CN1323213C - Method for implementing non-uniform rational B strip model of prestress beam - Google Patents

Abstract

The present invention relates to a method for implementing a non-uniform rational B spline model of a prestressed tendon, particularly to a function expression method of each point coordinates of prestressed reinforcement in a prestressed concrete beam bridge. When the existing method of polygonal approximation is described, the present invention overcomes the defects of large handling capacity of data and inconvenient adjustment of a curve shape. The present invention has the steps that an original shape line of a prestressed reinforcement tendon is drawn; characteristic points are chosen from the original shape line of the prestressed reinforcement; a segmented B spline primary function is determined between every two characteristic points according to a computing formula of a non-uniform rational B spline; a contrast shape line for connecting all characteristic points is drawn by using every segmented B spline primary function. The coincidence condition between the original shape line and the contrast shape line is detected, and non-coincidence points on the original shape line are confirmed as interpolation characteristic points; the contrast shape line for orderly connecting all characteristic points and interpolation characteristic points is drawn by using every segmented B spline primary function; an expression function of the prestressed reinforcement tendon is formed by seeking the congregation of every segmented B spline primary function.

Description

Method with non-uniform rational B-spline model measurement prestressing tendon

Technical field

The present invention relates to the measuring method of prestressed reinforcement in the prestressed concrete beam bridge.

Background technology

The numerical analysis of prestressed concrete beam is the important technical of carrying out the prestressed concrete beam design in the current field of civil engineering, adopting when analyzing.As a complicated technology system, the related theoretical algorithm of the numerical analysis of prestressed concrete beam is comparatively ripe mostly, and promptly calculation and thinking is comparatively clear and stable on theoretical and algorithm aspect.But, realizing on the technological layer, but there is comparatively outstanding technical barrier---a numerical expression problem of each point coordinates of prestressing force curve reinforcing bar (claiming prestressing tendon again) always, and this problem is to carry out the precondition that prestressing effect calculates, prestressing tendon in the actual engineering is linear comparatively complicated, is difficult to express with the normal function formula.Therefore, adopt which kind of Mathematical Modeling to express prestressing tendon, and carry out the correlation computations of prestressing tendon, become the prestressed concrete beam bridge structure and carry out one of major issue of prestressing effect analysis.In current engineering practice, prestressing force curve reinforcing bar shape is generally the geometric figure among the AutoCAD at the beginning of design, and numerical expression is by finite value coordinate point recording geometry, to finish the numerical analysis of postorder prestressed concrete beam.Function about each point coordinates of prestressing tendon is expressed model, and prior art is to use the space broken line of plurality of continuous to replace former space curve, promptly adopts segment approaching method that former space curve is dispersed and is some broken lines.Abundant in theory broken line can infinitely approach virgin curve, thereby give expression to the functional expression of this curve, but there are the following problems in actual applications: (1) prestressing tendon space curve generally has single order and Second Order Continuous, and replace the processing method of curved section to lose two rank continuitys with straightway, its direct result is equivalent load calculating all changing into a concentrated force, and this substituting will produce Algorithm Error; (2) in order to reach enough geometric accuracies, need more broken line, not only increased the pre-treatment workload, and the human error that broken line approaches is restive; (3) when structure design, the layout of prestressing tendon is to adjust to determine in the process of checking computations repeatedly, and the method for discrete straightway is not easy to the adjustment of prestressing tendon.

Summary of the invention

The purpose of this invention is to provide a kind of method, describe the large and small defective of being convenient to the curve shape adjustment of prestressing tendon space curve data processing amount to overcome with existing segment approaching method with non-uniform rational B-spline model measurement prestressing tendon.Technical scheme of the present invention comprises the steps: one, determines the shape and size of prestressing tendon reinforcing bar according to the stressing conditions of beam bridge, and is depicted as the original-shape line of prestressing tendon reinforcing bar with the spline curve order in the AutoCAD mapping software; It also comprises the steps: two, gets characteristic point on the original-shape line of prestressed reinforcement, get two end points as characteristic point for the straightway on the original-shape line, get three to seven characteristic points by the principle of dividing equally between the first and last end points for the curved section on the original-shape line; Three, determine the B spline base function of the sections between per two characteristic points according to the non-uniform rational B-spline design formulas, utilize the B spline base function of each sections to be depicted as the contrast shape line that connects all characteristic points; Four, detect the original-shape line of prestressing tendon reinforcing bar and the identical situation between the contrast shape line along its length, judge whether to exist the distance between certain extension position place original-shape line and the contrast shape line to surpass setting value; Five, the result of step 4 is for being then the point on this position original-shape line to be confirmed as the interpolation characteristic point; Six, determine the B spline base function of sections between the interpolation characteristic point characteristic point adjacent with left according to the non-uniform rational B-spline design formulas, determine the B spline base function of sections between interpolation characteristic point and the right-hand adjacent characteristic point simultaneously, utilize the B spline base function of each sections to be depicted as the contrast shape line of be linked in sequence all characteristic points and interpolation characteristic point; Return the starting end of step 4; Seven, the result of step 4 is not, and the set of the B spline base function of each sections of then asking for is the expression function of this prestressing tendon reinforcing bar.

The advantage of the inventive method is: 1, can reach higher fitting precision with less control point parameter, therefore the data volume of handling is beneficial to control accuracy, raises the efficiency than segment approaching method much less; 2, in the fit procedure of figure, can not change the calculating of equivalent load, not have Algorithm Error.3, local property.The weight factor of getting in touch as the control vertex or the change of mobile k nurbs curve will only influence the shape that fixes on that part of curve on the sections, and the other parts of nurbs curve are not exerted an influence.Though and the sectional broken line method can be adjusted indivedual line segments, lost whole continuity.Method step of the present invention is simple, reliable operation, have bigger promotional value.

Description of drawings

Fig. 1 is the schematic flow sheet of the inventive method.

The specific embodiment

The specific embodiment one: specify present embodiment below in conjunction with Fig. 1.Present embodiment is realized by following step: one, determine the shape and size of prestressing tendon reinforcing bar according to the stressing conditions of beam bridge, and be depicted as the original-shape line of prestressing tendon reinforcing bar with the spline curve order in the AutoCAD mapping software (spline order); Two, on the original-shape line of prestressed reinforcement, get characteristic point, get two end points as characteristic point, get three to seven characteristic points by the principle of dividing equally between the first and last end points for the curved section on the original-shape line for the straightway on the original-shape line; Three, determine the B spline base function of the sections between per two characteristic points according to the non-uniform rational B-spline design formulas, utilize the B spline base function of each sections to be depicted as the contrast shape line that connects all characteristic points; Four, detect the original-shape line of prestressing tendon reinforcing bar and the identical situation between the contrast shape line along its length, judge whether to exist the distance between certain extension position place original-shape line and the contrast shape line to surpass setting value; Described setting value can be set up on their own by the user, be generally ordinate difference maximum on the original-shape line two some distances 5%, also can estimate between original-shape line and the contrast shape line degree of agreement and estimate the size of determining setting value; Five, the result of step 4 is for being then the point on this position original-shape line to be confirmed as the interpolation characteristic point; Six, determine the B spline base function of sections between the interpolation characteristic point characteristic point adjacent with left according to the non-uniform rational B-spline design formulas, determine the B spline base function of sections between interpolation characteristic point and the right-hand adjacent characteristic point simultaneously, utilize the B spline base function of each sections to be depicted as the contrast shape line of be linked in sequence all characteristic points and interpolation characteristic point; Return the starting end of step 4; Seven, the result of step 4 is not, and the set of the B spline base function of each sections of then asking for is the expression function of this prestressing tendon reinforcing bar; Because after this method finishes, contrast shape line and original-shape line are basic identical in the geometry linear contrast, then its relevant mechanical property is also basic identical, and whether the determined contrast shape of the expression function line of therefore checking the prestressing tendon reinforcing bar satisfies the stressing conditions of beam bridge just can be finished the stressing conditions analysis.

Spline function (curve) is that civil engineering stock that personnel are familiar with is in the application and the research of SPLINE FINITE ELEMENT METHOD.Use spline curve to derive from such fact as the intention of prestressing tendon geometrical model: implement the prestressing tendon location in the bridge construction is that bellows or other class molded cannula method by the discrete point colligation is realized.No matter prestressing tendon is circular curve or parabola in design idea, finally in fact all becomes the multi-point fitting curve with certain slickness in implementation process.And, use spline function (curve) to become the comparatively ripe solution of field of Computer Graphics already for the multi-point fitting problem.Spline curve is divided into polynomial spline curve, Bezier (Brizer) curve, B-spline curves (Basic Spline) by structural form.Select B-spline curves to be as the main reason of prestressing tendon model: B-spline curves have an outstanding advantage---locality than the above two.Locality is one of critical nature of B-spline curves, by changing the position at control point, just can realize the local modification to curve, and this modification can not be diffused into entire curve, thereby can greatly improve the flexibility of design.

Be the convenience of narration, provide the definition of k B batten vector function herein earlier:

$\stackrel{\mathrm{UV}}{P}\left(u\right)=\frac{\underset{i=0}{\overset{n}{\mathrm{\Σ}}}{\mathrm{\ω}}_i{\stackrel{\mathrm{UV}}{d}}_i{B}_{i,k}\left(u\right)}{\underset{i=0}{\overset{n}{\mathrm{\Σ}}}{\mathrm{\ω}}_i{B}_{i,k}\left(u\right)}---\left(1\right)$

ω in the formula _i, i=0,1 ..., be power or weight factor (weights), d _i, i=0,1 ..., n is the control point, B _{I, k}(u) be k B spline base function of i knot vector, the B spline base function is by knot vector U=[u ₀, u ₁..., u _N+k+1] obtain by De Buer-Cox recurrence formula, suc as formula (2a) and formula (2b).

$B_{i,m}\left(x\right)=\frac{x-x_i}{x_{i+m-1}-x_i}\·B_{i,m-1}\left(x\right)---\left(2a\right)$

$+\frac{x_{i+m}-x}{x_{i+m-1}-x_{i+1}}\·B_{i+1,m-1}\left(x\right)$

$B_{i,l}\left(x\right)=\left\{\begin{array}{cc}1,& x\∈[x_i,x_{i+1}]\\ 0,& x\∉[x_i,x_{i+1}]\end{array}\right.---\left(2b\right)$

Three unreasonable B-spline functions habitually practising in the SPLINE FINITE ELEMENT METHOD, come down to a special shape of formula (1), be k=3, knot vector is for along parameter evenly or be spacedly distributed, and there is not denominator term in the expression formula, claim this kind form be even unreasonable B batten (Uniform Non-Rational B Spline, UNRBS).Otherwise, if the parameter unequal interval distributes, and denominator term is arranged in the basic function, then be called non-uniform rational B-spline (NURBS).

For the model problem of prestressing tendon, it is equally distributed cannot say for sure to demonstrate,prove knot vector on the one hand, because this means that the colligation point that requires in the construction is equally distributed on the parameter meaning, obviously is difficult to realize in engineering design and implementation process.On the other hand, when prestressing tendon was conical section such as circular arc, parabola, unreasonable B-spline function can't accurately be represented other conic section except that parabola ^[4]The non-uniform rational B-spline curves then do not have above-mentioned restriction, can accurately describe to comprise straight line, conical section and parametric cubic curve and free curve.Based on this, International Standards Organization (ISO) issued the STEP international standard about the industrial products exchanges data in 1991, the unique mathematical method of NURBS as definition industrial products geometry ^[5]This paper also finally selects the model of non-homogeneous B spline curve (NURBS) as prestressing tendon for use, is called the NURBS model of prestressing tendon.

The characteristic of NURBS makes it show more advantage as the prestressing tendon model: (1) local property.A control vertex d as mobile k nurbs curve _iOr change the weight factor got in touch and only to influence and fix on [u between node area _i, u _I+k+1] go up the shape of that part of curve, the other parts of nurbs curve are not made a difference.Prestressing tendon is comparatively convenient in the design adjustment process, and can guarantee C ²Continuity.Though and the sectional broken line method can be adjusted indivedual line segments, lost whole continuity.(2) calculation stability, speed is fast.Be compared to sectional broken line and cubic parametric spline curve (as the Hermite curve), the nurbs curve storage capability is little, and computational speed is fast, and the prestressing tendon adjustment that is fit to prestressed concrete beam bridge is calculated.(3) NURBS has strong how much supporting technologies, as inserts a node, the lifting of exponent number and division (splitting into the curve of two or many independent expressions) etc., and these technology are effective especially for the mesh discretization of prestressing tendon.

NURBS is comparatively ripe in area of computer graphics, and aspect the program realization, most CAD softwares all have the systematic function of NURBS.In order to be applied to graphics calculations easily, Philippe Lavoie has developed the NURBS++ class libraries, widely adopts for computer graphical and industrial design research field at present.The NURBS prestressing tendon model that this paper proposes promptly is based on the realization of NURBS++ class libraries.

The bottom data type of NURBS++ is vectorial class (Vector Class) and matrix class (Matrix Class), and both possess stronger mathematical operation function and Memory Allocation expanded function, and it is comparatively convenient to use.Formed class of a curve (Curve Class) and curved surface class (Surface Class) on this basis, wherein nurbs curve class (NurbsCurve Class) is the subclass of class of a curve.

Owing to comprise that all classes of NurbsCurve all define by template class, therefore when concrete the application, specifically defined floating type data and curves class (NurbsCurvef) and double data and curves class (NurbsCurved) in the class libraries, the present invention uses the former, and its precision is enough.

Claims (1)

1, uses the method for non-uniform rational B-spline model measurement prestressing tendon, it comprises the steps: one, determines the shape and size of prestressing tendon reinforcing bar according to the stressing conditions of beam bridge, and is depicted as the original-shape line of prestressing tendon reinforcing bar with the spline curve order in the AutoCAD mapping software; It is characterized in that it also comprises the steps: two, gets characteristic point on the original-shape line of prestressed reinforcement, get two end points as characteristic point for the straightway on the original-shape line, get three to seven characteristic points by the principle of dividing equally between the first and last end points for the curved section on the original-shape line; Three, determine the B spline base function of the sections between per two characteristic points according to the non-uniform rational B-spline design formulas, utilize the B spline base function of each sections to be depicted as the contrast shape line that connects all characteristic points; Four, detect the original-shape line of prestressing tendon reinforcing bar and the identical situation between the contrast shape line along its length, judge whether to exist the distance between certain extension position place original-shape line and the contrast shape line to surpass setting value; Five, the result of step 4 is for being then the point on this position original-shape line to be confirmed as the interpolation characteristic point; Six, determine the B spline base function of sections between the interpolation characteristic point characteristic point adjacent with left according to the non-uniform rational B-spline design formulas, determine the B spline base function of sections between interpolation characteristic point and the right-hand adjacent characteristic point simultaneously, utilize the B spline base function of each sections to be depicted as the contrast shape line of be linked in sequence all characteristic points and interpolation characteristic point; Return the starting end of step 4; Seven, the result of step 4 is not, and the set of the B spline base function of each sections of then asking for is the expression function of this prestressing tendon reinforcing bar.

Images