Calculation features of trapezoidal combined ribbed slabs on wooden frame Текст научной статьи по специальности «Строительство и архитектура»

DOI: 10.17516/1999-494X-0208 yflK 624.011

Calculation Features of Trapezoidal Combined Ribbed Slabs on Wooden Frame

Viktor I. Zhadanov*a, Ivan S. inzhutovb and Dmitrij A. Ukrainchenkoa

aOrenburg State University Orenburg, Russian Federation bSiberian Federal University Krasnoyarsk, Russian Federation

Received 18.11.2019, received in revised form 25.12.2019, accepted 21.01.2020

Abstract. The article describes the features of design and calculation of ribbed slabs on wooden frame with trapezoidal plan, where the cover is included in the general work of the structure on operational loads. The field of possible application of the investigated elements is shown. The features of trapezoidal slab design are described. The main provisions of the proposed method of the considered structures class calculation are reflected. The formula dependencies of trapezoidal slabs calculation on deformations have been determined due to the use of the calculation method of structures that consist of different module materials; the outcomes of previous numerical studies carried out by the authors of the paper as well as the algorithms of a numerical functions determination based on a particular integral method. The location of a weak slab section when calculating on a normal stress has been defined with the regard to a part of an operating cover, which varies along the span length. It is shown that the proposed method allows to assess adequately the stress-deformed state of trapezoidal slabs and to calculate them on strength and rigidity according to the suggested algorithm.

Keywords: trapezoidal ribbed slabs, combined structure, application area, rib, cover, diaphragm, calculation method, stress-deformed state, reduction factor, weak section, calculation algorithm, approximation.

Citation: Zhadanov V.I., Inzhutov I.S., Ukrainchenko D.A. Calculation features of trapezoidal combined ribbed slabs on wooden frame, J. Sib. Fed. Univ. Eng. & Technol., 2020, 13(1), 100-110. DOI: 10.17516/1999-494X-0208

© Siberian Federal University. All rights reserved

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0). Corresponding author E-mail address: organ-2003@bk.ru

Особенности расчета

трапециевидных совмещенных ребристых плит на деревянном каркасе

В.И. Жаданова, И.С. Инжутовб, Д.А. Украинченкоа

аОренбургский государственный университет Российская Федерация, Оренбург бСибирский федеральный университет Российская Федерация, Красноярск

Аннотация. В статье рассмотрены особенности конструирования и расчета ребристых плит на деревянном каркасе с трапециевидным планом, у которых обшивка включена в общую работу конструкции на восприятие эксплуатационных нагрузок. Показана область возможного применения исследуемых элементов. Описаны особенности конструкции трапециевидных плит. Отражены основные положения предлагаемой методики расчета рассматриваемого класса конструкций. С применением метода расчета конструкций, состоящих из разномодульных материалов, результатов ранее проведенных авторами статьи численных исследований, а также алгоритмов численного определения функций на основе методики определенного интеграла, установлены формульные зависимости расчета трапециевидных плит по деформациям. Местоположение опасного сечения плиты при расчете по нормальным напряжениям вычислено с учетом доли работающей обшивки, которая изменяется по длине пролета. Показано, что предложенная методика позволяет адекватно оценивать напряженно-деформированное состояние трапециевидных плит и производить их расчет на прочность и жесткость по предложенному алгоритму.

Ключевые слова: трапециевидные ребристые плиты, совмещенная конструкция, область применения, ребро, обшивка, диафрагма, методика расчета, напряженно-деформированное состояние, коэффициент приведения, опасное сечение, алгоритм расчета, аппроксимация.

Цитирование: Жаданов, В.И. Особенности расчета трапециевидных совмещенных ребристых плит на деревянном каркасе / В.И. Жаданов, И.С. Инжутов, Д.А. Украинченко // Журн. Сиб. федер. ун-та. Техника и технологии, 2020. 13(1). С. 100-110. DOI: 10.17516/1999-494Х-0208

Introduction

The construction volume growth with the use of wood and the development of the glued wooden structures base have resulted in a wide application of ribbed slabs on the wooden frame with the largest overall dimensions that are possible according to technological parameters and transportability conditions in buildings of various purposes. Such structures allow to increase the degree of factory elements readiness that are supplied to the construction site; to reduce the objects construction time and, at the same time, to improve the quality of installation and construction works. The highest degree of efficiency is provided by slabs, where the combination principle of bearing and enclosing functions is realized. This principle suggests that the main longitudinal ribs act as beams and covers together with auxiliary elements are fences of buildings [1-3]. In the slabs on the wooden frame, such combination is achieved by gluing the cover to the ribs, and due to that they are included into the overall operation of the structure. It significantly increases the geometric characteristics of the cross sections.

The stress-strain state features of the combined slabs on the wooden frame with a rectangular plan as well as the method of their calculation have been studied by Russian and foreign researchers. In Russia the studies were carried out by V.N. Bykowsky, A.B. Gubenko, P.A. Dmitriev, V.A. Ivanov, Y.M. Ivanov, L.M. Kowalchuk, S.V. Kolpakov, S.G. Lyakhnitsky, I.M. Linkov, D.V. Martinez, R.B. Orlovich, G.G. Rostovtsev, E.I. Svetozarova, E.N. Serov, B.S. Sokolovsky, Y.D. Strizhakov. Modern Russian normative documents on rectangular slabs calculation on wooden frame with the regard to joint work of ribs and plywood covers are based on the results of experimental and theoretical studies of A.B. Gubenko [4]. The calculation is carried out according to the method of transformed section. In this case, wood is brought to the cover material by means of the coefficient n = Ei/E0, which considers different modularity of materials in the calculated sections. In addition, the cover area in the calculation is introduced in terms of the reduction factor ko6, which considers the cover inclusion degree in the general operation of the structure [5-7].

In this respect, regarding the high architectural and aesthetic qualities of raised objects, the cover configurations in plan often differ from rectangular. It is predetermined by the use of trapezoidal elements. A perfect example of it is a building with polygonal, circular, ellipse plans in dome structures, etc. When using trapezoidal slabs in roof or floors in buildings, a designer deals with the problem that the width of their cross-section and the magnitude of the real load change linearly along the whole length of the span, and the height of the main ribs can also change, for example, in order to create the necessary roof slope. When changing the width of the cross section, the degree of cover involvement in the overall operation of the slab also changes. Such degree generally depends on the space between ribs, and it varies linearly, as well as the width along the length of the span. However, these features are considered neither in available methods or normative and technical sources. There are no clear methods for calculating combined rib slabs on a wooden frame with a trapezoidal plan. In this regard, the development of such methods is an urgent task, which will increase the competitiveness of wooden panel structures and contribute to the expansion of their implementation in construction practice.

research object

The object of the research is a trapezoidal ribbed slab on a wooden frame with a cover that is glued to the main ribs and involved in the overall operation of the structure.

research objective

The objective of the research is to investigate theoretically the features of the stress-deformed state of trapezoidal combined rib slabs on the wooden frame with the calculation suggestions.

research Methodology

When studying and experimenting, some tested and widely used methods of calculation of structures that consist of different module materials; outcomes of previous numerical studies; algorithms of numerical functions determination based on the methodology of a particular integral were used.

Results of Research

The features of the plan regarding the structural technological requirements determine a few issues that are necessary to be considered when developing the trapezoidal slabs. The issues are as follows:

- minimum width of the slab should be no less than 500 mm (the condition of structural connections with underlying constructions);

- maximum width of the slab is limited to 3000 mm (transportation condition);

- the cross section is generally U-shaped (condition of minimum distance between major ribs);

- orientation of auxiliary ribs is taken across the slab span (conditions for stability of compressed cover and manufacturing).

The example of design solution of trapezoidal slab and its parameters are given in Fig. 1.

In addition, when determining the overall dimensions of slabs with trapezoidal plan, it is necessary to consider the specified design data of the building or structure.

For example, let us consider a circular building in plan, where R, r - the dimensions of the outer and inner support contours, ln - the slab span (Fig. 2). The tolerances for values b0, b1, ln are given:

b0 min — b0 — b0 max

b1 min — b1 — b1 max (1)

l0 =6,0... 18,0 m.

r 1-1

1 V

<6000 \4- <6000 1 <6000

6000... 18000 ;

----<=t--—1

J S — 8 —

bx Yi_ 1

Fig. 1. The main parameters of trapezoidal slabs:1 - mainribs;2 - auxiliary ribs; 3 - cover; 4 - diaphragms

Trapezoidal

C)

J

Fig. 2. The trapezoidal slabs calculation: a - designation of building parameters with a circular plan; b - designation of slab parameters; c - design diagram

Asarule,thevalues/aftand — areinthe range of l,5<b0<3,0m; 0,5 <bs <7,0m for trapezoidal slabs.

The outer radius R is the main initial paraerater. 0hea,thbnumbrrof5verlapping slaOs "w" win be in the range:

(2)

where nmin = 27rR - rounded to an integerup; nmax = - rounded to anintegerdown.

b0max b0min

Accordingly, the required radius "r" of the inner support contour with the selected number of slabs is inthe range:

n ' b]mjn < r < n ' blmax (3)

2n 2n

The total tolerance interval for "r" is:

nmin ' b1min < r < nmax ' b1 max (4)

2n 2n

When the radius of the inner support contour is set, it must be checked for validity according to (4).

- ioa -

nmin — n — nmax>

The choice of the slabs size in the plan depends to a large extent on the conditions. When it is necessary to maximize the part of the plan area covered by the slabs (for example, for tower structures), the following should be acc epted:

r — r — nmin 'b/min (5)

' 'min t ' y '

ln = lmax = R — rmin ■ (6)

For example, let the outer radius of the tower-type structure be ^=18,0 m. Then, according to the above suggestions:

2-3,14-d8 nmin =-7-= 37,68 ^ nmm = 38,

38 • 0,5 rmin = 2 ^ 314 = 3,025m ,

i.e., in the structure ehher an adohtioual aewof columns tocatedalon-the mnecradmsora nentral column with 3.025 m canOHeaerr id —;quireq tu jere^e-ride the mnersuppost cnntoee.

According to thedesign dragram (Frg. 2et, tqe7 k-catio n of w, ak section otrl ciilcsuridt;(oos ar s^ scon deformations are the teaSuers of tsopecddq- sil^^.rsi catdeitption. ."Jce foUqwrng meinartotogq cs o iterstsd><^d to consider such features.

When determining she locntion of Iter woak lectin. tro Sracnooidal nlan n^^lus, in in nrcensary eo note that the step of mrtjor aitts is? rSneiiiiln. vaiiable akrng ttie lts-gtdi oi ihe slnbs. tti- ohie case, rFe reduction factor of plywoot t;o^t:d 7ttpenli onthe coordinate <tF the sechost it and ean ite determined according to the formulc:

ko6(e) = k2(e)' *eep +ki(e) • sp + ko(e)2 (7)

where, k2(x), k1(x), k0(x) - functions that can be found from quadratic approximation:

k2(e) e k2(bn(e)) = tt22 ■b2n(e) + tt21 ■bJe) + tt00 >

k1(e) e ki(bf( e)) = tir ■ bf;( e) + tn ■ bf( e) + tio, (8)

k0(x) = k0{bn(xj) = a02 ■ b2n(x) + a01 ■ bn(x) + a00,

8$ - plywood cover thickness, cm.

The dependencies for the coefficients of quadratic approximations in formulas (8) are easily found in terms of the quadratic approximations that are given [8] with respect to rectangular plan ribbed slabs. For example, tlie vaiurs of thecoefficientsof rhequndratlc approrimaiion/^S^Stl ran tie found by solving a system oflint ar ecuatitns:

a22 ■ 0,752 + a21 ■ i,75 + a2i = -0,042

a22 ■ 1,02 + a20 ■ 1,0 + a20 = -0,009 ■ (9)

a22 ■ 3,02 + a21 ■ 3,0 + a20 = -0,014

The approximation coefficientsfor k(bn(x)) and k0(bn(x)) cansimilarlybedefined.

The function bn(x) is linear,that is, it can be defined according to the symbols of Fig. 2 from the; expression:

b„(x)= bo+n--b0 c^ (10)

If we substitute (10) for (8), thefinal dependenciet fosthe apptoximationcoefficienis (7) rte obtained.

Then, the algorithm ot the weak sention location, iikn tlra sMi) of nonstanf widnh, is as follows.

1. Define the designwidth oe jpl^g^oo^

0 (c) = ko6 (x) • (b e^^-mj + b- mop. (11)

2. Calculate the area of the transfotmed section of the steb:

AP = A + n • A^ (act) =1 m npb(hin + i • x) + n • f (x). (12)

3. Calculate the reduced static moment of arbitrary rection:

1 , „ . . . „ . . i, . • . St

Sp ( x) = 2 h + i-x)2+r,-At (*) • \hon +i-x + -*-J. (13)

4. Determine thepositionofneutral axis ofarbitrarysection:

S„„ (X )

'tû (14)

5. Find the moment of inertiaof an arbitrary cross se ction:

J ( x) = m ■ b ■ (h + i ■ x)

np \ / op v on /

+ n■ bt(x)\ hon + i ■ x-y0 + y 1 .

(h + i ■x) I , „ (h + i ■x)

—-- +1 y0(x)- —--

12 ^ 0 12

2

(15)

Make the expression (tor algorithm) for determining normal stresses at the points of slab arbitrary section:

*(°,y) = M<*>■<»><X - y> ,

hp(x) (16)

where M(x) - bending moment in arbitrary section of slab 0 < x < ln. Define the maximum:

M (x)^(yo ( X) - y )

' Jnp(x )

(17)

0 < x < ln,0< y <(hon + i • xy+Si When only normal stresses are considered while determining the weak section which is quite true for combined slabs where the ratio of the maximum height of the main ribs cross section to the span is small and is about 1/15... 1/24, it is possible to check only the extreme (bottom and top) points of the section by normal stresses,thatis

Journal of Siberian Federal University. Engineering & Technologies2020 13(1): 100-110 0,

y hon + i • x + S<P '

The determinationof max diai(lFretcnnbe cabcutated by :ny atrengtFrheory. TFe normal, tangent and main stresses mustbedeaermineb ei the; pnints of the ceos r eection.

The maximum values of the main stresses maxx,y \a3Ke(x,y)\ or maxxya3Ke(x,y) can be determined by means of a search algorithm of an absolutely extreme problem solution, including by scanning over a selected grid x,y.

To determine trapezoidaO ila0 deynrmatiioni, it isadvioable Oouse diiferenyial equeition of oeams bending:

^JT (18)

H J np (X)

Using the methodology of a particular integral, let us build an epure u.

The numerical determination algorithm of u (x) based on the methodology of a particular integral is:

1. Choose a step cf numerical integration A and build the epure 00 (xy— it approximately

E ' J np (x)

coincides witl^the epure v"(x) (Fig. 3, a. where« i s t he nu rtdii^r ofspliiting sites of a slab, 0 < x < n • A,

X = A •j.

For design fc3eme with hinged supportM(0) = M(l)= 0.

3n M((X} «3nv '(x). (19)

E • Jnp (*)

1 !)•(/„: n jc

d;(O)=U;

Fig. 3.Na3ctreofepure-=— m Эп d"(=) at distributed load of intensity q(x) (a) and epure v'(x) (b)

E'Jnp(=)

2. Build epure v'(x) by using dependence (18) and performing step-by-step numerical integration (Fig. 3b).

3. Obtain the expression for the first derivatives by averaging the values of the second derivative in the split step:

i

u'(x e) = u'o +

d'(0) + u'(x,) {

2

i A

+ Tu"(xk ) k=i j

where j=l,... n; u'0 - initial acglao.rotation; A - splitting step;

Mix,)

U(Xj)=F I i ,■

E-Inp(x])

Introducethefollowing symtiols. Let itbe as follows:

(, E'

A V =

u"(0)+u"(x.)

1 A

2

k=1

■A,

where j=1,...,n.

Then formula(20)will be a si fo llows:

v'fXj e = v'0+ AV',

where j=1,...,n.

4. Similarlyto(n0),calculate values v(xj). As v(xc) = v(0)=v0=0, then:

u(x.) =

iu'n+u'fx,) ^

2 k=l /

Using the dependency (23),obtain:

(

u(xi) =

AV' i

i \

v'o-j + -VL+kLAVl

2

k=i

•A-

lf Xj in :

Therefore:

Ay' n-l

uOxn) raaijr | U'0-n+—n- + Yi-V |-r 0.

f-V' n-l \

a

+ Y-vk

, / - _v_k=l_I

U0 r .

n

Using dependencies(24- 27), buildy3dicy u, and

max\v\r max\yO xj)\,

where j=1,...,n-1.

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

Apart from this, the algorithm of trapezoidal slabs calculation is completely like the algorithm of combined slabs calculation on a wooden frame with a rectangular plan.

Conclusion

The proposed method of combined slabs calculation on a wooden frame with a trapezoidal plan allows to assess their stressed-deformed state adequately, to carry out practical calculations for strength and rigidity. It makes possible to use full-board trapezoidal elements in construction practice for roofs and floors of buildings and structures of different purposes successfully.

References

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[2] Пятикрестовский К.П. Вопросы дальнейшего совершенствования конструкций с применением древесины и новых плитных материалов. Пространственные конструкции, сб. трудов РААСН, 2007, 9, 49-51 [Pyatikrestovsky K.P. Issues of further improvement of structures with the use of wood and new tile materials. Spatial Constructions, In the collection of works of RAASN,

2007, 9, 49-51 (in Russian)]

[3] Гетц К.Г., Хоор Д., Мелер К., Наттерер Ю. Атлас деревянных конструкций. Пер. с нем. М.: Стройиздат, 1985, 272 с. [Getz K.G., Hoor D., Meler K., Natterer Y. Atlas of wooden structures. Translation from German. Moscow, Stroyizdat, 1985, 272 p. (in Russian)]

[4] Губенко А.Б. Клееные деревянные конструкции в строительстве. М.: Госстройиздат, 1957. 240 с. [Lubenko A.B. Glued wooden structures in construction. Moscow, Gosstroizdat, 1957, 240 p. (in Russian)]

[5] Торяник Н.Н., Корзон С.А. Расчет клеефанерных плит покрытия. Конструкции из клееной древесины и пластмасс. Межвузовский тематический сборник трудов ЛИСИ. 1978, 2, 58-64 [Toryanik N.N., Korzon S.A. Calculation of glued plywood roof slabs. Structures of glued wood and plastics, Inter-university thematic collection of LISI works, 1978, 2, 58-64 (in Russian)]

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2008, 7, 4-10 (in Russian)]

[8] Гребенюк, Г.И., Яньков Е.В., Ажермачев А.В. Оптимизация параметров большеразмерных ребристых плит на основе древесины. Проблемы оптимального проектирования сооружений: сборник докладов V Всероссийского семинара. Новосибирск: НГАСУ (Сибстрин), 2005, 110-119

[Grebenyuk G.I., Yankov E.V., Ajermachev A.V. Optimization of parameters of large-scale ribbed slabs on the basis of wood, Problems of optimal design of structures: collection of reports of the V All-Russian seminar. Novosibirsk, NGASU (Sibstrin), 2005, 110-119 (in Russian)]