Method for evaluating safety of cavity foundation under eccentric load

Abstract

The application discloses a method for evaluating the safety of a cavity foundation under eccentric load, and belongs to the field of geotechnical stability evaluation. The method comprises the following steps: A, defining a constant value and a variable initial value; B, iteratively calculating a strength reduction coefficient; and C, evaluating the safety of the cavity foundation under eccentric load. Newton-Raphson iteration method is adopted to solve the parameters of the failure mode. The application considers the situation of the cavity foundation under eccentric load, the influence of the self-weight load of the rock and soil, and the influence of the size of the ground load on the failure mode of the cavity foundation, and the technical scheme is convenient for engineering application.

Description

Technical Field

[0001] This invention belongs to the field of geotechnical stability evaluation technology, specifically a method for evaluating the safety of cavitary foundations under offset load. Background Technology

[0002] This invention relates to the theory of limit analysis. Coulomb proposed the friction-cohesion yield criterion {Coulomb C.A..Essai sur une Application des Règles des Maximis et Minimisàquelques Problèmes de Statique Relatifsàl'Architecture[J].Mémoires De L'Académie Royale des Sciences,1773.}. Culmann proposed the planar sliding surface hypothesis for analyzing foundation stability {Culmann C..Die Graphische Statik[M].Zurich:Meyer and Zeller,1865.}. Petterson proposed the circular sliding surface hypothesis for analyzing the stability of clay slopes {Petterson KE.The early history of circular sliding surfaces[J].Geotechnique,1955,5(4):275-296.}. Fellenius proposed the slice method for analyzing slope stability {Fellenius W. Calculation of the stability of earthdams[A]. Proceedings of the second congress of large dams[C]. Washington DC: vol 4, pp 445-463, 1936.}. Taylor calculated the safety factor of slopes based on the circular sliding surface method {Taylor D.W. Stability of earth slopes[J]. Journal of the Boston Society of Civil Engineers, 1937, 24(1): 197-246.}. Terzaghi derived the formula for the ultimate bearing capacity of foundation based on the principle of static field superposition {Terzaghi K. Theoretical Soil Mechanics[M]. New York: John Wiley and Sons, Inc., 1943.}.Kotter proposed the slip line method {Kotter F..Die Bestimmung des Druckes anGekummten Gleittlachen[M].eine Aufgabe aus der Lehre vom Erddruck,1903.}, and then Prandtl{Prandtl L.,Harte plashecher Korper[M].Gottingen:Nach.Ges.Wiss.,1920.}, Reissner{Reissner H..Zum Erddruckproblem[A].Proceedings of the first congress in applied mechanics[C].Delft:1924,pp.295-311.}, Novotortsev{ChenW.F..Limit Analysis and Soil Plasticity[M].Amsterdam:Elsevier ScientificPublishing Company,1975.}、Sokolovski{Sokolovski VV.Statics of soil The method was developed by DeJong, Spencer, Dembicki, and others {Chen W.F. Limit Analysis and Soil Plasticity [M]. Amsterdam: Elsevier Scientific Publishing Company, 1975.}. Drucker proposed the plastic limit theorem, which can be used to determine the upper and lower limits of the failure load in stable problems {Drucker DC. A more fundamental approach to plastic stress-strainrelations [A]. US National Congress for Applied Mechanics [C]. 1951.}. Chen Huifa systematically expounded on the upper and lower limit method of limit analysis {Chen WF. Limit Analysis and Soil Plasticity [M]. Amsterdam: Elsevier Scientific Publishing Company, 1975.}.Shen Zhujiang {Shen Zhujiang. Limit Equilibrium of Foundation under Horizontal Load [J]. Journal of Hydraulic Engineering, 1963(05):37-46.} Zhang Xueyan et al. {Zhang Xueyan, Chen Huan. Comparative Analysis of Limit Analysis Methods in Geotechnical Engineering [C] / / National Conference on Geosynthetics. China Geosynthetics Engineering Association, 2008.} Gong Xiaonan et al. {Wang Guoguang, Yan Ping, Gong Xiaonan. Upper Limit Estimation of Vertical Stress of Foundation Soil under Pile Load [J]. Chinese Journal of Geotechnical Engineering, 2003.} Chen Zuyu {Chen Zuyu. Slope Stability The analysis of the upper limit solution of plasticity mechanics [A]. Academic Conference on Soil Mechanics and Foundation Engineering of China Civil Engineering Society [C]. 1994.} Zheng Yingren et al. {Wang Jinglin, Zheng Yingren, Chen Yuyao, et al. Discussion on the upper limit method of limit analysis of geotechnical materials [J]. Rock and Soil Mechanics, 2003, 24(4): 538-544.} Luan Maotian et al. {Luan Maotian, Zhang Qiyi, Yang Qing, et al. Analysis of the upper limit of ultimate bearing capacity of shallow buried rectangular foundation on homogeneous foundation [J]. Ocean Engineering, 2008, 26(2): 9.} Yan Shuwang {Wang MC, Hsieh CW. Collapse Load of Strip Footing Above Circular Void [J]. Journal of Geotechnical Engineering, 1987, 113(5): 511–515.} etc. have successively developed the upper and lower limit methods.

[0003] To address the issue of bearing capacity of underlying cavitary foundations, numerous methods based on limit analysis theory have emerged. Wang et al. summarized three broken-line failure modes of cavitary foundations under central load and derived the upper limit solutions for the ultimate bearing capacity of cavitary foundations under these three failure modes {Wang MC, Hsieh CW. Collapse Load of Strip Footing Above Circular Void[J]. Journal of Geotechnical Engineering, 1987, 113(5): 511–515.}. Leca et al. summarized three conical failure modes of shallow circular tunnel excavation faces under the self-weight of soil and rock and derived the upper and lower bound solutions for the stability of the excavation face {Dormieux L., Leca E.. Upper and lower bound solutions for theface stability of shallow circular tunnels in frictional material[J]. Géotechnique, 1990, 40(4): 581-606.}. Fraldi et al. proposed a dormant collapse mode for the roof of a cavity {Fraldi M., Guarracino F.. Limit analysis of collapse mechanisms in cavities and tunnels according to the Hoek–Brown failure criterion[J]. International Journal of Rock Mechanics and Mining Sciences, 2009, 46(4): 665–673.}. Yang Xiaoli et al. extended the dormant collapse mode to three-dimensional states and obtained an upper bound solution {Yang XL, Huang F.. Three-dimensional failure mechanism of a rectangular cavity in a Hoek–Brown rock medium[J]. International Journal of Rock Mechanics and Mining Sciences, 2013, 61(1): 189–195.}.Liu Hui et al. assumed that the failure mode of the void foundation under central load is a logarithmic spiral and derived the upper limit solution of the ultimate bearing capacity of the foundation above the void {Liu Hui, Yang Feng, Yang Junsheng. Failure mode and ultimate bearing capacity analysis of shallow foundation above void [J]. Rock and Soil Mechanics, 2010, 31(11):3373-3378.}; Kiyosumi et al. assumed that the failure mode of the void foundation under eccentric load is a broken line and used the upper limit method to analyze the influence of the existence of void on the bearing capacity of the foundation {Kiyosumi M., Kusakabe O., Ohuchi M.. Model Tests and Analyses of Bearing Capacity of StripFooting on Stiff Ground with Voids [J]. Journal of Geotechnical and Geoenvironmental Engineering, 2011, 137(4):363-375.}; Lei Yong et al. assumed that the punching failure mode of the pile end rock layer above the cavity is an axisymmetric rotating body, and derived the calculation method of ultimate bearing capacity by the upper limit method {Lei Yong, Chen Qiunan, Ma Binhui. Punching analysis of pile end rock layer based on limit analysis [J]. Chinese Journal of Rock Mechanics and Engineering, 2014, 33(3):631-638.}; Lei Yong et al. assumed that the punching failure mode of the pile end rock layer above the karst cave is a frustum cone and an axisymmetric rotating body, and proposed the calculation method of ultimate bearing capacity of the top plate for these two failure modes {Lei Yong, Yin Junfan, Chen Qiunan, et al. Study on ultimate bearing capacity of karst cave top plate based on limit analysis [J]. Rock and Soil Mechanics, 2017, 38(7):7-13.}.Son assumes that the failure mode above the cavity under the self-weight of the soil and the central load is an inclined sliding surface, and extends the Terzaghi arch formula to inclined sliding surfaces and three-dimensional space {Son M..Three-Dimensional Expansion of the Terzaghi Arching FormulaConsidering Inclined Sliding Surfaces and Examination of Its Effects[J].International Journal of Geomechanics, 2017, 17(7): 06016043.}; Ma Shaokun et al. assumed that the failure mode of the small kiln goaf cavity under the self-weight of the rock and soil and the central load is a broken line shape, and proposed an analytical method to verify the stability of the overlying and sidewall rock and soil of the shallow buried roadway in the small kiln goaf {Ma Shaokun, Li Tao, Shao Yu. Analysis of soil nailing reinforcement of overlying rock and soil in the shallow buried roadway in the small kiln goaf [J]. Journal of Guangxi University (Natural Science Edition), 2018, 43(4): 1455-1465.} {Invention Patent 201710727470.1} {201810144469.0}; Yin Junfan et al. assumed that the top plate of the karst cave under axisymmetric and eccentric punching shear load The failure mode is a symmetrical rotating body. Based on the upper limit method, a calculation method for the ultimate bearing capacity of the cave roof is proposed, and an estimation method for the range of punching shear failure is given {Yin Junfan, Lei Yong, Chen Qiunan, et al. Analysis of the upper limit of punching shear failure of cave roof under eccentric loading [J]. Rock and Soil Mechanics, 2018, 39(8):2837-2843.}; Lei Yong et al. assumed that the failure mode of the cave roof under punching shear load is a rotating body with a helix as the generatrix, and derived the upper limit solution of the three-dimensional ultimate bearing capacity of the cave roof {Lei Yong, Liu Yixin, Deng Jiazheng, et al. Calculation of ultimate bearing capacity of cave roof under punching shear failure mode [J]. Chinese Journal of Rock Mechanics and Engineering, 2018, 37(9):8.}. Yuan Wei et al. assumed that the failure mode of the karst cave roof under punching shear load is a symmetrical rotating body, and derived the calculation formula of the critical thickness-to-diameter ratio of the karst cave roof using the upper limit method {Yuan Wei, Liu Shangge, Nie Qingke, et al. Research on the determination method of critical thickness of karst cave roof at the pile tip of rock-socketed pile based on punching shear failure mode[J]. Rock and Soil Mechanics, 2019, 40(7):2789-2798.}; Lei Yong et al. assumed that the failure mode of the hollow rock foundation under eccentric load is a double logarithmic spiral, and derived the calculation expression of the ultimate bearing capacity using the upper limit method {Lei Yong, Deng Jiazheng, Liu Zeyu, et al. Calculation method of ultimate bearing capacity of hollow rock foundation considering load position offset[J]. Rock and Soil Mechanics, 2020, 41(10):3326-3332.}.

[0004] The literature by Lei Yong et al. (2014, 2017, 2018), Yin Junfan et al. (2018), and Yuan Wei et al. (2019) addresses the case of punching shear at the top of the cavity, while the literature by Wang et al. (1987), Liu Hui et al. (2010), Son (2017), and Ma Shaokun et al. addresses the case of a cavity foundation with a central load on the ground surface. None of these studies meet the load conditions of an offset cavity foundation. The literature by Leca et al. (1990), Fraldi et al. (2009), and Yang Xiaoli et al. (2013) only considers the self-weight of the soil and rock, without considering the influence of eccentric load. Although the literature by Kiyosumi et al. (2011) considers the case of load offset from the cavity foundation, it does not consider the influence of the magnitude of the ground load on the failure mode of the cavity foundation. Although Lei Yong et al. (2020) considered the case of load offset void foundation, their theory requires the assumption that the load size and void span are consistent, ignores the influence of the soil and rock self-weight load, and does not consider the influence of the ground load on the failure mode of void foundation. Summary of the Invention

[0005] The technical problem to be solved by this invention is to consider the case of load offset in cavitary foundations, the influence of soil and rock self-weight loads, and the impact of ground load on the failure mode of cavitary foundations, and to provide a method for evaluating the safety of cavitary foundations under offset loads, which is also easy to apply in engineering.

[0006] To address the aforementioned problems, this invention provides a method for evaluating the safety of cavitary foundations under offset loads, comprising the following steps: A. Specifying constant values ​​and initial values ​​of variables.

[0007] The underground cavity is simplified as a rectangular cavity. The analysis and calculation are performed on the cavity foundation under a unit length offset load along the cavity axis. All analysis and calculation are limited to the plane perpendicular to the cavity axis.

[0008] Offset loads are distributed loads acting vertically on the ground. An offset load has two edges, with the edge closer to the cavity being called the near edge and the edge farther from the cavity being called the far edge.

[0009] The cavity has two sidewalls. The sidewall closer to the far edge of the offset load is called the near sidewall of the cavity, and the sidewall farther from the far edge of the offset load is called the far sidewall of the cavity.

[0010] The near edge of the offset load coincides with the near sidewall of the cavity in the horizontal projection direction, while the far edge of the offset load is located at infinity. The offset load is simplified to a uniformly distributed load.

[0011] Under offset load, the surrounding soil and rock of the cavity exhibit a top straight sliding surface and an outer broken sliding surface. The top straight sliding surface is an inclined straight line starting from the top of the far sidewall of the cavity, moving towards the direction of the offset load, and reaching the ground. The outer broken sliding surface starts from the bottom of the near sidewall, developing obliquely upward in a straight line, and after reaching the inflection point at the top height of the cavity, it develops towards the ground at a greater angle.

[0012] Defined constant values: c, γ, b, h, H, p, η,

[0013] Specify the initial value of the variable: R m ;

[0014] B. Iterative calculation of intensity reduction factor

[0015] Calculated according to formula (1),

[0016]

[0017] When |R m -R m+1 When | > ε, the following text will be executed in a loop.

[0018] Solve equation (2) to obtain θ.

[0019]

[0020] If the calculated value θ > 1, then the cutoff is θ = 1.

[0021] If the calculated value θ < 0, then it is truncated to θ = 0.

[0022] Calculate sequentially according to equations (3), (4), (5), (6), (7), (8), (9), and (10).

[0023]

[0024]

[0025]

[0026]

[0027] A1=γ(ηhθ+0.5ηHθ-e0θ+e0)H+p{[η(H+h)-e0]θ+e0} (7)

[0028]

[0029] A3=0.5γh[(ηh-e0)θ+e0] (9)

[0030]

[0031] At the end of the loop, execute expression (11).

[0032] R = R m+1 (11)

[0033] R is the strength reduction factor to be calculated;

[0034] C. Evaluating the safety of void foundations under offset loads

[0035] R is considered the safety factor of the cavitary foundation under offset load. If R > 1, the cavitary foundation under offset load is considered safe; if R < 1, the cavitary foundation under offset load is considered unsafe; if R = 1, the cavitary foundation under offset load is considered to be in a limit state.

[0036] End of step A to step C;

[0037] The symbols in the above formulas and steps have the following meanings: e0 is the horizontal distance from the inflection point of the outer broken sliding surface to the near sidewall of the cavity when the offset load is 0. Let be the internal friction angle of the soil and rock; c be the cohesion of the soil and rock; h be the height of the cavity; and R be the value of c and when the underlying cavity foundation reaches its limit state. Strength reduction factor; R m Let R be the value of R in the m-th incremental step during the iterative solution of R. m+1 In the process of iteratively solving R, R is the (m+1)th increment step; the subscript m is the iteration order variable; H is the vertical distance from the top of the cavity to the ground; γ is the unit weight of the rock and soil of each slider; η is 1; θ is the parameter describing the failure mode; p is the magnitude of the ground load per unit area; ε is an infinitesimal quantity; b is the width of the cavity; π is pi; B1, B2, B3, B4, A1, A2, and A3 are the variables defined by equations (3), (4), (5), (6), (7), (8), and (9), respectively.

[0038] The θ in equation (2) is solved using the Newton-Raphson iteration method, with the initial value of the variable specified as: θ n ,

[0039] When |θ n -θ n+1 When | > ε, the following text will be executed in a loop.

[0040] Calculate according to equations (12), (13), and (14) in sequence.

[0041]

[0042]

[0043] At the end of the loop, execute expression (15).

[0044] θ=θ n+1 (15)

[0045] In the loop: the subscript n is the iteration order variable; θ n In the process of solving θ in equation (2) using the Newton-Raphson iterative method, θ in the nth increment step; θ n+1 In the process of solving θ in equation (2) using the Newton-Raphson iterative method, the θ in the (n+1)th increment step; f(θ) n f′(θ) is a function defined by equation (12); n ) is the derivative of equation (12) with respect to θ.

[0046] The principle of this invention:

[0047] This invention is based on the upper limit theory of soil mechanics limit analysis.

[0048] For ease of analysis, the underground cavity is simplified as a rectangular cavity. The analysis and calculation are performed on the cavity foundation under a unit-length offset load along the cavity's axis. All analyses and calculations are limited to a plane perpendicular to the cavity's axis.

[0049] Offset loads are distributed loads that act vertically on the ground. An offset load has two edges, with the edge closer to the cavity being called the near edge of the offset load, and the edge farther from the cavity being called the far edge of the offset load.

[0050] The cavity has two sidewalls. The sidewall closer to the far edge of the offset load is called the near sidewall of the cavity, and the sidewall farther from the far edge of the offset load is called the far sidewall of the cavity.

[0051] For safety reasons, regulatory authorities typically prohibit applying ground loads within the horizontal projection range of a cavity; however, for profit, construction companies often place ground loads as close as possible to the edge of the cavity in the horizontal projection direction. Therefore, this invention stipulates that the near edge of the offset load coincides with the near sidewall of the cavity in the horizontal projection direction. Since the width of the ground load in actual engineering projects is much larger than the width of the cavity, this invention approximates the far edge of the offset load as being at infinity. This invention simplifies the offset load to a uniformly distributed load.

[0052] The simulation showed that, under offset load, the surrounding soil and rock of the cavity exhibited a straight sliding surface at the top and a broken sliding surface on the outer side. The starting point of the straight sliding surface at the top was at the top of the far sidewall of the cavity; the starting point of the broken sliding surface on the outer sidewall was at the bottom of the near sidewall of the cavity.

[0053] Simulations showed that when the offset load approached zero, the straight sliding surface of the top plate approached the vertical line; the outer polygonal sliding surface approached the Terzaghi sliding surface. The Terzaghi sliding surface begins near the bottom of the sidewall and forms an angle with the vertical line of [missing information]. It extends diagonally upwards in a straight line, then, after reaching the top of the cavity, it extends vertically towards the ground. Here, π represents the mathematical constant pi. The equivalent internal friction angle is approximated as the internal friction angle.

[0054] In the simulation, it was observed that when the offset load approaches infinity, the straight sliding surface of the top plate approaches a straight line from the top of the far sidewall of the cavity to the near edge of the offset load; the outer broken sliding surface approaches a straight line with an angle of π / 4 with the vertical reaching the ground.

[0055] Simulations showed that when the offset load was between 0 and infinity, the straight sliding surface of the top plate and the zigzag sliding surface of the outer side exhibited intermediate degrees between these two extremes. The straight sliding surface of the top plate was a sloping straight line starting from the top of the far sidewall of the cavity, moving towards the direction of the offset load, and reaching the ground. The zigzag sliding surface of the outer side started from the bottom of the near sidewall, developing obliquely upward in a straight line, and after reaching the inflection point at the top height of the cavity, it developed towards the ground at a greater angle. The changes in the magnitude of the offset load, the changes in the inclination of the straight sliding surface of the top plate, and the changes in the inclination of the zigzag sliding surface of the outer side were synchronous.

[0056] The above observations can be summarized as follows: the magnitude of the ground load is strongly correlated with the inclination of the straight sliding surface and the outer broken sliding surface of the top plate. The inventors describe the changes in the straight sliding surface and the outer broken sliding surface between these two extreme degrees as linear synchronization, i.e.:

[0057] d=(d1-d0)θ+d0 (16)

[0058] e = (e1 - e0)θ + e0 (17)

[0059] f = bθ (18)

[0060] Where: d0 is the horizontal distance from the intersection of the outer polygonal sliding surface and the ground to the near edge of the offset load when the offset load is 0; d1 is the horizontal distance from the intersection of the outer polygonal sliding surface and the ground to the near edge of the offset load when the offset load is infinite; d is the horizontal distance from the intersection of the outer polygonal sliding surface and the ground to the near edge of the offset load when the offset load is between 0 and infinity; e0 is the distance from the inflection point of the outer polygonal sliding surface to the cavity when the offset load is 0. The horizontal distance between the near sidewalls; e1 is the horizontal distance from the inflection point of the outer broken-line sliding surface to the near sidewall of the cavity when the offset load is infinite; e is the horizontal distance from the inflection point of the outer broken-line sliding surface to the near sidewall of the cavity when the offset load is between 0 and infinity; f is the horizontal distance from the intersection of the straight sliding surface of the top plate and the ground to the far sidewall of the cavity when the offset load is between 0 and infinity; θ is the parameter describing the failure mode; b is the width of the cavity.

[0061] Based on the geometric relationships of the cavity, the ground, and the outer broken-line slip surface, and the theory of the Terzaghi slip surface, we have:

[0062] d1=η(H+h) (19)

[0063] e1=ηh (20)

[0064] d0=e0 (21)

[0065]

[0066] in: denoted as θ, where θ is the internal friction angle of the soil and rock; h is the height of the cavity; H is the vertical distance from the top of the cavity to the ground surface; η is the ratio of d1 to H+h, and for a cavity foundation under offset load, η is 1. Substituting equations (19), (20), and (21) into equations (16) and (17) yields:

[0067] d=[η(H+h)-e0]θ+e0 (22)

[0068] e=(ηh-e0)θ+e0 (23)

[0069] Definition: j = de (24)

[0070] j1 = d1 - e1 (25)

[0071] Where: j is the difference between d and e; j1 is the difference between d1 and e1.

[0072] Substituting equations (19), (20), (22), and (23) into equations (24) and (25), we obtain:

[0073] j=ηHθ (26)

[0074] j1=ηH (27)

[0075] In the simulation, it was observed that during the loading process, the top of the cavity collapsed, and the near-side wall of the cavity underwent horizontal displacement. Therefore, two velocity discontinuities exist within the sliding body enclosed by the straight sliding surface of the top plate, the ground, the outer broken sliding surface, and the cavity. The first velocity discontinuity is a straight line connecting the top of the near-side wall of the cavity and the turning point of the outer broken sliding surface, called the "horizontal velocity discontinuity line of the cavity side"; the second velocity discontinuity is a straight line perpendicularly extending from the top of the near-side wall of the cavity to the ground, called the "vertical velocity discontinuity line of the cavity top". The area enclosed by the vertical velocity discontinuity line of the cavity top, the ground, the outer broken sliding surface, and the horizontal velocity discontinuity line of the cavity side is called the "loaded slider". The area enclosed by the horizontal velocity discontinuity line of the cavity side, the outer broken sliding surface, and the cavity is called the "cavity side slider". The area enclosed by the vertical velocity discontinuity line of the cavity top, the cavity, the straight sliding surface of the top plate, and the ground is called the "cavity top slider".

[0076] Based on the geometry of the loaded slider and the outer polygonal sliding surface:

[0077]

[0078] Where: v1 is the magnitude of the sliding velocity above the inflection point of the outer polygonal sliding surface; v 1v Let be the vertical component of the sliding velocity of the loaded slider. Combining equations (26) and (28), we get:

[0079]

[0080] Similarly, we can obtain:

[0081]

[0082] Where: v 1h Let be the horizontal component of the sliding velocity of the loaded slider. Since the loaded slider and the slider at the top of the tunnel are separated only by the vertical velocity discontinuity line at the top of the tunnel, and this line is perpendicular, the horizontal velocities of the two sliders are equal. Therefore:

[0083]

[0084] Where: v 2h Let be the horizontal component of the sliding velocity of the slider on the tunnel ceiling. This is derived from the geometric relationship between the slider and the linear sliding surface of the ceiling:

[0085]

[0086] Substituting equations (18) and (31) into equation (32) and rearranging, we get:

[0087]

[0088] Where: v2 is the modulus of the sliding velocity of the linear sliding surface of the top plate. The vertical component of the sliding velocity of the top slider is:

[0089]

[0090] Where: v 2v Let be the vertical component of the sliding velocity of the slider at the top of the tunnel. Substituting equations (18) and (33) into equation (34), we get:

[0091]

[0092] At the points where the vertical velocity discontinuity is located on the cave ceiling:

[0093]

[0094] Where: v 5v This represents the vertical component of the discontinuous velocity at the vertical velocity discontinuity line at the top of the tunnel.

[0095]

[0096] Where: v5 is the modulus of the discontinuous velocity at the vertical velocity discontinuity line at the top of the tunnel.

[0097] Because the loaded slider and the slider on the side of the tunnel are separated only by a horizontal velocity discontinuity line on the side of the tunnel, and this horizontal velocity discontinuity line is horizontal, the two sliders have the same velocity in the vertical direction. Therefore:

[0098] v 3v =v 1v (38)

[0099] Where: v 3v Let be the vertical component of the sliding velocity of the slider on the side of the hole. Based on the geometric relationship between the loaded slider and the outer polygonal sliding surface:

[0100]

[0101] Where: v3 is the magnitude of the sliding velocity of the portion below the inflection point of the outer polygonal sliding surface. Substituting equations (23) and (38) into equation (39) and rearranging, we obtain...

[0102]

[0103] Similarly, we can obtain:

[0104]

[0105] Where: v 3hLet be the horizontal component of the sliding velocity of the slider on the side of the hole. Substituting equations (23) and (40) into equation (41), we get:

[0106]

[0107] At the discontinuity of the horizontal velocity on the side of the tunnel:

[0108]

[0109] Where: v 4h This represents the horizontal component of the discontinuous velocity at the discontinuity line on the side of the tunnel.

[0110]

[0111] Where: v4 is the modulus of the discontinuous velocity at the horizontal velocity discontinuity line on the side of the hole. The gravitational force acting on each slider is:

[0112] G1=0.5γ(e+d)H (45)

[0113] G2=0.5γ(2b-f)H ​​(46)

[0114] G3=0.5γeh (47)

[0115] G p =pd (48)

[0116] Where: G1 is the weight of the loaded slider; G2 is the weight of the top slider; G3 is the weight of the side slider; G p γ represents the total ground load acting on the top of the loaded slider; p represents the magnitude of the ground load per unit area; and γ represents the unit weight of the soil and rock in each slider.

[0117] Substituting equations (18), (22), and (23) into equations (45), (46), and (47), we obtain:

[0118] G1=γ(ηhθ+0.5ηHθ-e0θ+e0)H (49)

[0119] G2=0.5γbH(2-θ) (50)

[0120] G3=0.5γh[(ηh-e0)θ+e0] (51)

[0121] Substituting the expression into equation (48), we get:

[0122] G p =p{[η(H+h)-e0]θ+e0} (52)

[0123] According to the upper bound theory of limit analysis, in the limit state we have:

[0124]

[0125]

[0126]

[0127]

[0128] in: This represents the sum of the external power experienced by each slider in the limiting state. This refers to the external power experienced by the loaded slider under extreme conditions. This represents the external power delivered to the slide block at the top of the tunnel under extreme conditions. This represents the external power experienced by the slider on the side of the hole under extreme conditions.

[0129] According to the upper limit theory of limit analysis, in the limit state, the sum of the internal power of each slider is:

[0130]

[0131] in: This is the sum of the internal power of each slider in the limiting state; The internal power is the portion above the inflection point of the outer polygonal sliding surface; The internal power of the linear sliding surface of the top plate; The internal power is the portion below the inflection point of the outer polygonal sliding surface; The internal power of the horizontal velocity discontinuity line on the side of the tunnel; The internal power of the vertical velocity discontinuity line at the top of the tunnel.

[0132] According to the upper limit theory of limit analysis and the Mohr-Coulomb strength theory, the internal power on each sliding surface in the limit state is:

[0133]

[0134]

[0135]

[0136] Where: c is the cohesion of the soil and rock; R is the value of c and R when the underlying cavity foundation reaches its limit state. The strength reduction factor. Substituting equations (29) and (26) into equation (58) yields:

[0137]

[0138] Substituting equations (33) and (18) into equation (59), we get:

[0139]

[0140] Substituting equations (40) and (23) into equation (60), we get:

[0141]

[0142] According to the upper limit theory of limit analysis and the Mohr-Coulomb strength theory, the internal power on each rate discontinuity line in the limiting state is:

[0143]

[0144]

[0145] Substituting equations (23) and (44) into equation (64), we get:

[0146]

[0147] Substituting equation (36) into equation (65), we get:

[0148]

[0149] Let be the internal power of the theoretical vertical velocity discontinuity at the tunnel top. Although a vertical velocity discontinuity at the tunnel top is observed in the simulation, it is not significant and therefore will not be considered in the subsequent derivation. According to the principle of virtual power:

[0150]

[0151] in: Let be the functional of the sum of the internal and external powers of each slider in the limiting state. According to the principle of minimum potential energy, we take the first variation of equation (68), and then find θ at the extreme point of the functional:

[0152]

[0153] Where: δ is the variational symbol. θ at the extreme point of the functional is the θ corresponding to the real sliding surface, but the solution process is extremely cumbersome and the expression of θ is very complicated, making it unsuitable for engineering practice. In the simulation, it was observed that the magnitude of the ground load is strongly correlated with the inclination of the straight sliding surface of the top plate and the outward broken sliding surface. In engineering practice, this correlation is usually simplified to a direct causal relationship, that is, the change in the magnitude of the ground load causes the change in the inclination of the part above the turning point of the straight sliding surface of the top plate, the part below the turning point of the outward broken sliding surface, and the interaction of each local area of ​​the underlying cavity foundation is ignored. In order to facilitate engineering application, the inventors proposed a hypothesis based on this conventional method: among all possible local area failure modes of the underlying cavity foundation, the failure mode that minimizes the potential power functional of the local area is approximately the real failure mode. That is, the principle of minimum potential power in the local area. From equation (69):

[0154]

[0155] in: It is a functional representing the sum of the internal and external power of the loaded slider under limiting conditions.

[0156] Substituting equations (54) and (61) into equation (70), and rearranging, we get:

[0157]

[0158] right Taking the second variation, we get:

[0159]

[0160] Where: ∝ represents the direct proportion sign. In practical engineering... It is usually greater than 0, meaning that in practical engineering, the functional... The minimum value is obtained by solving equation (71). However, R in equation (71) cannot be obtained in advance. Therefore, we first assume an R in the neighborhood of R=1 to calculate θ, then perform subsequent calculations to obtain R for the next round, and then substitute the R of the next round back into equation (71) to calculate θ for the next round. Then perform subsequent calculations to obtain R for the next round. This process is repeated iteratively until R tends to stabilize. The iterative form of equation (71) is:

[0161]

[0162] Where: subscript m is the iteration order variable; R m Let R be the value of R at the m-th increment during the iterative solution of R. m The initial value is chosen in R mThe neighborhood of θ = 1. Since θ is approximate, in actual calculations, the calculated value θ > 1 or θ < 0 may occur. If the calculated value θ > 1, then it is truncated to θ = 1; if the calculated value θ < 0, then it is truncated to θ = 0.

[0163] Combining equations (53), (54), (55), (56), (57), (61), (62), (63), (66), and (68), we obtain:

[0164]

[0165] Where: R m+1 Let R be the value of the (m+1)th increment in the iterative solution of R. Other variables are defined as follows:

[0166]

[0167]

[0168]

[0169]

[0170] A1=γ(ηhθ+0.5ηHθ-e0θ+e0)H+p{[η(H+h)-e0]θ+e0} (7)

[0171]

[0172] A3=0.5γh[(ηh-e0)θ+e0] (9)

[0173] Where: B1, B2, B3, B4, A1, A2, A3 are the variables defined in equations (3), (4), (5), (6), (7), (8), and (9), respectively.

[0174] Alternatively, the Newton-Raphson iteration method is used to solve for θ in equation (2), let:

[0175]

[0176] Where: subscript n is the iteration order variable; θ n In the process of solving θ in equation (2) using the Newton-Raphson iterative method, θ in the nth increment step; θ n+1 In the process of solving θ in equation (2) using the Newton-Raphson iteration method, θ is obtained in the (n+1)th increment step. n The initial value is chosen at θ n =0.5 neighborhood.

[0177] Find the equation (12) with respect to θn The derivative:

[0178]

[0179] Where: f(θ) n f′(θ) is a function defined by equation (12); n ) is the derivative of equation (12) with respect to θ. The Newton-Raphson iterative formula is:

[0180]

[0181] The beneficial effects of this invention are:

[0182] 1. Equations (2) and (7) are used to consider the case of load offset void foundation;

[0183] 2. The influence of the self-weight load of soil and rock was considered through equations (2), (7), (8) and (9);

[0184] 3. Equations (2) to (9) are used to consider the influence of ground load on the failure mode of the cavitary foundation.

[0185] 4. Formula (2) makes the technical solution easier to apply in engineering. Attached Figure Description

[0186] The specific embodiments of the present invention will be further described in detail below with reference to the accompanying drawings, wherein:

[0187] Figure 1 This is a schematic diagram of a cavity foundation under offset load.

[0188] Figure 2 A schematic diagram showing the failure modes corresponding to different unit area offset loads;

[0189] Figure 3 A schematic diagram showing the discontinuity lines between each slider and each speed;

[0190] Figure 4 This is a contour map of the plastic strain of a cavitary foundation under offset load simulated by finite element method when p = 25000 Pa.

[0191] Figure 5 This is a contour map of the plastic strain of a cavitary foundation under offset load in finite element simulation when p = 50000 Pa.

[0192] Figure 6 This is a contour map of the plastic strain of a cavitary foundation under offset load in finite element simulation when p = 100000 Pa.

[0193] Figure 7This is a contour map of the horizontal displacement of a cavity foundation under offset load simulated by finite element analysis when p = 50000 Pa.

[0194] Figure 8 This is a contour map of the vertical displacement of a cavity foundation under offset load in finite element simulation when p = 50000 Pa.

[0195] Figure 9 This is a comparison of contour maps of plastic strain in a cavitary foundation under offset load in finite element simulation with theoretical values ​​when p = 50000 Pa.

[0196] The diagram is labeled as follows: 1 represents the foundation; 11 represents the outer zigzag sliding surface; 12 represents the top plate straight sliding surface; 111 represents the turning point of the outer zigzag sliding surface; 14 represents the horizontal velocity discontinuity line on the side of the tunnel; 15 represents the vertical velocity discontinuity line on the top of the tunnel; 2 represents the cavity; 21 represents the near sidewall of the cavity; 22 represents the far sidewall of the cavity; 3 represents the ground; 4 represents the offset load; 41 represents the far edge of the offset load; 42 represents the near edge of the offset load; 5 represents the vertical line; 61 represents the loaded slider; 62 represents the top slider of the tunnel; 63 represents the side slider of the tunnel; e0 represents the horizontal distance from the turning point of the outer zigzag sliding surface to the near sidewall of the cavity when the offset load is 0; h represents the height of the cavity; H represents the vertical distance from the top of the cavity to the ground; b represents the width of the cavity. θ is the internal friction angle of the soil and rock; p is the magnitude of the ground load per unit area; π is pi; ∞ is infinity. Detailed Implementation

[0197] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0198] A method for evaluating the safety of cavitary foundations under offset loads includes the following steps.

[0199] A. Specify constant values ​​and initial values ​​for variables

[0200] The underground cavity is simplified as a rectangular cavity. The analysis and calculation are performed on the cavity foundation under a unit length offset load along the cavity axis. All analysis and calculation are limited to the plane perpendicular to the cavity axis.

[0201] Offset loads are distributed loads acting vertically on the ground. An offset load has two edges, with the edge closer to the cavity being called the near edge and the edge farther from the cavity being called the far edge.

[0202] The cavity has two sidewalls. The sidewall closer to the far edge of the offset load is called the near sidewall of the cavity, and the sidewall farther from the far edge of the offset load is called the far sidewall of the cavity.

[0203] The near edge of the offset load coincides with the near sidewall of the cavity in the horizontal projection direction, while the far edge of the offset load is located at infinity. The offset load is simplified to a uniformly distributed load.

[0204] Under offset load, the surrounding soil and rock of the cavity exhibit a top straight sliding surface and an outer broken sliding surface. The top straight sliding surface is an inclined straight line starting from the top of the far sidewall of the cavity, moving towards the direction of the offset load, and reaching the ground. The outer broken sliding surface starts from the bottom of the near sidewall, developing obliquely upward in a straight line, and after reaching the inflection point at the top height of the cavity, it develops towards the ground at a greater angle.

[0205] Defined constant values: c, γ, b, h, H, p, η,

[0206] c = 41000 Pa, γ = 18500 N / m 3 Given b = 2m, h = 3m, H = 3m, p = 50000Pa, η = 1, and specify the initial value of the variable as R. m ;R m =1,

[0207] B. Iterative calculation of intensity reduction factor

[0208] Calculated according to formula (1),

[0209]

[0210] e0 = 1.329m,

[0211] When |R m -R m+1 When | > ε, the following text will be executed in a loop.

[0212] Solve equation (2) to obtain θ.

[0213]

[0214] θ = 0.820,

[0215] If the calculated value θ > 1, then the cutoff is θ = 1.

[0216] If the calculated value θ < 0, then it is truncated to θ = 0.

[0217] Calculate sequentially according to equations (3), (4), (5), (6), (7), (8), (9), and (10).

[0218]

[0219] B1 = 357055.748, only the result of the last iteration is shown.

[0220]

[0221] B2 = 1540152.580, only the result of the last iteration is shown.

[0222]

[0223] B3 = 258616.239, only the result of the last iteration is shown.

[0224]

[0225] B4 = 19623.191, only showing the result of the last iteration.

[0226] A1=γ(ηhθ+0.5ηHθ-e0θ+e0)H+p{[η(H+h)-e0]θ+e0} (7)

[0227] A1 = 560381.007, only the result of the last iteration is shown.

[0228]

[0229] A2 = 959106.408, only the result of the last iteration is shown.

[0230] A3=0.5γh[(ηh-e0)θ+e0] (9)

[0231] A3 = 52453.610, only the result of the last iteration is shown.

[0232]

[0233] R m+1 =1.384, only showing the result of the last iteration.

[0234] At the end of the loop, execute expression (11).

[0235] R = R m+1 (11)

[0236] R = 1.384,

[0237] R is the strength reduction factor to be calculated;

[0238] C. Evaluating the safety of void foundations under offset loads

[0239] R is considered the safety factor of the cavitary foundation under offset load. If R > 1, the cavitary foundation under offset load is considered safe; if R < 1, the cavitary foundation under offset load is considered unsafe; if R = 1, the cavitary foundation under offset load is considered to be in a limit state and therefore safe.

[0240] End of step A to step C;

[0241] The symbols in the above formulas and steps have the following meanings: e0 is the horizontal distance from the inflection point of the outer broken sliding surface to the near sidewall of the cavity when the offset load is 0. Let be the internal friction angle of the soil and rock; c be the cohesion of the soil and rock; h be the height of the cavity; and R be the value of c and when the underlying cavity foundation reaches its limit state. Strength reduction factor; R m Let R be the value of R in the m-th incremental step during the iterative solution of R. m+1 In the process of iteratively solving R, R is the (m+1)th increment step; the subscript m is the iteration order variable; H is the vertical distance from the top of the cavity to the ground; γ is the unit weight of the rock and soil of each slider; η is 1; θ is the parameter describing the failure mode; p is the magnitude of the ground load per unit area; ε is an infinitesimal quantity; b is the width of the cavity; π is pi; B1, B2, B3, B4, A1, A2, and A3 are the variables defined by equations (3), (4), (5), (6), (7), (8), and (9), respectively.

[0242] The aforementioned method for evaluating the safety of cavitary foundations under offset loads uses the Newton-Raphson iteration method to solve for θ in equation (2), with the initial value of the variable specified as: θ n θ n =0.5,

[0243] When |θ n -θ n+1 When | > ε, the following text will be executed in a loop.

[0244] Calculate according to equations (12), (13), and (14) in sequence.

[0245]

[0246] f(θ n =32.832, only the result of the last iteration is shown.

[0247]

[0248] f′(θ n =602059.210, only the result of the last iteration is shown.

[0249]

[0250] θ n+1 =0.802, only showing the result of the last iteration.

[0251] At the end of the loop, execute expression (15).

[0252] θ=θ n+1 (15)

[0253] θ = 0.802, only the result of the last iteration is shown.

[0254] In the loop: the subscript n is the iteration order variable; θ n In the process of solving θ in equation (2) using the Newton-Raphson iterative method, θ in the nth increment step; θ n+1 In the process of solving θ in equation (2) using the Newton-Raphson iterative method, the θ in the (n+1)th increment step; f(θ) n f′(θ) is a function defined by equation (12); n ) is the derivative of equation (12) with respect to θ.

[0255] Finite element method simulation was performed on a cavitary foundation under offset load. The elastic modulus of the soil and rock was 16,000,000 N / m², and the Poisson's ratio was 0.25. c=41000N / ㎡,γ=18500N / m 3 The soil and rock element type used is an 8-node bi-quadratic plane strain quadrilateral, with reduced integration. Infinite elements are set at the left, right, and bottom edges of the soil and rock element, with the element type being a 5-node quadratic plane strain unidirectional infinite quadrilateral, with reduced integration. The first analysis step is stress equilibrium; the second analysis step is excavation of the cavity; the third analysis step is application of ground load; and the fourth analysis step is Mohr-Coulomb material strength reduction. When the finite element iteration fails to converge, the cavity foundation is considered to have reached its limit state. This embodiment lists simulation results and theoretical calculation results for different working conditions:

[0256] 1 100000 2x3 2 1.152 1.200 1 4.17％ 2 100000 2.5x2.5 3 1.182 1.221 1 3.30％ 3 100000 3x2 4 1.198 1.187 1 0.92％ 4 50000 2x3 3 1.491 1.554 0.82 4.23％ 5a 25000 2.5x2.5 4 1.659 1.334 0.63 19.59％ 5b 50000 2.5x2.5 4 1.486 1.384 0.802 6.86％ 5c 100000 2.5x2.5 4 1.226 1.249 1 1.88％ 6 50000 3x2 2 1.439 1.544 0.732 7.30％ 7 25000 2x3 4 1.600 1.636 0.664 2.25％ 8 25000 2.5x2.5 2 1.746 1.641 0.563 6.01％ 9 25000 3x2 3 1.617 1.278 0.55 20.96％

[0257] Only the simulation results for conditions 5a, 5b, and 5c are shown in detail:

[0258] When p = 25000 Pa, the contour map of plastic strain of cavitary foundation under offset load simulated by finite element method is as follows: Figure 4 As shown; when p = 50000 Pa, the contour map of plastic strain of cavitary foundation under offset load in finite element simulation is as follows. Figure 5 As shown; when p = 100000 Pa, the contour map of plastic strain of cavitary foundation under offset load in finite element simulation is as follows. Figure 6 As shown. From Figure 4 , Figure 5 , Figure 6 It was observed that the surrounding soil and rock of the cavity under offset load exhibited a linear sliding surface at the top and a zigzag sliding surface on the outer side. The linear sliding surface at the top began at the top of the far sidewall of the cavity; the zigzag sliding surface on the outer side began at the bottom of the near sidewall of the cavity. The linear sliding surface at the top was a sloping straight line starting from the top of the far sidewall of the cavity, moving towards the direction of the offset load, and reaching the ground. The zigzag sliding surface on the outer side began from the bottom of the near sidewall, developing in a straight line diagonally upwards, and after reaching the inflection point at the top of the cavity, it developed towards the ground at a greater angle. Observations were made sequentially. Figure 4 , Figure 5 , Figure 6 The changes in the magnitude of the offset load p, the inclination of the straight sliding surface of the top plate, and the inclination of the outer broken sliding surface are synchronous. In other words, the magnitude of the ground load is strongly correlated with the inclination of the straight sliding surface of the top plate and the outer broken sliding surface.

[0259] When p = 50000 Pa, the contour map of the horizontal displacement of the cavitary foundation under the offset load simulated by the finite element method is as follows: Figure 7 As shown; when p = 50000 Pa, the contour map of the vertical displacement of the cavitary foundation under the offset load simulated by the finite element method is as follows. Figure 8 As shown. During loading, the top of the cavity collapsed, and the near-side wall of the cavity underwent horizontal displacement. From Figure 7 A clear "discontinuity in the horizontal velocity along the tunnel side" was observed. Although from Figure 8 A "discontinuity line of vertical velocity at the top of the tunnel" was observed, but it was not obvious.

[0260] Contour maps of plastic strain in cavitary foundations under offset loads, compared with theoretical data, are shown below. Figure 9 As shown. The thin solid lines represent the simulated contour lines of plastic strain; the thick solid lines represent the theoretical outer broken-line sliding surface, the roof straight sliding surface, and the horizontal velocity discontinuity lines on the tunnel side. The theoretical outer broken-line sliding surface, the roof straight sliding surface, and the horizontal velocity discontinuity lines on the tunnel side roughly coincide with the simulated contour lines of plastic strain.

[0261] The data in the table shows that the safety factor R obtained from finite element simulation and theoretical calculation has a large error under conditions 5a and 9. However, the ground load p is small in these conditions, and this error represents an underestimation rather than an overestimation of the safety factor. Therefore, the technical solution of this invention will not lead to any safety hazards.

Claims

1. A method for evaluating the safety of cavitary foundations under offset loads, characterized in that: Includes the following steps, A. Specify constant values ​​and initial values ​​for variables The underground cavity is simplified as a rectangular cavity. The analysis and calculation are performed on the cavity foundation under a unit length offset load along the cavity axis. All analysis and calculation are limited to the plane perpendicular to the cavity axis. Offset loads are distributed loads acting vertically on the ground. An offset load has two edges, with the edge closer to the cavity being called the near edge and the edge farther from the cavity being called the far edge. The cavity has two sidewalls. The sidewall closer to the far edge of the offset load is called the near sidewall of the cavity, and the sidewall farther from the far edge of the offset load is called the far sidewall of the cavity. The near edge of the offset load coincides with the near sidewall of the cavity in the horizontal projection direction, while the far edge of the offset load is located at infinity. The offset load is simplified to a uniformly distributed load. Under offset load, the surrounding soil and rock of the cavity exhibit a top straight sliding surface and an outer broken sliding surface. The top straight sliding surface is an inclined straight line starting from the top of the far sidewall of the cavity, moving towards the direction of the offset load, and reaching the ground. The outer broken sliding surface starts from the bottom of the near sidewall, developing obliquely upward in a straight line, and after reaching the inflection point at the top height of the cavity, it develops towards the ground at a greater angle. Defined constant values: c, γ, b, h, H, p, η, Specify the initial value of the variable: R m ; B. Iterative calculation of intensity reduction factor Calculated according to formula (1), When |R m -R m+1 When | > ε, the following text will be executed in a loop. Solve equation (2) to obtain θ. If the calculated value θ > 1, then the cutoff is θ = 1. If the calculated value θ < 0, then it is truncated to θ = 0. Calculate sequentially according to equations (3), (4), (5), (6), (7), (8), (9), and (10). A1=γ(ηhθ+0.5ηHθ-e0θ+e0)H+p{[η(H+h)-e0]θ+e0} (7) A3=0.5γh[(ηh-e0)θ+e0] (9) At the end of the loop, execute expression (11). R=R m+1 (11) R is the strength reduction factor to be calculated; C. Evaluating the safety of void foundations under offset loads R is considered the safety factor of the cavitary foundation under offset load. If R > 1, the cavitary foundation under offset load is considered safe; if R < 1, the cavitary foundation under offset load is considered unsafe; if R = 1, the cavitary foundation under offset load is considered to be in a limit state. End of step A to step C; The symbols in the above formulas and steps have the following meanings: e0 is the horizontal distance from the inflection point of the outer broken sliding surface to the near sidewall of the cavity when the offset load is 0. Let be the internal friction angle of the soil and rock; c be the cohesion of the soil and rock; h be the height of the cavity; and R be the value of c and when the underlying cavity foundation reaches its limit state. Strength reduction factor; R m Let R be the value of R in the m-th incremental step during the iterative solution of R. m+1 In the process of iteratively solving R, R is the (m+1)th increment step; the subscript m is the iteration order variable; H is the vertical distance from the top of the cavity to the ground; γ is the unit weight of the rock and soil of each slider; η is 1; θ is the parameter describing the failure mode; p is the magnitude of the ground load per unit area; ε is an infinitesimal quantity; b is the width of the cavity; π is pi; B1, B2, B3, B4, A1, A2, and A3 are the variables defined by equations (3), (4), (5), (6), (7), (8), and (9), respectively.

2. The method for evaluating the safety of a cavitary foundation under offset load according to claim 1, characterized in that: In equation (2), θ is solved using the Newton-Raphson iteration method, with the initial value of the variable specified as: θ n , When |θ n -θ n+1 When | > ε, the following text will be executed in a loop. Calculate according to equations (12), (13), and (14) in sequence. At the end of the loop, execute expression (15). θ=θ n+1 (15) In the loop: the subscript n is the iteration order variable; θ n In the process of solving θ in equation (2) using the Newton-Raphson iterative method, θ in the nth increment step; θ n+1 In the process of solving θ in equation (2) using the Newton-Raphson iterative method, the θ in the (n+1)th increment step; f(θ) n f′(θ) is a function defined by equation (12); n ) is the derivative of equation (12) with respect to θ.

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