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A Contributon to the Analysis of Geotechnical
Senior Assistant, Geotechnical Department, University of Zagreb, Croatia Assistant, University of Zagreb, Varaždin, Croatia and Franjo Veric Professor, Geotechnical Department, University of Zagreb, Croatia |
ABSTRACT
In this paper a numerical procedure is presented which allows a safe and quick assessment of geotechnical anchor ultimate capacity, obtaining the distribution of all important statical values along the fixed anchor length (force, bond stresses, displacements). The back analysis of the results of measuring the distribution of strains and forces along the anchor was done in order to optimise the choice of anchor zone length. The results of the parameter analysis are meant to show the importance of the correct choice of the bond stress-displacement diagram to achieve the correct value of the anchor ultimate capacity.
KEYWORDS: geotechnical anchor, ultimate capacity, bond stress, displacement, peak and residual shear strength, yield
INTRODUCTION
Ultimate capacity of the geotechnical anchor depends on the activated bond stresses between the anchor body and surrounding ground. Deformation characteristics of wires, injection mass and surrounding ground dictate the displacement distribution, and thus, the bond stresses along the fixed anchor zone.
Regular assumption in everyday engineering practice [1] is that the contact bond stresses are even. In this case, the change of the fixed anchor length results in the linear change of anchor ultimate capacity. Experimental test results show this is not the case [5]. Generally there is a some distinction beetwen short and long anchor. Shorter anchors have much greater usability, i.e. the mean value of bond stresses on short anchor envelopes is greater in comparison to the longer anchor zones for the same anchor diameter and process technology.
The results of experiments conducted by Ostermayer and Scheele [2] in medium and well compacted sands show that the distribution of bond stresses is highly nonuniform (Figure 1).
Figure 1. Distribution of a) Tensile load, and b) Bond stresses along the anchor
(Ostermayer and Scheele, 1978)
For the small values of the forces (in relation to the anchor ultimate capacity), the ultimate bond stress is at the proximal end of the fixed anchor. By gradual increase of external tensile force, the position of ultimate bond stress is translated toward the distal end of the anchor.
The mentioned distribution of bond stresses can be explained using the bond stress- displacement relationship, which is reliable when assessing the anchor body-surrounding ground contact. Two types of curves are characteristic (Figure 2): one with only the peak value of bond stress (u, which does not decrease with the increase of displacement, and the curve with both peak and residual values of bond stress tr. In the former case, the maximum resistance force is gained by reaching the limit value of bond stress. The problem, however, is in activating the displacement, which can reach values too large even before reaching the ultimate bond stresses along the whole fixed anchor. When dealing with curves of pronounced peak and residual values of bond stresses, the ultimate resistance force is proportional to the maximum value of the surface area beneath the curve t-u (Figure 3.) In this case Ostermayer and Scheele [2], Sommerville [4], et.al. have shown that larger displacement on the contact of anchor zone with the surrounding ground results in lesser bond stresses on residual values.
Figure 2. Characteristic t-u curves on the contact of the fixed anchor body with surrounding ground
Figure 3 shows how the boundary anchor displacements (the proximal end in Point 1 and distal end in point 2 of the fixed anchor length) determine the anchor ultimate capacity, but also the distribution of bond stresses along the entire anchor zone. The range of proximal and distal end will depend, the force being the same, on the anchor stiffness and the t-u relationship. The problem is finding such a pair of boundary displacements which will give the ultimate force of the anchor.
Figure 3. Displacements and bond stresses on the boundaries of the fixed anchor zone
The problem of calculating the distribution of bond stresses along the fixed anchor zone can be solved using various numerical procedures. One of these is the finite elements method using the interface elements on the contact part of the anchor body and the surrounding ground [6]. Such elements have in themselves embedded various constitutive relations which allow solving a wide spectre of problems. Using the method of finite elements, it is possible to analyse the activation of bond stresses under various values of external loads, resulting in the force-displacement curves at the proximal end of the anchor, and the displacement analysis along the anchor length. However, such sophisticated analyses have not yet become a practical solution for problems in everyday engineering practice. The proposed process of calculating the force distribution, displacement, and bond stresses along the fixed anchor length will be presented, which can fill the void between the rough calculations, based on crude empirical knowledge, and sophisticated analyses, for which one must have the appropriate exactness of input parameters.
PROPOSED NUMERICAL PROCEDURE
The basic assumptions of the proposed procedure are the following.
- The tendon and surrounding cement injection mixture work uniquely, without relative displacements, which means the surrounding mixture deforms elastically, just like iron, with possible appearance of radial micro-cracks in the mixture, but without influencing the distribution of stresses and displacements.
- There is a defined cylindrical body of injection mixture which is in contact with surrounding intact ground.
- At the contact of the cylinder with the surrounding ground, where a normal pressure acts, the relationship t-u is known.
The unknowns in this model are defined as three statical values: force, displacement, and bond stress along the fixed anchor zone (Figure 4). Based on the mutual dependency of each of the three statical values, it is possible to construct the equations to be solved by iteration in the finite number of points.
Figure 4. Discretization of fixed anchor length with statical unknowns
Equations (1) and (2) can be written for point i of anchor zone for relations between the unknown values (according to Figure 4):
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(1) |
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(2) |
Equation (1) represents the magnitude of the force in point i after the bond stresses have been activated on the interface of the anchor zone to the point in question. The left part of equation (2) is the condition of deformation, while the entire equation (2) represents the condition of compatibility of the displacement of the tendon and the anchor zone envelope.
Values Ki are secant modules in diagram t-u which is showed in Figure 5. In the first iteration step, the values Ki are equal for all points, and meant as a tangent module for the nil-displacement on the given curve t-u. In all the other steps, they are equal to the values reached in the previous step of the calculation.
Figure 5. Secant moduli in curve t-u
When values Ki are known the problem can be reduce to determining the force vector along the anchor zone via the relations given by (1) and (2). Using (2), the bond stresses (i can be expressed as a function of unknown forces Fj, where j=n, …, i. It follows for point i:
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(3) |
If the following sub-equation is added: (inline-eq.gif), equation (3) can be written as:
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(4) |
The procedure analogue to the one just shown can be done for all discrete points, and the system of algebraic equations obtained by this method can then be shown as a matrix:
| [B] {p} = {f} | (5) |
where
[B] is a non-dimensional matrix
{p} is the vector of tensile forces along the anchor Fi, i = 2, … n,
{f} is the vector of external tensile forces.
Values for forces F2 to Fn along the anchor are obtained by solving the system (5), which, along with the known forces F1 = F give the distribution of external force on the fixed anchor. When external forces are also known, equation (2) is used to calculate the displacement along the anchor. Knowing the displacement values allows us to calculate bond stresses via the adopted relation t-u, and the values of new secant modules Ki, which we use for further calculations. Iteration for the increment of load force ends when the difference between two consecutive steps is smaller than that of the given criterion.
The calculations of the proposed procedure:
- 1) defining input values d, E, A, L, Dx, F, DF,
- 2) defining the function t-u,
- 3) adding the load increment,
- 4) iterative procedure for calculating forces, displacements, and bond stresses,
- 5) back to 3),
- 6) ending the calculation after the last load increment.
-U RELATIONSHIP
A relatively complicated relationship between bond stresses and displacements is approximated linearly by segments (Figure 6), where three characteristic points can be noticed: u1 - displacement resulting in reaching the peak bond stress value tu, u2 - start of yielding, and u3 - displacement resulting in reaching the residual value of bond stress tr.
Figure 6. Calculatory relation between bond stresses and displacements
Measured Distribution of Force and
Bond stresses along the Anchor
The results of measurement analysis of force value and displacement along the fixed anchor length of the diaphragm wall used for erecting the Branimir Centre in Zagreb, 2001 [3] are shown in order to explain the proposed procedure.
Figure 6 a) shows the schematics of the measuring chain and the positions of tensometre pairs along the anchor. Figure 7 b) shows the cross-section of strain gauges positions.
Figure 7. Position of measuring chains a) and cross-section of strain gauges positions b)
Testing the anchor continued to the limits of anchor ultimate capacity. The anchor was a BBRV CONA-Multi CM 706 (7Ø0.6") with 8.0 m fixed and 9.0 m free anchor length. Anchoring is performed in a layer of clay gravel GC. Calculated diameter of the anchor was d = 0.16 m, the tendon area A = 971 mm2, elasticity modulus E = 1.985x108 kN/m2. Figure 8 shows the values of force along the anchor obtained using mean values of measured deformations along the anchor. Figure 9 shows the values of distributions for displacements according to Figure 8. Figure 10 shows the distribution of bond stresses on the envelope of fixed anchor obtained using the results of force distribution along the anchor.
Figure 8. Distribution of force along the anchor
It is apparent from looking at Figure 8 that gradually increasing external tensile force brings on the change in line slope along the anchor. This means the values of bond stresses along the anchor will be changing.
Figure 9. Distribution of displacements along the anchor plate
Figure 10. Distribution of bond stresses along the anchor
Figure 10 shows the maximum value of bond stresses translated toward the distal end of the anchor. In this way the stronger force means a gradual widening of the zone with higher values of bond stresses along the anchor. Distribution of bond stresses is visibly nonuniform. Activation of the entire anchor length is possible at a relatively high force with value 401.9˜0.7x577.3. When using smaller value of the force, the anchor with observed length of anchor zone of 8.0 m is effectively unloaded.
Using Figure 9 and Figure 10 it is possible to arrange the pairs of displacements and bond stresses in observed anchor points, as is shown on Figure 10.
Figure 11. Pairs t-u along the anchor and linear approximation
Back analysis (Figure 11) gives the following values of curve t-u: (u = 250 kN/m2, (r = 25 kN/m2, u1 = 0.035 m, u2 = 0.040 m and u3 = 0.13 m. Linear interpolation for defining the final calculations diagram was used between these values.
Results of Parametric Analyses
When the rule for activation of bond stress as a displacement function is known, the analysis of the optimal choice of anchor length can be made. By varying lengths of anchor zone, it is possible to determine the anchor ultimate capacity and give recommendations about possible changes to anchor length. Table 1 gives ultimate forces values of anchor for various anchor lengths. Average values of bond stresses (m on the anchor zome for ultimate forces Fu are also shown.
Table 1. Forces of anchor ultimate capacity and
average bond stresses for various anchor lengths
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| L [m] | Fu [kN] | (m [kN/m2] |
| ||
| 8.0 | 577 | 143 |
| 7.5 | 569 | 151 |
| 7.0 | 564 | 160 |
| 6.5 | 555 | 170 |
| 6.0 | 542 | 180 |
| 5.5 | 525 | 190 |
| 5.0 | 502 | 200 |
| ||
Table 1 shows there is no drastic lowering of anchor ultiamte capacity if the anchor gets shortened, while the average values of bond stress change considerably. This anchor behaviour is a result of numeric value of residual and peak values tr/tu , in this case equalling 25/250 =0.10. Shorter anchor results in higher relative ratio of anchor part with ultimate value of bond stress and total length of anchoring. For higher values of quotient tr/tu there will be a higher decrease of ultimate anchor force for equal shortening of anchoring length, because of lower average bond stresses.
Figure 11 shows parameters encompassed by multi-parameter analysis. Calculation of anchor ultimate capacity force (Fu) was done for various lengths of anchoring and various values of quotient tr/tu = 0.05, 0.25, 0.5, and 0.75 with the same value of peak shear strains (u = 250 kN/m2 and with the value of displacement u1, u2, and u3 as in the back analysis.
Figure 12. Lenghts of fixed anchor zone and t-u diagrams used in parametar analysis
Figure 13 shows relative values of anchor holding capacity forces divided by the force obtained for the greatest anchoring length (L0 = 8.0 m).
Figure 13. Anchor ultimate capacity as a function of anchoring length and the ratio tr/tu
The dashed line in Figure 13 represents directly proportional relationship of anchor lengthening and anchor ultimate capacity reached when tr/tu = 1.0. The lower the ratio tr/tu, the higher the effect of shortening the length of anchoring zone. For example, for tr/tu = 0.05 shortening of anchoring length to 75% of starting length of anchor results in less than 5% lower anchor ultimate capacity. In this way, curves with smaller residual values of holding capacity have a significantly milder impact on shorter anchors. These same reasons hold when values of average bond stresses are raised when using shorter anchors. Figure 14. shows relative bond stresses for various anchoring lengths. The normalization of the bond stresses is done using the value of mean bond stress (m for the shortest anchor (L'0 = 6.0 m).
Figure 14. Normalized mean bond stresses in function of normalized anchor lengths related with tr/tu
Figure 14 clearly shows how incorrect it can be to assume the same value of bond stress for different lengths of anchoring zone. The values of mean bond stresses decreases when the length of anchoring zone rises. The smaller the ratio tr/tu the greater the decrease.
CONCLUSIONS
Distribution of bond stresses along the anchoring section is highly nonuniform, which applies to all values of external tensile forces. There is a special case when bond stresses can be uniform for the t-u relationship with only the peak value of bond stress showing; for force equalling the anchor ultimate capacity. Deviation from uniform distribution grows with the length of anchoring zone and the difference between peak and residual values of bond stresses in calculation curve tu.
Generally qualitative form of external tensile force distribution, but also displacements along the anchoring section, does not significantly change with the increase of external force for various anchoring lengths. Maximum force and displacement are activated at the proximal end of anchor, and gradually decrease toward the distal end because of bond stresses having an effect on the contact of injection mass with the surrounding ground. At small external forces (in relation to the anchor ultimate capacity), force is exerted only on a part of anchoring zone. Increasing force on the limits of anchor ultimate capacity results in activation of whole anchoring section along with displacement of bond stresses toward the distal end of the anchor.
The results of back analyses show the importance of correct assessment of relations t-u on the contact of anchor body with surrounding ground. Using the regular assumption of uniform bond stresses along the anchoring section often leads to irrational solutions, especially when using long anchors. In such cases there is an activation of residual shear strains along a sizable portion of the anchor, and the value of mean bond stress is lower than in the case of shorter anchoring section.
REFERENCES
1. Littlejohn, G.S.: Design Estimation of the Ultimate Load-Holding Capacity of Ground Anchors, Ground Eng. Found. Publ., Essex, England, 1980.
2. Ostermayer, H.; Scheele, F.: Research on Ground Anchors in Non-Cohesive Soils, Revue Francaise de Geotechnique 3 (1978), 92-97.
3. Results of measurements along the anchor, Project of safety construction pit for the Branimir centre - garage and shopping centre, Internal archives of the Geotechnical Institute, Faculty of Civil Engineering, Zagreb, 2001.
4. Somerville, M.A.: A Design for Inclined Ground Anchor Fixed Length in Cohesionless Soil, Ground Eng., 14 (1981) 2, 26-28.
5. Weerasinghe, R.B.; Littlejohn, G.S.: Load Transfer and Failure of Anchorages in Weak Mudstone, Ground anchorages and anchored structures, Proc. of the Int. Conf., Institution of CE, Thomas Telford, London, 1997, 23-24.
6. Woods, R.I.; Barkhordari, K.: The Influence of Bond Stress Distribution on Ground Anchor Design, Ground anchorages and anchored structures, Proc. of the Int. Conf., Institution of CE, Thomas Telford, London, 1997, 55-65.
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