CN106758738A - The method that elastic restraint against rotation border steel reinforced concrete combination deck-molding web longitudinal stiffener is set - Google Patents

Abstract

The present invention relates to the method that a kind of steel reinforced concrete combination deck-molding web longitudinal stiffener in elastic restraint against rotation border is set.Initially set up displacement function equation and boundary condition with elastic restraint against rotation border, the buckling stress calculation expression that rigidity is put more energy into ribbed stiffener under flexible two states of putting more energy into is derived based on energy method, the computing formula of elastic restraint against rotation border longitudinal stiffener critical conception is obtained；Then, it is considered to external work and minimum potential energy principal, the Critical Buckling Stress calculation formula of non-homogeneous compression plate under elastic restraint against rotation border is proposed；Finally, according to flexing degree of safety equal principle, the optimal location of longitudinal stiffener setting under rotational restraint border is obtained.The method can exactly carry out the setting that steel reinforced concrete combines deck-molding web longitudinal stiffener according to actual loading situation, the big problem of current steel reinforced concrete combined bridge ribbed stiffener over-designed, rolled steel dosage is solved, reasonability and economy that web ribbed stiffener high is set is significantly improved.

Description

Elastic restraint against rotation border steel-mixed combination deck-molding web longitudinal stiffener is set Method

Technical field

The present invention relates to transportation bridge engineeting field, a kind of elastic restraint against rotation border steel-mixed is specifically related to The method that combination deck-molding web longitudinal stiffener is set.

Background technology

Steel-mixed combined bridge has good stress performance and economy, and engineer applied is progressively obtained in recent years.With bridge Girder span footpath constantly increases, and the ratio of height to thickness of its web constantly increases, and Web Stability high has turned into the major issue in Bridge Design, The longitudinal stiffener rigidity and its position for influenceing Web Stability high are set, also as the key issue in design.

At present, frequently with classical elastic stability theory, its boundary condition is four for steel-mixed combination deck-molding Web Stability analysis Side freely-supported, but steel-mix the special border that combination beam boundary of web is the constraint between freely-supported and fixed edge, i.e. elastic rotation about Beam border.In recent years, the plate that Qiao, Chattopadhyay, Vescovini etc. are analyzed under elastic restraint against rotation is critical in the wrong Transverse stress, obtains the rule that plate Critical Buckling stress increases with the increase of elastic restraint against rotation rigidity, but not yet consider Elastic restraint against rotation border carries out the calculating of web longitudinal stiffener critical conception high and its setting of rational position.Steel-mixed group Closing put more energy into design of the Boundary Stiffness of beam on web high influences larger, and ignoring boundary of web rigidity will cause web stress high and reality There is larger error in border, and produce unnecessary vertically and horizontally ribbed stiffener.Therefore, elastic restraint against rotation border web longitudinal direction high is derived The computational methods of ribbed stiffener critical conception, setting for steel-mixed combination beam longitudinal stiffener is proposed according to web actual loading situation high Method is put, the big problem of current steel-mixed combined bridge ribbed stiffener over-designed, rolled steel dosage is effectively solved, web high is improved and is added Reasonability and economy that strength rib is set, are very necessary.

The content of the invention

The technical problems to be solved by the invention are：A kind of accurate elastic restraint against rotation border steel-mixed combination beam is provided The method that web longitudinal stiffener high is set, according to the method can obtain that longitudinal stiffener under rotational restraint border sets it is optimal Position, can reduce material usage, and the Buckling Loads of stiffener are improved using the inventive method.

The technical solution adopted for the present invention to solve the technical problems is：

The method that a kind of elastic restraint against rotation border steel-mixed combination deck-molding web longitudinal stiffener is set, it is characterised in that： Initially set up displacement function equation and boundary condition with elastic restraint against rotation border, based on energy method derive rigidity put more energy into and Flexibility is put more energy into the buckling stress calculation expression of ribbed stiffener under two states, is obtained elastic restraint against rotation border longitudinal stiffener and is faced The computing formula of boundary's rigidity；Then, it is considered to external work and minimum potential energy principal, it is non-homogeneous under proposition elastic restraint against rotation border to receive The Critical Buckling Stress calculation formula of pressing plate；Finally, according to flexing degree of safety equal principle, longitudinal direction under rotational restraint border is obtained The optimal location (rational position) that ribbed stiffener is set.

The method that heretofore described elastic restraint against rotation border steel-mixed combination deck-molding web longitudinal stiffener is set, It is characterized in that comprising the following steps：

(1) stiffener buckling stress in rotational restraint border is calculated

Classical plate stable theory is the border of simply supported on four sides, using double series as plate flexural function；To calculate Steel-mixed combination beam is put more energy into the buckling stress of web, and web is separated with the edge of a wing, and edge of a wing border is equivalent to two opposite side and has rotate effect The restricted spring answered；In order to construct the border with elastic restraint against rotation effect, displacement function equation uses multinomial and triangle The product form of function：

In formula：A, b are the width and height of web；M is the numbers of half wave in x directions；a₁、a₂、a₃For displacement function is undetermined Coefficient, is determined by boundary condition；Boundary condition is as follows：

Wherein：k₀、k_bIt is the constraint rigidity of boundary of web, D is web bending rigidity；Bring boundary condition into displacement function In equation, dimensionless border rotational restraint coefficient is made：X₀、X_bThe rotation at y=0 and y=b is represented respectively Constraint factor.

Simultaneous displacement function and boundary condition, obtaining each undetermined coefficient is：

Border rotates and suffers restraints, and equivalent strain energy is：

The set-up mode of ribbed stiffener；According to minimum potential energy principal, and by variation computing, when ribbed stiffener adds for flexible During strength rib, shown in the buckling stress such as formula (1) of stiffener：

Wherein：The length-width ratio of plateThe ratio of rigidity of ribbed stiffenerThe section ratio of ribbed stiffenerI in formula^LTable Show the rigidity of ribbed stiffener, A^LRepresent the area of ribbed stiffener, B₁、B₂、B₃、B₄、B₅、B₆It is the ginseng relevant with boundary stiffness coefficient Number, without practical significance；T represents the thickness of web, σ₁The buckling stress of stiffener when representing that ribbed stiffener is flexible ribbed stiffener；

B₄=X₀(X_b+6)², B₅=X_b(X₀+6)², B₆=(X₀X_b+8X₀+8X_b+60)²

In formula：X₀、X_bThe rotational restraint coefficient at y=0 and y=b is represented respectively.

When ribbed stiffener is rigid ribbed stiffener, (plate unit is for the plate unit that the flexing of stiffener occurs between rigid ribbed stiffener Plate split by ribbed stiffener after area's lattice), now shown in the buckling stress of stiffener such as formula (2)：

In formula：b_LIt is the maximum height of the plate unit between rigid ribbed stiffener；B₇=B₂+5B₄+5B₅, B₇Represent and boundary constraint The relevant parameter of stiffness coefficient, without practical significance, uses, σ to simplify formula₂Stiffener when representing that ribbed stiffener is rigid ribbed stiffener Buckling stress.

Understood according to formula (1) and (2)：When plate is put more energy into by flexibility, the area ratio of plate the ratio of width to height and floor and When Boundary Stiffness keeps constant, as ribbed stiffener rigidity increases, Buckling of Web stress also gradually increases；Plate is rigidly put more energy into When, Buckling of Web stress increases and increases with boundary constraint coefficient.

(2) longitudinal stiffener critical conception in rotational restraint border is calculated

Classical plate stable theory it is determined that ribbed stiffener critical conception when using making ribbed stiffener meet rigid flexing simultaneously Rigidity corresponding with two kinds of flexing situations of flexible flexing, i.e. σ₁=σ₂；Then under rotational restraint border, the critical conception of ribbed stiffener is expired Foot：

Then under elastic rotation boundary constraint, the critical conception of ribbed stiffener is：

Wherein：

M is used without practical significance to simplify formula in formula；γ₀Represent the critical conception of ribbed stiffener.

(3) the non-homogeneous compression plate Critical Buckling stress in rotational restraint border

Plate is subject to external force load action work：

In formula：V display plates are subject to external force load action work, N₀Represent the load at y=0.

According to minimum potential energy principal, the Critical Buckling Stress calculation of non-homogeneous compression plate under elastic restraint against rotation border can be obtained Formula is：

Wherein：σ_crThe Critical Buckling stress of non-homogeneous compression plate under expression elastic restraint against rotation border, λ is stress gradient, λ =(σ_{It is high}-σ_{It is low})/σ_{It is high}；σ_{It is high}Larger stress, σ in expression unequal stress_{It is low}Represent less stress in unequal stress.

(4) calculating of longitudinal stiffener optimal location (rational position)

When a) arranging two ribbed stiffeners on web：

The boundary condition of each plate unit：1. first plate unit is one side elastic restraint against rotation border, and another side is simple supported edge；Second 2. plate unit is simply supported on four sides border.

OrderFor characterizing ribbed stiffener relative to web height position, then the rigid flexing for putting more energy into place's web should Power：

Represent ribbed stiffener relative to web height position, b₁Distance of first ribbed stiffener apart from compression area edge is represented, b₀Represent the height of compressive region；σ₂The buckling stress of stiffener when representing that ribbed stiffener is rigid ribbed stiffener.

According to the equal principle of flexing degree of safety：

In formula：σ_cr1Represent the first plate unit Critical Buckling stress 1., σ_cr2Represent the second plate unit Critical Buckling stress 2.；

I.e.：

According to above formula, solveSo that it is determined that b₁、b₀, you can obtain steel-mixed combination beam under elastic restraint against rotation border Two particular locations of ribbed stiffener of web, are shown in Table 1.

The different boundary of table 1 constrains longitudinal stiffener optimal location

When b) for n bar longitudinal stiffeners：

Web is divided into n+1 plate unit, σ by longitudinal stiffener₁,σ₂,L,σ_n+1There is proportionate relationship with the position of ribbed stiffener, σ_cr1Limit stress, σ are calculated according to rotational restraint_cr2,L,σ_crn+1With reference to simply supported on four sides theoretical calculation limit stress, each plate unit need to expire Sufficient below equation group, solution can obtain the rational position of a plurality of ribbed stiffener：

In formula：σ₃Represent the external load stress level at the 2nd article of ribbed stiffener, σ_cr3Represent that the Critical Buckling of the 2nd piece of plate unit should Power σ_nRepresent the external load stress level at (n-1)th ribbed stiffener, σ_crnRepresent (n-1)th piece of Critical Buckling stress of plate unit, σ_n+1 Represent the external load stress level at nth bar ribbed stiffener, σ_crn+1Represent n-th piece of Critical Buckling stress of plate unit.

The present invention has following major advantage：

One, the influence for considering web elastic restraint against rotation, can accurately calculate plate Critical Buckling stress and obtain The optimal location that ribbed stiffener is set.

Secondly, Critical Buckling stress is greatly improved compared to specification so that improve reasonability that web ribbed stiffener high sets with Economy, can reduce material usage, and the Buckling Loads of stiffener are improved using the inventive method.

Thirdly, the present invention can be suitably used for any bar web longitudinal stiffener optimal location calculating.

Four, ribbed stiffener setting calculating process of the present invention is simple, easy, is worth with larger practical engineering application.

Brief description of the drawings

Fig. 1 is that simplified model and the bucking of plate of putting more energy into calculate schematic diagram.

Fig. 2 is longitudinal stiffener boundary constraint rotation coefficient schematic diagram.

Fig. 3 is the setting schematic diagram of vertically and horizontally ribbed stiffener.

Fig. 4 is two longitudinal stiffener layout drawings.

Fig. 5 is I-shaped combination beam size layout drawing.

In figure：1. steel web；Rotational restraint coefficient at 2.y=0；Rotational restraint coefficient at 3.y=b；4. laterally add Strength rib；5. longitudinal stiffener；6. first ribbed stiffener；7. Article 2 ribbed stiffener.

Specific embodiment

With reference to embodiment and accompanying drawing, the invention will be further described, but does not limit the present invention.

The method that a kind of elastic restraint against rotation border steel-mixed combination deck-molding web longitudinal stiffener is set, initially sets up tool The displacement function equation and boundary condition on flexible rotational restraint border, derive that rigidity is put more energy into and flexibility puts more energy into two based on energy method The buckling stress calculation expression of ribbed stiffener under the state of kind, obtains the meter of elastic restraint against rotation border longitudinal stiffener critical conception Calculate formula；Then, it is considered to external work and minimum potential energy principal, the critical of non-homogeneous compression plate under elastic restraint against rotation border is proposed Buckling stress computing formula；Finally, according to flexing degree of safety equal principle, longitudinal stiffener is set under obtaining rotational restraint border Optimal location (rational position).

The method includes following steps：

Determine elastic rotation boundary constraint coefficient X₀、X_b：

Referring to 2 in Fig. 2,3, concrete slab size, WELDING STUDS and its arrangement, the wing are mainly considered for steel-mixed combination beam Listrium size etc. influences, for clean steel beam, it is only necessary to consider the influence of the size of flange plate and web, specific computing formula is There are Patents to support, China Patent No. discloses a kind of determination steel-mixed composite beam bridge boundary of web for 201310686380.4 The computational methods of elastic restraint against rotation coefficient.

(1) stiffener buckling stress in rotational restraint border is calculated

Classical plate stable theory is the border of simply supported on four sides, using double series as plate flexural function；To calculate Steel-mixed combination beam is put more energy into the buckling stress of web, and web is separated with the edge of a wing, and edge of a wing border is equivalent to two opposite side and has rotate effect The restricted spring answered；In order to construct the border with elastic restraint against rotation effect, displacement function equation uses multinomial and triangle The product form of function：

In formula：A, b are the width and height of web；M is the numbers of half wave in x directions；a₁、a₂、a₃For displacement function is undetermined Coefficient, is determined by boundary condition；Boundary condition is as follows：

Wherein：k₀、k_bIt is the constraint rigidity of boundary of web, D is web bending rigidity；Bring boundary condition into displacement function In equation, dimensionless border rotational restraint coefficient is made：X₀、X_bThe rotation at y=0 and y=b is represented respectively Constraint factor；

Simultaneous displacement function and boundary condition, obtaining each undetermined coefficient is：

Border rotates and suffers restraints, and equivalent strain energy is：

The set-up mode of ribbed stiffener；According to minimum potential energy principal, and by variation computing, when ribbed stiffener adds for flexible During strength rib, shown in the buckling stress such as formula (1) of stiffener：

Wherein：The length-width ratio of plateThe ratio of rigidity of ribbed stiffenerThe section ratio of ribbed stiffenerI in formula^LTable Show the rigidity of ribbed stiffener, A^LRepresent the area of ribbed stiffener, B₁、B₂、B₃、B₄、B₅、B₆It is the ginseng relevant with boundary stiffness coefficient Number, without practical significance.T represents the thickness of web, σ₁The buckling stress of stiffener when representing that ribbed stiffener is flexible ribbed stiffener；

B₄=X₀(X_b+6)², B₅=X_b(X₀+6)², B₆=(X₀X_b+8X₀+8X_b+60)²

X in formula₀、X_bThe rotational restraint coefficient at y=0 and y=b is represented respectively.

When ribbed stiffener is rigid ribbed stiffener, (plate unit is for the plate unit that the flexing of stiffener occurs between rigid ribbed stiffener Plate split by ribbed stiffener after area's lattice), now shown in the buckling stress of stiffener such as formula (2)：

In formula：b_LIt is the maximum height of the plate unit between rigid ribbed stiffener；B₇=B₂+5B₄+5B₅, B₇Represent and boundary constraint The relevant parameter of stiffness coefficient, without practical significance, uses, σ to simplify formula₂Stiffener when representing that ribbed stiffener is rigid ribbed stiffener Buckling stress.

Understood according to formula (1) and (2)：When plate is put more energy into by flexibility, the area ratio of plate the ratio of width to height and floor and When Boundary Stiffness keeps constant, as ribbed stiffener rigidity increases, Buckling of Web stress also gradually increases；Plate is rigidly put more energy into When, Buckling of Web stress increases and increases with boundary constraint coefficient；

(2) longitudinal stiffener critical conception in rotational restraint border is calculated

Classical plate stable theory it is determined that ribbed stiffener critical conception when using making ribbed stiffener meet rigid flexing simultaneously Rigidity corresponding with two kinds of flexing situations of flexible flexing, i.e. σ₁=σ₂；Then under rotational restraint border, the critical conception of ribbed stiffener is expired Foot：

Then under elastic rotation boundary constraint, the critical conception of ribbed stiffener is：

Wherein：

M is used, γ without practical significance to simplify formula in formula₀Represent the critical conception of ribbed stiffener.

(3) the non-homogeneous compression plate Critical Buckling stress in rotational restraint border

Plate is subject to external force load action work：

In formula：V display plates are subject to external force load action work, N₀Represent the load at y=0.

According to minimum potential energy principal, the Critical Buckling Stress calculation of non-homogeneous compression plate under elastic restraint against rotation border can be obtained Formula is：

Wherein：σ_crThe Critical Buckling stress of non-homogeneous compression plate under expression elastic restraint against rotation border, λ is stress gradient, λ =(σ_{It is high}-σ_{It is low})/σ_{It is high}；σ_{It is high}Larger stress, σ in expression unequal stress_{It is low}Represent less stress in unequal stress.

(4) calculating of longitudinal stiffener optimal location (rational position)

When a) arranging two ribbed stiffeners on web：

The boundary condition of each plate unit：1. first plate unit is one side elastic restraint against rotation border, and another side is simple supported edge；Second 2. plate unit is simply supported on four sides border.

OrderFor characterizing ribbed stiffener relative to web height position, then the rigid flexing for putting more energy into place's web should Power：

Represent ribbed stiffener relative to web height position, b₁Distance of first ribbed stiffener apart from compression area edge is represented, b₀Represent the height of compressive region；σ₂The buckling stress of stiffener when representing that ribbed stiffener is rigid ribbed stiffener.

According to the equal principle of flexing degree of safety：

In formula：σ_cr1Represent the first plate unit Critical Buckling stress 1., σ_cr2Represent the second plate unit Critical Buckling stress 2.；

I.e.：

According to above formula, solveSo that it is determined that b₁、b₀, you can obtain steel-mixed combination beam under elastic restraint against rotation border Two particular locations of ribbed stiffener of web, are shown in Table 1.

The different boundary of table 1 constrains longitudinal stiffener optimal location

When b) for n bar longitudinal stiffeners：

Web is divided into n+1 plate unit, σ by longitudinal stiffener₁,σ₂,L,σ_n+1There is proportionate relationship with the position of ribbed stiffener, σ_cr1Limit stress, σ are calculated according to rotational restraint_cr2,L,σ_crn+1With reference to simply supported on four sides theoretical calculation limit stress, each plate unit need to expire Sufficient below equation group, solution can obtain the rational position of a plurality of ribbed stiffener：

In formula：σ₃Represent the external load stress level at the 2nd article of ribbed stiffener, σ_cr3Represent that the Critical Buckling of the 2nd piece of plate unit should Power σ_nRepresent the external load stress level at (n-1)th ribbed stiffener, σ_crnRepresent (n-1)th piece of Critical Buckling stress of plate unit, σ_n+1 Represent the external load stress level at nth bar ribbed stiffener, σ_crn+1Represent n-th piece of Critical Buckling stress of plate unit.

Application of the invention is described in further details with concrete application embodiment below in conjunction with the accompanying drawings.

Concrete application embodiment：

Known certain I-shaped combination beam, girder steel uses Q345qd, and concrete slab is C50, specific size (chi as shown in Figure 5 Very little unit is mm).It is assumed that shearing peg is uniformly distributed and connects good (not occurred level and vertical sliding), the combination is determined The longitudinal stiffener rigidity of web and position.

The constraint factor of steel web lower flange is 2.81, and WELDING STUDS connection in the edge of a wing is reliable, and concrete slab rigidity is big, top flange Border is thought of as approximate fixed boundary.

Longitudinal stiffener rigidity value, longitudinal stiffener position are respectively arranged according to the present invention and method for normalizing, arrange result It is shown in Table 2.

The embodiment combination beam critical conception of table 2 and position result of putting more energy into

In the present embodiment, in ribbed stiffener critical conception value, the present invention can reduce material usage about compared to method for normalizing 36%.The two methods of embodiment are carried out with buckling analysis using Ansys finite element softwares to understand, is calculated using the inventive method Stiffener Buckling Loads coefficients be 44.344, code requirement method is 27.065, and stiffener is improved using the inventive method Buckling Loads about 63.8%.

Claims (2)

1. the method that a kind of elastic restraint against rotation border steel-mixed combination deck-molding web longitudinal stiffener is set, it is characterised in that：It is first First set up with elastic restraint against rotation border displacement function equation and boundary condition, based on energy method derive rigidity put more energy into it is soft Property put more energy into the buckling stress calculation expression of ribbed stiffener under two states, obtain elastic restraint against rotation border longitudinal stiffener critical The computing formula of rigidity；Then, it is considered to external work and minimum potential energy principal, non-homogeneous compression under elastic restraint against rotation border is proposed The Critical Buckling Stress calculation formula of plate；Finally, according to flexing degree of safety equal principle, acquisition rotational restraint longitudinally adds under border The optimal location that strength rib is set.

2. a kind of elastic restraint against rotation border steel according to claim 1-mixed combination deck-molding web longitudinal stiffener is set Method, it is characterised in that：Comprise the following steps：

(1) stiffener buckling stress in rotational restraint border is calculated

Classical plate stable theory is the border of simply supported on four sides, using double series as plate flexural function；To calculate steel-mixed Combination beam is put more energy into the buckling stress of web, and web is separated with the edge of a wing, and edge of a wing border is equivalent to two opposite side has turning effect Restricted spring；In order to construct the border with elastic restraint against rotation effect, displacement function equation uses multinomial and trigonometric function Product form：

$\mathrm{\ω}(x,y)=\left[\begin{array}{c}\frac{y}{b}+a_1{\left(\frac{y}{b}\right)}^2+\\ a_2{\left(\frac{y}{b}\right)}^3+a_3{\left(\frac{y}{b}\right)}^4\end{array}\right]\underset{m=1}{\overset{\mathrm{\∞}}{\Σ}}{A}_{m}sin\left(\frac{m\mathrm{\π}x}{a}\right)$

In formula：A, b are the width and height of web；M is the numbers of half wave in x directions；a₁、a₂、a₃It is displacement function system undetermined Number, is determined by boundary condition；Boundary condition is as follows：

$\left\{\begin{array}{c}\mathrm{\ω}(x,0)=0\\ \mathrm{\ω}(x,b)=0\\ M(x,0)=-D{(\frac{{\∂}^2\mathrm{\ω}}{\∂y^2}+\mathrm{\υ}\frac{{\∂}^2\mathrm{\ω}}{\∂x^2})}_{y=0}=-k_0{\left(\frac{\∂\mathrm{\ω}}{\∂y}\right)}_{y=0}\\ M(x,b)=-D{(\frac{{\∂}^2\mathrm{\ω}}{\∂y^2}+\mathrm{\υ}\frac{{\∂}^2\mathrm{\ω}}{\∂x^2})}_{y=b}=k_b{\left(\frac{\∂\mathrm{\ω}}{\∂y}\right)}_{y=b}\end{array}\right.$

Wherein：k₀、k_bIt is the constraint rigidity of boundary of web, D is web bending rigidity；Bring boundary condition into displacement function equation In, make dimensionless border rotational restraint coefficient：X₀、X_bThe rotational restraint at y=0 and y=b is represented respectively Coefficient.

Simultaneous displacement function and boundary condition, obtaining each undetermined coefficient is：

$a_1=\frac{X_0}{2},a_2=-\frac{X_0{X}_b+5X_0+3X_b+12}{X_b+6},$

$a_3=\frac{1}{2}\frac{X_0{X}_b+4X_0+4X_b+12}{X_b+6}$

Border rotates and suffers restraints, and equivalent strain energy is：

$U_s=\frac{1}{2}\∫\[k_0{\left(\frac{\∂\mathrm{\ω}}{\∂y}{|}_{y=0}\right)}^2+k_b{\left(\frac{\∂\mathrm{\ω}}{\∂y}{|}_{y=b}\right)}^2\]d_x$

The set-up mode of ribbed stiffener；According to minimum potential energy principal, and by variation computing, when ribbed stiffener is flexible ribbed stiffener When, shown in the buckling stress such as formula (1) of stiffener：

${\mathrm{\σ}}_1=\frac{{\mathrm{D\π}}^2}{b^{2}t}\frac{\left(\begin{array}{c}\frac{{\mathrm{\π}}^2{B}_1}{{\mathrm{\α}}^{3}5040}{m}^4+\frac{{\mathrm{\αB}}_2}{10{\mathrm{\π}}^2}+\frac{B_3}{210\mathrm{\α}}{m}^2+\\ \frac{(B_4+B_5)\mathrm{\α}}{2{\mathrm{\π}}^2}+\mathrm{\γ}\frac{B_6{\mathrm{\π}}^2}{2048{\mathrm{\α}}^3}{m}^4\end{array}\right)}{(\frac{{\mathrm{\π}}^2{B}_1}{5040\mathrm{\α}}{m}^2+\frac{{\mathrm{\ηB}}_6{\mathrm{\π}}^2}{2048\mathrm{\α}}{m}^2)}---\left(1\right)$

Wherein：The length-width ratio of plateThe ratio of rigidity of ribbed stiffenerThe section ratio of ribbed stiffenerI in formula^LRepresent and add The rigidity of strength rib, A^LRepresent the area of ribbed stiffener, B₁、B₂、B₃、B₄、B₅、B₆It is the parameter relevant with boundary stiffness coefficient, t Represent the thickness of web, σ₁The buckling stress of stiffener when representing that ribbed stiffener is flexible ribbed stiffener；

$B_1=X_0^2{X}_b^2+17X_0^2{X}_b+17X_0{X}_b^2+76X_0^2+272X_0{X}_b+76X_b^2+1140X_0+1140X_b+4464$

$B_2=X_0^2{X}_b^2+8X_0^2{X}_b+8X_0{X}_b^2+36X_0^2+36X_0{X}_b+36X_b^2+216X_0+216X_b+864$

$B_3=X_0^2{X}_b^2+15X_0^2{X}_b+15X_0{X}_b^2+72X_0^2+210X_0{X}_b+72X_b^2+936X_0+936X_b+3672$

B₄=X₀(X_b+6)², B₅=X_b(X₀+6)², B₆=(X₀X_b+8X₀+8X_b+60)²

In formula：X₀、X_bThe rotational restraint coefficient at y=0 and y=b is represented respectively.

When ribbed stiffener is rigid ribbed stiffener, the plate unit between rigid ribbed stiffener in the flexing of stiffener, now stiffener there is Buckling stress such as formula (2) shown in：

${\mathrm{\σ}}_2=\frac{{\mathrm{D\π}}^2}{{b_L}^{2}t}\frac{24}{B_1{\mathrm{\π}}^2}(2B_3+\sqrt{14B_1{B}_7})---\left(2\right)$

In formula：b_LIt is the maximum height of the plate unit between rigid ribbed stiffener；B₇=B₂+5B₄+5B₅, B₇Represent and boundary stiffness The relevant parameter of coefficient, σ₂The buckling stress of stiffener when representing that ribbed stiffener is rigid ribbed stiffener.

Understood according to formula (1) and (2)：When plate is put more energy into by flexibility, the area ratio of plate the ratio of width to height and floor and border When rigidity keeps constant, as ribbed stiffener rigidity increases, Buckling of Web stress also gradually increases；When plate is subject to rigidity to put more energy into, Buckling of Web stress increases and increases with boundary constraint coefficient.

(2) longitudinal stiffener critical conception in rotational restraint border is calculated

Classical plate stable theory it is determined that ribbed stiffener critical conception when using making ribbed stiffener meet rigid flexing and soft simultaneously The property corresponding rigidity of two kinds of flexing situations of flexing, i.e. σ₁=σ₂；Then under rotational restraint border, the critical conception of ribbed stiffener meets：

$\frac{{\mathrm{D\π}}^2}{b^{2}t}\frac{\frac{{\mathrm{\π}}^2{B}_1}{{\mathrm{\α}}^{3}5040}+\frac{{\mathrm{\αB}}_2}{10{\mathrm{\π}}^2}+\frac{B_3}{210\mathrm{\α}}+\frac{(B_4+B_5)\mathrm{\α}}{2{\mathrm{\π}}^2}+{\mathrm{\γ}}_0\frac{B_6{\mathrm{\π}}^2}{2048{\mathrm{\α}}^3}}{(\frac{{\mathrm{\π}}^2{B}_1}{5040\mathrm{\α}}+\frac{{\mathrm{\ηB}}_6{\mathrm{\π}}^2}{2048\mathrm{\α}})}=\frac{{\mathrm{D\π}}^2}{{b_L}^{2}t}\frac{24}{B_1{\mathrm{\π}}^2}(2B_3+\sqrt{14B_1{B}_7})$

Then under elastic rotation boundary constraint, the critical conception of ribbed stiffener is：

${\mathrm{\γ}}_0=\frac{8M}{315b_L^2{B}_1{B}_6{\mathrm{\π}}^4}---\left(3\right)$

Wherein：

M is to simplify formula to use in formula, γ₀Represent the critical conception of ribbed stiffener；

(3) the non-homogeneous compression plate Critical Buckling stress in rotational restraint border

Plate is subject to external force load action work：

$V=\frac{1}{2}{N}_0\∫\∫(1-\mathrm{\λ}\frac{y}{b}){\left(\frac{\∂\mathrm{\ω}}{\∂x}\right)}^2{d}_x{d}_y$

In formula：V display plates are subject to external force load action work, N₀Represent the load at y=0；

According to minimum potential energy principal, the Critical Buckling Stress calculation formula of non-homogeneous compression plate under elastic restraint against rotation border can be obtained For：

${\mathrm{\σ}}_{cr}=\frac{{\mathrm{D\π}}^2}{b^{2}t}\frac{12(\sqrt{14}{B}_1{B}_2+5\sqrt{14}{B}_1{B}_4+5\sqrt{14}{B}_1{B}_5+2B_3\sqrt{B_1{B}_7})}{B_8{\mathrm{\π}}^2\sqrt{B_1{B}_7}}---\left(4\right)$

$\begin{array}{c}{B}_8=\[(1-\mathrm{\λ}/2)X_b^2+(17-9\mathrm{\λ})X_b+(76-43\mathrm{\λ})\]X_0^2+\[(17-8\mathrm{\λ})X_b^2+(272-136\mathrm{\λ})X_b+(1140-612\mathrm{\λ})\]X_0+\\ \[(76-33\mathrm{\λ})X_b^2+(1140-528\mathrm{\λ})X_b+(4464-2232\mathrm{\λ})\]\end{array}$

Wherein：σ_crRepresent the Critical Buckling stress of non-homogeneous compression plate under elastic restraint against rotation border, λ is stress gradient, λ= (σ_{It is high}-σ_{It is low})/σ_{It is high}；σ_{It is high}Larger stress, σ in expression unequal stress_{It is low}Represent less stress in unequal stress；(4) longitudinal direction The calculating of ribbed stiffener optimal location

When a) arranging two ribbed stiffeners on web：

The boundary condition of each plate unit：1. first plate unit is one side elastic restraint against rotation border, and another side is simple supported edge；Second plate unit 2. it is simply supported on four sides border；

OrderFor characterizing ribbed stiffener relative to web height position, the then buckling stress of place's web of rigidly putting more energy into：

Represent ribbed stiffener relative to web height position, b₁Represent distance of first ribbed stiffener apart from compression area edge, b₀Table Show the height of compressive region；σ₂The buckling stress of stiffener when representing that ribbed stiffener is rigid ribbed stiffener；

According to the equal principle of flexing degree of safety：

$\frac{{\mathrm{\σ}}_1}{{\mathrm{\σ}}_{cr1}}=\frac{{\mathrm{\σ}}_2}{{\mathrm{\σ}}_{cr2}}$

In formula：σ_cr1Represent the first plate unit Critical Buckling stress 1., σ_cr2Represent the second plate unit Critical Buckling stress 2.；

I.e.：

According to above formula, solveSo that it is determined that b₁、b₀, you can obtain steel-mixed combination web under elastic restraint against rotation border Two particular locations of ribbed stiffener；

When b) for n bar longitudinal stiffeners：

Web is divided into n+1 plate unit, σ by longitudinal stiffener₁,σ₂,L,σ_n+1There is proportionate relationship, σ with the position of ribbed stiffener_cr1 Limit stress, σ are calculated according to rotational restraint_cr2,L,σ_crn+1With reference to simply supported on four sides theoretical calculation limit stress, each plate unit needs to meet Below equation group, solution can obtain the rational position of a plurality of ribbed stiffener；

$\left\{\frac{{\mathrm{\σ}}_1}{{\mathrm{\σ}}_{cr1}}=\frac{{\mathrm{\σ}}_2}{{\mathrm{\σ}}_{cr2}},\frac{{\mathrm{\σ}}_2}{{\mathrm{\σ}}_{cr2}}=\frac{{\mathrm{\σ}}_3}{{\mathrm{\σ}}_{cr3}},L,\frac{{\mathrm{\σ}}_n}{{\mathrm{\σ}}_{crn}}=\frac{{\mathrm{\σ}}_{n+1}}{{\mathrm{\σ}}_{crn+1}}\right\}---\left(6\right)$

In formula：σ₃Represent the external load stress level at the 2nd article of ribbed stiffener, σ_cr3Represent the 2nd piece of Critical Buckling stress σ of plate unit_n Represent the external load stress level at (n-1)th ribbed stiffener, σ_crnRepresent (n-1)th piece of Critical Buckling stress of plate unit, σ_n+1Represent External load stress level at nth bar ribbed stiffener, σ_crn+1Represent n-th piece of Critical Buckling stress of plate unit.