RU2645318C1 - Pentagonal formed hollow profile - Google Patents

Abstract

FIELD: construction.

SUBSTANCE: proposed technical solution relates to the field of construction and can be used as rod elements in the development of load-bearing structures of buildings and structures of various purposes. In the particular case, these can be the rod elements of the chords of run-through and non-run-through coatings. This technical result is achieved by the fact that in a formed hollow section of a pentagonal contour with one horizontal face (shelf), two vertical faces (walls) and two inclined faces, as well as with gussets from the crimps of blanks docked together along the entire length, the section has the pentagon shape with an angle of 90° between inclined faces, the height of which is twice the size of the gusset, or the pentagon shape with an angle of 120° between oblique faces, which height is 1.1415 times larger than the size of the gusset.

EFFECT: technical result of the proposed solution is equal stability (equistability) of the formed hollowed sections from the plane and in the plane of the construction, an increase in the bearing capacity, a reduction in the construction height.

1 cl, 10 dwg

Description

The proposed technical solution relates to the field of construction and can be used as bar elements in the development of load-bearing structures of buildings and structures for various purposes. In the particular case, these may be the core elements of the belts of trusses of run-and-drive-free coatings.

A technical solution is known in the form of a frame T-shaped profile with one edge made of a continuous strip. The profile is made with a lower horizontal shelf, a hollow upper reinforcing capsule-shaped extension and a vertical rib extending upward from the shelf to the extension. To minimize lateral eccentricity, the rib is made in the form of a single layer of a strip and is formed with a pair of vertically spaced offsets. Displacements occupy most of the single rib layer in the nominal mid-plane of the profile, which divides the flange and expansion in half [Rakhil M.M., Likhein J.J.M.L., Lalond P. Frame T-profile with one rib made from one strip . - Patent No. 2481442, 05/10/2013, bull. No. 13]. Such a profile is rational enough to be used as a suspended ceiling run. However, its shape and bearing capacity limit the possibility of using coatings and other supporting structures in trusses.

Another well-known (adopted as an analogue) is a technical solution, which is a bent closed profile made in a cross section of a square or rectangular shape with a joint approximately in the middle of one of the faces. Each part of the face on which the joint is located has a continuation in the form of a G- or I-shaped rib [Levin E.V. Bent closed profile. - Patent No. 98155, 10/10/2010, bull. No. 28]. Such a profile effectively competes with I-beams. However, as a rod element, equally stable both from the plane and in the plane of the supporting structure, it requires some refinement.

Closest to the proposed (adopted as a prototype) is the technical solution of the rod elements, the multifaceted cross section of which, including pentagonal, is formed by bending along the length of both edges of the sheet blank (strip) in the opposite directions with the formation of a chamfer along the entire length of the profile and closing its section with installation of coupling bolts. Of the two polyhedral cross-sections of a pentagonal shape, the prototype is one that consists of one horizontal face (shelf), two vertical faces (walls) and two inclined faces. Such sections are recommended as truss belts with gratings made of galvanized steel profiles [Salakhutdinov MA, Kuznetsov I.L., Sayanov S.F. Steel trusses with belts from pipes of multifaceted cross-section. - Proceedings of KSASU, 2016, No. 4 (38). - S. 236-242, Fig. 2, c]. The use of sections with a gusset along the entire length is rational in non-stop coatings, when stability from the truss plane is ensured by laying and fastening the profiled flooring directly along the upper belts. Here, the sections under consideration are sufficiently developed in the truss plane to provide effective resistance to the combined action of bending moments and compressive forces. For run coatings, sections that are equally stable both from the plane and in the truss plane are more preferable. Therefore, in such cases, a multifaceted section with a gusset needs additional elaboration.

The technical result of the proposed solution is the same stability (equidistance) of bent closed profiles from the plane and in the plane of the structure, an increase in bearing capacity, a decrease in building height.

The specified technical result is achieved by the fact that in the bent profiles of closed sections of pentagonal outlines with one horizontal face (shelf), two vertical faces (walls) and two inclined faces, as well as with gussets from bent edges of the workpieces joined together along the entire length, the profiles have the shape of a pentagon with an angle of 90 ° between the inclined faces, the height of which is 2 times the size of the gusset, or a pentagon with an angle of 120 ° between the inclined faces, whose height is 1.1415 times the size of the gusset.

The proposed bent closed (bent closed) profiles have a fairly universal technical solution, with the implementation of which welded, bolted or riveted joints can be used for their manufacture. If at the same time the height of the pentagonal section with an angle of 90 ° between the inclined faces is twice the size of the gusset, then such profiles are equally stable, that is, they have the same stability from the plane and in the plane of the supporting structure. The same equilibrium is ensured if the height of the pentagonal section with an angle of 120 ° between the inclined faces is 1.1415 times the size of the gusset. The equidistance of bent closed profiles contributes to the effectiveness of their use in the belts of roof trusses and trusses of run-through coatings. With respect to truss belts of non-chromed coatings, it is rational to lengthen the sizes of double-thickness gussets of bent closed profiles depending on the values of jointly acting bending moments and compressive forces, developing their calculated cross section in the force plane of the supporting structure and keeping the tubular (closed) parts of single thickness unchanged.

The pentagonal shape of the cross section of the tubular parts of single thickness of bent closed profiles provides the necessary rigidity, and the shaped (rib) parts of double thickness of bent closed profiles increase the crushing area, which can contribute to a certain increase in the bearing capacity of the joints of thin-walled elements working mainly on shear [I. Kuznetsov , Fakhrutdinov A.F., Ramazanov PP The results of experimental studies of the work of compounds of thin-walled elements in shear. - Vestnik MGSU, 2016, No. 12. - S. 34-43].

The proposed technical solution is illustrated by graphic materials, where in FIG. 1 shows a cross section of pentagonal bent closed profiles with an angle of 90 °, optimized by the criterion of equidistance; in FIG. 2 - section of pentagonal bent closed profiles with an angle of 90 ° and gussets elongated by 1 gusset size; in FIG. 3 is a cross section of pentagonal bent closed profiles with an angle of 90 ° and gussets elongated by 2 gusset sizes; in FIG. 4 is a cross section of pentagonal bent closed profiles with an angle of 90 ° and cuts elongated by 3 sizes of cut; in FIG. 5 shows a section of pentagonal bent closed profiles with an angle of 120 °, optimized by the criterion of equidistance; in FIG. 6 is a cross section of pentagonal bent closed profiles with an angle of 120 ° and gussets elongated by 1 gusset size; in FIG. 7 is a cross section of pentagonal bent closed profiles with an angle of 120 ° and cuts elongated by 2 sizes of cut; in FIG. 8 is a cross section of pentagonal bent closed profiles with an angle of 120 ° and cuts elongated by 3 sizes of cut; in FIG. 9 shows a fragment of a pentagonal bent closed profile with an angle of 90 ° and an optimized section in a perspective view; in FIG. 10 is a fragment of a pentagonal bent closed profile with an angle of 120 ° and an optimized section in axonometry.

The proposed bent closed profiles according to the analogue example can be formed in 5 ... 10 passes, depending on the yield strength and elongation of the material.

To derive the given ratios of the parameters of pentagonal bent closed profiles with equal stability from the plane and in the plane of the supporting structure, as well as quantifying their bearing capacity, it is advisable to calculate the moments of inertia of the cross section I _X and I _Y relative to the main central axes and equate them to each other using the stepwise approximation method to a friend. In this case, it is permissible to carry out computational calculations along the midline of a thin-walled section without taking into account numerical values containing the values of the thickness 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].

Pentagonal bent closed profiles with an angle of 90 °, optimized according to the criterion of equidistance, have the following characteristics of the cross section:

- overall height

H = 3.0V;

- overall width

U = 3.4286V;

- cross-sectional area

A = 10.8488tV;

- ordinate of the center of gravity of the section relative to the upper face

y _s = 0.9792V;

- moments of inertia

I _X = 9,80070678tV ³ ;

I _Y = 9.7878655tV ³ ;

I _X / I _Y = 9.80070678 / 9.7878655 = 1.0013119≈1

with relative error

100 (9.80070678-9.7878655) / (9.80070678 ... 9.7878655) = 0.1310 ... 0.1312%;

- moments of resistance

W _{X, max} = 9.80070678tV ³ / (0.9792V) = 6.67424tV ² ;

W _{X, min} = 9.80070678tV ³ / (3.0V-0.9792V) = 3.28493tV ² ;

W _Y = 9.7878655tV ³ / (3.4286V / 2) = 3.62132tV ² ;

- radii of inertia

i _X = V (9.80070678 / 10.8488) ^1/2 = 0.9504687V;

i _Y = V (9.7878655 / 10.8488) ^1/2 = 0.9498458V;

i _X / i _Y = 0.9504687 / 0.9498458 = 1,0006557≈1

with relative error

100 (0.9504687-0.9498458) / (0.9504687 ... 0.9498458) = 0.06553 ... 0.06557%,

where V is the size of the gusset (rib part) of the section.

Similarly, pentagonal bent closed profiles with an angle of 120 °, optimized by the criterion of equidistance, have the following cross-sectional characteristics:

- overall height

H = 2.1415V;

- overall width

U = 2.2830V;

- cross-sectional area

A = 7.8841294tV;

- ordinate of the center of gravity of the section relative to the upper face

y _c = 0.7174V;

- moments of inertia

I _X = 3.4060621tV ³ ;

I _Y = 3.3938988tV ³ ;

I _X / I _Y = 3.4060621 / 3.3938988 = 1.0035838≈1

with relative error

100 (3.4060621-3.3938988) / (3.4060621 ... 3.3938988) = 0.3571 ... 0.3584%;

- moments of resistance

W _{X, max} = 3.4060621tV ³ / (0.7174V) = 4.7475959tV ² ;

W _{X, min} = 3.4060621tV ³ / (2.1415V-0.1174V) = 2.3917779tV ² ;

W _Y = 3.3938988tV ³ / (2.2830V / 2) = 2.97319218tV ² ;

- radii of inertia

i _X = V (3.4060621 / 7.8841294) ^1/2 = 0.6572784V;

i _Y = V (3.3938988 / 7.8841294) ^1/2 = 0.6561038V;

i _X / i _Y = 0.6572784 / 0.6561038 = 1.0017902≈1

with relative error

100 (0.6572784-0.656103 8) / (0.6572784 ... 0.6561038) = 0.1787 ... 0.1790%,

where V is the size of the gusset (rib part) of the section.

With the combined action of bending moments and compressive forces that take place in the belts of trusses of drive-free coatings, it is rational to develop bent-closed profiles in the force planes of load-bearing structures. For this, it is advisable to take the obtained ratios of the parameters of equidistant cross sections as the base so that for each design case, to develop the height of the section in series by one step size of the profile (rib part) of the section.

Then, if you increase the height by 1 dimension of the gusset and repeat all the calculations, then pentagonal bent closed profiles with an angle of 90 ° will have the following cross-sectional characteristics:

- overall height dimension H = 4V;

- overall dimension in width U = 3.4286V;

- cross-sectional area A = 12.8488tV;

- ordinate of the center of gravity of the section relative to the upper face

y _s = 1.3716V;

- moments of inertia I _X = 20.69807tV ³ , I _Y = 9.7878655tV ³ ;

- moments of resistance

W _{X, max} = 15.090457tV ² , W _{X, min} = 7.8747797tV ² , W _Y = 5.7095406tV ² ;

- radii of inertia i _X = 1.2692104V, i _Y = 0.8727959V.

If you develop the height by 2 sizes of the gusset, then pentagonal bent closed profiles with an angle of 90 ° will have the following characteristics of the cross section:

- overall dimension in height H = 5V;

- overall dimension in width U = 3.4286V;

- cross-sectional area A = 14.8488tV;

- ordinate of the center of gravity of the section relative to the upper face

y _c = 1.7929V;

- moments of inertia I _X = 37.802386tV ³ , I _Y = 9.7878655tV ³ ;

- moments of resistance

W _{X, max} = 21.084492tV ² , W _{X, min} = 11.787093tV ² , W _Y = 5.7095406tV ² ;

- radii of inertia i _X = 1,5955628V, i _Y = 0,811892V.

If you develop the height by 3 sizes of the gusset, then pentagonal bent closed profiles with an angle of 90 ° will have the following characteristics of the cross section:

- overall height dimension H = 6V;

- overall dimension in width U = 3.4286V;

- cross-sectional area A = 16.8488tV;

- ordinate of the center of gravity of the section relative to the upper face

y _c = 2.233V;

- moments of inertia I _X = 62.191082tV ³ , I _Y = 9.7878655tV ³ ;

- moments of resistance

W _{X, max} = 27.850909tV ² , W _{X, min} = 16.509445tV ² , W _Y = 5.7095406tV ² ;

- radii of inertia i _X = 1.9212309V, i _Y = 0.7621833V.

Similarly, if you increase the height by 1 dimension of the gusset and repeat all the calculations, then pentagonal bent closed profiles with an angle of 120 ° will have the following cross-sectional characteristics:

- overall height dimension H = 3.1415V;

- overall dimension in width U = 2.2830V;

- cross-sectional area A = 9.8841294tV;

- ordinate of the center of gravity of the section relative to the upper face

y _c = 1.1068V;

- moments of inertia I _X = 9.4786495tV ³ , I _Y = 3.393898tV ³ ;

- moments of resistance

W _{X, max} = 8.5643673tV ² , W _{X, min} = 4.6583949tV ² , W _Y = 2.9731921tV ² ;

- radii of inertia i _X = 0.9792735V, i _Y = 0.5859765V.

If you increase the height by 2 sizes of the gusset, then pentagonal bent closed profiles with an angle of 120 ° will have the following cross-sectional characteristics:

- overall height dimension H = 4.1415V;

- overall dimension in width U = 2.2830V;

- cross-sectional area A = 11.8841294tV;

- ordinate of the center of gravity of the section relative to the upper face

y _s = 1.5333V;

- moments of inertia I _X = 20.33266tV ³ , I _Y = 3.393898tV ³ ;

- moments of resistance

W _{X, max} = 13.260452tV ² , W _{X, min} = 7.7957595tV ² , W _Y = 2.9731921tV ² ;

- radii of inertia i _X = 1.308017V, i _Y = 0.5343991V.

If you develop the height by 3 sizes of the gusset, then the pentagonal bent closed profiles with an angle of 120 ° will have the following characteristics of the cross section:

- overall height dimension H = 5.1415V;

- overall dimension in width U = 2.2830V;

- cross-sectional area A = 13.8841294tV;

- ordinate of the center of gravity of the section relative to the upper face

y _s = 1.9775V;

- moments of inertia I _X = 37.037699tV ³ , I _Y = 3.393898tV ³ ;

- moments of resistance

W _{X, max} = 18.729941tV ² , W _{X, min} = 11.705822tV ² , W _Y = 2.9731921tV ² ;

- radii of inertia i _X = 1.6332876V, i _Y = 0.4944131V.

To compare pentagonal bent closed profiles (new technical solution) with the prototype, the panel of the upper truss belt made of steel of class C255, with a calculated length of 3 m in the plane, internal forces N = 412 kN and M = 16.7 kN⋅m, was adopted as the basic object. The waist profile (prototype) has a cross-sectional area of 24.6 cm ² and includes the tubular part of the pentagonal shape, consisting of one horizontal face (shelf) of 56 mm in size, two vertical faces (walls) of 54 mm in size and two inclined faces of 54 mm in size also a gusset (rib part) measuring 260 mm [Salakhutdinov MA, Kuznetsov I.L., Sayanov S.F. Steel trusses with belts from pipes of multifaceted cross-section. - Proceedings of KSASU, 2016, No. 4 (38). - S. 237, 238].

The prototype is represented by a bent closed profile with the following cross-sectional characteristics:

- overall height dimension H = 260 + 46.2 + 54 = 360.2 mm,

where (54 ² - (56/2) ² ) ^1/2 = 46.174≈46.2 mm is the projection of the inclined faces onto the vertical;

- overall width U = 56 mm;

- the perimeter of the section along the median line P = 56 + 2 (54 + 54 + 260) = 792 mm:

- sectional area A = 24.6 cm ² ;

- reduced section thickness t = A / P = 24.6 / 79.2 = 0.310606 cm≈3.106 mm;

- ordinate of the center of gravity of the section relative to the upper face

y _c = Hy _n = 360.2-19.49 = 16.53 cm,

where y _n - the distance from the center of gravity of the section to the lower edge,

y _n = 0.106 (5.6 × 36.02 + 2 (5.4 × 33.32 + 5.4 × 28.31 + 26 × 13)) / 24.6 = 19.485932≈19.49 cm,

360.2-54 / 2 = 333.2 mm - the distance from the center of gravity of the wall section (vertical face) to the lower edge,

360.2-54-46.2 / 2 = 283.1 mm - the distance from the center of gravity of the cross section of the inclined face to the lower edge,

260/2 = 130 mm - the distance from the center of gravity of the section of the gusset (rib) to the lower edge,

- moments of inertia

I _X = 0,3106 ³ × 5,6 / 12 + 0,3106 × 5,6 × 16,53 ² + 2 × 0,3106 × 5,4 ^3/12 + 2 × 0,3106 × 5,4 ( 33.32-19.49) ² + 2 × 0.3106 × 5.4 ³ × 0.731976 / 12 + 2 × 0.3106 ³ × 5.4 × 0.268863 / 12 + 2 × 0.3106 × 5.4 (28,31-19,49) ² + ² × 0,3106 × 26 ^3/12 + 0,3106 × 26 (19,49-13) ² = 2982.1048 cm ⁴

I _Y = 0,3106 × 5,6 ^3/12 + 2 × 0,3106 ³ × 5,4 / 12 + 2 × 0,3106 × 5,4 (5,6 / 2) ² + 2 × 0,3106 ³ × 5.4 × 0.731976 / 12 + 2 × 0.3106 × 5.4 ³ × 0.268863 / 12 + 2 × 0.3106 × 5.4 (5.6 / 4) ² + 2 × 0 , 3106 ³ × 26/12 + 2 × 0.3106 × 26 (0.3106 / 2) ² = 40.913022 cm ⁴ ,

where cos ² α = (46.2 / 54) ² = 0.731976, sin ² α = ((56/2) / 54) ² = 0.268862, α is the angle between the inclined face and the vertical (cos ² α + sin ² α = 1,000838≈1);

- resistance moment W _X = 2982.1048 / 16.53 = 180.40561 cm ³ ;

- radius of inertia i _X = (2982.1048 / 24.6) ^1/2 = 11.010166 cm.

Then the calculated voltage from the combined action of internal forces in the section of the panel from the profile of the prototype will be:

σ = N / (φA) + M / W X = 41200 / (0,9345 × 24,6) + 167000 / 180.40561 = 1792.2 + 925.7 = 2717.9 kgf / cm ² _y = 0,971R ,

where the design flexibility of the panel is λ = l / i _X = 300 / 11.010166 = 27.25; conditional (reduced) panel flexibility λ * = λ (R _y / E) ^1/2 = 27.25 (2800/2100000) ^1/2 = 0.995 <2.5; design resistance of steel of class C285 R _y = 2800 kgf / cm ² ; modulus of elasticity of steel E = 2100000 kgf / cm ² ; coefficient of longitudinal bending ϕ = 1-0.066 (λ *) ^3/2 = 1-0.066 (0.995) ^3/2 = 0.9345.

The new technical solution is represented by a pentagonal bent closed profile with an angle of 90 °, developed in height by 3 step sizes of the gusset, with the following parameters:

- sectional area A = 16.8488tV = 24.6 cm ² ;

- calculated step size

V = A / (16.8488t) = 24.6 / (16.8488 × 0.3106) = 4.7007231≈4.701 cm;

- overall height dimension H = 6V = 6 × 4.701 = 28.206 cm;

- overall width dimension U = 3.4286V = 3.4286 × 4.701 = 16.12 cm;

- ordinate of the center of gravity of the section relative to the upper face

y _c = 2.233V = 2.233 × 4.701 = 10.497 cm;

- moments of inertia

I _X = 62.191082tV ³ = 62.191082 × 0.3106 × 4.701 ³ = 2006.7824 cm ⁴ ;

I _Y = 9.7878655tV ³ = 9.7878655 × 0.3106 × 4.701 ³ = 315.83494 cm ⁴ ;

- moment of resistance

W _X = 27.850909tV ² = 27.850909 × 0.3106 × 4.701 ² = 191.17069 cm ³ ;

- radius of inertia i _X = 1.9212309V = 1.9212309 × 4.701 = 9.0317064 cm.

Then, the calculated voltage from the combined action of internal forces in the section of the panel from an equally stable profile according to the new technical solution will be:

σ = N / (φA) + M / W _X = 41200 / (0,9118 × 24,6) + 167000 / 191.17069 = 1836.8 + 873.6 = 2710.4 kgf / cm ² _y = 0,968R ,

where the design flexibility of the panel is λ = 300 / 9.0317064 = 33.22; conditional flexibility of the panel λ * = 32.22 (2800/2100000) ^1/2 = 1.213 <2.5; coefficient of longitudinal bending ϕ = 1-0.066 (1.179) ^3/2 = 0.9118.

As you can see, the calculated voltage in the new technical solution is 100 (2717.9-2710.4) / (2717.9 ... 2710.4) = 0.2759 ... 0.2767% less than in the prototype, and the prototype cross section is 36 , 02 / 28,206 = 1,277 times more than the new solution.

The cross-sectional characteristics in the new solution were calculated using formulas obtained by the method of stepwise approximations, therefore, their repeated (test) calculation along the midline of the thin-walled section is of practical interest, including taking into account numerical values containing the thickness values raised to the second and third degree (t ² , t ³ ):

- ordinate of the center of gravity of the section relative to the upper face

y _c = Hy _n = 28.206-17.712 = 10.494 cm

with relative error

100 (10.497-10.494) / (10.497 ... 10.494) = 0.02857 ... 0.02858%,

where y _n - the distance from the center of gravity of the section to the lower edge,

y _n = 0.3106 (16.12 × 28.206 + 2 (1.343 × 27.5345 + 11.3986 × 22.8335 + 18.804 × 9.402)) / 24.6 = 17.711522≈17.712 cm,

282.06-13.43 / 2 = 275.345 mm - the distance from the center of gravity of the wall section (vertical face) to the lower edge,

188.04+ (94.02-13.43) / 2 = 228.335 mm - the distance from the center of gravity of the section of the inclined face to the lower edge,

188.04 / 2 = 94.02 mm - the distance from the center of gravity of the section of the gusset (rib) to the lower edge,

- moments of inertia

I _X = 0,3106 ³ × 16,12 / 12 + 0,3106 × 16,12 × 10,494 + ² 2 × 0,3106 × 1,343 ^3/12 + 2 × 0,3106 × 1,343 (27,5345-17,712) ² + 2 × 0.3106 × 11.3986 ³ × 0.5 / 12 + 2 × 0.3106 ³ × 11.3986 × 0.5 / 12 + 2 × 0.3106 × 11.3986 (22.834-17.712) ² + 2 × 0,3106 × 18,804 ^3/12 + 0,3106 × 18,804 (17,712-94,02) ² = 2006.9992 cm ⁴

with relative error

100 (2006.9992-2006.7824) / (2006.9992 ... 2006.7824) = 0.0108%,

I _Y = 0,3106 × 16,12 ^3/12 + 0.3106 2 × ³ × 1,343 / 2 × 12 + 0.3106 × 1,343 (16,12 / 2) 2 × ² + 0.3106 × 11 ^3, 3986 × 0.5 / 12 + 2 × 0.3106 × 11.3986 ³ × 0.5 / 12 + 2 × 0.3106 × 11.3986 (16.12 / 4) ² + 2 × 0.3106 ³ × 18,804 / 12 + 2 × 0,3106 × 18,804 (0,3106 / 2) ² = 316,36144 cm ⁴ ,

with relative error

100 (316.36144-315.83494) / (316.36144 ... 315.83494) = 0.1664 ... 1.1667%,

where cos ² α = sin ² α = 0.5, α = 45 ° is the angle between the inclined face and the vertical.

The results obtained confirm the rationality and prospects of the new technical solution, as well as the correctness of its design parameters. It is advisable to continue the comparison with the prototype of the new technical solution, presenting it as a pentagonal bent closed profile with an angle of 120 °, developed in height by 3 step sizes of the shape, with the following parameters:

- sectional area A = 13.8841294tV = 24.6 cm ² ;

- calculated step size

V = A / (13.8841294t) = 24.6 / (13.8841294 × 0.3106) = 5.7044661≈5.7045 cm;

- overall height dimension H = 5.1415V = 5.1415 × 5.7045 = 29.33 cm;

- overall width dimension U = 2.283V = 2.283 × 5.7045 = 13.023 cm;

- ordinate of the center of gravity of the section relative to the upper face

y _c = 1.977454V = 1.97745 × 5.7045 = 11.28 cm;

- moments of inertia

I _X = 37.037699tV ³ = 37.037699 × 0.3106 × 5.7045 ³ = 2135.493 cm ⁴ ;

I _Y = 3.3938988tV ³ = 3.3938988 × 0.3106 × 5.7045 ³ = 195.68298 cm ⁴ ;

- moment of resistance

W _X = 18.729941tV ² = 18.729941 × 0.3106 × 5.7045 ² = 189.30975 cm ³ ;

- radius of inertia i _X = 1.6332876V = 1.6332876 × 5.7045 = 9.1370891 cm.

Then the calculated voltage from the combined action of internal forces in the section of the panel from an equally stable profile according to the new technical solution will be

σ = N (ϕA) + M / W _X = 41200 / (0.916 × 24.6) + 167000 / 189.300975 = 1828.4 + 882.2 = 2710.6 kgf / cm ² = 0.968R _y ,

where the design flexibility of the panel is λ = 300 / 9.1370891 = 32.30; panel conditional flexibility λ * = 32.30 (2800/2100000) ^1/2 = 1.176 <2.5; coefficient of longitudinal bending ϕ = 1-0.066 (1.126) ^3/2 = 0.916.

As you can see, the calculated voltage in the new technical solution is 100 (2717.9-2710.6) / (2717.9 ... 2710.6) = 0.2686 ... 0.2693% less than in the prototype, and the prototype cross section is 36 , 02 / 29.33 = 1.228 times more than the new solution.

To summarize, one more repeated (test) calculation is required along the midline of the thin-walled section:

- ordinate of the center of gravity of the section relative to the upper face

y _c = Hy _n = 29.33-18.712 = 11.300139 cm

with relative error

100 (11,30139-11,28) / (11,30139 ... 11,28) = 0,1893 ... 0,1896%,

where y _n - the distance from the center of gravity of the section to the lower edge,

y _n = 0.3106 (13.023 × 29.33 + 2 (2.7521 × 27.95395 + 7.519 × 24.6977 + 22.818 × 11.409)) / 24.6 = 18.02861 cm,

293.3-27.521 / 2 = 279.5395 mm - the distance from the center of gravity of the wall section (vertical face) to the lower edge,

228.18+ (65.115-27.521) / 2 = 246.977 mm - the distance from the center of gravity of the cross-section of the inclined face to the lower edge,

228.18 / 2 = 114.09 mm - the distance from the center of gravity of the section of the chamfer (rib) to the lower edge,

- moments of inertia

I _X = 0,3106 ³ × 13,023 / 12 + 0,3106 × 13,023 × 11,30139 ² + 2 × 0,3106 × 2,7521 ^3/12 + 2 × 0,3106 × 2,7521 (27,95395- 18.02861) ² + 2 × 0.3106 × 7.519 ³ × 0.25 / 12 + 2 × 0.3106 ³ × 7.519 × 0.75 / 12 + 2 × 0.3106 × 7.519 (24.6977-18, 02861) ² + 2 × 0,3106 × 22,818 ^3/12 + 0,3106 × 22,818 (18,02861-11,409) ² = 2135.5545 cm ⁴

with relative error

100 (2135.5545-2135.493) / (2135.5545 ... 2135.493) = 0.0029%,

I _Y = 0,3106 × 13,023 ^3/12 + 2 ³ × 0,3106 × 2,7521 / 12 + 2 × 0,3106 × 2,7521 (13,023 / 2) ² + 2 × 0,3106 × 7,519 × ³ 0.25 / 12 + 2 × 0.3106 ³ × 7.519 × 0.25 / 12 + 2 × 0.3106 × 7.519 ³ × 0.75 / 12 + 2 × 0.3106 × 7.519 ³ × 0.75 / 12 + 2 × 0.3106 × 7.519 (13.023 / 4) ² + 2 × 0.3106 ³ × 228.18 / 12 + 2 × 0.3106 × 228.18 (0.3106 / 2) ² = 196.14774 cm ⁴

with relative error

100 (196.14774-195.68298) / (196.14774 ... 195.68298) = 0.2369 ... 0.2375%.

where cos ² α = 0.25, sin ² α = 0.75, α = 60 ° is the angle between the inclined face and the vertical.

Thus, the above calculation calculations confirm their correctness and rationality of the proposed triangular bent closed profiles. At the same time, the universality of their technical solution, if necessary, allows having a section optimized according to the criterion of equilibrium stability and starting from it, as from the basic one, to select derived sections according to the parameters specified by the project.

Claims (1)

A bent profile of a closed section of a pentagonal shape with one horizontal face (shelf), two vertical faces (walls) and two inclined faces, as well as gussets from bent edges of workpieces joined to each other along the entire length, characterized in that the profile has the shape of a pentagon with an angle of 90 ° between the inclined faces, the height of which is two times the size of the gusset, or a pentagon with an angle of 120 ° between the inclined faces, whose height is 1.1415 times the size of the gusset.

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