Optimization of soil anchorages M. Teschner, C. Mattheck Kernforschungszentrum Karlsruhe GmbH, Institut fur Materialforschung II, W-7500 Karlsruhe 1, Postfach 3640, Germany ABSTRACT A new optimization method for soil anchorages is presented, which is based on the SKO method (Soft Kill Option) and was developed at the Karlsruhe Nuclear Research Centre. With the aid of this method, which simulates the mechanical growth of tree roots, the designer is able to increase the loading capacity under the demand of a lightweight design. The layout procedure considers the special failing behaviour of soils. The steps of the optimization algorithm are presented and also the necessary fundamental principles of soil mechanics are given. The optimization method applies the mechanical growth of tree roots into the field of anchorage design. INTRODUCTION One possibility to increase the loading capacity of soil anchorages is the optimal shaping of the anchor geometry. Tree roots for example are biological soil anchorages which achieve a maximum loading capacity with a minimum of material. This quality is a result of the tough competition in nature for energy and living space. The success and the efficiency of evolution is also demonstrated in the design of biological load carriers. The mechanical rule
4 Optimization of Structural Systems which determines the design of biological load carriers is the axiom of constant stress [1]. Biological load carriers tend to grow into a state of even stress distribution on the surface on a time average [2]. Soil anchor optimization by SKO The presented optimization method for soil anchorages is based on the SKO method [3]. The main difference lies in the choice of the equivalent stress. With the SKO method an optimum structure topology can be found by simulating adaptive bone mineralization. Bones are well optimized biological load carriers which are adapted to the actual load case by shape optimization and adaptive mineralization. Adaptive mineralization means that areas subjected to higher stresses become stiffer than parts subjected to lower loads. This can easily be simulated by varying the Young's modulus according to the stresses. The failure behaviour of soils is quite different from other materials. Problems of soil mechanics are often treated as problems of plasticity. The soil is replaced by an idealised material which is assumed to be perfectly plastic. This means that the material behaves elastically up to some state of stress at which yielding occurs. The failure of the soil depends on the shearing stresses in the soil. The Mohr-Coulomb hypothesis [4] is a good description of the failure, which gives the relation between the shear stress and normal pressure on the slip surface. In the Mohr-Coulomb rule the shear strength depends linearly upon the normal pressure on the slip surface (see Fig. 1). This linear function is defined by cohesion c and the friction angle <p. The cohesion c is the shear strength when the normal stress to the plane is zero. The friction angle <p is the gradient of the straight line in the Mohr- Coulomb hypothesis. The shear strength of the soil is defined by the formula: Tf =c +a tan<p (1) Here, a is the normal pressure on the slip surface.
Optimization of Structural Systems Fig. 1: Mohr-Coulomb hypothesis (positive in pressure) If the limit load is reached, the Mohr-Coulomb rule can be written as: (Oi -o,) +((?i +cr,)sin<p -2ccos<? =0 (2) where a\ and 03 are the maximum and the minimum principal stresses. The outof-plane stress is ignored in this model. To characterise failure in the multiaxial state of stress the introduction of a yield function is necessary. Drucker and Prager [5] suggested a yield function which is a good generalisation of the Mohr-Coulomb hypothesis: f = (3) Here, a and k are positive constants, Jj is the sum of the principal stresses and J2 is the second invariant of the stress deviation. In case of plane strain the yield function (3) can be reduced to the Mohr-Coulomb rule. The following relations between a, k and the constants of the Mohr-Coulomb rule can be found as: a = k =cvl -12or
6 Optimization of Structural Systems Fig. 2 shows the yield function in case of plane stress for several a-values (positive in pressure). For a = 0 the yield surface f = k is identical with the Mises criterion. 0=0.20 Fig. 2: Yield function in case of plane stress With knowledge of the state of stress in the soil and the equivalent stress, areas in which failure occurs can be found. The SKO method can be subdivided in the following steps: 1. Calculation of the equivalent stresses in the soil (Drucker Prager model) for a constant Young's modulus over the whole soil area, where the stresses are induced by the anchor forces. 2. Variation of Young's modulus as a function of the equivalent stresses in the soil. The FEM (Finite Element Method) code ABAQUS [6] has an option to
Optimization of Structural Systems 7 define the Young's modulus as a function of the temperature. To use this option, the stresses in the soil have to be expressed as a distribution of temperatures. This is only a help and is of no physical importance. The new Young's modulus is then varied between E^m (Young's modulus of the soil) and Emax (Young's modulus of the anchor) according to the following equation: -a.,) (4) Areas of the soil where the limit load is reached become stiffer than parts subjected to lower stresses. Consequently the Young's modulus of regions of the soil where slip occurs are set to the maximum Young's modulus and become a part of the anchor. The shape of the anchor is therefore extended by the areas of the soil where failure occurs. 3. Here, the equivalent stresses in the soil are calculated with the new Young's modulus distribution by FEM. Steps 2 and 3 are repeated until there is no significant change in the Young's modulus distribution. The new anchor layout is the area with a maximum Young's modulus in the soil. Fig. 3 shows the procedure of the modified SKO method.
Optimization of Structural Systems DESIGN PROPOSAL 1 i 1 aoas of symmetry i f V i CONSTANT YOUNG'S MODULUS FOR SOIL AND AhJCHOR FE CALCULATION OF EQUIVALENT STRESSES FOR THE SOIL ^ VARIATION OF THE YOUNG'S MODULUS AS A FUNCTION OF THE EQUPS/ALhNI SIRhSShS *" V ELASTIC FE RUN WITH THEE NEW YOUNG'S MODULUS DISTRII3UTION i ITERATION CYCLES FINAL DESIGN «# A! if new anchor Fig. 3: SKO procedure
Example of an anchor optimization Optimization of Structural Systems 9 The presentation example is a pipe anchor (see Fig. 4), for which a two dimensional Finite-Element model was built (with 4 node, linear displacement elements), using symmetry in geometry and loading. I Or soil A F anchor- <l Fig. 4: Model of pipe anchor with boundary and loading conditions. (half structure modeled only!) The following material parameters were chosen: ANCHOR; Young's modulus =70000 MPa poison's ratio v y=0.3 SOIL: Young's modulus E=200 MPa poison's ratio v y=0.3 cohesion c c=0.02 MPa
10 Optimization of Structural Systems friction angle y? <p=15 For linear elastic material behaviour the Drucker equivalent stress was calculated. The friction between anchor and soil is assumed to be perfect. Fig. 5 shows the stresses in the soil with the maximum at the lateral vertex of the pipe. With this calculated stress distribution the SKO method starts to vary the Young's modulus as a linear function of the Drucker stresses. The Young's modulus increases where high stresses are calculated and decreases where low stresses are calculated..317e-1.278e-1.238e-1.199e-1.160e-1.12be-1.809e-2.416e-2 Fig. 5: Equivalent stress distribution (Drucker) After 25 iteration cycles the new anchor geometry is found. Fig. 6 shows the new Young's modulus distribution calculated by SKO. A clear distinction be-
Optimization of Structural Systems 11 tween regions of high and low Young's modulus can now be made. The light area has a low Young's modulus while the dark part has the Young's modulus of the anchor. new anchor Fig. 6: New anchor layout To investigate the success of the optimization, three soil anchors (see Fig. 7) have been tested with a tensile testing machine. The anchor at position a) is a good approximation of the optimized anchor design. The three soil anchors, which have the area projection, are installed in a receptacle with sand and then loaded by the testing machine. Fig. 8 shows the loading capacity for the different soil anchor types. The triangle pipe has the best loading capacity with
12 Optimization of Structural Systems 50 % more than the plate and 30 % more than the pipe. The conical anchor leads to a better lateral compression of the soil than the pipe and the plate. The larger failure area which is developed by the triangle, leads to a decreasing of the volume of the pulled out soil. The result is a better anchorage. r a) b) c) Fig. 7: Cross sections for the different anchors a) TRIANGLE PIPE b) PIPE c) PLATE Fig. 8: Loading capacity of the testet soil anchors
CONCLUSIONS Optimization of Structural Systems 13 The use of the optimization method SKO (adapted for soil problems) in the design process for soil anchors, leads to a significant increasing of the loading capacity. SKO creates new design proposals for different soil types and loading conditions under the demand of a lightweight design. Experimental investigations confirm the success of the optimization. The optimized anchor has a loading capacity with 30 % more than the pipe (design proposal) and 50 % more than the plate with same area projection. SKO works well for two and threedimensional problems. The SKO method can be applicate to the most commercial Finite Element codes. References 1. Mattheck, C, 'Design in der Natur* (German), Rombach Verlag Freiburg 1992 2. Mattheck, C, Trees- the mechanical design', Springer Heidelberg, New York 1991 3. Baumgartner, A., Harzheim, L., Mattheck, C, 'SKO: Soft Kill Option. The biological way to find an optimum structure topology*, Int. J. Fatigue 14 No 6, 1992, pp 387-393. 4. Lang, H.J., Huder, J., 'Bodenmechanik und Grundbau.' Chapter 7, Festigkeitseigenschaften der Boden', Vol.4, pp. 69-84, Springer- Verlag, Berlin Heidelberg, 1990 5. Drucker, DC and Prager, W., 'Soil Mechanics and Plastic Analysis or Limit Design', Quarterly of Applied Mathematics, Vol. 10, pp 157-165, 1952. 6. Hibitt, Karlsson, Sorensen, 'ABAQUS User's Manual Version 4.8', Providence R.I., U.S.A.