CN111046480B - Method for calculating seismic soil pressure time course of retaining wall based on energy dissipation principle - Google Patents
Method for calculating seismic soil pressure time course of retaining wall based on energy dissipation principle Download PDFInfo
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Abstract
The invention provides a method for calculating the seismic soil pressure time course of a retaining wall based on an energy dissipation principle, which comprises the following steps: establishing a calculation model; respectively calculating to obtain the seismic inertia force and the gravity of the sliding soil wedge body and the retaining force of the wall to the sliding soil wedge body; respectively calculating the power and internal energy dissipation of the external force sliding the soil wedge body based on an energy dissipation principle; establishing a balance equation based on an energy conservation principle; calculating to obtain the maximum time course of the retaining force of the corresponding retaining wall to the sliding soil wedge under the continuous action of the earthquake, namely the solved earthquake soil pressure time course; and drawing to obtain a seismic soil pressure time-course curve. The method solves a series of problems that the slip surface is assumed to be an oblique straight line in the traditional earthquake soil pressure calculation, the earthquake action is simplified into an inertia force with constant size and direction, the influence of earthquake waves on the earthquake soil pressure when the earthquake waves are transmitted in a slope body is not considered, the obtained earthquake soil pressure is mostly a fixed value and is distorted, and how the earthquake soil pressure changes along with time cannot be obtained.
Description
Technical Field
The invention belongs to the technical field of seismic soil pressure analysis, and particularly relates to a method for calculating a seismic soil pressure time course of a retaining wall based on an energy dissipation principle.
Background
The gravity type retaining wall is the most common retaining wall form at present due to the advantages of simple form, easy local material taking, convenient construction and the like, and is characterized in that the self gravity of the retaining wall is used for maintaining the stability of the retaining wall under the action of soil pressure, but the existing anti-seismic design of the structure is too simple, practical evidence for the safety and economy of the design is lacked, specific research on how the soil pressure of the gravity type retaining wall changes along with time under the continuous action of an earthquake is lacked, and therefore the earthquake soil pressure time course of the gravity type retaining wall needs to be researched.
There are some existing methods for calculating the earthquake soil pressure of a gravity retaining wall under earthquake conditions, such as: quasi-static method, Newmark method, etc. However, the existing method simplifies the seismic load into an inertia force with constant magnitude and invariable direction, does not consider the influence of seismic waves on the seismic soil pressure when the seismic waves propagate in a slope body, mostly assumes that a slip surface is a straight line and the obtained seismic soil pressure is mostly a fixed value, and cannot obtain the change of the seismic soil pressure along with the time, which is inconsistent with the actual engineering. The method still leaves a blank about how to calculate the seismic soil pressure time course of the curvilinear slip surface gravity type retaining wall under the seismic condition through the form of power based on the energy dissipation principle. Therefore, a simple, convenient and practical gravity type retaining wall earthquake soil pressure time interval calculation method based on the energy dissipation principle is urgently needed, so that the earthquake-proof design of the gravity type retaining wall is accurately and reliably carried out, the safety and the economical efficiency of the design are both considered, and the method has important significance for the development of the field of earthquake-proof and disaster-reduction of the gravity type retaining wall.
Disclosure of Invention
Aiming at the defects in the prior art, the method for calculating the seismic soil pressure time interval of the retaining wall based on the energy dissipation principle solves a series of problems that the slip surface is assumed to be an oblique line, the seismic action is simplified into an inertia force with constant size and direction, the influence of seismic waves on the seismic soil pressure when the seismic waves are transmitted in a slope is not considered, the obtained seismic soil pressure is mostly a constant value, and how the seismic soil pressure changes along with time cannot be obtained.
In order to achieve the above purpose, the invention adopts the technical scheme that:
the scheme provides a method for calculating the seismic soil pressure time course of a retaining wall based on an energy dissipation principle, which comprises the following steps:
s1, establishing a calculation model according to the soil body curve slip crack surface behind the retaining wall;
s2, calculating and obtaining the seismic inertia force and the gravity of the sliding soil wedge body and the retaining force P of the retaining wall to the sliding soil wedge body by utilizing the calculation modela；
S3, according to the earthquake inertia force, the gravity and the retaining force P of the retaining wall to the sliding soil wedge bodyaCalculating the power of the external force of the sliding soil wedge body based on an energy dissipation principle, and calculating the internal energy dissipation of the sliding soil wedge body based on the energy dissipation principle by utilizing the cohesive force and the internal friction angle of the soil body;
s4, establishing a balance equation according to the power and internal energy dissipation of the external force of the sliding soil wedge;
s5, calculating according to the balance equation to obtain a seismic soil pressure time course;
and S6, drawing a time course curve according to the seismic soil pressure time course, and completing the calculation of the seismic soil pressure of the gravity retaining wall.
Further, the expression for calculating the tangent slope tan ω of the cycloid in the model in step S1 is as follows:
wherein the content of the first and second substances,
the derivative of y to x is expressed, x and y both represent the equation of the curvilinear slip surface, theta represents the radian traversed by the radius of the circle generating the cycloid, and R represents the reaction force of the soil body acting on the slip surface.
Still further, the step S2 includes the following steps:
s201, setting a curve slip surface by using the calculation model;
s202, calculating and obtaining the seismic inertia force and gravity of the sliding soil wedge body and the retaining force P of the retaining wall to the sliding soil wedge body based on the curve sliding crack surfacea。
Still further, the seismic inertia force F of the sliding soil wedge in the step S202shThe expression of (a) is as follows:
wherein H represents a height, θaRepresenting the angle of rotation of the cycloid through the wall toe, a0Representing the base acceleration amplitude, gamma, of the input wavesThe gravity of the sliding soil wedge body is shown, g is the gravity acceleration, z is the depth of any point on the slip surface, c is the cohesive force, and dz is the thickness of the micro-element body at any point on the slip surface.
Still further, the expression of the gravity W of the sliding soil wedge in step S202 is as follows:
wherein, γsIndicating the severity of the sliding soil wedge, θaThe method is characterized in that the method represents the rotation angle of a cycloid passing through a wall toe, x and y both represent equations of a curvilinear slip surface, d represents integral derivation, R represents a counter force of a soil body acting on the slip surface, and theta represents a radian passed by the radius of a circle generating the cycloid.
Still further, in step S3, the expression of the power Q' made by the external force of the sliding soil wedge calculated based on the energy dissipation principle is as follows:
Q′＝QW+QVP+QhP+QF
wherein Q iswRepresenting the power done by gravity, QFRepresenting the power, Q, made by horizontal seismic inertia forcesVPRepresents the power of the retaining wall to the sliding soil wedge in the vertical direction, QhPRepresents the power of the retaining wall to the sliding soil wedge in the horizontal direction, H represents the height of the retaining wall, msg represents the gravity borne by the differential unit, v represents the sliding surface of the differential unit body on the curve
The strain rate of an upper point P, theta' represents the corresponding rotation angle of an optional differential unit body on a slip crack surface, dz represents the thickness of the unit body at any point on the slip crack surface, W represents the gravity borne by the sliding soil wedge body, q represents the weight of the sliding soil wedge bodyshRepresenting the horizontal seismic inertia force, F, experienced by the body of the differential unitshRepresenting the seismic inertia force, p, of a sliding earth wedgeaThe retaining force of the retaining wall to the differential unit cell is shown, and delta represents the back normal angle of the vertical retaining wall.
Still further, the expression of the internal energy dissipation Q ″ of the sliding soil wedge calculated based on the energy dissipation principle in the step S3 is as follows:
wherein Q iscRepresenting internal energy dissipation, Q, caused by frictional forces in the soilfIndicating the dissipation of internal energy caused by cohesive forces in the soil,
denotes the integrated internal friction angle, v denotes the differential unit cell on the curved sliding surface
Strain rate at point P, H height, θ' rotation angle of optional differential unit cell on the slip surface, dz thickness of the unit cell at any point on the slip surface, and PaThe retaining force of the retaining wall to the differential unit body is shown, delta represents the angle of the normal line of the back of the vertical retaining wall, FshRepresenting the earthquake inertia force of the sliding soil wedge body, c representing the cohesive force of the soil body, R representing the counter force of the soil body acting on the slip crack surface, thetaaRepresenting the angle of rotation of the cycloid through the wall toe, dsDenotes the arc length, m, of the differential unit cellsRepresenting the mass of the differentiating element, do represents the integral derivative of theta, g represents the gravitational acceleration, paRepresenting the retaining force of the retaining wall on the differential unit body, qshRepresenting the horizontal seismic inertia force, W, experienced by the body of the differential unitsRepresenting the force of gravity of the differentiating unit cell.
Still further, the expression of the balance equation in step S4 is as follows:
wherein the content of the first and second substances,
denotes the integrated internal friction angle, v denotes the differential unit cell on the curved sliding surface
The strain rate at point P, H is the height, theta' is the rotation angle of any differential unit cell on the slip surface, dz is the thickness of the unit cell at any point on the slip surface, PaThe retaining force of the retaining wall to the sliding soil wedge is shown, delta represents the angle of the normal line of the back of the vertical retaining wall, FshRepresenting the earthquake inertia force of the sliding soil wedge body, c representing the cohesive force of the soil body, R representing the counter force of the soil body acting on the slip crack surface, thetaaThe rotation angle of the cycloid passing through the wall toe is shown, v represents the differential unit body on the curve sliding surface
Strain rate at upper P point, PaShowing the retaining force of the retaining wall on the differential unit body, FshThe earthquake inertia force of the sliding soil wedge body is shown, and W shows the gravity borne by the sliding soil wedge body.
Still further, the step S5 includes the following steps:
s501, obtaining the retaining force P of the retaining wall to the sliding soil wedge under the earthquake action according to the balance equationaThe expression of (1);
s502, order
And calculating the rotation angle theta of the cycloid passing through the wall and toe by utilizing Mathemica softwarea；
S503, according to the supporting and retaining force P of the retaining wall on the sliding soil wedge body under the earthquake actionaAnd a rotation angle thetaaCalculating to obtain the supporting and retaining force P of the retaining wall to the sliding soil wedge body under the action of the earthquakeaAn extreme value of (d);
s504, comparing the retaining force P of the retaining wall to the sliding soil wedge under the action of earthquakeaObtaining the holding force PaIs measured.
S505, respectively calculating the retaining force P of the corresponding retaining wall to the sliding soil wedge under the continuous action of the earthquakeaThe maximum value of (1) is the required seismic soil pressure time course.
Still further, the step S501 supporting force P of retaining wall to sliding soil wedge under earthquake actionaThe expression of (c) is as follows:
wherein W represents the gravity borne by the sliding soil wedge body, H represents the height, theta' represents the corresponding rotation angle of any differential unit body on the sliding surface, dz represents the thickness of the micro unit body at any point on the sliding surface, FshRepresenting the seismic inertia force of the sliding soil wedge,
representing the integrated internal friction angle, c representing the cohesive force of the soil mass, R representing the counter force of the soil mass acting on the slip surface, thetaaThe angle of rotation of the cycloid passing through the toe of the wall is shown, delta represents the angle of the normal line of the back of the vertical retaining wall, z represents the depth of any point on the slip crack surface, and W represents the gravity borne by the sliding soil wedge body.
The invention has the beneficial effects that:
the invention aims to provide a gravity type retaining wall seismic soil pressure time interval calculation method based on an energy dissipation principle. By the aid of a vibration table test, theoretical calculation results and actual measurement results of the method are compared, so that the theoretical calculation results of the method have high precision, and applicability of the calculation method is verified. The calculation method can calculate the seismic soil pressure time course of the gravity type retaining wall with the curved sliding surface and analyze the seismic soil pressure response characteristic of the gravity type retaining wall, so that the seismic design of the gravity type retaining wall is more accurately and reliably performed, the safety and the economy of the design are both considered, and the method has important significance for the development of the field of seismic mitigation of the gravity type retaining wall.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
FIG. 2 is a diagram of a computational model in the present invention.
FIG. 3 is a model diagram of calculating seismic inertia force of the sliding soil wedge according to the present invention.
FIG. 4 is a graph of the base acceleration amplitude time course of the input wave in the present invention.
FIG. 5 is a diagram of the force applied to the differential unit cell according to the present invention.
FIG. 6 is a schematic view of a seismic earth pressure time course curve at 0.4g calculated by the present analysis method.
FIG. 7 is a diagram of a test model according to the present invention.
Fig. 8 is a schematic diagram of the experimental results of the present invention.
Fig. 9 is a schematic diagram illustrating comparison between experimental data and theoretical data in this embodiment.
Detailed Description
The following description of the embodiments of the present invention is provided to facilitate the understanding of the present invention by those skilled in the art, but it should be understood that the present invention is not limited to the scope of the embodiments, and it will be apparent to those skilled in the art that various changes may be made without departing from the spirit and scope of the invention as defined and defined in the appended claims, and all matters produced by the invention using the inventive concept are protected.
Examples
As shown in FIG. 1, the invention discloses a method for calculating the seismic soil pressure time course of a retaining wall based on an energy dissipation principle, which is characterized by comprising the following steps of:
and S1, establishing a calculation model according to the soil mass curve slip crack surface behind the retaining wall.
In this embodiment, when the earth filling behind the retaining wall slides down along the wall back and one of the sliding surfaces in the earth filling to reach the limit balance state, the sliding surface passes through the curve of the toe of the wall instead of the straight line theoretically assumed to be proposed by coulomb's law, and the curve is assumed to be a cycloid. The invention assumes the gravity typeThe retaining wall is rigid, the filling soil behind the wall is non-cohesive soil, and a concrete research model is shown in figure 2. Wherein W is the gravity borne by the sliding soil wedge body, R is the counterforce acted on the slip crack surface by the soil body, and the included angle between the counterforce and the normal line of the tangent line of the slip crack surface is the internal friction force of the soil
PaThe supporting and retaining force of the retaining wall to the sliding soil wedge body under the action of earthquake, PaThe included angle formed by the normal line of the wall back is equal to the included angle delta between the wall back and the filling soil, theta is the radian (rotation angle) passed by the radius of the circle generating the cycloid, and when the cycloid passes through the wall toe, the rotation angle is set as thetaaOmega is the included angle between the tangent line of one point on the cycloid and the horizontal direction, and when the cycloid passes through the wall toe, the included angle between the tangent line of the cycloid and the horizontal direction is assumed to be omegaa。
In this embodiment, a coordinate axis about a cycloid is established in fig. 2 with O as an origin, and an equation of a slip crack surface generated by filling soil behind a wall is assumed as:
when theta is equal to thetaaWhen y is H, the radius R of the runner can be obtained according to the equation of the slip surface as H/(1-cos theta)a). The slope of the tangent at a point on the cycloid can be expressed as tan ω, then:
Wherein the content of the first and second substances,
the derivative of y to x is expressed, x and y both represent the equation of the curvilinear slip surface, theta represents the radian traversed by the radius of the circle generating the cycloid, and R represents the reaction force of the soil body acting on the slip surface.
S2, calculating and obtaining the seismic inertia force, gravity and retaining soil of the sliding soil wedge body by using the calculation modelWall retaining force P for sliding soil wedgeaThe realization method comprises the following steps:
s201, setting a curve slip crack surface by using the calculation model;
s202, calculating and obtaining the seismic inertia force and gravity of the sliding soil wedge body and the retaining force P of the retaining wall to the sliding soil wedge body based on the curve sliding crack surfacea。
In this embodiment, the seismic inertial force calculation herein uses the following basic assumptions:
(1) when the earth retaining wall overturns to be in a critical state, the earth behind the wall reaches the ultimate balance and forms a sliding wedge body, and a curved sliding crack surface passing through the earth retaining wall is formed in the earth;
(2) the retaining wall is a rigid body, and the filling behind the wall is homogeneous and isotropic sandy soil, but has certain cohesiveness.
In the embodiment, under the action of an earthquake, when the back of the wall is away from the filling soil, the soil body behind the wall slides downwards along a certain cycloid curved surface to form a cycloid-shaped sliding soil wedge body. As shown in fig. 3, a computational model map is created, and a minute element having a thickness dz at a point P on the slip surface, where the depth of the point P is z and the corresponding rotation angle θ' is taken as a study object. In fig. 3, the gravity retaining wall is upright on the back and has a height H. The filling behind the wall is sandy soil with certain cohesiveness and the density is rhosThe gravity of the sliding soil wedge body is gammasA cohesive force of csInternal angle of friction of
Poisson ratio is upsilons，VsThe top surface of the filled soil is horizontal and the surface is not subjected to other loads for the shear wave velocity of the filled soil. Under the action of earthquake, the sliding soil wedge body is subjected to horizontal earthquake force FshThe function of (1).
In this embodiment, the vertical acceleration response of the retaining structure makes the structure produce the action of throwing under the effect of seismic wave for the ground slope of granular structure and flexible structures such as stock frame structure can produce great destruction effect, and is then less to gravity type barricade and stake plate type barricade's destruction effect. Therefore, when the acceleration dynamic response characteristics of the retaining structure are researched, only the acceleration dynamic response characteristics in the horizontal direction of the gravity retaining wall and the pile-plate retaining wall are analyzed and discussed. The Mononobe-Okabe theory of the traditional quasi-static method is proposed for sandy soil, but in actual engineering, the filling soil behind the retaining wall is not all sandy soil but has cohesiveness, so in order to make the calculation method more suitable for the actual engineering, the internal friction angle in the calculation is taken as the comprehensive internal friction angle of the filling soil, as shown in formula (2):
wherein the content of the first and second substances,
indicating the integrated internal friction angle of the fill,
denotes the internal friction angle, γsIndicating the severity and H the height.
Mass m of the thin layer units(z) is:
wherein g is gravity acceleration, R is the reaction force of the soil body acting on the slip crack surface, and thetaaThe rotation angle is represented as z, which represents the depth of any point on the slip surface, and dz, which represents the thickness of the infinitesimal body at any point on the slip surface.
The weight W of the whole sliding soil wedge body is as follows:
wherein, γsIndicates the degree of gravity,. thetaaRepresenting the angle of rotation of a cycloid through a wall toe, x and y both representing equations for curvilinear slip planes, d representing integral derivatives, and R representingThe reaction force of the soil body acting on the slip surface, and θ represents the radian through which the radius of the circle generating the cycloid curve passes.
Under the action of earthquake, the sliding soil wedge body is subjected to horizontal earthquake inertia force FshCan be expressed as:
wherein H represents a height, ashRepresents the horizontal seismic acceleration of the sliding soil wedge body at any position along the wall height, z represents the depth of any point on the slip fracture surface, t represents time, msThe mass of the differential unit cell is shown, and dz is the thickness of the unit cell at any point on the slip plane.
Neglecting the amplification effect generated when the seismic wave propagates in the soil body, so the horizontal seismic acceleration a of the filling soil sliding wedge bodyshCan be respectively expressed as:
ash＝a0(z＝H) (6)
wherein, a0The base acceleration amplitude of the input wave varies with time t, H represents the height, and z represents the depth of any point on the slip surface, as shown in fig. 4. The formula (3) and (6) may be taken into the formula (7):
finishing to obtain:
wherein H represents a height, θaRepresenting the angle of rotation of the cycloid through the wall toe, a0Representing the base acceleration amplitude, gamma, of the input wavesThe gravity of the sliding soil wedge body is shown, g is the gravity acceleration, z is the depth of any point on the slip crack surface, c is the cohesive force of the soil body, and dz is the thickness of the micro-element body at any point on the slip crack surface.
S3, according to the earthquake inertia force, the gravity and the retaining force P of the retaining wall to the sliding soil wedge bodyaBased on the energy dissipation principle, the power of the external force of the sliding soil wedge body is calculated, and based on the energy dissipation principle, the internal energy dissipation of the sliding soil wedge body is calculated by utilizing the cohesive force and the internal friction angle of the soil body.
(1) Calculating the power of the external force based on the energy dissipation principle:
in this embodiment, the external force acting on the differential unit body of the sliding soil wedge has a gravity msg. Seismic inertia force qshAnd the supporting and retaining force p of the retaining wall to the differential unit bodyaWhen the soil body is in a plastic flow state, the differential unit body is assumed to be on a curve sliding surface
The strain rate of the upper point P is v, the upper point P is inclined downwards along the tangential direction of the point P, the stress schematic diagram of the differential unit body is shown in figure 5, and the external force acting on the sliding soil wedge body is respectively calculated.
1) External power by gravity:
in this embodiment, the gravity m applied to the differential unit bodysThe acting direction of g is consistent with the component direction of the strain velocity direction v in the vertical direction, and the power is regulated to be positive along the velocity direction and negative along the velocity direction, so that the gravity does positive work, and the gravity magnitude Q'wComprises the following steps:
thus, over the entire slip plane, the power done by gravity is: qw
2) External power of the retaining wall to the sliding soil wedge body:
in this embodiment, the retaining wall supports the differential unit cellaForming an angle delta with the normal of the back of the vertical retaining wall,respectively calculate paThe power made on the vertical component of the strain rate v by the vertical component and the power made on the horizontal component of the strain rate v by the horizontal component of paThe direction points to the filling soil, so as to do negative work, and the vertical power and the horizontal power are respectively as follows:
the vertical direction is as follows:
horizontal direction:
thus, for the entire sliding soil wedge, the power made is:
the vertical direction is as follows:
horizontal direction:
3) external power made by horizontal seismic inertia force:
in this embodiment, the horizontal seismic inertia force q applied to the differential unit bodyshThe component direction of the strain rate v in the horizontal direction is consistent, so that the positive work is done, and the power magnitudes are respectively as follows:
thus, for a sliding soil wedge, the power made by the seismic inertia force is:
the external power of all external forces obtained from equations (18), (21), (22) and (24) is:
Q′＝QW+QVP+QhP+QF
wherein Q iswRepresenting the power done by gravity, QFRepresenting power, Q, done by horizontal seismic inertia forcesVPRepresents the power of the retaining wall to the sliding soil wedge in the vertical direction, QhPRepresents the power of the retaining wall to the sliding soil wedge in the horizontal direction, H represents the height of the retaining wall, msg represents the gravity borne by the differential unit, v represents the sliding surface of the differential unit body on the curve
The strain rate of an upper point P, theta' represents the corresponding rotation angle of an optional differential unit body on a slip crack surface, dz represents the thickness of the unit body at any point on the slip crack surface, W represents the gravity borne by the sliding soil wedge body, q represents the weight of the sliding soil wedge bodyshRepresenting the horizontal seismic inertia force, F, experienced by the body of the differential unitshRepresenting the seismic inertia force, p, of a sliding earth wedgeaThe retaining force of the retaining wall to the differential unit cell is shown, and delta represents the back normal angle of the vertical retaining wall.
(2) Calculating power dissipated by internal energy on a slip surface based on an energy dissipation principle:
in this example, the internal energy dissipation and the curvilinear slip surface at the point P are studied on the arbitrarily selected differential unit body
The energy consumed by any point on the soil body respectively comprises cohesive force c and friction force of the soil body
Power of induced cohesion and friction:
1) frictional force
Caused byPower made by friction:
frictional force F at point Pf' is defined as:
wherein μ represents a friction factor, FNIndicating a positive pressure acting at point P,
denotes the combined internal friction angle, m, of the fillsg represents the gravity borne by the differential unit, theta' represents the corresponding rotation angle of an optional differential unit body on the slip fracture surface, and paRepresenting the retaining force of the retaining wall on the differential unit body, qshRepresenting the horizontal seismic inertia force to which the differential unit cell is subjected.
Power Q 'by friction at point P'fComprises the following steps:
therefore, the temperature of the molten metal is controlled,
the resulting internal energy dissipation is:
2) the power of cohesive force caused by the cohesive force c of the soil body is as follows:
energy loss Q caused by c at point Pc′：
Qc′＝c·ν·ds (21)
Wherein the differential unit arc length
Thus, over the entire slip plane:
the energy loss for all external forces can be obtained from the equations (20) and (22):
wherein Q iscRepresenting internal energy dissipation, Q, caused by frictional forces in the soilfIndicating the dissipation of internal energy caused by cohesive forces in the soil,
denotes the integrated internal friction angle, v denotes the differential unit cell on the curved sliding surface
Strain rate at point P, H height, θ' rotation angle of optional differential unit cell on the slip surface, dz thickness of the unit cell at any point on the slip surface, and PaThe retaining force of the retaining wall to the differential unit body is shown, delta represents the angle of the normal line of the back of the vertical retaining wall, FshRepresenting the earthquake inertia force of the sliding soil wedge body, c representing the cohesive force of the soil body, R representing the counter force of the soil body acting on the slip crack surface, thetaaRepresenting the angle of rotation of the cycloid through the wall toe, dsDenotes the arc length, m, of the differential unit cellsRepresenting the mass of the differentiating cell, d θ representing the derivative on θ, g representing the gravitational acceleration, paRepresenting the retaining force of the retaining wall on the differential unit body, qshRepresenting the horizontal seismic inertia force, W, experienced by the body of the differential unitsRepresenting the gravitational force of the differential unit cell.
S4, establishing a balance equation according to the power and internal energy dissipation of the external force of the sliding soil wedge;
s5, calculating the seismic soil pressure time course according to the balance equation, wherein the implementation method comprises the following steps:
s501, obtaining the retaining force P of the retaining wall to the sliding soil wedge under the earthquake action according to the balance equationaThe expression of (1);
s502, order
And calculating the rotation angle theta of the cycloid passing through the wall and toe by utilizing Mathemica softwarea；
S503, according to the supporting and retaining force P of the retaining wall on the sliding soil wedge body under the earthquake actionaAnd a rotation angle thetaaCalculating to obtain the supporting and retaining force P of the retaining wall to the sliding soil wedge body under the action of the earthquakeaAn extreme value of (d);
s504, comparing the retaining force P of the retaining wall to the sliding soil wedge under the action of earthquakeaObtaining the holding force PaMaximum value of (d);
s505, respectively calculating the retaining force P of the corresponding retaining wall to the sliding soil wedge under the continuous action of the earthquakeaThe maximum value of (1) is the required seismic soil pressure time course.
And S6, drawing a time course curve according to the seismic soil pressure time course, and completing the calculation of the seismic soil pressure of the gravity retaining wall.
In the present embodiment, it is basically assumed that:
a. the material presents ideal plasticity, namely the material is an ideal rigid plastic body and does not generate strain hardening or strain softening;
b. the material, when yielding, obeys the Mohr-Coulomb yield criterion and associated flow laws;
c. the deformation of the object under the limit load is very small, and the virtual work equation can be satisfied.
When carrying out retaining wall design, in order to guarantee the stability of filling out soil behind the wall, then need guarantee under the earthquake action time, its interior energy dissipation is greater than the power that is done equal to external force:
Qc+Qf≥Qw+QPa+QF (24)
considering the critical state, when the power done by the external force is equal to the energy dissipated internally:
wherein the content of the first and second substances,
denotes the integrated internal friction angle, v denotes the differential unit cell on the curved sliding surface
The strain rate at point P, H is the height, theta' is the rotation angle of any differential unit cell on the slip surface, dz is the thickness of the unit cell at any point on the slip surface, PaThe retaining force of the retaining wall to the sliding soil wedge is shown, delta represents the angle of the normal line of the back of the vertical retaining wall, FshRepresenting the earthquake inertia force of the sliding soil wedge body, c representing the cohesive force of the soil body, R representing the counter force of the soil body acting on the slip crack surface, thetaaDenotes the rotation angle of cycloid passing through wall toe, ν denotes the differential unit body on the curve sliding surface
Strain rate at point P, FshRepresenting the seismic inertia force of the sliding soil wedge.
Further obtaining:
from the curve equation:
therefore:
wherein W represents a sliding wedgeThe applied gravity, H represents the height, theta' represents the corresponding rotation angle of an arbitrary differential unit body on the slip crack surface, dz represents the thickness of the unit body at any point on the slip crack surface, FshRepresenting the seismic inertia force of the sliding soil wedge,
representing the integrated internal friction angle, c representing the cohesive force of the soil mass, R representing the counter force of the soil mass acting on the slip surface, thetaaThe angle of rotation of the cycloid through the toe of the wall is shown, δ is the angle of the normal line of the back of the vertical retaining wall, and z is the depth of any point on the slip crack surface.
As can be seen from the formula (25), if
δ、γsIn the known manner, it is known that,
the angle between the normal line of the normal line and the tangent line of the slip surface is represented as the internal friction of the soil, delta represents the angle of the normal line of the back of the vertical retaining wall, and gamma represents the anglesIndicating the severity of the sliding soil wedge, PaIs only dependent on thetaaThen P isaThere is a maximum value, which is the calculated earthquake active earth pressure when calculating PaWhen extreme value of (1), order
Will find out thetaaSubstituting the formula (26) to obtain the earthquake active soil pressure.
In this embodiment, the PGA is 0.4 g: the seismic soil pressure calculation method based on the energy principle substitutes the expression and each parameter in the formula (26), and then calculates by utilizing Mathemica software, namely a0＝3.92m/s2The calculation process of (a) is as follows:
1) first to PaThe final expression of (2) is derived;
2) to obtain thetaaAnd making it equal to zero to solve the extreme point;
3) theta to be obtainedaSubstitution of value of (1)PaIn the original formula of (1), solving for PaAn extreme value of (d);
4) comparison PaObtaining P from the extreme value ofaThe maximum value of (1) is the magnitude of the seismic soil pressure, and finally a seismic soil pressure time-course curve is obtained, as shown in fig. 6.
In order to verify the precision of the theoretical calculation method, Wenchuan earthquake survey data are combined, a large-scale vibration table test is taken as a research means, the soil pressure resultant force distribution of the gravity type embankment retaining wall under the earthquake condition is obtained, comparison research is carried out on the soil pressure resultant force distribution and a result of theoretical calculation, and the research result verifies the applicability of the theoretical calculation method.
The reliability of the model test results depends on whether the test model actually reproduces the actual working state of the prototype structural system. In the research, a gravity distortion model and a dimension analysis method are adopted to design the similarity relation of the model, and the similarity constants of the physical quantities are shown in the following table 1:
TABLE 1
As shown in fig. 7 to 8, the dynamic response characteristics of the gravity retaining wall with a height of 9.6m were simulated in the test prototype, and the model dimensions of the retaining wall were 1.6m in height and 6m in width, 0.33m in top width, 0.505m in bottom width, 0.204m in toe height and 0.102m in toe width according to the geometric similarity ratio. The start point of the slope of the soil wedge body is flush with the top of the wall, and the transverse slope angle of the ground is 0 degree. The filler behind the wall is standard sand, the internal friction angle is 33 degrees, and the gravity is 17kN/m3(ii) a The internal friction angle of the foundation soil is 30.4 degrees, the cohesive force is 6.9kPa, and the gravity is 20.26kN/m3The water content is 3.6 percent, and the unit in the graph is mm. The method is characterized in that strain type soil pressure sensors are selected to test soil pressure strength, the wall back is a concrete wall surface, the friction coefficient is large, relative slippage between a soil body and a retaining wall does not occur, the strain type soil pressure sensors are arranged on one surface of the gravity type retaining wall, which is in contact with filling soil, and the strain type soil pressure sensors are arranged at intervals of 250mm along the height of the wall and are respectively arranged at the positions 0.05m, 0.30m, 0.55m, 0.80m, 1.05m and 1.3m away from the wall bottom, and the total number of the strain type soil pressure sensors is 6. From the wall backAfter the soil pressure intensity at each geometric height measured by the strain type soil pressure sensor, the area enclosed by the distribution curve and the coordinate axis is calculated, and then the resultant force of the seismic soil pressure can be obtained. As shown in fig. 9, the results of the actual measurement and the results of the theoretical calculation were compared and studied, and the comparison results confirmed the applicability of the proposed analysis method. By the aid of a vibration table test, theoretical calculation results and actual measurement results of the method are compared, so that the theoretical calculation results of the method have high precision, and the applicability of the analysis method of the invention is verified.
Claims (7)
1. A method for calculating the seismic soil pressure time course of a retaining wall based on an energy dissipation principle is characterized by comprising the following steps:
s1, establishing a calculation model according to the soil body curve slip crack surface behind the retaining wall;
the expression of the tangent slope tan ω of the cycloid in the calculation model in step S1 is as follows:
wherein the content of the first and second substances,
expressing the derivation of y to x, wherein x and y both express the equation of a curvilinear slip surface, theta expresses the radian passed by the radius of a circle generating a cycloid, and R expresses the counter force of the soil body acting on the slip surface;
s2, calculating and obtaining the seismic inertia force and the gravity of the sliding soil wedge body and the retaining force P of the retaining wall to the sliding soil wedge body by utilizing the calculation modela；
S3, according to the earthquake inertia force, the gravity and the retaining force P of the retaining wall to the sliding soil wedge bodyaCalculating slip based on energy dissipation principlesThe power is made by the external force of the soil wedge body, and the internal energy dissipation of the sliding soil wedge body is calculated by utilizing the cohesive force and the internal friction angle of the soil body on the basis of the energy dissipation principle;
in step S3, an expression of the power Q' made by the external force sliding the soil wedge based on the energy dissipation principle is calculated as follows:
Q′＝QW+QVP+QhP+QF
wherein Q iswRepresenting the power done by gravity, QFRepresenting power, Q, done by horizontal seismic inertia forcesVPRepresents the power of the retaining wall to the sliding soil wedge in the vertical direction, QhPRepresents the power of the retaining wall to the sliding soil wedge in the horizontal direction, H represents the height of the retaining wall, msg represents the gravity borne by the differential unit, v represents the sliding surface of the differential unit body on the curve
The strain rate of an upper point P, theta' represents the corresponding rotation angle of an optional differential unit body on a slip crack surface, dz represents the thickness of the unit body at any point on the slip crack surface, W represents the gravity borne by the sliding soil wedge body, q represents the weight of the sliding soil wedge bodyshRepresenting horizontal seismic inertia force to which a differential unit body is subjected，FshRepresenting the seismic inertia force, p, of a sliding earth wedgeaThe supporting and retaining force of the retaining wall to the differential unit body is represented, delta represents a vertical retaining wall back normal line angle, and W represents the gravity borne by the sliding soil wedge body;
in step S3, an expression of the internal energy dissipation Q ″ of the sliding soil wedge is calculated based on the energy dissipation principle as follows:
wherein QcIndicating internal energy dissipation, Q, caused by friction of the soilfIndicating the dissipation of internal energy caused by cohesive forces in the soil,
denotes the integrated internal friction angle, v denotes the differential unit cell on the curved sliding surface
Strain rate at point P, H height, θ' rotation angle of optional differential unit cell on the slip surface, dz thickness of the unit cell at any point on the slip surface, and PaThe retaining force of the retaining wall to the differential unit body is shown, delta represents the angle of the normal line of the back of the vertical retaining wall, FshRepresenting the seismic inertia force of the sliding soil wedge body, c representing the cohesive force of the soil body, R tableReaction force of soil mass acting on slip surface, thetaaRepresenting the angle of rotation of the cycloid through the wall toe, dsDenotes the arc length, m, of the differential unit cellsRepresenting the mass of the differentiating element, do represents the integral derivative of theta, g represents the gravitational acceleration, paRepresenting the retaining force of the retaining wall on the differential unit body, qshRepresenting the horizontal seismic inertia force, W, experienced by the body of the differential unitsThe gravity of the differential unit body is shown, and W shows the gravity borne by the sliding soil wedge body;
s4, establishing a balance equation according to the power and internal energy dissipation of the external force of the sliding soil wedge;
s5, calculating according to the balance equation to obtain a seismic soil pressure time course;
and S6, drawing a time course curve according to the seismic soil pressure time course, and completing the calculation of the seismic soil pressure of the gravity retaining wall.
2. The method for calculating the seismic earth pressure time course of a retaining wall based on the energy dissipation principle as claimed in claim 1, wherein the step S2 comprises the steps of:
s201, setting a curve slip crack surface by using the calculation model;
s202, calculating and obtaining the seismic inertia force and gravity of the sliding soil wedge body and the retaining force P of the retaining wall to the sliding soil wedge body based on the curve sliding crack surfacea。
3. The method for calculating the seismic soil pressure time course of a retaining wall based on the energy dissipation principle as claimed in claim 2, wherein the seismic inertia force F of the sliding soil wedge in the step S202shThe expression of (a) is as follows:
wherein H represents a height, θaRepresenting the angle of rotation of the cycloid through the wall toe, a0Representing the base acceleration amplitude, gamma, of the input wavesIndicating the severity of the sliding soil wedgeG represents the gravity acceleration, z represents the depth of any point on the slip fracture surface, c represents the cohesive force of the soil body, and dz represents the thickness of the micro-element at any point on the slip fracture surface.
4. The method for calculating the seismic soil pressure time course of a retaining wall based on the energy dissipation principle according to claim 2, wherein the expression of the gravity W of the sliding soil wedge in the step S202 is as follows:
wherein, γsIndicating the severity of the sliding soil wedge, θaThe method is characterized in that the method represents the rotation angle of a cycloid passing through a wall toe, x and y both represent equations of a curvilinear slip surface, d represents integral derivation, R represents a counter force of a soil body acting on the slip surface, and theta represents a radian passed by the radius of a circle generating the cycloid.
5. The method for calculating the seismic soil pressure time course of a retaining wall based on the energy dissipation principle as claimed in claim 1, wherein the expression of the balance equation in the step S4 is as follows:
wherein the content of the first and second substances,
denotes the integrated internal friction angle, v denotes the differential unit cell on the curved sliding surface
The strain rate at point P, H is the height, theta' is the rotation angle of any differential unit cell on the slip surface, dz is the thickness of the unit cell at any point on the slip surface, PaThe retaining force of the retaining wall to the sliding soil wedge is shown, delta is verticalNormal angle of retaining wall back, FshRepresenting the earthquake inertia force of the sliding soil wedge body, c representing the cohesive force of the soil body, R representing the counter force of the soil body acting on the slip crack surface, thetaaThe rotation angle of the cycloid passing through the wall toe is shown, v represents the differential unit body on the curve sliding surface
Strain rate at upper P point, PaShowing the retaining force of the retaining wall on the differential unit body, FshRepresenting the seismic inertia force of the sliding soil wedge.
6. The method for calculating the seismic earth pressure time course of a retaining wall based on the energy dissipation principle as claimed in claim 1, wherein the step S5 comprises the steps of:
s501, obtaining the retaining force P of the retaining wall to the sliding soil wedge under the earthquake action according to the balance equationaThe expression of (1);
s502, order
And calculating the rotation angle theta of the cycloid passing through the wall and toe by utilizing Mathemica softwarea；
S503, according to the supporting and retaining force P of the retaining wall on the sliding soil wedge body under the earthquake actionaAnd a rotation angle thetaaCalculating to obtain the supporting and retaining force P of the retaining wall to the sliding soil wedge body under the action of the earthquakeaAn extreme value of (d);
s504, comparing the retaining force P of the retaining wall to the sliding soil wedge under the action of earthquakeaObtaining the holding force PaThe maximum value of (a);
s505, respectively calculating the retaining force P of the corresponding retaining wall to the sliding soil wedge under the continuous action of the earthquakeaThe maximum value of (1) is the required seismic soil pressure time course.
7. The method for calculating the seismic soil pressure time course of a retaining wall based on the energy dissipation principle of claim 6, wherein the step S501 of blocking under the action of earthquakeSupporting and retaining force P of soil wall on sliding soil wedge bodyaThe expression of (a) is as follows:
wherein W represents the gravity borne by the sliding soil wedge body, H represents the height, theta' represents the corresponding rotation angle of any differential unit body on the sliding surface, dz represents the thickness of the micro unit body at any point on the sliding surface, FshRepresenting the seismic inertia force of the sliding soil wedge,
representing the integrated internal friction angle, c representing the cohesive force of the soil mass, R representing the counter force of the soil mass acting on the slip surface, thetaaThe angle of rotation of the cycloid through the toe of the wall is shown, δ is the angle of the normal line of the back of the vertical retaining wall, and z is the depth of any point on the slip crack surface.
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