An Analysis for Cross Beam-Ground Anchor Reinforcement

1. REVIEW

With the rapid development of water facilities, hydroelectric projects, highways and railways in China, beam-anchor reinforcement has been widely used to stabilize slopes in recent years(Hu et al., 1999; Huang, 1999; Wu et al., 1998; Cheng and Wang, 1997; Hu, 1997). Beam-anchor reinforcement is a new kind of slope protection structure combining cross beams and ground anchors, which can ensure reliable reinforcement for both deep foundations and slopes(Li et al., 1997). No excavation is needed during the installation and the slope is not disturbed. The installation can be safely and quickly completed and great benefits achieved by afforestation and environment beautification.

The theory for the design of beam-anchor reinforcement is far behind the application. At present there is no widely accepted design theory and the design is still based on the method for pre-stressed anchors. Research on a beam-anchor reinforcement mechanism is limited. Previous investigation is focused on the mechanism of pre-stressed anchors rather than on the reinforcement beams or on the interaction between beams and soils. Research on beams is very important for the determination of design parameters such as size and the spacing of beam-anchor reinforcement.

The beam-anchor reinforcement discussed in this paper is a combined structure of cast-in-situ concrete beams and pre-stressed anchors, which is mainly used for slopes with lower bearing capacity and sliding slopes with loose deposits. The tensile force of the pre-stressed anchor is transferred to the surface of the slope through the beams. Since the contact area between the beam and the slope is larger, the reaction from the slope surface would be reduced. Meanwhile the force acting on, and the deformation of the slope surface are more uniform due to the greater stiffness of the beams. Hence reasonably designed beam-anchor reinforcement can satisfy the need for the bearing capacity of the slope surface. Nevertheless, the current reinforcement design usually analyzes the beam based on experience or an approximated computation, which results in an unreasonable beam structure. A detailed analysis of the loading condition of the beam is very important to the development of a design theory for the beam. Theoretical analysis and finite element modeling are used in this paper to discuss the reasonable form of the reinforcement beam and to explore the optimum design method for the beam.

2. THEORETICAL ANALYSIS OF REINFORCEMENT BEAMS

The Winkler model is used to analyze the beams. It is assumed that concrete beams are elastic; the pre-stress of the anchor is treated as a concentrated force applied to the beam; the stiffness of the longitudinal beam is much larger than that of the transverse beam and only the longitudinal beam is considered to transfer load.

2.1 Analysis of a Beam on Winkler Foundation

The Winkler model assumes that the force at any point on the surface of a foundation soil is proportional to the deformation at that point

where k is the modulus of sub-grade reaction of the foundation(kN/m³).Based on this assumption and mechanic modeling, a differential equation for the curvature of an elastic beam can be written as

where E_b, I_b are elastic modulus and moment of inertia of the beam respectively; b is the width of the beam; q(x)is the load on the beam.

The generalized solution for this equation is

where a variable(1/m); y is the deflection of the beam; the integration constants c₁-c₄ can be obtained by considering loading position and boundary conditions.

Considering the compatibility of the deformation of the beam, the bending moment and shear force of the beam M and V are given by

In the slope treatment projects, the beam is usually 15 m long and the distance between the loaded point and the end of the beam is generally less than π /λ, which means the beam is of finite length(Hetenyi, 1946). Hence the beam of finite length will be discussed.

The force due to the pre-stressed anchor can be simplified as a concentrated load. When tied by n anchors the deflection and force of the beam can be expressed by

where I₁, I₂, I₃ may be found in Hetenyi(1946). It can be seen that the deflection and force of the beam are related to λ_l. The relative flexibility of the soilbeam system, which is a dimensionless parameter, determines to certain extent the deflection of the beam on the foundation soil, the magnitude of beam curvature and the speed of the shrinking effect of external load along the beam. The following sections will discuss the influence of λ, spacing and beam length on the deflection and force.

2.2 Examples

The reinforcement beam shown in Fig. 1 is tied by 5 anchors. Each anchor carries a load of 500 kN and the spacing between the anchors is 3 m. The cantilevered sections of the beam are 0, 1.5 m and 3 m long respectively. The cross section of the beam is 0.4 m× 0.5 m and k for the foundation soil is 3× 10⁴ kN/m³. When λ varies between 0.1 -0.6, the curves of bending moment and deflection of the beam are shown in Figs. 2-4.

Figure  1.  Sketch of a reinforcement beam subjected to loads.

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Figure  2.  The curves of bending moment and deflection of the beam(a= 1.5 m).(a)the curves of bending moment; (b)the curves of deflection.

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Figure  3.  The curves of bending moment and deflection of the beam with different cantilevered lengths(l_s= 3 m, λ= 0.4).(a)the curves of bending moment; (b)the curves of deflection.

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Figure  4.  The curves of bending moment and deflection of the beam with different cantilevered lengths(l_s= 6 m, λ= 0.3).(a)the curves of bending moment; (b)the curves of deflection.

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2.3 The effect of λ

It can be found from Fig. 1 that the magnitude of deflection change lessens with the decrease of λ. The smaller λ is, the greater the relative stiffness of the beam, the smoother the deflection curves become, and the more uniform the deflection. The curves of bending moment show that the smaller λ is, the larger the bending moment is within the length ofπ / λ. The bending moment is basically not affected by λ beyond the length ofπ / λ. For such a case, the beam can be treated as infinitely long and the bending moment is basically not affected by λ. The value of λ should be chosen as 0.3 -0.5 or in order to ensure a uniform deflection and smaller bending moment for the beam.

2.4 The effect of the cantilevered length(a)of the beam

The cantilevered length (a) affects the deflection and bending moment of the beam. The effect on the deflection mainly occurs at the end of the beam. The effect of a on the bending moment is large and the affected section is within a distance of π / λ to the end of the beam. When λ is smaller, a large a will generate large M_max. A smaller a will result in a bigger negative bending moment(Fig. 3). Therefore load should not be applied at the end of the beam. When the spacing between the applied forces is 3 m, a value of 1.5 m for a is suitable.

In another example, the spacing between anchors is set to 6 m and 3 anchors are used to hold the beam. Each anchor has a capacity of 833 kN, and λ is set to be 0.3. Other conditions are the same as those in the above example. Figure 4 shows curves of bending moment and deflection for various values of a. The effect of a on the bending moment and deflection for a spacing of 6 m is similar to the case with a spacing of 3 m, but the reasonable value of a should be 2 m. Compared to the case where the spacing was 3 m and the reasonable value of a was 1 -1.5 m, it can be seen that the reasonable value of a is related to the spacing. The suitable value of a should be(0.3 0.5)lsin cases involving commonly used materials such as reinforced concrete, soils and rocks by completely considering the deflection and force of the beam.

2.5 The effect of spacing l_s

If conditions are the same as those given in the above examples and the value of a and total capacity of anchors are assumed to be 1.5 m and 2 500 kN, three cases for spacing of 3 m, 4 m, and 6 m with corresponding anchor capacity of 500 kN, 625 kN and 833 kN respectively will be considered. Figure 4 illustrates the deflection and bending moment of the beam for λ= 0.4. If the values of a and λ remain unchanged, the magnitude of the change of bending moment and deflection rises significantly with the increase of the spacing. The deflection increases by 6.3 times and the bending moment by 5.2 times. It is therefore recommended that small spacing between anchors with smaller capacity should be adopted in the design for anchors.

It has been found by comparing Fig. 5 with Fig. 2 that under a reasonable loading condition the requirement of the beam with different spacings for λ is different. The selection of lsis related to that of λ and the beam should be reasonably designed to satisfy ls λ < π/2.

Figure  5.  The curves of bending moment and deflection of the beam with different spacings.(a)the curves of bending moment; (b)the curves of deflection.

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3. THREE-DIMENSIONAL FINITE ELEMENT MODELING

When a reinforcement beam applies forces to foundation soils, the loading condition is complex. The analytical approach simplifies some factors but has certain limitations. The three-dimensional finite element method can model the change of stress and strain during the reinforcement and reflect the loading condition of the beam and anchor. A commonly

3.1 The Determination of Computer Modeling

There were some basic simplifications and assumptions for the modeling.(1)The slope ratio was not considered and so the beam was horizontally installed; (2) The pre-stress of the anchor was assumed as a concentrated load applied at the joint of the longitudinal and transverse beams; (3) The Drucker-Prager elastic and plastic model commonly used in geotechnical engineering was adopted for the simulation.

The modeling is based on the beam-anchor reinforcement design for a slope treatment in the Three Gorges area. The lengths of longitudinal and transverse beams are 14 m and 17 m respectively; the spacing between the longitudinal beams and between the transverse beams is 3 m; the value of a is 1 m; and the size of the cross-section and other material parameters of the longitudinal beams are equal to those of the transverse beams. Each anchor has a capacity of 400 kN which is applied at the joint of longitudinal and transverse beams and, in total, 30 anchors were installed(Fig. 6).

Figure  6.  3D FEM mesh. used software program ANSYS was adopted to conduct three-dimensional finite element analysis of beam-soil interaction and to further analyze the loading condition of the beam.

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The parameters of the beams are assumed to be: cross section b× h= 0.4 m× 0.5 m; elastic modulus E_b= 25.5 GPa; Poisson's ratio ν_b= 0.167; density ρ_b= 2.45 g/cm³.

The parameters for soil are adopted as follows: deformation modulus E_s= 160 MPa; Poisson's ratio ν_s= 0.39; density ρ_s= 2.0 g/cm³; cohesion c= 30 kPa; friction angle φ= 17°.

3.2 Analysis of Results

The bending moment of the beam was obtained by analyzing the results. By comparing the bending moment of the external and internal longitudinal beams, the curves of bending moment for half of the beams(symmetric about the center line)can be plotted as shown in Fig. 7. It can be seen that the bending moment of the internal beam is slightly smaller than that of the external beam, which demonstrates that the overall properties of the cast-in-situ beams are good and that the loading conditions for the internal beam and the external one are similar. Hence the same design method can be used for all longitudinal beams and so can do for all transverse beams.

Figure  7.  The curves of bending moment of the beam(T= 400 kN, E_s= 160 MPa).

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When E_s= 160 MPa, the curves of the bending moment for different pre-stresses are given in Fig. 8. With the increase of the pre-stress applied on the beam, the bending moment rises and the change is nearly linear, which reveals that the beam is under the elastic condition and the foundation soils are beginning to enter the plasticity process that does not affect the beam too much.

Figure  8.  The curves of bending moment of the beam with different pre-stresses(E_s= 160 MPa).

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A comparison of the bending moment of the beam for a different deformation module of soils is shown in Fig. 9. It indicates that for a case involving a very weak soil(e.g. E_s= 50 MPa)the deformation of the loaded soil becomes greater, which results in a large bending moment of the beam, and in particular, the section in the middle of the beam undergoes the maximum bending moment. With the rise of the deformation modulus of the soil, the deformation of the loaded soil becomes smaller; the bending moments of the beam at different sections are similar and the bending moment in the edge section is larger than that in the middle section. These results are identical to the analytical solutions, which means that the analytical solutions are reliable.

Figure  9.  The curves of bending moment of the beam of different deformation modulus(T= 400 kN).

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The comparison of the finite element results with the analytical solutions can guide the design. The analytical solutions are for a single beam under T= 400 kN while the finite element modeling has taken into account the joint actions of the longitudinal and transverse beams with the same stiffness. So the analytical solutions should be compared with those values obtained by the finite element approach for T= 800 kN as shown in Fig. 10. It has been found by comparison that the positive bending moment by the analytical solutions is slightly smaller and the negative bending moment a little larger, which means the design for the positive bending moment obtained by the analytical method is slightly unsafe. If the influence of longitudinal and transverse beams is considered at the same time, a coefficient of safety 1.3 is recommended for the design based on the analytical solutions.

Figure  10.  The curves of bending moment of the beam compared with analytic solution and FEM solution.

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4. CONCLUSIONS AND RECOMMENDATIONS

The following conclusions and recommendations can be presented based on the theoretical and numerical results for the beam: (1)It is better to use longer continuous beams with smaller λ which can satisfy l λ≥ 2π; (2)The spacing between anchors has a large influence on the loading and deflection of the beam and it is recommended that beam and anchor with smaller lsand capacity should be used to satisfy ls λ < π /2;(3) It is suggested to adopt the value of a as (0.3 -0.5)ls; (4) The numerical results show that the application of the analytical method for reinforcement beam analysis is acceptable. The same design can be used for all beams in different directions and the design for the internal beam is also the same as that for the external beam. For safety, a coefficient of safety 1.3 is recommended for the design based on the analytical solutions.