δ 

(45) 


T x = T1 − (D∕2) − a − x 

a 


(46) 

Using this expression of the geomembrane tension, it is then possible to calculate the 

geomembrane elongation by integrating the strain between x =0andx =(D/2) - a: 

e = D 2 −a 

0 

ε x 

dx 

(47) 

The value of ε x for x =(D/2) - a is the strain, ε, in the unsupported portion of the geomembrane. 

The relationship between ε and T is given by Equation 39. 

If the tension-strain curve of the geomembrane is linear, combining Equations 36 and 

47 gives: 

e = 1 2 −a 

J D T x dx 

(48) 

0 



Combining Equations 39, 46 and 48, and integrating, give: 

e = ε[(D∕2) − a]1 − (D∕2) − a 

2a 


(49) 

It should be noted that Equations 40 and 49 are identical when L F (which is given by 

Equation 25) is equal to (D/2) - a, i.e. at the boundary between the first and the second 

case of integration of Equation 33. 

In the case of a right-angle corner, the angle θ is equal to π/2 and Equation 49 becomes: 

[(D∕2) − a]ε 

e = 2 − (D∕2) − a 

(50) 

2 a tan δ 

4.2 Deformation of the Geomembrane 

The liquid in the tank applies a pressure, p, on the geomembrane at the considered 

level. As a result, there is a tension, T, and a strain, ε, in the unsupported portion of the 

geomembrane. An incremental pressure change, dp, causes incremental changes in tension, 

dT,andstrain,dε. As a result of the strain change, the distance, a, between the corner 

and the point, A or B, where the geomembrane is tangent with the wall becomes 

a +da (Figure 8, where da is negative, i.e. a decrease of a,ifdp is positive, i.e. an increase 

of p). At the same time, there are incremental elongations, de A and de B ,ofthe 

a 

--- da a+da 

Ai 

Li 

θ 

L 

Figure 8. Incremental deformation of the geomembrane considered in the theoretical 

analysis. (Note that da < 0 as the geomembrane moves toward the wall, whereas the 

increments of pressure, dp,tension,dT,andstrain,dε,arepositive.) 


987


geomembrane, at A and B respectively, as a result of a change in the mobilization of 

the shear stresses between the geomembrane and the wall. 

The change in the geomembrane configuration can be described by the following 

equation: 

− 2da + L′ =L(1 + dÁ) + de A + de B 

(51) 

where: L = length of the unsupported portion of the geomembrane when the pressure 

is p; Li = length of the unsupported portion of the geomembrane when the pressure is 

p +dp;dε = strain change in the unsupported portion of the geomembrane that corresponds 

to the pressure change, dp; and de A and de B = incremental elongations of the portions 

of the geomembrane beyond A and B, respectively, where shear stresses are mobilized 

in response to the tension change, dT, due to a pressure change, dp. 

Herein only the case of tanks with identical corners is considered; therefore: 

e A = e B 

(52) 


L = 

aθ 

tan(θ∕2) 

(53) 

Similarly: 

L′ = 

(a + da)θ 

tan(θ∕2) 

(54) 

Combining Equations 51, 52, 53 and 54 gives: 

θ 

tan(θ∕2) − 2da = θa dε + 2de 

tan(θ∕2) 

(55) 

Regarding the term de in Equation 55, two cases can be considered, depending on the 

condition expressed by Equation 30 or 31. 

First Case. If the condition expressed by Equation 30 or 31 is satisfied, the term de 

of Equation 55 can be expressed as follows by calculating the derivative of Equation 

40: 

de = 

ε da + a dε 

2tan(θ∕2) tan δ 

(56) 

It should be noted that Equation 56 was established assuming that the tension-strain 

curve of the geomembrane is linear. 




1 da 

1 + θ tan δ a =− 

dε 

[2 tan(θ∕2) − θ]tanδ + ε 

(57) 

Equation 57 is a differential equation where the two variables, a and ε, are separate. 

Therefore, the two sides can be integrated separately, which gives: 

1 

(58) 

1 + θ tan δ ln(a 2∕a 1 ) = ln [2 tan(θ∕2) − θ]tanδ + ε 1 

[2 tan(θ∕2) − θ]tanδ + ε 2 

 

where: a 1 = value of the distance, a, between the corner and the point where the geomembrane 

is tangent to the wall when p = p 1 ; a 2 = value of a when p = p 2 ; ε 1 =value 

of the strain, ε, in the geomembrane when p = p 1 ;andε 2 = value of ε when p = p 2 . 

It should be noted that, according to Equation 5: 

a 2 ∕a 1 = d 2 ∕d 1 

(59) 

Equation 58 can also be written as: 

a 2 

a 1 

= 

[2 tan(θ∕2) − θ]tanδ + ε 1 

[2 tan(θ∕2) − θ]tanδ + ε 2 

 

1+ θ tan δ 

(60) 

In the case of a right-angle corner, the angle θ is equal to π/2, and Equation 60 becomes: 

a 2 

a = 

1 

(2 − π∕2) tan δ + ε 1 

(2 − π∕2) tan δ + ε 2 

 

1+ π 2 tan δ 

If there is perfect adhesion between the geomembrane and the wall, Equation 55 can 

be rewritten with de = 0 and, then, solved. Alternatively, the limit of Equation 60 for 

δ =90_ can be calculated, which gives: 

a 2 

ε 

a = (62) 

1 

exp 2 − ε 

1 

1 − [tan(θ∕2)]∕(θ∕2) 

In the case of a right-angle corner, the angle θ is equal to π/2 and Equation 62, i.e. 

the equation for the case of perfect adhesion, becomes: 

(61) 

a 2 

a 1 

= exp ε 2 − ε 1 

1 − 4∕π = exp− ε 2 − ε 1 

0.2732 

(63) 


989


Second Case. If the condition expressed by Equation 30 or 31 is not satisfied, the term 

de of Equation 55 can be expressed as follows by calculating the derivative of Equation 

49: 

de = (D∕2) − a 

a 

1 − (D∕2) − a tan(θ∕2) tan δa dε 

2a 

−1 − (D∕2) − a 

2a 

tan(θ∕2) tan δε da + D 4a (D∕2) − a 

a 

tan(θ∕2) tan δεda 

(64) 

Combining Equations 55 and 64 results in a differential equation (not shown here) 

where the variables, a and ε, are not separate, unlike in the first case. Therefore, integration 

of the differential equation would be complicated. For the sake of simplicity, only 

the first case (i.e. Equations 57 to 63) is considered herein. 

However, the second case (i.e. the case where the condition expressed by Equation 

30 or 31 is not satisfied) is not left unsolved. A conservative solution of the second case 

can be obtained by replacing the actual interface friction angle, δ, by a conservative 

friction angle, δ conserv , obtained by writing that the condition expressed by Equation 30 

or 31 is just met: 

tan δ conserv = 

1 

tan(θ∕2)[(D∕2a install ) − 1] 

(65) 

In the case of a right-angle corner, θ = π/2 and: 


1 

(D∕2a install ) − 1 

(66) 

The value of δ conserv thus obtained is then used in Equations 58 to 61 to obtain a conservative 

solution to the considered problem. The solution is conservative because, δ conserv 

being greater than the actual δ, the geomembrane would be less likely to elongate when 

the pressure is applied if the interface friction angle were δ conserv instead of δ, and, therefore, 

the geomembrane tension would be less likely to be released. (Also, see the comment 

on the influence of δ in Section 4.3, in the last paragraph before Example 1, and 

in Section 4.4, at the end of Example 3.) 

4.3 Relationship between Liquid Pressure and Geomembrane Deformation 

In the analysis presented below, a geomembrane is characterized by its allowable 

strain, ε all , and its allowable tension, T all . The liquid pressure that causes the geomembrane 

strain to be ε all and the geomembrane tension to be T all is called the allowable pressure, 

p all . This pressure depends not only on the geomembrane allowable strain and tension 

but also on the tank corner angle and the distance between the geomembrane and 

the tank corner as shown below. When the pressure p all is applied, the distance between 

the tank corner and the point where the geomembrane is tangent to the wall is given the 



notation a min . A relationship between T all , p all and a min is obtained by writing Equation 

18 as follows: 

T all = p all a min 

tan(θ∕2) 

(67) 

In Equation 67, a min is unknown and will be eliminated in the next step. Combining 

Equations 5 and 67 with Equation 60, where ε 1 = 0 (i.e. strain at installation), a 1 = a install 

(i.e. value of a at installation), ε 2 = ε all and a 2 = a min ,gives: 

a install p all 

T all tan(θ∕2) = d install p all cos(θ∕2) 

T all [1 − cos(θ∕2)] =1 + 

ε all 

[2 tan(θ∕2) − θ]tanδ 

1+θ 

tan δ 

(68) 

In the case of a right-angle corner, Equation 68 becomes: 

1+ π 2 tan δ 


= d install p all 

T all T all ( 2 

ε all 

− 1) = ⎪ ⎡ ⎣ 1 + 

2 − π 2 tan δ⎪⎤ ⎦ 

(69) 

If there is perfect adhesion between the geomembrane and the wall, δ =90_, then 

Equation 68 becomes: 


T all tan(θ∕2) 

= d install p all cos(θ∕2) 

exp 

T all [1 − cos(θ∕2)] = ε 

all 

[tan(θ∕2)]∕(θ∕2) − 1 

In the case of a right-angle corner, the angle θ is equal to π/2 and Equation 70, i.e. 

the equation for the case of perfect adhesion, becomes: 

(70) 


= d install p all 

ε 

= 

T all T all ( 2 

− 1) 

exp all 

(4∕π) − 1 = exp ε all 

0.2732 

(71) 

Equations 68 to 71 can be used as follows: 

S If the allowable pressure, p all , is known (e.g. the pressure to which the geomembrane 

will be subjected, multiplied by a factor of safety), Equations 68 to 71 can be used 

to determine the maximum value of d install and a install (d install being the distance between 

the geomembrane and the corner at the time of geomembrane installation, and a install 

being the distance, at the time of geomembrane installation, between the tank corner 

and the point where the geomembrane is tangent to the wall). 

S If the distance, d install , between the geomembrane and the tank corner at the time of 

installation is known (or if the corresponding value a install is known), Equations 68 to 

71 can be used to determine the allowable pressure, p all . Then, Equation 1 can be used 

to determine the allowable depth of liquid. 


991


The pressure p all should not be exceeded. If the maximum pressure, p max , expected 

to be applied on the geomembrane is greater than p all , the corners of the tank should 

be redesigned as explained in Section 5. 

Using a factor of safety to multiply the pressure, p, does not imply that there is any 

significant uncertainty on the value of the pressure. In fact, the pressure is known with 

great accuracy because all of the terms of Equation 1 are well known. The factor of safety 

is intended to account for uncertainties in parameters such as the interface friction 

angle and the geomembrane tensile characteristics. The factor of safety is applied herein 

to the pressure (or the depth of liquid) only because it is convenient. Another approach 

could be used wherein factors of safety are applied to the parameters with significant 

uncertainties, such as the interface friction angle and the geomembrane tensile 

characteristics, rather than the pressure which is known with great accuracy. 

Graphs were established using Equations 68 to 71: these graphs are presented in Figure 

9a for an exterior corner angle θ =45_ (i.e. an interior corner angle ω = 135_), and 

in Figure 9b for a right-angle corner (θ = ω =90_). For a given geomembrane (defined 

by T all and ε all ) placed in a tank with right-angle corners, Figure 9b shows that the term 

d install p all increases as δ decreases. In other words, the lower the interface friction angle, 

the farther from the wall the geomembrane can be installed, or the greater the allowable 

pressure. This result was expected since the geomembrane elongates over a longer 

length if the interface friction angle is lower (see Equation 26) and, therefore, moves 

more toward the corner. The same trend appears on the graph for θ =45_ (Figure 9a), 

except for δ =90_ (case of perfect adhesion) when the allowable strain is more than 

15%. No explanation has been found for this difference between Figure 9a and Figure 

9b. 

Example 1. Water is stored in a 6 m × 6 m square tank that is 3.4 m deep. The geomembrane 

has an allowable strain of 5% and an allowable tension of 23 kN/m. The interface 

friction angle between the geomembrane and the wall is 20_. At what distance 

from the corner should the geomembrane be installed to meet the allowable strain and 

tension with a factor of safety of 1.5 on the water pressure when the tank is full? 

When the tank is full, the water pressure is given as follows by Equation 1: 

p = (1, 000)(9.81)(3.4) = 33, 354 Pa 

The allowable pressure is then derived from the above pressure using a factor of safety 

of 1.5: 

p all = (33, 354)(1.5) = 50, 031 Pa ≈ 50 kPa 

Then, Equation 69 gives: 

d install = ( 2 

− 1) 

23 

50⎪ ⎡ ⎣ 1 0.05 

2 − π 2 tan 20⎪⎤ _ 

⎦ 

_ 

1 + π 2 tan 20 = 0.29 m 



(a) 

6 

θ =45_ 

δ =90_ 

d install p all /T all 

4 

2 

δ =10_ 

δ =15_ 

δ =20_ 

δ =40_ 

(b) 

0 

0 10 20 30 40 50 

Allowable geomembrane strain, ε all (%) 

6 

θ =90_ 

d install p all /T all 

4 

2 

δ =10_ 

δ =15_ 

δ =20_ 

0 

δ =90_ 

δ =40_ 

0 10 20 30 40 50 


Figure 9. Graphs providing relationships between design parameters for two values of the 

corner angle: (a) exterior angle, θ =45_ and interior angle,ω =135_; (b) exterior and interior 

angles, θ = ω =90_. 

(Notes: θ and ω are defined in Figure 7. These graphs were established using Equations 68 to 71. For δ = 

90_, it was checked that the same curve was obtained using Equation 68 or 70 with δ = 89.9_, on one hand, 

and using Equation 69 or 71, on the other hand. The value of d install p all /T all for ε = 0 is 0.082 for θ =45_, 

according to Equations 68 and 70, and 0.414 for θ =90_, according to Equations 69 and 71.) 


993


Equation 69 also gives a install =0.71m. 

Alternatively, Figure 9 can be used as follows. For ε all =5%andδ =20_, one may 

read approximately on Figure 9b: 

d install p all 

T all 

≈ 0.6 

hence d install ≈ 0.28 m for p all =50kPaandT all = 23 kN/m. 

To make sure this solution is valid, it is necessary to check if the condition expressed 

by Equation 31 is satisfied: 

a install < a crit = 

6 

21 + 

tan20 1 = 0.80 m 

_ 

The condition is satisfied: 0.71 m < 0.80 m. 

If the interface friction angle had been 10_, the following values would have been 

obtained: 

d install = 0.36 m a install = 0.88 m a crit = 0.45 m 

In this case, the condition a install < a crit is not satisfied, and the calculated values of d install 

and a install are not correct. A conservative solution can be obtained by using δ conserv instead 

of δ. The value of δ conserv can be obtained by trial and error using Equation 66 as follows, 

starting with a install = 0.6 m (which is between 0.45 m and 0.80 m): 


1 

6∕(2 × 0.6) − 1 = 0.25 

Then, Equation 69 gives a install =0.78m. 

A second iteration is performed with a install = 0.70 m (which is between 0.6 m and 0.78 

m). Equation 66 gives tanδ conserv = 0.304 and Equation 69 gives a install =0.74m. 

A third iteration is performed with a install = 0.73 m. Equation 66 gives tanδ conserv = 0.322 

andEquation69givesa install = 0.73 m. This is the sought conservative value of a install 

since the iteration process has converged. The value of d install can then be derived from 

the value of a install using Equation 8. The value obtained is d install =0.30m. 

END OF EXAMPLE 1 

Example 2. The same tank and the same geomembrane as in Example 1 are considered. 

The geomembrane is installed at 0.30 m from the tank corner. To what depth of 

water can the tank be filled with a factor of safety of 1.5? 

First, it is necessary to check that the condition expressed by Equation 31 is satisfied: 



a crit = 

6 

21 + 

tan20 1 = 0.80 m 

_ 

The value of a install can be derived from the known value of d install using Equation 8: 

a install = 0.30(1 + 2 ) = 0.72 m 

The condition a install < a crit is satisfied and Equation 69 as well as Figure 9 can be used. 

Equation 69 gives: 

p all = 2 − 1 

0.3⎪ ⎡ 23 

⎣ 1 + 0.05 

2 − π 2 tan 20⎪⎤ _ 

⎦ 

1+ π 2 tan 20 = 49.13 kPa = 49, 130 Pa 

The depth of water to which the tank can be filled is then calculated using Equation 

1 as follows: 

h = 

49, 130 

(1, 000) (9.81) = 5.01 m 

which becomes 3.34 m after a factor of safety of 1.5 is applied. 

With an interface friction angle of 30_, the maximum height for a factor of safety of 

1.5 is 4.60/1.5 = 3.06 m. This illustrates that a low friction angle is beneficial. (Note: 

If δ =10_, the problem is a little more complex because the condition expressed by 

Equation 31 is not met as shown in Example 1; it is then necessary to use δ conserv as also 

shown in Example 1.) 

This example will be expanded in Example 3. 

_ 


4.4 Strain and Tension in the Geomembrane as it Moves Toward the Corner 

As shown in Figure 2, the geomembrane moves toward the tank corner as the liquid 

pressure increases. Assuming that the required amount of liquid pressure is applied, can 

a geomembrane elongate enough to reach the corner without failing? To answer this 

question, Equation 60 should be used with a 1 = a install , ε 1 =0,a 2 =0andε 2 = ε cor ,which 

gives: 

0 


a = 

install 

 

[2 tan(θ∕2) − θ]tanδ + ε cor 

 

1+θ tan δ 

(72) 


995


where ε cor is the strain in the geomembrane when the geomembrane reaches the corner. 

If δ > 0, which is always the case, the only solution of Equation 72 is ε cor = ∞,which 

demonstrates that any geomembrane, regardless of its tensile characteristics, can reach 

a corner only if it undergoes an infinite strain, which indicates that the geomembrane 

would have failed before it reached the corner. 

As the depth of liquid increases in the tank, the pressure, p, increases (as shown by 

Equation 1), the geomembrane strain and tension increase, and the distance between 

the geomembrane and the corner decreases. The following equation derived from Equations 

59 and 60 can be used to evaluate the distance between the geomembrane and the 

corner (expressed by a or d defined in Figure 5) as a function of the geomembrane strain, 

ε: 

a 

a install 

= 

d 

d install 

= 


[2 tan(θ∕2) − θ]tanδ + ε 

1+θ 

tan δ 

The relationship between a and d, ora install and d install , is, according to Equation 5: 

(73) 

a 

d = a install 

= 

sin(θ∕2) 

d install 1 − cos(θ∕2) 

(74) 

If the geomembrane tension-strain curve is linear, the geomembrane tension, T, is 

derived from its strain, ε, using the tensile stiffness, J, as follows: 

T = ε J 

(75) 

The liquid pressure at the considered geomembrane elevation is given by the following 

equation derived from Equation 18: 

p = T tan(θ∕2) 

a 

(76) 

The height of liquid above the considered geomembrane elevation is given by the following 

equation derived from Equation 1: 

h = p 

ρg 

(77) 

In the case of a tank with right-angle corners, θ = π/2 and the above Equations 73, 

74 and 76, respectively, become: 



a 

a install 

= 

d 

d install 

= 

(2 − π∕2) tan δ 

(2 − π∕2) tan δ + ε 

a 

d = a install 

= 1 + 2 

d install 

p = T∕a 

1+(π∕2) 

tan δ 

(78) 

(79) 

(80) 

If there is perfect adhesion between the geomembrane and the wall, δ =90_, and 


a 

a install 

= 

d = 

d install 

exp 

− ε 

 

[tan(θ∕2)]∕(θ∕2) − 1 

(81) 

and Equation 78 for the case of a right-angle corner becomes: 

a 

a = 

d = 

install d install 

exp 

− ε 

(4∕π) − 1 

(82) 

Graphs were established using Equations 73, 78, 81 and 82: these graphs are presented 

in Figure 10a for an exterior corner angle θ =45_ (i.e. an interior corner angle 

ω = 135_), and in Figure 10b for a right-angle corner (θ = ω =90_). Both Figures 10a 

and 10b show that the geomembrane strain increases as the distance, d, between the geomembrane 

and the corner decreases, and show that the geomembrane strain tends toward 

infinity as the distance, d, tends toward zero (which was demonstrated using Equation 

72). Figure 10b shows that, for θ =90_, the geomembrane strain is smaller for a 

smaller interface friction angle. However, Figure 10a shows that, when θ =45_ and 

d/d install is small, a smaller geomembrane strain may be obtained for δ =90_ (case of 

perfect adhesion between the geomembrane and the wall of the tank) than for other values 

of δ. No explanation has been found for the difference in behavior between the case 

θ =90_ and the case θ =45_. Finally, comparing Figures 10a and 10b shows that the 

geomembrane strain is significantly less with θ =45_ than with θ =90_, which illustrates 

the beneficial effect of chamfered corners. (This will be discussed in detail in Section 

5.) 

Example 3. A geomembrane with a linear tension-strain curve and a tensile stiffness 

of 460 kN/m is installed 0.3 m from the corner of a square tank. The interface friction 

angle between the geomembrane and the tank walls is 20_. What are the geomembrane 

strain, tension and distance to the corner, as a function of the height and pressure of water 

above the considered elevation of the geomembrane? 

Equations 75, 77, 78, 79 and 80 were used to calculate the values of T, a, d, p and h 

given in Table 2 for a series of values of ε. Then, the curves for δ =20_ presented in 

Figures 11a and 11b were established from the values given in Table 2. Similar calcula- 


997


(a) 

1 

0.8 

θ =45_ 

d / d install 

0.6 

0.4 

0.2 

δ =15_ 

δ =20_ 

δ =40_ 

δ =90_ 

0 

δ =10_ 

0 10 20 30 40 50 

Geomembrane strain, ε (%) 

(b) 

1 

0.8 

θ =90_ 

d / d install 

0.6 

0.4 

δ =40_ 

δ =90_ 

0.2 

0 

δ =10_ 

δ =15_ 

δ =20_ 

0 10 20 30 40 50 


Figure 10. Graphs giving the geomembrane strain as a function of its distance from the 

corner: (a) exterior angle θ =45_ (interior angle ω = 135_); (b) exterior and interior angles 

θ = ω =90_. 

(Notes: θ and ω are defined in Figure 7. The graphs were established using Equations 73, 78, 81, and 82. 

For δ =90_, it was checked that the same curve was obtained using Equation 73 or 81 with δ = 89.9_,and 

using Equation 78 or 82.) 



(a) 

30 

Height of liquid, h (m) 

25 

20 

15 

10 

5 

δ =10_ 

δ =20_ 

δ =90_ 

0 

0 5 10 15 


(b) 

30 

Height of liquid, h (m) 

25 

20 

15 

10 

5 

δ =10_ 

δ =20_ 

δ =90_ 

0 

0 0.1 0.2 0.3 

Distance to corner, d (m) 

Figure 11. Influence of liquid height for the case of Example 3 where the geomembrane was 

initially installed 0.3 m from the right-angle corner: (a) relationship between liquid height 

and geomembrane strain; (b) relationship between liquid height and distance between 

geomembrane and corner. 

tions were performed with δ =10_ and δ =90_ (using Equation 82), and the curves thus 

obtained are also shown in Figures 11a and 11b. The curves in Figure 11a confirm a 

statement made earlier in the paper that successive increments in pressure (or height 

of liquid) of equal magnitude cause increments in geomembrane strain (and consequently 

tension) of decreasing magnitude. Figure 11b shows that successive increments 

in pressure (or height of liquid) of equal magnitude cause increments of movement of 

the geomembrane toward the corner of decreasing magnitude. 


999


Table 2. Values of geomembrane strain, ε, tension,T, and distance to corner, a and d, asa 

function of liquid pressure, p, and liquid height, h, for the case of Example 3. 

ε 

(%) 

T 

(kN/m) 

a 

(m) 

d 

(m) 

p 

(kPa) 

h 

(m) 

0 

1 

2 

0 

4.6 

9.2 

0.724 

0.657 

0.599 

0.300 

0.272 

0.248 

0.00 

7.00 

15.35 

0.00 

0.71 

1.56 

3 

4 

5 

13.8 

18.4 

23.0 

0.550 

0.506 

0.468 

0.228 

0.210 

0.194 

25.11 

36.35 

49.13 

2.56 

3.71 

5.01 

6 

7 

8 

27.6 

32.2 

36.8 

0.435 

0.405 

0.378 

0.180 

0.168 

0.157 

63.52 

79.56 

97.32 

6.47 

8.11 

9.92 

9 

10 

11 

41.4 

46.0 

50.6 

0.354 

0.333 

0.313 

0.147 

0.138 

0.130 

116.86 

138.23 

161.48 

11.91 

14.09 

16.46 

12 

13 

14 

55.2 

59.8 

64.4 

0.296 

0.280 

0.265 

0.122 

0.116 

0.110 

186.67 

213.85 

243.08 

19.03 

21.80 

24.78 

15 

100 

∞ 

69.0 

460 

∞ 

0.251 

0.031 

0.000 

0.104 

0.013 

0.000 

274.39 

14,763.04 

∞ 

27.97 

1,504.90 

∞ 

Notes: This table was established using Equations 75, 77, 78, 79 and 80 with δ =20_. The tabulated values 

are presented graphically in Figure 11. 

The curves for friction angles of 10_,20_, and 90_ (perfect adhesion) shown in Figure 

11 illustrate the beneficial effect of a small friction angle: for a given height of liquid, 

the smaller the interface friction angle, the closer the geomembrane moves toward the 

corner, and the smaller the geomembrane strain (and, therefore, the smaller the geomembrane 

tension). 


5 CORNER DESIGN 

5.1 Overview 

Some geomembranes are rather stiff and it may not be possible to install them close 

enough to tank corners to satisfy the requirements expressed by Equations 68 to 71. A 

solution consists of changing the shape of the corner. Hereinafter, two cases are considered: 

a rounded corner and a chamfer. 



5.2 Rounded Corner 

A rounded corner can be obtained by filling with any appropriate material, such as 

concrete, the space between the tank wall and the location of the geomembrane when 

the allowable strain is reached, thereby transforming the angular corner into a rounded 

corner. 

The required radius of the rounded corner, R c , is expressed using the following equation 

derived from Equation 3 (see Figure 5): 

R c = 

a min 

tan(θ∕2) 

(83) 

where a min is the distance between the initial (angular) corner and the point where the 

geomembrane is tangent to the wall when the geomembrane strain is equal to the allowable 

strain. According to Equation 67: 

a min = T all tan(θ∕2)∕p all 

(84) 


R c = T all ∕p all 

(85) 

If the pressure, p, applied by the liquid on the geomembrane is equal to p all , the geomembrane 

is in contact with the material (such as concrete) that fills the corner. The 

pressure, p all , applied by the liquid is then entirely supported by the geomembrane and 

no pressure is transmitted to the material that fills the corner. Then, if the pressure, p, 

increases beyond p all , the geomembrane only supports p all and transmits the difference, 

p - p all , to the material that fills the corner. If the maximum pressure, p max , that will be 

applied to the geomembrane is less than p all , the geomembrane alone can support p max 

and no filling material is required in the corner. 

Combining Equation 5, Equation 60 (with a 1 = a install , ε 1 =0,a 2 = a min and ε 2 = ε all )and 

Equation 83 gives the required radius, R c , of the rounded corner, as follows: 

R c tan(θ∕2) 

a install 

= R c[1 − cos(θ∕2)] 

d install cos(θ∕2) 

= 


[2 tan(θ∕2) − θ]tanδ + ε all 

 

1+θ tan δ 

(86) 

In the case of a right-angle corner, Equation 86 becomes: 

R c 

a = R c( 2 

install 

− 1) (2 − π∕2) tan δ 

= 

d install 

 

(2 − π∕2) tan δ + ε all 

 

1+ π 2 tan δ 

(87) 


1001


A conservative (i.e. large) value of R c can be obtained with δ =90_ (perfect adhesion). 

In this case, the following equation is obtained either by combining Equations 5, 62 and 

83, or by calculating the limit of Equation 86 for δ =90_: 

R c tan(θ∕2) 

a = R c[1 − cos(θ∕2)] 

ε 

= (88) 

install d install 

exp 

all 

cos(θ∕2) 1 − tan(θ∕2)∕(θ∕2) 

In the case of a right-angle corner with δ =90_, Equation 88 becomes: 

R c 

a = R c( 2− 1) 

= exp− ε all 

install d install (4∕π) − 1 

(89) 

Equations 86 to 89 were used to establish graphs that give the required radius of the 

rounded corner as a function of the following parameters: the distance between the geomembrane 

and the corner at the time of installation, d install ; the allowable geomembrane 

strain, ε all ; the interface friction angle between the geomembrane and the wall of the 

tank, δ; and the corner angle, θ =45_ (Figure 12a) and θ =90_ (Figure 12b). 

According to Equation 86, the maximum value of R c occurs when ε all =0,andisas 

follows: 

R cmax = 

cos(θ∕2) 

1 − cos(θ∕2) d install 

(90) 

hence, for θ =90_: 

R cmax = (1 + 2 ) dinstall = 2.41 d install 

(91) 

and, for θ =45_: 

R cmax = 12.14 d install 

(92) 

These two values of R cmax can be seen in Figure 12: 12.14 in Figure 12a and 2.41 in 

Figure 12b. 

Also, in each particular case, there is an obvious minimum value for R c , which is the 

value of R c for the case where the maximum pressure, p max , is equal to the allowable 

pressure, p all . Therefore, from Equation 85: 

R cmin = T all ∕p max 

(93) 

In each case, the engineer designing a rounded corner should check that the radius, 

R c , obtained for the rounded corner is greater than the value of R cmin given by Equation 

93. In fact, the condition expressed by Equation 93 is identical to the condition expressed 

by Equations 68 and 69. This is illustrated in the following example. 



(a) 

R 

c 

/ dinstall 

(b) 


R 

c 

/ dinstall 


Figure 12. Radius of the rounded corner, R c , as a function of the distance between the 

geomembraneand thecorner atthe timeof installation,d install ,the allowablegeomembrane 

strain, ε all , and the interface friction angle, δ: (a) corner with exterior angle θ =45_; (b) 

corner with a right angle, θ =90_. 

(Notes: These graphs were established using Equations 86 to 89. The value of R c /d install for ε all =0is12.14 

for θ =45_ and2.41forθ =90_.) 


1003


Example 4. A 1.5 mm thick HDPE geomembrane is used to line a tank with 90_ corners 

(θ = π/2). The maximum applied liquid pressure, p max , the geomembrane liner will 

be subjected to in the tank is 80 kPa. The geomembrane has an allowable strain, ε all , 

of 5% at an allowable tension, T all , of 23 kN/m, and the interface friction angle between 

the geomembrane and the wall isδ =20_. The installer believes that it would be possible 

to install the geomembrane such that the maximum initial distance, d install , between the 

corner and the geomembrane would be 0.3 m. What is the radius R c of the rounded corner 

that is required to ensure that the allowable geomembrane strain and tension are not 

exceeded at the maximum applied liquid pressure? 

The radius R c can be calculated using Equation 87 as follows: 

R c = 0.3 (2 − π∕2) tan 20_ 

2 

2 

− 1 

tan 20 (2 − π∕2) tan 20 _ + 0.05 1+π = 0.47 m 

_ 

Alternatively, Figure 12b with ε all =5%andδ =20_ gives R c /d install ≈ 1.6, hence R c 

≈ (1.6) (0.3) ≈ 0.48 m. 

Then, it should be checked that this value is greater than R cmin , which is calculated 

using Equation 93 as follows: 

R cmin = 23∕80 = 0.29 m 

The fact that R c is greater than R cmin indicates that it is necessary to fill the corner (e.g. 

rounded corner, or chamfer). Alternatively, a methodical design engineer would have 

checked, before calculating R c , that filling the corner was necessary. This would have 

been done using Equation 69 as follows: 

p all = ( 2 − 1)(23) 

2 tan 20 

1 + 

0.05 

0.3 (2 − π∕2) tan 20_ 

1+π = 49 kPa 

The maximum pressure, p max = 80 kPa, being greater than the allowable pressure, 

p all = 49 kPa, it appears that filling the corner is necessary. 

The geomembrane as installed will reach the allowable strain, ε all = 5%, for a liquid 

pressure equal to the allowable pressure, i.e. 49 kPa. To increase the liquid pressure beyond 

that value, it is necessary to fill the corner. If this is achieved using a rounded corner, 

this rounded corner should have a radius of 0.47 m, as calculated above. This 

rounded corner is shown in Figure 13. 

The pressure applied by the liquid at the moment when the geomembrane comes in 

contact with the rounded corner is p all , which was calculated above or is alternatively 

given by Equation 85 as follows: 

p all = 23∕0.47 = 49 kPa 

_ 



0.72 m 

0.47 m 

Geomembrane 

as installed 

Rounded corner 

and position of the 

geomembrane at 

pressure p ≥ 49 kPa 

Figure 13. Rounded corner of Example 4. 

In this design example, the maximum expected pressure is 80 kPa. When this maximum 

pressure is applied, the pressure transmitted to the material (such as concrete) that 

fills the rounded corner is 31 kPa and the geomembrane is only subjected to 49 kPa. 

Therefore, the tension and strain in the geomembrane remain T all and ε all , respectively, 

regardless of the applied pressure beyond 49 kPa (assuming that no tension relaxation 

occurs with time). On the other hand, when the applied pressure is less than 49 kPa, the 

geomembrane does not touch the rounded corner. 

It should be noted that the radius of the rounded corner does not depend on the maximum 

pressure the geomembrane will be subjected to (e.g. in this design example, R c 

does not depend on 80 kPa). Therefore, the designed rounded corner is applicable to 

any liquid pressure in the considered tank, provided the geomembrane is installed, as 

assumed, 0.3 m from the original right-angle corner. 



1005


5.3 Chamfer 

Instead of constructing a rounded corner, it may be more convenient to construct a 

chamfer (Figure 1b). Like the rounded corner, the chamfer does not need to be in contact 

with the geomembrane at the time of geomembrane installation to be effective. As the 

liquid pressure increases, the geomembrane deforms from its initial position (Figure 

14a) until it reaches the chamfer (Figure 14b). During this first phase, the applicable 

equations are those related to the initial corner (i.e. the corner without chamfer). After 

(a) 

a install 

(b) 

(c) 

(d) 

Figure 14. Successive positions of the geomembrane in a right-angle corner with a 45_ 

chamfer: (a) geomembrane initial position (p = 0); (b) the geomembrane when it reaches 

the chamfer (p = p 1 ); (c) the geomembrane deforms toward the two angles of the chamfer 

(p = p 2 ); (d) the geomembrane resists the maximum pressure (p = p max ) and does not fail. 



the geomembrane has reached the chamfer, it deforms toward the two angles between 

the chamfer and the walls (points C and D in Figure 14c). Finally, either the geomembrane 

fails before the maximum pressure, p max , has been reached (which happens if the 

chamfer is too small) or the geomembrane is able to resist the maximum pressure, p max 

(which is the case if the chamfer has been properly sized) (Figure 14d). It is, therefore, 

necessary to determine the minimum size that the chamfer should have. The equations 

given below are for the case of a right-angle corner with a 45_ chamfer. 

When the geomembrane reaches the chamfer, its strain is ε c . The following relationship, 

derived from Equations 59 and 61, with d 1 = d install , d 2 = d c , ε 1 =0,andε 2 = ε c , 

exists between the strain ε c and the initial distance, d install , between the geomembrane 

and the corner without chamfer: 

d c 

= 

d install 

 

(2 − π∕2) tan δ 

(2 − π∕2) tan δ + ε c 

 

1+π 2 tan δ 

(94) 

From Figure 14b: 

a c − c = d c = c 2 

(95) 

Combining Equations 94 and 95 gives the following relationship between the chamfer 

size, c, and the strain in the geomembrane when it reaches the chamfer, ε c : 

c 

(2 − π∕2) tan δ 

= 

2 

dinstall 

 

(2 − π∕2) tan δ + ε c 

 

1+ π 2 tan δ 

(96) 

Then, as the liquid pressure continues to increase, an equation for 45_ exterior angles 

(θ = π/4) should be used. However, it should be noted that the length FG in Figure 14c 

is too short to fully provide for one of the two de terms in Equation 51 (i.e. if corner C 

in Figure 14c is considered, de E ≠ 0 and de F ≈ 0). Therefore, Equation 55 becomes: 

 

θ 

tan(θ∕2) − 2da = θa dε + de 

tan(θ∕2) 

(97) 

The solution of Equation 97 is similar to Equation 60, which is the solution of Equation 

55, except that tanδ is replaced by 2tanδ (as a result of the replacement of 2de by 

de, in accordance with the expression of de in Equation 56), hence: 

a 2 

a 1 

= 

2[2 tan(θ∕2) − θ]tanδ + ε 1 

2[2 tan(θ∕2) − θ]tanδ + ε 2 

 

1+2θ tan δ 

(98) 


1007


According to the way it has been generated, Equation 98 is applicable to corners such 

as C and D in Figure 14c where the geomembrane can move (interface friction angle 

δ) on one side of the corner, but is almost unable to move on the other side (which is 

approximated by δ =90_). In the case of a 45_ chamfer, the following equation is obtained 

with a 1 = a c - c (i.e. the value of a when the geomembrane reaches the chamfer, 

according to Figure 14b), ε 1 = ε c (i.e. the value of the strain in the geomembrane when 

it reaches the chamfer), a 2 = a min (i.e. the value of a that corresponds to the allowable 

strain in the geomembrane), ε 2 = ε all (i.e. the allowable strain), and θ = π/4: 

a min 

a c − c = 2[2 tan(π∕8) − π∕4] tan δ + ε c 

2[2 tan(π∕8) − π∕4] tan δ + ε all 

 

1+π 2 tan δ 

(99) 


T all 

p allc c(1 + 2 

= [4( 2 − 1) − π∕2] tan δ + εc 

∕2) 

[4( 2 

− 1) − π∕2] tan δ + εall 

1+ π 2 tan δ 

(100) 

It should be noted that the symbol p allc is used to differentiate the allowable pressure 

with a chamfer, p allc , from the allowable pressure without a chamfer, p all , which is less 

than p allc . (The symbol p allc was not needed in the case of rounded corners where p allc 

= ∞, asmentionedinSection5.2.) 

The next step consists of eliminating ε c between Equation 96, which is valid at the 

time when the geomembrane comes in contact with the chamfer, and Equation 100, 

which gives the relationship between the condition of the geomembrane when it reaches 

the chamfer (Figure 14c) and its condition when the allowable pressure is reached. 

Eliminating ε c requires lengthy calculations that lead to the following equation: 

c 

d install 

= m 1 T all 

p allc d install 

⎪ ⎪⎪ 

⎡ 

⎣ 

m 3 


m 4 T all 

1 

1+ π 2 tan δ − (ε all ∕ tan δ) − m 2 

m 5 

⎪ ⎪⎪ 

⎤ 

⎦ 

1+ π 2 tan δ 

(101) 

where m 1 ,m 2 ,m 3 ,m 4 and m 5 are numerical constants defined as follows: 

m 1 = 2 

∕(1 + 2) 

= 0.5858 

(102) 

m 2 = 4( 2 

− 1) − π∕2 = 0.08606 (103) 



(104) 

m 3 = 2 − π∕2 = 0.4292 

m 5 = 2(3 − 2 2 ) = 0.3431 (106) 

m 4 = 2 

− 1 = 0.4142 

(105) 

If there is perfect adhesion between the geomembrane and the wall, δ =90_, and 


c 

= expln 2 

d install 

+ 1 − π∕4 

3 − 22 

− 1 − π∕4 

T 

− 1 ln 

3 − 22 


πε all 

− 1 ln(1 + 2 ) − 

4(3 − 2 2 ) 

(107) 

It should be noted that lengthy analytical developments including complicated truncated 

series expansions are necessary to derive Equation 107 from Equation 101. Alternatively, 

Equation 107 can be established directly by solving Equation 97 with de =0 

and using Equation 63 instead of Equation 61 to establish an equation similar to Equation 

96 but with δ =90_. This direct method for establishing Equation 107 also requires 

lengthy analytical developments. 

The solution expressed by Equations 101 and 107 is not valid for c/d install greater than 

2 

because, in this case, the chamfer is in contact with the geomembrane at the time 

of installation (Figure 15a). Also, it is possible to consider a limit case when the developed 

length of the chamfer (ACDB in Figure 15b) is equal to the developed length of 

the geomembrane (arc AB in Figure 15b) if the geomembrane were installed in a corner 

without chamfer. This results in: 

c max = (2 + 3∕ 2 )(2− π∕2) dinstall = 1.769 d install 

(108) 

No chamfer greater than c max is needed, because, when c = c max , the geomembrane is 

totally supported as installed. 

A new solution should be developed for the cases where c/d install is greater than 2 

and less than 1.769. However, the solution expressed by Equations 101 and 107 is conservative 

when c/d install is greater than 2 

since this case corresponds to a geomembrane 

partly supported by the chamfer. Accordingly, graphs were established using Equations 

101 and 107 for 0 < c ≤ c max . These graphs are presented in the Appendix. The values 

of c given by these graphs are accurate for c ≤ 2 

d install and larger than the accurate 

values for 2 

d install < c ≤ c max . 

The minimum value, c min /d install , shown in the graphs was obtained as follows. Equation 

69 gives the relationship between d install , p all , T all ,andε all for the limit case where 


1009


(a) 

c = 2 d install 

(b) 

c max = 1.769 d install 

A 

A 

C 

d install 

d install 

D 

B 

B 

Figure 15. Chamfer: (a) limit case where the geomembrane is just in contact with the 

chamfer at the time of installation; (b) case of the “maximum chamfer”, where the length 

of ACDB is equal to the length of arc AB, i.e. the position where the geomembrane would 

be installed if there was no chamfer. 

the geomembrane does not need to be supported. In this case, p all = p allc . Combining 

Equation 69 and Equation 101, and performing calculations, give: 

c min 

 

= 2 

d install 1 + 2 

T all 


(109) 

Equation 109 is the equation of a straight line in the graphs given in the Appendix. 

This straight line extends from the origin of the axes to the point defined by the abscissa 

T all /(p allc d install )=1+ 2 

and the ordinate c/d install = 2 . No chamfer is needed below this 

straight line, which is a simple and important result. 

Example 5. The same case as in Example 4 is considered, but, instead of a rounded 

corner, a chamfered corner is designed. What chamfer dimension, c, is required to ensure 

that the allowable geomembrane strain and tension are not exceeded at the maximum 

applied liquid pressure? 

A first possibility consists of using the graph for δ =20_ given in the Appendix (Figure 

A-4). To use the graph, the following dimensionless parameter must be calculated: 

T all 


= 23 

(80) (0.3) = 0.9583 

The graph gives c/d install ≈ 1.18, hence c = (1.18) (0.3) = 0.35 m. 

Alternatively, Equation 101 can be used as follows: 



⎡ 

⎪ 

⎣ 

c 

d install 

= (0.5858) (0.9583) 

1 

1+ 

0.4292∕[(0.4142)(0.9583)] 

π 2 tan 20 _ 

− 0.05∕ tan 20_ 

− 0.08606 

0.3431 ⎪ ⎤ ⎦ 

1+ π tan 20_ 

2 

which gives: 

c 

d install 

= 1.176 

hence: 

c = (1.176) (0.3) = 0.35 m 

In Figure 16, this chamfer is compared to the rounded corner obtained in Example 

4 for the same tank (see Figure 13). 

0.47 m 

0.35 m 

Rounded corner 

(Example 4) 

Chamfer 

(Example 5) 

Figure 16. 

Example 5. 

Comparison between the rounded corner of Example 4 and the chamfer of 


1011


To be complete, it is necessary to check that the chamfer is required. This is done easily 

using the graph for δ =20_ given in the Appendix (Figure A-4) by checking that the 

point with abscissa T all /(p allc d install ) = 0.9583 and ordinate c/d install = 1.18 is above the 

straight line for c min /d install . This can also be done analytically by calculating c min /d install 

using Equation 109 as follows: 

c min 

 

= 2 

d install 1 + 2 

(0.9583) = 0.56 

hence: 

c min = (0.56) (0.3) = 0.17 m 

It appears that c = 0.35 m is greater than c min , which confirms that a chamfer is needed. 

A methodical engineer would have checked, before calculating c, that filling the corner 

was necessary. This would have been done using Equation 69 as follows: 

p all = ( 2 − 1)(23) 

1 + 

0.3 

2 tan 20 

0.05 

(2 − π∕2) tan 20 1+π = 49 kPa 

_ 

The maximum pressure, p max = p allc = 80 kPa, being greater than the allowable pressure 

(without chamfer), p all = 49 kPa, it appears that a chamfer is necessary. 

_ 


5.4 Comparison Between Chamfer and Rounded Corner 

A chamfer may be easier to construct than a rounded corner. This appears to be the 

only reason for preferring a chamfer to a rounded corner. 

Figure 16 shows the rounded corner of Example 4 and the chamfer of Example 5 that 

correspond to the same conditions. It appears that the area between the chamfer and the 

original right-angle corner (0.061 m 2 ) is 30% larger than the area between the rounded 

corner and the original right-angle corner (0.047 m 2 ). Therefore, the chamfer requires 

significantly more filling material, e.g. concrete, than the rounded corner. 

Furthermore, the chamfer is less effective than the rounded corner. As indicated in 

Section 5.2, a properly designed rounded corner is effective for every pressure. For example, 

the rounded corner shown in Figures 13 and 16 is effective for every pressure 

(see Example 4). In contrast, the chamfer shown in Figure 16 is effective up to 80 kPa 

only (see Example 5). In Example 5, to be effective for every pressure the chamfer size 

should be c max given as follows by Equation 108: 

c max = (1.769) (0.3) = 0.53 m 



The area between such a chamfer and the original right-angle corner is 0.14 m 2 which 

is nearly 200% more than the area between the rounded corner and the original rightangle 

corner. 

Clearly, the above discussion shows that a rounded corner is preferable to a chamfer. 

6 CONCLUSIONS 

This paper presents an analysis of the tensions and strains developed in a geomembrane 

liner subjected to a liquid pressure in the corner of a tank. The analysis leads to 

equations that can be used to evaluate if a given geomembrane can resist the liquid pressure 

in a tank corner and to determine whether or not the tank corners should be rounded 

or chamfered. For cases where a rounded corner or a chamfer is required, equations and 

graphs are provided to select the dimensions and geometry of the rounded corner or the 

chamfer. The determination of the required chamfer dimension is illustrated by a design 

example and is shown to depend on the following parameters: the allowable tension and 

strain in the geomembrane, the maximum pressure exerted by the liquid on the geomembrane, 

the interface friction angle between the geomembrane and the walls of the 

tank, the angle of the corner, and the initial distance between the geomembrane and the 

corner. The determination of the radius of a rounded corner is also illustrated by a design 

example. It appears that a rounded corner is more effective than a chamfer in preventing 

geomembrane failure in tank corners. The appropriate use of the equations presented 

in this paper should lead to safer design of geomembrane liner applications for tanks, 

which is important from a practical standpoint since a number of geomembrane failures 

have been observed in tank corners. 

From a theoretical standpoint, an important result has been demonstrated: the required 

interface friction angle between the geomembrane and the wall decreases as the 

liquid pressure increases and the geomembrane moves toward the corner. In other 

words, it has been demonstrated that the geomembrane does not slide with respect to 

the wall, because the increase in interface shear strength resulting from the increase in 

liquid pressure is greater than the increase in geomembrane tension due to the increase 

in liquid pressure. (It should be noted that these results were demonstrated with the assumption 

that the interface shear strength between the geomembrane and the wall is due 

to friction only, not adhesion, and that the interface friction angle does not decrease with 

time and/or as the pressure increases.) Another important theoretical result is that any 

geomembrane, regardless of its tensile characteristics, can reach the corner only if it 

undergoes an infinite strain, which indicates that the geomembrane would fail before 

reaching the corner. (It should be noted that this result is applicable to angular corners 

only; in the case of a properly designed rounded corner, the geomembrane does reach 

the corner without excessive strain.) 

The study presented in this paper was initiated in the early 1980s when the senior author 

analyzed failures of geomembranes in tank corners. The equations presented in this 

paper should help engineers prevent geomembrane failures. Also, the experience 

gained in solving the complex problems presented in this paper can be used to solve 

other problems involving geomembrane movements. 


1013


ACKNOWLEDGMENTS 

The authors are grateful to G. Saunders, A. Mozzar, and S.M. Berdy for their assistance 

in the preparation of this paper. 

REFERENCE 

Giroud, J.P., Beech, J.F. and Soderman, K.L., 1994, “Yield of Scratched Geomembranes”, 

Geotextiles and Geomembranes, Vol. 13, No. 4, pp. 231-246. 

NOTATIONS 

Basic SI units are given in parentheses. 

a = distance between the corner and the point where the geomembrane is in 

contact with the wall (Figures 5 and 6) (m) 

a c = value of a when the geomembrane first contacts the chamfer (Figure 14b) 

(m) 

a crit = critical value of a definedbyEquation30or31(m) 

a install = value of a at installation, i.e. when p =0(m) 

a max = value of a required to prevent geomembrane failure (m) 

a min = value of a when ε = ε all and T = T all (m) 

c = length of side of chamfer (Figure 14) (m) 

c max = maximum length of side of chamfer (Figure 15b) (m) 

c min = minimum length of side of chamfer (m) 

D = distance between two tank corners (m) 

D min = minimum required value of D to provide enough space for L F (m) 

d = distance between the geomembrane and the tank corner along the 

bisector, CCi, of the interior angle of the corner (Figure 5) (m) 

d c = dimension of chamfer along the bisector of the interior angle of the corner 

(Figure 14) (m) 

d install = value of d at installation, i.e. when p =0(m) 

d max = value of d required to prevent geomembrane failure (m) 

e = elongation of the geomembrane (m) 

F AB = force per unit height exerted by the pressure p on arc AB (Figure 5) (N/m) 

g = acceleration of gravity (m/s 2 ) 

h = height of liquid above the considered point of the geomembrane (m) 

J = geomembrane tensile stiffness (N/m) 

L = length of arc AB, i.e. length of the unsupported portion of the 

geomembrane, when the pressure is p (Figures 5 and 8) (m) 



Li = length of AB when the pressure is p +dp (Figure 8) (m) 

L AB = length of chord AB (Figure 5) (m) 

L F = length over which the geomembrane/tank wall interface friction is 

developed (Figure 6) (m) 

L Fmax = maximum required value of L F (m) 

p = pressure exerted by the liquid on a given point of the geomembrane (Pa) 

p all = allowable pressure, i.e. the liquid pressure that causes the 

geomembrane strain to be equal to the allowable strain and the 

geomembrane tension to be equal to the allowable tension (Pa) 

p allc = allowable pressure in the case of a chamfered corner (Pa) 

p max = maximum pressure expected in a tank (Pa) 

R = radius of curvature of the geomembrane (Figure 5) (m) 

R c = radius of curvature of a rounded corner (m) 

R cmax = maximum radius of curvature of a rounded corner (m) 

R cmin = minimum radius of curvature of a rounded corner (m) 

T = tension in the unsupported portion of the geomembrane (N/m) 

T all = allowable geomembrane tension (N/m) 

T M = geomembrane tension at midway between two identical tank corners 

(Figure 6b) (N/m) 

T x = geomembrane tension at abscissa x (Figure 6) (N/m) 

x = horizontal distance along geomembrane (Figure 6) (m) 

δ = interface friction angle between geomembrane and tank wall (_) 

δ conserv = value of δ used in a special case and defined by Equations 65 and 66 (_) 

ε = strain in the unsupported portion of the geomembrane (dimensionless) 

ε all = allowable geomembrane strain (dimensionless) 

ε c = strain in the geomembrane when it comes in contact with a chamfer 

(dimensionless) 

ε cor = strain in the geomembrane when the geomembrane reaches the corner 

(dimensionless) 

ε x = geomembrane strain at abscissa x (dimensionless) 

θ = exterior angle of tank corner (Figures 5 and 7) (radians or _) 

ρ = density of the liquid (kg/m 3 ) 

τ = interface shear stress between the geomembrane and the tank wall (Pa) 

ω = interior angle of tank corner (Figure 7) (radians or _) 


1015


APPENDIX 

The following are design graphs for 45_ chamfers in right-angle corners (Figure 14). 

These graphs were established using Equations 101 and 107. 

δ =5_ 

c/d install 

2 

ε all =1% 

ε all =5% 

ε all =0% 

c min /d install 

ε all = 20% 

ε all = 50% 

ε all = 10% 

T all /(p allc d install ) 

Figure A-1. 

Chamfer design graph for interface friction angle δ =5_. 

δ =10_ 

2 

c/d install 

ε all =5% 

ε all =1% 

ε all =0% 


ε all = 20% 

ε all = 50% 

ε all = 10% 


Figure A-2. 




2 

δ =15_ 

ε all =0% 

c/d install 


ε all =1% 

ε all =5% 

ε all = 10% 

ε all = 50% 

ε all = 20% 


Figure A-3. 


c/d install 

2 

ε all = 20% 

ε all = 50% 

δ =20_ 

ε all =0% 

ε all =1% 


ε all =5% 

ε all = 10% 


Figure A-4. 



1017


c/d install 

2 

ε all = 10% 

δ =30_ 

ε all =0% 

ε all =1% 


ε all =5% 

ε all = 50% 

ε all = 20% 


Figure A-5. 


c/d install 

2 

ε all =5% 

ε all = 10% 

ε all = 20% 


ε all = 50% 

δ =90_ 

ε all =0% 

ε all =1% 


Figure A-6.

Technical Paper by JP Giroud, KL Soderman, T. Pelte and JF Beech

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