CN1724985A - Calculating method and calculating formula of equivalent bending moment caused by pressure in bend - Google Patents

Abstract

The surface area of the radial outward-bending radius pipe wall is larger than that of the radial inward-bending radius pipe wall on the neutral line of a bend, and under the action of inner pressure, the radial outward acting force is greater than the radial outward acting force, and the two acting forces are not balanced and thus the resultant force produces an equivalent bending moment and a corresponding bending stress in the bend; the invention establishes a mechanical model for quantitatively analyzing the equivalent bending moment, derivates a calculation formula of equivalent radial bending moment (along the central line of the bent tube) Mthita is about equal to 0.032ri2RP and a calculation formula of equivalent ring bending moment (along the circumference of cross section of the bent tube) Mphi=pi(3pi-10)/1080Prgamai2, thus able to further calculate the bending stresses of the two equivalent bending moments, and known by comparison, the ring bending stress caused by the inner pressure is 1-3 times greater than the stress of the ring thin film, becoming an assignable important stress composition, and the superposition of the bending stresses and the traditional calculated thin film stress is the whole ring (circumferential) stress.

Description

Computing method and the computing formula of pressure caused equivalent bending moment in elbow

Elbow (bend pipe) is widely used common accessory in the various pipelines such as petrochemical unit pipeline, oil gas long distance pipeline, cities and towns domestic life resource pipeline, have and change the functions such as stress that pipeline internal medium flows to, mitigation pipeline expands with heat and contract with cold and produces, its planform is shown in shown in accompanying drawing 1.(for example bear in practice interior pressure or external pressure, subsea pipeline), the effect of the external load such as moment of flexure, moment of torsion, the medium of carrying due to pipeline often has inflammable and explosive characteristic, thus to the force analysis of pipe fitting whether accurately, whether can be directly involved in the safety of the country and people's lives and properties comprehensively.

Ring (week) when for a long time, elbow bears interior pressure is respectively Soviet Union A. Б. A Yingbinjieer and A. Γ to the traditional calculations formula of stress. the bend pipe circumference stress formula that the scholars such as Camael Shi Jieying, S. iron Mo Xinke (Timoshenko) derive is the calculating formula of membrane stress part in fact

${\mathrm{\σ}}_{\mathrm{\φ}}=\frac{{\mathrm{Pr}}_i}{2t}\frac{2R+\sin \mathrm{\φ}}{R+r\sin \mathrm{\φ}}---\left(1\right)$

In the experiment of monitor strain, to the actual measuring and calculating value of stress and the result of calculation of above-mentioned formula, there is certain error in the ring (week) of elbow when researchist finds to bear interior pressure very early, the ultimate bending moment of the ultimate bending moment of equal-diameter elbow in interior pressure with while closing curved synergy than interior pressure with while opening curved synergy is larger, but, still the report quantitatively not calculating about qualitative analysis and the error of this source of error at present, the mechanical mechanism of these differences is unclear, therefore, the security with elbow pipeline does not still have well theoretical guarantee.

Patent inventor's (being also patent applicant) finds by analysis and research, traditional calculations formula be take membrane (shell) theory and is derived as basis, traditional theory is regarded elbow as a part for toroidal shell as and don't is considered that straight tube will unavoidably cause error to just the derive plastic limit pressure type of bend pipe of the impact of elbow extreme pressure, in fact, elbow while bearing interior press is also subject to the effect of equivalent moment, the membrane (shell) theory of take as basis, derive the traditional calculations formula come be can not comprehensive representation the stressing conditions of elbow now.Particular content is as follows.

1 elbow is poor along the area pressure of bending radius direction

The poor equivalent bending moment mechanical model of interior pressure acting force is shown in accompanying drawing 2, and the radially inside and outside inside pipe wall surface area ratio of equal-diameter elbow midline is shown in accompanying drawing 3 compared with audio-visual picture.Press accompanying drawing 2 and accompanying drawing 3, on the bend pipe neutral line, the radially outer tube wall surface of bending radius is long-pending long-pending larger than radially inner tube wall surface on the neutral line, and under same interior pressure effect, radially outer acting force is greater than radially inner acting force, therefore on footpath, be, unbalanced with joint efforts.The 2 hoop bending stresses that the area pressure along bending radius direction is poor and warp-wise moment of flexure causes indirectly

For the bend loss of a warp-wise angular unit (π/180 radian) shown in accompanying drawing 2, pipe internal surface product moment radially inside and outside on center line is caused, poorly along the acting force of Y direction be

${\mathrm{\ΔF}}_y=P\×\mathrm{\Δ}A\sin \mathrm{\φ}=2P{\∫}_0^{\mathrm{\π}/2}{r}_i\mathrm{d\φ}\×[(R+r_i\sin \mathrm{\φ})-(R-r_i\sin \mathrm{\φ})]\frac{\mathrm{\π}}{180}\×\sin \mathrm{\φ}$

$=2P{\∫}_0^{\mathrm{\π}/2}\frac{\mathrm{\π}}{90}{r}_i^2{\sin }^2\mathrm{\φd\φ}=\frac{\mathrm{\π}}{45}P{r}_i^2(\frac{\mathrm{\φ}}{2}-\frac{\sin 2\mathrm{\φ}}{4}){|}_{\mathrm{\φ}=0}^{\mathrm{\φ}=\mathrm{\π}/2}=\frac{\mathrm{\π}}{45}P{r}_i^2\×\frac{\mathrm{\π}}{4}=\frac{1}{180}{\mathrm{\π}}^2{r}_i^{2}P---\left(2\right)$

The plane of bending of making a concerted effort to be parallel to bend pipe due to this difference acting force, and radially outward, it is the moment of flexure of M that its effect to elbow internal wall is equivalent to size, under interior pressure effect, with the finite element analysis model of straight tube elbow, see accompanying drawing 4, accompanying drawing 4 (a) is the finite element analysis model deformation tendency arrow arrow figure of a two ends band straight tube reducing elbow while bearing interior press, accompanying drawing 4 (b) be the amplification after distortion show with distortion before profile schematic diagram relatively.From these figure, when at one end straight tube does not have foreign object constraint, elbow has out curved trend.But due to the engineering limited case of bend pipe straight tube that two ends connect in fact, this moment of flexure can be equivalent to a pair of elbow that makes here and close curved end face moment of flexure.Due to the adjection of this equivalence end face moment of flexure, the buckling stress σ of swan neck_{θ W}meeting is respective change with bent angle θ, still, and equivalent end face moment of flexure and the axial stress σ causing thereof_{θ W}the flexural center O of all can take claims a little as mid point along axis, and both maximal values all occur on the bend pipe xsect at O place, flexural center, therefore, and when limit load analysis, as long as analyze the moment of flexure in this cross section.In θ=0～π/4, the horizontal component of interior Δ F and vertical component are respectively the moment of flexure of flexural center O

$={\∫}_0^{\mathrm{\π}/4}\frac{1}{180}{\mathrm{\π}}^2{r}_i^{2}P\sin \mathrm{\θ}\·R(\cos \mathrm{\θ}-\frac{\sqrt{2}}{2})\mathrm{d\θ}$

$=\frac{1}{360}{\mathrm{\π}}^2{r}_i^2\mathrm{RP}{\∫}_0^{\mathrm{\π}/4}(\sin 2\mathrm{\θ}-\sqrt{2}\sin \mathrm{\θ})\mathrm{d\θ}$

$=\frac{1}{360}{\mathrm{\π}}^2{r}_i^2\mathrm{RP}({\sin }^2\mathrm{\θ}+\sqrt{2}\cos \mathrm{\θ}){|}_0^{\mathrm{\π}/4}$

$=\frac{1}{36}{\mathrm{\π}}^2{r_i}^2\mathrm{RP}({\sin }^2\frac{\mathrm{\π}}{4}+\sqrt{2}\cos \frac{\mathrm{\π}}{4}-\sqrt{2})---\left(3\right)$

$=\frac{1}{360}(\frac{3}{2}-\sqrt{2}){\mathrm{\π}}^2{r}_i^2\mathrm{RP}$

$={\∫}_0^{\mathrm{\π}/4}\frac{1}{180}{\mathrm{\π}}^2{r}_i^{2}P\cos \mathrm{\θ}\·R(\frac{\sqrt{2}}{2}-\sin \mathrm{\θ})\mathrm{d\θ}$

$=\frac{1}{360}{\mathrm{\π}}^2{r}_i^2\mathrm{RP}{\∫}_0^{\mathrm{\π}/4}(\sqrt{2}\cos \mathrm{\θ}-\sin 2\mathrm{\θ})\mathrm{d\θ}$

$=\frac{1}{360}{\mathrm{\π}}^2{r}_i^2\mathrm{RP}(\sqrt{2}\sin \mathrm{\θ}-{\sin }^2\mathrm{\θ}){|}_0^{\mathrm{\π}/4}---\left(4\right)$

$=\frac{1}{360}{\mathrm{\π}}^2{r}_i^2\mathrm{RP}(\sqrt{2}\sin \frac{\mathrm{\π}}{4}-{\sin }^2\frac{\mathrm{\π}}{4})$

$=\frac{1}{720}{\mathrm{\π}}^2{r}_i^2\mathrm{RP}$

In whole 90 ° of equal-diameter elbows, Δ F_ythe equivalent bending moment M total to flexural center O_θfor

By M_θ(6) formula that the hoop bending stress causing can be respectively proposes by T Kalman or P carat and gram/(7) formula that И leu Si Nieer proposes calculates.

${\mathrm{\σ}}_{\mathrm{\φW}}^{\′}=\frac{18M_{\mathrm{\θ}}\mathrm{\λ}{r}_o{\cos }^2\mathrm{\φ}}{I(1+12{\mathrm{\λ}}^2)}=\frac{18\×0.032r_i^2\mathrm{RP}\×\frac{\mathrm{tR}}{r^2}{r}_o{\cos }^2\mathrm{\φ}}{\frac{\mathrm{\π}}{4}(r_o^4-r_i^4)(1+12{\mathrm{\λ}}^2)}\≈\frac{0.736P{R}^2{r}_i^2{r}_{o}t{\cos }^2\mathrm{\φ}}{r^2(r_o^4-r_i^4)(1+12{\mathrm{\λ}}^2)}---\left(6\right)$

${\mathrm{\σ}}_{\mathrm{\φw}}^{\′}=\frac{1.8M_{\mathrm{\θ}}{r}_o}{I{\mathrm{\λ}}^{2/3}}{\cos }^2\mathrm{\φ}=\frac{1.8\×0.032r_i^2\mathrm{RP}\×r_o}{\frac{\mathrm{\π}}{4}(r_o^4-r_i^4)\×{\mathrm{\λ}}^{2/3}}{\cos }^2\mathrm{\φ}\≈\frac{0.0736\mathrm{PR}{r}_i^2{r}_o}{(r_o^4-r_i^4){\mathrm{\λ}}^{2/3}}{\cos }^2\mathrm{\φ}---\left(7\right)$

The poor circumferential moment causing at hoop of 2 area pressure along bending radius direction and hoop bending stress

Interior pressure causes that unit angle bend pipe cross section circumferential moment mechanical model is shown in accompanying drawing 5, for the pipeline section (middle line length π R/180) of a warp-wise angular unit shown in accompanying drawing 2, the poor component along X-direction of the caused acting force of pipe internal surface product moment radially inside and outside on center line is

$\mathrm{\Δ}{F}_x=P\×\mathrm{\Δ}A\cos \mathrm{\φ}=P{\∫}_0^{\mathrm{\φ}}{r}_i\mathrm{d\φ}\×[(R+r_i\sin \mathrm{\φ})-(R-r_i\sin \mathrm{\φ})]\frac{\mathrm{\π}}{180}\×\cos \mathrm{\φ}$

$=\frac{\mathrm{\π}}{90}{\mathrm{Pr}}_i^2{\∫}_0^{\mathrm{\φ}}\sin \mathrm{\φd}\left(\sin \mathrm{\φ}\right)=\frac{\mathrm{\π}}{90}{\mathrm{Pr}}_i^2\frac{{\sin }^2\mathrm{\φ}}{2}{|}_0^{\mathrm{\π}/2}=\frac{\mathrm{\π}}{180}{\mathrm{Pr}}_i^2---\left(8\right)$

φ on tube section₀the moment of flexure that place, ∈ (pi/2～pi/2) angle causes has the equation of comptability

$={\∫}_{{\mathrm{\φ}}_0=-\mathrm{\π}/2}^{\mathrm{\π}/2}\left(\frac{1}{2}{\mathrm{\ΔF}}_y\right)\×r\cos {\mathrm{\φ}}_0+{\∫}_{{\mathrm{\φ}}_0=\mathrm{\π}/2}^{\mathrm{\π}/2}{\∫}_{\mathrm{\φ}=0}^{\mathrm{\π}/2}(P\×\mathrm{\Δ}A\sin \mathrm{\φ})\×(r-r\cos {\mathrm{\φ}}_0)$

$-{\∫}_{{\mathrm{\φ}}_0=-\mathrm{\π}/2}^{\mathrm{\π}/2}\left(\mathrm{\Δ}{F}_x\right)\×[2r-(r+r\sin {\mathrm{\φ}}_0)]-{\∫}_{{\mathrm{\φ}}_0=\mathrm{\π}/2}^{\mathrm{\π}/2}{\∫}_{\mathrm{\φ}=0}^{\mathrm{\π}/2}(P\×\mathrm{\Δ}A\cos \mathrm{\φ})\×r(\sin \mathrm{\φ}-\sin {\mathrm{\φ}}_0)$

$={\∫}_{{\mathrm{\φ}}_0=-\mathrm{\π}/2}^{\mathrm{\π}/2}\frac{{\mathrm{\π}}^2}{360}{\mathrm{Pr}}_i^2{\×r\cos \mathrm{\φ}}_0$

$+P{\∫}_{{\mathrm{\φ}}_0=\mathrm{\π}/2}^{\mathrm{\π}/2}{\∫}_{\mathrm{\φ}=0}^{\mathrm{\π}/2}{r}_i\mathrm{d\φ}\×[R+r_i\sin \mathrm{\φ}-(R-r_i\sin \mathrm{\φ})]\frac{\mathrm{\π}}{180}\×\sin \mathrm{\φ}\×r(1-\cos {\mathrm{\φ}}_0)$

$-{\∫}_{{\mathrm{\φ}}_0=-\mathrm{\π}/2}^{\mathrm{\π}/2}\left(\frac{\mathrm{\π}}{180}{\mathrm{Pr}}_i^2\right)\×r(1-\sin {\mathrm{\φ}}_0)$

$-P{\∫}_{{\mathrm{\φ}}_0=\mathrm{\π}/2}^{\mathrm{\π}/2}{\∫}_{\mathrm{\φ}=0}^{\mathrm{\π}/2}{r}_i\mathrm{d\φ}\×[(R+r_i\sin \mathrm{\φ})-(R-r_i\sin \mathrm{\φ})]\frac{\mathrm{\π}}{180}\×\cos \mathrm{\φ}\×r(\sin \mathrm{\φ}+\sin {\mathrm{\φ}}_0)$

$={\∫}_{{\mathrm{\φ}}_0=-\mathrm{\π}/2}^{\mathrm{\π}/2}\left(\frac{\mathrm{\π}}{360}\mathrm{Pr}{r}_i^2\right)\×(-2+\mathrm{\π}\cos {\mathrm{\φ}}_0+2\sin {\mathrm{\φ}}_0)$

$+\frac{\mathrm{\π}}{90}\mathrm{Pr}{r}_i^2{\∫}_{{\mathrm{\φ}}_0=\mathrm{\π}/2}^{\mathrm{\π}/2}{\∫}_{\mathrm{\φ}=0}^{\mathrm{\π}/2}{\sin }^2\mathrm{\φ}(1-\cos {\mathrm{\φ}}_0)\mathrm{d\φ}-{\∫}_{{\mathrm{\φ}}_0=\mathrm{\π}/2}^{\mathrm{\π}/2}{\∫}_{\mathrm{\φ}=0}^{\mathrm{\π}/2}\left[\sin \mathrm{\φ}\cos \mathrm{\φ}(\sin \mathrm{\φ}-\sin {\mathrm{\φ}}_0)\right]\mathrm{d\φ}$

$=\frac{\mathrm{\π}}{90}\mathrm{Pr}{r}_i^2\left[\frac{1}{4}{\∫}_{{\mathrm{\φ}}_0=-\mathrm{\π}/2}^{\mathrm{\π}/2}(-2+\mathrm{\π}\cos {\mathrm{\φ}}_0+2\sin {\mathrm{\φ}}_0)\right]$

$+{\∫}_{{\mathrm{\φ}}_0=\mathrm{\π}/2}^{\mathrm{\π}/2}\frac{1}{2}(\mathrm{\φ}-\frac{1}{2}\sin 2\mathrm{\φ})(1-\cos {\mathrm{\φ}}_0){|}_{\mathrm{\φ}=0}^{\mathrm{\φ}-\mathrm{\π}/2}-{\∫}_{{\mathrm{\φ}}_0=\mathrm{\π}/2}^{\mathrm{\π}/2}\frac{1}{3}{\sin }^3\mathrm{\φ}{|}_{\mathrm{\φ}=0}^{\mathrm{\φ}=\mathrm{\π}/2}+{\∫}_{{\mathrm{\φ}}_0}^{\mathrm{\φ}}\×\frac{1}{2}{\sin }^2\mathrm{\φ}{|}_{\mathrm{\φ}=0}^{\mathrm{\φ}=\mathrm{\π}/2}$

$=\frac{\mathrm{\π}}{90}\mathrm{Pr}{r}_i^2[\frac{1}{4}{\∫}_{{\mathrm{\φ}}_0=-\mathrm{\π}/2}^{\mathrm{\π}/2}(-2+\mathrm{\π}\cos {\mathrm{\φ}}_0+2\sin {\mathrm{\φ}}_0)-{\∫}_{{\mathrm{\φ}}_0=-\mathrm{\π}/2}^{\mathrm{\π}/2}[\frac{\mathrm{\π}}{4}(1-\cos {\mathrm{\φ}}_0)+\frac{1}{3}-\frac{1}{2}\sin {\mathrm{\φ}}_0]$

$=\frac{\mathrm{\π}}{90}\mathrm{Pr}{r}_i^2[\frac{1}{4}(-2+\mathrm{\π}\cos {\mathrm{\φ}}_0+2\sin {\mathrm{\φ}}_0)+\frac{\mathrm{\π}}{4}(1-\cos {\mathrm{\φ}}_0)-(\frac{1}{3}-\frac{1}{2}\sin {\mathrm{\φ}}_0)]{|}_{{\mathrm{\φ}}_0=-\mathrm{\π}/2}^{{\mathrm{\φ}}_0=\mathrm{\π}/2}$

$=\frac{\mathrm{\π}}{180}\mathrm{Pr}{r}_i^2(-\frac{5}{3}+\frac{\mathrm{\π}}{2}+2\sin {\mathrm{\φ}}_0){|}_{{\mathrm{\φ}}_0=-\mathrm{\π}/2}^{{\mathrm{\φ}}_0=\mathrm{\π}/2}---\left(9\right)$

At tube section neutral line place, φ₀=0, so the moment of flexure at this place is

$M_{\mathrm{\φ}}=\frac{\mathrm{\π}(3\mathrm{\π}-10)}{1080}\mathrm{Pr}{r}_i^2---\left(10\right)$

By deflection of beam stress-type, the bending stress that this moment of flexure causes is

${\mathrm{\σ}}_{\mathrm{\φw}}^{\′\′}=\frac{M_{\mathrm{\φ}}t}{2I_x}=\frac{\mathrm{\π}(3\mathrm{\π}-10)}{1080}r{r}_i^{2}P\×\frac{t}{2\×\frac{\mathrm{\πR}}{180}\×\frac{t^3}{12}}=(3\mathrm{\π}-10)\frac{{\mathrm{rr}}_i^2}{t^{2}R}P\≈-0.575\frac{{\mathrm{rr}}_i^2}{t^{2}R}P---\left(11\right)$

In fact, to the such variable cross-section circular arc of accompanying drawing 5 (b), structure has very large flexibility, and the moment of flexure of one end cannot be delivered to the other end completely, and it is inappropriate by deflection of beam stress-type, asking for the bending stress that local moment of flexure causes in whole segmental arc.From (6) formula and (7) formula, can find out the circumference stress of bend pipe and cos²φ is in direct ratio, so, (11) formula is multiplied by cos²φ, becomes the final bending stress formula directly causing at hoop about acting force that will ask for

${\mathrm{\σ}}_{\mathrm{\φW}}^{\′\′}=-0.575\frac{\mathrm{Pr}{r}_i^2}{t^{2}R}{\cos }^2\mathrm{\φ}---\left(12\right)$

The geometrical variations of the bend pipe xsect being caused by interior pressure, on the one hand, the stepless action being inside pressed in radially makes xsect become round, on the other hand, the additional bending moment that interior pressure causes makes xsect become oval, both effects cancel out each other after the Geometrical change of xsect very small, remain the circular section that radius is r, therefore, hoop membrane stress still can calculate by (1) formula.

In 3, be pressed in the hoop total stress that 45 ° of 90 ° of equal-diameter elbows are located

For 90 ° of elbows in actual pipeline, from its circumference stress of end to end of elbow, gradually change, the stress of central cross-section is maximum and become the starting point of inefficacy, it is main research object that 45 ° of bend pipes are located cross section, according to above-mentioned analysis, corresponding to formula (6) and (7), be inside pressed in 45 ° of isometrical 90 ° of bend pipes and locate the hoop total stress formula that tube wall causes and can be respectively

${\mathrm{\σ}}_{\mathrm{\φ}}={\mathrm{\σ}}_{\mathrm{\φB}}+{\mathrm{\σ}}_{\mathrm{\φW}}^{\′}+{\mathrm{\σ}}_{\mathrm{\φW}}^{\′\′}+{\mathrm{\σ}}_{\mathrm{\φT}}$

$=\frac{{\mathrm{Pr}}_i}{2t}\frac{2R+r\sin \mathrm{\φ}}{R+r\sin \mathrm{\φ}}-\frac{0.736P{R}^2{r}_i^2{r}_{o}t}{r^2(r_o^4-r_i^4)(1+12{\mathrm{\λ}}^2)}{\cos }^2\mathrm{\φ}-0.575\frac{\mathrm{Pr}{r}_i^2}{t^{2}R}{\cos }^2\mathrm{\φ}---\left(13\right)$

Or

${\mathrm{\σ}}_{\mathrm{\φ}}={\mathrm{\σ}}_{\mathrm{\φB}}+{\mathrm{\σ}}_{\mathrm{\φW}}^{\′}+{\mathrm{\σ}}_{\mathrm{\φW}}^{\′\′}+{\mathrm{\σ}}_{\mathrm{\φT}}$

$=\frac{{\mathrm{Pr}}_i}{2t}\frac{2R+r\sin \mathrm{\φ}}{R+r\sin \mathrm{\φ}}-\frac{0.0736\mathrm{PR}{r}_i^2{r}_o}{(r_o^4-r_i^4){\mathrm{\λ}}^{2/3}}{\cos }^2\mathrm{\φ}-0.575\frac{\mathrm{Pr}{r}_i^2}{t^{2}R}{\cos }^2\mathrm{\φ}---\left(14\right)$

4 discuss

For (13) and (14) two formulas, more known as calculated, the hoop bending stress that interior pressure causes reaches to three times of hoop membrane stress, becomes the important stress composition of can not ignore.

Certainly, if the connected straight tube in elbow two ends does not suffer restraints, can freely stretch, above-mentioned various middle σ_{φ W}' character can change.

In fact, its xsect of elbow that any process is manufactured is general all there is to a certain degree non-circular, often be approximately the ellipse of ovality ρ=1%～10%, the non-circular elbow of bend pipe xsect also can produce another kind of hoop additional bending stress under the effect of interior pressure, obviously, for manufacture, cause xsect transverse direction at the elbow of neutral line direction, the poor component in bending radius direction of area pressure is larger, thereby formula (5) and formula (10) have larger end value.

In addition, as R → ∞, have${\mathrm{\σ}}_{\mathrm{\φ}}=\frac{{\mathrm{Pr}}_i}{t},$Identical with the circumference stress formula of straight tube.

The described equivalent bending moment M of formula (5)_θ≈ 0.032r_i²rP is that the two ends straight tube that is connected exists a kind of quantitative description of impact on 90 ° of iso-diameter elbow extreme pressures, and the mechanism of action of this phenomenon is the restriction that straight tube channel angular elbow is opened curved trend under effect.And formula (5) M_θ≈ 0.032r_i²rP and formula (10)$M_{\mathrm{\φ}}=\frac{\mathrm{\π}(3\mathrm{\π}-10)}{1080}\mathrm{Pr}{r}_i^2$And be the ultimate bending moment larger a kind of quantitative description of the ultimate bending moment of equal-diameter elbow in interior pressure and while closing curved synergy than interior pressure and while opening curved synergy.

When bearing external pressure, elbow does the used time, the feature equally with above-mentioned equivalent bending moment, and the numerical values recited of equivalent bending moment still can be calculated by derived formula, only the action direction of acting force and equivalent bending moment all with interior pressure effect under opposite direction, simultaneously, the calculating of area difference will be take elbow outside surface as reference field, and the inside radius in formula replaces external radius.

Below the mark in this instructions is further described:

The center line bending radius of R elbow, mm

The angle of the θ turning radius and left end disc, degree or °

The circumferential angle of pipe radius and neutral axis on φ tube section, degree or °

R_ithe inside radius of swan neck xsect, mm

R_othe external radius of swan neck xsect, mm

T bend pipe tools shell thickness, mm

In P, press, MPa (10⁶n/m²)

$\mathrm{\λ}=\frac{\mathrm{Rt}}{r^2}$Tortuosity factor

5 accompanying drawing explanations

Accompanying drawing 1 is the solid model of elbow (bend pipe) planform.

The equivalent bending moment mechanical model that it is poor that accompanying drawing 2 is interior pressure acting force is shown in.

Accompanying drawing 3 is radially comparison diagrams directly perceived of inside and outside inside pipe wall surface area of equal-diameter elbow midline.

Accompanying drawing 4 is with the finite element analysis model of straight tube elbow under interior pressure effect, wherein accompanying drawing 4 (a) is the finite element analysis model deformation tendency arrow arrow figure of a two ends band straight tube reducing elbow while bearing interior press, accompanying drawing 4 (b) be the amplification after distortion show with distortion before profile schematic diagram relatively.

Accompanying drawing 5 is that interior pressure causes unit angle bend pipe xsect circumferential moment mechanical model, take the center of circle of cross circular section to have set up rectangular coordinate system XOY as initial point.Wherein accompanying drawing 5 (a) is that complete bend pipe xsect bears the mechanical model of interior pressure, and accompanying drawing 5 (b) is that bend pipe xsect is cut left and right two halfs open along interior extrados again, and half bears the mechanical model of interior pressure and internal force right side.

Claims (6)

1. about the computing method of pressure caused equivalent bending moment in elbow, the relevant principle analysis of the authorship area pressure poor warp-wise equivalent bending moment computing formula that cause of elbow along bending radius direction of having derived$M_{\mathrm{\θ}}\≈0.032r_i^2\mathrm{RP}$With hoop equivalent bending moment computing formula$M_{\mathrm{\φ}}=\frac{\mathrm{\π}(3\mathrm{\π}-10)}{1080}\mathrm{Pr}{r}_i^2,$These two formula are that under pressure-acting, the two ends straight tube that is connected exists a kind of quantitative description of impact on 90 ° of iso-diameter elbows, the restriction that to be straight tube channel angular elbow open curved trend under effect of the mechanism of action of this phenomenon is also the ultimate bending moment larger a kind of quantitative description of the ultimate bending moment of equal-diameter elbow in interior pressure and while closing curved synergy than interior pressure and while opening curved synergy.

2. the computing method of pressure according to claim 1 caused equivalent bending moment in elbow, is characterized in that: the neutral line of elbow both sides of take is boundary, according to the internal surface area (m dividing near outer arch in elbow internal surface area²) be greater than the internal surface area (m dividing near internal arch part²) reality, even if press under (MPa) effect in uniformly same, the summation (N) of the acting force that the inside surface dividing near outer arch in also can qualitative explanation elbow bears is greater than the summation (N) of the acting force that inside surface that close internal arch part divides bears.

3. the computing method of pressure according to claim 1 caused equivalent bending moment in elbow, is characterized in that: the neutral line of elbow both sides of take is boundary, utilizes the geometry feature of elbow and concrete size thereof to calculate two-part area discrepancy (m²), can Qualitative calculate go out same and press in uniformly under (MPa) act on, by this area difference (m²) caused acting force (N) poor.

4. the computing method of pressure according to claim 1 caused equivalent bending moment in elbow, it is characterized in that: poor due to the caused acting force of area difference, can judge qualitatively that the effect that the difference of this acting force plays in elbow is equivalent to the effect of opening winding square (kNm) of elbow in its plane of bending.

5. the computing method of pressure according to claim 1 caused equivalent bending moment in elbow, is characterized in that: utilize the physical dimension of elbow and the interior pressure (MPa) of bearing can quantitatively calculate the equivalent value (mutually on duty) of opening winding square (kNm) acting on because area difference is caused in elbow plane of bending.

6. the computing method of pressure according to claim 1 caused equivalent bending moment in elbow, it is characterized in that: when the used time of doing that elbow bears external pressure (MPa), there are equally the claims 2 to feature claimed in claim 5, only the action direction of acting force (N) and Equivalent Moment (kNm) all with interior pressure (MPa) effect under opposite direction, Equivalent Moment is now to close winding square, still can be by formula$M_{\mathrm{\θ}}\≈0.032r_i^2\mathrm{RP}$With$M_{\mathrm{\φ}}=\frac{\mathrm{\π}(3\mathrm{\π}-10)}{1080}\mathrm{Pr}{r}_i^2$Quantitatively calculate warp-wise moment of flexure in elbow plane of bending and the equivalent value of hoop equivalent bending moment (kNm).