Mechanical Model of Tunnel Support Pipe Support
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Mechanical Model of Tunnel Support Pipe Support Zuo Zhuo1 Fu Helin3 Peng Wenxuan4 1,2 Changsha University of Science and technology,410007 3,4 School of Civil Engineering e of Central South University Changsha 41007 China e-mail: [email protected] ABSTRACT In this paper, based on the principle analysis of tunnel lining structure function, starting from the angle of structural mechanics, the elastic foundation beam theory to build the tunnel support function of piperoof mechanical model, and the application of combined with the specific project. It provides a new approach for stress analysis of tunnel support. KEYWORDS: pipe shed support mechanical model of foundation beam theory INTRODUCTION In tunnel support, the pipe roof is an extremely effective means, and the mechanical model is shown in Figure 1. Figure 1: pipe roof beam on elastic foundation model Based on this, the excavation process can be divided into two cases, shown in Table 1, and the elastic foundation beam model can be established. - 257 - Vol. 24 [2019], Bund. 01 258 Table 1: Tunnel excavation and the corresponding elastic foundation beam model No. Force characteristic Mechanical schema Beam model of semi infinite foundation The excavation section of AB length a for himself, has excavation without support BC segment length is s, the working range of the front CD relaxation length was D, the tunnel excavation height is h. BC shed pipe under only the rock pressure above Q (x) A and pass it to the rear of the initial support and the surrounding rock; CD can withstand the rock pressure Q (x), but also by the elastic resistance of P (x) DE segment; surrounding rock pressure is negligible, only under the pipe roof and surrounding rock deformation caused by the ground reaction force P(x) With pipe roof and initial support of the connecting end of the B as the origin of coordinates, which has been excavated section of supporting AB segment length is a, excavation without support BC segment length is s, CD is in front of the working range of relaxation length is D, h is the height of B tunnel excavation. The BC section pipe roof only withstands the surrounding rock pressure Q (x) and passes it to the rear primary support and the front wall rock; the CD not only withstands the surrounding rock pressure Q (x), but also is subjected to the elastic resistance of P (x). BEAM MODEL OF ELASTIC FOUNDATION WITH TWO PARAMETERS As shown in Figure 2, a layer of shear layer existing in the spring unit in the Kerr foundation beam model, and the layer can only produce shear deformation and incompressible shear, so there is interaction between the spring element, can be in good agreement with the surrounding rock stress. The theory is referred to as the "PAP" theory, and the "pipe to pipe" model is analyzed. Vol. 24 [2019], Bund. 01 259 Figure 2: B model The shear layer is assumed to be isotropic in the X and Y planes and its shear modulus Gx=Gy=Gp,then ∂w τ xz = Gpγ xz = Gp ∂x (1) ∂w τ yz = Gpγ yz = Gp ∂y (2) The total shear force per unit length of shear layer is: 1 ∂w N = τ dz = G s ∫ xz p ∂x 0 (3) 1 ∂w N = τ dz = G y ∫ yz p ∂y 0 (4) The equilibrium condition of the force in the direction of Z is obtained : ∂ ∂N x N y + + q − r0 = 0 ∂x ∂y (5) In the formula, r0=kw, the formula (3) (4) will be given to the people q(x, y) = kw(x, y) − G ∇2w(x, y) p (6) Here:q(x,y)—Subgrade reaction ,,KN/m2 ;k—Coefficient of subgrade reaction ,KN/m3 ;Gp —Characterization of transfer shear parameters ;KN/mm, w—Foundation settlement ,m ;When Gp=0,i.e Winker foundation model Vol. 24 [2019], Bund. 01 260 IDENTIFICATION OF CALCULATION MODEL FOR TWO PARAMETER ELASTIC FOUNDATION BEAM The method of distinguishing two parameter foundation beam is shown in table 2. Table 2: beam pattern discrimination Concentrated load position Length of beam Calculation type of beam (to beam endless x) Infinite beam Both ends of the beam are satisfied L ≥ 2π / λ x ≥ π / λ Semi infinite beam Only a section from the beam is satisfied L ≥ 2π / λ x ≥ π / λ Finite length beam Both ends of the beam are satisfied π/(4λ)< L < 2π / λ x < π / λ ≤ π λ L /(4 ) Rigid beam No connection λ = 4 k noted: 4EI ,It is related to the relative stiffness of the beam and foundation . GOVERNING DIFFERENTIAL EQUATIONS OF BEAMS ON ELASTIC FOUNDATIONS Based on the Pasternak elastic foundation beam, the force model of the pipe roof is established, and the force is shown in Figure 3. Figure 3: beam pattern discrimination From balance condition of FG ∑ Y = 0 then: dQ(X ) = b[ p(x) − q(x)] (4) dx To F section, from ∑ M = 0 then: (dx)2 (dx)2 M − (M + dM ) + (Q + dQ)dx + q(x) − p(x) = 0 (5) 2 2 Vol. 24 [2019], Bund. 01 261 Omitting two orders of differentiation: dM (x) Q(x) = dx (6) Bring in (4): dQ(x) dM 2 (x) = = − 2 b[ p(x) q(x)] dx dx (7) From Bernoulli-Euler typical theory, the force、displacement and inner force have following relationship: dω(x) θ(x)= dx 2 dω (x) = − M (x) EI 2 dx dM (x) dω 3 (x) Q(x) = = −EI 3 dx dx (8) The differential equation of beam deflection is brought into (7) type dω 4 (x) dω 2 (x) − * + *ω = EI 4 Gpb 2 kb (x) bq(x) dx dx (9) b* = b[1+ (G / k)1/ 2 / b] Here:b*--as equivalent width of beam,,satisfied p ;b- equivalent width of beam;I-as the moment of inertia of foundation beam ;;E- as The elastic modulus of the beam on the elastic foundation ; Gp-as foundation shear modulus; ω (x) -as Deflection of foundation beam ;k-as Coefficient of subgrade reaction. On the basis of formula (7) and (8), the deflection curve of two cases can be obtained by analyzing the model of pipe roof. 1) the governing equations of type A (1) in the BC section, the surrounding rock pressure is q0, the foundation reaction force is p (x) =0, and the governing equation is: dω 4 (x) = EI 4 bq0 dx (10) (2)CD section, the surrounding rock pressure is q0, the foundation reaction dω 2 (x) = ω − p(x) k (x) Gp 2 dx ,The governing equations are: dω 4 (x) dω 2 (x) EI − G b* + kb*ω(x) = bq (11) dx4 p dx2 0 Vol. 24 [2019], Bund. 01 262 (3)CD section, the surrounding rock pressure is q(x)=0,the foundation reaction dω 2 (x) = ω − p(x) k (x) Gp 2 dx ,The governing equations are: dω 4 (x) dω 2 (x) − * + *ω = EI 4 Gpb 2 kb (x) 0 dx dx (12) 2)the governing equations of type B (1)in the BC section, the surrounding rock pressure is q0, the foundation reaction force is p(x)=0,The governing equations are: dω 4 (x) EI = bq dx4 0 (13) (2)CD section, the surrounding rock pressure is q0, the foundation reaction dω 2 (x) = ω − p(x) k (x) Gp 2 dx ,The governing equations are: dω 4 (x) dω 2 (x) − * + *ω = EI 4 Gpb 2 kb (x) bq0 dx dx (14) CONTROL DIFFERENTIAL EQUATIONS 1) A type (1)(1) the governing equation (10) of the DE section pipe roof is a homogeneous equation, that is, the deflection equation of the foundation beam ,,assum kb* λ4 = 4EI ,then: 4 4 2 1 dω (x) Gpλ dω (x) − + λ4ω = 4 2 (x) 0 4 dx k dx (15) Its characteristic equation is: 4 1 Gpλ r 4 − r 2 + λ4 = 0 4 k (16) G λ2 p <1 Normally,k is much bigger than Gp,and λ <1,so k ,So the characteristic equation has two pairs of plural root α = λ[1+ (G λ2 / k)]1/ 2 , β = λ[1− (G λ2 / k)]1/ 2 ±α ± iβ ,Here p p ,The general solution of the equation (16) is: Vol. 24 [2019], Bund. 01 263 ω(x)= eαx (B cos βx + B sin βx) + e−αx (B cos βx + B sin βx) 3 1 2 3 4 (17) Here B1,B2,B3,B4 are the undetermined coefficients can be derived from the boundary conditions of the pipe roof. ( ) 2 The characteristic equation of the BC segment control equation (11) is 喷射混凝土 4 轴力N 内压力 p r = 0 ,There are 4 roots 1 ,the general solution of the equation is P σ P σ ,When the load of pipe shed is uniformly distributed load P σ P σ q0, the special solution is ,the equation (11) is the general solution : P σ P σ (18) Here A1,A2,A3,A4 are are the undetermined coefficients . (3)It is obvious that the governing equation (13) of the DE section pipe roof is the homogeneous equation of the differential control equation () of the CD section, so that the general solution of the CD section control equation (12) can be obtained: ω(x)= eαx (B cos βx + B sin βx) + e−αx (B cos βx + B sin βx) +ω * 3 1 2 3 4 2 (19) ω * In the formula, the particular solution 2 is related to the load q(x) and boundary conditions. When the load on the pipe roof is uniformly distributed load q0, the boundary condition can be obtained by continuity at the D point: ′ ′ ω = θ = θ = ω ω = ω 2 2 x=s+d 3 x=s+d 3 2 x=s+d 3 x=s+d , x=s+d x=s+d ,take special solution : * q ω = 0 − β − − β − − 2 * [1 cosh (x s d)cos (x s d)] kb 6 DEFLECTION EQUATION OF PIPE SHED 1) B type (1) in the BC section, the control differential equation (14) is the same as the control equation (12) of the A type, so the general solution is (16).