RU2701404C1 - Cantilever beam of regularly variable height from shaped tubes (bent-welded sections) - Google Patents

Abstract

FIELD: construction.

SUBSTANCE: invention relates to construction, in particular to cantilever beam design. Cantilever beam of regular-variable height from shaped pipes (bent-curved profiles) has a section on the cantilever of round ring shape and in the closing – a flat-oval shape with height equal to 3.064 of the ring diameter, or cross-section of the beam on the cantilever has square shape and in closing – rectangular shape with height equal to 3.0 of the square size.

EFFECT: reduced consumption of structural material (steel), reduced level of stress concentration, increased efficiency of thin-walled profiles.

1 cl, 8 dwg

Description

The proposed technical solution relates to the field of construction, as well as the field of mechanical engineering, and can be demanded in relation to metal structures of beam type. In particular, they can be cantilever beams of coatings and ceilings, industrial shelves, bridges and overpasses, cargo booms and elevators, racks and other beam systems of buildings (structures) or machines (mechanisms).

A known method of manufacturing lightweight metal structures Kaminsky, in which the zigzag cut line of the vertical wall of the beam (including the cantilever) is placed at an angle to the longitudinal axis, and then one of the parts of the beam is rotated 180 ° [Kaminsky MB A method of manufacturing lightweight metal structures Kaminsky. - Copyright certificate No. 548402, 02/28/1977, bull. No. 8]. The main disadvantage of such a technical solution is the limited lateral stiffness of the beam in the case of an extended overhang of the console.

A well-known solution is also a method of manufacturing a steel support of a regular sectional height-variable section of two sections formed by the formed parts of a square pipe cut obliquely on each side in the form of equal-angle corners [Kuznetsov I.L., Sabitov L.S. A method of manufacturing a steel support of regular variable cross-section. - Copyright certificate No. 2495213, 10/10/2013, bull. No. 28]. Such a solution can be considered as an analogue, however, it is advisable to modify it to increase the rigidity in the force plane of the supporting structure of the beam type.

The cantilever beam of equal resistance of variable cross-section of a rectangular shape, the height of which on the cantilever and in the seal, has certain correlations with the height of the cross section found from the greatest (maximum) bending moment [Stravinskas S.S.] has the necessary and sufficient rigidity in the force plane. Cantilever beam. - Copyright certificate No. 1596034. - 09/30/1990, bull. 36]. The above solution is closest to the proposed technical solution, which allows us to take it as a prototype, which needs some refinement for metal structures. If solid sections of a rectangular shape are very effective for reinforced concrete or wooden structures [Karamysheva AA, Yazyev BM, Chepurenko AS, Yazyeva SB Optimization of geometric parameters of a gable beam of rectangular cross section. - Engineering Journal of the Don, 2015, No. 3 (ivdon.ru/ru/magazine/archive/n3y2015/3138)], then in steel beams shaped and thin-walled profiles are more rational.

The technical result of the proposed solution is to reduce the consumption of structural material (steel), reduce the level of stress concentration, increase the efficiency of thin-walled profiles.

The specified technical result is achieved by the fact that in cantilever beams of regularly variable height from profile pipes (bent sections) in one embodiment, the section of the beam on the console has the shape of a round ring and in the seal, a flat oval shape with a height equal to 3.064 of the diameter of the ring, and in another embodiment, the cross section the beams on the console have the shape of a square and in the embedment - the shape of a rectangle with a height equal to 3.0 the size of the square.

The proposed cantilever beams of regularly variable height from profile pipes (bent sections) have a completely universal technical solution, with the implementation of which it is possible to use them not only inside buildings, but also in open buildings such as flyovers and bridges, industrial shelves, cargo booms, ship masts, antenna structures, pylons of power lines, racks and other load-bearing systems of beam type.

Comparing the proposed technical solution with its analogue, it is not difficult to notice that it is much easier to dissolve one closed bent welded profile of rectangular section into two channels, or one plane-oval profile pipe into two U-shaped profiles, than into four corners. It is no less obvious that the continuous cross-section of the rectangular shape from the prototype in metal structures, including steel beams, is significantly inferior in terms of efficiency and rationality to profile pipes (bent-welded profiles). At the same time, the thin-walled and closed cross-sectional shape contributes to a decrease in the consumption of structural material (steel) and an increase in corrosion resistance, and the round profile shape leads to a decrease in the stress concentration level. An additional increase in the efficiency of thin-walled profiles is also ensured by the fact that the regularly variable height of the cantilever beams fully corresponds to the shape of the diagrams of bending moments from a concentrated and distributed load.

The proposed technical solution is illustrated by graphic materials, where

FIG. 1 shows a perspective view of a cantilever beam of regularly variable height from a flat oval profile pipe;

FIG. 2 is a perspective view of a cantilever beam of a regularly variable height from a rectangular profile pipe;

FIG. 3 shows a design diagram of a cross section of a flat oval profile pipe;

FIG. 4 is a design diagram of a cross section of a rectangular profile pipe;

FIG. Figure 5 shows graphs of changes in the moments of resistance of a section of a flat oval shaped pipe, as well as the moments of resistance of a section with constant voltage (equal to the calculated resistance of the material) at a concentrated and distributed load, depending on the ratio between the height of the section and its width;

FIG. 6 - graphs of changes in the moments of resistance of the cross section of a rectangular profile pipe, as well as the moments of resistance of the section with constant voltage (equal to the calculated resistance of the material) at a concentrated and distributed load, depending on the ratio between the height of the section and its width;

FIG. 7 is a diagram of an example of forming a cantilever beam of regularly variable height from a rectangular shaped pipe;

FIG. 8 is a diagram of an example of shaping a cantilever beam of regularly variable height from a flat oval profile pipe.

From the load concentrated on the console, the diagram of bending moments has a straight line with a maximum value in the seal and zero on the console [Reference for resistance of materials / Resp. ed. Pisarenko G.S. - Kiev: Naukova Dumka, 1988. - S. 117]:

M _z = -Fz;

M _z = 0 at z = 0;

F - нагрузка, сосредоточенная на консоли;

- длина вылета консольной балки; M_max - наибольший (максимальный) изгибающий момент в заделке.where M _z is the bending moment in the considered section of the cantilever beam; z is the coordinate of the considered section,

F is the load concentrated on the console;

- the length of the extension of the cantilever beam; M _max - the largest (maximum) bending moment in the seal.

Obviously, the graph (plot) of the required moments of resistance of the cantilever beam section will also be straightforward:

where W _xF is the required moment of resistance of the considered section from the concentrated load; R _y is the design resistance of the material of the cantilever beam (steel).

From the load distributed over the entire length of the cantilever beam, the bending moment diagram has a parabolic concave outline with a maximum value in the embedment and zero on the console [Handbook of material resistance / Res. ed. Pisarenko G.S. - Kiev: Naukova Dumka, 1988. - S. 118]:

M _z = -qz ^2/2;

M _z = 0 at z = 0;

where q is the intensity of the load distributed evenly over the entire length of the extension of the cantilever beam.

The graph (plot) of the required moments of resistance of the section of the cantilever beam will also be curved:

W _xq = 0 at z = 0;

where W _xq is the required moment of resistance of the considered section from the distributed load.

To both graphs (diagrams), a cantilever beam of equal resistance of a regularly variable height from a profile pipe (bent-welded profile), the cross section of which on the console has the shape of a circular ring, and in the seal, a flat-oval shape with a height equal to 3.064 of the diameter of the ring, completely corresponds. With this ratio of height and width, the flat-oval pipe has a maximum moment of resistance of the section [Marutyan AS A new way of reprofiling round pipes and calculating its optimal parameters for beam structures. - Structural mechanics and calculation of structures, 2018, No. 1. - S. 66-72]:

A _o = tU (2 / n + 1.14);

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

W _xo = tU ² (0.3333333 / n ² + 0.57 / n + 0.0216012n-0.13982);

W _xo = 0.7851145tU ² at V / U = 1;

W _xo = 1.0421807tU ² with V / U = 1.2 = 1 / 0.8333333;

W _xo = 1.3269426tU ² with V / U = 1.4 = 1 / 0.71428857;

W _xo = 1.6390138tU ² with V / U = 1.6 = 1 / 0.6250;

W _xo = 1.9781808tU ² with V / U = 1.8 = 1 / 0.5555555;

W _xo = 2,3443139tU ² with V / U = 2.0 = 1 / 0.50;

W _xo = 2.7373322tU ² with V / U = 2.2 = 1 / 0.4545454;

W _xo = 3.1571809tU ² with V / U = 2.4 = 1 / 0.4166666;

W _xo = 3.6038223tU ² with V / U = 2.6 = 1 / 0.3846153;

W _xo = 4.0772285tU ² with V / U = 2.8 = 1 / 0.3571428;

W _xo = 4.5773803tU ² with V / U = 3.0 = 1 / 0.3333333;

W _xo = 4.7431359tU ² with V / U = 3.064 = 1 / 0.3264,

where t is the thickness of the profile pipe; U is the width of the profile pipe along the midline of its thin-walled section; n is the ratio of the width to the height of the profile pipe along the midline of its section, n = U / V; V is the height of the profile pipe along the midline of its section.

A rectangular bent-welded profile is more compact than a flat oval pipe if they are profiled from the same sheet blanks (strips), so in cantilever beams of equal resistance of a regularly variable height, it can be no less rational and effective. Its cross section can be considered composite of four rectangular sections: two horizontal faces (shelves) and two vertical faces (walls). In this case, it is permissible to perform computational calculations along the midline of a thin-walled section without taking into account numerical values containing thicknesses raised to the second and third degree (t ² , t ³ ), as well as without taking into account angular roundings [Marutyan A.S. 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].

When calculating the axial moments of inertia of the cross section of a rectangular pipe, you can use the rule of parallel transfer of axes and, after appropriate transformations, obtain the calculation formulas in the following form:

I _x = tU ³ (0.1666666 / n + 0.5) / n ² ;

I _y = tU ³ (0.1666666 + 0.5 / n),

where the value of the larger side, which is the height of the wall, V is replaced by its ratio to the smaller side, which is the width of the shelf, U, that is, V = U / n, and t is the thickness of the rectangular pipe.

The cross-sectional area of a thin-walled rectangular profile can be calculated along the length of the midline:

A = 2tU (1 + 1 / n).

Cross-section moments:

W _x = 2I _x / V = tU ² (0.3333333 / n + 1) / n;

W _y = 2I _y / U = tU ² (0.3333333 + 1 / n) / n.

If, to continue the computational calculations, as the initial data, we take the section area A = const and the thickness t = const of the strip (sheet blank), then the section parameters can be rewritten in the following form:

U = (A / t) (0.5 / (1 + 1 / n));

V = (A / t) (0.5 / (1 + n));

I _x = (A ³ / t ² ) (0,0208333 / n + 0,0625) / (n ² (1 + 1 / n) ³ );

I _y = (A ³ / t ² ) (0.0208333 + 0.0625 / n) / (1 + 1 / n) ³ ;

W _x = (A ² / t) (0.0833333 / n + 0.25) / (n (1 + 1 / n) ² );

W _y = (A ² / t) (0.0833333 / n + 0.25) / (1 + 1 / n) ² .

To find the extreme value of the moment of resistance of the cross section W _x, its formula must be differentiated with respect to the variable n and equating the derivative (dW _x / dn = 0) to zero, we obtain an equation of the second degree

n ² + 0.6666666n-0.3333333 = 0

with roots

n ₁ = -0.9999999; n ₂ = 0.3333333.

Of these roots, the second is of practical interest, the value of which can be rounded to

n = 0.3333333≈1 / 3.

Thus, the obtained graphs (diagrams) W _xF and W _xq correspond to a cantilever beam of equal resistance of a regularly variable height from a rectangular profile pipe (bent-welded profile), the cross section of which on the console has the shape of a square, and in the seal - the shape of a rectangle with a height of 3, 0 square size:

W _xq = tU ² (0.3333333 / n + 1) / n;

W _xq = 1.3333333tU ² at V / U = 1;

W _xq = 1.680tU ² with V / U = 1.2 = 1 / 0.8333333;

W _xq = 2.0533333tU ² with V / U = 1.4 = 1 / 0.71428857;

W _xq = 2.4533333tU ² with V / U = 1.6 = 1 / 0.6250;

W _xq = 2.880tU ² with V / U = 1.8 = 1 / 0.5555555;

W _xq = 3.3333333tU ² with V / U = 2.0 = 1 / 0.50;

W _xq = 3.8133333tU ² with V / U = 2.2 = 1 / 0.4545454;

W _xq = 4.320tU ² with V / U = 2.4 = 1 / 0.4166666;

W _xq = 4.8503333tU ² with V / U = 2.6 = 1 / 0.3846153;

W _xq = 5.4133333tU ² with V / U = 2.8 = 1 / 0.3571428;

W _xq = 6.0tU ² with V / U = 3.0 = 1 / 0.3333333.

An example of the formation of a cantilever beam of a regularly variable height from a rectangular profile pipe can be presented on the basis of a closed bent welded profile with a section of 160 × 80 × 4 mm according to TU 67-2287-80 (A = 18.56 cm ² , I _x = 623.5 cm ⁴ , W _x = 77.9 cm ³ ) [Metal structures. In 3 t. T. 1. General. (Designer Handbook) / Ed. V.V. Kuznetsov (TsNIIproektstalkonstruktsiya im.N.P. Melnikova) - M.: Publishing House ASV, 1998. - S. 135]:

n = U / V = (80-4) / (160-4) = 76/156 = 0.4871794 = 1 / 2.0526315;

A = 2tU (1 + 1 / n) = 2 × 0.4 × 7.6 (1 + 1 / 0.4871794) = 18.559999≈18.56 cm ² ;

W _x = 623.002 / (16.0 / 2) = 77.87525≈77.872 cm ³ .

With a double oblique cut, this profile can be disassembled into two channels, the resulting channels are turned 180 ° relative to each other and superimposed double weld beads by arranging a cantilever beam of regular height with a square section of 80 × 80 × 4 mm on a console and a rectangular section of 240 × 80 × 4 mm in the seal:

n is 1;

A = 2 × 0.4 × 7.6 (1 + 1/1) = 12.16 cm ² ;

I _x = 0.4 × 7.6 ³ (0.1666666 / 1 + 0.5) / 1 ² = 117.01025 cm ⁴ ;

W _x = 117.01025 / (8.0 / 2) = 29.265062≈29.265 cm ³ ;

n = (80-4) / (240-4) = 76/236 = 0.3220338 = 1 / 3.1052631;

A = 2 × 0.4 × 7.6 (1 + 1 / 0.3220338) = 24.96 cm ² ;

I _x = 0.4 × 7.6 ³ (0.1666666 / 0.3220338 + 0.5) / 0.3220338 ² = 1722.8638 cm ⁴ ;

W _x = 1722.8638 / (24.0 / 2) = 143.57198≈143.572 cm ³ .

м с нагрузкой F=8 кН на консоли:If the cantilever beam of the given arrangement is made of mild steel of class C235 (R _y = 2300 kgf / cm ² ), then its bearing capacity is quite enough for the length

m with a load of F = 8 kN on the console:

where σ _max is the normal stress from the maximum moment, σ _max = M _max / W _x .

The maximum transverse force Q _max = 8 kN acts on the console; therefore, the strength of the square section must be checked by the Zhuravsky formula:

τ _max / R _s = Q _max S _x / (I _x t _s R _s ) = 800 × 17.328 / (117.01025 × 0.8 × 1350) = 0.1097 <1,

where τ _max is the shear stress from the maximum transverse force, Q _max = F; S _x is the static moment of the movable part of the section relative to the neutral axis, S _x = 2 × 0.4 × 3.8 × 1.9 + 0.4 × 7.6 × 3.8 = 17.328 cm ³ ; t _s is the thickness of the movable wall, t _s = 2 × 0.4 = 0.8 cm; R _s is the calculated steel shear resistance, for steel of class C235 R _s = 1350 kgf / cm ² .

It remains to check the deflection on the console, for which you can apply the formula of the cantilever beam of variable cross-section of a rectangular shape [Reference for resistance of materials / Fesik SP - Kiev: Budivelnik, 1982. - S. 144]:

where E is the modulus of elasticity of the structural material, for steel E = 2100000 kgf / cm ² ; b _ƒ and h are the width and height of the beam section on the console, respectively.

The given (conditional) value of the cross-sectional width on the console can be determined by taking as a prerequisite the equality between the moment of inertia of the profile pipe and the moment of inertia of the rectangular section:

b _ƒ = 12I _x / h ³ = 12 × 1722.8638 / 24 ³ = 1.4955414≈1.496 cm.

As can be seen, the cantilever beam of a regularly variable height from a bent welded profile of rectangular section has the necessary and sufficient margin of bearing capacity, therefore, its comparison with a similar beam from a flat oval profile pipe is of practical interest.

For a more correct comparison, the initial bent welded profile of a rectangular shape with a section of 160 × 80 × 4 mm must be conditionally re-profiled into a flat oval pipe:

n = 1.5 / 3.064 = 0.4895561 = 1 / 2.0426666;

U = A _o / (t (2 / n + 1.14) = (18.56 / (0.4 (2 / 0.4895561 + 1.14)) = 8.8798158 cm;

V = 8.8798158 / 0.4895561 = 18.138505 cm;

W _x = 2I _x / (V + t) = 2 × 693.9256 / (18.138505 + 0.4) = 74.863167 cm ³ .

Conditionally reprofiled pipe has a flat oval section with dimensions

18.138505 + 0.4 = 18.538505≈185.4 mm;

8.8798158 + 0.4 = 9.2798158≈92.8 mm,

that is, its dimensions increased 185.4 / 160 = 1.1588 and 92.8 / 80 = 1.160 times, the moment of inertia increased 693.9256 / 623.5 = 1.113 times, and the moment of resistance decreased 77.9 / 74 , 863167 = 1.0406 times.

With a double oblique cut, a conditionally re-shaped pipe with a plano-oval cross-section of 185.4 × 92.8 × 4 mm can be split into two U-shaped profiles, turned the resulting profiles 180 ° relative to each other, and double welded beads are applied, arranging the cantilever beam regularly - a variable height with a round cross-section of 92.8 × 4 mm on the console and a flat oval cross-section of 278 × 92.8 × 4 mm in the seal:

n is 1;

U = 92.8-4 = 88.8 mm;

A = 0.4 × 8.88 (2/1 + 1.14) = 11.15328 cm ² ;

W _x = 109.95166 / (9.28 / 2) = 23.696478 cm ³ ;

n = (92.8-4) / (278-4) = 88.8 / 274 = 0.3240875 = 1 / 3.0855855;

A = 0.4 × 8.88 (2 / 0.3240875 + 1.14) = 25.969286≈25.97 cm ² ;

W _x = 2074.0032 / (278.0 / 2) = 149.20886≈149.209 cm ³ .

м с нагрузкой F=8 кН на консоли с большим запасом:If the cantilever beam of the given arrangement is made of mild steel of class C235, then its bearing capacity will be enough for the length

m with a load of F = 8 kN on a console with a large margin:

τ _max / R _s = Q _max S _x / (I _x t _s R _s ) = 800 × 15.7723 / (109.95166 × 0.8 × 1350) = 0.1063 <1,

where S _x is the static moment of the shifted part of the section in the form of a round half ring relative to the neutral axis [Marutyan AS, Abovyan AG Calculation of the optimal parameters of flat oval pipes for trusses. - Building mechanics, 2017, No. 4. - S. 18],

S _x = 1.57 × 0.4 × 8.88 × 0.3185 × 8.88 = 15.772299≈15.7723 cm ³ .

Checking deflection on the console is as follows

where the given (conditional) value of the section width on the console is

b _ƒ = 12I _x / h ³ = 12 × 2074.0032 / 27.8 ³ = 1.1583939≈1.158 cm.

As you can see, the bearing capacity resource of the cantilever beam of regularly variable height from the same sheet billet (strip) with the transition from a rectangular closed bent section to a flat oval profile pipe at the maximum normal stress increased in 0.7268 / 0.6993 = 1.03933 times, by the maximum tangential stress - 0.1097 / 0.1063 = 1.03198 times, and by the maximum deflection - 3.98 / 3.31 = 1.2024 times.

Thus, summing up some results, we can come to the main conclusion that the proposed cantilever beams of regularly variable height from profile pipes (bent-welded profiles) are quite effective, rational and promising for use as part of the supporting structures of buildings (structures) or machines (mechanisms).

Claims (1)

A cantilever beam of regularly variable height from profile pipes (bent-welded profiles), characterized in that the section of the beam on the console has the shape of a round ring and in embedment is a flat oval shape with a height equal to 3.064 of the diameter of the ring, or the section of the beam on the console has the shape of a square and in the seal - the shape of a rectangle with a height equal to 3.0 of the size of the square.

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