CN106168986A - Optimization method based on dynamic trait i-shaped roof beam Section Design - Google Patents

Abstract

The invention discloses the optimization method of a kind of i-shaped roof beam cross dimensions design, first this method tries to achieve potential energy of deformation and the kinetic energy of I-shaped beam bridge, and then utilize energy variation method to obtain control differential equation and the natural boundary conditions of i-shaped roof beam vibration, work out related application based on this, utilize Matlab software to launch the analysis that becomes more meticulous of such structure self-vibration characteristic.Last with I-shaped beam bridge natural frequency value as criterion, i.e. natural frequency value is big, and malformation potential energy is little, and i-shaped roof beam cross dimensions is excellent, and such structure static analysis further demonstrates effectiveness of the invention.Owing to this method mechanical analysis is accurate, thus the I-shaped beam bridge after cross dimensions optimization will be in good mechanical state, be conducive to avoiding the bad diseases such as beam body cracking, rigidity reduction and the excessive downwarp of span centre.This method mechanical concept is clear, it is simple to calculate, and has good using value, is the useful supplement to existing I-shaped beam bridge design theory.

Description

Optimization method based on dynamic trait i-shaped roof beam Section Design

Technical field

The present invention relates to the optimization method of a kind of I-shaped beam bridge Section Design, belong to technical field of bridge engineering.

Background technology

Owing to I-shaped beam bridge has a good mechanical property, and it is at the superiority easily such as prefabricated and construction, because of And this class formation is widely used in the railway of China's middle span, highway and urban viaduct.But rational cross section size Selecting, i-shaped roof beam bridge can be made to be in good operation state, this not only reduces the maintenance and reinforcement expense of this class formation, and The suitability and the durability of such bridge will be effectively improved.The design theory of the most I-shaped beam bridge is the most perfect, should in operation The bridge defect of class formation is the most serious.

Present stage, owing to concreter's font beam bridges a lot of in operation there occurs serious bridge defect, such as wing plate and abdomen The reduction of plate serious cracking, beam body rigidity and the excessive downwarp of span centre etc., these are all by the safety of this class formation of serious threat.So On the premise of meeting bridge structure use function, how to adapt to its mechanics feature, filter out the I-shaped of good mechanical performance Section becomes the target of vast bridge worker's unremitting effort.Because reasonably form of fracture, is possible not only to improve this class formation Mechanical characteristic, and the span ability of I-shaped beam bridge will be improved, and then be prevented effectively from bridge defect, and make bridge be in good Good duty.Thus, the research of this aspect, invent more theory significance and engineering practical value.

Summary of the invention

The present invention seeks to: for the problems referred to above, the present invention proposes the optimization method of a kind of i-shaped roof beam Section Design, should Method has considered the mechanics feature of I-shaped beam bridge, is selected by rational cross dimensions, makes I-shaped beam bridge be in good Good mechanical state.

The technical scheme is that the optimization method of a kind of I-shaped beam bridge Section Design, it is characterised in that the method For:

First the new differential equation about W (x) is derived

$W^{\left(4\right)}+\frac{{\mathrm{\ρ\ω}}^2}{E}{W}^{\′\′}+\frac{{\mathrm{\ρA\ω}}^2}{EI}w+\frac{1}{EI}{q}_0=0---\left(9\right)$

And then, build following equation

$\begin{array}{c}W\left(x\right)=c_1\cosh ({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_2\sinh ({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_3\cosh ({\mathrm{\α}}_2+{\mathrm{\β}}_{2}i)x+\\ c_4\sinh ({\mathrm{\α}}_2+{\mathrm{\β}}_{2}i)x-\frac{1}{{\mathrm{\ρA\ω}}^2}{q}_0\end{array}---\left(10\right)$

θ (x)=c₁(α₁+β₁i)sinh(α₁+β₁i)x+c₂(α₁+β₁i)cosh(α₁+β₁i)x+

c₃(α₂+β₂i)sinh(α₂+β₂i)x+c₄(α₂+β₂i)cosh(α₂+β₂i)x (11)

In formula:

X, z, y are respectively by the axial, vertical of i-shaped roof beam cross-section centroid and lateral coordinates；θ (x) is that i-shaped roof beam cuts The vertical corner in face；I is imaginary unit；Sinh is hyperbolic sine function；Cosh is hyperbolic cosine function；A is that i-shaped roof beam cuts Face area；ρ is the mass density of i-shaped roof beam material；q₀For vertical uniform simple harmonic quantity power amplitude；ω is i-shaped roof beam frequency of vibration； α₁,α₂,β₁,β₂For the coefficient about W (x) characteristic equation solution；c₁；c₂；c₃；c₄For constant coefficient, can be according to i-shaped roof beam dependence edge Boundary's condition solves；

Above-mentioned equation (10) and equation (11) are substituted into corresponding boundary condition, tries to achieve I-shaped beam bridge under this boundary condition Natural frequency value, and with its natural frequency value as criterion, i-shaped roof beam cross dimensions b can be optimized₁,b₂,t_w,t₁,t₂,h,h₁, h₂；And then, by i-shaped roof beam cross dimensions b₁,b₂,t_w,t₁,t₂,h,h₁,h₂Reasonable selection, to improving I-shaped beam bridge Mechanical property；

Wherein, b₁；b₂It is respectively the half of lower wing plate length on i-shaped roof beam；t_wFor webs in I-girders thickness；t₁；t₂Point Wei lower wing plate thickness on i-shaped roof beam；H is i-shaped roof beam height；h₁For i-shaped roof beam upper flange centre-to-centre spacing neutral axis distance；h₂ For i-shaped roof beam lower wing plate centre-to-centre spacing neutral axis distance.

The invention have the advantage that first this method tries to achieve potential energy of deformation and the kinetic energy of I-shaped beam bridge, and then utilize energy The calculus of variations obtains control differential equation and the natural boundary conditions of i-shaped roof beam vibration, works out related application, profit based on this The analysis that becomes more meticulous of such structure self-vibration characteristic is launched with Matlab software.Last with I-shaped beam bridge natural frequency value for sentencing According to, i.e. natural frequency value is big, and malformation potential energy is little, and I-shaped beam bridge cross dimensions is excellent, and such structure static analysis enters One step demonstrates effectiveness of the invention.Owing to this method mechanical analysis is accurate, thus the i-shaped roof beam after cross dimensions optimization Bridge will be in good mechanical state, be conducive to avoiding the bad diseases such as beam body cracking, rigidity reduction and the excessive downwarp of span centre.This Method mechanical concept is clear, it is simple to calculate, and has good using value, is useful to existing I-shaped beam bridge design theory Supplement.

Accompanying drawing explanation

In order to more clearly illustrate the technical scheme of the embodiment of the present invention, required use in embodiment being described below Accompanying drawing be briefly described, below describe in accompanying drawing be only some embodiments of the present invention, skill common for this area From the point of view of art personnel, on the premise of not paying creative work, it is also possible to obtain other accompanying drawing according to these accompanying drawings.

Fig. 1 is I-shaped beam bridge cross dimensions and coordinate system figure in the embodiment of the present invention；

Fig. 2 is I-shaped beam bridge longitudinal size and coordinate system figure in the embodiment of the present invention；

Fig. 3 is the 1st group of i-shaped roof beam direct stress comparison diagram in the embodiment of the present invention；

Fig. 4 is the 2nd group of i-shaped roof beam direct stress comparison diagram in the embodiment of the present invention；

Fig. 5 is the 3rd group of i-shaped roof beam direct stress comparison diagram in the embodiment of the present invention

Detailed description of the invention:

Below in conjunction with specific embodiment, such scheme is described further.Should be understood that these embodiments are for illustrating The present invention and and unrestricted the scope of the present invention.The implementation condition used in embodiment can be according to Specific construction and designing unit Condition do further adjustment, not marked implementation condition is usually the condition in normal experiment.

Specific embodiment mode:

1, the i-shaped roof beam vibration control differential equation and natural boundary conditions

The setting of 1.1 mechanics parameters

To the I-shaped cross-section beam shown in Fig. 2, if the span of structure is L, under symmetric bend state, (x is t) that cross section is erected to w To dynamic deflection, (x t) is vertical dynamic corner to θ.

During i-shaped roof beam steady-state vibration, its every potential energy is

I-shaped roof beam is by external force potential energy π time curved_pFor

${\mathrm{\π}}_p=-{\∫}_0^{l}M(x,t)\frac{\∂\mathrm{\θ}}{\∂x}dx---\left(1\right)$

The dynamic strain energy π of i-shaped roof beam_yFor

${\mathrm{\π}}_y=\frac{1}{2}{\∫}_0^{l}EI{\left(\frac{\∂\mathrm{\θ}}{\∂x}\right)}^{2}dx---\left(2\right)$

Total potential energy π is

π=π_p+π_y (3)

Structure total kinetic energy T is

$T=\frac{1}{2}\underset{0}{\overset{l}{\∫}}{\left(\frac{\∂w}{\∂t}\right)}^2\mathrm{\ρ}Adx+\frac{1}{2}\mathrm{\ρ}I{\∫}_0^l{\left(\stackrel{\·}{\mathrm{\θ}}\right)}^{2}dx---\left(4\right)$

In formula: x, z, y are respectively by the axial, vertical of cross-section centroid and lateral coordinates；(x is t) that i-shaped roof beam cuts to θ The dynamic corner in face；I is the moment of inertia of total cross-section centering axle；E, G are respectively Young's modulus of elasticity and the shearing elasticity mould of material Amount；(x t) is x cross section dynamical bending moment to M；K is cross section shape coefficient；A is I-beam area of section；ρ is the mass density of material；q (x t) is vertical simple harmonic quantity power.

The 1.2 i-shaped roof beam vibration control differential equation and the acquisitions of natural boundary conditions

By Hamiton's principleThe i-shaped roof beam dynamic Control differential equation and natural boundary can be derived Condition is

$\mathrm{\ρ}A\stackrel{\·\·}{w}+{\mathrm{EI\θ}}^{\left(3\right)}-\mathrm{\ρ}I{\left(\stackrel{\·\·}{\mathrm{\θ}}\right)}^{\′}-q(x,t)=0---\left(5\right)$

θ=w'(6)

$\[-M(x,t)+{\mathrm{EI\θ}}^{\′}\]{|}_0^l\mathrm{\δ}\mathrm{\θ}=0---\left(7\right)$

$\[Q(x,t)-{\mathrm{EI\θ}}^{\′\′}\]{|}_0^l\mathrm{\δ}w=0---\left(8\right)$

In formula (4)～(8), symbol ". " and " ' " represent time t respectively and coordinate x sought partial derivative.And " .. " is right 2 derivations of time t, θ⁽³⁾It is then 3 derivations to coordinate x, and " θ " and " w " is that (x, t) with w (x, t) abbreviated formula for θ.

2, the solving of i-shaped roof beam oscillatory differential equation

If i-shaped roof beam frequency of vibration is ω, then can make AndEquation (5) is substituted into equation (6), and collated conversion can obtain the new differential equation and be

$W^{\left(4\right)}+\frac{{\mathrm{\ρ\ω}}^2}{E}{W}^{\′\′}+\frac{{\mathrm{\ρA\ω}}^2}{EI}w+\frac{1}{EI}{q}_0=0---\left(9\right)$

Analyzing equation (9) and understand, its characteristic equation solution can be following form:

r_1,2=± (α₁+β₁i),r_3,4=± (α₂+β₂i)

Therefore the general solution of equation (9) is

$\begin{array}{c}w\left(x\right)=c_1\cosh ({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_2\sinh ({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_3\cosh ({\mathrm{\α}}_2+{\mathrm{\β}}_{2}i)x+\\ c_4\sinh ({\mathrm{\α}}_2+{\mathrm{\β}}_{2}i)x-\frac{1}{{\mathrm{\ρA\ω}}_0^2}{q}_0\end{array}---\left(10\right)$

The form that θ (x) solves is assume that, by equation (10) and θ (x) generation according to ordinary differential system character and equation (10) Enter equation (6), try to achieve the constant coefficient of θ (x) according to identity principle, then the solution of θ (x) is represented by

θ (x)=c₁(α₁+β₁i)sinh(α₁+β₁i)x+c₂(α₁+β₁i)cosh(α₁+β₁i)x+

c₃(α₂+β₂i)sinh(α₂+β₂i)x+c₄(α₂+β₂i)cosh(α₂+β₂i)x (11)

In formula:

c₁；c₂；c₃；c₄For constant coefficient, can solve according to relevant border condition.And work as q₀When=0, can be to I-shaped beam bridge Self-vibration characteristic be analyzed.I is imaginary unit；Sinh is hyperbolic sine function；Cosh is hyperbolic cosine function；q₀For vertically Uniform simple harmonic quantity power amplitude；ω is i-shaped roof beam frequency of vibration；α₁,α₂,β₁,β₂For the coefficient about W (x) characteristic equation solution.

3, the natural boundary conditions of I-shaped beam bridge

According to the differential equation (7)-(8), I-shaped beam bridge is commonly used natural boundary conditions and can be reduced to

(1) displacement and the mechanic boundary condition of freely-supported i-shaped roof beam is

A, simple harmonic quantity even distributed force

$w\left(x\right){|}_0^l=0;{\mathrm{\θ}}^{\′}\left(x\right){|}_0^l=0---\left(12\right)$

B, simple harmonic quantity concentration power

For freely-supported i-shaped roof beam, if across institute's stress be a simple harmonic quantity concentration power, and concentration power Left and right adjacent boundary distance is l₁And l₂, now forming two coordinate systems, its zero is respectively 0₁With 0₂.As in figure 2 it is shown, Then 0₂Must also introduce following continuity boundary conditions at Dian is

$w_1\left(l_1\right)=w_2\left(0\right);w_1^l\left(l_1\right)=w_2^{\′}\left(0\right);{\mathrm{\θ}}_1^{\′}\left(l_1\right)={\mathrm{\θ}}_2^{\′}\left(0\right);{\mathrm{\θ}}_1^{\′\′}\left(l_1\right)-{\mathrm{\θ}}_2^{\′\′}\left(0\right)=\frac{-p_0}{kGA}---\left(13\right)$

4, i-shaped roof beam statics Analysis

For equation (5) and (6), whenTime, equation group is statics Analysis equation group, solves it Can obtain:

$w\left(x\right)=c_1^s{x}^3+c_2^s{x}^2+c_3^{s}x+c_4^s+\frac{q}{24EI}{x}^4---\left(14\right)$

$\mathrm{\θ}\left(x\right)=c_1^s\left(3x^2\right)+c_2^s\left(2x\right)+c_3^s+\frac{q}{6EI}{x}^3---\left(15\right)$

Through analyzing, i-shaped roof beam force model boundary condition is identical, and during static analysisConstant coefficient can root Solve according to relevant border condition.

5, the solving of i-shaped roof beam self-vibration characteristic

By equation (10), (11) substitute into corresponding boundary condition, can consolidating in the hope of beam bridge I-shaped under the conditions of corresponding edge circle There is a frequency values, and cross dimensions b of its natural frequency value and i-shaped roof beam₁,b₂,t_w,t₁,t₂,h,h₁,h₂Relevant.Then with work Font beam bridge natural frequency value is criterion, optimizes the cross dimensions of i-shaped roof beam based on this.And then by i-shaped roof beam section chi Very little reasonable selection, to improving the mechanical characteristic of I-shaped beam bridge.I.e. for different section form under the conditions of certain mechanics I-shaped beam bridge, if its natural frequency value increases, then its potential energy of deformation reduces, then the cross dimensions of this I-shaped beam bridge more becomes Excellent.It is W (0)=0 as beam bridge I-shaped for freely-supported can obtain equation group；W (L)=0；θ ' (0)=0；θ ' (L)=0, then This homogeneous equation group perseverance is made to set up, its coefficient (c₁；c₂；c₃；c₄) determinant must be zero, now will only unknown number in the equation ω, can try to achieve the natural frequency value of freely-supported i-shaped roof beam based on this.Equally, by above-mentioned equation (14) and (15) or its derivative formula generation Enter corresponding boundary condition, I-shape construction can be carried out statics Analysis.This process passes through factorization, utilizes Matlab software solves.

6, the optimization of i-shaped roof beam cross dimensions

Under certain mechanical state, the energy of structural system is made up of potential energy of deformation and kinetic energy, and in same system, kinetic energy is big then Potential energy of deformation is little.The present invention is solved by different section form I-shaped beam bridge natural frequency value, i.e. natural frequency value is big, work Tee beam potential energy of deformation is little, then this malformation is the least, thus its mechanical property becomes excellent.The present invention is intrinsic with I-shaped beam bridge Frequency values is criterion, and based on this, I-shaped beam bridge cross dimensions is optimised.(note: when certain span I-shaped beam bridge quality one Regularly, its basal area must be definite value.Based on this, the applicable elements of this screening technique is: no matter this i-shaped roof beam cross dimensions is such as What change, its sectional area is equal.)

Actual application and compliance test result:

Now selecting 3 groups of I-shaped cross-section beams, its material parameter and geometric parameter are respectively as follows:

(1) ρ=2500kg/m is respectively for the 3 groups of I-shaped cross-section beams selected, its material parameter and geometric parameter³；E =3.5 × 10⁴MPa；G=1.5 × 10⁴MPa；t_w=0.2m；t₁=0.15m；b₁=0.9m；t₂=0.15m；b₂=0.4m, beam A height of h=1.5m.(2) ρ=2500kg/m³；E=3.5 × 10⁴MPa；G=1.5 × 10⁴MPa；t_w=0.3m；t₁=0.1m；b₁ =0.85m；t₂=0.1m；b₂=0.35m, its deck-molding is h=1.5m.(3) ρ=2500kg/m³；E=3.5 × 10⁴MPa；G= 1.5×10⁴MPa；t_w=0.25m；t₁=0.18m；b₁=0.75m；t₂=0.2m；b₂=0.3m, its deck-molding is h=1.2m.Root According to this paper derivation formula, the natural frequency that boundary condition is the I-shaped beam bridge of freely-supported can be calculated, and filter out mechanical property accordingly The i-shaped roof beam cross section that energy is excellent.

The natural frequency (L=20m) (unit: Hz) of table 1 freely-supported i-shaped roof beam

Table 1 shows, the size of i-shaped roof beam natural frequency value is ordered as the 1st group, the 2nd group, the 3rd group.Based on this, I-shaped The trap queuing of beam mechanical property should be the 1st group, the 2nd group, the 3rd group.The i.e. mechanical property of the 1st group of i-shaped roof beam is 3 groups of I-shapeds The superior in ellbeam, on the premise of meeting other design condition, certainly this group i-shaped roof beam should be in the design preferred ?.

Above 3 groups of I-shaped cross-section beams, its material parameter and geometric parameter are the most above-mentioned, and span centre concentration power is p₀= 14700N.Then according to simple boundary condition, i-shaped roof beam spaning middle section static characteristic is analyzed, proves with I-shaped with this Ellbeam bridge natural frequency value is the effectiveness of criterion method for optimizing.

With reference to shown in Fig. 3, Fig. 4 and Fig. 5, analyzed by freely-supported I-shaped beam bridge static characteristic, equally must go to work The trap queuing of tee beam mechanical property is the 1st group, the 2nd group and the 3rd group.And this conclusion with according to I-shaped beam bridge natural frequency The judgement of value is consistent, thus statics Analysis further demonstrates the effectiveness of the inventive method.

Certainly, above-described embodiment only for technology design and the feature of the present invention are described, its object is to make people's energy much of that Solve present disclosure and implement according to this, can not limit the scope of the invention with this.All according to major technique of the present invention Equivalent transformation that the spirit of scheme is done or modification, all should contain within protection scope of the present invention.

Claims (1)

1. method for optimizing based on dynamic trait i-shaped roof beam Section Design, it is characterised in that the method is:

First the new differential equation about W (x) is derived

$W^{\left(4\right)}+\frac{{\mathrm{\ρ\ω}}^2}{E}{W}^{\′\′}+\frac{{\mathrm{\ρA\ω}}^2}{EI}w+\frac{1}{EI}{q}_0=0---\left(9\right)$

And then, build following equation

$\begin{array}{c}W\left(x\right)=c_1\cosh ({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_2\sinh ({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_3\cosh ({\mathrm{\α}}_2+{\mathrm{\β}}_{2}i)x+\\ c_4\sinh ({\mathrm{\α}}_2+{\mathrm{\β}}_{2}i)x-\frac{1}{{\mathrm{\ρA\ω}}^2}{q}_0\end{array}---\left(10\right)$

$\begin{array}{c}\mathrm{\θ}\left(x\right)=c_1({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)\sinh ({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_2({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)\cosh ({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+\\ c_3({\mathrm{\α}}_2+{\mathrm{\β}}_{2}i)\sinh ({\mathrm{\α}}_2+{\mathrm{\β}}_{2}i)x+c_4({\mathrm{\α}}_2+{\mathrm{\β}}_{2}i)\cosh ({\mathrm{\α}}_2+{\mathrm{\β}}_{2}i)x\end{array}---\left(11\right)$

In formula:

X, z, y are respectively by the axial, vertical of i-shaped roof beam cross-section centroid and lateral coordinates；θ (x) is i-shaped roof beam cross section Vertical corner；I is imaginary unit；Sinh is hyperbolic sine function；Cosh is hyperbolic cosine function；A is face, i-shaped roof beam cross section Long-pending；ρ is the mass density of i-shaped roof beam material；q₀For vertical uniform simple harmonic quantity power amplitude；ω is i-shaped roof beam frequency of vibration；α₁, α₂,β₁,β₂For the coefficient about W (x) characteristic equation solution；c₁；c₂；c₃；c₄For constant coefficient, can be according to i-shaped roof beam relevant border Condition solves；

Above-mentioned equation (10) and equation (11) are substituted into corresponding boundary condition, tries to achieve the intrinsic of I-shaped beam bridge under this boundary condition Frequency values, and with its natural frequency value as criterion, i-shaped roof beam cross dimensions b can be optimized₁,b₂,t_w,t₁,t₂,h,h₁,h₂；Enter And, by i-shaped roof beam cross dimensions b₁,b₂,t_w,t₁,t₂,h,h₁,h₂Reasonable selection, to improving the power of I-shaped beam bridge Learn performance；

Wherein, b₁；b₂It is respectively the half of lower wing plate length on i-shaped roof beam；t_wFor webs in I-girders thickness；t₁；t₂It is respectively Lower wing plate thickness on i-shaped roof beam；H is i-shaped roof beam height；h₁For i-shaped roof beam upper flange centre-to-centre spacing neutral axis distance；h₂For work Tee beam lower wing plate centre-to-centre spacing neutral axis distance.