Technical Paper by J.H. Greenwood - IGS - International ...
Technical Paper by J.H. Greenwood - IGS - International ... Technical Paper by J.H. Greenwood - IGS - International ...
GREENWOOD D Designing to Residual Strength Instead of Stress-Rupture where: a = crack length; A = a constant; n = index; and, K = stress concentration factor which is equal to σYa ½ for simple geometries, where σ = applied stress and Y =geometrical factor (Evans and Wiederhorn 1974). For any value of crack length, a, thereisa corresponding residual strength σ t with an initial value of σ 0 , corresponding to a critical value of K=K IC . By expressing σ t as a function of time and integrating, it can be shown that σ t is given by the following expression: σ n−2 t = σ n−2 0 − σ n t B where B =2/((n - 2) AY 2 K n−2 IC ), omitting any effects of crack growth during tensile testing. The stress-rupture curve is given by setting σ = σ t which requires solving the following equation: (2) σ n−2 t = σ n−2 0 − σ n t t B Since n is usually large and σ 0 > σ t , Equation 3 can be approximated using the following expression: σ n−2 0 = σ n t t B which plots as a double logarithmic stress-rupture diagram, and has a gradient of -(1/n). As mentioned above, for polyester the stress-rupture gradient is of the order of -1/40 and thus n is equal to 40. Thus, if the values σ 0 =1,B =10 - 6 and n = 40 are substituted into Equation 2, then for an applied load of σ = 0.6 (i.e. 60% of tensile strength), the time to failure is 748 hours. After 720 hours, the residual strength is 91.7%, after 730 hours it is 90.6%, and after 740 hours it is 88.7%, before reducing to 60% at failure. This example illustrates how a stress-rupture curve with a low gradient ( -1/n), as observed in geotextiles, may be associated with a high residual strength over most of the design life. This index implies that the majority of crack growth takes place just before failure, and thus the residual strength remains close to the original tensile strength. A complete model would have to incorporate the processes of primary and secondary creep, with decreasing and constant strain rates, respectively, as well as the chemical aspects of the fracture of polymer chains (Horrocks 1996). (3) (4) 5 REVISED DESIGN PROCEDURE 5.1 Residual Strength Due to Sustained Load Alone It is proposed that instead of using the stress-rupture curve directly, the design strength could be based on the residual strength curve, T R , corresponding to the anticipated design strength, T D .AvalueofT D must be found such that at the end of the design life, a factor of safety, f m , will remain between the design strength and the residual or available strength, T RD , such that T D =T RD /f m (Figure 5). In the absence of an analytical 6 GEOSYNTHETICS INTERNATIONAL S 1997, VOL. 4, NO. 1
GREENWOOD D Designing to Residual Strength Instead of Stress-Rupture Stress-rupture curve Residual strength curve at T = TD Residual strength at t = t d Load, T T CR T D Unfactored strength Design strength or applied load T RD Log t (time) Figure 5. Redefinition of T D such that T D =T RD /f m . Design life, t d function to describe the residual strength curves, an iterative process will be required to derive T D . 5.2 Residual Strength Due to Sustained Load, Mechanical Damage and the Environment A procedure based on residual strength has the further advantage that it can easily incorporate degradations due to other effects. Figure 6 shows the residual strength due to sustained load, mechanical damage and environmental effects individually. If all of the effects - load, mechanical damage and environmental - apply simultaneously but without any synergistic effects, such as environmental stress cracking, the degradation curves expressed as fractions of the initial tensile strength may be multiplied to provide a set of reduced residual strength curves as shown in Figure 7. The design strength T D may then be calculated such that, at time t d , T RD /T D =f m in the same manner as shown in Figure 4 but using the reduced curves. In dealing with stress-rupture or residual strength, it is important to recognise that where the geosynthetic reinforcement is under variable load along its length (e.g. due to interlock with the soil) one should always consider the point of maximum load, as this is the point where the geosynthetic will break. However, when calculating the total creep strain, it is necessary to calculate the distribution of load arising from an applied load T D along the whole load-bearing length of the geosynthetic reinforcement, taking into account any reduction due to a transfer of load from the geosynthetic reinforcement to the soil. From this it is possible to derive the total time-dependent strain, which may then be compared with the criterion for serviceability or maximum strain. GEOSYNTHETICS INTERNATIONAL S 1997, VOL. 4, NO. 1 7
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GREENWOOD D Designing to Residual Strength Instead of Stress-Rupture<br />
Stress-rupture curve<br />
Residual strength curve at T = TD<br />
Residual strength at t = t d<br />
Load, T<br />
T CR<br />
T D<br />
Unfactored strength<br />
Design strength or applied load<br />
T RD<br />
Log t (time)<br />
Figure 5. Redefinition of T D such that T D =T RD /f m .<br />
Design life, t d<br />
function to describe the residual strength curves, an iterative process will be required<br />
to derive T D .<br />
5.2 Residual Strength Due to Sustained Load, Mechanical Damage and the<br />
Environment<br />
A procedure based on residual strength has the further advantage that it can easily<br />
incorporate degradations due to other effects. Figure 6 shows the residual strength due<br />
to sustained load, mechanical damage and environmental effects individually. If all of<br />
the effects - load, mechanical damage and environmental - apply simultaneously but<br />
without any synergistic effects, such as environmental stress cracking, the degradation<br />
curves expressed as fractions of the initial tensile strength may be multiplied to provide<br />
a set of reduced residual strength curves as shown in Figure 7. The design strength T D<br />
may then be calculated such that, at time t d , T RD /T D =f m in the same manner as shown<br />
in Figure 4 but using the reduced curves.<br />
In dealing with stress-rupture or residual strength, it is important to recognise that<br />
where the geosynthetic reinforcement is under variable load along its length (e.g. due<br />
to interlock with the soil) one should always consider the point of maximum load, as<br />
this is the point where the geosynthetic will break. However, when calculating the total<br />
creep strain, it is necessary to calculate the distribution of load arising from an applied<br />
load T D along the whole load-bearing length of the geosynthetic reinforcement, taking<br />
into account any reduction due to a transfer of load from the geosynthetic reinforcement<br />
to the soil. From this it is possible to derive the total time-dependent strain, which may<br />
then be compared with the criterion for serviceability or maximum strain.<br />
GEOSYNTHETICS INTERNATIONAL S 1997, VOL. 4, NO. 1<br />
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