Slope frame variation model in spring slip model

The mechanical model of the sliding plane instability of the slope is different in the stress level, material composition and structure of different parts of the weak interlayer. The interlayer itself may contain a variety of media with different mechanical properties, such as elastic brittleness, strain hardening and strain softening properties. In order to simplify the analysis, it is considered that the interlayer medium consists of two kinds of media, that is, the medium 1

The mechanical model of the sliding plane instability of the slope is different in the stress level, material composition and structure of different parts of the weak interlayer. The interlayer itself may contain a variety of media with different mechanical properties, such as elastic brittleness, strain hardening and strain softening properties. In order to simplify the analysis, it is considered that the interlayer medium consists of two media, that is, the medium 1 has complex elastic brittleness (such as hard rock or rock bridge) or strain hardening (such as hard clay or loose sand), and the medium 2 has strain. Softening characteristics, the constitutive curves of the two media.

The cusp catastrophe model For the system shown, the total potential energy is the sum of the strain energy and the sliding potential energy of the slip surface medium: V = lsu0Gsuhexp-(u0)mdu+12Ge1lehu2+G(au)22B(ls+le)-Wgusinu It is assumed here that ls and le are much larger than u and remain unchanged during the sliding process. If le is much larger than ls, the slope is stable. In the analysis below, only the case where le is much smaller than ls is considered.

From V=0, we can determine that the equilibrium surface is V=Gslshuexp-(u0)m+Geleh+GaB(ls+le)u-Wgsin+GaB(ls+le)=0 when u The cusp catastrophe model performs the Taylor expansion of the equilibrium surface with respect to the state variable value u1 at the cusp, intercepts it to the 3rd term and substitutes the variable to obtain the standard form of the cusp mutation: x3+px+q=0 where x=u -u1u1 p=6(m+1)2(k-1) q=6m(m+1)2(mk+1-) k=1mGslsGele+GahB(ls+le)exp(m+1m) hWgsin+GaB (ls+le)Gslsu1exp(m+1m) The parameter k is called the stiffness ratio. The parameters are related to the rock mass force Wg, the geometry of the system, the mechanical parameters of the medium, etc., which are called geometric-mechanical parameters. Substituting the formula into the bifurcation set equation yields D = 4p3 + 27q2 = 43(k - 1) 3 + 27 (m) 2 (1 + mk - ) 2 = 0 where = 6 / (m + 1) 2 .

As shown, the coordinates of the three-dimensional space are the control parameters p, q and the state variable x. For example: starting from point B, with the continuous change of the control parameters, the system state evolves along path BB to B, and the state variable continuously changes, Mutation occurs (D>0); and from point A, along the path AA, when approaching the edge of the folded wing, as long as the control parameters have a slight change, the system state will be abrupt, from the lower wing of the folding wing to the folding wing Upper leaves. This shows that the system can only be abrupt when it crosses the bifurcation set, so the formula is the criterion of the necessary mechanical condition for the sudden instability of the plane sliding type slope.

It is known from the formula that when a0, ie k1, D may be equal to zero, the necessary condition for the system to be abrupt is k1, that is, the system instability has a greater correlation with the stiffness than k. It can be seen from the formula that, with other parameters unchanged, k decreases with increasing m, and the larger the m value (the smaller the stiffness ratio), that is, the higher the uniformity or brittleness of the material, the more likely the mutation is caused. Water has an important influence on the instability of the slope. It is generally believed that in addition to chemical corrosion, water also exerts the effect of hydrostatic pressure (floating force) and hydrodynamic pressure. It can be seen from <7> that as the water content of the rock increases, the slope of the post-peak curve becomes steeper, and the stiffness after the peak increases, that is, m increases and the stiffness ratio k decreases, which is prone to mutation. This shows that water has a more important role in increasing material uniformity (brittleness) and reducing stiffness ratio.

Conclusion (1) The relationship between the weak zone medium and the upper rock mass is analyzed, and a spring-slider model is proposed. The inside of the weak medium is connected to each other by a spring KC, and the weak medium is connected to the upper rock body by a spring KL. (2) Considering the strain hardening and strain softening properties of weak zone media under different environmental factors, a cusp catastrophe model of slope sliding along a single soft structural plane is established by using the catastrophe theory.

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