RU2725340C1 - C-shaped curved closed profile with perforated wall - Google Patents

Abstract

FIELD: construction.SUBSTANCE: invention relates to construction, namely to profiles of rod elements of buildings. In the C-shaped curved closed profile, the perforated wall and the tubular flanges are conjugated to each other by means of tooth short-circuiting of the longitudinal edges and mutual support in contact zones of its two sheet workpieces. Inner face of upper flange and inner face of lower flange in its cross section have the shape of round semi-ring with diameter equal to width of flanges, and the inner and outer edges of the wall throughout the vertical sections between the flanges are provided with cutouts with toothed faults of the edges along the entire perimeter of each of said cutouts.EFFECT: technical result is higher stability of profile.1 cl, 7 dwg

Description

The proposed technical solution relates to the field of construction and can be used as rod and beam elements in the development of load-bearing structures of buildings and structures for various purposes. In particular, these can be waist and lattice elements of coating trusses, floor beams, wall beams or roof runs.

A well-known technical solution can be attributed to the C-shaped thermal profile of the cross section with wall perforation along the entire length from four to eight rows, which allows to reduce thermal conductivity along the profile by 70 ... 80% when compared with the same profile without perforation [V. Chernoivan , Chernoivan N.V., Horovets V.V., Chernoivan A.V. The construction and reconstruction of residential buildings using light steel thin-walled structures (LSTK) - Bulletin of the Brest State Technical University, 2018, No. 1. - S. 115-118]. However, multi-row perforation, reducing the thermal conductivity and self-weight (weight) of the structure, negatively affects its local stability and bearing capacity.

Another well-known technical solution is a modified Atlant profile, the distinguishing feature of which is the presence of reinforced trapezoidal cutouts on the wall [1. Kashevarova G.G., Kosykh P.A. Comparative analysis of the results of field and numerical experiments to determine the ultimate bearing capacity of thin-walled profiles "Atlant". - International Journal for Computational Civil and Structural Engineering, 14 (3) 50-58 (2018); 2. Torokhova Ya.B. Frame-monolithic building structure "Atlas". - Patent No. 188669, 04/19/2019, bull. No. 11]. Such cuts reduce the susceptibility to local loss of stability and the adverse effect of thermal conductivity, but complicate the geometry of the profile with a cross-sectional shape that is not constant in length, which causes additional costs in its calculation, as well as in manufacturing and installation.

A well-known technical solution is also a bent welded profile channel shape, the wall of which is paired with tubular shelves of rectangular or half-flat section of certain sizes [US patent US 20080028720 A1, 02/07/2008]. The parameters of the wall and shelves are proportional to the thickness of such a profile and are selected very rationally. However, the presence of two welds in its composition causes additional costs, limits the minimum thickness of the welded elements and does not allow the use of galvanized steel.

Another well-known solution (taken as an analogue) is a C-shaped bent profile with a perforated wall, the cutouts on which are reinforced similarly to Atlant profiles [US patent US 6691478 B2, 17.02.2004]. The ratio of overall dimensions in width and height of such a profile is close to optimal parameters when the values of the moment of resistance of the cross section are maximum. However, in this case, the bearing capacity is limited by the thinness of a single sheet blank and the open (open) contour of its cross section.

Closest to the proposed (adopted as a prototype) is a technical solution in the form of a channel bent profile, the wall and tubular shelves of which are interconnected by means of gear closures of the longitudinal edges and mutual support in the contact zone of two of its sheet blanks, where the inner face of the wall and shelves in its cross section has the shape of a circular semicircle with a diameter equal to the height of the wall [Marutyan AS Channel bent profile. - Patent No. 2685013, 06/08/2018, bull. No. 11]. Such a profile has a closed, but quite compact cross-sectional shape, quite stable from the plane and in the plane of the supporting structure. However, a fixed ratio of its overall dimensions in width and height, equal to 1/2, differs from the optimal parameters at which the calculated values of the moment of resistance of the cross section are maximum. Therefore, it is advisable to further elaborate the channel bent closed profile by developing it in height and equipping each perforated wall with serrated locks to ensure the integrity of the composite section without welded, bolted or riveted joints, as well as giving it a C-shaped outline to increase lateral rigidity and stability from the plane of the supporting structure.

The indicated transformation of channel bent closed profiles into the same profiles of C-shaped outlines with perforated walls, supplemented by optimization of the design parameters of the cross sections, can help expand the field of their rational use.

The technical result of the proposed solution is sufficient local (local) and overall stability of the C-shaped bent closed profile with a perforated wall from the plane and in the plane of the supporting structure, expanding the field of rational use, as well as reducing additional costs and consumption of structural material.

The specified technical result is achieved by the fact that in the C-shaped bent profile, the perforated wall and the tubular shelves are interconnected by means of serrated closures of the longitudinal edges and mutual support in the contact zones of its two sheet blanks, where the inner edge of the upper shelf and the inner edge of the lower shelf are in their the cross section has the shape of a circular semicircle with a diameter equal to the width of the shelves, and the inner and outer faces of the wall along the entire length of the vertical sections between the shelves are provided with cutouts with serrated closures of the edges along the entire perimeter of each of these cutouts. When the height of the cutouts reaches 0.8 of the profile height, the ratio of the overall dimensions of the profile in width and height is 1/5, and in the absence of cuts - 1/10.

The proposed C-shaped bent closed profile has a fairly universal technical solution, with the implementation of which for its manufacture it is possible to use not only gear locks, but also welded, bolted or riveted joints. In the manufacture of bent closed profiles without welded, bolted or riveted joints, it is advisable to select the parameters of the serrated longitudinal edges of their sheet blanks so that they form the edges of two blanks with one zigzag cut. At the same time, production costs will be minimal, which will reduce additional costs. The bends of the gear mounts of bent-closed profiles increase the thickness of the collapse, which can contribute to a certain increase in the bearing capacity of the joints of thin-walled elements, working mainly on shear [Kuznetsov I.L., Fakhrutdinov AF, Ramazanov P.P. The results of experimental studies of the work of compounds of thin-walled elements in shear. - Vestnik MGSU, 2016, No. 12. - S. 34-43]. In addition, the bends of the gear mounts ensure the preservation of local (local) stability and the cross-sectional shape of thin-walled elements until the limit state is reached, which makes it possible to calculate not reduced sections, but net sections [Bely G.I. To the determination of reduced sections of the core elements of light steel thin-walled structures. - Bulletin of Civil Engineers, 2017, No. 6. - S. 33-37]. It should be added that the bends of gear mounts distributed evenly over the wall and shelves throughout the C-shaped bent closed profile retain its integrity and solidity to a greater extent than the contact zones with friction of the seam type connection in the rectangular beam profile with the ends attached to the hooks, which made it possible to consider the beam cross-section as monolithic [Yakovleva E.L., Atavin I.V., Kazakova Yu.D., Maksudov I.Kh. Strength characteristics of thin-walled elements. - Construction of unique buildings and structures, 2017, No. 12 (63). - S. 125-139]. In some cases, to reduce the additional costs, the integrity of the composite section of the bent closed profile is achievable due to the use of similar zones of contact with friction without perforation of its wall.

The proposed technical solution is illustrated by graphic materials, where

FIG. 1 shows a side view of a C-shaped bent closed profile with a solid wall without cutouts;

FIG. 2 is a side view of a C-shaped bent closed profile with cutouts in a perforated wall;

FIG. 3 is a transverse section of a C-shaped bent closed profile within one of the cutouts of its perforated wall;

FIG. 4 presents a design diagram of the cross section of a C-shaped bent closed profile within one of the cutouts of its perforated wall;

FIG. 5 is a perspective view of a fragment of a C-shaped bent closed profile with a solid wall without cutouts in an unassembled form;

FIG. 6 is an exploded perspective view of a fragment of a C-shaped bent closed profile with a perforated wall;

FIG. Figure 7 shows graphs of changes in the main design parameters of a C-shaped bent closed profile depending on the growth of the relative height h / V of the cuts of its perforated wall (h is the height of the cuts, V is the profile height).

The C-shaped bent closed profile according to the proposed technical solution consists of two sheet blanks (strips or molding strips) of thickness t. The outer (external) blank 1 has a C-shaped cross-section with angular roundings of flat faces with dimensions: 2 (t × U + t × 0.5U) - two shelves with flanges parallel to the wall; t × V is the wall, where U is the width of the bent closed profile; V is the height of the same profile. The inner blank 2 also has a C-shaped cross section and consists of a vertical face (wall) of dimensions t × (VU) and two semicircular shelves of thickness t and diameter D = U. Both blanks 1 and 2 are made along the entire length with serrated longitudinal edges, the teeth of which are staggered relative to each other and mutually bent in grooves between themselves after closing a bent profile. The perforated blanks 1p (outer) and 2p (inner) are additionally made with the same serrated edges along the entire perimeter of each of the cutouts. The teeth of these edges are exactly the same in relation to each other in a checkerboard pattern and mutually bent in grooves between themselves after closing a bent profile. At the same time, the height of the cut-outs h of the perforated blanks 1n and 2n is limited by the dimension h _{max of} their flat sections, limited in height by the rounding of the inner faces, that is, h <h _max = VD = VU.

To quantify the resources of the bearing capacity of a C-shaped bent closed profile, it is advisable to calculate the area A, as well as the moments of inertia of its cross section I _x and I _y relative to the main central axes. Here, it is obvious that the cross section of such a profile can be considered a composite figure, including a pair of round semirings of thickness t and radius R = 0.5U, a pair of horizontal rectangles of size t × U, a pair of vertical rectangles of size t × 0.5U, and a vertical rectangle of dimensions t × V and a vertical rectangle of dimensions t × (VU). In this case, it is permissible to carry out computational calculations along the midline of a thin-walled section without taking into account its angular curves and without taking into account numerical values containing the thicknesses raised to the second and third degree (t ² , t ³ ) [Marutyan AS Optimization of structures from tubular (bent-welded) profiles of square (rectangular) and rhombic sections. - Structural mechanics and calculation of structures, 2016, No. 1. - S. 30-38].

The cross section of the semicircular faces of the C-shaped profile is a circular semicircle, to which the calculation formulas are fully applicable, tested during optimization of the design parameters of half-flat pipes for truss and beam structures [Marutyan AS Calculation of optimal parameters for half-flat pipe for truss and beam structures. - Structural mechanics and calculation of structures, 2019, No. 2. - S. 68-74]:

y _{c, min} = 0.1815U;

y _{c, max} = 0.3185U;

A _c = 1.57tU,

I _xc = 0.0370143tU ³ ;

I _yc = 0.196250tU ³ ,

where y _{c, min} , y _{c, max} , A _c , I _xc , I _yc are respectively the ordinates of the center of gravity, the cross-sectional area, the moments of inertia with respect to the axes x-x and y-y of the half ring.

The estimated net cross-sectional area of the C-shaped bent closed profile is the sum of the calculated net cross-sectional areas of the semicircular and rectangular sections of its wall and shelves:

A = tU (1 / n + (1 / n-1) +2 (1 + 0.5) + 2 × 1.57) = tU (2 / n + 5.14),

where are the overall dimensions

U = (A / t) / (2 / n + 5.14) and V = (A / t) / (2 + 5.14n).

Moment of inertia of the calculated net cross section of a C-shaped bent closed profile relative to the central abscissa axis:

I _x = tU ³ ((1 / n) ^3/12 + (1 / n-1) ^3/12 + 2 (1 (0.5 / n) ² 0.5 ^3/12 + 0.5 (0 , 5 / n-0.25) ² ) +2 (0.0370143 + 1.57 (0.5 / n-0.1815) ² ) = tU ³ (0.1666666 / n ³ + 1.285 / n ² - 0.56991 / n + 0.1774671).

The obtained calculation formulas can be checked if the C-shaped profile is schematically represented as a double shell of a rectangular shape along the outer contour and a flat oval shape along the inner contour, from which it is necessary to conditionally cut off the corresponding rectangular section of double thickness. The length of this section is equal to the distance between the ends of the shelves (belts) of the C-shaped profile.

The calculated values of the area and moment of inertia of the cross section of a rectangular contour:

The calculated values of the area and moment of inertia of the cross section of the plane-oval contour [Marutyan AS, Abovyan AG Calculation of the optimal parameters of flat oval pipes for trusses. - Structural mechanics and calculation of structures, 2017, No. 4. - S. 17-22]:

A ₀ = tU (2 / n + 1.14);

I _x0 = tU ^{3 (0.0740286} + 0.785 (1 / n-0,363) ² + (1 / n-1) ^3/6) / n ^2.

The calculated values of the area and moment of inertia of the cross section of a rectangular section of double thickness:

A _I = 2tU (1 / n-1);

I _xI = tU ³ (1 / n-1) ^3/6.

The calculated values of the area and moment of inertia of the cross section of the C-shaped profile:

It is obvious that the tested formulas are quite suitable for calculating the geometric (static) characteristics of the C-shaped profile.

The moment of resistance of the calculated net section of the C-shaped bent closed profile relative to the central abscissa axis:

W _x = I _x / y _max = 2I _x / V = 2I _x / (U / n) = tU ² (0,3333333 / n ² + 2,57 / n-1,13982 + 0,3549342n) = (A ² / t) (0.3333333 / n ² + 2.57 / n-1.13982 + 0.3549342n) / (2 / n + 5.14) ² ,

Where

U ² = (A / t) ² / (2 / n + 5.14) ² .

To find the extreme value of the moment of resistance W _x its formula must be differentiated with respect to the variable n:

dW _x / dn = d ((A ² / t) (0.3333333 / n ² + 2.57 / n-1.13982 + 0.3549342n) / (4 / n ² + 20.56 / n + 26, 4196)) / (dn).

Equating the derivative to zero (dW _x / dn = 0), we can obtain an equation of the fourth degree

9.3772195n ⁴ + 14.594894n ³ -87.07.073862n ² -26.731624n + 3.426667 = 0

with roots

n ₁ = -3.7973563, n ₂ = -0.3891050, n ₃ = 0.0976609, n ₄ = 2.5323802.

The third root is of practical importance when the C-shaped profile is a beam element with the largest absolute value of the strength characteristics of the cross section (W _x = W _{x, max} ):

n = 0.0976609 = 1 / 10.239512;

A = tU (2 / 0.0976609 + 5.14) = 25.619024tU;

U = 0.0390334A / t;

V = 0.3996829A / t;

h is 0;

h _max = VU = 9.239512U = 9.239512 × 0.0390334A / t = 0.3606495A / t;

I _x = tU ³ (0.1666666 / 0.0976609 ³ + 1.285 / 0.0976609 ² -0.56991 / 0.0976609 + 0.1774671) = 308.00238tU ³ = 308.00238t (0.0390334A / t) ³ = 0.0183173A ³ / t ² ;

W _x = 2I _x / V = 2 (0.0183173A ³ / t ² ) / (0.3996829A / t) = 0.0916591A ² / t;

i _x = (I _x / A) ^1/2 = ((0.0183173A ³ / t ² ) / A) ^1/2 = 0.1353414A / t;

I _y = tU ³ (2 (0.196250 + 1.57 (0.5-0.1393495) ² ) +2 (1/12 ³ +1 (0.5-0.1393495) ² ) +10.239512 × 0.1393495 ² + (10.239512-1) 0.1393495 ² + 2 × 0.5 (1-0.1393495) ² ) = 2.3420354tU ³ = 2.3420354t (0.0390334A / t) ³ = 0.0001392A ³ / t ² ;

i _y = (I _y / A) ^1/2 = ((0.0001392A ³ / t ² ) / A) ^1/2 = 0.0117983A / t.

In a C-shaped bent closed profile with a ratio of the dimensions of its section n = 1 / 10,239512≈1 / 10,24≈1 / 10 and a double wall without cutouts (h = 0), welded, bolted, riveted or other compounds to ensure the integrity of the composite section of thin-walled waist elements and the same thickness of the wall elements. Therefore, a perforated wall (h _min <h <h _max ) in which all cuts are framed by the same serrated edges as the shelf (waist) elements is more preferable here.

Further numerical calculations are more obvious when compared with the corresponding design parameters of the prototype (channel bent profile), adopted as reference (100 percent) values:

n = U / V = 1/2;

A = 7.14tU;

U = 0.140056A / t (100%);

V = 0.280112A / t (100%);

h is 0;

h _max = 0;

I _x = 4.2366666tU ³ = 4.2366666t (0.140056A / t) ³ = 0.0116389A ³ / t ² (100%);

W _x = 4.2366666tU ² = 4.2366666t (0.140056A / t) ² = 0.0831047A ² / t (100%);

i _x = (0.0116389A ² / t ² ) ^1/2 = 0.1078837A / t (100%);

I _y = 0.7352826tU ³ = 0.7352826t (0.140056A / t) ³ = 0.0020199A ³ / t ² (100%);

W _{y, min} = 1,0500137tU ² = 1,0500137t (0,140056A / t) ² = 0,0205966A ² / t (100%);

W _{y, max} = 2.4350679tU ² = 2.435067t (0.140056A / t) ² = 0.0477653A ² / t (100%);

i _y = (0.0020199A ² / t ² ) ^1/2 = 0.0449432A / t (100%),

where t and A are the thickness and total area of strips (sheet blanks or formed strips, respectively), t = const and A = const.

The calculated parameters of the new technical solution (C-shaped bent closed profile),

n = 0.5 = 1/2;

A = tU (2 / 0.5 + 5.14) = 9.14tU;

U = 0.1094091A / t (78.12%);

V = 0.2188182A / t (78.12%);

h is 0;

h _max = VU = U = 1 × 0.1094091A / t = 0.1094091A / t;

I _x = tU ³ (0.1666666 / 0.5 ³ + 1.285 / 0.5 ² -0.56991 / 0.5 + 0.1774671) = 5.5109799tU ³ = 5.5109799t (0.1094091A / t) ³ = 0.0072175A ³ / t ² (62.01%);

W _x = 2 (0.0072175A ³ / t ² ) / (0.2188182A / t) = 0.065968A ² / t; (62.01%);

i _x = (I _x / A) ^1/2 = ((0.0072175A ³ / t ² ) / A) ^1/2 = 0.1353414A / t (79.38%);

I _y = tU ³ (2 (0.196250 + 1.57 (0.5-0.3905908) ² ) +2 (1/12 ³ +1 (0.5-0.3905908) ² ) + 2 × 0 , 3905908 ² + (2-1) 0.3905908 ² + 2 × 0.5 (1-0.3905908) ² ) = 1.4497566tU ³ = 1.49975664t (0.0390334A / t) ³ = 0.0018986A ³ / t ² (94.0%);

W _{y, min} = (0.0018986A ³ / t ² ) / ((0.1094091-0.0427341) A / t) = 0.0284754A ² / t (138.3%);

i _y = (I _y / A) ^1/2 = ((0.0018986A ³ / t ² ) / A) ^1/2 = 0.0435729A / t (96.95%).

As expected, in the presence of a margin of lateral rigidity, the C-shaped profile turned out to be more compact than the channel profile. Therefore, its further optimization is rather promising, accompanied by an increase in the height of the perforated wall.

To complete the calculation of the optimized section of a C-shaped bent closed profile without perforation of the wall, as a prerequisite, you can take the integrity of its composite section:

n = U / V = 1/10 = 0.1;

A = tU (2 / 0.1 + 5.14) = 25.14 tU;

U = 0.0397772A / t (28.40%);

V = 0.397772A / t (142.0%);

h is 0;

h _max = VU = 9U = 9 × 0.0397772A / t = 0.3579948A / t;

I _x = tU ³ (0.1666666 / 0.1 ³ + 1.285 / 0.1 ² -0.56991 / 0.1 + 0.1774671) = 289.64496tU ³ = 289.64496t (0.0397772A / t) ³ = 0.0182292A ³ / t ² (156.6%);

W _x = 2 (0.0189466A ³ / t ² ) / (0.397772A / t) = 0.0916565A ² / t (110.3%);

i _x = ((0.0182292A ³ / t ² ) / A) ^1/2 = 0.1350155A / t (125.15%);

I _y = tU ³ (2 (0.196250 + 1.57 (0.5-0.1420047) ² ) +2 (1/12 ³ +1 (0.5-0.1420047) ² ) +10.239512 × 0.1420047 ² + (10.239512-1) 0.1420047 ² + 2 × 0.5 (1-0.1420047) ² ) = 2.3468689tU ³ = 2.3468689t (0.0397772A / t) ³ = 0.0001477A ³ / t ² (7.31%);

W _{y, min} = (0.0001477A ³ / t ² ) / ((0.0397772-0.0056485) A / t) = 0.0043277A ² / t (21.01%);

W _{y, max} = (0.0001477A ³ / t ² ) / (0.0056485A / t) = 0.0269279A ² / t (56.38%);

i _y = ((0.0001477A ³ / t ² ) / A) ^1/2 = 0.0121531A / t (27.04%).

As can be seen, with an increase in the overall size in height by 1.420 times in the force plane of the supporting structure, the moment of inertia of the section increased by 1.566 times and the moment of resistance by 1.103 times. Moreover, the rounding of the ratio of the overall dimensions of the C-shaped profile to an integer value (n = U / V = 1/10) is accompanied by a decrease in the height of its cross section by 0.3996829 / 0.397772 = 1.004804 times, the moment of inertia in the force plane of the supporting structure - 0.0183173 / 0.0182292 = 1.0048329 times, and the moment of resistance - 0.0916591 / 0.0916565 = 1.0000283 times.

It is necessary to consider in no less detail those design cases when the perforated wall of the C-shaped bent closed profile has small cutouts necessary and sufficient to accommodate gear mounts that ensure the integrity of the composite section and reduce the waste of structural material, including galvanized steel.

The optimization calculation of a C-shaped bent closed profile with a perforated wall in cross section with a cutout at h / V = 0.1 has the following form:

h = 0.1V;

I _x = tU ³ (0,1666666 / n ³ + 1,285 / n ² -0,56991 / n + 0,1774671-2 (0,1 / n) ^3/12) = tU ³ (0,16650 / n ³ + 1.285 / n ² -0.56991 / n + 0.1774671);

dW _x / dn = d ((A ² / t) (0.3330 / n ² + 2.57 / n-1.13982 + 0.3549342n) / (4 / n ² + 20.56 / n + 26, 4196)) / (dn);

9.3772195n ⁴ + 14.594894n ³ -87.07.073862n ² -26.714013n + 3.433352 = 0;

n ₁ = -3.7974324, n ₂ = -0.3891051, n ₃ = 0.0978590, n ₄ = 2.5322583;

n = 0.0978590 = 1 / 10.218784;

A = tU (2 / 0.0978590 + 5.14) = 25.579448tU;

U = 0.0390938A / t (27.91%);

V = 0.3994911A / t (142.6%);

h = 0.1 × 0.3994911A / t = 0.0399491A / t;

h _max = VU = 9.218784U = 9.218784 × 0.0390938A / t = 0.3603972A / t;

I _x = tU ³ (0.16650 / 0.0978590 ³ + 1.285 / 0.0978590 ² -0.56991 / 0.0978590 + 0.1774671) = 306.207002tU ³ = 306.207002t (0.0390938A / t) ³ = 0.0182952A ³ / t ² (157.2%);

W _x = 2 (0.0182952A ³ / t ² ) / (0.3994911A / t) = 0.0915925A ² / t (110.2%);

i _x = ((0.0182952A ³ / t ² ) / A) ^1/2 = 0.1352597A / t (125.4%);

I _y = tU ³ (2 (0.196250 + 1.57 (0.5-0.1395651) ² ) +2 (1/12 ³ +1 (0.5-0.1395651) ² ) +10.218784 × 0.1395651 ² + (10.218784-1-0.1 × 10.218784) 0.1395651 ² + 2 × 0.5 (1-0.1395651) ² ) = 2,3259774tU ³ = 2,3259774t ( 0.0390938A / t) ³ = 0.0001389A ³ / t ² (6.88%);

W _{y, min} = (0.0001389A ³ / t ² ) / ((0.0390938-0.0054561) A / t) = 0.0041292A ² / t (20.05%);

W _{y, max} = (0.0001389A ³ / t ² ) / (0.0054561A / t) = 0.054577A ² / t (53.30%);

i _y = ((0.0001389A ³ / t ² ) / A) ^1/2 = 0.0117855A / t (26.22%).

As can be seen, with an increase in the overall size in height by 1.426 times in the force plane of the supporting structure, the moment of inertia of the section increased by 1.572 times and the moment of resistance by 1.102 times.

If we repeat all the numerical calculations in the range from h = 0.1V to h = 0.9V, then the dynamics of their main results can be more graphically presented as a reaction to changes in the relative height of the cutouts h / V, from which the most optimal parameters are quite obvious:

n = U / V = 0.2008236 = 1 / 4,9794944≈0,2 = 1/5 at h = h _max = 0,8V,

where n is the value of the ratio of overall dimensions, optimized in each calculation case according to the criterion of maximum strength (W _x = W _{x, max} ), and the calculation at 0.8 <h / V is conditional, and its main results are graphically shown in dashed lines.

The optimization calculation of a C-shaped bent closed profile with a perforated wall in cross section with a cutout at h / V = 0.8 has the following form:

h = 0.8V;

I _x = tU ³ (0,1666666 / n ³ + 1,285 / n ² -0,56991 / n + 0,1774671-2 (0,8 / n) ^3/12) = tU ³ (0,0813333 / n ³ + 1.285 / n ² -0.56991 / n + 0.1774671);

dW _x / dn = d ((A ² / t) (0,1626666 / n ² + 2,57 / n-1,13982 + 0,3549342n) / (4 / n ² + 20,56 / n + 26, 4196)) / (dn);

9.3772195n ⁴ + 14.594894n ³ -87.07.073862n ² -17.713733n + 6.935575 = 0;

n ₁ = -3.8357491, n ₂ = -0.3891051, n ₃ = 0.2008236, n ₄ = 2,4676103

n = 0.2008236 = 1 / 4.9794944;

A = tU (2 / 0.2008236 + 5.14) = 15.098988tU;

U = 0.0662298A / t (47.29%);

V = 0.3297899A / t (117.7%);

h = 0.8 × 0.3297899A / t = 0.2638319A / t;

h _max = VU = 3.9794944U = 3.9794944 × 0.0662298A / t = 0.2635603A / t;

I _x = tU ³ (0.16650 / 0.2008236 ³ + 1.285 / 0.2008236 ² -0.56991 / 0.2008236 + 0.1774671) = 39.243737tU ³ = 39.243737t (0.0662298A / t) ³ = 0.0114005A ³ / t ² (97.95%);

W _x = 2 (0.0114005A ³ / t ² ) / (0.3297899A / t) = 0.0691379A ² / t (83.19%);

i _x = ((0.0114005A ³ / t ² ) / A) ^1/2 = 0.1067731A / t (98.97%);

I _y = tU ³ (2 (0.196250 + 1.57 (0.5-0.2232662) ² ) +2 (1/12 ³ +1 (0.5-0.2232662) ² ) +4.9794944 × 0.2232662 ² + (4.9794944-1-0.8 × 4.9794944) 0.2232662 ² + 2 × 0.5 (1-0.2232662) ² ) = 1.8041231tU ³ = 1.8041231t ( 0.0662298A / t) ³ = 0.0005241A ³ / t ² (25.95%);

W _{y, min} = (0.0005241A ³ / t ² ) / ((0.0662298-0.0147868) A / t) = 0.0101879A ² / t (49.46%);

W _{y, max} = (0.0005241A ³ / t ² ) / (0.0147868A / t) = 0.0354437A ² / t (74.20%);

i _y = ((0.0005241A ³ / t ² ) / A) ^1/2 = 0.0228932A / t (50.94%).

As you can see, with an increase in the overall size in height by 1.177 times in the force plane of the supporting structure, the moment of inertia of the section decreased by 2.05%, and the moment of resistance - by 16.81%, which is a consequence of a decrease in the estimated cross-sectional area due to the cutout of the perforated wall 100 × 0.8 × 2 × 4.9794944 / 15.098988 = 52.77%.

Since the height of the cutouts of the perforated wall actually reached the limit value

h / h _max = 0.2638319 / 0.2635603 = 1.0010305≈1,

Of practical interest is the generalization of the optimization calculation with the rounding of the ratio of the overall dimensions of the C-shaped bent closed profile to an integer value:

n = 0.2 = 1/5 for h = h _max = 0.8V;

A = tU (2 / 0.2 + 5.14) = 15.14tU;

U = 0.0660501A / t (41.29%);

V = 0.3302505A / t (117.7%);

h = 0.8 × 5U = 4U = 4 × 0.0660501A / t = 0.2642004A / t;

h _max = VU = 4U = 4 × 0.0660501A / t = 0.2642004A / t;

I _x = tU ³ (0.16650 / 0.2 ³ + 1.285 / 0.2 ² -0.56991 / 0.2 + 0.1774671) = 39.61958tU ³ = 39.61958t (0.0660501A / t) ³ = 0.0114164A ³ / t ² (98.09%);

W _x = 2 (0.0114164A ³ / t ² ) / (0.3302505A / t) = 0.0691378A ² / t (83.19%);

i _x = ((0.0114164A ³ / t ² ) / A) ^1/2 = 0.1067731A / t (98.97%);

I _y = tU ³ (2 (0.196250 + 1.57 (0.5-0.2357992) ² ) +2 (1/12 ³ +1 (0.5-0.2357992) ² ) + 5 × 0 , 2357992 ² + (5-1-0.8 × 5) 0.2357992 ² + 2 × 0.5 (1-0.2357992) ² ) = 1.7799575tU ³ = 1.7799575t (0.0660501A / t) ³ = 0.0005128A ³ / t ² (25.39%);

W _{y, min} = (0.0005128A ³ / t ² ) / ((0.0660501-0.0155745) A / t) = 0.0101593A ² / t (49.33%);

W _{y, max} = (0.0005128A ³ / t ² ) / (0.0155745A / t = 0.0329256A ² / t (68.93%);

i _y = ((0.0005128A ³ / t ² ) / A) ^1/2 = 0.022645A / t (50.94%).

According to the calculation results, the rounding of the ratio of the overall dimensions of the C-shaped profile to an integer value (n = U / V = 1/5) is accompanied by an increase in the height of its section by 0.3302505 / 0.3297899 = 1.0013966 times, the moment of inertia in the carrier force plane construction - by 0.0114164 / 0.0114005 = 1.0013946 times, and the moment of resistance by a decrease of 0.0691379 / 0.0691378 = 1.0000014 times.

Practical interest cannot but cause a C-shaped bent closed profile with a rounded value of the ratio of overall dimensions, but without perforation of its wall:

n = 0.2 = 1/5 for h / V = 0;

A = tU (2 / 0.2 + 5.14) = 15.14tU;

U = 0.0660501A / t (47.29%);

V = 0.3302505A / t (117.7%);

h is 0;

h _max = VU = 4U = 4 × 0.0660501A / t = 0.2642004A / t;

I _x = tU ³ (0.1666666 / 0.2 ³ + 1.285 / 0.2 ² -0.56991 / 0.2 + 0.1774671) = 50.286243tU ³ = 50.286243t (0.0660501A / t) ³ = 0.011490A ³ / t ² (98.72%);

W _x = 2 (0.011490A ³ / t ² ) / (0.3302505A / t) = 0.0695835A ² / t (83.73%);

i _x = ((0.011490A ³ / t ² ) / A ² ) ^1/2 = 0.1203744A / t (111.6%);

I _y = tU ³ (2 (0.196250 + 1.57 (0.5-0.2357992) ² ) +2 (1/12 ³ +1 (0.5-0.2357992) ² ) + 5 × 0 , 2357992 ² + (5-1) 0.2357992 ² + 2 × 0.5 (1-0.2357992) ² ) = 2.0023623tU ³ = 2.0023623t (0.0660501A / t) ³ = 0.0005769A ³ / t ² (25.39%);

W _{y, min} = (0.0005769A ³ / t ² ) / ((0.0660501-0.0155745) A / t) = 0.0114292A ² / t (56.58%);

W _{y, max} = (0.0005769A ³ / t ² ) / (0.0155745A / t) = 0.0370413A ² / t (77.55%);

i _y = ((0.0005769A ³ / t ² ) / A) ^1/2 = 0.0240187A / t (53.44%).

As you can see, with an increase in the overall dimension in height by 1.177 times in the force plane of the supporting structure, the moment of inertia of the section decreased by 1.28%, and the moment of resistance - by 16.27%, which can be explained by the double difference between the ratio of overall dimensions and its optimal value. However, in this case, you can save on additional labor costs necessary for perforating the walls of a C-shaped bent closed profile.

The above numerical calculations show that C-shaped bent-closed profiles with perforated and non-perforated walls are quite promising for use in load-bearing structures. Therefore, further clarification of their design characteristics with the addition of gear mounts instead of welded, bolted or riveted joints is of practical importance. To do this, in the considered profile, it is necessary to select the sizes of the gear fastening elements (teeth), which should be at least 1/10 of the overall cross-sectional size [SP 260.1325800.2016. Thin-walled steel structures made of cold-formed galvanized profiles and corrugated sheets. Design rules. - M., 2016. - S. 16, formula (7.2)]. In this case, this size is 0.1U, where U is the width of the bent closed profile.

In the calculated section of the C-shaped bent closed profile with a non-perforated wall, the parameter of the tooth fastenings (the size of the teeth) will be reflected in a 4-fold way, since this profile has a composite section of 2 sheet blanks (strips or molding strips) with longitudinal edges of the gear shape:

A _gross = A + ΔA = A + 2 × 2 × 0.1tU = A + 0.4tU.

In the calculated section of the C-shaped bent-closed profile with a perforated wall, the parameter of the gear mounts will be reflected 8-fold, since its workpieces are additionally made with the same gear edges around the entire perimeter of each of the cutouts. On the serrated edges of the cuts, a part of the structural material is consumed, which is removed during the manufacture of the perforated wall; therefore, the above formula for A _gross can be extended to the second design case.

Then, with reference to the calculated sections of the considered profiles, we can write:

with n = U / V = 1/10 and h / V = 0

A _gross = A + 0.4tU = 25.14tU + 0.4tU = 25.54tU;

A / A _gross = 25.14 / 25.54 = 0.9843;

for n = U / V = 1/5 and h / V = 0, as well as for n = U / V = 1/5 and h / V = 0.8

A _gross = A + 0.4tU = 15.14tU + 0.4tU = 15.54tU;

A / A _gross = 15.14 / 15.54 = 0.9743.

Approximately the same relations have the geometric characteristics of bent C-shaped cross-section profiles, which, like other bar elements of light steel thin-walled structures (LSTC), are also characterized by a reduction coefficient (or reduction coefficient):

ρ = A _red / A (or ρ = A _cf / A _p ),

where A _red (A _sr ) is the estimated area of the reduced section (or the estimated cross-sectional area under compression), A (or A _p ) is the estimated total cross-sectional area (or the estimated cross-sectional area under tension).

In particular, for C-shaped bent profiles of grades ПГС75С, ПГС100С and ПГС150С, the values of the reduction coefficient are [Recommendations on the design, manufacture and installation of enclosing and supporting structures from steel bent profiles of increased rigidity. - M.: TSNIIPSK them. N.P. Melnikova, 1999. - S. 8-11]:

ρ = 0.696 ... 0.950.

To evaluate the results as a base object, for comparison, you can use the geometric (static) characteristics of the Atlant thin-walled profiles with a ratio of overall dimensions of the cross section n = U / V = 41.3 / 203 = 1 / 4.9152542≈1 / 5 [Kashevarova G.G., Kosykh P.A. Determination of equivalent geometric characteristics of Atlant profiles. - Nizhny Novgorod: Bulletin of the PTO RAASN, 2016, No. 19. - S. 207-214]:

profile 203 × 41.3 × 1 mm (design section without cut)

A = 325 mm ² (100%);

I _x = 1848260 mm ⁴ , I _x / A = 5686.9538 mm ⁴ / mm ² (100%);

I _y = 61 150 mm ⁴ , I _y / A = 188.15384 mm ⁴ / mm ² (100%);

profile 203 × 41.3 × 1 mm (calculated cross-section with cut-out)

A = 231 mm ² (100%);

I _x = 1445007 mm ⁴ , I _x / A = 6255.4415 mm ⁴ / mm ² (100%);

I _y = 51608 mm ⁴ , I _y / A = 223.411125 mm ⁴ / mm ² (100%);

profile 203 × 41.3 × 1.5 mm (design section without cut)

A = 482 mm ² (100%);

I _x = 2711201 mm ⁴ , I _x / A = 5624.8983 mm ⁴ / mm ² (100%);

I _y = 87367 mm ⁴ , I _y / A = 181,25933 mm ⁴ / mm ² (100%);

profile 203 × 41.3 × 1.5 mm (calculated cross-section with cut-out)

A = 342 mm ² (100%);

I _x = 2148423 mm ⁴ , I _x / A = 6281.9385 mm ⁴ / mm ² (100%);

I _y = 73548 mm ⁴ , I _y / A = 215.05263 mm ⁴ / mm ² (100%);

profile 203 × 41.3 × 2 mm (design section without cut)

A = 635 mm ² (100%);

I _x = 3537466 mm ⁴ , I _x / A = 5570.8125 mm ⁴ / mm ² (100%);

I _y = 110974 mm ⁴ , I _y / A = 174.7622 mm ⁴ / mm ² (100%);

profile 203 × 41.3 × 2 mm (calculated cross-section with cut-out)

A = 448 mm ² (100%);

I _x = 2820120 mm ⁴ , I _x / A = 6294.9107 mm ⁴ / mm ² (100%);

I _y = 93083 mm ⁴ , I _y / A = 218.93526 mm ⁴ / mm ² (100%).

The calculated parameters of C-shaped bent closed profiles with a perforated wall, that is, the proposed (new) technical solution, with respect to the overall dimensions of the cross section n = U / V = 40.6 / 203 = 1/5 are:

profile 203 × 40.6 × 1 mm (design cross-section without cut)

t = A _gross / (15.54U) = 325 / (15.54 × 40.6) = 0.5151174 mm≈0.50 mm;

A _gross = 15.54 × 0.50 × 40.6 = 315.462 mm ² (97.07%);

A = 0.9743A _gross = 0.9743 × 315.462 = 307.355 mm ² (94.57%);

I _x = 50.286243tU ³ = 50.286243 × 0.50 × 40.6 ³ = 1682664 mm ⁴ ;

I _x / A = 5474.6595 mm ⁴ / mm ² (96.27%);

I _y = 2.0023623tU ³ = 2.0023623 × 0.50 × 40.6 ³ = 67002 mm ⁴ ;

I _y / A = 212.46531 mm ⁴ / mm ² (112.9%);

profile 203 × 40.6 × 1 mm (calculated cross section with a cutout h / V = 0.8)

A = 307.355-2 × 0.50 × 0.8 × 203 = 144.955 mm ² (62.75%);

I _x = 39.61958tU ³ = 39.61958 × 0.50 × 40.6 ³ = 1325739 mm ⁴ ;

I _x / A = 9145.8659 mm ⁴ / mm ² (146.2%);

I _y = 1.7799575tU ³ = 1.7799575 × 0.50 × 40.6 ³ = 59,560 mm ⁴ ;

I _y / A = 410.88902 mm ⁴ / mm ² (154.1%);

profile 203 × 40.6 × 1.5 mm (design section without cut)

t = A _gross / (15.54U) = 482 / (15.54 × 40.6) = 0.7639588 mm≈0.75 mm;

A _gross = 15.54 × 0.75 × 40.6 = 473.193 mm ² (98.17%);

A = 0.9743A _gross = 0.9743 × 481.352 = 461.032 mm ² (95.65%);

I _x = 50.286243tU ³ = 50.286243 × 0.75 × 40.6 ³ = 2523995 mm ⁴ ;

I _x / A = 5474.6633 mm ⁴ / mm ² (97.33%);

I _y = 2.0023623tU ³ = 2.0023623 × 0.75 × 40.6 ³ = 100504 mm ⁴ ;

I _y / A = 217.99788 mm ⁴ / mm ² (120.3%);

profile 203 × 40.6 × 1.5 mm (design cross section with a cutout h / V = 0.8)

A = 461.032-2 × 0.75 × 0.8 × 203 = 217.432 mm ² (63.58%);

I _x = 39.61958tU ³ = 39.61958 × 0.75 × 40.6 ³ = 1988608 mm ⁴ ;

I _x / A = 9145.8846 mm ⁴ / mm ² (145.6%);

I _y = 1.7799575tU ³ = 1.7799575 × 0.75 × 40.6 ³ = 89341 mm ⁴ ;

I _y / A = 410.89168 mm ⁴ / mm ² (191.1%);

profile 203 × 40.6 × 2 mm (design cross-section without cut)

t = A _gross / (15.54U) = 635 / (15.54 × 40.6) = 1.0064603 mm≈1.0 mm;

A _gross = 15.54 × 1.0 × 40.6 = 630.924 mm ² (99.36%);

A = 0.9743A _gross = 0.9743 × 577.622 = 614.709 mm ² (96.80%);

I _x = 50.286243tU ³ = 50.286243 × 1.0 × 40.6 ³ = 3365327 mm ⁴ ;

I _x / A = 5474.6668 mm ⁴ / mm ² (98.27%);

I _y = 2.0023623tU ³ = 2.0023623 × 1.0 × 40.6 ³ = 134005 mm ⁴ ;

I _y / A = 217.99745 mm ⁴ / mm ² (124.7%);

profile 203 × 40.6 × 2 mm (design cross section with a cutout h / V = 0.8)

A = 614.709-2 × 1.0 × 0.8 × 203 = 246.709 mm ² (55.07%);

I _x = 39.61958tU ³ = 39.61958 × 1.0 × 40.6 ³ = 2651478 mm ⁴ ;

I _x / A = 10,747.390 mm ⁴ / mm ² (170.7%);

I _y = 1.7799575tU ³ = 1.7799575 × 1.0 × 40.6 ³ = 119121 mm ⁴ ;

I _y / A = 482.8401 mm ⁴ / mm ² (220.5%),

where the accepted thickness values of thin-sheet elements (t = 0.50 mm, t = 0.75 mm and t = 1.0 mm) meet the requirements of GOST 19904-90 “Cold-rolled sheet metal. Assortment ”, which reduces additional costs.

As can be seen, the results of the comparative calculation once again confirmed the sufficient effectiveness of the proposed C-shaped bent closed profiles for future use in the supporting structures of buildings and structures.

Claims (1)

C-shaped bent-closed profile formed by one outer and one inner blanks having a cross-section in the form of C-shaped with shelves and perforated or solid walls mating with each other by means of serrated closures of their longitudinal edges, characterized in that the shelves of the outer blank are made flat and the shelves of the inner workpiece are bent in the form of a circular semicircle with a diameter equal to the width of the shelves, while the outer and inner workpieces with perforated walls are provided with cutouts with serrated edges along the entire perimeter of each of these cutouts, and when the height of the cuts reaches 0.8 profile height , the ratio of the overall dimensions of the profile in width and height is 1/5, and in the absence of cutouts - 1/10.

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