DYNAMIC DEFORMATION OF A BEAM AT SUDDEN STRUCTURAL TRANSFORMATION OF FOUNDATION Текст научной статьи по специальности «Строительство и архитектура»

Magazine of Civil Engineering. 2019. 91(7). Pp. 129-144 Инженерно-строительный журнал. 2019. № 7(91). С. 129-144

Magazine of Civil Engineering

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

ISSN 2071-0305

DOI: 10.18720/MCE.91. 12

Dynamic deformation of a beam at sudden structural transformation of foundation

V.I. Travusha, V.A. Gordonb, V.I. Kolchunovc*, E.V. Leontiev"

a Russian academy of architecture and construction science, Moscow, Russia b Orel State University named after I.S. Turgenev, Orel, Russia c Southwest State University, Kursk, Russia d Main State Expertise of Russia, Moscow, Russia * E-mail: asiorel@mail.ru

Keywords: beam, foundation, natural vibrations, forced vibrations, mode, frequency, accidental impact, structural transformation

Abstract. The article presents a methodic for analytical determining forces, displacements, modes and frequencies of natural flexural vibrations of a beam on elastic foundation. The beam consists of two sections: the first one supports on Winkler elastic foundation, and next one is free. Equations for flexural natural and forced vibrations were written in dimensionless variables and parameters and solved using the initial parameters method and Krylov functions. At the same time second and higher frequencies of natural vibrations of the beam were determined assuming unknown frequency is higher than "conventional" frequency which characterizes generalized stiffness of a system "beam-foundation". Using numerical analysis, authors showed dependencies between the first three dimensionless frequencies of natural vibrations of the beam and a generalized stiffness of the system "beam-foundation" when foundation suddenly partially failure under the beam. Investigation established that effect of a sudden structural transformation leads to five-time moment increasing in the system "beam-foundation" at sudden foundation failure under the second half of the beam.

Number of investigations on defense of buildings and structures against progressive destruction increases permanently [1] and most of these works deal with load redistribution in structural systems when a constructive element is removed from a building frame [2-4]. Investigations on deformation features of structures in a system "structure-foundation" under accidental impacts caused by sudden damage of a foundation are practically absent [13, 24]. A negligible number of studies [5-12] describes dynamic performance of beams and piles partially supported by an elastic base or partially imbedded into such a base. At the same time, it is usually assumed that a free structural segment is initially designed as a quasi-static body disregarding the inertia force [7-12]. In this regard, a problem of dynamical effects appearing when damage of a part of the system "beam-foundation" (such a partial destruction, boundary condition changing, crack formation, layers separation, reinforcement rupture) occurs suddenly, is of theoretical and practical interest [2, 25-27]. In this paper, we formulate and solve a problem on determination of dynamical force increment in a beam supported by an elastic base of Winkler's type during forced transverse oscillations of the beam due to sudden partial damage of the base. The paper presents results on analytical determination of forces, modes, and frequencies of a beam supported by elastic foundation when sudden partial destruction of foundation does occur.

Travush, V.I., Gordon, V.A., Kolchunov, V.I., Leontiev, E.V. Dynamic deformation of a beam at sudden structural transformation of foundation. Magazine of Civil Engineering. 2019. 91(7). Pp. 129-144. DOI: 10.18720/MCE.91.12

Травуш В.И., Гордон В.А., Колчунов В.И., Леонтьев Е.В. Динамическое деформирование балки при внезапном структурном изменении упругого основания // Инженерно -строительный журнал. 2019. № 7(91). С. 129-144. DOI: 10.18720/MCE.91.12

1. Introduction

This work is licensed under a CC BY-NC 4.0

2. Methods

The paper presents a formulation of the problem on determination of modes and frequencies of natural transverse oscillations for a beam of length L with flexural stiffness EI and distributed mass рА and consisting of two segments (Figure 1): the first segment of length Li is supported by an elastic foundation of Winkler's type and the second one of length L - Li is free. Solutions of the vibration problem for these two segments are constructed separately. Integration constants for proper differential equation can be determined owing to the conjunction conditions between beam's segments along with the boundary conditions at beam's endpoints.

У

xi

XXXXlxXx xxXXXXXXXXXXXXXXX

Li

X2

L

x

Figure 1. Beam partially supported by elastic foundation.

Transverse oscillations of the first segment 0 < £1 < v Let us introduce dimensionless variables and parameters

^ x, 1 „N V L IK

£ = j= i,2); w,= j; t = ° v = j ° = J-p:

a

Kj_.

4EI ''

_ o

o = —

where Xi - axial coordinate for i-th segment (i = 1, 2);

Vi = Vi(xi, t) is deflection field of i-th segment (i = 1, 2); v is relative length of the segment supported by foundation; K = kb is foundation stiffness; k is modulus of subgrade reaction, b is width of cross section;

a is generalized stiffness of the system «beam is foundation»;

00 is "conventional" frequency parameter that takes dimension of frequency [sis1];

t is physical time;

0 is frequency of natural oscillations.

The equation of natural oscillations for the first section takes the form [25, 26]

d4 d^4

+ 4az

w +

dT

= o.

(1)

У

Let us find the solution of the equation (1) assuming that oscillations are harmonic and using separation of variables:

Wi (£, t ) = Wi(£i ) sin 00 t,

(2)

_ o

where o = — is dimensionless frequency;

o

— o

o0 = —° is dimensionless "conventional" frequency;

o

rne = —7 is "reference" frequency. Further, we use the dimensionless «conventional» frequency

_i EL Pa

o0, as a generalized stiffness of the system "beam-foundation" instead of parameter a which has a physical meaning. Reducing the base stiffness K by means of the relationships

4

K KL

o0 = - and a = 4-,

0 \pA \4EL

and taking into account the reference frequency o^ we obtain

4a4 = ®o

2 f m \

1 EI

V®e J

= ®o2-

L4 pA

Substituting (2) into (1), we obtain the equation for modes of natural oscillations

WIV+ (5 -52 )w=0. (3)

The structure of equation (3) yields the following three possible solutions: 1) if o > o0, then, writing the equation (3) in the form

WIV-(o2-52 )W = 0 (4)

and solving the last equation by Euler's substitution

w = Ae^1, (5)

we obtain the characteristic equation

r4 -(o2 -o0) = 0,

the roots of which are as follow:

r,2 = ±A; r?,4 = ±ft = 4lo2-o2. (6)

Then the solution of equation (4) takes the form

Wi (Zi) = W10 R4 ("Zi)+W0 R3 ("Zi)+W'0 R2 ("Zi)+WR ("Zi), (7)

where Wi0, Wi'0, W"0, W0 are initial parameters; Ri = Ri ("ZZ) is Krylov's function;

= sh"^- sin^i ; = ch^^, - cos " ;

Ri ; R2 0 «2 ;

2P3 2"

R3--~-' R4--2-; R4 = " Ri.

2 "i 2

The matrix equation for the state of arbitrary cross section Zi in the first segment reads as

Wi (Zi) = Vii (" )Wi0, (8)

where Wi (Zi) = {W (Zi) Wi'(Zi) Wi''(Zi) Wi"(Zi )}T is state vector of an arbitrary cross section Zi W0 = {W0 W0 W0 Wi0}T is initial parameters vector;

HH®eHepH0-CTp0HTe^bHhiH ^ypHaa, №2 7(91), 2019

' r4 ) R3 №) R №) R №1

№1 (№) R4 (№£) R3 (№£) R2 №)

№4R2 №) №4R №) R4 №) R3 (№1)

№4R3 (№£) №R2 №) №R (№) R4 (№),

(9)

is a functional matrix describing the initial parameters influence on the state of cross section £1 in the first segment.

2) if b < b0, then, substituting equation (5) into (3), we obtain the characteristic equation

r 4+(b2-b2) =

with the complex roots

2)=0

rK4= (±i ±1)^2; №2 =

= 4I b -b

and the solution of the equation (3) reads as

Wx= W10K4 №) + W1O K3 ) + W1O K2 ) + W0K (№1), where Ki = Ki (№2£) are Krylov's functions that are of the form

_ sh№£1ch№£1 - Cos№£1shA£1. v _ sinP1£l - sh№£1

K1---; K 2--r^s-;

(10)

(11)

K3 =

№

_ sh^2^1Ch№2£1 + COsA£1shA£1 .

2№

2№

^cos^+ch^; k4 = -4&X

In this case, the state of an arbitrary cross section £1 of the first segment can be described in the following matrix form:

W (£1)=V12 (£1 )Ww,

where

f K4 ) K3 №£1) K2 №) K1

-4№X №1) K4 №) K3 №) K2 №)

-№K (№£) -4№4K1 №) K4 №) K3 )

-4№X №1) -4№X №) -4№X №£1) K4 №)

^ (£1) =

(12)

3) if bb = b0, then by using the serial integration of the equation

w iv=0,

we obtain the function

and the matrix equation where

£2 £

W = W + W' £ + W" — + w "— 2 6

W (£1) = ^ (£1 )W10,

(14)

V13 (£1) =

f 1 £1 £2 2 £31 6

0 1 £1 £2 2

0 0 1 £1

v 0 0 0 1J

4

Magazine of Civil Engineering, 91(7), 2019

Thus, the strain-stress state of the first segment is described by equations (8), (11) and (14) that all together can be expressed in the form

Wj (¿) = Vj (¿1 )Ww, j = 1,2,3. (15)

Transverse vibrations of 2-nd segment 0 < ¿2 < 1 - v

Natural transverse oscillations of this segment can be described by the equation [25-27]

d V , ^ 4 d2w2

—l + 4a4—-f = 0. (16)

¿4 dz

Separating variables by the representation

W2 (¿2,T) = W2 (¿2 ) Sin GT, (17)

we obtain

w2iv -g 2w2 = 0,

from where, assuming

we deduce the characteristic equation

W2 (¿2 ) = (18)

/-G2 = 0,

the roots of which are of the form

su = ±A; ^3,4 = ±№; №3 ^ Vbb. (19)

Express the state of an arbitrary cross section £2 of the 2-nd segment by the corresponding vector

W2 = {W2 (£2) W2 (£2) w;(£2 ) w2\£2 )}T

and the matrix equation

W2= V11 №£2 )W20, (20)

where w20 = {w20 w2'0 W2"0 W20}r is initial parameters vector for the 2-nd segment. Using the conjunction condition between the segments, we obtain

w20= w2(0) = w1(V),

since the matrix vn(0) is unit. Then the vector W20 = V1 j (v)W10 (j = 1,2,3) and the state vector for both the segments can be expressed via the initial parameters for the first segment

W (£1)=V11 № w (b > b)

W (£1 ) = V12 №1 W (bb < b0 ) - - (21)

W1 (£1) = V13 (£1 )Ww (b = b0)

W2 (£2) = ^ (№£2 )V j (v)W10.

Transverse oscillations of a beam with free endpoints

A beam resting on an elastic foundation without restrictions at the endpoints can be described as a proper model of spread footing. In this case, the boundary conditions read:

W" =W'" =0

"10 "10 u

(1 -v) = <(1 -v) = 0,

From here it follows that

W10 = K Wi'o 0 of (1-v) = {w(1-v) w'(1-v)oof .

(23)

1. At first, we accept a condition according to which the unknown frequency cC for a partially supported beam equals to "conventional" frequency c0. Then, according to (21) and (14), we have

10

(24)

W (6) = Vl3 (6 )W W2 6) = Vu 6 )vu (v)Ww. Let us write the second equation (24) in the expanded form by taking into account (23) and 62 = 1 - v

( W(1 -v) ^ ' R4 (P3(1 - v) ) R3 (P(1 - v) ) R2 (P3(1 - v) ) R (P(1

w2(1 -v) P34 R1 (P(1- -v) ) R4 (P(1 - v) ) R3 (PP(1 - v) ) R2 (P(1

0 pp R2 (P(1 -v) ) P34 R1 (P(1 -v) ) R4 (PP(1 - v) ) R3 (PP(1

v 0 V PR3 (P(1 -v) ) P34R2 (PP(1 -v) ) P34 R1 (PP(1 -v) ) R4 (PP(1

v2 v ]

2 6 (W \ "10

v v2 2 W ' '' 10 0

1 v v 0 ,

0 1 ,

0 1 v —

From here, we obtain the homogenous system of equations relatively the unknown initial parameters

W10 and W/0 :

' R (P(1 - v) ) W10 + (vR2 (P(1 - v) ) + R (P(1 - V) )) W0 = 0 R3 (p(1 -v) )W10+ (VR3 (P3(1 -v) ) + R2 (P(1 -v) ))W0 = 0.

The condition of existing of nonzero roots for this system is the equality to zero the determinant R2 (p(1 -v)) R2 (P(1 -v)) + R (p(1 -v)) R3 (P(1 -v) ) vR3 (P3(1 -v) ) + R2 (P3(1 -v) ) Expanding this determinant, we obtain the frequency equation

ch (p(1 -v) ) cos (p(1 -v) ) = 1,

the roots of which are [17]

= 0.

Pn(1 -v) = 0; P32(1 -v) = 4,73; P33(1 -v) = 7,853; P34(1 -v) =

2n + 1

n at n > 3 gives

physically impossible results at v= 1

lim P32 = limVC = lim 4,73 = œ and etc.

v—>1

v—>1

v—1 1 -v

Consequently, the accepted condition CC = c0 is not realized.

2. As is known [25], a free (i.e., without foundation) beam without constrains at its endpoints has two null frequencies corresponding to translational and rotational motion of the beam as a rigid body in addition to

X

Magazine of Civil Engineering, 91(7), 2019

the frequencies of free oscillations that coincide with the frequencies of the beam with clamped endpoints. Consequently, the rigid body motion should be added to the deflections caused by beam's vibrations. Such complex motion is described by the function

W = C1+

Following to the accepted model of the system «beam-foundation», the existence of small length v ^ 0 of beam's part interacting with foundation excludes the possibility of beam's motion as a rigid body. At the same time, calculation of the basic first frequency of natural oscillations can be performed according to (10)-

(11) that is when the condition 3 < m0, holds, beginning from m = 0 at v= 0 and m0 ^ 0. Let us accept this

condition, that is the unknown frequency m is smaller than the «conventional» frequency m0. Then, according to (21) and (12) we have:

W (¿1) = ^ )Ww; W2 (£2) = Vu (P)Vi2 {Pv)Ww. (25)

Let us write the second equation (25) using the expanded form for ¿2 =1- v

' W2(1 — V) ' ' R4 (PP(1 — V) ) R3 (P3(1 — V) ) R2(P(1— V) ) R (PP(1 — -v))]

w2(1 — V) _ P4R (PP(1- — v)) R4(P3(1— V) ) R3 (P(1— V) ) R2(P(1— -v)) v

0 P34 R2 (P(1 — V)) P34 R P(1 — V)) R4P(1— V) ) R3 (P3(1 — X -V))

V 0 y vP34R3 (PP(1 — v)) P34 R2 (P(1 — V) ) P34 R (PP(1 — V) ) R4(P(1— V) )J

K2 (&v) k (Av)Yw

( K4 (p2v) K3 (P2v)

-4#K (P2v) K4 (p2v) K3 (p2v) K2 (P2v)

-4ftK2 (P2v) -4P24Kj (P2V) K4 (P2V) K3 (P2V)

-4PX (P2V) —4PPK2 (P2V) -4P24K1 (P2v) K4 (P2v)

(26)

W'

"10

V 0 y

Also, the two linear homogenous equations relatively the unknown initial parameters W10 and W' can be found from the matrix equation (26) as it was done in the Section 1 of the paper:

[tfW0+U2W'0 = 0

1^0+ U4W0 = 0,

(27)

where

U = P34 R (X) K4 (Y)—4P24 (P34 R (X) K (Y)+R4 (X) K2 (Y))+R3 (X) K3 Y));

U2 = P34 (R (X) K4 (i!)+r ( X ) K4 (i!))—4P24 (R4 ( X ) K (Y)+R3 ( X ) K2 (Y)); U3 = P34 (R3 (X) K4 (Y) — 4P4 (R2 (X) K (Y) + R1 (X) K2 (Y))) — 4P4 R1 (X) K3 (Y); U4 = P34 (R3 (X K (Y) + R2 (X) K4 (Y) — 4P4 R1 (X) K ft)) — 4P4 R4 (X) K2 (Y);

X = P3O — v); Y = P2v. Now, obtain the frequency equation by equating the determinant value of the system (27) with zero:

UU4—u2U3 = 0.

(28)

The deflection functions wi (¿i) (i = 1, 2) along with the bending moments w"(£i) in arbitrary cross sections are found from the matrix equations (25). For the first segment (0 < ¿1 < V) we obtain

W (£) = Ww (K4 P) — UK3 P)); w1 (£) = Ww (— K2 P) — UK1 P))

X

and the same for the second one (0 < ¿2 < 1 - v):

W2

(4) = W10XRвР; ) = W10XК(вР

n=1 n=1

where

P = 4/?24 (-K3 (Y) + UK2 (Y)); P2 = 4A2 (-K2 ( Y) + UK, ( Y ));

P = -4PX (Y)-UK4 (Y); P4= K4 (Y)-UK3 (Y;); U = ^.

U 4

The second and higher frequencies and modes of natural vibrations of a beam with free endpoints resting on elastic foundation should be evaluated by assuming that the "conventional" frequency b0 is smaller

than the unknown one bb in accordance with the variant (6)-(7). It is determined by the requirement that the second frequency (following the first null) of a free beam (i.e., a beam without foundation, when b0 = 0) equals to 4.73 [25]. Then, according to (21) and (9) we have:

W (6 ) = Vn (M ) W,; W2 (£ ) = V;; (P3Z2 )VU (Av)Ww. (29)

From here, we obtain the frequency equation by performing the actions similar to ones in Section 1 and 2

Z,Z4 - Z2Z3 = 0, (30)

where

Z, = b4 R (X) R4 (Y2) + b;4 (b4 R (X) R (Y2) + R4 (X) R (Y2) + R (X )R (Y2)); Z2 = b4 (R2 (X )R (Y2) + R (X )R (Y2)) + b;4 (R4 (X) R (Y2) + R3 (X) R2 (Y2)); Z3 = A4 (R3 (X )R (Y2) + A4 (R2 (X )R (Y2) + R (X) R2 (Y2))) + AX (X) R3 (Y2); Z4 = A34 ( R (X) R (Y2) + R (X) R4 (Y2) + A;4 R (X) R (Y2) ) + A;4 R (X) R2 Y );

Y2 = Av.

The deflection functions wi (6 ) (i = 1,2) and the bending moments ) in arbitrary cross section of the first segment (0 < 6; < V are found from equation (29) as this was done in Section 1:

w (6 ) = w (r (A)- zr3 (A6 ))

w;(6) = w (r (A6)- ZR (A6))

and the same for the second segment (0 < 62 < 1 - v)

(62) = W;o±Rn(A362)s„; <(62) = W;o±R':(A362)S„ ,

W2'

where

S; = A;4 (R (Y2) - ZR2 (Y2)); S2 = A4 (R (Y2) - ZR; (Y2)); S3 = A4R(Y:)-zr4(Y2); S4 = R4(Y2)-ZR3(Y2); Z = ^

Z4

3. Results and Discussion

Table 1 shows the values of the first three dimensionless natural frequencies which are obtained from

equations (28) and (30) for different combinations of "conventional" frequency values b0, that characterize

generalized stiffness of the system "beam-foundation" and v is segment's length after partial destruction of the foundation.

Table 1. The first three frequencies of beam's natural oscillations.

_ 31 (3 < 30 ) 33 (3 > 30 ) 33 (3 = 30 )

30 ____________

0 V = 0.25 V = 0.5 V = 0.75 V = 0.25 V = 0.5 V = 0.75 V = 0.25 V = 0.5 V = 0.75

2 0.038 0.268 0.889 22.4 22.42 22.44 61.68 61.685 61.7

6 0.195 1.245 3.309 22.61 22.77 22.94 61.75 61.83 61.89

10 0.412 2.31 4.991 23.08 23.48 23.961 61.86 62.08 62.28

14 0.664 3.25 6.985 23.87 24.53 25.48 62.05 62.47 62.87

18 0.937 4.04 8.174 25.11 25.92 27.46 62.25 63 63.65

22 1.219 4.69 9.078 26.94 27.67 29.83 62.57 63.68 64.61

26 1.498 5.25 9.772 29.27 29.87 32.52 63.04 64.51 65.76

30 1.772 5.76 10.24 32.03 32.59 35.47 63.52 65.5 67.06

34 2.031 6.25 11.15 35.16 35.8 38.6 64 66.63 68.53

Figure 2 shows the dependencies of frequency on parameter <m0 for various lengths v of the supported segment.

1-st frequency

15.00 ■

0 5 10 15 20 25 30 35 40

so

—■v=025 -B-v=0_5 -±-v=0.75

2-nd frequency

40 35 a 30 25

20 -|-1-1-1-1-1-1-1-1

0 5 10 15 20 25 30 35 40

BO

—*—'v=025 ■v=0J vM].75

3-id frequency

69

63 67 66

(3 65

64 63 62

61 -\-1-1-1-1-1-1-1-1

0 5 10 15 20 25 30 35 40

(DO

—v=025 -B-v=0J —v=0.75 Figure 2. Dependency between frequency of vibrations and generalized stiffness m.

Figure 3 presents modes of natural oscillations corresponding to these frequencies for ©0 = 18 and v = 0.5.

Figure 3. The first three modes of beam's vibrations.

The following equation describes beam's forced oscillations caused by sudden partial damage of a foundation which supports a loaded beam [27]

d4

f

- + 4a4

w. +

d2 wy^

дт2

= q,

(31)

_ ql3

where q = is dimensionless intensity of an evenly distributed load;

Wdyn = Wdyn

(£, т) is deflection function for an arbitrary cross section £ (0 < £<1);

ris physical time. Let us separate variables in equation (31) using the series

w

dyn

= £ Qn (t)W (т),

(32)

where Wn = Wn (¿) is eigen function obtained by conjunction of eigen functions W1m ) and W2m (¿); for both the segments;

Qn = Qn(r) is unknown time function.

We obtain the equations that allow determining the functions Qn(r):

dQ, -

dT

+®lQn = Rn,

(33)

where

R =

. 1 qwn (t) dt

A n

n —2 1 C0

1W (t)dt

The common solution of the equation (31) takes the form [25]

wdyn =

s

n=1

R

^

D1n COsCT + D2n sln CT + ^T V ®ny

W (t).

(34)

The integration constants D\„ and Din can be determined from the initial conditions

(£0) = w„ (£),

dw

dyn

dT

= 0,

(35)

t,0

where wst(£) is static deflection of a beam entirely supported by an elastic foundation. This deflection can be determined from the equation [27] taking into account the constraints at the endpoints of the beam:

at4

+4a wst = q •

For a beam simply supported by an elastic foundation of Winkler's type and loaded with evenly distributed load q = const, the deflection in the foundation (without flexure) descends versus the depth according to the law

q

w.

(t)=

4a

4 '

From 2-nd condition (35), it follows From 1-st condition (35), we obtain

D2n = 0.

s

D + Rn-

Mn ^ —2 V cn y

w.

(t) = w„ •

Multiplying both the parts of (37) by Wn(£) and integrating by £from 0 to 1 we obtain

R

i

1 wstwn (t)dt

D1n = Bn -"T' Bn =

c

1 w2 (t)dt

(36)

(37)

(38)

(39)

Substituting (37) and (39) into the series (34) and taking into account the equality

we obtain

w

1 - cos cor = 2sin — t, 2

dyn (t, r) = s [Bn cos mHr + Cn sin2 CTJ Wn (t),

(40)

where

C =

2q

J W (ç) dç

'n ~2 1 G)

J W2 (ç) dç

Using a similar transformation, we obtain the series for bending moments

Mdyn = <yn =X| Bn cosG„r + C sin2Grjw;'(Ç).

A

n=1

(41)

Figure 4 presents the bending moment behavior in the cross section % = 0.43 at the beginning of the dynamic process in the beam after sudden foundation destruction under a half of the beam (v = 0.5) when the generalized stiffness of the system "beam-foundation" is m0 = 18 (a = 3) (Figure 4 a) and the graph of

stationary vibrations at t> 14 (Figure 4 b). The bending moment reaches its maximum value M^a" = 0.666.

a) The beginning of bending moment increasing b) stationary oscillations of bending moment

in the cross section g = 0.43 in the same cross section

Figure 4. Bending moment after sudden damage of foundation (v = 0.5).

W

0.02 0

-0.02 -0.04 -0.06 -0.08 -0.1

0.6

0.3

1.2

y = -0.043 k2 - 0.0553k + 0.0111

Figure 5. Distribution of deflection and bending moment cased by quasi-static damaging of foundation.

When quasi-static appearing of such a damage, the bending moment reaches it maximum value in the same cross section £ = 0.43 and is equal to M^ = 0.143 (Figure 5) while the external load q = 1 is unit. One can see that the effect of sudden damage exhibits as the five-time increase in the internal bending moment.

4. Conclusions

The obtained analytical solution of the problem on determination forces, modes and frequencies of natural and forced flexural vibrations of a beam supported by an elastic foundation can be applied to verification of mathematical models of static-dynamic and quasi-static deforming of complex structural systems «beam - foundation» under accidental impacts caused by sudden destruction a part of foundation. Besides, this analytical solution can be applied to problems on defense of buildings and structures against progressive destruction when, according to the scenario of a special accidental impact, an additional dynamic force due to sudden subsidence of a base in the system «strip footing - structure» plays an important role.

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25. Travush, V.I., Gordon, V.A., Kolchunov, V.I., Leontiev, E. V. The response of the system 'beam - Foundation' on sudden changes of boundary conditions. IOP Conference Series: Materials Science and Engineering. 2018. 456(1). Pp. 012130. doi: 10.1088/1757-899X/456/1/012130

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27. Travush, V.I., Gordon, V.A., Kolchunov, V.I., Leontiev, Y.V. Dynamic effects in the beam on an elastic foundation caused by the sudden transformation of supporting conditions. International Journal for Computational Civil and Structural Engineering. 2018. 14(4). Pp. 27-41. doi:10.22337/2587-9618-2018-14-4-27-47

Contacts:

Vladimir Travush, +7(495)625-79-67; travush@mail.ru

Vladimir Gordon, +7(486)2419802; gordon@ostu.ru

Vitaly Kolchunov, +7(471)2222461; asiorel@mail.ru

Evgeny Leontiev, +7(495)6259595; e.leontyev@gge.ru

© Travush, V.I., Gordon, V.A., Kolchunov, V.I., Leontiev, E.V., 2019

Инженерно-строительный журнал ISSN

r 7 2071-0305

сайт журнала: http://engstroy.spbstu.ru/

DOI: 10.18720/MCE.91.12

Динамическое деформирование балки при внезапном структурном изменении упругого основания

В.И. Травушa, В.А. ГopdoHb, В.И. Колчуновc', Е.В. Леонmьевd

a Российская академия архитектуры и строительных наук, г. Москва, ул. Россия b Орловский государственный университет имени И. С. Тургенева, г. Орел, Россия c Юго-Западный государственный университет, г. Курск, Россия d Главгосэкспертиза России, г. Москва, Россия * E-mail: asiorel@mail.ru

Ключевые слова: система «балка - основание», собственные и вынужденные колебания, формы и частоты, аварийное воздействие, структурная перестройка

Аннотация. Приведена методика аналитического определения усилий, перемещений, форм и частот собственных поперечных колебаний балки на упругом основании, состоящей из двух участков: один опирается на упругое основание Винклера, второй свободен. Уравнения поперечных собственных и вынужденных колебаний балки записаны в безразмерных координатах и решены методом начальных параметров с использованием функций Крылова. При этом вторая и высшие частоты и формы собственных колебаний балки определяются в предположении, что искомая частота больше «условной» частоты, характеризующей обобщенную жесткость системы «балка - основание». Численным анализом показаны зависимости трех первых безразмерных частот собственных колебаний балки от обобщенной жесткости системы «балка - основание» после частичного разрушения основания под балкой. Установлено, что при внезапном разрушении основания под половиной балки при некотором значении обобщенной жесткости системы «балка - основание» эффект внезапной структурной перестройки системы приводит почти к пятикратному увеличению момента.

Литература

1. Pearson C., Delatte N. Ronan Point Apartment Tower Collapse and its Effect on Building Codes // J. Perform. Constr. Facil. 2005. № 2(19). Pp. 172-177.

2. Травуш В.И., Федорова Н.В. Живучесть конструктивных систем сооружений при особых воздействиях // Инженерно-строительный журнал. 2018. № 5(81). С. 73-80. doi:10.18720/MCE.81.8

3. Fedorova N., Savin S. Structural transformation of reinforced concrete structural system at sudden loss of stability of one of its elements // IOP Conf. Ser. Mater. Sci. Eng. 2018. № 5(365). 052018.

4. Li J., Zhou H. Energy-based collapse assessment of complex reinforced concrete structures with uncertainties // Procedia Eng. 2017. № 199. Pp. 1246-1251.

5. Уткин В.С. Расчет надежности грунтовых оснований фундаментов зданий и сооружений по критерию деформации при ограниченной информации о нагрузках и грунтах // Инженерно-строительный журнал. 2016. № 1(61). С. 4-13. doi:10.5862/MCE.61.1

6. Уткин В.С., Борисова О.Л. Расчет надежности щелевого фундамента по критерию несущей способности грунта основания на стадии эксплуатации // Строительство уникальных зданий и сооружений. 2017. № 6(57). С. 7-17. doi: 10.18720/CUBS.57.1

7. Motaghian S.E., Mofid M., Alanjari P. Exact solution to free vibration of beams partially supported by an elastic foundation // Sci. Iran. 2011. № 4 A(18). Pp. 861-866.

8. Motaghian S., Mofid M., Akin J.E. On the free vibration response of rectangular plates, partially supported on elastic foundation // Appl. Math. Model. 2012. № 9(36). Pp. 4473-4482.

9. Rezaei E., Dahlberg T. Dynamic behaviour of an in situ partially supported concrete railway sleeper // Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit. 2011. № 5(225). Pp. 501-508.

10. Eisenberger M., Yankelevsky D.Z., Clastornik J. Stability of beams on elastic foundation // Comput. Struct. 1986. № 1(24). Pp. 135-139.

11. Liu Y., Shu D.W. Analytical Solution of the Vibration of Delaminated Bimaterial Beams Fully or Partially Supported by Elastic Foundation // Appl. Mech. Mater. 2013. № 394. Pp. 75-79.

12. Attar M., Karrech A., Regenauer-Lieb K. Dynamic response of cracked Timoshenko beams on elastic foundations under moving harmonic loads // JVC/Journal Vib. Control. 2017. № 3(23). Pp. 432-457.

13. Tsai M.H., Huang T.C. Progressive collapse analysis of an RC building with exterior non-structural walls // Procedia Eng. 2011. № 14. Pp. 377-384.

14. Amiri S., Saffari H., Mashhadi J. Assessment of dynamic increase factor for progressive collapse analysis of RC structures // Eng. Fail. Anal. 2018. № 84. Pp. 300-310.

15. Amiri S., Saffari H., Mashhadi J. Assessment of dynamic increase factor for progressive collapse analysis of RC structures // Eng. Fail. Anal. 2018. № 84. Pp. 300-310.

16. Weng J., Tan K.H., Lee C.K. Adaptive superelement modeling for progressive collapse analysis of reinforced concrete frames // Eng. Struct. 2017. № 151. Pp. 136-152.

17. Fedorova N. V, Savin S.Y. Ultimate State Evaluating Criteria of RC Structural Systems at Loss of Stability of Bearing Element // IOP Conf. Ser. Mater. Sci. Eng. 2018. № 463. 032072.

18. Bazant Z.P., Verdure M. Mechanics of Progressive Collapse: Learning from World Trade Center and Building Demolitions // J. Eng. Mech. 2007. № 3(133). Pp. 308-319.

19. Khandelwal K., El-Tawil S. Pushdown resistance as a measure of robustness in progressive collapse analysis // Eng. Struct. 2011. № 9(33). Pp. 2653-2661.

20. Szyniszewski S., Krauthammer T. Energy flow in progressive collapse of steel framed buildings // Eng. Struct. 2012. № 42. Pp. 142-153.

21. Botez M., Bredean L., loani A.M. Improving the accuracy of progressive collapse risk assessment: Efficiency and contribution of supplementary progressive collapse resisting mechanisms // Comput. Struct. 2016. № 174. Pp. 54-65.

22. Belostotsky A.M. Correct numerical methods of analysis of structural strength and stability of high-rise panel buildings part 2: Results of modelling // Key Eng. Mater. 2016. № 685. Pp. 221-224.

23. Travush V. I., Fedorova N.V. Survivability parameter calculation for framed structural systems // Russ. J. Build. Constr. Archit. 2017. № 1(33). Pp. 6-14.

24. Gei M., Misseroni D. Experimental investigation of progressive instability and collapse of no-tension brickwork pillars // Int. J. Solids Struct. 2018. № 155. Pp. 81-88.

25. Travush V.I., etc. The response of the system «beam - Foundation» on sudden changes of boundary conditions // IOP Conf. Ser. Mater. Sci. Eng. 2018. № 1(456). 012130.

26. Gordon V.A., Pilipenko O. V., Trifonov V.A. The reactions of the «beam - foundation» system to the sudden change of the boundary conditions // MATEC Web Conf. 2018. № 188. 03008.

27. Travush V.I., Gordon V.A., Kolchunov V.I., Leontiev Y.V. Dynamic effects in the beam on an elastic foundation caused by the sudden transformation of supporting conditions // International Journal for Computational Civil and Structural Engineering. 2018. 14 (4). Pp. 27-41. DOI: 10.22337/2587-9618-2018-14-4-27-47

Контактные данные:

Владимир Ильич Травуш, +7(495)625-79-67; travush@mail.ru Владимир Александрович Гордон, +7(486)2419802; gordon@ostu.ru Виталий Иванович Колчунов, +7(471)2222461; asiorel@mail.ru Евгений Владимирович Леонтьев, +7(495)6259595; e.leontyev@gge.ru

© Травуш В.И., Гордон В.А., Колчунов В.И., Леонтьев Е.В., 2019