CN103866664B - A kind of defining method of Ballast track high-speed railway foundation bed structure thickness - Google Patents

Abstract

A kind of Ballast track high-speed railway foundation bed structure thickness defining method, its step mainly: determine based on normal distribution curve the wheel load power that adjacent sleeper is born and determine that the average pressure at the end rested the head on by sleeper 3; When considering vehicular load stress going down, both considered (left side) at the bottom of the pillow of side immediately below wheel and born the vertical diffusion of wheel load power, have also contemplated that opposite side (right side) at the bottom of this pillow bore the oblique stress diffusing to position immediately below (left side) center at the bottom of this pillow of wheel load power; Also contemplate the adjacent sleeper left and right side pillow end bears the oblique stress diffusing to position immediately below (left side) center at the bottom of this pillow of wheel load power simultaneously, thus draws the relation of depth location gross vehicle load stress and the degree of depth immediately below side (left side) wheel; And then draw foundation bed structure thickness.This method calculates simple, and the result drawn is more accurate, thus has both ensured the operation life-span of railway and the safe operation of train, can reduce building railway, maintenance cost again, avoids waste.

Description

A kind of defining method of Ballast track high-speed railway foundation bed structure thickness

Technical field

The present invention relates to railway construction technical field, particularly relate to a kind of method that Ballast track high-speed railway foundation bed structure thickness is determined.

Background technology

The load that railway bed bears refers to the stress acted on road bed, comprising track structure weight and act in dead load on road bed and train operation the dynamic load be delivered to by wheel, track on road bed, is the important evidence determining road structure.Along with the increase of train speed in high-speed railway, the continuous dynamic load amplitude that roadbed bears will obviously increase, and significantly change also will occur embankment load feature, thus cause the increase of subgrade deformation.High-speed railway requires more strict to the settlement after construction of roadbed, thus has higher requirement and new challenge to high-speed railway subgrade.Bedding is the most important key position of railway bed, is also stressed the most complicated, the position of most cracky.Thickness of subgrade bed is less than normal, and it is easily damaged that bedding intensity and support stiffness are not enough, reduces the operation life-span of railway and affects the safe operation of train.Thickness of subgrade bed is blocked up, easily causes building railway high cost, and construction efficiency reduces, and forms waste.Therefore need reasonably to analyze dynamic load feature and depth of interaction thereof to determine the thickness of bedding.Existing thickness of subgrade bed defining method committed step mainly contains two steps: what wheel load power born by first clearly adjacent sleeper shares effect; The wheel load power born according to sleeper immediately below wheel again, along the transfer characteristic of the roadbed degree of depth, draws the impact of vehicular load stress on roadbed, and then draws the thickness of foundation bed structure.Its Problems existing is:

One, determine that wheel load power mainly contains following four kinds: Winkler continuous elastic ground Infinite Beam model analyzing computing method at the sharing method of each sleeper; Field measurement and indoor model test method; Numerical analysis model computing method; Load share ratio engineering experience subjunctive.

Wherein, rail need be assumed to the Infinite Beam model be placed on elastic sleeper by Winkler continuous elastic ground Infinite Beam method, and set up the differential equation and carry out analytical Calculation analysis, the method is set up equation and solves calculating all very loaded down with trivial details, the shear stress in ground is have ignored during calculating, thus the diffusion of stress cannot be considered, and the parameter such as rail support stiffness, ballast coefficient, subgrade modulus is not easily determined when calculating, the variation of these parameters simultaneously can affect to result of calculation, makes result of calculation there is larger fluctuation.Field measurement and indoor model test method need be carried out a large amount of field measurements and model testing and be shared wheel load power rule to sleeper and study, and human and material resources and financial resources expend huge, and cost is higher.Numerical analysis method carries out numerical computations by the interaction model setting up rail and sub rail foundation, although the method calculates comparatively accurate, modeling comparison bothers, larger by parameter influence, there is higher requirement to computational resource, be therefore of little use in common engineering application.For empirical analysis method, the empirical model being shared wheel load power by 5 sleepers is often supposed in practical engineering application, the ratio that every sleepers shares wheel load power is respectively 0.1,0.2,0.4,0.2,0.1, although the method is simply clear and definite, its loading sharing numerical value and actual conditions have larger difference.

Two, determine adjacent sleeper bear wheel load power share effect after, the wheel load power born by sleeper immediately below wheel is counted as the certain earth pillar thickness identical with foundation bed structure packed bulk density and the strip load wide with road bed surface by the letter of conversion earth pillar method, by the stress diffusion angle mode of experience, vehicular load is delivered on road bed again, to determine the impact of vehicular load stress on road bed.But traditional method also exists many deficiencies: 1, only considered the Stress transmit bearing wheel load power sleeper immediately below wheel during stress spread, and do not consider adjacently to bear that wheel load power sleeper is oblique diffuses to immediately below wheel that position is on the impact of stress immediately below wheel, its vehicular load stress calculating results is on the low side; Consider the pattern of the sleeper pillow end total cross-section load-bearing when 2, calculating sleeper pillow end average pressure, rest the head on the end with sleeper and stressedly concentrate on that vehicle is left and right takes turns position immediately below rail, the situation that sleeper rests the head on end zone line stressed is not hardly inconsistent.3, because angle of flare size and diffusion way during stress spread need artificial experience to determine, therefore result in Stress transmit to road bed, the difference of road bed load assignment form, road bed load assignment form is unreasonable will cause larger impact to result of calculation, and computational accuracy cannot ensure.

More than analyzing known, for meeting design and the construction needs of high-speed railway subgrade, needing a kind of defining method calculating simple, result Ballast track high-speed railway foundation bed structure thickness accurately of exploitation badly.

Summary of the invention

The object of this invention is to provide a kind of defining method of Ballast track high-speed railway foundation bed structure thickness, the method calculates simple, the foundation bed structure thickness results drawn is more accurate, for the design of Ballast track high-speed railway subgrade, construction and maintenance provide more accurately, foundation reliably, thus both ensured the operation life-span of railway and the safe operation of train, the building of railway, maintenance cost can be reduced again, avoid waste.

The present invention realizes the technical scheme that its goal of the invention adopts, a kind of defining method of Ballast track high-speed railway foundation bed structure thickness, and its step is as follows:

Sharing of A, wheel load power

The revolver that 5 sleepers i below calculating wheel bear carries power P _ia, right wheel load power P _ib:

$P_{\mathrm{ia}}=P_{\mathrm{ib}}=\frac{1}{2}P(1+0.003v)\·e^{-\frac{{(i-3)}^2}{3.125}}/\underset{i=1}{\overset{5}{\mathrm{\Σ}}}{e}^{-\frac{{(i-3)}^2}{3.125}}$

In formula, i is the sequence number of 5 sleepers below wheel, is followed successively by 1,2,3,4,5 from front to back, and wherein sleeper 3 is the sleeper immediately below wheel, and P is the axle weight of wheel, and v is designed driving speed, and e is the truth of a matter of natural logrithm;

B, the average pressure at the pillow end calculate

Calculate the average pressure p that the left half at the end rested the head on by sleeper 3 _3a:

$P_{3a}=\frac{P_{3a}}{l\·b}$

In formula, l is 2 times of sleeper end to rail axis distance, and b is pillow bottom width degree;

The determination of the maximum vehicular load stress of C, sleeper lower position and depth relationship

Local area cornerpoints method is adopted to calculate the average pressure p that the left half at the end rested the head on by sleeper 3 _3ato the vehicular load stress σ that degree of depth z place immediately below its bottom centre produces _z3a,

${\mathrm{\σ}}_{z3a}=\frac{{2p}_{3a}}{\mathrm{\π}}[\frac{2\mathrm{lzb}}{\sqrt{b^2+{4z}^2+l^2}}\·\frac{b^2+l^2+{8z}^2}{(b^2+{4z}^2)(l^2+{4z}^2)}+\arctan \frac{\mathrm{lb}}{2z\sqrt{b^2+{4z}^2+l^2}}];$

Power P is carried according to the revolver that sleeper i bears _ia, sleeper i rests the head on the distance that the left half center at the end rested the head on by the left half center at the end and sleeper 3, revolver that sleeper i bears carries power P to adopt vertical point load mode computation to go out _iathe vehicular load stress σ that immediately below the left half center of sleeper 3 being rested the head on the end of to, degree of depth z place produces _zia, and i ≠ 3, wherein d is adjacent two sleeper pitches;

According to the right wheel load power P that sleeper i bears _ib, sleeper i rests the head on the distance that the left half center at the end rested the head on by the right half center at the end and sleeper 3, the right wheel load power P adopting vertical point load mode computation to go out sleeper i to bear _ibthe vehicular load stress σ that immediately below the left half center of sleeper 3 being rested the head on the end of to, degree of depth z place produces _zib, ${\mathrm{\σ}}_{\mathrm{zib}}=\frac{3}{2}\·\frac{P_{\mathrm{ib}}}{\mathrm{\π}}\·\frac{z^3}{{[{(3-i)}^2{d}^2+{1.435}^2+z^2]}^{5/2}},$ Wherein 1.435 is standard gauge;

By above all vehicular load stress σ _z3a, σ _zia, σ _zibsummation obtains the gross vehicle load stress σ at degree of depth z place immediately below left half center that sleeper 3 rests the head on the end _za, the gross vehicle load stress σ at this place _zabe the maximum vehicular load stress of the same degree of depth z each position in side under sleeper pillow;

The dead load stress q of degree of depth z immediately below the left half center that the end rested the head on by D, the sleeper 3 obtained immediately below wheel _za, q _za=q _c+ γ _m(z-c); In formula, q _cfor selfweight stress, the γ of track structure _mfor foundation bed structure bankets unit weight, c is thickness of ballast bed, and z>=c;

The determination of E, foundation bed structure thickness

The gross vehicle load stress σ at degree of depth z place immediately below the left half center of resting the head on the end according to sleeper 3 _zaand dead load stress q _zaobtain foundation bed structure thickness, that is: make solve this equation, obtain the root z of degree of depth z ₀, by root z ₀deduct the thickness H that thickness of ballast bed c is foundation bed structure, i.e. H=z ₀-c.

Principle of the present invention is:

In continuous elastic supporting beam model, according to Winkler supposition, reaction at basement is directly proportional to displacement.When to calculate under wheel load effect stressed, only need consider the wheel impact in certain distance, single-wheel is carried masterpiece and is generally shared by 5 sleepers for time directly over sleeper.Meanwhile, according to calculating gained steel rail displacement deformation curve and rail M curve, on pillow, pressure history is identical with steel rail displacement deformation curve.Applicant finds: on pillow, the form of pressure history and normal distyribution function curve have similitude, can be similar to employing normal distyribution function describe subgrade reaction breadth coefficient, the distance of xi to be sequence number the be pitch of sleepers sleeper 3 of i, its normal distyribution function form as shown in Figure 1.Subgrade reaction within the scope of the sleeper pitch of both sides 1/2, sleeper center is born by this sleeper, and the present invention is similar to the subgrade reaction breadth coefficient p (x adopting single-wheel to carry sleeper position in the longitudinal influence basin of power _i) bear wheel load power with the product of sleeper pitch d as this sleeper initially share ratio cc _i=d p (x _i).

Single-wheel is carried masterpiece and is about 95.45%, with p (x for the wheel load force rate value that time directly over sleeper is born within the scope of 5 times of sleeper pitches _i) integrated value of function in (-2 σ ,+2 σ) scope be identical, so p (x _i) form parameter σ should meet 4 σ=5d, i.e. σ=1.25d in function.

Because each sleeper supporting force initially shares ratio cc _isummation does not meet vertical force equilibrium conditions, need to α _imethod of weighting is carried out load inversion and is obtained each sleeper and share the final ratio of wheel load power in proportion

The high-speed railway driving wheel power of carrying equals quiet wheel load power and is multiplied by the coefficient of impact (1+0.003v), then the revolver that each sleeper is born carries power P _ia, right wheel load power P _ibfor $P_{\mathrm{ia}}=P_{\mathrm{ib}}=\frac{1}{2}P(1+0.003v)\·e^{-\frac{{(i-3)}^2}{3.125}}/\underset{i=1}{\overset{5}{\mathrm{\Σ}}}{e}^{-\frac{{(i-3)}^2}{3.125}}.$

Under actual conditions, wheel load power born by sleeper is not that sleeper pillow end total cross-section is stressed, but bearing load at the bottom of the pillow of effective area, generally with 2 times of sleeper end to rail axis distance for long, sleeper pillow bottom width degree is wide pillow floor space bearing load, and sleeper pillow end central area does not then bear load.Therefore, the average pressure of the left half at the end rested the head on by sleeper 3

Vehicular load stress from the bottom of sleeper, consider during calculating wheel load do the sleeper of use numbering i=3 and adjacent 5 sleepers totally 10 loads bear the impact in region.Be positioned at immediately below the sleeper and rail crossover location that wheel load makes use according to the dynamic stress distribution maximum value of existing data and the known road bed of field measurement and foundation bed structure thereof.Therefore, when calculating vehicular load stress transmits along the degree of depth, first adopt local area cornerpoints method to calculate the average pressure p that the left half at the end rested the head on by sleeper 3 _3ato the vehicular load stress σ that depth location z immediately below its bottom centre produces _z3a, consider that vertical point load revolver that all the other sleepers are born carries power and right wheel load power rests the head on to sleeper 3 the vehicular load stress σ that immediately below the left half center at the end, depth location z produces simultaneously _zia, σ _zib, finally obtain depth location z and gross vehicle load stress σ immediately below left half center that sleeper 3 rests the head on the end _zarelational expression the gross vehicle load stress σ at this place _zabe the maximum vehicular load stress of the same degree of depth each position in side under sleeper pillow.

Train dynamic stress reaches road bed by track structure, then decays gradually along the degree of depth.Generally the larger part of dynamic stress impact is defined as subgrade bed.The Dynamic triaxial test of compacted soil shows, when the ratio of dynamic and static stress is below 0.2, loads the accumulative distortion of the plasticity produced for 100,000 times below 0.2%, and reaches stable very soon.Therefore, the present invention adopts dynamic and static stress than being 0.2 as the foundation determining thickness of subgrade bed.

Compared with prior art, the invention has the beneficial effects as follows:

One, applicant finds, the single-wheel that Winkler continuous elastic ground Infinite Beam model calculates is carried the vertical deflection curve form of rail and subgrade reaction distribution curve form under power effect and is all had this feature of similitude with normal distyribution function tracing pattern, thus gives based on normal distyribution function expression the Analytic Calculation Method that a kind of sleeper bears wheel load power.This Analytic Calculation Method comparatively Winkler continuous elastic ground Infinite Beam model analyzing computing method and numerical analysis model computing method simplifies greatly, and what wheel load power born by its adjacent sleeper determined shares ratio precision again far above load share ratio engineering experience subjunctive, its computational methods are simple simultaneously, and result accurately, reliably.

Two, determine adjacent sleeper bear wheel load power share effect after, concentrated on immediately below side (left side) wheel the sleeper force part immediately below wheel, and the zone line of sleeper does not bear power, the actual loading situation of resting the head on the end with sleeper more meets.When considering to rest the head on the wheel load power going down born side, the end (left side) immediately below wheel, both considered the vertical diffusion that wheel load power born by this sleeper pillow end (left side), have also contemplated that this sleeper pillow end opposite side (right side) was born the oblique of wheel load power and diffused to the impact that immediately below this sleeper pillow end (left side) center, position is brought; Also contemplate oblique this sleeper that diffuses to of the wheel load power born position, adjacent sleeper left and right side pillow bottom simultaneously and rest the head on the impact that immediately below the end (left side) center, position is brought.Immediately below its side determined (left side) wheel, the result of the gross vehicle load stress at depth location place and actual conditions more meet.Vehicular load stress is counted from the bottom of sleeper, comparatively conventional method first by stress spread to road bed, then by the downward calculation from road bed of road bed load assignment form, mechanical analysis is more clear and definite, result of calculation more accurately, reliable.

It is more accurate that visible the inventive method calculates foundation bed structure thickness results that is simple, that draw, more accurate, reliable foundation can be provided for the design of Ballast track high-speed railway foundation bed structure, construction and maintenance, thus both ensured the operation life-span of railway and the safe operation of train, the building of railway, maintenance cost can be reduced again, avoid waste.

Below in conjunction with the drawings and specific embodiments, the present invention will be further described in detail.

Accompanying drawing explanation

Fig. 1 be the inventive method wheel load masterpiece for directly over sleeper time sleeper share wheel load power model computational analysis schematic diagram.

Fig. 2 is the left and right part bearing area schematic diagram bearing load in the inventive method sleeper pillow end.

Fig. 3 is the distribution schematic diagram of the inventive method vehicular load at roadbed.

Detailed description of the invention

Embodiment

Fig. 1-3 illustrates, a kind of detailed description of the invention of the present invention is, a kind of defining method of Ballast track high-speed railway foundation bed structure thickness, and its step is as follows:

Sharing of A, wheel load power

The revolver that 5 sleepers i below calculating wheel bear carries power P _ia, right wheel load power P _ib:

$P_{\mathrm{ia}}=P_{\mathrm{ib}}=\frac{1}{2}P(1+0.003v)\·e^{-\frac{{(i-3)}^2}{3.125}}/\underset{i=1}{\overset{5}{\mathrm{\Σ}}}{e}^{-\frac{{(i-3)}^2}{3.125}}$

In formula, i is the sequence number of 5 sleepers below wheel, is followed successively by 1,2,3,4,5 from front to back, and wherein sleeper 3 is the sleeper immediately below wheel, and P is the axle weight of wheel, and v is designed driving speed, and e is the truth of a matter of natural logrithm;

B, the average pressure at the pillow end calculate

Calculate the average pressure p that the left half at the end rested the head on by sleeper 3 _3a:

$P_{3a}=\frac{P_{3a}}{l\·b}$

In formula, l is 2 times of sleeper end to rail axis distance, and b is pillow bottom width degree;

The determination of the maximum vehicular load stress of C, sleeper lower position and depth relationship

Local area cornerpoints method is adopted to calculate the average pressure p that the left half at the end rested the head on by sleeper 3 _3ato the vehicular load stress σ that degree of depth z place immediately below its bottom centre produces _z3a,

${\mathrm{\σ}}_{z3a}=\frac{{2p}_{3a}}{\mathrm{\π}}[\frac{2\mathrm{lzb}}{\sqrt{b^2+{4z}^2+l^2}}\·\frac{b^2+l^2+{8z}^2}{(b^2+{4z}^2)(l^2+{4z}^2)}+\arctan \frac{\mathrm{lb}}{2z\sqrt{b^2+{4z}^2+l^2}}];$

Power P is carried according to the revolver that sleeper i bears _ia, sleeper i left half center and the sleeper 3 of resting the head on the end rest the head on the distance at the left half center at the end, adopt vertical point load mode computation to go out revolver that sleeper i bears carries the vehicular load stress σ of degree of depth z place generation immediately below power Pia rests the head on end left half center to sleeper 3 _zia, and i ≠ 3, wherein d is adjacent two sleeper pitches;

According to the right wheel load power P that sleeper i bears _ib, sleeper i rests the head on the distance that the left half center at the end rested the head on by the right half center at the end and sleeper 3, the right wheel load power P adopting vertical point load mode computation to go out sleeper i to bear _ibthe vehicular load stress σ that immediately below the left half center of sleeper 3 being rested the head on the end of to, degree of depth z place produces _zib, ${\mathrm{\σ}}_{\mathrm{zib}}=\frac{3}{2}\·\frac{P_{\mathrm{ib}}}{\mathrm{\π}}\·\frac{z^3}{{[{(3-i)}^2{d}^2+{1.435}^2+z^2]}^{5/2}},$ Wherein 1.435 is standard gauge;

By above all vehicular load stress σ _z3a, σ _zia, σ _zibsummation obtains the gross vehicle load stress σ at degree of depth z place immediately below left half center that sleeper 3 rests the head on the end _za, the gross vehicle load stress σ at this place _zabe the maximum vehicular load stress of the same degree of depth z each position in side under sleeper pillow;

The dead load stress q of degree of depth z immediately below the left half center that the end rested the head on by D, the sleeper 3 obtained immediately below wheel _za, q _za=q _c+ γ _m(z-c); In formula, q _cfor selfweight stress, the γ of track structure _mfor foundation bed structure bankets unit weight, c is thickness of ballast bed, and z>=c;

The determination of E, foundation bed structure thickness

The gross vehicle load stress σ at degree of depth z place immediately below the left half center of resting the head on the end according to sleeper 3 _zaand dead load stress q _zaobtain foundation bed structure thickness, that is: make solve this equation, obtain the root z of degree of depth z ₀, by root z ₀deduct the thickness H that thickness of ballast bed c is foundation bed structure, i.e. H=z ₀-c.

Provide the deterministic process and the result that adopt the example method to have tiny fragments of stone, coal, etc. high-speed railway foundation bed structure thickness to one below.

The present embodiment high speed railway Ballast track structural parameters are 60kg/m rail, and tread width is 73mm; Type III reinforced concrete sleeper, adjacent two sleeper pitches are 0.6m, length of sleeper 2600mm, sleeper pillow bottom width degree 320mm; Gauge is 1435mm; Ballast bed structure rubble railway ballast unit weight γ _g=17.5kN/m ³, thickness of ballast bed c=0.35m.Roadbed Soil unit weight γ _m=20.5kN/m ³.Design vehicle axle heavy P=200kN, designed driving speed v=300km/h.

The formula walked by A according to above parameter obtains the revolver that each sleeper bears and carries power P _ia, right wheel load power P _ib, the results are shown in Table listed by 1.

The revolver that each sleeper of table 1 is born carries power P _iaand right wheel load power P _ib

Sleeper i	1	2	3	4	5
						Revolver carries power P ia（kN）	17.556	45.866	63.156	45.866	17.556
Right wheel load power P ib（kN）	17.556	45.866	63.156	45.866	17.556
						Actual measurement wheel load power P i（kN）	17.1	41.8	64.6	41.8	17.1

Absolute error | P ia-P i|（kN）	0.456	4.066	1.444	4.066	0.456
						Relative error | P ia-P i|/P i	2.67%	9.73%	2.24%	9.73%	2.67%

Known by table 1, the wheel load power that the determined each sleeper of the present invention is born and measured result comparatively close, absolute error within the scope of 0.456 ~ 4.066kN, average is only 2.0976kN, relative error between 2.24% ~ 9.73%, average is about 5.4046%, can meet engineer applied demand.

And then the formula to be walked by B obtains the average pressure p that the left half at the end rested the head on by sleeper 3 _3a=180.73kPa.

Again by above parameter and table 1 data through C, D two step computing obtain degree of depth z position and gross vehicle load stress σ immediately below left half center that sleeper 3 rests the head on the end _zarelation and with dead load stress q _zarelation.The wherein selfweight stress q of track structure itself _c=17.6kPa, q _za=17.6+20.5 (z-0.35).Finally, make solve this equation, obtain root z ₀=2.95m, H=2.95-0.35=2.6m, then this Ballast track high-speed railway foundation bed structure thickness H=2.6m.

Visible, adopt the inventive method can according to the thickness of different classification of track and carrier performance determination foundation bed structure, foundation bed structure thickness fixing in the relevant roadbed specification of contrast, its form of structure tallies with the actual situation more, economic performance is more superior.

Claims (1)

1. a defining method for Ballast track high-speed railway foundation bed structure thickness, its step is as follows:

Sharing of A, wheel load power

The revolver that 5 sleepers i below calculating wheel bear carries power P _ia, right wheel load power P _ib:

$P_{ia}=P_{ib}=\frac{1}{2}P(1+0.003v)\·e^{-\frac{{(i-3)}^2}{3.125}}/\underset{i=1}{\overset{5}{\Σ}}{e}^{-\frac{{(i-3)}^2}{3.125}}$

In formula, i is the sequence number of 5 sleepers below wheel, is followed successively by 1,2,3,4,5 from front to back, and wherein the 3rd sleeper is the sleeper immediately below wheel, and P is the axle weight of wheel, and v is designed driving speed, and e is the truth of a matter of natural logrithm;

B, the average pressure at the pillow end calculate

Calculate the average pressure p of the left half at the 3rd sleeper pillow end _3a:

$p_{3a}=\frac{P_{3a}}{l\·b}$

In formula, l is 2 times of sleeper end to rail axis distance, and b is pillow bottom width degree;

The determination of the maximum vehicular load stress of C, sleeper lower position and depth relationship

Local area cornerpoints method is adopted to calculate the average pressure p of the left half at the 3rd sleeper pillow end _3ato the vehicular load stress σ that degree of depth z place immediately below its bottom centre produces _z3a, ${\mathrm{\σ}}_{z3a}=\frac{2p_{3a}}{\mathrm{\π}}\[\frac{2lzb}{\sqrt{b^2+4z^2+l^2}}\·\frac{b^2+l^2+8z^2}{(b^2+4z^2)(l^2+4z^2)}+arctan\frac{lb}{2z\sqrt{b^2+4z^2+l^2}}\];$

Power P is carried according to the revolver that sleeper i bears _ia, sleeper i rests the head on the distance that the left half center at the end rested the head on by the left half center at the end and the 3rd sleeper, revolver that sleeper i bears carries power P to adopt vertical point load mode computation to go out _iato the vehicular load stress σ that degree of depth z place immediately below the left half center at the 3rd sleeper pillow end produces _zia, and i ≠ 3, wherein d is adjacent two sleeper pitches;

According to the right wheel load power P that sleeper i bears _ib, sleeper i rests the head on the distance that the left half center at the end rested the head on by the right half center at the end and the 3rd sleeper, the right wheel load power P adopting vertical point load mode computation to go out sleeper i to bear _ibto the vehicular load stress σ that degree of depth z place immediately below the left half center at the 3rd sleeper pillow end produces _zib, ${\mathrm{\σ}}_{zib}=\frac{3}{2}\·\frac{P_{ib}}{\mathrm{\π}}\·\frac{z^3}{{\[{(3-i)}^2{d}^2+{1.435}^2+z^2\]}^{5/2}},$ Wherein 1.435 is standard gauge;

By above all vehicular load stress σ _z3a, σ _zia, σ _zibthe gross vehicle load stress σ at degree of depth z place immediately below the left half center that summation obtained for the 3rd sleeper pillow end _za, the gross vehicle load stress σ at this place _zabe the maximum vehicular load stress of the same degree of depth z each position in side under sleeper pillow;

D, obtain the 3rd sleeper pillow end immediately below wheel left half center immediately below the dead load stress q of degree of depth z _za, q _za=q _c+ γ _m(z-c); In formula, q _cfor selfweight stress, the γ of track structure _mfor foundation bed structure bankets unit weight, c is thickness of ballast bed, and z>=c;

The determination of E, foundation bed structure thickness

According to the gross vehicle load stress σ at degree of depth z place immediately below the left half center at the 3rd sleeper pillow end _zaand dead load stress q _zaobtain foundation bed structure thickness, that is: make solve this equation, obtain the root z of degree of depth z ₀, by root z ₀deduct the thickness H that thickness of ballast bed c is foundation bed structure, i.e. H=z ₀-c.