CN105040569B - The method for optimizing that a kind of simply supported steel box girder presstressed reinforcing steel is linear - Google Patents

Abstract

The invention discloses the method for optimizing that a kind of simply supported steel box girder presstressed reinforcing steel is linear, theoretical owing to introducing Thin-walled Box Girder, thus its mechanical analysis is more accurate, presstressed reinforcing steel is linear compares with tradition, has played it very well and has obtained camber and the double utility of Shi Hanzhang.Presstressed reinforcing steel after optimization is arranged and is made steel box-girder be in better mechanical state, thus is conducive to avoiding the bad diseases such as beam body cracking, local buckling, and then improves the durability of this class formation.Meanwhile, its mechanical concept is clear, it is simple to calculate, and has application prospect and the practical value of broadness, and the present invention is possible not only to optimize the mechanical property of steel box-girder, and can saving steel consumption further.

Description

The method for optimizing that a kind of simply supported steel box girder presstressed reinforcing steel is linear

Technical field

The present invention relates to the method for optimizing that a kind of simply supported steel box girder presstressed reinforcing steel is linear.

Background technology

Prestressing technique is widely used in modern structure engineering, particularly science of bridge building, prestress steel case The combinative structure that beam mainly arranges external curved prestressing tendon in steel case and formed, for steel box-girder, The introducing of presstressed reinforcing steel greatly improves the mechanical characteristic of structure, improves the span ability of bridge, with general Logical steel construction is compared and can be saved steel about 30%, such as construction and the Ha Er of Shenyang City's Second Ring Road overpass The reparation etc. of shore city workers, peasants and soldiers' bridge all have employed steel box girder with prestressed tendons.The layout of presstressed reinforcing steel is the most internal Presstressed reinforcing steel and external prestressing steels, prestressing with bond muscle is mainly used in concrete-bridge, and external Presstressed reinforcing steel is then mostly applied to Bridge repair and reinforcement and steel box girder with prestressed tendons.

In steel box-girder, the presstressed reinforcing steel of employing is linear varied, such as linear, fold-line-shaped and curve Shape etc., be wherein most widely used or shaped form.But, according to existing document, for prestressing force The linear selection of muscle does not carry out systematic research, thus the effect of presstressed reinforcing steel fails to give full play to, But present stage is due to the extensive application of steel box girder with prestressed tendons, then research and the invention of this aspect more must The property wanted.But, owing to being affected by Shear Lag Effect, such structure mechanics analysis is more complicated, with mixed Solidifying soil box-girder bridge compares, and the Liang Bi of steel box girder with prestressed tendons is thinner, then it is more careful not carry out it Mechanical analysis, be likely to result in the Local Cracking of this class formation, local buckling, or more serious bridge Beam disease.

Summary of the invention

The present invention seeks to: provide a kind of mechanical analysis accurate, and its acquisition camber can be played very well With the dual function of Shi Hanzhang, make steel box-girder be in better mechanical state, be conducive to avoiding The bad diseases such as beam body cracking, local buckling, and then the simply supported steel box girder improving such structure durability is pre- The method for optimizing that stress rib is linear.

The technical scheme is that the method for optimizing that a kind of simply supported steel box girder presstressed reinforcing steel is linear, including Following steps:

Step 1): introduce 3 generalized displacements when the vertical bending analyzing steel box-girder, i.e. w (x) is box beam vertical deflection, u (x) For the maximum lonitudinal warping displacement difference function of box beam wing plate, θ (x) is the box-girder vertical corner about z-axis, and x is beam Across direction, then the length travel of box-girder wing plate is represented by:

$U(x,z)=\±h\[\mathrm{\θ}\left(x\right)+(1-\frac{z^2}{b^2})u\left(x\right)\]---\left(1\right)$

In formula: b is the half of box beam clear span；

H is the vertical y-coordinate in box-girder cross section；

Step 2): (1) web potential energy of deformation

$V_1=\frac{1}{2}{\mathrm{EI}}_w{\∫}_0^l{\left({\mathrm{\θ}}^{\′}\right)}^{2}dx---\left(2\right)$

(2) wing plate potential energy of deformation

$V_2=\frac{1}{2}{\mathrm{EI}}_1{\∫}_0^l\[{\left({\mathrm{\θ}}^{\′}\right)}^2+\frac{4}{3}{\mathrm{\θ}}^{\′}{u}^{\′}+\frac{8}{15}{\left(u^{\′}\right)}^2\]dx+\frac{1}{2}{\∫}_0^l\frac{4{\mathrm{GI}}_1}{3b^2}{u}^{2}dx---\left(3\right)$

(3) hophornbeam pungent Ke detrusion potential energy

$V_3=\frac{1}{2}{\∫}_0^{l}kGA{(\mathrm{\θ}-w^{\′})}^{2}dx---\left(4\right)$

(4) load potential energy

$V_p=-{\∫}_0^{l}q\left(x\right)w\left(x\right)dx-\[Q\left(x\right)w\left(x\right)+M_1\left(x\right)u\left(x\right)+M\left(x\right)\mathrm{\θ}\left(x\right)\]{|}_0^l---\left(5\right)$

So total potential energy of structural system is:

V=V₁+V₂+V₃+V_p (6)

In formula: M₁(x) be box beam wing plate Girder with Shear Lag Effect produce about z-axis moment of flexure；M (x) is that beam section end produces vertical corner About z-axis moment of flexure during θ (x)；Q (x), q (x) are vertical distributed force in beam section end vertical shear and box beam；E, G are the poplar of material Family name's elastic modelling quantity and the coefficient of rigidity；K is cross section shape coefficient；I_w,I₁For box girder web and wing plate about the moment of inertia of z-axis； A is box section area.

Step 3): according to variation principleObtaining its differential equation is:

${\mathrm{EI\θ}}^{\′\′}+\frac{2}{3}{\mathrm{EI}}_1{u}^{\′\′}-kGA(\mathrm{\θ}-w^{\′})=0---\left(7\right)$

$\frac{2}{3}{\mathrm{EI}}_1{\mathrm{\θ}}^{\′\′}+\frac{8}{15}{\mathrm{EI}}_1{u}^{\′\′}-\frac{4{\mathrm{GI}}_1}{3b^2}u=0---\left(8\right)$

KGA (θ '-w ")-q (x)=0 (9)

Its boundary condition is:

$\[{\mathrm{EI\θ}}^{\′}+\frac{2}{3}{\mathrm{EI}}_1{u}^{\′}-M\]{|}_0^l\mathrm{\δ}\mathrm{\θ}=0---\left(10\right)$

$\[\frac{2}{3}{\mathrm{EI}}_1{\mathrm{\θ}}^{\′}+\frac{8}{15}{\mathrm{EI}}_1{u}^{\′}-M_1\]{|}_0^l\mathrm{\δ}u=0---\left(11\right)$

$\[kGA(\mathrm{\θ}-w^{\′})+Q\left(x\right)\]{|}_0^l\mathrm{\δ}w=0---\left(12\right)$

Step 4): being converted by the arrangement between equation (7), (8) and (9), trying to achieve new Solutions of Ordinary Differential Equations is:

$u^{\left(3\right)}+\frac{2G}{{\mathrm{Emb}}^2}u-\frac{q}{E\mathrm{Im}}=0---\left(13\right)$

Wherein:I=I_w+I₁, and u⁽³⁾3 derivation formulas for u；

Then its characteristic equation solution is:

γ_1,2=± (α₁+β₁i)；γ₃=0.

Finally, the solution of u (x), θ (x) and w (x) is respectively as follows:

$u\left(x\right)=c_{1}sh({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_{2}ch({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_3+\frac{b^2}{2GI}qx---\left(14\right)$

$\mathrm{\θ}\left(x\right)=c_1{B}_{1}sh({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_2{B}_{1}ch({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_3\frac{m_2}{2}{x}^2+c_{4}x+c_5+\frac{q}{6EI}{x}^3---\left(15\right)$

$\begin{array}{c}w\left(x\right)=c_1-\frac{B_1}{{\mathrm{\α}}_1+{\mathrm{\β}}_{1}i}ch({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_2\frac{B_1}{{\mathrm{\α}}_1+{\mathrm{\β}}_{1}i}sh({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_3(\frac{m_2}{6}{x}^3\\ -\frac{m_{2}EI}{kGA}x)+\frac{c_4}{2}{x}^2+c_{5}x+c_6+\frac{q}{24EI}{x}^4-\frac{q}{2kGA}{x}^2\end{array}---\left(16\right)$

Wherein: $m_1=-\frac{4}{5};m_2=\frac{2G}{{\mathrm{Eb}}^2};B_1=\frac{m_2+m_1{({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)}^2}{{({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)}^2};$ And c₁；c₂；c₃；c₄；c₅；c₆For constant coefficient；

Step 5) curvilinear equation of assuming presstressed reinforcing steel is:Wherein l is steel box-girder span, and f is The sag of presstressed reinforcing steel span centre, h is the distance of presstressed reinforcing steel deviation neutral axis, and q is the pre-applied force uniform vertical load of equivalence；

Step 6) according to specific border condition, equation w (x), θ (x), u (x) or its derivative formula are substituted into boundary condition, first counts Calculate undetermined constant c₁；c₂；c₃；c₄；c₅；c₆, then calculate w (x), the size of steel box-girder camber can be obtained, simultaneously can Calculate the size of lower wing plate prestress, it may be assumed thatAfter considering, screening Go out simply supported steel box girder optimum prestress muscle (rope) and arrange linear.

The method for optimizing that simply supported steel box girder presstressed reinforcing steel the most according to claim 1 is linear, its feature exists In, step 6) in filter out simply supported steel box girder optimum prestress muscle and arrange that linear concrete grammar is as follows: first First according to the curvilinear equation of presstressed reinforcing steelH is divided into some deciles as circulation item, Then, under the conditions of specific border condition and certain prestressing force, form matrix equation according to boundary condition, calculate based on this Undetermined constant c₁；c₂；c₃；c₄；c₅；c₆, and then constant term is inputted w (x), θ ' (x), u'(x) camber and the wing can be obtained Board prestress value, finally according to camber size, takes into account flange plate stress value and filters out the equation that presstressed reinforcing steel is linear.

The invention have the advantage that

1. present invention introduces Thin-walled Box Girder theoretical, its mechanical analysis is more accurate, with tradition presstressed reinforcing steel Linear compare, play it very well and obtain camber and double utility of Shi Hanzhang, pre-after optimization Stress rib is arranged and is made steel box-girder be in better mechanical state, is conducive to avoiding beam body to ftracture, locally The bad disease such as unstability, and then improve the durability of this class formation；

Mechanical concept the most of the present invention is clear, it is simple to calculate, and has application prospect and the practical value of broadness, It is possible not only to optimize the mechanical property of steel box-girder, and can saving steel consumption further.Meanwhile, right Maintenance and reinforcement in concrete box girder has directive significance (using external prestressing steels (rope)).

Accompanying drawing explanation

Below in conjunction with the accompanying drawings and embodiment the invention will be further described:

Fig. 1 is the linear schematic diagram of steel box-girder presstressed reinforcing steel of the present invention；

Fig. 2 is steel box-girder schematic cross-section of the present invention (O is the box beam centre of form)；

Fig. 3 is the equivalent load schematic diagram of presstressed reinforcing steel of the present invention；

Fig. 4 is steel box-girder transverse section of the present invention cloth muscle schematic diagram；

Fig. 5 is steel box-girder span centre transverse section cloth muscle schematic diagram in embodiment 1；

Detailed description of the invention

Embodiment 1: as shown in Fig. 2, Fig. 5, geometric parameter and the material parameter of steel box-girder are respectively as follows: t₁=t₂=t₃=3mm；t₄=4mm；B=0.25m；α b=0.25m；H=0.4m；L=5m；E=2.01 × 10⁵Mpa； G=7.9 × 10⁴Mpa, and the pre-applied force of every presstressed reinforcing steel is 50KN, wherein E₁；F₁；G₁And E₂；F₂；G₂It is respectively Pressure detection point or calculating point.Finally according to the pre-applied force applied, and different reinforcing plan can obtain difference Camber and different flange plate stress distribution forms, the layout that can obtain optimum prestress muscle accordingly is linear.

The main purpose that presstressed reinforcing steel is arranged is to make bridge structure obtain appropriate camber and prestress, enters And making structure have good mechanical property, steel box-girder is thin-wall construction, has the mechanical characteristic of uniqueness, by In being affected by Shear Lag Effect, being unevenly distributed of lower wing plate direct stress on it, thus in advance should Under power effect, it is possible to cause upper flange and web junction (such as E₁；G₁Near Dian) cracking, and lower wing plate and abdomen Plate junction is (such as E₂；G₂Near Dian) there is local buckling.

Through circulation tentative calculation, comprehensive every factor, this experiment beam selects h to be 3cm, and the most whole steel box-girder section is all pressed Stress influence, and camber is sufficiently large, thus the screening of presstressed reinforcing steel is linear should beBut tradition choosing Selecting and should be h=8cm, its linear equations isNow steel box-girder upper flange still has bigger drawing to answer Power, thus fail to reach good prestressing force effect.

Table 1 simply supported steel box girder camber and wing plate direct stress sample calculation (wherein 3 groups of data)

Note: negative stress is compressive stress, direct stress is tension.

Claims (2)

1. the method for optimizing that a simply supported steel box girder presstressed reinforcing steel is linear, it is characterised in that include following Step:

Step 1): introduce 3 generalized displacements when the vertical bending analyzing steel box-girder, i.e. w (x) is box beam vertical deflection, u (x) For the maximum lonitudinal warping displacement difference function of box beam wing plate, θ (x) is the box-girder vertical corner about z-axis, and x is beam Across direction, then the length travel of box-girder wing plate is represented by:

$U(x,z)=\±h\[\mathrm{\θ}\left(x\right)+(1-\frac{z^2}{b^2})u\left(x\right)\]---\left(1\right)$

In formula: b is the half of box beam clear span；

H is the vertical y-coordinate in box-girder cross section；

Step 2): (1) web potential energy of deformation

$V_1=\frac{1}{2}{\mathrm{EI}}_w{{\∫}_0^l\left({\mathrm{\θ}}^{\′}\right)}^{2}dx---\left(2\right)$

(2) wing plate potential energy of deformation

$V_2=\frac{1}{2}{\mathrm{EI}}_1{\∫}_0^l\[{\left({\mathrm{\θ}}^{\′}\right)}^2+\frac{4}{3}{\mathrm{\θ}}^{\′}{u}^{\′}+\frac{8}{15}{\left(u^{\′}\right)}^2\]dx+\frac{1}{2}{\∫}_0^l\frac{4{\mathrm{GI}}_1}{3b^2}{u}^{2}dx---\left(3\right)$

(3) hophornbeam pungent Ke detrusion potential energy

$V_3=\frac{1}{2}{\∫}_0^{l}kGA{(\mathrm{\θ}-w^{\′})}^{2}dx---\left(4\right)$

(4) load potential energy

$V_p=-{\∫}_0^{l}q\left(x\right)w\left(x\right)dx-\[Q\left(x\right)w\left(x\right)+M_1\left(x\right)u\left(x\right)+M\left(x\right)\mathrm{\θ}\left(x\right)\]{|}_0^l---\left(5\right)$

So total potential energy of structural system is:

V=V₁+V₂+V₃+V_p (6)

In formula: M₁(x) be box beam wing plate Girder with Shear Lag Effect produce about z-axis moment of flexure；M (x) is that beam section end produces vertical corner About z-axis moment of flexure during θ (x)；Q (x), q (x) are vertical distributed force in beam section end vertical shear and box beam；E, G are the poplar of material Family name's elastic modelling quantity and the coefficient of rigidity；K is cross section shape coefficient；I_w,I₁For box girder web and wing plate about the moment of inertia of z-axis； A is box section area；

Step 3): according to variation principleObtaining its differential equation is:

${\mathrm{EI\θ}}^{\′\′}+\frac{2}{3}{\mathrm{EI}}_1{u}^{\′\′}-kGA(\mathrm{\θ}-w^{\′})=0---\left(7\right)$

$\frac{2}{3}{\mathrm{EI}}_1{\mathrm{\θ}}^{\′\′}+\frac{8}{15}{\mathrm{EI}}_1{u}^{\′\′}-\frac{4{\mathrm{GI}}_1}{3b^2}u=0---\left(8\right)$

KGA (θ '-w ")-q (x)=0 (9)

Its boundary condition is:

$\[{\mathrm{EI\θ}}^{\′}+\frac{2}{3}{\mathrm{EI}}_1{u}^{\′}-M\]{|}_0^l\mathrm{\δ}\mathrm{\θ}=0---\left(10\right)$

$\[\frac{2}{3}{\mathrm{EI}}_1{\mathrm{\θ}}^{\′}+\frac{8}{15}{\mathrm{EI}}_1{u}^{\′}-M_1\]{|}_0^l\mathrm{\δ}u=0---\left(11\right)$

$\[kGA(\mathrm{\θ}-w^{\′})+Q\left(x\right)\]{|}_0^l\mathrm{\δ}w=0---\left(12\right)$

Step 4): being converted by the arrangement between equation (7), (8) and (9), trying to achieve new Solutions of Ordinary Differential Equations is:

$u^{\left(3\right)}+\frac{2G}{{\mathrm{Emb}}^2}u-\frac{q}{E\mathrm{Im}}=0---\left(13\right)$

Wherein:I=I_w+I₁, and u⁽³⁾3 derivation formulas for u；

Then its characteristic equation solution is:

γ_1,2=± (α₁+β₁i)；γ₃=0；

Finally, the solution of u (x), θ (x) and w (x) is respectively as follows:

$u\left(x\right)=c_{1}sh({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_{2}ch({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_3+\frac{b^2}{2GI}qx---\left(14\right)$

$\mathrm{\θ}\left(x\right)=c_1{B}_{1}sh({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_2{B}_{1}ch({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_3\frac{m_2}{2}{x}^2+c_{4}x+c_5+\frac{q}{6EI}{x}^3---\left(15\right)$

$\begin{array}{c}w\left(x\right)=c_1-\frac{B_1}{{\mathrm{\α}}_1+{\mathrm{\β}}_{1}i}ch({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_2\frac{B_1}{{\mathrm{\α}}_1+{\mathrm{\β}}_{1}i}sh({\mathrm{\α}}_1+{\mathrm{\β}}_{1}i)x+c_3(\frac{m_2}{6}{x}^3\\ -\frac{m_{2}EI}{kGA}x)+\frac{c_4}{2}{x}^2+c_{5}x+c_6+\frac{q}{24EI}{x}^4-\frac{q}{2kGA}{x}^2\end{array}---\left(16\right)$

Wherein:And c₁；c₂；c₃；c₄；c₅；c₆For constant coefficient；

Step 5) curvilinear equation of assuming presstressed reinforcing steel is:Wherein l is steel box-girder span, and f is The sag of presstressed reinforcing steel span centre, h is the distance of presstressed reinforcing steel deviation neutral axis, and q is the pre-applied force uniform vertical load of equivalence；

Step 6) according to specific border condition, equation w (x), θ (x), u (x) or its derivative formula are substituted into boundary condition, first counts Calculate undetermined constant c₁；c₂；c₃；c₄；c₅；c₆, then calculate w (x), the size of steel box-girder camber can be obtained, simultaneously can Calculate the size of lower wing plate prestress, it may be assumed thatAfter considering, screening Go out simply supported steel box girder optimum prestress muscle and arrange linear.

The method for optimizing that simply supported steel box girder presstressed reinforcing steel the most according to claim 1 is linear, it is characterised in that Step 6) in filter out simply supported steel box girder optimum prestress muscle and arrange that linear concrete grammar is as follows: first basis The curvilinear equation of presstressed reinforcing steelH is divided into some deciles as circulation item, then specific Under the conditions of boundary condition and certain prestressing force, form matrix equation according to boundary condition, calculate undetermined constant based on this c₁；c₂；c₃；c₄；c₅；c₆, and then constant term is inputted w (x), θ ' (x), u'(x) camber and wing plate prestress value can be obtained, Finally according to camber size, take into account flange plate stress value and filter out the equation that presstressed reinforcing steel is linear.