CN108846212B - Rigid frame pile internal force and displacement design calculation method - Google Patents
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Abstract
A rigid frame pile internal force and displacement design calculation method is characterized in that a rigid frame pile structure is divided into a plurality of rod pieces constrained by joint ends according to a structure calculation displacement method; each rod piece is used as a module for analysis, rod end counter force and a rod piece unit node displacement value of the rod piece when the constraint end generates unit displacement under the conditions of specific stratum mechanical parameters and load action are calculated through a numerical method, then according to a displacement method of structural analysis, the rod end displacement is used as an unknown number, the rod end displacement is calculated according to structural node bending moment and a shear force balance condition, the obtained displacement is multiplied by the displacement value of the rod piece unit node when the constraint end generates the unit displacement, an algebraic sum is obtained, the displacement of each rod piece unit node is obtained, the algebraic sum is obtained, the displacement is substituted into a rod piece bending moment-deflection differential expression and a shear force-deflection differential expression, and structural internal force is calculated. The design calculation method has strong universality; and the rigidity matrix dimension of the generated rigid frame pile structure is greatly reduced by modularized processing, and the calculation efficiency is improved.
Description
Technical Field
The invention belongs to the technical field of anti-sliding structures, and relates to a rigid frame pile internal force and displacement design calculation method.
Background
The anti-sliding structure comprises a rigid frame pile, a miniature pile combined structure and the like, and internal force and deformation analysis can be carried out by utilizing an underground rigid frame displacement method based on the elastic foundation beam. However, in this method, the foundation coefficient on the pile body is assumed to be a fixed value, and the displacement of the top is generally the largest according to the deformation rule of the anti-sliding structure, so that the foundation reaction force on the top of the structure is the largest and unreasonable.
This problem can be solved by using the m-method or C-method of the elastic foundation beam, assuming that the foundation coefficient increases linearly with the depth of the ground or changes according to the power exponent. Based on the m method and the C method of the elastic foundation beam, the deflection equation of the pile body is as follows:
wherein: y is the pile body deflection;kis the foundation coefficient;mthe variation coefficient of the foundation coefficient along the depth of the stratum;cis the power exponent of the variation of the foundation coefficient along the depth of the stratum,𝒸if the pile depth is not less than 1, the method is m, and the foundation coefficient is linearly increased along the pile depth; c =1/2, C method, foundation coefficient is expressed according to fraction with pile depthA number change;q(x) To distribute the load acting on the pile.
However, the search for the analytic solution of the differential equation shows that the equation has no analytic solution, and cannot be solved according to the existing underground rigid frame displacement method, so that a new solution needs to be found.
Of course, the finite difference method or the member finite element method can be adopted to perform the overall analysis and calculation on the structure. For a common slide-resistant pile, a numerical method can be conveniently adopted for analysis; however, for the rigid frame piles with the double-row, three-row and multi-row structures, if the differential method is adopted to analyze the whole structure, the established model is lack of generality or very complex due to the consideration of various conditions such as different distributed loads, foundation coefficients of different stratums, variation coefficients of the foundation coefficients and different lengths of each section. How to solve the rigid frame piles with double-row or three-row and multi-row structures, and meanwhile, the universality of the established model and the complexity of calculation reduction are considered, and the method is the key for establishing the design calculation method of the combined anti-skid structure. .
Disclosure of Invention
The invention aims to provide a design and calculation method for internal force and displacement of double-row, three-row and multi-row rigid frame piles bearing horizontal loads, which is clear in concept, simple, convenient and practical.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows: a rigid frame pile internal force and displacement design calculation method is characterized in that according to a displacement method of structure calculation, a rigid frame pile structure is divided into a plurality of rod pieces constrained by node ends according to nodes; analyzing each rod as a module, calculating rod end counter force and displacement values of rod unit nodes when a constraint end generates unit displacement by using a numerical method under the conditions of specific stratum mechanical parameters and load action, then calculating rod end displacement according to a structural analysis displacement method by using the rod end displacement as an unknown number and according to a structural node bending moment and shear force balance condition, multiplying the obtained displacement and the displacement values of the rod unit nodes when the constraint end generates the unit displacement and solving the algebraic sum to obtain the displacement of each rod unit node, and then substituting the displacement into a rod bending moment-deflection differential expression and a shear force-deflection differential expression to calculate the internal force of the structure.
Compared with the method of analyzing the whole structure by adopting a finite difference method, a finite element method and other numerical methods, the design calculation method has clear concept and convenient realization, realizes modular combination and reduces the complexity of calculation.
If a finite difference method or a finite element method and other numerical methods are adopted to analyze and calculate the rigid frame pile bearing the horizontal load, element numbering is carried out in sequence, then discretization of a differential equation or selection of an element fitting function is carried out to establish a rigidity matrix, and then calculation of displacement and the like is carried out by combining boundary conditions. If conditions such as pile length, load, and ground coefficient change, numbering needs to be performed again, and the boundary position conversion needs to be performed, so that the universality is poor and the complexity is high.
The design calculation method of the invention utilizes modularization thought and a displacement method of structure calculation to disperse double-row, three-row and multi-row rigid frame piles into rod piece modules, divides units for single rod pieces, analyzes by utilizing a finite difference method or a finite element method, and calculates the counter force generated by the rod piece which accords with the theoretical assumption of the elastic foundation beam at the constraint end of the rod piece and the displacement value of each node of the rod piece unit under the load action when the constraint end of the rod piece generates unit displacement; setting the displacement of the split node of the structure as unknown quantity, calculating the displacement of the structure node according to the balance of the force of the structure node, multiplying the obtained displacement by the displacement value of the rod unit node when the constraint end generates unit displacement, solving the algebraic sum to obtain the displacement of each rod unit node, substituting the displacement into a differential equation of the relationship between load and deflection, and calculating to obtain the calculation result of the internal force and the displacement of the rigid frame pile. The rod pieces are used as modules, discrete numbering is carried out according to fixed starting points and fixed end points, different pile section foundation coefficients, different types of loads acting on the pile sections and different types of unit displacement (turning angles or deflection) of the constraint end parts of the discrete rod pieces are considered, the end part counterforce of the constraint rod pieces is respectively calculated, and the rod piece module rod end constraint counterforce is formed by stacking the constraint rod pieces. For the split straight rod, the pile sections can be conveniently divided to take the change of the coefficient distribution of the foundation into consideration, the change of the load distribution of different forms, the change of the numerical value and the like into consideration, and the universality of the method is enhanced. And each rod piece is taken as a module, so that the dimension of a rigidity matrix generated by calculating the structural node of the generated integral rigid frame pile is greatly reduced, and the calculation efficiency is improved.
Drawings
Fig. 1 is a schematic view of a double row rigid frame pile.
Fig. 2 is a schematic diagram of splitting the double-row rigid frame pile shown in fig. 1 into end-restraining rods in the design calculation method of the present invention.
Detailed Description
The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.
An improved differential equation generation method is provided in limited differential method of structural analysis (Wang Lei, Li Jia Bao, Ren traffic Press, 1982.12), which can avoid the establishment of a virtual differential point outside a pile, and the differential form of pile deflection-load differential equation is established as follows:
first from the relationship between bending moment and deflection
(wherein E is the modulus of elasticity of the pile material, I is the pile section moment of inertia, M is the pile bending moment,ythe deflection of the pile body), writing a differential formula of the differential equation, and then according to the relation between the load and the bending moment
To obtain a differential equation shown in the background art
The derivation process, regardless of the difference equation of the ground coefficient term, is as follows:
according to the central difference formula, the relationship between the bending moment and the deflection can be written as the following difference equation:
M i -1 、M i andM i+1 are respectivelyi-1、iAndi+1 bending moments at the three differential nodes,Iis the inertia moment of the pile body,λin order to be the difference step size, yis the deflection of the pile body. The equation of the relation between the load and the bending moment isiCenter difference formula of points:
substituting the formula (2), the formula (3) and the formula (4) into the formula (5), and obtaining a difference equation after arrangement:
the term of the ground coefficient in the formula (1) ((1))k+mx c )yWritten according to the principle of difference
And brings this term into difference equation (6), resulting in the difference format of equation (1):
the right distributed load of the formula (7) is expressed as differential node concentration forceP i =q i λThen, a final difference equation of equation (1) is obtained:
when designing the anti-sliding structure of the double-row rigid frame pile shown in fig. 1: according to a displacement method of structural calculation, the rigid frame pile structure is split into a plurality of rod pieces with fixedly restrained ends at the nodes. If the double-row rigid frame pile shown in fig. 1 is split into three sections of rod pieces AB, CD and BD, fixed support constraints for limiting horizontal displacement and rotation of the rod end are respectively applied to a node B of the rod piece AB and the rod piece BD and a node D of the rod piece BD and the rod piece CD, as shown in fig. 2; then, applying boundary conditions of unit displacement (deflection and corner) to the end of the rod piece separated from the node and replaced by constraint (such as the end of the rod piece corresponding to the B, D node in fig. 2), and writing a differential expression of the displacement boundary conditions by using a differential form of a load-deflection differential equation; the expression is combined with a difference equation (8) representing the pile body load-deflection relation and a difference format equation of the boundary condition of a free end (such as the end of a rod A, C in fig. 2) of the rod, and the constraint counter force of the constrained rod end and the displacement value of the node of the rod unit are calculated when the rod end generates unit displacement under the load action condition. Then, setting the displacement of the node end as an unknown quantity by utilizing the balance conditions of the bending moment and the shearing force of each structural node in the displacement method to obtain a rigidity matrix and a balance equation of the structure, thereby calculating to obtain the displacement of the node; multiplying the obtained displacement by the displacement value of the rod unit node when the constraint end generates unit displacement, solving the algebraic sum to obtain the displacement of each rod unit node, and substituting the displacement of the displacement unit node into a rod internal force-deflection differential expression, such as a formula (3) to obtain the internal force of the rod differential node, thereby obtaining the internal force and the displacement (deflection) of the structure. The design calculation method of the invention is to analyze the rod module obtained by splitting according to a finite difference method, and can also analyze the split rod by using a finite element method.
According to the principle, according to the stress characteristics, the stratum mechanical properties and the constraint characteristics of the discrete rod pieces of the rigid frame pile structure, the difference form of the deflection differential equation of the rod pieces under various conditions needs to be established, and the constraint counter force of the end part under different end part boundary conditions and different load action conditions is calculated. Taking the double-row rigid frame piles shown in fig. 1 as an example, the rigid frame piles are separated into boundary conditions formed by corresponding end constraint rod pieces according to nodes, and the types of the rod pieces needing to establish a differential equation are divided into the following seven types according to the requirements of a displacement method: 1) one end is fixedly restrained, the other end is free of a rod piece, and the fixed restrained end generates unit displacement (deflection); 2) one end of the rod is fixedly restrained, the other end of the rod is free, and the fixed restrained end generates a unit rotation angle; 3) one end of the rod piece is fixedly restrained, and the other end of the rod piece is free, and the local section is under the action of uniformly distributed load; 4) one end of the rod piece is fixedly restrained, and the other end of the rod piece is free, and the local section of the rod piece is under the action of triangular load; 5) one end of the rod piece is fixedly restrained, and the other end of the rod piece is free and is acted by concentrated load; 6) the two ends of the rod piece are fixedly restrained, and the left end of the rod piece generates a unit corner; 7) the two ends of the rod piece are fixed and restrained, and the right end of the rod piece generates a unit corner.
The following description will be given to the process of establishing a differential boundary condition by taking a differential format for establishing a fixed boundary as an example:
the boundary conditions of the fixed end are as follows:y 0 =0，θ 0 =0
pile cornerθAnd deflectionyThere is a relationship:
rewrite to differential format at the rod end and bring in boundary conditions:
to obtainy -1 =y 1 Whereiny -1 Is the adjacent differential virtual node outside the stub end.
Substituting the boundary conditions into the differential equations (2) to (4) to obtain an expression of the bending moment of the node near the rod end:
then, the difference expressions are respectively brought into the expression (5), and the node is obtained by referring to the expression (8) (the form of the comparison expression (8) and each coefficient)i=1 andifinal difference equation when = 2:
in addition to the above boundary conditions, there is a class of loading conditions, i.e., the establishment of a differential format when a unit displacement (deflection) and a unit rotation angle are applied to a fixed constraint end, respectively. In these two cases, two boundary conditions can be also classified, and the specific derivation is as follows:
1. applying a unit angle of rotation
When a unit angle of rotation is applied to the rod ends, it is equivalent to a boundary conditiony 0 =0，θ 0 =1, i.e.y 0 =0，y -1 = y 1 -2 λ, substituting the difference equations (2) to (4) to obtain an expression of the bending moment of the node near the rod end:
the two differential expressions are respectively taken into the formula (5), and the formula (8) is compared to obtain the nodei=1 andifinal difference equation of = 2:
2. applying unit displacement (deflection)
When a displacement (deflection) is applied to the rod end, it is equivalent to a boundary conditiony 0 =1，θ 0 Is =0, i.e.y 0 =1，y -1 = y 1 And (3) substituting the difference equations (2) to (4) to obtain an expression of the bending moment of the node near the rod end:
the two differential expressions are respectively taken into the formula (5), and the formula (8) is referred to obtain the nodei=1 andifinal difference equation of = 2:
similarly, the boundary condition difference format can be obtained when the rod end is a free end:
the following is a difference equation for the rod tip free end boundary condition:
wherein, the parameter coefficient 0 represents the rod top difference starting point number.
Difference equation with the rod bottom end being the free end:
wherein, the parameter coefficient n represents the rod top difference starting point number.
And (3) according to the actual boundary conditions of the rod piece, simultaneously solving the differential equation of the corresponding boundary conditions and the differential equation of other differential nodes on the rod piece (namely the equation (8)), so as to obtain the bending moment and the shearing force of the calculation constraint rod end.
Still taking the double-row rigid frame pile as an example, according to the bending moment and shear force balance conditions of the nodes:
b, D is a double row rigid frame pile node.
The balance equation of the structural nodes with the node displacement as an unknown number can be obtained, so that the node displacement value is obtained through calculation, and the internal force and the displacement of the rod piece and the whole structure can be calculated through back substitution.
Taking the analysis of double rows of piles as an example, on the basis of not considering the universality of the calculation method, if the calculation is carried out by adopting a structural integral difference method, if the number of nodes of the front pile rod piece and the rear pile rod piece is 100 and the number of nodes of the top beam rod piece is 50, 250 equations are solved; by adopting the method, the solution is divided into two steps, 100 and 50 solutions are respectively firstly carried out, and the solution steps are increased when the solution is carried out for 3 solutions, so that the total occupation of computing resources is reduced, and the efficiency is improved.
Claims (2)
1. A rigid frame pile internal force and displacement design calculation method is characterized in that according to a displacement method of structure calculation, a rigid frame pile structure is divided into a plurality of rod pieces constrained by node ends according to nodes; analyzing each rod as a module, calculating rod end counter force and displacement values of rod unit nodes when a constraint end generates unit displacement by using a numerical method under the conditions of specific stratum mechanical parameters and load action, then calculating rod end displacement according to a displacement method of structural analysis by using the rod end displacement as an unknown number and according to the bending moment and shear force balance conditions of a structural node, multiplying the obtained displacement and the displacement values of the rod unit nodes when the constraint end generates the unit displacement and solving the algebraic sum to obtain the displacement of each rod unit node, and then substituting the displacement into a differential expression of bending moment-deflection and a differential expression of shear force-deflection of the rod to calculate the internal force of the structure;
the types of the rod pieces in the differential form of the deflection differential equation under different boundary conditions and different load actions need to be established are as follows: 1) one end of the rod piece is fixedly restrained, the other end of the rod piece is free, and the fixed restrained end generates unit displacement; 2) one end of the fixed rod is fixedly constrained, the other end of the fixed rod is free, and the fixed constrained end generates a unit corner; 3) one end of the rod piece is fixedly restrained, and the other end of the rod piece is free, and the local section is under the action of uniformly distributed load; 4) one end of the rod piece is fixedly restrained, and the other end of the rod piece is free, and the local section of the rod piece is under the action of triangular load; 5) one end of the rod piece is fixedly restrained, and the other end of the rod piece is free and is acted by concentrated load; 6) the two ends of the rod piece are fixedly restrained, and the left end of the rod piece generates a unit corner; 7) the two ends of the rod piece are fixedly restrained, and the right end of the rod piece generates a unit corner;
then, according to actual problems, the rod piece differential equations in the independent forms are combined and superposed to obtain the differential format of the deflection differential equation of the specific rod piece;
the design calculation method specifically comprises the following steps:
the boundary conditions of the fixed end are as follows:y 0 =0，θ 0 =0
pile cornerθAnd deflectionyThere is this relationship:
rewrite to differential format at the rod end and bring in boundary conditions:
to obtainy -1 =y 1 Whereiny -1 The differential virtual nodes are adjacent except the pile end;
the above boundary conditions are substituted into the following difference equation:
M i -1 、M i andM i+1 are respectivelyi-1、iAndi+1 bending moments at the three differential nodes,Iis the pile section inertia moment of the pile body,λin order to be the difference step size, yis pile body deflection, E is the elastic modulus of the pile body material, and M is pile body bending moment;
obtaining an expression of bending moment of a node near the rod end:
respectively substituting the obtained expressions of the bending moments of the nodes near the rod piece ends into formulas
And refer to the formula
Get nodei=1 andifinal difference equation when = 2:
and the bending moment and the shearing force of the constrained rod end can be calculated.
2. The method for designing and calculating the internal force and displacement of the rigid frame pile according to claim 1, wherein a type of loading condition is provided in addition to the fixed boundary condition, namely, the establishment of a differential format when a unit displacement and a unit rotation angle are respectively applied to the fixed constraint end; in these two cases, two boundary conditions can be also classified, and the specific derivation is as follows:
1) applying a unit angle of rotation
When a unit angle of rotation is applied to the rod ends, it is equivalent to a boundary conditiony 0 =0，θ 0 =1, i.e.y 0 =0，y -1 = y 1 -2 λ, into a difference equation
Obtaining an expression of bending moment of a node near the rod end:
respectively taking the two differential expressions into
And comparing type
To obtain an on-sectionDoti=1 andifinal difference equation of = 2:
2) applying a unit displacement
When a displacement is applied to the rod end, it is equivalent to a boundary conditiony 0 =1，θ 0 =0, i.e.y 0 =1，y -1 = y 1 Equation of difference
Obtaining an expression of bending moment of a node near the rod end:
respectively taking the two differential expressions into
And are of the formula
Get at the nodei=1 andifinal difference equation of = 2:
similarly, the boundary condition difference format can be obtained when the rod end is a free end:
the following is a difference equation for the rod tip free end boundary condition:
wherein, the parameter coefficient 0 represents the number of the difference starting points at the top end of the rod piece;
difference equation with the rod bottom end being the free end:
wherein the parameter coefficient n represents the rod top difference starting point number.
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| CN112329156A (en) * | 2019-07-31 | 2021-02-05 | 深圳市建筑设计研究总院有限公司 | Loading and internal force calculation method for axial stress member |
| CN112329157A (en) * | 2019-07-31 | 2021-02-05 | 深圳市建筑设计研究总院有限公司 | Loading and internal force calculation method for secondary internal force structure |
| CN112446100A (en) * | 2019-08-14 | 2021-03-05 | 深圳市建筑设计研究总院有限公司 | Pre-internal force of torsion member and calculation method thereof |
| CN112507503A (en) * | 2019-09-16 | 2021-03-16 | 深圳市建筑设计研究总院有限公司 | Loading and pre-reaction force calculation method for secondary self-reaction force structure |
| CN111046552A (en) * | 2019-12-10 | 2020-04-21 | 中铁西北科学研究院有限公司 | Displacement loading method for calculating internal force of pile anti-sliding structure |
| CN111460544B (en) * | 2020-03-05 | 2022-06-17 | 北京交通大学 | Structure calculation method of elastic foundation beam considering shearing effect |
| CN111809537B (en) * | 2020-07-22 | 2021-12-07 | 中国科学院、水利部成都山地灾害与环境研究所 | Design method of collapse landslide blocking energy dissipation pile array |
| CN117973165B (en) * | 2024-04-02 | 2024-06-28 | 湖南大学 | Method for calculating pile internal force and displacement of horizontal loaded pile in deep soft soil area |
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