The methods for analysis of the ferroconcrete construction for a particular case are proposed in
the present work. The underground construction is considered and analyzed as a double-drift frame
situated on linearly-deformed basis. The formulas of analysis are given herein

Kikava, Temuri R

Transactions of the VŠB - Technical University of Ostrava Civil Engineering Series

The methods for analysis of the ferroconcrete construction for a particular case are proposed in the present work. The underground construction is considered and analyzed as a double-drift frame situated on linearly-deformed basis. The formulas of analysis are given herein

Kikava Temuri R.

Directory of Open Access Journals

The Analysis of the Tunnel with the Middle Soil-Based Wall

Abstract
				 The methods for analysis of the ferroconcrete construction for a particular case are proposed in the present work. The underground construction is considered and analyzed as a double-drift frame situated on linearly-deformed basis. The formulas of analysis are given herein.</jats:p

Temuri R. Kikava

Crossref

DSpace at VSB Technical University of Ostrava

25 Transactions of the VŠB – Technical University of Ostrava Civil Engineering Series, Vol. 16, No. 1, 2016 paper #4 Temuri R. KIKAVA1 THE ANALYSIS OF THE TUNNEL WITH THE MIDDLE SOIL-BASED WALL Abstract The methods for analysis of the ferroconcrete construction for a particular case are proposed in the present work. The underground construction is considered and analyzed as a double-drift frame situated on linearly-deformed basis. The formulas of analysis are given herein. Keywords Frame, ground pressure, underground structure, reinforced concrete, linearly-deformed basis.   1 INTRODUCTION The underground constructions like tunnels are commonly used in construction practice. They may be intended for different types of transport as well as for laying gas, petrol and water pipelines. Many of these constructions serve the purposes of the frame constructions while vary sufficiently in outline and complexity.  The purpose of this paper is to consider joint action of the frame’s lower girder and the soil basis, the analysis of a beam separately on the basis of the linearly-deformed basis (lower girder) and the frame regardless of this and thereafter the consideration of joint deformation of the lower girder and the frame’s upper part. Let us first consider the transverse section of a double-drift tunnel in the form of a ferroconcrete frame (Figure 1a). Let us analyze mentioned frame in two versions (Figure 1b). The analysis shall be made per unit of the tunnel’s length, i.e. b = 1 m. The tunnel is affected by equally distributed over ground load and the soil’s specific weight from above and by soil’s pressure from the side surfaces.   2 THE FIRST VERSION OF THE ANALYSIS We divide the frame (tunnel) cutout per unit of the frame (tunnel)’s length in separate parts and consider each of them separately (Figure 1). From conditions of statics, in the result of symmetry and load of the construction:   |Mୡ| ൌ |Mୈ|   Yୡ ൌ Yୈ ൌ 0.5ሺqሺୡ୮ሻ ∙ L െ Y୉ሻ (1)                                                                                                                                                                    1  Temuri T. Kikava, Professor, MSc, PhD., Engineering and Construction Department, Faculty of Technologies, Batumi Shota Rustaveli State University, 35 Ninoshvili, 6010 Batumi, Georgia, phone: +995 (422) 271 780, e-mail: temurikikava@yahoo.com. 26  Figure 1 By using universal formula of the beam’s elastic curve we find turning angle φсሺେୈሻ  of the upper girder CD in C node point:   φେሺେୈሻ ൌ ୑ౙ∙୐ଶ୉మ∙୍మ ൅୐మ଼୉మ∙୍మ ∙ ቀଢ଼ుଶ െ୯ሺిీሻ∙୐ଷ ቁ , (2)  where  УЕ - independent longtitutional force in the middle wall EF; qሺେୈሻ _ equally distributed load on the upper girder (own mass of land and over ground load). 27 We find turning angles φେሺ୅େሻ  and φ୅ሺ୅େሻ of the left hand column being under action of the land’s side pressure. We receive the side pressure of the land in the form of trapezium (Figure 1) and by using the universal formula of the beam’s elastic curve we receive:  φ୅ሺ୅େሻ ൌ ଵ୉భ∙୍భ ቂ୦଺ ሺ2M୅ᇱ ൅ Mେᇱ ሻ െ୦యଷ଺଴ ሺ8qଶୟ ൅ 7qଵୟሻቃ   (3)  φେሺ୅େሻ ൌ ଵ୉భ∙୍భ ቂെ୦଺ ሺM୅ᇱ ൅ 2Mେᇱ ሻ ൅୦యଷ଺଴ ሺ7qଶୟ ൅ 8qଵୟሻቃ .   By using the universal formula of the beam’s elastic curve we find bilge of the upper girder CD in point E:   f୉ሺେୈሻ ൌ ୐మଷ଼ସ∙୉మ∙୍మ ൫48Mୡ ൅ 8Y୉ ∙ L െ 5qሺେୈሻ ∙ Lଶ൯ . (4)  We find turning curve of the lower girder AB:   	φ୅ሺ୅୆ሻ ൌ ଵ஠୉బୠ୐మ ቄെφഥଶ୅ሺ୅ሻ ∙ M୅ ൅ φഥଶ୆ሺ୅ሻ ∙ M୆		 ൅ L ቂφഥଷ୅ሺ୅ሻ ∙ Y୅ ൅ φഥଷ୊ሺ୅ሻ ∙ Y୊ ൅ φഥଷሺ୆ሻሺ୅ሻ ∙ Y୆ ൅൅∑φഥଷ୧ሺ୅ሻ ∙ P୧ቃቅ		.							  (5)  It is supposed that the forces P1= P2= P3= P4=P and they are placed symmetrically on the lower girder.  We find deflection f୊ሺ୅୆ሻ  in the middle of the lower girder:   f୊ሺ୅୆ሻ ൌ ଵ஠∙୉బ∙ୠ∙୐ ൝െyതଶ୅ሺ୊ሻ ∙ M୅ ൅ yതଶ୆ሺ୊ሻ ∙ M୆ ൅ L ൥yതଷ୅ሺ୊ሻ ∙ Y୅ ൅ yതଷ୊ሺ୊ሻ ∙ Y୊ ൅ yതଷ୆ሺ୊ሻ ∙ Y୆ ൅൅∑yതଷ୧ሺ୊ሻ ∙ P୧൩ൡ . (6)  For the case under review:   |Mୡ| ൌ |Mୈ|;   |M୅| ൌ |M୆|;    Y୅ ൌ Y୆; 			Y୉ ൌ Y୊;		Yେ ൌ Yୈ;  X୅ᇱ ൌ ୑ఽᇲ ି୑ᇲి୦ ൅୦଺ ሺqଵୟ ൅ 2qଶୟሻ   (7)  Xେᇱ ൌ ୑ఽᇲ ି୑ᇲి୦ ൅୦଺ ሺ2qଵୟ ൅ qଶୟሻ      28 From conditions φେሺେୈሻ ൌ φେሺ୅େሻ   ;    φ୅ሺ୅େሻ ൌ φ୅ሺ୅୆ሻ    and     f୉ሺେୈሻ ൌ f୊ሺ୅୆ሻ  we receive equation for defining the unknown quantities M୅, Mс, Y୅  and Y୊  :   ୑ి∙୐ଶ୉మ∙୍మ ൅୐మସ଼୉మ∙୍మ ൫3Y୉ െ 2qሺେୈሻ ∙ L൯ ൌ ଵ୉భ୍భ ቂെ୦଺ ሺM୅ ൅ 2Mେሻ ൅୦యଷ଺଴ ሺ7qଶ୅ ൅ 8qଵ୅ሻቃ (8)   െ ଵ஠∙୉బ∙ୠ∙୐మ ቄെφഥଶ୅ሺ୅ሻM୅ ൅ φഥଶ୆ሺ୅ሻM୆ ൅ L ቀφഥଷ୅ሺ୅ሻ ∙ Y୅ ൅ φഥଷ୊ሺ୅ሻ ∙ Y୊ ൅ φഥଷ୆ሺ୅ሻ ∙ Y୆ ൅ ∑φഥଷ୧ሺ୅ሻ ∙ P୧ቁቅ ൌଵ୉భ∙୍భ ቂ୦଺ ሺ2M୅ ൅ Mେሻ െ୦యଷ଺଴ ሺ8qଶ୅ ൅ 7qଵ୅ሻቃ  (9)  ୐మଷ଼ସ୉మ∙୍మ ൫48Mେ ൅ 8Y୉ ∙ L െ 5Qሺେୈሻ ∙ Lଶ൯ ൌ ଵஈ∙୉బ∙୆୐ ቄെYഥଶ୅ሺ୊ሻ ∙ M୅ ൅ Yഥଶ୆ሺ୊ሻ ∙ M୆ ൅L ቂYഥଷ୅ሺ୊ሻ ∙ Y୅ ൅ Yഥଷ୊ሺ୊ሻ ∙ Y୊ ൅ Yഥଷ୆ሺ୊ሻY୆ ൅ ∑Yഥଷ୍ሺ୊ሻ ∙ P୍ ቃቅ      (10)  We solve these equations together with the equations of statics, find unknown M୅, 	Mେ,Y୅		and		Y୊.  After defining the unknown quantities for the given frame according to table (3) it shall be possible to construct the epures of reactive pressure of soil p, intersecting forces Q and bending moments, M.   3 THE SECOND VERSION OF THE ANALYSIS Let us consider the closed frame allocated to transverse direction (Figure 1b) and the same frame divided into separate elements (Figure 1d).  Formulas (2) and (4) are true for defining the turning angle of the upper girder φେሺେୈሻ in node point C and deflection f୉ሺେୈሻ in the middle of the upper girder CD in section E. By accepting approximately module of soil’s deformation by depth as constant, we consider the side walls AC and BD as the beams on the linearly-deformed basis  relied on the rigid bearers at the same time (wall AC relies on the points A and C and wall BD – on the points B and D) (Kikava, T. 2007.) We use the following formulas for defining the turning angles φେሺ୅େሻ and  φ୅ሺ୅େሻ :   φେሺ୅େሻ ൌ ଵ஠∙୉బୠ୦మ ቄെφഥଶେሺେሻ ∙ Mେ ൅ φଶ୅ሺେሻ ∙ M୅ ൅ h ቂφഥଷେሺେሻ ∙ Xେ ൅ φഥଷ୅ሺେሻ ∙ X୅ቃቅ (11)   φ୅ሺ୅େሻ ൌ ଵ஠∙୉బୠ୦మ ቄെφഥଶେሺ୅ሻ ∙ Mେ ൅ φଶ୅ሺ୅ሻ ∙ M୅ ൅ h ቂφഥଷେሺ୅ሻ ∙ Xେ ൅ φഥଷ୅ሺ୅ሻ ∙ X୅ቃቅ (12)  The lower girder works as a beam on the linearly-deformed basis (Simvulidi, I. Kikava, T. Bulatov, V. 1986.)  and formulas (5) and (6) are true for defining the turning angle  φ୅ሺ୅୆ሻ and deflection f୊ሺ୅୆ሻ . Therefore, by using conditions:  φେሺେୈሻ ൌ φେሺ୅େሻ;		φ୅ሺ୅େሻ ൌ φ୅ሺ୅୆ሻ; 		 f୉ሺେୈሻ ൌ f୊ሺ୅୆ሻ 29  as well the conditions of statics (1) and by solving the received equations related to the unknown quantities M୅, 	Mେ, Y୅		and		Y୊ jointly, we receive possibility to construct the epures p, Q and M for the whole frame (Simvulidi, I. 1987). Let us consider concrete example for the beam calculation (lower cross bar) on the linearly-deformed basis loaded with the concentrated forces (Fig. 2). It is required to determinate values of curving moments M and reaction pressures of the ground ρ, if : P1=P2=P3=P4=100kN; PF=200kN; MA=14kNm; the width b=2m; the index of flexibility α=200 the module of deformation E0=320∙102kN/m2.  Figure 2 Solution. According to the condition of the problem: βଵ ൌ lଷଵܮ ൌ 0.17; βଶ ൌ lଷଶܮ ൌ 0.33; β୊ ൌ lிయܮ ൌ 0.5; βଷ ൌ lଷଷܮ ൌ 0.67; βସ ൌ lସଷܮ ൌ 0.83;  According to conditions of symmetry │MA│= │MB│ According to conditions of statics: YA= YB = YC = YD = 0 30  For the determination of ordinate ρ1; ρ2; ρF; ρ3; ρ4; from force P1; P2; PF; P3 and P4; we use the values of ρതଵ;	ρതଶ; ρതଵ; ρത୊; ρതଷ; ρതସ; corresponding α=200 and βଵ ൌ 0.17; βଶ ൌ 0.33; β୊ ൌ 0.5; βଵଷ ൌ0.67; βସ ൌ 0.83;  for all values ρതଵ;	ρതଶ; ρതଵ; ρത୊; ρതଷ; and ρതସ; from ξ=0 till ξ=1, where ξ is considered cross-section (Simvulidi, I.1987). Having multiplied each ordinate according to the formula:  ρ ൌ ρത ∙ P b⁄ ∙ ܮ	         (13) where 				P bL⁄ ൌ 100 2⁄ ∙ 12 ൌ 4.2	kN/mଶ 				P୊ bL⁄ ൌ 200 2⁄ 4 ൌ 8.4	kN/mଶwe’ll get values ρ in various points. Then we’ll get values ρത୑ಲ and  ρത୑ಳ for α=200; βଶଵ ൌ 0 and βଶଶ ൌ 1.0 multiplied each ordinate to MA/bL2= 14/2∙122 = 0.05 kN/m2 and MB/bL2= 0.05 kN/m2 Table 1: ξ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ρതଵ 3.845 2.793 1.988 1.388 0.956 0.648 0.424 0.246 0.068 -0.145 -0.437 ρ1 16.149 11.731 8.35 5.83 4.02 2.72 1.781 1.033 0.29 -0.61 -1.84 ρതଶ 2.386 2.05 1.736 1.434 1.157 0.901 0.669 0.464 0.28 0.132 0.01 ρ2 10.02 8.61 7.29 6.023 4.86 3.784 2.81 1.95 1.18 0.55 0.04 ρതி 0.666 0.846 0.987 1.087 1.147 1.167 1.147 1.087 0.987 0.846 0.666 ρF 5.594 7.106 8.29 9.13 9.64 9.802 9.64 9.13 8.29 7.106 5.594 ρതଷ 0.01 0.132 0.28 0.464 0.669 0.901 1.157 1.434 1.736 2.05 2.386 ρ3 0.04 0.55 1.18 1.95 2.81 3.784 4.86 6.023 7.29 8.61 10.02 ρതସ -0.437 -0.145 0.068 0.246 0.424 0.648 0.956 1.388 1.988 2.793 3.845 ρ4 -1.84 -0.61 0.29 1.033 1.781 2.723 4.02 5.83 8.35 11.731 16.149 ρതெಲ 15.323 7.758 2.572 -0.655 -2.343 -2.911 -2.781 -2.373 -2.106 -2.402 -3.679 ρതெಳ -3.679 -2.402 -2.106 -2.373 -2.781 -2.911 -2.343 -0.655 -2.572 -7.758 -15.323 ρതெಲ൅ ρതெಳ 11.644 5.356 0.466 -3.028 -5.124 -5.822 -5.124 -3.028 0.466 5.356 11.644 ρெಲ൅ ߩெಳ 0.582 0.268 0.023 -0.151 -0.256 -0.291 -0.256 -0.151 0.023 0.268 0.582 ρ 30.545 27.655 25.423 23.815 22.355 22.519 22.355 23.815 25.423 27.655 30.545   The values ρ give in kN/m2; M-kN∙m. Using the independence principle of force activity for getting values ρ we take algebraic sum: ρ= ρ1+ ρ2+ ρF+ ρ3+ ρ4+ ρMA+ ρMB For the determination of ordinates curving moment M from the forces P1; P2; PF; P3; P4; and from moments MA and MB , we use the values of ܯഥଵ; ܯഥଶ; ܯഥி; ܯഥଷ; ܯഥସ; ܯഥெಲand ܯഥெಳfor α=200 and βଵ ൌ 0.17 ; βଶ ൌ 0.33 ; β୊ ൌ 0.5 ; βଷ ൌ 0.67 ; βସ ൌ 0.83 ; βெಲ ൌ 0 and βெಳ ൌ 1.0 for all values ܯഥ  from ξ=0 till ξ=1 (Simvulidi, I.1987). Having multiplied each ordinate to: 31  ܯ௉ ൌ ܯഥ௉ ∙ ܲ ∙ ܮ		 (14)  ܯெಲ ൌ ܯഥெಲ ∙ ܯ஺   (15)  ܯெಳ ൌ ܯഥெಳ ∙ ܯ஻   where	 P∙L=100∙12=1200 kN∙m ; PF∙L=200∙112=2400 kN∙m Using the independence principle of force activity for getting values M we take algebraic sum: ܯ ൌ ܯ௉భ ൅ ܯ௉మ ൅ ܯ௉ಷ ൅ܯ௉య ൅ ܯ௉ర ൅ ܯ஺ ൅ܯ஻ Table 2: ξ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ܯഥ௉భ 0 0.018 0.063 0.029 0.008 -0.003 -0.007 -0.007 -0.005 -0.001 0 ܯ௉భ 0 21.6 75.6 34.8 9.6 -3.6 -8.4 -8.4 -6 -1.2 0 ܯഥ௉మ 0 0.012 0.044 0.093 0.057 0.032 0.016 0.007 0.003 0.001 0 ܯ௉మ 0 14.4 52.8 111.6 68.4 38.4 19.2 8.4 3.6 1.2 0 ܯഥ௉ಷ 0 0.004 0.015 0.037 0.07 0.114 0.07 0.037 0.015 0.004 0 ܯ௉ಷ 0 9.6 36 88.8 168 273.6 168 88.8 36 9.6 0 ܯഥ௉య 0 0.001 0.003 0.007 0.016 0.032 0.057 0.093 0.044 0.012 0 ܯ௉య 0 1.2 3.6 8.4 19.2 38.4 68.4 111.6 52.8 14.4 0 ܯഥ௉ర 0 -0.001 -0.005 -0.007 -0.007 -0.003 0.008 0.029 0.063 0.018 0 ܯ௉ర 0 1.2 -6 -8.4 -8.4 3.6 9.6 34.8 75.6 21.6 0 ܯഥ஺ -1 -0.937 -0.796 -0.625 -0.460 -0.318 -0.204 -0.119 -0.054 -0.015 0 ܯഥ஻ 0 -0.015 -0.054 -0.119 -0.204 -0.318 -0.460 -0.625 -0.796 -0.937 -1 ܯഥெಲ൅ܯഥெಳ-1 -0.952 -0.85 -0.744 -0.664 -0.636 -0.664 -0.744 -0.85 -0.952 -1 ܯெಲ൅ܯெಳ -14 -13.33 -11.9 -10.42 -9.3 -8.9 -9.3 -10.42 -11.9 -13.33 -14 ܯ -14 -34.67 -150.1 224.8 247.5 341.5 247.5 224.8 -150.1 -34.67 -14   32  4 CONCLUSIONS The methods for analysis of the ferroconcrete construction for a particular case are proposed in the present work. The underground construction is considered and analyzed as a double-drift frame situated on linearly-deformed basis. The formulas of analysis are given herein. Example of calculation is given.  REFERENCES [1] SIMVULIDI, I.  KIKAVA, T.  BULATOV, V. 1986. Approximate method for analysis of a beam on elastic foundation. Moscow., VINITI 2341-86. p.1-7. [2] SIMVULIDI, I. 1987. Analysis of the engineering constructions on elastic foundation. Moscow., Higher school. p.79-99. [3] KIKAVA, T. 2007. The analysis of the rectangular closed rigid frame supported by trestle bents. Tbilisi., Gournal“Energy” №1. p. 109-111. 

The analysis of the tunnel with the middle soil-based wall

http://dspace.vsb.cz/bitstream/10084/112075/1/04_kikava.pdf

The analysis of the tunnel with the middle soil-based wall

Abstract

Similar works

Full text

Available Versions

Directory of Open Access Journals

Crossref

DSpace at VSB Technical University of Ostrava