RU2336397C1 - Metal whole beam - Google Patents

Abstract

FIELD: construction.

SUBSTANCE: metal whole beam of rectangular, box or double-T cross-section includes beam itself made of rolled or fabricated sections, support devices manufactured according to hinge support or rigid fixing of its ends scheme. The beam is made stepped-variable rigid, at that sections with higher bending rigidity are located under epure of bending moments with extreme ordinates and rational ratios of section lengths and their bending stiffness are selected from condition of equality of absolute values of allowed normal stresses in cross-sections with maximal and minimal values of bending moments.

EFFECT: reduction of metal whole beam materials consumption due to change of its bending stiffness along length and bending stiffness of there sections.

5 cl, 5 dwg

Description

The invention relates to the field of construction and is intended for use as load-bearing elements of the overlap of buildings and structures.

Known metal single-span (or multi-span) continuous beam of constant cross-section (rectangular, box-shaped or double-tee), including the beam itself, made of rolled or welded profiles, support devices made according to the scheme of articulated support or rigid jamming of its ends, and intermediate articulated movable support devices in the case of a multi-span beam [1, p.114-116, 120-122].

The disadvantage of such a beam is uneconomical due to overspending of the metal, since most of it remains under-stressed due to the uneven distribution of bending moments along the span (s) of the beam.

The problem to which the invention is directed, is to reduce the material consumption of a continuous metal beam by changing its bending stiffness along the length in individual sections and a rational choice of the ratio of lengths and bending stiffnesses of these sections.

This is achieved by the fact that the metal continuous beam of rectangular, box-shaped or I-sections, including the beam itself, made of rolled or welded profiles, supporting devices made according to the scheme of articulated bearing or rigid jamming of its ends, is made of stepwise variable stiffness, while sections with greater bending stiffness are located under the plot of bending moments with extreme ordinates, and rational ratios of the lengths of sections and their bending stiffnesses are selected from the condition avenstva absolute values permitted by normal stress in sections with the highest and lowest values of the bending moments.

где М_max и М_min - экстремальные значения изгибающих моментов в наиболее опасных сечениях балки. При этом в случае действия по всей длине балки равномерно распределенной нагрузки рациональные соотношения длин участков k и их изгибных жесткостей n определяют с помощью аппроксимирующих функций, построенных по результатам точного аналитического расчета:In the case of equal height and symmetry of all sections of the continuous metal beam relative to their neutral axis, the ratio of the bending stiffnesses of adjacent sections of the beam n is determined by the formula

where M _max and M _min - extreme values of bending moments in the most dangerous sections of the beam. Moreover, in the case of a uniformly distributed load acting along the entire length of the beam, rational ratios of the lengths of sections k and their bending stiffness n are determined using approximating functions constructed from the results of an exact analytical calculation:

- for a single-span beam, one end of which is rigidly pinched, and the other pivotally supported, according to the formula

- for rigidly clamped at the ends of a single-span beam according to the formula

- for a two-span beam with equal spans and articulation of the end and intermediate supports according to the formula

n (k) = - 7.9788 · k ³ + 5.3285 · k ² -0.6077 · k + 0.5816.

The invention is illustrated by drawings, presented in figure 1 ... 5.

Figure 1 shows a single-span continuous beam of constant bending stiffness, one end of which is rigidly pinched, and the other pivotally supported, with possible variants of cross sections.

Figure 2 shows variants of continuous beams of stepwise variable stiffness: in scheme a), a beam, one end of which is rigidly pinched, and the other pivotally supported; on the scheme b) - a beam with two rigidly clamped ends; on the diagram c) - pivotally supported two-span beam with equal spans. All schemes show the ratio of the lengths of sections with different bending stiffnesses and the ratio of these stiffnesses.

Figure 3 graphically presents the stages of the exact solution of the problem of determining bending moments in a single-span beam of stepwise variable stiffness, one end of which is rigidly clamped and the other is articulated.

Figure 4 shows the final diagrams of bending moments in three continuous beams of stepwise variable stiffness: in scheme a) - a beam, one end of which is rigidly clamped, and the other is articulated; on the scheme b) - a beam with both rigidly clamped ends; in diagram c) - two-span articulated beam with equal spans.

Figure 5 presents graphs of approximating functions n (k) for the selection of the optimal ratios of the lengths of sections k and their bending stiffness n for all three beam schemes under consideration.

The continuous metal beam (Fig. 1) includes the beam 1 itself, made of rolled or welded profiles, supporting devices 2 and 3, made according to the scheme of articulated bearing 3 or rigid pinching 2 of its ends.

The proposed design of continuous beam of stepwise variable stiffness differs from the beam of constant cross section in that it has two sections with different bending stiffnesses. In the area with the greatest bending moments, the cross-sectional area is taken larger than in the area with the lower bending moments. The implementation of these design features of continuous beams leads to a more economical consumption of material. It is only necessary to choose rational ratios of the lengths of sections with different bending stiffnesses k and the ratio of these stiffnesses n.

Let us consider the theoretical solution to the problem of selecting the optimal ratios n and k using the example of a continuous beam of stepwise variable stiffness, one end of which is rigidly clamped and the other is articulated, loaded with a uniformly distributed load of intensity q.

Divide the beam into two characteristic sections (figure 3, scheme a). The bending stiffness of the section from the side of the hard seal is denoted by EI (E is the elastic modulus of the material, I is the moment of inertia of the section), and the other section is nEI, where n is the variable parameter determined by the calculation. The length of the section from the side of hard termination is denoted by kl, where k is a parameter also determined by calculation.

To solve this problem, we use the displacement method [2] and write the system of canonical equations for the main system shown in Fig. 3 (scheme b)

Hereinafter, the notation generally accepted in structural mechanics is used [2]. To determine the coefficients and free terms of these equations, we construct the diagrams of the bending moments of single and load states (Fig. 3, schemes c, d, e). Cutting the corresponding nodes of the beam from the diagrams of bending moments of single and load states, from the equilibrium conditions of these nodes (Fig. 3, scheme e) we find the coefficients and free terms in equations (1):

We substitute these expressions into equations (1):

Solving the resulting system of equations, we find:

To build the final plot of bending moments (figure 3, scheme g), you should calculate the bending moments in the characteristic sections of the beam according to the formula:

The extreme ordinates in this moment plot will be in sections 1-1 and 5-5:

We write the condition for the optimal design of the beam in the form of the equality of the largest normal stresses in sections 1-1 and 5-5:

Considering that σ = M / W and W = 2I / h, where W is the moment of resistance of the section, h is its height, this condition will take the following form:

With a constant cross-sectional height h and the accepted stiffness ratio for characteristic sections, expression (8) can be rewritten as

Substituting this equality into the corresponding expressions for bending moments in sections 1-1 and 5-5 after transformations, we obtain:

This formula is inconvenient to use; therefore, using the exact values of n and k obtained by formula (10), an approximating function is constructed

on the domain of definitions n∈ (0.3; 0.55), which with an error of up to 1.5% satisfies the exact solution by formula (10).

The graph of function (11) is built in Fig. 5. The physical meaning of this graph is that for each of its points with abscissa k and ordinate n, stresses in sections with maximum and minimum values of bending moments with corresponding ratios of bending stiffness and lengths of sections will be equal.

By the formula (11) or the graph shown in Fig. 5, given the value of the parameter k, you can find the value of the parameter n and, conversely, by the parameter n - find the parameter k. These relations will be the most rational, since they follow from the exact analytical solution of the problem under consideration.

The same calculations were performed for a single-span beam with rigidly clamped ends and for a two-span articulated beam with equal spans. The following results were obtained

- for a single-span beam:

- for a two-span beam with equal spans:

The analytical expression of the bending moment M ₅ in both cases was found by minimizing the function M, written in general form in the considered section.

An example implementation of the invention. Since the order of selection of rational ratios of the parameters n and k is the same for all three beam schemes under consideration, we consider only one example of the invention for a single-span metal beam, one end of which is rigidly clamped and the other is articulated (Fig. 3, scheme a).

For the metal beam made from Art. 3, the following initial data are accepted: the main beam section is made of two channels No. 30C according to GOST 8240-97 (I = 2 · 6045 = 12090 cm ⁴ = 1.209 · 10 ^-4 m ⁴ , W = 2 · 403 = 806 cm ³ = 8.06 · 10 ^-4 m ³ ), beam length l = 6 m, modulus of elasticity of the material E = 2 · 10 ⁵ MPa; allowable stress [σ] = 30 MPa. Shelves of channels are located outward (figure 2, section 2-2), the distance between the walls of the channels is 250 mm On the reinforced section of the beam at the top and bottom between the channels, metal strips 28 mm thick and 250 mm wide are welded (Fig. 2, section 1-1).

The moment of inertia of the obtained composite section is found by the formula:

I = 2I ₁ + 2I ₂ = 2I ₁₀ + 2I ₂₀ + 2A ₂ · a ² ,

where 2I ₁₀ = 1,209 · 10 ^-4 m ⁴ - the moment of inertia of two channels No. 30C (taken from the assortment); 2I ₂₀ - moment of inertia of two bands relative to their horizontal central axis:

A ₂ - sectional area of the strip; and - the distance between the horizontal central axes of the entire section and strip:

2A ₂ · a ² = 2 · 0.25 · 0.028 · 0.136 ² = 2.589 · 10 ^-4 m ⁴ ;

I = 1.209 · 10 ^-4 + 0.0092 · 10 ^-4 + 2.589 · 10 ^-4 = 3.8072 · 10 ^-4 m ⁴ .

Find the ratio of the moments of inertia of the main and composite sections:

From this value of the parameter n using the graph in figure 5 (scheme a) we find the value of the parameter k = 0.26. Given these parameters, using formulas (5), (6), we find the values of extreme bending moments:

Substituting the obtained values of M ₁ and M ₅ into expression (9), we verify the equality of normal stresses in the corresponding sections:

The maximum normal stress in section 5-5 should satisfy the condition of strength σ = M ₅ / W ₅ ≤ [σ], where W ₅ is the resistance moment of two channels No. 30С found above. Then we find the allowable bending moment from the inequality M ₅ ≤ [σ] · W ₅ . Expressing M ₅ from expression (6) by substituting the parameters n and k into it and equating it to the permissible value, we find the intensity of the uniformly distributed load q:

For further calculations, we take q = 12.36 kN / m.

Let us compare the material consumption of a beam of variable stiffness with a beam of constant stiffness, perceiving a load q = 12.36 kN / m.

The maximum bending moment in a beam of constant cross section is determined by the formula

and the required moment of resistance of the section is according to

According to GOST 8239-89, we take two I-beams No. 40, W = 1906 cm ³ , linear mass m = 57 kg.

Compare the material consumption of the options:

- the mass of the cross section composed of two channels No. 30C and two strips 0.26 · l = 0.26 · 6 = 1.56 m long, 0.25 m wide and 0.028 m thick

m = 2 · 6 · 34.44 + 2 · 1.56 · 54.95 = 584.73 kg;

- the mass of two I-beams No. 40

m = 2 · 6 · 57 = 684 kg.

Material saving is

Thus, the technical result - the reduction of material consumption of metal beams - is achieved due to their amplification in the most dangerous sections. Moreover, rational ratios of the bending stiffnesses of beams in adjacent sections and the lengths of these sections are determined by the analytical dependences obtained in the present invention.

Information sources

1. Vasiliev A.A. Metal constructions. - M .: Stroyizdat, 1975 .-- 424 p.

2. Snitko N.K. Structural mechanics. - M.: Higher School, 1972. - 488 p.

Claims (5)

1. Metal continuous beam of rectangular, box-shaped or double-tee section, including the beam itself, made of rolled or welded profiles, support devices made according to the scheme of articulated bearing or rigid jamming of its ends, characterized in that the beam is made of stepwise variable stiffness, while sections with greater bending stiffness are located under the plot of bending moments with extreme ordinates, and rational ratios of the lengths of sections and their bending stiffnesses are selected from Via equal absolute values permitted by normal stress in sections with the highest and lowest values of the bending moments. 2. The metal continuous beam according to claim 1, characterized in that for equal height and symmetry of all sections of the beam relative to their neutral axis, the ratio of the bending stiffnesses of the sections is determined by the formula n = M _max / | M _min |, where M _max and M _min - extreme values of bending moments in the most dangerous sections of the beam. 3. The metal continuous beam according to claim 2, characterized in that in the case of rigid jamming of one end of the beam and articulated support of the other and the action of the load uniformly distributed over its entire length, the rational ratios of the lengths of the sections and their bending stiffnesses are determined by the formula

where k is the ratio of the lengths of adjacent sections of the beam. 4. The metal continuous beam according to claim 2, characterized in that in the case of hard jamming of both ends of the beam and the action of the load uniformly distributed over its entire length, the rational ratios of the lengths of the sections and their bending stiffnesses are determined by the formula

where k is the ratio of the lengths of adjacent sections of the beam. 5. The metal continuous beam according to claim 2, characterized in that in the case of a pivotally supported two-span beam with equal spans and the action of the load uniformly distributed along its entire length, the rational ratios of the lengths of the sections and their bending stiffnesses are determined by the formula n (k) = - 7.9788K ³ + 5.3285K ² -0.6077K + 0.5816, where k is the ratio of the lengths of adjacent sections of the beam.

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