STRESS CONDITION OF ORTHOTROPIC VAULT STRUCTURE WITH CYLINDRICAL ANISOTROPY Текст научной статьи по специальности «Строительство и архитектура»

Magazine of Civil Engineering. 2022. 116(8). Article No. 11605

Magazine of Civil Engineering issn

2712-8172

journal homepage: http://engstroy.spbstu.ru/

Research article UDC 624.074.3:539.3 DOI: 10.34910/MCE.116.5

Stress condition of orthotropic vault structure with cylindrical anisotropy

A-Kh.B. Kaldar-ool1 " , E.K. OpbuP

1 Tuvan State University, Kyzyl. Republic of Tuva, Russia, oorzhaka-h@mail.ru, 0000-0002-5304-3747

2 St. Petersburg State University of Architecture and Civil Engineering, St. Petersburg, Russia M oorzhaka-h@mail.ru

Keywords: barrel vaults, brickwork, anisotropy, elasticity modulus, elasticity constants, stress condition

Abstract. This work considers analytical calculation of brickwork barrel vault, material structure of which has a pronounced variability of elastic constants. In normative documents, brickwork is considered as a complex two-component building material with elastoplastic properties. However, there are no clear recommendations that consider the variability of the elastic properties of brickwork. This article considers influence of anisotropic properties of a brickwork three-centered flat-arched vault on its stress condition on the basis of the elasticity theory. The calculation of a flat-arched vault is based on the classic theory of bending a curved curvilinearly-anisotropic beam in view of the properties of brickwork materials with cylindrical anisotropy. We cite a mathematical solution of a differential equation of fourth order in partial derivative with two variables for an anisotropic orthotropic body in polar coordinates for creation of mathematical models describing changes in the vault material elasticity modulus. Based on the solution to the curved orthotropic body anisotropy problem, we obtained correlations between elastic constants in the main anisotropy directions.

Citation: Kaldar-ool, A-Kh.B., Opbul, E.K. Stress condition of orthotropic vault structure with cylindrical anisotropy. Magazine of Civil Engineering. 2022. 116(8). Article no. 11605. DOI: 10.34910/MCE.116.5

1. Introduction

Study object is brickwork barrel vault of complex curvilinear outline. Brickwork is an orthotropic two-component material with elastoplastic properties. In vaulted structures, depending on the direction, in a three-dimensional coordinate system, brickwork has different elastic properties. In the case of two-dimensional problem, variability manifests itself in tangential (circumferential) and radial directions.

It is well known that elastic constants include the values of the modulus of elasticity and Poisson's ratio of the masonry. Ignoring the elastic properties variability factor in calculations, for example, at the design stage, can lead to serious errors or accidents during operation. Moreover, in brickwork vaulted ceilings, anisotropy is observed not only in the material itself, but also in structural layout. In view of more frequent progressive collapses, theoretical studies based on calculation of the strength and stress state of stone vaulted structures, taking into account the properties of anisotropy, are extremely critical.

According to Professor S.G. Lekhnitsky's theory [1], brickwork barrel vaults [2] can be viewed as an orthotropic body with cylindrical anisotropy [3] for which laws of mechanics of anisotropic bodies are valid.

In the classical practice of constructing buildings and structures, including brickwork ceilings, load-bearing vaults of different geometric shapes and layouts were used. Methods of their calculation were mainly based on the laws of elasticity theory, considering the vault as a hinged-supported elastic body. Approximate methods of limiting equilibrium, proposed by Maurice Levy, and a graphical method for determining the pressure curve, based on experiments on the destruction of vaults, are used.

© Kaldar-ool, A-Kh.B., Opbul, E.K., 2022. Published by Peter the Great St. Petersburg Polytechnic University.

A particularly noteworthy study [4] examines the technical condition of historical buildings, where strength characteristics of brickwork vaulted structures is assessed on the basis of numerical modeling. The study [5] considers flat circular double-hinged vaults, statically loaded with a uniformly distributed load in the form of pressure. The authors managed to analytically solve the geometrically nonlinear deformability and stability of the vaults. Behavior of Prussian brick-concrete vaults based on numerical model built in the Abaqus software packageis shown in a scientific paper [6]. Finite element analysis of the behavior of a stonework single-span arched bridge is presented in a scientific study [7]. Laboratory and numerical analysis of the destruction of stone arches on models with and without reinforcement can be found in a scientific study [8]. In the original paper [9], the behavior of a segment of an arch from a dry joint of masonry subjected to a lateral load was investigated using the finite element method and rigid blocks based on a micromodeling strategy, a nonlinear analysis was also presented taking into account various combinations of displacement of the supports (vertical, horizontal and oblique). The article [10] presents the original results of experimental studies to reduce the seismic vulnerability of stone vaults using a composite reinforced mortar. The authors of work [11] argue that displacements, observed in many historical masonry structures are concentrated at the joints of stonework and can be significant before collapse becomes a global problem. Therefore the stability modeling using discrete element modeling is particularly relevant. In order to reuse the building complex of Monte Pio port and the upper warehouses of the Palazzo Monte di Pieta, in work [12], numerical studies of two-level stone vaults were carried out and the technical condition was checked for the stability of building structures in accordance with the current Italian building codes. In scientific work [13], based on analytical formulation of the problem, the angle of friction is investigated as a geometric constraint of brickwork sample in order to find a possible range of minimum values of the thickness of round and elliptical vaults from masonry under static loads based on the limit state analysis theorem. Numerical studies of the stability of masonry vaults are presented in works [14, 15] including a unified formulation of historical stone structures modeled as 2D assemblies of rigid blocks interacting on frictional contact surfaces without stress. Experimental evaluation and development of numerical and analytical modeling of ancient masonry vaults and vaults, reinforced with composite systems, are given in work [16]. The study [17] examines the effect of stereotomy on the value of the minimum thickness of a semicircular brickwork vault: a material with low tensile strength. In the scientific work [18], on the basis of historical, experimental and numerical analysis and the results of field studies using our own software LiABlock_3D, a spatial rigid block model of the entire structural unit was built. The work [19] proposes combined precast and cast-in-sit construction of metro station in Shanghai in the form of a large-span underground vault of 56 elements of ribbed arched segments to be assembled as triple-hinged arches. In the construction of large diameter tunnels, a new mechanized technology for the construction of precast arches has been proposed [20]. For a preliminary assessment of damage in the form of cracks, for example, in the process of mechanized construction, numerical and laboratory experiments were carried out on an arch made of volcanic rock, i.e. tuff; the research results are given in work [21]. Work [22] represents experimental studies of stonework cylindrical vaults made of thin brick tiles; the vaulted structure is considered as a permanent formwork with subsequent reinforcement based on reinforced concrete.

The purpose of the study is to create a calculation a system based on the well-known theory of bending of a curved anisotropic beams, given in work [1].

To achieve this goal, the following problems are solved:

1. Determining the elasticity moduli of bricks and mortar.

2. Determining the elasticity modulus of brickwork and the anisotropic index depending on the main anisotropy directions.

3. Determining the elasticity parameters of brickwork that allow to obtain elastic constants of the vault material.

4. Analytical calculation for evaluating the stress condition of a barrel vault in the form of a curved orthotropic beam with a cylindrical anisotropy.

2. Methods

According to regulatory document [23], the elastic modulus of soft-mud bricks EQnck is determined on the basis of deformation of cubes or prisms cut from bricks using the formula:

E0rck = 200 1200 • Rbrick, (1)

where Rbrick is brick compressive strength [2, 3].

_ yyt ]>*

In a residential buildings design manual [25], the elastic modulus of mortar E0 in a compressed bed joint is determined using the formula:

T-^mortar _ tmortar /o\

Eo — г-> (2)

mortar

where tmortar is the joint thickness; Tmortar is compression compliance of a manually laid horizontal mortar joint under short-term loads determined according to the formula:

_ 2

г — 15.10_3 . R 3 . t

mortar ~ ' ^mortar mortar ■>

Rmortar is the mortar strength limit [2, 3].

Then within the framework of the task at hand, we use the phenomenological method of rheology in order to find out the brickwork elasticity modulus on the main axes of anisotropy.

According to, we can reconstruct the real pattern of material behavior under load using more complicated schemes combining elastic and viscous elements. If we take elastic (bricks) and viscous (mortar) materials and connect them in parallel (at the vault head) and in series (at the vault abutments), we obtain rheological models: Kelvin and Maxwell bodies in the following form [3].

For parallel connection of elements:

77 T?brick . T-^mortar ,0ч

Er = E0 + E0 . (3)

r

For series connection of elements:

т-^brick т-^mortar

E = E_E_ (4)

t brick mortar E0 + E0

where Er, Et are constant elasticity modulus of the brickwork anisotropy, respectively, in the radial and tangential (circular) directions.

A partial fourth-order differential equation in polar coordinates for anisotropic body has the following form[1]:

1 d4F ( 1 „Vrt Л 1 d4 F 11 54 F

Et dr4

Grt Er

„2-^2^2 + j7 ' 4-^4 +

+ Et r dr3

vGrt Er j r2 dr2Ш2 Er r4 Ш4

1 d2 F

+ (5)

2 1 d3F ( 1 _ 2 Л 1 d2F 1 1 52F

r3 dree1 Er r2 dr2

V Grt Er J r urue Er

Гл.. d2 F 1 1 dF

2^1 + 1

- +---= 0,

r Grt J r4 de2 Er r3 dr

v Er G

where , GH are Poisson's ratio and elastic modulus, respectively.

When solving equation (5) in plane elastic problem for a circular plate with cylindrical anisotropy, the stress function was taken as a sum of polynomials proposed by E.K. Ashkenazi:

F —ix'-fk (У ), (6)

i-1

where fk (y) is an unknown function satisfying differential equation (5).

Solving equation (5) with the substitution of the corresponding derivatives of the stress function (6) and rearrangement result in second-order algebraic equation (7), the roots of which are respectively equal [26]:

B2 - 2 (5 + k2 ) B - -k4 +k2 +1 = 0.

3 V / 3 3

(7)

Professor V.N. Glukhikh states in his studies [26] that cylindrically anisotropic materials differ by elasticity parameters (B(i), B(2)) and can be divided into 2 groups:

• for the first group, elasticity parameter is characterized by three extremes as axes turn from the radial direction to the tangential one, i.e. from 0° to 90°:

B(i)= 3 - k

the second group with two extremes:

B( 2) =

1 + 5kz

3

(8)

(9)

The studying the plane stress condition of an anisotropic material often encounters the following problem: the elastic constants are known for a certain coordinate system x, y and it is required to find the

elastic constants for a new coordinate system X, y'. For an orthotropic body, it is inconvenient to use the

main coordinate system and recalculate the elastic constants [1].

The known formulas as per [26] in view of elasticity parameter of anisotropic bodies for elasticity

modulus Ex:, shear modulus Gxy, Poisson's ratio pxy, will have a simpler form, which is of prime importance for engineering calculations:

1 cos4 0 sin4 0 B(1) n . 2a +-+ ^^-cos2 0-sin2 0;

Ex

Er Et

E

G ' '

w xy

Mxy = -Ex

»( k2 -1)

Et

2 ( k2 -1)

t

sin2 0-cos2 0+ 1

G

rt

•sin2 0-cos2 0 + -^

E

(10)

(11)

(12)

E

where Grt =-r—t- is shear modulus; Gxy and pxy respectively, the shear modulus and

3 - k2 + 2-prt

Poisson's ratio in a new coordinate system.

Similarly, studies [26] recommend calculation formulas for anisotropic materials belonging to the 2nd group 2):

1 cos4 0 sin4 0

Ex Er

1 2 2 + B( ---sin2 0-cos2 0;

Et (2) Et '

Gx'y'

3 • Et

•sin2 0-cos2 0 + —;

Grt

(13)

(14)

MXy' = -EX

2-(1 - k2 )

3 • Et

sin2 0-cos2 0--^-

E

(15)

where Grt =

3 • Et

1 + 5 •k + 6 •prt Poisson's ratio in a new coordinate system.

is shear modulus; Gxy and pxy respectively, the shear modulus and

"xy

Figure 1. General view of the barrel vault.

A body with cylindrical anisotropy can be formed artificially by constructing it from homogeneous (rectilinear-anisotropic) elements that have the same elastic properties. If we consider that vault consists of a large number of homogeneous anisotropic elements with the same elastic properties, then the structure as a whole will have the property of a body with cylindrical anisotropy. Equivalent axial directions of elements in the vault will be radial directions [1].

For an orthotropic beam with a cylindrical anisotropy in application to a barrelvault we adopt a design diagram presented in Fig. 2.

Figure 2. Design diagram of a vault in the form of a curved orthotropic beam with a cylindrical anisotropy: 0 - center of anisotropy coinciding with the center of curvature, 0' - determining the position of abutment surfaces.

For determining the main stresses in the crest part a barrel vault under the impact of dead (q) and

temporary additional loads (F) according to the theory of S.G. Lekhnitsky [1], we use stress function in the following form:

F (r) = fo (r) + fi (r)• cos (0) + /i*(r)• sin (0), (16)

where the first addend has the following form:

f0 (r ) = A + B • r2 + C • r1+k + D • r1-k,

k = E .

yEr

The second addend for an orthotropic beam is determined by the following formula:

(17)

f1 (r )• cos (0) = ( A-r1+ßl + B• r1-ß + C-r + D-r • ln r )• cos 0 + + ( A'-r1+ß1 + B'-r1-ß + C-r + D'-r-ln r )• sin 0,

(19)

where A, B, C, D are arbitrary complex constant determined from boundary conditions and free-end conditions; A', B', C', D' are conjugate values.

Stresses ar, a0, Tr0 are expressed through the stress function:

Boundary conditions:

CTfl =

1 dF(r) 1 d F(r)

— •--1--•-

r dr r2 502

d 2 F (r )

dr 2

Tr0 ="

d ( 1 dF(r)

dr I r 50

when r = rdor = 0, ir6 = 0;

when r = rtor =-(q + F)cos9Tre = 0.

(20)

(19)

Then the condition is true:

r . R f a0dr = —-ä-

J 6 h

rd

f U0r3r = 0 rd r

(21)

1 R

Kea- = ± R-

rd

Arbitrary constants included in expression (19) are determined on the basis of boundary conditions (21) and (22).

Normal stresses in the main directions are determined according to [1] by formulas (23), (24) using the Mathcad software system:

P + Q

( r \

V rt

k -1

+ R •' r

k+1

q ft

<Ja= — '

P + Q • k

( r ^

V rt y

k-1

rt • b • g1 r

k+1

V rt

+ c ß-i^

ß1

-(1 + cß )

• cos0

cos ^

+ R• k-I ^

r

q rt

Tre ="

q ri

rt • b • g1 r

rt • b • g1 r

( ^ß r

V r y

(1 + ß1 )•(r ^ß+(1 + ß1 )• c ß1 ( r f -(1 + cß1 )

+ c ß( f ^'-(1 + c ß )

cos y/

• cos0

is ($-y)

cos^

(23)

where

=

r

P =

Q =

R =

1

2 • (k2 -1)•(1 2k - c ) • g

1

2 • (k - 1) • (1 - 2k c ) • g

1

•[2• k•(k -l)-(l-ck+1 )+ 2• k •(k +1).ck+1 -(l-ck-1 )-

-1 K - I I'll - i I-

- (k2-1) • (1 + c) • (1 - cZk ) • m],

„2 A k-A , ,, 1 W1 ,

• (1 - ck+1) • m],

2 • (k +1) • (1 - c 2k ) • g

•[(k +1>c2k -c2)-2• k• ck+1 -ck+1 )-(k +1)•

_rd m _ sin ^ sin (^-vX _ (1 - c2) k (1-ck+1)2 k • c2 (1-ck-1)2 _

c = — ; m =

r

cos y

; g =

(1 + c) • cZk • (1 - cl-k ) • m] k • c c

+

2 k+1 1-c2 k -1 1-

2k

gj = • (1 -CPi) + (1 + CPi) • lnc ; b = 1 - a single element width.

Pi

It is stated in study [1] that coefficient Pj included in formula (23):

ß1 ^ 1 + E • (1 - 2 •prt ) + Et

Grt

E,

(24)

(25)

3. Results and Discussion

Initial data for practical calculation:

Elastic modulus of a brick of grade Mbrick = 102 and Bbrick = 10.03 MPa according to [2], obtained by the formula (1) is E0nck = 12036 MPa.

Elastic modulus of the solution of grade Mmortar = 22 and Bmortar = 2.14 MPa, according to the

formula (2) is Eim°rtar = 2140 MPa.

Radial and tangential moduli of elasticity of brickwork according to formulas (3, 4), respectively Er = 14176 MPa and Et = 1816 MPa.

Poisson's ratio according to K.P. Yakovlev [27] - fat = 0.15

For a practical calculation example, Table 1 shows the collection of loads.

Table 1. Collection of loads per 1 running meter barrel vault in section at the crest part.

№

Name

Regulatory load, kN/m

Coefficient reliability yf

Estimated load, kN/m

Curb self weight qc.s.w. (Y= 18 kN/m3 ) 18 h in the crest part

Backfill weight qb.w. (Y= 18 kN/m3)

Ground floor weight qg.f. t=55 m: cement-sand mortar t=28.4 mm (y=18 kN/m3) marble chips t =15 mm (y = 16 kN/m3)

15.3 Variable

0.511 0.24

1.1 1.3

1.3

16.83

0.976

live load [8.2, table.8.3, p. 3]

1.2

2.4

2

3

4

2

Experimental live load Total:

10.0

30.206

The values of the vault loads in the crest part per running meter at an angle p = 0° (see Fig. 3) and according to Table 1 are q = 30.206 kN/m.

If the supporting surfaces are assumed to be horizontal (Fig. 3), then we have p = 0,y = 0, 0 = n/2, cos(^ - v) = 0.

Bending of curvilinearly anisotropic beam: the radius axis of slope part of vault r = 12.16 m; vault inner radius rd = 11.73 m, outer radius rt = 12.58 m.

1. Using known formulas (10, 13), we find out the elastic constants of brick and mortar, including for brickwork.

The values of theoretical elastic moduli of brick and mortarare presented in Tables 2-5.

Table 2. Theoretical elastic modulus of brick at B(1) = 3 - k2 according to the formula (1),

(MPa).

9° 0° 15° 30° 45° 60° 75° 90°

k2=0.5 24072 20190 14810 12036 11330 11700 12036

1 + 5k 2

Table 3. Theoretical elastic modulus of brick at B(2) =- according to the formula (1),

3

(MPa).

15° 30° 45° 60° 75° 90°

k2=0.5 24072 23480 21400 18050 14810 12730 12036

Table 4. Theoretical elastic modulus of mortar at B(1) = 3 - k2 according to formula (2), (MPa).

9° 0° 15° 30° 45° 60° 75° 90°

k 2=0.5 4280 3591 2634 2140 2014 2080 2140

1 + 5k 2

Table 5. Theoretical elastic modulus of mortar at B(2) = —-— according to formula (2),

(MPa).

9° 0° 15° 30° 45° 60° 75° 90°

k 2=0.5 4280 4174 3804 3210 2634 2263 2140

The values of theoretical brickwork elasticity moduli are presented in Tables 6, 7.

Table 6. Theoretical brickwork elasticity modulus at Bm = 3 - k2 (E0=12036 MPa).

e° 0° 15° 30° 45° 60° 75° 90°

According to formulas (10, 13) k2=0.128 14176 6146 2698 1816 1638 1729 1816

Table 7. Theoretical brickwork elasticity modulus at B{2) = 1 + 5k2 = 3 (MPa).

e° 0° 15° 30° 45° 60° 75° 90°

According to formulas (10, 13) k2=0.128 14176 12092 7660 4337 2698 2006 1816

2. The coefficient is determined Pj.

According to [1], for parameter B(j we have:

Gr -2-i^Er=B(i). (26)

Grt Er

With regard for expression (18), (26) and (8) obtained in study [26], from formula (25) we get a parameter equal to:

p1 =>/1 + k2 + 3 - k2 = 2. At Pj = 2 stress distribution is analogous to that in an isotropic beam. Studies [26] confirmed relationship (8) mathematically.

If elastic constants meet the condition:

fH'^+GT=3 (27)

r rt

then Pj = 2 and stress distribution will be exactly the same as in an isotropic beam.

If we transform the radical expression (23) for Pj, we obtain from solution [1] the same root from

E EE E

formula (8), i.e. —^-2-|rt ■—- + —L = 3 or if—^L = k2 we shall obtain: k2 -2-|rt-k2 = 3-k2, what

Er Er Grt Er

we have obtained B(1) = 3 - k2 (8) earlier as a result of theoretical studies [1].

Or we may use this formula ß1 II + k2

1 + 5 • k2 2>/l + 2 • k2

V3

2

If k = 1 we get the same result ß1 =

2V1 + 2 • k2

S

Due to substitution of known ratio so felastic moduli in equation (27), we obtain the same expression (8), which was similarly obtained by S.G. Lekhnitskiy in equation [1] for a curvi linear cylindrically anisotropic orthotropic beam. As to the second elastic parameter B( 2), there are no corresponding studies.

It should be noted that to find the ratio p1, which is necessary for the theoretical study of the stress condition of the brickwork of vaults in the form of cylindrically anisotropic orthotropic bodies, the authors first used the second elastic parameter B( 2).

In particular, the elasticity parameter B(2) is recommended to be used to improve the methods for

calculating the stress condition and slope part of flat-arched vault. At the same time, orthotropic materials with cylindrical anisotropy can be conventionally divided into two groups.

For the first group of materials that satisfy condition (8), the change inelastic modulus from 0 ° to 90 ° (from the radial to the tangential direction) occurs through an intermediate extreme point when the layers are tilted at an angle of 30 ° to force line.

For the second group, there is no intermediate bending point and elastic modulus changes from 0 ° to 90 ° smoothly.

Fig. 4 shows the obtained numerical values of the distribution of radial ar and tangential oe stresses depending on the radius r, considering calculated elastic moduli and ratio Pi.

Figure 4. Stress distribution or and Oq (MPa) in the crest part of the vault depending on radius r at the vault head at angle y = 0.

4. Conclusion

Using the example of solving the problem of bending a curved beam with cylindrical anisotropy, an approximate method for calculating the stress state of masonry duct vaults is proposed, taking into account the properties of anisotropy.

obtained

Findings:

Based on the phenomenological method of rheology and elastic moduli (E0nck ,E0ioriflr

through the experimental strength limits of brick and mortar (Bbrick = 10.03 MPa, Bmortar = 2.14 MPa), the elastic moduli of brickwork in the radial and tangential directions are determined, equal to Er = 14176 MPa v\ Et = 1816 MPa, respectively.

As a result of taking into account the properties of anisotropy at an angle of 90 ° using the MathCad software package, we obtained:

- elasticity parameters: Ba) = 2.872 with coefficient Pj = 2 and Bi2\ = 0.547 with coefficient

5(i)

P1 = 1.392;

( 2)

- modules for parameters B^ and Bi2\, equal: for brick Eh^lck = 12036 MPa, mortar

5(i)

12)

E,

mortar 0

= 2140 MPa and brickwork Ex = 1816 MPa;

- maximum stress in the locking part, equal to: cr = 0.031 MPa and ct = 0.447164 MPa.

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Information about authors:

Anay-Khaak Kaldar-ool, PhD in Technical Science ORCID: https://orcid.org/0000-0002-5304-3747 E-mail: oorzhaka-h@mail. ru

Eres Opbul, PhD in Technical Science ORCID: https://orcid.ora/0000-0002-7796-2350 E-mail: fduecnufce@mail.ru

Received 22.08.2021. Approved after reviewing 31.05.2022. Accepted 31.05.2022.