CN105005660A - Stress calculation method for non-linear flexible member close to practical running state - Google Patents

Abstract

The invention discloses a stress calculation method for a non-linear flexible member close to a practical running state. The method comprises the following steps: selecting a structural model and a material model according to the structural composition and material of an iron tower, and generating a rigidity matrix, a node displacement array and a node load array of iron tower member units respectively specific to each member unit and node displacement of the iron tower; correcting an acquired stress matrix and displacement matrix with a finite incremental method and a Newton iteration method; and performing Lagrange interpolation on a strain matrix and the displacement matrix of the iron tower to obtain the overall load and displacement of the iron tower in order to obtain the stress magnitude at any point in an iron tower structure. The stresses of non-linear flexible members of a power transmission iron tower are calculated effectively through a mathematical method, so that an important scientific basis can be laid for safety evaluation of the iron tower structure.

Description

The calculation method for stress of the non-linear flexible member of closing to reality running status

The application is application number: the divisional application of 201410425896.8, the applying date: 2014.8.26, title " calculation method for stress of the non-linear flexible member of electric power pylon ".

Technical field

The present invention relates to and a kind ofly structure mechanics analysis is carried out to electric power pylon flexible member and accurately can obtain the method for arbitrfary point stress on steel tower, belong to transmission line of electricity running status safety evaluation field.

Background technology

Transmission line of electricity is the passage that State Grid's industry and relevant enterprise are depended on for existence, and the safety evaluation for transmission line of electricity has vital effect for whole power system stability operation.Due to development and the innovation of power industry, the grade of transmission line of electricity is more and more higher, the landform of process also complexity all the more, be easily subject to the interference of the series of complex such as physical geography condition and social condition situation, the stable operation of circuit receives great harm.Therefore, experts and scholars both domestic and external, evaluate circuit various aspects also in continuous advance this research.

For 08 year especially big ice disaster, it was one of major reason of power outage generation that industry and educational circles expert generally believe that electrical network lacks scientific and effective accident early warning combined system etc. of preventing and reducing natural disasters.The key setting up this system gives scientific evaluation to the safe condition of transmission line of electricity.Current correlative study is mainly through applying pulling force sensor, online design monitor for stress.Because hardware cost, device body are subject to the influence factors such as outside destroy, the application of its further genralrlization receives restriction.Develop a kind of zero hardware cost, become the task of top priority of electric power netting safe running without environmental constraints, reusable in the safety appraisement of structure system of all operation transmission lines of electricity, wherein carrying out that accurate stress distribution calculates to steel tower rod member is key wherein.

For the stress distribution analytical calculation of electric power pylon, the technology path being seen in open report is all repeatedly simplify equivalence to iron tower structure, load suffered by steel tower is simply decomposed or is superposed, and ignores the elastic deformation of iron tower structure.The result that this kind of analytical approach provides, cannot provide stress data exactly for safety evaluation.Consider from the angle of mechanical analysis, out of true.And be only based on some assumed condition for the safety evaluation of electric power pylon, and carry out linearization approximate process, and on node after steel tower stressed and Displacement-deformation everywhere is all summed up in the point that iron tower model.And the situation of reality is complicated a lot, being not the inearized model simulated, is not that trouble spot is just on shaft tower node yet.Therefore, be necessary to carry out more accurate and rational modeling to the shaft tower situation of reality, adopt each position situation of more accurate method to whole shaft tower to carry out A+E.

Based on above-mentioned analysis, the present invention proposes the model of more closing to reality shaft tower structure and running status, by adopting stiffness matrix and the loading matrix of finite element incremental method and Newton iteration method correction iron tower construction member unit, and introduce Lagrangian difference functions, to obtain more accurate iron tower construction member structural mechanics.

Summary of the invention

In order to solve the problem, the present invention proposes the calculation method for stress of the non-linear flexible member of electric power pylon, for the stress of the every root rod member of analytical calculation steel tower, comprises the steps:

Step 1: structural model and the material model of determining electric power pylon;

Step 2: for each member unit of steel tower and nodal displacement, respectively generation unit stiffness matrix [k] ^(e)with nodal displacement array and uniformly distributed load suffered by steel tower and not a node load-transfer mechanism are displaced on node, form panel load array

Step 3: the stiffness matrix and the loading matrix that adopt the iron tower construction member unit obtained in finite element incremental method and Newton iteration method correction step 2, and calculate the transposed matrix of iron tower construction member unit;

Step 4: stability and strength checking are carried out to the stressed of electric power pylon rod member;

Step 5: according to the transposed matrix of loading matrix and acquired iron tower construction member, carry out Lagrange's interpolation calculating respectively, obtain load and the displacement of steel tower entirety.

Wherein, the structural model of the electric power pylon in step 1 is that shaft tower is considered as three dimensions truss, and the bar unit of space truss is all two power bar units, only by axial force in structure stress, and using non-linear unit as analytic target; Material model is the complicated steel tower that there is flexible member in iron tower structure.

Nodal displacement array in step 2 with element stiffness matrix [k] ^(e)and panel load array be respectively:

1) displacement array is put expression formula be:

$\begin{array}{cc}{\left[\stackrel{\‾}{\mathrm{\δ}}\right]}^{\left(e\right)}=\left[\begin{array}{c}{\left[\stackrel{\‾}{\mathrm{\δ}}\right]}_1\\ {\left[\stackrel{\‾}{\mathrm{\δ}}\right]}_2\\ {\left[\stackrel{\‾}{\mathrm{\δ}}\right]}_3\\ .\\ .\\ .\\ {\left[\stackrel{\‾}{\mathrm{\δ}}\right]}_n\end{array}\right]& {\left[\stackrel{\‾}{\mathrm{\δ}}\right]}_i=\left[\begin{array}{c}{u}_i\\ v_i\\ w_i\\ {\mathrm{\θ}}_{\mathrm{xi}}\\ {\mathrm{\θ}}_{\mathrm{yi}}\\ {\mathrm{\θ}}_{\mathrm{zi}}\end{array}\right]\end{array},i=\mathrm{1,2},...,n$

Wherein, for the nodal displacement array in global coordinate system; it is the displacement array of the 1st node; it is the displacement array of the 2nd node; By that analogy it is the displacement array of the n-th node; u _i, v _i, w _iit is the displacement of the lines in the i-th node three directions in local coordinate system; θ _xi, θ _yi, θ _zibe the rotation of the i-th Nodes cross section around 3 coordinate axis, θ _xirepresent the torsion in cross section, θ _yi, θ _zirepresent the rotation of cross section in xz and xy coordinate surface respectively.

2) element stiffness matrix [k] ^(e)expression formula be:

Wherein, [k] ^(e)for the stiffness matrix of bar unit in unit local coordinate system; A is bar unit cross-sectional area; I _yfor cross sectional moment of inertia in xz face; I _zfor the cross sectional moment of inertia in xy face; I _pfor the torsional moment of inertia of unit; L is length; E and G is respectively elastic modulus and the modulus of shearing of material.

3) panel load matrix expression formula be:

$\begin{array}{cc}{\[\stackrel{\‾}{R}\]}^{\left(e\right)}=\left[\begin{array}{c}{\[\stackrel{\‾}{R}\]}_1\\ {\[\stackrel{\‾}{R}\]}_2\\ {\[\stackrel{\‾}{R}\]}_3\\ .\\ .\\ .\\ {\[\stackrel{\‾}{R}\]}_n\end{array}\right]& {\[\stackrel{\‾}{R}\]}_i=\left[\begin{array}{c}{N}_{xi}\\ N_{yi}\\ N_{zi}\\ M_{xi}\\ M_{yi}\\ M_{zi}\end{array}\right],i=1,2,\mathrm{...}n\end{array}$

Wherein, for panel load arrays all in overall coordinate; for the load column of i-th node in overall coordinate; N _xibe the axial force of i-th node, N _yi, N _zibe respectively the shearing of i-th node in xy and xz face; M _xibe the moment of torsion of i-th node, M _yi, M _zibe the moment of flexure of i-th node in xz and xy face.

The expression formula of the stiffness matrix and loading matrix of revising the iron tower construction member unit obtained in step 2 in step 3 is:

$\left\{\begin{array}{c}{\∫}_{tv}{\left\{{\mathrm{\δ}}_t\mathrm{\ϵ}\right\}}^T\[C_t\]{\left\{\mathrm{\ϵ}_t\right\}}^{t}dCV+{\∫}_{tv}{\left\{{\mathrm{\δ}}_t\mathrm{\η}\right\}}^T{\left\{\mathrm{\σ}_t^t\right\}}^{t}dV{=}_t^{t+\mathrm{\Δ}t}W+{\∫}_{tv}{\left\{{\mathrm{\δ}}_{t}l\right\}}^T{\left\{\mathrm{\σ}_t^t\right\}}^{t}dV\\ {\∫}_{tv}{\left\{{\mathrm{\δ}}_{t}l\right\}}^T\[C_t\]{\left\{l_t\right\}}^{t}dCV+{\∫}_{tv}{\left\{{\mathrm{\δ}}_t\mathrm{\η}\right\}}^T{\left\{\mathrm{\σ}_t^t\right\}}^{t}dV{=}_t^{t+\mathrm{\Δ}t}W+{\∫}_{tv}{\left\{{\mathrm{\δ}}_{t}l\right\}}^T{\left\{\mathrm{\σ}_t^t\right\}}^{t}dV\\ ({\[K_t^t\]}_L+{\[K_t^t\]}_{NL})\left\{{u_t}^k\right\}=\left\{R_t^{t+\mathrm{\Δ}t}\right\}-\[F_t^t\]\end{array}\right.$

Wherein, _tε } be the strain of t iron tower construction member, [ _tc] be the constitutive relation of steel tower material, _tη } be increment non-linear partial, for t iron tower construction member stress intensity, _tl} is increment linear segment, for acting on the virtual work that iron tower construction member external force is done, for Green's Lagrange strain component, for iron tower construction member linear strain increment rigidity matrix, for iron tower construction member strain increment stiffness matrix, for t is equivalent to the joint forces vector of iron tower construction member element stress, for t+ Δ t acts on the load vector on iron tower construction member unit node.

In step 4, the stressed formula carrying out stability and strength checking of electric power pylon rod member is respectively:

1) Stability Checking:

2) strength checking:

S=(T or N)/(m (A-2d ₀t))

Wherein, σ is the stability coefficient of steel tower bar element, and s is strength factor, and T is pulling force suffered by rod member, and A is bar element cross-sectional area, and N is rod member pressure, d ₀for bolt aperture, m is service factor, and φ is conversion factor.

For the transposed matrix of loading matrix and acquired iron tower construction member in step 5, carrying out Lagrange's interpolation computing formula is respectively:

$L\left(x_i\right)=\underset{j=0}{\overset{k}{\Σ}}{\mathrm{\ϵ}}_{xi}{l^{\′}}_i\left(x_i\right)$

$l_i\left(x_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{x-x_i}{x_j-x_i}=\frac{(x-x_0)}{(x_i-x_0)}\mathrm{...}\frac{(x-x_{j-1})}{(x_j-x_{j-1})}\frac{(x-x_{j+1})}{(x_j-x_{j+1})}\mathrm{...}\frac{(x-x_k)}{(x_j-x_k)}$

$L\left(y_i\right)=\underset{j=0}{\overset{k}{\Σ}}{\mathrm{\ϵ}}_{yi}{l^{\′}}_i\left(y_i\right)$

$l_i\left(y_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{y-y_i}{y_j-y_i}=\frac{(y-y_0)}{(y_i-y_0)}\mathrm{...}\frac{(y-y_{j-1})}{(y_j-y_{j-1})}\frac{(y-y_{j+1})}{(y_j-y_{j+1})}\mathrm{...}\frac{(y-y_k)}{(y_j-y_k)}$

$L\left(z_i\right)=\underset{j=0}{\overset{k}{\Σ}}{\mathrm{\ϵ}}_{zi}{l^{\′}}_i\left(z_i\right)$

$l_i\left(z_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{z-z_i}{z_j-z_i}=\frac{(z-z_0)}{(z_i-z_0)}\mathrm{...}\frac{(z-z_{j-1})}{(z_j-z_{j-1})}\frac{(z-z_{j+1})}{(z_j-z_{j+1})}\mathrm{...}\frac{(z-z_k)}{(z_j-z_k)}$

$L\left(x_i\right)=\underset{j=0}{\overset{k}{\Σ}}{u}_{xi}{l^{\′}}_i\left(x_i\right)$

$l_i\left(x_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{x-x_i}{x_j-x_i}=\frac{(x-x_0)}{(x_i-x_0)}...\frac{(x-x_{j-1})}{(x_j-x_{j-1})}\frac{(x-x_{j+1})}{(x_j-x_{j+1})}...\frac{(x-x_k)}{(x_j-x_k)}$

$L\left(y_i\right)=\underset{j=0}{\overset{k}{\Σ}}{u}_{yi}{l^{\′}}_i\left(y_i\right)$

$l_i\left(y_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{y-y_i}{y_j-y_i}=\frac{(y-y_0)}{(y_i-y_0)}...\frac{(y-y_{j-1})}{(y_j-y_{j-1})}\frac{(y-y_{j+1})}{(y_j-y_{j+1})}...\frac{(y-y_k)}{(y_j-y_k)}$

$L\left(z_i\right)=\underset{j=0}{\overset{k}{\Σ}}{u}_{zi}{l^{\′}}_i\left(z_i\right)$

$l_i\left(z_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{z-z_i}{z_j-z_i}=\frac{(z-z_0)}{(z_i-z_0)}\mathrm{...}\frac{(z-z_{j-1})}{(z_j-z_{j-1})}\frac{(z-z_{j+1})}{(z_j-z_{j+1})}\mathrm{...}\frac{(z-z_k)}{(z_j-z_k)}$

Wherein, x, y, z represent the coordinate of iron tower construction member node respectively; ε _xi, ε _yi, ε _zirepresent that steel tower i-th component is at x, y respectively, the strain on z-axis direction; μ _xi, μ _yi, μ _zirepresent that steel tower i-th component is at x, y respectively, the displacement on z-axis direction; L is Lagrangian basis function; L is Lagrangian differential polynomial, and i represents the number of iron tower construction member node.

Technique effect of the present invention:

1) this method is not subject to the restriction of hardware cost and outside environmental elements, is applicable to the safety appraisement of structure of all operation iron tower of power transmission line.

2) this method considers the actual conditions of steel tower, more closing to reality electric power pylon structure and running status model.

3) this method is by introducing stiffness matrix and the loading matrix of finite element incremental method and Newton iteration method correction iron tower construction member unit, and calculates through Lagrange's interpolation, can obtain more accurate steel tower overall load and displacement.

Accompanying drawing explanation

Fig. 1 is " doing " font steel tower schematic diagram.

Embodiment

The calculation method for stress of the non-linear flexible member of electric power pylon of the present invention, can be summarized as four-stage: process in early stage, finite element analysis, post-processed and improvement project calculate.Process in early stage comprises sets up iron tower structure model and material model; Namely FEM mechanics analysis is analyzed the finite element model of iron tower structure, in conjunction with relation between an iron tower structure material, carry out Nonlinear Superposition, according to finite increment method and Newton iteration method, revise the stress-strain matrix of steel tower and the displacement loading matrix of each unit; Namely post-processed carries out the verification of stability and intensity to the matrix of respectively trying to achieve, guarantee that computation process and result do not have mistake; Namely improvement project carries out Lagrange's interpolation to each cell data, obtains the calculation expression of each parts of Lifting Method in Pole Tower Integral Hoisting, and releases position the weakest in shaft tower accordingly.The method mainly comprises the steps:

Step 1: structural model and the material model of determining electric power pylon;

Step 2: for each member unit of steel tower and nodal displacement, respectively generation unit stiffness matrix [k] ^(e)with nodal displacement array and uniformly distributed load suffered by steel tower and not a node load-transfer mechanism are displaced on node, form panel load array

Step 3: the stiffness matrix and the loading matrix that adopt the iron tower construction member unit obtained in finite element incremental method and Newton iteration method correction step 2, and calculate the transposed matrix of iron tower construction member unit;

Step 4: stability and strength checking are carried out to the stressed of electric power pylon rod member;

Step 5: according to the transposed matrix of loading matrix and acquired iron tower construction member, carry out Lagrange's interpolation calculating respectively, obtain load and the displacement of steel tower entirety.

Below each step is described in further detail:

In step 1: structural model and the material model of determining electric power pylon, its specific implementation process is:

The material that transmission of electricity uses is generally angle steel, bar steel, round steel and wire rope.For self-supporting tower, Guywire tower, the moment of flexure caused by the Lateral Wind etc. on eccentric load, rod member is little, and therefore, shaft tower can be considered desirable three dimensions truss.The bar unit of space truss is all two power bar units, only by axial force in structure stress.The present invention is directed to complicated large-scale steel tower, the feature of this steel tower distortion produces Large travel range, small strain, simultaneously, for the complicated steel tower that there is flexible member, its flexible member can not bear pressure, it strains with stress and nonlinear relationship, and therefore, in analyzing iron tower structure, this method adopts non-linear unit as analytic target.For the steel tower not having flexible member, material is the linear elastic material only determined by elastic modulus.The complicated shaft tower existed for there being flexible member, in finite element nonlinear analysis, is divided into two groups: the rigid element 1) bearing tension and compression by material: material is the linear elastic materials only determined by elastic modulus; 2) flexible unit of pulling force is only born: setting nonlinear material pattern processes and can only bear pulling force, the material that can not bear rod member (band steel, round steel, drag-line etc.) flexible member of pressure is assumed to be nonlinear elastic material, and its characteristic is that the piecewise linear function by stress being expressed as current strain defines.Therefore, total stress and tangent modulus are directly determined by overall strain.Material ideal is turned to nonlinear elastic material, and when iron tower construction member tension, tangent modulus is large, and shaft tower is normally stressed; When iron tower construction member pressurized, tangent modulus is very little, and the distortion regardless of rod member is much, and stress is close to null value, and rod member is stressed small, can be considered as not stressing.

In step 2: for each member unit of steel tower and nodal displacement, generation unit stiffness matrix [k] respectively ^(e)with nodal displacement array and uniformly distributed load suffered by steel tower and not a node load-transfer mechanism are displaced on node, form panel load array its specific implementation process is:

After iron tower structure discretize, mechanical characteristic analysis to be carried out to unit, the relation namely between determining unit nodal force and nodal displacement.In order to analyze and determine this relation, need to select displacement model, displacement function is displacement that unit the is put function to the coordinate of point, this method polynomial expression of the coordinate of unit internal point represents, rod member in space, each node has 6 degree of freedom, and namely rod member is except bearing the effect of one dimension axle power, bidimensional shearing and bidimensional moment of flexure, also may bear the effect of one dimension moment of torsion.And, space framed rods bears one dimension axle power, bidimensional shearing, bidimensional moment of flexure, one dimension moment of torsion, namely correspond to 6 degree of freedom of node, be respectively the displacement of the lines on 3 directions and the rotation in Nodes cross section around 3 coordinate axis, because the polynomial expression of the coordinate of this element internal point can be expressed as δ=k ₁u+k ₂v+k ₃w+k ₄θ _x+ k ₅θ _y+ k ₆θ _z, accordingly, the displacement array of all nodes can be formed

$\begin{array}{cc}{\[\stackrel{\‾}{\mathrm{\δ}}\]}^{\left(e\right)}=\left[\begin{array}{c}{\[\stackrel{\‾}{\mathrm{\δ}}\]}_1\\ {\[\stackrel{\‾}{\mathrm{\δ}}\]}_2\\ {\[\stackrel{\‾}{\mathrm{\δ}}\]}_3\\ .\\ .\\ .\\ {\[\stackrel{\‾}{\mathrm{\δ}}\]}_n\end{array}\right],& {\[\stackrel{\‾}{\mathrm{\δ}}\]}_i=\left[\begin{array}{c}{u}_i\\ v_i\\ w_i\\ {\mathrm{\θ}}_{xi}\\ {\mathrm{\θ}}_{yi}\\ {\mathrm{\θ}}_{zi}\end{array}\right],i=1,2,\mathrm{...},n\end{array}$

Wherein, for the nodal displacement array in global coordinate system; it is the displacement array of the 1st node; it is the displacement array of the 2nd node; By that analogy it is the displacement array of the n-th node; u _i, v _i, w _iit is the displacement of the lines in the i-th node three directions in local coordinate system; θ _xi, θ _yi, θ _zibe the rotation of the i-th Nodes cross section around three coordinate axis, θ _xirepresent the torsion in cross section, θ _yi, θ _zirepresent the rotation of cross section in xz and xy coordinate surface respectively.

The basic step setting up stiffness equation is: on the basis of assuming unit displacement function, according to theory of elastic mechanics, sets up strain, relational expression between stress and nodal displacement.Then according to the principle of virtual displacement, try to achieve the relation between cell node power and nodal displacement, thus draw following element stiffness matrix [k] ^(e):

Wherein, [k] ^(e)for the stiffness matrix of bar unit in unit local coordinate system; A is bar unit cross-sectional area; I _yfor cross sectional moment of inertia in xz face; I _zfor the cross sectional moment of inertia in xy face; I _pfor the torsional moment of inertia of unit; L is length; E and G is respectively elastic modulus and the modulus of shearing of material.

Again, uniformly distributed load, not a node load-transfer mechanism suffered by steel tower are displaced on node, form panel load array rod member in space, each node has 6 degree of freedom, and namely rod member is except bearing the effect of axle power, shearing and moment of flexure, also may bear the effect of moment of torsion.Further, space framed rods bears one dimension axle power, bidimensional shearing, bidimensional moment of flexure, one dimension moment of torsion, namely correspond to 6 degree of freedom of node.The bar unit space framed rods just of electric power pylon.

$\begin{array}{cc}{\[\stackrel{\‾}{R}\]}^{\left(e\right)}=\left[\begin{array}{c}{\[\stackrel{\‾}{R}\]}_1\\ {\[\stackrel{\‾}{R}\]}_2\\ {\[\stackrel{\‾}{R}\]}_3\\ .\\ .\\ .\\ {\[\stackrel{\‾}{R}\]}_n\end{array}\right],& {\[\stackrel{\‾}{R}\]}_i=\left[\begin{array}{c}{N}_{xi}\\ N_{yi}\\ N_{zi}\\ M_{xi}\\ M_{yi}\\ M_{zi}\end{array}\right],i=1,2,\mathrm{...},n\end{array}$

Wherein, for panel load arrays all in overall coordinate; for the load column of i-th node in overall coordinate; N _xibe the axial force of i-th node, N _yi, N _zibe respectively the shearing of i-th node in xy and xz face; M _xibe the moment of torsion of i-th node, M _yi, M _zibe the moment of flexure of i-th node in xz and xy face.

Step 3: the stiffness matrix and the loading matrix that adopt the iron tower construction member unit obtained in finite element incremental method and Newton iteration method correction step 2, and calculate the transposed matrix of iron tower construction member unit, its specific implementation process is:

The present invention needs the problem solved to be embodied in the nonlinear elasticity of electric power pylon large deformation and material, therefore, Nonlinear FEM principle need be adopted to process the problems referred to above.For large deformation, adopt geometrical nonlinear analysis, for nonlinear elastic material, adopt material Nonlinearity Analysis method.For nonlinear problem, can not adopt the method for a step direct solution, nonlinear problem must be divided into several and load step, solve gradually stage by stage to problem, what namely adopt increment solves scheme.Steel tower large deformation feature is: the displacement of structure is fully large, but the elongation of bar unit is very little.When analyzing, be used as large deformation and small strain nonlinear problem to process.In nonlinear problem, the balance equation of structure must write out with the geometric position after distortion, and the reference configuration got is different, and the result obtained is also different.In the analysis to electric power pylon structure, adopt updated Lagrange description method, namely with the state of t for measuring standard, consider the state in the moment of t+ Δ t.Updated Lagrange description method is:

First the following constitutional balance equation adopting virtual work to represent is introduced:

${\∫}_{tv}{\left\{{\mathrm{\δ}}^{t+\mathrm{\Δ}t}\mathrm{\ϵ}_t\right\}}^T{\left\{s_t^{t+\mathrm{\Δ}t}\right\}}^{t}dv=W_t^{t+\mathrm{\Δ}t}---\left(1\right)$

Wherein, δ is electric power pylon nodal displacement array, for Equations of The Second Kind is than the thunder kirchhoff components of stress difficult to understand, for the virtual work that external force is done, for Green's Lagrange strain component.

The displacement of the above-mentioned components of stress, the components of strain and t+ Δ t is expressed as incremental form:

$\left\{s_t^{t+\mathrm{\Δ}t}\right\}=\left\{\mathrm{\σ}_t^t\right\}+\left\{s_t\right\}---\left(2\right)$

$\left\{\mathrm{\ϵ}_t^{t+\mathrm{\Δ}t}\right\}=\left\{\mathrm{\ϵ}_t^t\right\}---\left(3\right)$

$\left\{u_t^{t+\mathrm{\Δ}t}\right\}=\left\{u_t^t\right\}---\left(4\right)$

The stress of t+ Δ t, strain, displacement are considered as the stress of t, strain and displacement and incremental stress, strain, displacement sum.Incremental strain can be expressed as further linear segment _tl} and non-linear partial _tη } sum.

{ _tε}＝{ _tl}+{ _tη} (5)

So, describe the relation between ess-strain according to Lagrange, following formula can be obtained:

${\∫}_{tv}{\left\{{\mathrm{\δ}}_t\mathrm{\ϵ}\right\}}^T\[C_t\]{\left\{\mathrm{\ϵ}_t\right\}}^{t}dCV+{\∫}_{tv}{\left\{{\mathrm{\δ}}_t\mathrm{\η}\right\}}^T{\left\{\mathrm{\σ}_t^t\right\}}^{t}dV{=}_t^{t+\mathrm{\Δ}t}W+{{\∫}_{tv}\left\{{\mathrm{\δ}}_{t}l\right\}}^T{\left\{\mathrm{\σ}_t^t\right\}}^{t}dV---\left(6\right)$

Formula (6) be one about displacement increment _tthe nonlinear equation of l}, when processing, must carry out linearization process by above-mentioned equation.

If _tε }= _tl}, then { δ _lε }={ δ _ll}, the Lagrange's equation that therefore can obtain the correction of incremental form is:

${\∫}_{tv}\left\{{\mathrm{\δ}}_{t}l\right\}\[C_t\]{\left\{l_t\right\}}^{t}dCV+{\∫}_{tv}{\left\{{\mathrm{\δ}}_t\mathrm{\η}\right\}}^T{\left\{\mathrm{\σ}_t^t\right\}}^t{\mathrm{dV}}_t^{t+\mathrm{\Δ}t}W+{{\∫}_{tv}\left\{{\mathrm{\δ}}_{t}l\right\}}^T{\left\{\mathrm{\σ}_t^t\right\}}^{t}dV---\left(7\right)$

Nonlinear elasticity sex chromosome mosaicism, be embodied in material constitutive relation [ _tc] in, GEOMETRICALLY NONLINEAR, be then embodied in strain non-linear partial _tη }.

According to above-mentioned balance equation, utilize the model after structural separation, the non-linear incremental bending forming fundamental equation of following updated Lagrange description can be derived:

$({\[K_t^t\]}_L+{\[K_t^t\]}_{NL})\left\{{u_t}^k\right\}=\left\{R_t^{t+\mathrm{\Δ}t}\right\}-\[F_t^t\]---\left(8\right)$

Wherein, for iron tower construction member linear strain increment rigidity matrix, for iron tower construction member strain increment stiffness matrix, for t is equivalent to the joint forces vector of iron tower construction member element stress, for t+ Δ t acts on the load vector on iron tower construction member unit node, _tε } be the strain of t iron tower construction member, [ _tc] be the constitutive relation of steel tower material, _tη } be increment non-linear partial, for t iron tower construction member stress intensity, _tl} is increment linear segment, for acting on the virtual work that iron tower construction member external force is done, for Green's Lagrange strain component.

Step 4: carry out stability and strength checking to the stressed of electric power pylon rod member, its implementation process is as follows:

The inventive method have employed nonlinear finite element method and analyzes the stress that the result of iron tower structure is steel tower rod member, therefore, be necessary counter stress analysis, the factors such as comprehensive stability, security, economy, the stressed of rod member is tested, and carry out automatic group of material of material, generate the combination of materials required for user.In stress checking calculation, main compressive strength and the Compression Stability (flexible member does not consider Compression Stability) considering rod member.Therefore, adopt following formula to verify stability and intensity:

1) Stability Checking:

2) strength checking:

S=(T or N)/(m (A-2d ₀t)) (10) wherein, σ is the stability coefficient of steel tower bar element, and s is strength factor, and T is pulling force suffered by rod member, and A is bar element cross-sectional area, and N is rod member pressure, d ₀for bolt aperture, m is service factor, and φ is conversion factor.

Step 5: according to the transposed matrix of loading matrix and acquired iron tower construction member, carry out Lagrange's interpolation calculating respectively, obtain load and the displacement of steel tower entirety, its specific implementation process is:

Bright day interpolation is drawn respectively to the stress-strain matrix of the revised iron tower construction member obtained in step 3 and displacement loading matrix, unit is carried out entirety synthesis, obtain the computation model formula of whole shaft tower, the parameter of any point on steel tower can be calculated accordingly, and can obtain the extreme value of parameter on whole shaft tower, this is of great significance for the safe tool of analyzing rod tower structure.

According to the position coordinates of Lagrangian difference and iron tower construction member, can obtain:

$L\left(x_i\right)=\underset{j=0}{\overset{k}{\Σ}}{\mathrm{\ϵ}}_{xi}{l^{\′}}_i\left(x_i\right)$

$l_i\left(x_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{x-x_i}{x_j-x_i}=\frac{(x-x_0)}{(x_i-x_0)}...\frac{(x-x_{j-1})}{(x_j-x_{j-1})}\frac{(x-x_{j+1})}{(x_j-x_{j+1})}...\frac{(x-x_k)}{(x_j-x_k)}$

$L\left(y_i\right)=\underset{j=0}{\overset{k}{\Σ}}{\mathrm{\ϵ}}_{yi}{l^{\′}}_i\left(y_i\right)$

$l_i\left(y_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{y-y_i}{y_j-y_i}=\frac{(y-y_0)}{(y_i-y_0)}...\frac{(y-y_{j-1})}{(y_j-y_{j-1})}\frac{(y-y_{j+1})}{(y_j-y_{j+1})}...\frac{(y-y_k)}{(y_j-y_k)}$

$L\left(z_i\right)=\underset{j=0}{\overset{k}{\Σ}}{\mathrm{\ϵ}}_{zi}{l^{\′}}_i\left(z_i\right)$

$l_i\left(z_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{z-z_i}{z_j-z_i}=\frac{(z-z_0)}{(z_i-z_0)}\mathrm{...}\frac{(z-z_{j-1})}{(z_j-z_{j-1})}\frac{(z-z_{j+1})}{(z_j-z_{j+1})}\mathrm{...}\frac{(z-z_k)}{(z_j-z_k)}$

$L\left(x_i\right)=\underset{j=0}{\overset{k}{\Σ}}{u}_{xi}{l^{\′}}_i\left(x_i\right)$

$l_i\left(x_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{x-x_i}{x_j-x_i}=\frac{(x-x_0)}{(x_i-x_0)}\mathrm{...}\frac{(x-x_{j-1})}{(x_j-x_{j-1})}\frac{(x-x_{j+1})}{(x_j-x_{j+1})}\mathrm{...}\frac{(x-x_k)}{(x_j-x_k)}$

$L\left(y_i\right)=\underset{j=0}{\overset{k}{\Σ}}{u}_{yi}{l^{\′}}_i\left(y_i\right)$

$l_i\left(y_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{y-y_i}{y_j-y_i}=\frac{(y-y_0)}{(y_i-y_0)}...\frac{(y-y_{j-1})}{(y_j-y_{j-1})}\frac{(y-y_{j+1})}{(y_j-y_{j+1})}...\frac{(y-y_k)}{(y_j-y_k)}$

$L\left(z_i\right)=\underset{j=0}{\overset{k}{\Σ}}{u}_{zi}{l^{\′}}_i\left(z_i\right)$

$l_i\left(z_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{z-z_i}{z_j-z_i}=\frac{(z-z_0)}{(z_i-z_0)}\mathrm{...}\frac{(z-z_{j-1})}{(z_j-z_{j-1})}\frac{(z-z_{j+1})}{(z_j-z_{j+1})}\mathrm{...}\frac{(z-z_k)}{(z_j-z_k)}$

Wherein, x, y, z represent the coordinate of iron tower construction member node respectively; ε _xi, ε _yi, ε _zirepresent that steel tower i-th component is at x, y respectively, the strain on z-axis direction; μ _xi, μ _yi, μ _zirepresent that steel tower i-th component is at x, y respectively, the displacement on z-axis direction; L is Lagrangian basis function; L is Lagrangian differential polynomial, and i represents the number of iron tower construction member node.

To the L (x) in above-mentioned formula, L (y), L (z), L (xy), the independent variable of L (yz), L (yz) carries out first differential, obtains the point that its derivative equals 0, even L ' (x)=0, L ' (y)=0, L ' (z)=0, L ' (xy)=0, L ' (yz)=0, L ' (xz)=0; Its solution is designated as x ' respectively, y ', z ', xy ', yz ', and xz ' obtains L (x ') respectively, L (y '), L (z '), L (xy '), L (yz '), L (xz ').Now, the limit of the strain and displacement in steel tower all directions and maximal value thereof can obtain, and can in the hope of the maximum displacement point on shaft tower with this.

The Lagrange's interpolation expression formula of its stress and load is as follows:

$L^1\left(x_i\right)=\underset{j=0}{\overset{k}{\Σ}}{F}_{xi}{l^{\′}}_i\left(x_i\right)$

${l^1}_i\left(x_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{x-x_i}{x_j-x_i}=\frac{(x-x_0)}{(x_i-x_0)}\mathrm{...}\frac{(x-x_{j-1})}{(x_j-x_{j-1})}\frac{(x-x_{j+1})}{(x_j-x_{j+1})}\mathrm{...}\frac{(x-x_k)}{(x_j-x_k)}$

$L^1\left(y_i\right)=\underset{j=0}{\overset{k}{\Σ}}{F}_{yi}{l^{\′}}_i\left(y_i\right)$

${l^1}_i\left(y_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{y-y_i}{y_j-y_i}=\frac{(y-y_0)}{(y_i-y_0)}...\frac{(y-y_{j-1})}{(y_j-y_{j-1})}\frac{(y-y_{j+1})}{(y_j-y_{j+1})}...\frac{(y-y_k)}{(y_j-y_k)}$

$L^1\left(z_i\right)=\underset{j=0}{\overset{k}{\Σ}}{F}_{zi}{l^{\′}}_i\left(z_i\right)$

${l^{\′}}_i\left(z_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{z-z_i}{z_j-z_i}=\frac{(z-z_0)}{(z_i-z_0)}\mathrm{...}\frac{(z-z_{j-1})}{(z_j-z_{j-1})}\frac{(z-z_{j+1})}{(z_j-z_{j+1})}\mathrm{...}\frac{(z-z_k)}{(z_j-z_k)}$

$L^1\left(x\right)=\underset{j=0}{\overset{k}{\Σ}}{R}_{xi}{l^{\′}}_i\left(x\right)$

${l^{\′}}_i\left(x_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{x-x_i}{x_j-x_i}=\frac{(x-x_0)}{(x_i-x_0)}\mathrm{...}\frac{(x-x_{j-1})}{(x_j-x_{j-1})}\frac{(x-x_{j+1})}{(x_j-x_{j+1})}\mathrm{...}\frac{(x-x_k)}{(x_j-x_k)}$

$L^1\left(y\right)=\underset{j=0}{\overset{k}{\Σ}}{R}_{yi}{l^{\′}}_i\left(y\right)$

${l^1}_i\left(y_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{y-y_i}{y_j-y_i}=\frac{(y-y_0)}{(y_i-y_0)}\mathrm{...}\frac{(y-y_{j-1})}{(y_j-y_{j-1})}\frac{(y-y_{j+1})}{(y_j-y_{j+1})}\mathrm{...}\frac{(y-y_k)}{(y_j-y_k)}$

$L^1\left(z\right)=\underset{j=0}{\overset{k}{\Σ}}{R}_{zi}{l^{\′}}_i\left(z\right)$

${l^1}_i\left(z_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{z-z_i}{z_j-z_i}=\frac{(z-z_0)}{(z_i-z_0)}\mathrm{...}\frac{(z-z_{j-1})}{(z_j-z_{j-1})}\frac{(z-z_{j+1})}{(z_j-z_{j+1})}\mathrm{...}\frac{(z-z_k)}{(z_j-z_k)}$

F _ifor the stress vector of node, x, y, z is respectively the direction of node stress; R _ifor the load vector of node, x, y, z represents its direction.

In like manner, l ¹ _ifor Lagrangian fundamental polynomials (Lagrangian basis function), L ¹for Lagrange interpolation polynomial, it has existence and uniqueness.

To L wherein ¹(x), L ¹(y), L ¹(z), L ¹(xy), L ¹(yz), L ¹(xz) independent variable carries out first differential, obtains the point that its derivative equals 0, even L ¹' (x)=0, L ¹' (y)=0, L ¹' (z)=0, L ¹' (xy)=0, L ¹' (yz)=0, L ¹' (xz)=0; Its solution is designated as x respectively ¹', y ¹', z ¹', xy ¹', yz ¹', xz ¹', obtain L respectively ¹(x ¹'), L ¹(y ¹'), L ¹(z ¹'), L ¹(xy ¹'), L ¹(yz ¹'), L ¹(xz ¹') value.Now, can obtain stress in steel tower all directions and limit and maximal value.

Embodiment:

Below for " doing " font steel tower, to verify the contrast situation of the present invention and actual conditions.Left side result is answered for " do " font steel tower minimax utilizing this programme institute extracting method and calculate, and right side is " doing " font steel tower minimax stress result of actual measurement.

As seen from the above table, the result of calculation that institute of the present invention extracting method obtains and actual result are identical, and show validity and the Technology Potential of this method.

The content be not described in detail in this manual belongs to the known technology of those skilled in the art.

Claims (1)

1. the calculation method for stress of the non-linear flexible member of closing to reality running status, is characterized in that: comprise the steps:

Step 1: structural model and the material model of determining electric power pylon;

Step 2: for each member unit of steel tower and nodal displacement, respectively generation unit stiffness matrix [k] ^(e)with nodal displacement array and uniformly distributed load suffered by steel tower and not a node load-transfer mechanism are displaced on node, form panel load array

Step 3: the stiffness matrix and the loading matrix that adopt the iron tower construction member unit obtained in finite element incremental method and Newton iteration method correction step 2, and calculate the transposed matrix of iron tower construction member unit;

Step 4: stability and strength checking are carried out to the stressed of electric power pylon rod member;

Step 5: according to the transposed matrix of loading matrix and acquired iron tower construction member, carry out Lagrange's interpolation calculating respectively, obtain load and the displacement of steel tower entirety;

In step 1: structural model and the material model of determining electric power pylon, its specific implementation process is:

The material that transmission of electricity uses is angle steel, bar steel, round steel and wire rope; For the steel tower not having flexible member, material is the linear elastic material only determined by elastic modulus; The complicated shaft tower existed for there being flexible member, in finite element nonlinear analysis, is divided into two groups: the rigid element 1) bearing tension and compression by material: material is the linear elastic materials only determined by elastic modulus; 2) flexible unit of pulling force is only born: setting nonlinear material pattern processes and can only bear pulling force, the material that can not bear the rod member flexible member of pressure is assumed to be nonlinear elastic material, and its characteristic is that the piecewise linear function by stress being expressed as current strain defines; Material ideal is turned to nonlinear elastic material, and when iron tower construction member tension, tangent modulus is large, and shaft tower is normally stressed; When iron tower construction member pressurized, tangent modulus is very little, and the distortion regardless of rod member is much, and stress is close to null value, and rod member is stressed small, is considered as not stressing;

For the transposed matrix of loading matrix and acquired iron tower construction member in step 5, carrying out Lagrange's interpolation computing formula is respectively:

$L\left(x_i\right)=\underset{j=0}{\overset{k}{\Σ}}{\mathrm{\ϵ}}_{xi}{l^{\′}}_i\left(x_i\right)$

$l_i\left(x_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{x-x_i}{x_j-x_i}=\frac{(x-x_0)}{(x_i-x_0)}\mathrm{...}\frac{(x-x_{j-1})}{(x_j-x_{j-1})}\frac{(x-x_{j+1})}{(x_j-x_{j+1})}\mathrm{...}\frac{(x-x_k)}{(x_j-x_k)}$

$L\left(y_i\right)=\underset{j=0}{\overset{k}{\Σ}}{\mathrm{\ϵ}}_{yi}{l^{\′}}_i\left(y_i\right)$

$l_i\left(y_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{y-y_i}{y_j-y_i}=\frac{(y-y_0)}{(y_i-y_0)}\mathrm{...}\frac{(y-y_{j-1})}{(y_j-y_{j-1})}\frac{(y-y_{j+1})}{(y_j-y_{j+1})}\mathrm{...}\frac{(y-y_k)}{(y_j-y_k)}$

$L\left(z_i\right)=\underset{j=0}{\overset{k}{\Σ}}{\mathrm{\ϵ}}_{zi}{l^{\′}}_i\left(z_i\right)$

$l_i\left(z_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{z-z_i}{z_j-z_i}=\frac{(z-z_0)}{(z_i-z_0)}\mathrm{...}\frac{(z-z_{j-1})}{(z_j-z_{j-1})}\frac{(z-z_{j+1})}{(z_j-z_{j+1})}\mathrm{...}\frac{(z-z_k)}{(z_j-z_k)}$

$L\left(x_i\right)=\underset{j=0}{\overset{k}{\Σ}}{u}_{xi}{l^{\′}}_i\left(x_i\right)$

$l_i\left(x_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{x-x_i}{x_j-x_i}=\frac{(x-x_0)}{(x_i-x_0)}\mathrm{...}\frac{(x-x_{j-1})}{(x_j-x_{j-1})}\frac{(x-x_{j+1})}{(x_j-x_{j+1})}\mathrm{...}\frac{(x-x_k)}{(x_j-x_k)}$

$L\left(y_i\right)=\underset{j=0}{\overset{k}{\Σ}}{u}_{yi}{l^{\′}}_i\left(y_i\right)$

$l_i\left(y_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{y-y_i}{y_j-y_i}=\frac{(y-y_0)}{(y_i-y_0)}\mathrm{...}\frac{(y-y_{j-1})}{(y_j-y_{j+1})}\frac{(y-y_{j+1})}{(y_j-y_{j+1})}\mathrm{...}\frac{(y-y_k)}{(y_j-y_k)}$

$L\left(z_i\right)=\underset{j=0}{\overset{k}{\Σ}}{u}_{zi}{l^{\′}}_i\left(z_i\right)$

$l_i\left(z_i\right)=\underset{i=0,i\≠j}{\overset{k}{\Σ}}\frac{z-z_i}{z_j-z_i}=\frac{(z-z_0)}{(z_i-z_0)}\mathrm{...}\frac{(z-z_{j-1})}{(z_j-z_{j-1})}\frac{(z-z_{j+1})}{(z_j-z_{j+1})}\mathrm{...}\frac{(z-z_k)}{(z_j-z_k)}$

Wherein, x, y, z represent the coordinate of iron tower construction member node respectively; ε _xi, ε _yi, ε _zirepresent that steel tower i-th component is at x, y respectively, the strain on z-axis direction; μ _xi, μ _yi, μ _zirepresent that steel tower i-th component is at x, y respectively, the displacement on z-axis direction; L is Lagrangian basis function; L is Lagrangian differential polynomial, and i represents the number of iron tower construction member node.