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10Magazine of Civil Engineering. 2019. 89(5). Pp. 39-51 Инженерно-строительный журнал. 2019. № 5(89). С. 39-51
Magazine of Civil Engineering
journal homepage: http://engstroy.spbstu.ru/
ISSN
2071-0305
DOI: 10.18720/MCE.89.4
Nonlinear deformation and stability of geometrically exact elastic arches
V.V. Lalin, A.N. Dmitriev*, S.F. Diakov
Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia
Ключевые слова: stability of structures, buckling, geometrically exact theory, dead load, round arch, stiffness, stationary point, Lagrange functional
Abstract. In the present paper a plane round double-hinged arch under the potential dead load is investigated. To describe the stress-strain state and the equilibrium stability the geometrically exact theory is used. According to this theory every point of the bar has two translational degrees of freedom and one rotational, which is independent from the previous two. To solve the problem no displacements are simplified and all the stiffnesses are used: axial, shear and bending. Exact nonlinear differential equations are found for the static problem. A variational definition for the problem is defined as finding a stationary point of Lagrange functional. The match of the differential and variational formulations is shown. Exact stability equations accounting non-linear geometric deformations in pre-buckling state were worked out. The problem of the equilibrium stability of the round arch under the potential dead load was solved using the obtained equations regarding all the bar's stiffnesses. The characteristic transcendental equation and its asymptotic solution as simple formulas, suitable for practical application, were worked out. The comparison of described solution which regards all the bar's stiffnesses and classical solution, based on bending stiffness, was made.
Arches are one of the most widespread structural systems. On the one hand, this is due to their architectural expression; on the other hand, they are efficient at mechanics due to their curvature which can neglect the effect of the bending moment. As a result, the arch can be rather flexible as the size of the cross section can be relatively small. That is why the problem of arch equilibrium stability is one of the main problems for engineers to consider.
Historically, the most popular problem in the theory of stability of arches is the problem of stability of round arch under the radial pressure. This implication is typical for different underground structures - tunnels, pipelines and hull ribs of the submarines. The solution for this problem for the semi-ring was derived according to the fact that the loads, despite the bending of the axis of the bar, maintain the line of action. Besides, lines of action don't move in case of buckling [1-4]:
The solution for the problem of the stability of the plane arch under the radial pressure, when the load maintains the line of action, but the application points move with the axis of the arch were worked out by N.V. Kornoukhov [5] and A.N. Dinnik [6]. In case of the semi-ring the critical force is:
Lalin, V.V., Dmitriev, A.N., Diakov, S.F. Nonlinear deformation and stability of geometrically exact elastic arches. Magazine of Civil Engineering. 2019. 89(5). Pp. 39-51. DOI: 10.18720/MCE.89.4
Лалин В.В., Дмитриев А.Н., Дьяков С.Ф. Геометрически нелинейное деформирование и устойчивость упругих арок // Инженерно-строительный журнал. 2019. № 5(89). С. 39-51. DOI: 10.18720/MCE.89.4
1. Introduction
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This open access article is licensed under CC BY 4.0 (https://creativecommons.org/licenses/bY/4.0/)
In practice arches usually suffer different loads. A lot of problems were solved about the non-linear stability and the post-buckling deformation of round arches under the single force in the center [7-9]; under vertical or horizontal pressure, distributed on whole length [10-14] or located only on the part of the arch [15, 16]. Some other problems and their solutions can be found in the papers of V.N. Paimushin [17, 18], I.A. Karnovsky [19, 20], V.V. Galishnikova [21].
The critical force value is influenced not only by external loads but by the other parameters: flexibility of the supports, material properties and shape of the axis of the arch. The effect of the horizontal and vertical support stiffness on the stability of the round arch and frames was analyzed in [22, 23]. The effect of the physical properties, inconstant through the cross-section was analyzed with the help of functionally graded materials in [9, 14, 24, 25]. The problems of linear stability of plane parabolic arches can be found in [12, 13, 26]. Experimental researches of pre-buckling deformation, ultimate equilibrium and the failure behavior in case of buckling are described in [27-30].
Despite the big amount of analytical study, all the mentioned researches consider only the bending stiffness of the bar, and for this reason should be considered as approximate. There are no problem formulations and their solutions about the stability of arches, considering both axial and shear stiffness.
Variational method as the principle of virtual displacements is the most popular method to research the problems of stability [16-18, 31]. However, the variational definition for the problem as finding a stationary point of some functional was used only for straight bars [32], not for the arches. Note, that exact stability equations can be obtained from the second variation of the functional [31].
Thus, the purpose of this paper is to solve the problem of stability of plane double-hinged arch under the potential dead load regarding all the stiffnesses of the bar: axial, shear and bending by variational approach.
The aims of this paper are:
1. to work out a variational definition for the problem of deformation of the geometrical non-linear plane elastic round arch regarding axial, shear and bending stiffnesses as finding the stationary point of Lagrange functional;
2. to work out the stability equations as the result of the second variation of the Lagrange functional;
3. solving the problem of the stability of the arch with the help of the obtained equations under the potential dead load regarding all the stiffnesses of the bar: axial, shear and bending.
4. comparing the obtained solution with the Kornoukhov-Dinnik solution (2), where only bending stiffness was considered.
2. Methods
This paper is based on geometrically exact bar theory [32-37], whereby each point of the plane bar has two translational degrees of freedom - displacements u, w and one rotational - angle p, which is independent form the previous ones.
Consider a plane round double-hinged arch with radius R under the potential dead load: uniformly applied forces and moments. Each point of the arch can be described with the local trihedron (t, n, k): tangent basis vector t is directed towards the increasing 6, normal basic vector n is away from the center of curvature C. The direction of the basic binormal vector k can be found using the vectoral product t x n = k. All the unknowns, describing the stress-strain state of the bar can be found via angular coordinate 6, 0 <6 < ©, where 0 is the central angle (Figure 1).
с
Figure 1. Arch state in basic condition (condition before deformation).
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Magazine of Civil Engineering, 89(5), 2019
Definition of the problem of geometrically non-linear deformation of the round arch consists of three groups of differential equations: static (equilibrium equations), geometrical and physical. The equations for the static problem in the vector form were derived in [32-33]. The scalar form for the vector equations in the curvilinear coordinates will be derived below.
Equilibrium equations for the problem of the plane non-linear deformation of the arch are:
(Ncosp-Q sinp) +(Nsinp + Qcosp) + Rqt = 0;
* (Nsinp + Qcosp) -(Ncosp-Q sinp) + Rqn = 0; M' + (u' + w + R)(N sinp + Q cosp)-(w' - u)(N cosp- Q sinp) + Rm = 0,
where N is axial force; Q is shear force; M is bending moment;
qt, qn are projection of the distributed loads on the tangential and normal (radial) direction; m is distributed moment load; u, w are tangential and normal displacements;
(pis rotating angle. Henceforward differentiation is made according to angle 0( ) = d( )/d6 . Geometrical equations for the plane problem are:
s = -1 ( u' + w + R) cos p + -1 (w'- u ) sin p-1, R R
Y = - R (u' + w + R) sinp + R (w' - u) cosp,
R
¥ = R <P',
(4)
where s, y, y/are axial, shear and bending deformations. Physical equations for the linear elastic material are
N = ks; Q = k2y; M = ky,
(5)
where k\ = EA is axial stiffness; ki = GAk is shear stiffness; ki = EI is bending stiffness; E is Young's modulus; A is cross-section area of the bar; G is shear modulus; k is cross-section form coefficient; I is moment of inertia.
The equations (3)-(5) are the exact equations of geometrical non-linear round arch, taking into account all stiffnesses of round arch. To get the closed system on each end of the arch three boundary conditions are needed. For the double-hinged arch they are as follows:
0 = 0: u ( 0 ) = 0, w ( 0 ) = 0, M ( 0 ) = 0; 6 = 0: u (0) = 0, w (0) = 0,M (0) = 0.
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3. Results and Discussion
3.1. Variational formulation of non-linear static problem Lagrange functional can be written as follows:
C
0p
(u, w, (p) = rJI 2(kls2 + k2y2 + k3^2 )- qu
■ qnw - m(
d6.
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Инженерно-строительный журнал, № 5(89), 2019
It can be proved, that variational definition for the problem, defined as finding a stationary point of functional с
С ^ stat (8)
in case the fulfillment of the essential boundary conditions
u
( 0 ) = w ( 0 ) = u (0) = w (0) = 0 (9)
is equivalent to the initial problem (3)-(6). The first variation of the functional (7) is:
SC (u, w, p) = - j"{uv ((N cos p - Q sin p) + (Q cos p + N sin p) + Rqt) +
0 V
+wv ((N sinp + Q cosp) +(Q sinp- N cosp) + Rqn)+ (10)
+pv (M' + (u ' + w + R )(Q cosp + N sinp) + ( w ' - u )(Q sinp- N cosp) + Rm )} d6 + + [uv (N cos p - Q sin p) + wv (N sin p + Q cos p) + pM]©,
where the variations are labeled as follows:
uv =Su, wv = Sw, pv = Sp. (11)
The solution of the variational problem are the functions u, w, p, satisfying the essential boundary conditions (6), that SC = 0 for any variations uv, wv, pv. Initial nonlinear equilibrium equations (3) are the Euler equations of the variational problem (8)-(9), according to (10).
As it can be seen from (9), variations of the displacements on the boundaries equal to zero:
uv (0) = wv (0) = uv (0) = wv (0) = 0. (12)
Considering (12), the terms outside the integral (10) are:
M (0)pv (0)-M ( 0 )pv ( 0 ). (13)
From the stationary condition of the functional for any pv (0) and pv (0) are, it can be seen, that their factors should equal to zero. So, the natural boundary conditions are:
M (0) = 0, M (0 ) = 0. (14)
Thus, the equivalence of differential (3)-(6) and variation (8)-(9) formulations is proved.
3.2. Stability problem formulation
The second variation of the functional (7) is:
©
:(u,w,p) = -R Jjk [((u'v + wv)cosp-pv (u' + w + R)sinp + (w' -^)sinp +
0
\2
S2C (
+ pv (w' - u) cos p) + ((u' + w + R) cos p + (w' - u) sin p - R) (-2pv (u^ + wv ) sin p - p2 (u' + w + R) cos p + 2pv (w'v - uv ) cos p - pp (w' - u) sin p) +
(- (u'v + wv ) sin p-pv (u' + w + R) cos p + (w'v - uv ) cos p-pv (w' - u) sin p)
+ (-(u' + w + R) sin p + (w' - u) cos p) (-2pv (+ wv ) cos p + pp (u' + w + R) sin p
+ k2
2
+
(15)
- p (wv - uv ) sin p - p1 (w'- u) cos p) + k3p }
Let us label 2S2С = FS*T (uv, wv, pv), where FS*T (uv, wv, pv ) is static stability functional.
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Magazine of Civil Engineering, 89(5), 2019
To derive Euler equations for the variational problem of finding a stationary point of functional FT ^ STAT in carrying out essential boundary conditions (12) the first variation of the stability functional should be computed:
^fs*t (uv, wv, Vv ) = }[öuv (-(Nv cos <P - <PvNsin <P - Qv sin <P - %Q cos <P)' -
0 ^ ^
- Nv sin p-pvN cos p - Qv cos p + pvQ sin p) + öwv (- (Nv sin p + pvN cos p +
+ Qv cos p - pvQ sin p) + Nv cos p - pvN sin p- Qv sin p - pvQ cos p) + + Spv (-M'v -(u' + w + R) Nv sinp + ( w' - u) Nv cosp-(u'v + wv) N sin p - (i 6)
-pv (u' + w + R) N cos p + (w'v - uv) N cos p-pv (w' - u) N sin p -- (u' + w + R) Qv cos p - (w' - u) Qv sin p - (uv + wv) Q cos p + pv (u' + w + R) Q sin p -- (w'v - uv) Q sin p - pv (w' - u) Q cos p) dO + \j5uv (Nv cos p - pvN sin p - Qv sin p -
pQ cos p) + ^wv (Nv sin p + pvN cos p + Qv cos p - pvQ sin p) + ÖpMv ,
where the following labels are introduced:
Nv = klsv, Qv = k2Yv, Mv = k3Vv; £v = R ((wv - uv ) cosp-pv (u' + w + R ) sinp + ( wv - uv ) sinp + pv ( w' - u ) cosp), Yv = R (-(u'v + wv ) sinp-pv (u' + w + R) cosp + ( wv - uv ) cosp-pv ( w' - u ) sinp),
Wv = R p'v ■
Euler equations resulting from the condition öfS*t = 0 are the further equations:
pv (N cos p- Q sin p) + (Nv sin p + Qv cos p)-(pv (N sin p + Q cos p) -
-(Nv cosp- Qvsin p)) = 0;
pv (N sin p + Q cos p) - (Nv cos p- Qv sin p) + (pv (N cos p- Q sin p) + + (Nv sinp + Qv cosp)) '= 0;
Mv +(K + wv )( N sinp + Q cosp)-( w'v - uv )( N cosp- Q sinp) + + (u' + w + R )(pv (N cosp- Q sinp) + ( Nv sinp + Qv cosp)) + + (w'-u)(pv (Nsinp + Qcosp)-(Nv cosp-Qv sinp)) = 0.
Expression (18) is the system of equations involving functions uv, wv, pv. Functions u, w, p, N, Q, M are known and are the solution of nonlinear static problem, the stability of which is being researched.
Equations (18) are the exact stability equations of the elastic round arch under the potential dead load, taking into account all stiffnesses of round arch. To get these equations no hypothesis was made about the value of displacements and type of stress-strain state of the bar.
The natural boundary conditions can be derived from the terms outside the integral (16) regarding the essential boundary conditions (12):
(18)
Mv (0) = 0, Mv (0) = 0. (19)
Thus, the formulation of the stability problem consists of stability equations (18) and six boundary conditions (12), (19). The exact solution for the problem of stability can be derived if the exact solution of the nonlinear static problem (3)-(6) is put in system (18). There are no exact analytical solutions for the nonlinear static problems of the curvilinear bars. That is why the solution of the stability problem is derived in a linearized formulation [31]. This means, that solution of the original static problem in linear formulation is put in system (18).
3.3. Solving the problem of arch equilibrium stability
Consider the problem of half ring equilibrium (an arch with central angle 0 = П of radius R, under the dead radial pressure (Figure 2).
Figure 2. Structural model of the half-ring under the radial pressure.
Statically acceptable solution in linear formulation [19] can be written as follows:
N = -qR, Q = 0, M = 0.
(20)
As the physical equations (5) and functionals (7) and (15) are valid only for the elastic material, then the distributed pressure shouldn't outnumber the following values:
q <-
OyA
R '
(21)
where Oy is elastic limit.
Substitution of the static solution (20) into stability equations (18) leads to the following system:
(Npv + Qv) + (Nv )'= 0; - Nv +(Npv + Qv )'= 0;
Mv - N (wv - uv) + R (Npv + Qv ) = 0, where, according to the (17), are made the following labels:
Nv = R(<+ wv), Qv = R(-Rpv +(wv- uv)), Mv= R p'v. The solution of the system (22) are the following functions: uv = iGQcos6 + -2GC16sin6 + -2GC26cos6 + HC3 cos4A6 + HC4 sin4A6
+ C5 cos 6 + C6 sin 6; wv = -1GC16 cos 6 + R C1 sin 6 + R C2 cos 6 + 2 OC26 sin 6 - 2 GC2 cos 6 + + 4AhC, sin 4a6 - VAhC4 cos 4A6 + C5 sin 6 - C6 cos 6;
(22)
(23)
pv =RHC1 cos6-RHC2 sin6 + C3 cos VA6 + C4 sin VA6,
k3 k3
where Ci is integration constants and the following labels are used:
a=f1+qR1 qR
1+qR 1R
H =
1+qR 1 qR3 -1
G =
r1++-h i 1+qR 1
^2
k^ j k
3 j
(24)
Boundary conditions (12), (19) lead to the system of linear equations involving integration constants. After the equivalent transformations it can be derived:
GCX = 0;
4Ah sin () C3 = 0;
HRC2 +4A sin (4An) C4 = 0;
2 Gn4A sin (yfAn) c2+i4ah sin (JAn^ cos (4An)+1) c4 = 0.
According to the numerical test, a critical load, calculated from the first two equations of the set (25), outnumbers the critical load from the last system of two equations. Hence the transcendental equation relative to the minimal value of the critical load can be derived:
(25)
4AH2 R(cos (VAn) + l)-n AG sin i^JAn)
= 0,
(26)
Using the labels from (24) and trigonometric transformations, (26) can be rewritten:
f
tg
nil+^
qR 1 qR
kk2 J k^
i+-
qR 1 R2
n
1+
qR 1 qR3
kk2 J k^
qR l2 r2 ^
l+
f+f+-'
ki k2 i-ii+qR 1qR3
k2 J k3
1 -I 1 +
qR 1 qR
3 Y
k2 J k3
(27)
Transcendental equation (27) makes it possible to determine the value for the critical load q„ for the circle arch under the dead pressure considering all the stiffnesses of the bar. Equation (27) can be solved numerically for any arch with any cross-section, though you can't get a general correlation between loading and bar stiffness. Moreover, it is very uncomfortable to use such an equation in practice. To get the simple form for the critical load an asymptotic solution for the equation (27) will be done.
Consider the labels for the non-dimensional values:
_ [qR3 „ k
b =
k
-, 6 =
3
k2 R2
P = k3
62 = k R
2 •
(28)
For the cross-sections, widely used in practice, shear and axial stiffnesses are much more than bending stiffness, that is why £1, £2 can be considered as small parameters: £1, £2 << 1.
According to (28), (27) can be written as follows:
tg (f bj (1+b P ) ) =
(1 + b2P )2
n
+b2p1 )
p2 +p1 +;
(1 + bP )
2 "N
1 -
(1 + b2P )i
(1 -(1 + b2p1 ) b2 )2
(29)
The parameter £2 can be considered to be dependent on £1 through the k coefficient, where k is a certain constant value.
P2 =61 k.
(30)
According to (30), (29) can be written as follows:
_ 4
tg (n ^/¡1+^
(1+b% )2
n _(
by/ (l + b2Z ) 1
& ( k+1) +
(i+b2^i )2 1 -(l + b2& ) bb
(l-(l + b&i ) b2 )2
(31)
The unknown b can be estimated as an asymptotic series with a small parameter £1:
b = b0 + + + ..., (32)
where b0 = yjqKR3/ k3 relatives the value of the critical force qK neglecting axial and shear yielding, i.e. when ki ^ ao, ki ^ co, which is equivalent to £2 ^ 0, £1 ^ 0. In fact, when £1 ^ 0, £2 ^ 0, (29) transforms into
tg (f b0 )= 4 1
bo (1 - bo )'
(33)
The minimal positive root of the transcendental equation (33) is b0 «1.80866. Using the first expression in (28) a Kornoukhov-Dinnik solution (2) can be derived:
q =1.808662 % « 3.27.
cr R3 R3
(34)
An approximate formula for the critical force can be derived by substitution of the asymptotic series with a small parameter (32) into the equation (31), expanding both parts of the equation into a series, setting coefficients of the same powers equal and considering only terms of the first order of smallness:
qcr = qK
1 - 0.223
qK R ki
-1.223
qK R
(35)
The solution for the model neglecting axial stiffness (Timoshenko beam theory) can be obtained from the (35) by letting k1 ^ a
qcr = qK
1 -1.223
qK R
(36)
The solution for the model regarding only bending stiffness can be obtained from the (35) by letting ki ^ a, k2 ^ 0.. In that case the Kornoukhov-Dinnik solution (2) is derived.
Using the value for the critical load from (27) or (35) the mode of buckling can be found:
uB = 2 Gcr C2d cos 0 + H cr C4 sinVA>;
w = — e 2
i GcrC20 sin 0 - i GcrC2 cos 0 + R C2 cos 0 -JATHcrC4 cos 7A0; ' 2 ki
(37)
(e = -Hcr R C2 sin 0 + C4 sin VA0
where Acr, Hcr, Gcr are labels from (24), where the critical force value was substituted.
It is easy to prove, that each of the terms in (37) has no left-to-right symmetry, so the buckling mode is antisymmetry. This goes with the results of the experiments [38], shown in Figure 3.
The comparison of the numerical exact solution (27), asymptotic solutions (35), (36) and a Kornoukhov-Dinnik solution (2) can be made. Consider an arch with radius R = 12 m, cross-section is a thin-walled tube with a thickness of 10 mm and a variety of diameters: from 355.6 mm till 1420 mm. Geometrical and stiffness values can be found in Table 1.
3
a b
Figure 3. Buckling mode of the arch under the dead radial load: a - analytical model; b - experiment. Table 1. Geometrical and stiffness values of the cross-section.
Cross-section Outer diameter, [mm] Cross-section area A, [cm2] Moment of inertia I, [m4] Axial stiffness k1 = EA, [N] Shear stiffness k2 = GAk, [N] Bending stiffness k3 = EI, [N - m2]
355.6x10 355.60 108.57 1.62E-04 2.1714E+09 4.1758E+08 1.5698E+08
377x10 377.00 115.29 1.94E-04 2.3058E+09 4.4342E+08 1.9832E+08
406.4x10 406.40 124.53 2.45E-04 2.4906E+09 4.7896E+08 2.6780E+08
426x10 426.00 130.69 2.83E-04 2.6138E+09 5.0265E+08 3.2332E+08
478x10 478.00 147.02 4.03E-04 2.9404E+09 5.6546E+08 5.1252E+08
530x10 530.00 163.36 5.52E-04 3.2672E+09 6.2831E+08 7.7465E+08
630x10 630.00 194.77 9.36E-04 3.8954E+09 7.4912E+08 1.5465E+09
720x10 720.00 223.05 1.41E-03 4.4610E+09 8.5788E+08 2.6383E+09
820x10 820.00 254.46 2.09E-03 5.0892E+09 9.7869E+08 4.4387E+09
920x10 920.00 285.88 2.96E-03 5.7176E+09 1.0995E+09 7.0332E+09
1020x10 1020.00 317.29 4.05E-03 6.3458E+09 1.2203E+09 1.0627E+10
1120x10 1120.00 348.71 5.37E-03 6.9742E+09 1.3412E+09 1.5448E+10
1220x10 1220.00 380.12 6.96E-03 7.6024E+09 1.4620E+09 2.1749E+10
1420x10 1420.00 442.95 1.10E-02 8.8590E+09 1.7037E+09 3.9917E+10
Consider the non-dimensional values of critical forces (27), (35), (36) by dividing them by qK and construct a plot (Figure 4). An equation (27) is solved using the iterative Newton's method with the help of Wolfram Mathematica software.
x
£ P5
o o o o o o o o o o o o o
X X X X X X X X X X X X X
h- o o o o o o o o
[•» CO a •a- CM (■» n cn :\ CM CM CM CM
(O Lil CO r- CO Oi o CM
Exact solution with all stiffness (formula (27))
.....Kornoukhov-Dinniksolution with onlybending stiffness (formula (2))
Asymptotic solution with all stiffness (formula (35))
Asymptotic solution with bending and shearstiffness (formula (36))
Figure 4. The comparison for the critical load values.
According to the comparison of the critical loads, the one, that considers all stiffnesses is tends to be less, than the one, that regards only bending stiffness. It is worthwile noticing, that inaccuracy between the Kornoukhov-Dinnik solution (2) and transcendental equation (27) increases as the size of the cross-section grows. Critical loads, estimated using the asymptotic formulas (35) and (36) are always smaller, than the exact values, so they improve the margin of safety. Moreover, inaccuracy between the asymptotic formulas and the solution (27) is less than 0.3 %.
The results can be used in the analytical defining of the stress-strain state of structures, where tensile-compression and shear stiffnesses make a significant contribution. Such structures include masonry and concrete arches [39-42], curvilinear elements of dams [43, 44] and long-span steel roofs [45, 46].
4. Conclusions
1. An analytical model of a geometrical non-linear deformation and stability of the plane elastic round arch taking into account all stiffnesses was worked out. This model contains:
1.1. exact non-linear equilibrium equations;
1.2. variational formulation for the problem is defined as finding stationary point of Lagrange functional;
1.3. static stability functional;
1.4. exact stability equations.
2. Based on the derived equations the problem of the equilibrium stability of the round arch under the dead radial load was solved. The characteristic transcendental equation and its asymptotic solution as a number of simple formulas, suitable for practical application, were worked out.
3. The comparison of described solution which regards all the bar's stiffnesses and classical Kornoukhov-Dinnik solution based on bending stiffness, was made. It was shown, that considering axial and shear stiffnesses leads to decreasing the values of the critical forces.
References
1. Timoshenko, S.P., Gere, G.M. Theory of Elastic Stability. 17th ed. McGraw-Hill International Book Company. New York, 1985. 541 p.
2. Eslami, M.R. Buckling and Postbuckling of Beams, Plates and Shells. 1st ed. Springer. New York, 2018. 588 p.
3. Farshad, M. Stability of Structures. 1st ed. Elsevier Science. New York, 1994. 425 p.
4. Simitses, G.J., Hodge, D.H. Fundamentals of Structural Stability. 1st ed. Elsevier Science. New York, 2006. 379 p.
5. Kornoukhov, N.V. Prochnost i ustoychivost sterzhnevykh system [Strength and stability of rod systems]. Moscow: Gosstroyizdat, 1949. 376 p. (rus)
6. Dinnik, A.N. Ustoychivost arok [Stability of arches]. Moscow: Gostekhteorizdat, 1946. 128 p. (rus)
7. Pi, Y.L., Bradford, M.A. Non-linear buckling and postbuckling analysis of arches with unequal rotational end restraints under a central concentrated load. International Journal of Solids and Structures. 2012. 49(26). Pp. 3762-3773. DOI 10.1016/j.ijsolstr.2012.08.012.
8. Bradford, M.A., Uy, B., Pi, Y.-L. In-Plane Elastic Stability of Arches under a Central Concentrated Load. Journal of Engineering Mechanics. 2002. 128(7). Pp. 710-719. DOI 10.1061/(ASCE)0733-9399(2002)128:7(710)
9. Bateni, M., Eslami, M.R. Non-linear in-plane stability analysis of FGM circular shallow arches under central concentrated force. International Journal of Non-Linear Mechanics. 2014. No. 60. Pp. 58-69. DOI 10.1016/j.ijnonlinmec.2014.01.001
10. Pi, Y.L., Bradford, M.A., Uy, B. In-plane stability of arches. International Journal of Solids and Structures. 2002. 39(1). Pp. 105-125. DOI 10.1016/S0020-7683(01)00209-8
11. Pi, Y.L., Trahair, N.S. Non-linear buckling and postbuckling of elastic arches. Engineering Structures. 1998. 20(7). Pp. 571-579. DOI 10.1016/S0141 -0296(97)00067-9
12. Cai, J., Feng, J. Buckling of parabolic shallow arches when support stiffens under compression. Mechanics Research Communications. 2010. 37(5). Pp. 467-471. DOI 10.1016/j.mechrescom.2010.05.004
13. Cai, J., Xu, Y., Feng, J., Zhang, J. In-Plane Elastic Buckling of Shallow Parabolic Arches under an External Load and Temperature Changes. Journal of Structural Engineering. 2012. 138(11). Pp. 1300-1309. DOI 10.1061/(ASCE)ST.1943-541X.0000570.
14. Bateni, M., Eslami, M.R. Non-linear in-plane stability analysis of FG circular shallow arches under uniform radial pressure. Thin-Walled Structures. 2015. No. 94. Pp. 302-313. DOI 10.1016/j.tws.2015.04.019
15. Lu, H., Liu, A., Pi, Y.L., Bradford, M.A., Fu, J., Huang, Y. Localized loading and nonlinear instability and post-instability of fixed arches. Thin-Walled Structures. 2018. No. 131. Pp. 165-178. DOI 10.1016/j.tws.2018.06.019
16. Xu, Y., Gui, X., Zhao, B., Zhou, R. In-Plane Elastic Stability of Arches under a Radial Concentrated Load. Engineering. 2014. 09(06). Pp. 572-583. DOI 10.4236/eng.2014.69058
17. Paimushin, V.N., Polyakova, N.V. The consistent equations of the theory of plane curvilinear rods for finite displacements and linearized problems of stability. Journal of Applied Mathematics and Mechanics. 2009. 73(2). Pp. 220-236. DOI 10.1016/j.jappmathmech.2009.04.012
18. Paimushin, V.N., Polyakova, N.V. The stability of a ring under the action of a linear torque, constant along the perimeter. Journal of Applied Mathematics and Mechanics. 2011. 75(6). Pp. 691-699. DOI 10.1016/j.jappmathmech.2012.01.009
19. Karnovsky, I.A. Theory of Arched Structures: Strength, Stability, Vibration. 1st ed. Springer. New York, 2012. DOI 10.1007/978-14614-0469-9
20. Karnovsky, I.A., Lebed, O. Advanced Methods of Structural Analysis. 1st ed. Springer. New York, 2010. DOI 10.1007/978-1-44191047-9
21. Galishnikova, V.V., Pahl, P.J. Analysis of frame buckling without sidesway classification. Structural Mechanics of Engineering Constructions and Buildings. 2018. 14(4). Pp. 299-312. DOI 10.22363/1815-5235-2018-14-4-299-312
22. Han, Q., Cheng, Y., Lu, Y., Li, T., Lu, P. Nonlinear buckling analysis of shallow arches with elastic horizontal supports. Thin-Walled Structures. 2016. No.109. Pp. 88-102. DOI 10.1016/j.tws.2016.09.016
23. Zhou, Y., Yi, Z., Stanciulescu, I. Nonlinear Buckling and Post-buckling of Shallow Arches with Vertical Elastic Supports. Journal of Applied Mechanics. 2019. 86(6). Pp. 1-16. DOI 10.1115/1.4042572
24. Ghayesh, M.H., Farokhi, H. Mechanics of tapered axially functionally graded shallow arches. Composite Structures. 2018. No. 188. Pp. 233-241. DOI 10.1016/j.compstruct.2017.11.017
25. Kiss, L.P. Nonlinear stability analysis of FGM shallow arches under an arbitrary concentrated radial force. International Journal of Mechanics and Materials in Design. 2019. No. 2. Pp. 1-18. DOI 10.1007/s10999-019-09460-2
26. Cai, J., Zhou, Y., Feng, J. Post-buckling behavior of a fixed arch for variable geometry structures. Mechanics Research Communications. 2013. No. 52. Pp. 74-80. DOI 10.1016/j.mechrescom.2013.07.002
27. Lu, Y., Cheng, Y., Han, Q. Experimental investigation into the in-plane buckling and ultimate resistance of circular steel arches with elastic horizontal and rotational end restraints. Thin-Walled Structures. 2017. No. 118. Pp. 164-180. DOI 10.1016/j.tws.2017.05.010
28. Guo, Y.L., Yuan, X., Bradford, M.A., Pi, Y.L., Chen, H. Strength design of pin-ended circular steel arches with welded hollow section accounting for web local buckling. Thin-Walled Structures. 2017. No. 115. Pp. 100-109. DOI 10.1016/j.tws.2017.02.010
29. Guo, Y.-L., Chen, H., Pi, Y.-L., Dou, C., Bradford, M.A. In-Plane Failure Mechanism and Strength of Pin-Ended Steel I-Section Circular Arches with Sinusoidal Corrugated Web. Journal of Structural Engineering. 2016. 142(2). Pp. 15-21. DOI 10.1061/(ASCE)ST.1943-541X.0001393
30. Guo, Y.L., Chen, H., Pi, Y.L. In-plane failure mechanisms and strength design of circular steel planar tubular Vierendeel truss arches. Engineering Structures. 2017. No. 151. Pp. 488-502. DOI 10.1016/j.engstruct.2017.08.055
31. Perelmuter, A.V., Slivker, V.I. Handbook of Mechanical Stability in Engineering. Vol. 1 : General Theorems and Individual Members of Mechanical Systems. World Scientific. New York, 2013. DOI 10.1142/8372
32. Lalin, V.V., Rozin, L.A., Kushova, D.A. Variational functionals for two-dimensional equilibrium and stability problems of Cosserat-Timoshenko elastic rods. Magazine of Civil Engineering. 2013. 36(1). Pp. 87-96. DOI 10.5862/MCE.36.11 (rus)
33. Simo, J.C. A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Comput. Methods Appl. Mech. Eng. 1985. 49(1). Pp. 55-70. DOI 10.1016/0045-7825(85)90050-7
34. Lalin, V.V., Zdanchuk, E.V., Kushova, D.A., Rozin, L.A. Variational formulations for non-linear problems with independent rotational degrees of freedom. Magazine of Civil Engineering. 2015. 56(04). Pp. 54-65. DOI 10.5862/MCE.56.7 (rus)
35. Lang, H., Linn, J., Arnold, M. Multi-body dynamics simulation of geometrically exact Cosserat rods. Multibody System Dynamics. 2011. 25(3). Pp. 285-312. DOI 10.1007/s11044-010-9223-x
36. Jelenic, G., Crisfield, M.A. Geometrically exact 3D beam theory: Implementation of a strain-invariant finite element for statics and dynamics. Computer Methods in Applied Mechanics and Engineering. 1999. 171(1-2). Pp. 141-171. DOI 10.1016/S0045-7825(98)00249-7
37. Gerstmayr, J., Shabana, A.A. Analysis of thin beams and cables using the absolute nodal co-ordinate formulation. Nonlinear Dynamics. 2006. 45(1-2). Pp. 109-130. DOI 10.1007/s11071-006-1856-1
38. Dmitriev, A.N., Semenov, A.A., Lalin, V.V. Stability of the equilibrium of elastic arches with a deformed axis. Construction of Unique Buildings and Structures. 2018. 2(67). Pp. 19-31. DOI 10.18720/CUBS.67.2 (rus)
39. Bespalov, V., Semenova, M. Influence of masonry adhesion on mechanical performance of arches-walls. MATEC Web Conf. 245 02002. 2018. DOI 10.1051/matecconf/201824502002
40. Kaldar-ool, A-Kh.B., Babanov, V.V., Allahverdov, B.M., Saaya, S.S. Additional load on barrel vaults of architectural monuments. Magazine of Civil Engineering. 2018. 84(8). Pp. 15-28. DOI 10.18720/MCE.84.2.
41. Zubkov, S.V., Ulybin, A.V., Fedotov, S.D. Assessment of mechanical properties of brick masonry by flat-jack method. Magazine of Civil Engineering. 2015. 60(8). Pp. 20-29. DOI 10.18720/MCE.60.3.
42. Orlovich, R.B., Nowak, R., Vatin, N.I., Bespalov V.V. Natural oscillations of a rectangular plates with two adjacent edges clamped. Magazine of Civil Engineering. 2018. 82(6). Pp. 95-102. DOI 10.18720/MCE.82.9
43. Hirkovskis, A., Serdjuks, D., Goremikims, V., Pakrastins, L., Vatin, N.I. Behaviour analysis of load-bearing aluminium members. Magazine of Civil Engineering. 2015. 57(5). Pp. 86-96. DOI 10.5862/MCE.57.8
44. Gusevs, J., Serdjuks, D., Artebjakina, G.I, Afanasjeva, E.A, Goremikins, V. Behaviour of load-carrying members of velodromes' longspan steel roof. Magazine of Civil Engineering. 2016. 65(5). Pp. 3-16. DOI 10.5862/MCE.65.1
45. Eigenson, S.N., Korikhin, N.V., Golovin, A.I. Experimental investigation considering the stressed state of some essential constructions of large hydropower buildings. Magazine of Civil Engineering. 2014. 45(1). Pp. 59-70. DOI 10.5862/MCE.45.7
46. Kolosova, G.S., Lalin, V.V., Kolosova, A.V. The effect of construction joints and cracks on the stress-strain state of the arch-gravity dam. Magazine of Civil Engineering. 2013. 40(5). Pp. 76-85. DOI 10.5862/MCE.40.9
Contacts:
Vladimir Lalin, +7(921)319-98-78; vllalin@yandex.ru Andrey Dmitriev, +7(999)249-09-00; dmitriefan@outlook.com Stanislav Diakov, +7(921)300-89-17; stass.f.dyakov@gmail.com
© Lalin, V.V., Dmitriev, A.N., Diakov S.F., 2019
Инженерно-строительный журнал
сайт журнала: http://engstrov.spbstu.ru/
ISSN
2071-0305
DOI: 10.18720/MCE.89.4
Геометрически нелинейное деформирование и устойчивость
упругих арок
В.В. Лалин, А.Н. Дмитриев*, С.Ф. Дьяков
Санкт-Петербургский политехнический университет Петра Великого, Санкт-Петербург, Россия
Keywords: устойчивость конструкций, потеря устойчивости, геометрически точная теория, «мертвая» нагрузка, круговая арка, жесткость, точка стационарности, функционал Лагранжа
Аннотация. В статье рассматривается плоская круговая двухшарнирная арка, нагруженная потенциальной «мертвой» нагрузкой. Для описания напряженно-деформированного состояния и устойчивости равновесия используется геометрически точная теория, в соответствии с которой каждая точка стержня имеет две трансляционные степени свободы и одну вращательную, не зависящую от трансляционных. Для получения решения не используются никакие упрощения о величинах перемещений и углов поворота, а также учитываются все жесткости стержня - продольная, сдвиговая и изгибная. Получены точные нелинейные дифференциальные уравнения статической задачи. Сформулирована вариационная постановка в виде задачи поиска точки стационарности функционала типа Лагранжа. Доказана эквивалентность дифференциальной и вариационной постановок. Получены точные уравнения устойчивости, учитывающие геометрически нелинейное деформирование в докритическом состоянии. На основе полученных уравнений решена задача устойчивости равновесия круговой арки при действии «мертвого» радиального давления с учетом всех жесткостей стержня. Получено характеристическое трансцендентное уравнение, а также асимптотическое решение этого уравнения в виде простых формул, пригодных для практического применения. Выполнено сравнение полученного решения, учитывающего все жесткости стержня, с классическим решением, учитывающим только изгибную жесткость.
Литература
1. Timoshenko S.P., Gere G.M. Theory of Elastic Stability. 17th ed. McGraw-Hill International Book Company. New York, 1985. 541 p.
2. Eslami M.R. Buckling and Postbuckling of Beams // Plates and Shells. 1st ed. Springer. New York, 2018. 588 p.
3. Farshad M. Stability of Structures. 1st ed. Elsevier Science. New York, 1994. 425 p.
4. Simitses G.J., Hodge D.H. Fundamentals of Structural Stability. 1st ed. Elsevier Science. New York, 2006. 379 p.
5. Корноухов Н.В. Прочность и устойчивость стержневых систем. М.: Госстройиздат, 1949. 376 с.
6. Динник А.Н. Устойчивость арок. М.: Гостехтеориздат, 1946. 128 с.
7. Pi Y.L., Bradford M.A. Non-linear buckling and postbuckling analysis of arches with unequal rotational end restraints under a central concentrated load // International Journal of Solids and Structures. 2012. 26(49). Pp. 3762-3773.
8. Bradford M.A., Uy B., Pi Y.-L. In-Plane Elastic Stability of Arches under a Central Concentrated Load // Journal of Engineering Mechanics. 2002. No. 7 (128). Pp. 710-719. DOI 10.1061/(ASCE)0733-9399(2002)128:7(710)
9. Bateni M., Eslami M.R. Non-linear in-plane stability analysis of FGM circular shallow arches under central concentrated force // International Journal of Non-Linear Mechanics. 2014. No. 60. Pp. 58-69. DOI 10.1016/j.ijnonlinmec.2014.01.001
10. Pi Y.L., Bradford M.A., Uy B. In-plane stability of arches // International Journal of Solids and Structures. 2002. No. 1(39). Pp. 105125. DOI 10.1016/S0020-7683(01 )00209-8
11. Pi Y.L., Trahair N.S. Non-linear buckling and postbuckling of elastic arches // Engineering Structures. 1998. No. 7(20). Pp. 571-579. DOI 10.1016/S0141 -0296(97)00067-9
12. Cai J., Feng J. Buckling of parabolic shallow arches when support stiffens under compression // Mechanics Research Communications. 2010. No. 5(37). Pp. 467-471. DOI 10.1016/j.mechrescom.2010.05.004
13. Cai J., Xu Y., Feng J., Zhang J. In-Plane Elastic Buckling of Shallow Parabolic Arches under an External Load and Temperature Changes // Journal of Structural Engineering. 2012. No. 11(138). Pp. 1300-1309. DOI 10.1061/(ASCE)ST.1943-541X.0000570.
14. Bateni M., Eslami M.R. Non-linear in-plane stability analysis of FG circular shallow arches under uniform radial pressure. Thin-Walled Structures. 2015. No. 94. Pp. 302-313. DOI 10.1016/j.tws.2015.04.019
15. Lu H., Liu A., Pi Y.L., Bradford M.A., Fu J., Huang, Y. Localized loading and nonlinear instability and post-instability of fixed arches // Thin-Walled Structures. 2018. No. 131. Pp. 165-178. DOI 10.1016/j.tws.2018.06.019
16. Xu Y., Gui X., Zhao B., Zhou R. In-Plane Elastic Stability of Arches under a Radial Concentrated Load // Engineering. 2014. No. 6(9). Pp. 572-583. DOI 10.4236/eng.2014.69058
17. Paimushin V.N., Polyakova N.V. The consistent equations of the theory of plane curvilinear rods for finite displacements and linearized problems of stability // Journal of Applied Mathematics and Mechanics. 2009. No. 2(73). Pp. 220-236. DOI 10.1016/j.jappmathmech.2009.04.012
18. Paimushin V.N., Polyakova N.V. The stability of a ring under the action of a linear torque, constant along the perimeter // Journal of Applied Mathematics and Mechanics. 2011. No. 6(75). Pp. 691-699. DOI 10.1016/j.jappmathmech.2012.01.009
19. Karnovsky I.A. Theory of Arched Structures: Strength, Stability, Vibration. 1st ed. Springer. New York, 2012. DOI 10.1007/978-14614-0469-9
20. Karnovsky I.A., Lebed O. Advanced Methods of Structural Analysis. 1st ed. Springer. New York, 2010. DOI 10.1007/978-1-4419-1047-9
21. Галишникова В.В., Паль П.Я. Анализ устойчивости рам без учета классификации по возможности поперечных смещений // Строительная механика инженерных конструкций и сооружений. 2018. No. 4(14). С. 299-312. DOI: 10.22363/1815-5235-201814-4-299-312.
22. Han Q., Cheng Y., Lu Y., Li T., Lu P. Nonlinear buckling analysis of shallow arches with elastic horizontal supports // Thin-Walled Structures. 2016. No. 109. Pp. 88-102. DOI 10.1016/j.tws.2016.09.016
23. Zhou Y., Yi Z., Stanciulescu I. Nonlinear Buckling and Post-buckling of Shallow Arches with Vertical Elastic Supports // Journal of Applied Mechanics. 2019. No. 6(86). Pp. 1-16. DOI 10.1115/1.4042572
24. Ghayesh M.H., Farokhi H. Mechanics of tapered axially functionally graded shallow arches // Composite Structures. 2018. No. 188. Pp. 233-241. DOI 10.1016/j.compstruct.2017.11.017
25. Kiss L.P. Nonlinear stability analysis of FGM shallow arches under an arbitrary concentrated radial force // International Journal of Mechanics and Materials in Design. 2019. No. 2. Pp. 1-18. DOI 10.1007/s10999-019-09460-2
26. Cai J., Zhou Y., Feng J. Post-buckling behavior of a fixed arch for variable geometry structures // Mechanics Research Communications. 2013. No. 52. Pp. 74-80. DOI 10.1016/j.mechrescom.2013.07.002
27. Lu Y., Cheng Y., Han Q. Experimental investigation into the in-plane buckling and ultimate resistance of circular steel arches with elastic horizontal and rotational end restraints // Thin-Walled Structures. 2017. No. 118. Pp. 164-180. DOI 10.1016/j.tws.2017.05.010
28. Guo Y.L., Yuan X., Bradford M.A., Pi Y.L., Chen, H. Strength design of pin-ended circular steel arches with welded hollow section accounting for web local buckling // Thin-Walled Structures. 2017. No. 115. Pp. 100-109. DOI 10.1016/j.tws.2017.02.010
29. Guo Y.-L., Chen H., Pi Y.-L., Dou C., Bradford M.A. In-Plane Failure Mechanism and Strength of Pin-Ended Steel I-Section Circular Arches with Sinusoidal Corrugated Web // Journal of Structural Engineering. 2016. No. 2(142). Pp. 15-21. DOI 10.1061/(ASCE)ST.1943-541X.0001393
30. Guo, Y.L., Chen, H., Pi, Y.L. In-plane failure mechanisms and strength design of circular steel planar tubular Vierendeel truss arches // Engineering Structures. 2017. No. 151. Pp. 488-502. DOI 10.1016/j.engstruct.2017.08.055
31. Перельмутер А.В., Сливкер В.И. Устойчивость равновесия конструкций и родственные проблемы. Т. 1. Общие теоремы. Устойчивость отдельных элементов механических систем. М.: Изд-во СКАД СОФТ, 2010. 68l с.
32. Лалин В.В., Розин Л.А., Кушова Д.А. Вариационная постановка плоской задачи геометрически нелинейного деформирования и устойчивости упругих стержней // Инженерно-строительный журнал. 2013. № 1(36). С. 87-96. DOI 10.5862/MCE.36.11
33. Simo J.C. A finite strain beam formulation // The three-dimensional dynamic problem. Part I. Comput. Methods Appl. Mech. Eng. 1985. No. 1(49). Pp. 55-70. DOI 10.1016/0045-7825(85)90050-7
34. Лалин В.В., Зданчук Е.В., Кушова Д.А., Розин Л.А Вариационные постановки нелинейных задач с независимыми вращательными степенями свободы // Инженерно-строительный журнал. 2015. № 4(56). С. 54-65. DOI 10.5862/MCE.56.7
35. Lang H., Linn J., Arnold M. Multi-body dynamics simulation of geometrically exact Cosserat rods // Multibody System Dynamics. 2011. No. 3(25). Pp. 285-312. DOI 10.1007/s11044-010-9223-x
36. Jelenic G., Crisfield M.A. Geometrically exact 3D beam theory: Implementation of a strain-invariant finite element for statics and dynamics // Computer Methods in Applied Mechanics and Engineering. 1999. No. 1-2(171). Pp. 141-171. DOI 10.1016/S0045-7825(98)00249-7
37. Gerstmayr J., Shabana A.A. Analysis of thin beams and cables using the absolute nodal co-ordinate formulation // Nonlinear Dynamics. 2006. No. 1-2(45). Pp. 109-130. DOI 10.1007/s11071-006-1856-1
38. Дмитриев А.Н., Семенов А.А., Лалин В.В. Устойчивость равновесия упругих арок с учетом искривления оси // Строительство уникальных зданий и сооружений. 2018. № 4(67). С. 19-31. DOI:10.18720/CUBS.67.2.
39. Bespalov V., Semenova M. Influence of masonry adhesion on mechanical performance of arches-walls // MATEC Web Conf. 245 02002. 2018. DOI 10.1051/matecconf/201824502002
40. Калдар-оол А.Б., Бабанов В.В., Аллахвердов Б.М., Саая С.С. Дополнительная нагрузка на коробовый свод в памятнике архитектуры // Инженерно-строительный журнал. 2018. № 84(8). С. 15-28. DOI:10.18720/MCE.84.2.
41. Зубков С.В., Улыбин А.В., Федотов С.Д. Исследование механических свойств кирпичной кладки методом плоских домкратов // Инженерно-строительный журнал. 2015. № 8(б0). С. 20-29. DOI:10.5862/MCE.60.3.
42. Орлович Р.Б., Новак Р., Ватин Н.И., Беспалов В.В. Оценка прочности кирпичебетонных Прусских сводов // Инженерно-строительный журнал. 2018. № 6(82). С. 95-102. DOI:10.18720/MCE.82.9.
43. Хирковский А., Сердюк Д.О., Горемыкин В.В., Пакрастиньш Л., Ватин Н.И. Анализ работы несущих элементов из алюминиевых сплавов // Инженерно-строительный журнал. 2015. № 5(57). С. 86-96. DOI 10.5862/MCE.57.8
44. Гусев Е., Сердюк Д.О., Артебякина Г.И., Афанасьева Е.А., Горемыкин В.В. Behaviour of load-carrying members of velodromes' long-span steel roof // Инженерно-строительный журнал. 2016. № 5(65). С. 3-16. DOI:10.5862/MCE.65.1.
45. Эйгенсон С.Н., Корихин Н.В., Головин А.И. Экспериментальное исследование напряженного состояния некоторых ответственных конструкций крупных гидроэнергетических сооружений // Инженерно-строительный журнал. 2014. № 1(45). С. 59-70. DOI:10.5862/MCE.45.7.
46. Колосова Г.С., Лалин В.В., Колосова А.В. Влияние строительных швов и трещин на напряженно-деформированное состояние арочно-гравитационной плотины // Инженерно-строительный журнал. 2013. № 5(40). С. 76-85. DOI 10.5862/MCE.40.9
Контактные данные:
Владимир Владимирович Лалин, +7(921)319-98-78; эл. почта: vllalin@yandex.ru Андрей Николаевич Дмитриев, +7(999)249-09-00; эл. почта: dmitriefan@outlook.com Станислав Федорович Дьяков, +7(921)300-89-17; эл. почта: stass.f.dyakov@gmail.com
© Лалин В.В., Дмитриев А.Н., Дьяков С.Ф., 2019
