Three-dimensional Axisymmetric Analytical Method For Whole Pull-out Failure Process Of Anchor

Anchorage technique has been widely used in the strengthening of concrete structural engineering and rock engineering owing to its high bearing capacity of anchorage systems,simple construction and low cost.The anchorage technique generally refers to drilling holes on the surface of concrete or rock mass and then anchoring the anchor with adhesive.The failure of adhesive anchors is often associated with the following failure modes:anchor failure,interfacial debonding,shear failure of adhesive layer,matrix cone failure,and combined matrix cone-bond failure.Therefore,it is of practical significance to predict the ultimate load capacity and pullout failure process of adhesive anchors under different failure modes.As an effective tool to understand the stress transfer mechanism and failure modes of anchorage systems,analytical method has always been concerned by researchers and engineers.Because of the symmetry of geometries and loading conditions,the anchor pullout problem can be considered as a three-dimensional axisymmetric problem.However,most of existing analytical researches are based on the simplified one-dimensional shear-lag model,such that the real stress state in anchorage systems cannot be well predicted.In addition,the existing three-dimensional axisymmetric analytical studies are still in the range of linear elasticity and ignores the matrix deformation and the nonlinear behavior of interface and adhesive layer.Thus,the development of more accurate and effective three-dimensional axisymmetric analytical method is still an urgent need in the field of anchorage engineering.Based on the concept of fracture mechanics and the three-dimensional axisymmetric theory of elasticity,this paper develops a three-dimensional axisymmetric analytical method for the nonlinear pullout behavior of anchors,and presents three-dimensional axisymmetric analytical solutions for the whole pullout process of anchors under different failure modes.The main contents and conclusions are summarized as follows:(1)According to the concept of fracture mechanics,the debonding process of anchor-adhesive interface is considered as the interfacial shear crack propagation,a three-dimensional axisymmetric analytical method is proposed for the anchor pullout process considering interfacial debonding based on the interfacial trilinear bond-slip model.The analytical expressions are derived for the stress field,displacement field,ultimate pullout load,and load-displacement curve of adhesive anchors under different shear crack propagation stages.The validity and superiority of the proposed analytical method are verified with the existing pull-out test data and existing axisymmetric elastic theoretical solution.On this basis,the influence of the anchorage length on the interfacial debonding process,load-displacement curve and ultimate pull-out load is analyzed.The results show that the calculated load-displacement curve and ultimate pull-out load are in good agreement with the test results,and the calculation accuracy is obviously higher than the existing axisymmetric elastic method.The interfacial debonding process is governed by the characteristic anchorage length,and there exsits a minimum anchorage length and a critical anchorage length.When the anchorage length is smaller than the minimum anchorage length,the load-displacement response shows a significant post-peak softening behavior;when the anchorage length is larger than the minimum anchorage length,the snapback phenomenon appears and becomes more obvious with the increase of the anchorage length.When the anchorage length is smaller than the crtical anchorage length,the ultimate pullout load increases markedly with the anchorage length;when the anchorage length is larger than the critical anchorage length,the growth rate of the ultimate pullout load decreases significantly as the anchorage length increases.Given the basic mechanical parameters and interfacial bond parameters,the proposed three-dimensional axisymmetric analytical method can be used to accurately predict the anchor pullout process when interfacial debonding occurs and to calculate the characteristic anchorage length,the minimum anchorage length and the critical anchorage length without any calibration,and therefore can provide theoretical basis for the design and parameter selection of anchorage systems in practical engineering;(2)In view of the matrix cone failure during interfacial debonding,a three-dimensional axisymmetric analytical method is proposed for the anchor pullout process considering the combined matrix cone-bond failure based on the stress function approach.With the method,the possibility and location of matrix cone failure during interfacial debonding process are determined based on the maximum principal tensile stress.The stress components,the matrix cracking load,the cone angle,the cone height,and the ultimate pullout load are derived for the whole anchor pullout process.The accuracy and superiority of the proposed method are verified by comparing with the existing pull-out test data and axisymmetric elastic theoretical solution.The difference between the proposed three-dimensional axisymmetric analytical solution and the one-dimensional analytical solution is investiagaed so as to theoretically demonstrate the feasibility and applicability of the exsiting one-dimensional analytical method.The results show that the ultimate pullout load,the axial stress and interface shear stress distribution calculated by the proposed method agree well with the experimental results,and the calculation accuracy is significantly higher than the existing axisymmetric elastic method.The difference between the three-dimensional axisymmetric solution and the one-dimensional solution increases with the decrease of the elastic modulus of the matrix.When the elastic modulus of the matrix is rather small,such as soil,sand,etc.,the influence of Poisson's ratio on the anchor pullout behavior cannot be ignored.Therefore,the influence of Poisson's ratio should be considered reasonably in the design and calculation of anchors so as to capture the non-linear anchor pullout behavior more accurately.The proposed three-dimensional axisymmetric solution satisfies the equilibrium equation,stress boundary conditions and continuity conditions,and is not limited to the specific interfacial constitutive relationship and material properties,and can be easily reduced to the one-dimensional solution without considering the effect of Poisson's ratio,and therefore has the advantages of strong universality and wide practical application;(3)For the case of shear failure of cement grout in grouted anchors,the shear failure process of grout is considered as the shear crack propagation parallel to the anchor-grout interface based on the concept of fracture mechanics,and the shear crack propagation process in grout is investigated based on the three-dimensional axisymmetric analytical method.The analytical solutions for the stress field,displacement field,ultimate load and load-displacement relationship are derived for different shear crack propagation stages.The accuracy,validity and applicability of the presented solution are verfied by comparing with the exsiting experimental,numerical,and axisymmetric theoretical results.On this basis,the effect of anchorage length on the anchor pullout behavior is studied,and the prediction method for the failure mode of grouted anchors is proposed.The results show that the anchorage length has a significant effect on the shear crack growth and load displacement curve of grout.When the anchorage length is small(e.g.,five times the anchor diameter),the shear stress in grout is almost uniformly distributed along the anchorage length while the shape of the load-displacement curve is similar to that of the shear stress-displacement curve of grout,and the three-dimensional axisymmetric analytical solution is basically consistent with the one-dimensional analytical solution.With the increase of the anchorage length,the non-uniformity of the shear stress distribution of grout increases gradually.When the anchorage length is larger than the characteristic anchorage length,the load displacement curve shows an obvious nonlinear response,and there exists a significant difference between the three-dimensional axisymmetric analytical solution and the one-dimensional analytical solution,such that the influence of Poisson's ratio of anchor cannot be ignored.Given the basic mechanical parameters,the shear parameters of grout,and the interfacial bond parameters,the analytical solution proposed in this paper can be used to predict the failure modes of grouted anchor and the anchor pull-out process with reasonable accuracy,and therefore can provide a reliable basis for the design and calculation of grouted anchors.