Bar optimal design with the use of out-form

The paper describes steel girders optimal design. Steel girders optimal design is based on the strength theory introduced by M. Huber, R. Mises, H. Hencky and on the theory of mirror functions. With the help of out-form volume we can either calculate the optimal form for a girder with variable cross-section or the optimal height for a girder of uniform crosssection. The paper introduces girders calculations according to the authors' methodology. For use this methodology effect of steel saving is about 1020% compared with a girder or a beam with uniform cross-section. An example of determining optimal shape of a compressed-bend girder the paper describes.


Introduction
There are several well-known methods of building structures optimal design [1,2,5,6,7,8] that can be applied while designing building structures cross-beams or girders.In this paper, the authors present an example of determining optimal shape of a compressed-bend.The paper introduces girders calculations according to the authors' methodology.

Bar optimal design with the use of out-form
According to theory [3], strength in this case is provided on condition that: For a plane stress problem, direct stress in i point of the bar can be determined from the formula: Specific potential energy of the shape change in i point of the bar is As a result of transformations we can introduce a new equation from the strength condition.This equation is mathematically determined as non-linear modular form i bar capacity [4][5][6]: where Shape coefficients αi и βi are cross sections accessible characteristics and can be easily derived from gauge material or calculated.With given cross-sectional height, cross-section form coefficients α and β are taken as constant.
For a statically indeterminate bar with uniform cross-section the volume function can be presented as it follows: In general terms, Formula 5 can be given as where If this statically determinate bar is variable in its cross-section along its length, we can find the optimal shape or height for a uniform cross-section bar while using the form coefficient α and β.For that we need to find the dependence α and β from the crosssectional height h [9].If we take I-beams (according to All Union State Standard -GOST -8239-89) and compare all the coefficients α and β for outer fibers, fibers on flange and wall junction and neutral fibers we'll see that there is an approximate hyperbolic dependence between 3 and α and linear dependence between h and β (as Fig. 1 shows).
Paper [7] introduces statistical research proving that the radius of cross-section gyration of a defined height h-const in flexural and compressed flexural elements does not depend on their other characteristics if their construction and calculation constraints are taken into account.The radius of gyration changes only if cross-sectional height is changed.We can see almost the same dependence in respect of the width of (flange) section b.Tables 1 and 2 show α, β coefficients for rolled sections of different types.
Cross-section form coefficients for rolled sections, fabricated sections of simple and complex types, U-sections, etc can be approximate by functions like the following: For example, the following approximations correspond to wide-flange I-beam gauge: Let us put functions (10) into the strength condition (3).Here we get a modular form of the volume as regard to h in k point of the bar.
For a part of the bar with d z length the elementary capacity can be presented as an outform for n fiber points:

An example of determining optimal shape of a compressedbend girder
Let us analyze Figure 2, which shows the process of determining optimal shape of a hingesupported compressed-bend girder of a hydraulic gate [11,12,13]  Axial force in a hinge-supported girder is usually applied on the pier level.As Force N for a beam with uniform cross-section changes the bending moment in the middle of each cross-section, let us determine cross-sectional height in the middle of the span according to strength condition:  (14) Gauge material for such a moment corresponds to cross-section 60SH2, with W x =4490cm 3 .
From the condition of deflection limitation in the middle equal to 1/400L=3 cm we have:  3.Here we use gauge material according to GOST 26020-83.Points 1, 2, 3 consequently correspond to outer fibers, fibers on flange and wall junction and neutral fibers.We take into account the symmetry of the girder and analyze cross-sections spaced at intervals of 2 m.
1) Optimal design of a hinged beam with variable cross-section starts with determining its bearing cross-section.In bearing cross-section 1 we have N x =100 kN; M x =N x •h/2=100•h/2; Q y =300 kN.Let us take h=30cm cross-sectional height.Then M x =100•15=1500 kN•cm.The area we need is calculated according to equations (4).We take h=30cm and area F 1 =79.78cm 2 along neutral fiber where both shearing force and axial force are quite strong.

Conclusion
In real life, the construction also requires the account of the cross-section flanges and walls local buckling resistance, bending limitations, limitations connected with width, depth and size unification of flanges and walls.Still if we take all these limitations into consideration, the whole effect of steel saving is about 10-20% compared with a girder or a beam with uniform cross-section.

Table 3 .
Cross-section form coefficients for the line h.