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Dynamics of layered reinforced concrete beam on visco-elastic foundation with different resistances of concrete and reinforcement to tension and compression To cite this article: Y V Nemirovsky and S V Tikhonov 2018 J. Phys.: Conf. Ser. 973 012014
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AMSCM IOP Conf. Series: Journal of Physics: Conf. Series 973 (2018) 1234567890 ‘’“”012014
IOP Publishing doi:10.1088/1742-6596/973/1/012014
Dynamics of layered reinforced concrete beam on visco-elastic foundation with different resistances of concrete and reinforcement to tension and compression Y V Nemirovsky1 , S V Tikhonov2 1
Physics of Fast Processes Laboratory, Khristianovich Institute of Theoretical and Applied Mechanics Siberian Branch of the Russian Academy of Sciences, Institutskaya str., 4/1, Novosibirsk, 630090, Russia 2 Faculty of Information and Computer Systems, I. Ulianov Chuvash State University, 15, Moskovskiy pr., Cheboksary, Chuvash Republic, 428015, Russia E-mail:
[email protected] Abstract. Originally, fundamentals of the theory of limit equilibrium and dynamic deformation of building metal and reinforced concrete structures were created by A. A. Gvozdev [1] and developed by his followers [4, 5, 6, 7, 11, 12]. Forming the basis for the calculation, the model of an ideal rigid-plastic material has enabled to determine in many cases the ultimate load bearing capacity and upper (kinematically possible) or lower (statically valid) values for a wide class of different structures with quite simple methods. At the same time, applied to concrete structures the most important property of concrete to significantly differently resist tension and compression was not taken into account [10]. This circumstance was considered in [3] for reinforced concrete beams under conditions of quasistatic loading. The deformation is often accompanied by resistance of the environment in construction practice [8, 9]. In [2], the dynamics of multi-layered concrete beams on visco-elastic foundation under the loadings of explosive type is considered. In this work we consider the case which is often encountered in practical applications when the loadings weakly change in time.
1. Introduction In the scientific literature the calculation of reinforced concrete structures is often limited to the case of the simplest forms of the rod cross-section and simplest conditions of loading and fixing. Modern technological capabilities to create flexible sets of hybrid laminated reinforced structures, where various grades of concrete, reinforcing elements and reinforcement structures can be implemented in layers of cross-section, are not taken into account. In this paper, the problem of limit equilibrium and dynamic deformation of arbitrary reinforced rods based on concrete is examined by the ideas of the model of perfectly plastic deals. 2. Methods In this paper we used a classical model of ideal rigid-plastic body for all materials (concrete and reinforcement) in accordance with the diagram shown in Figure 1.
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AMSCM IOP Conf. Series: Journal of Physics: Conf. Series 973 (2018) 1234567890 ‘’“”012014
IOP Publishing doi:10.1088/1742-6596/973/1/012014
σ σ+ο
ε
ο - σ-ο
Figure 1. The diagram of concrete and reinforcement deformation, where the σ + 0 is a tensile − yield strength, the σ 0 is the yield strength in compression We assume that the structure of the reinforcement in layers 1 and 3 of the beam cross section varies according to the intensity and properties of the phase materials. For all considered sections we assume the classical kinematical hypotheses of Kirchgoff-Lyav to be valid, according to which the deformation will have the expression ε(x, z) = e0 (x) + zκ(x), e0 (x) =
du0 , dx
κ(x) = −
d2 w , dx2
where the u0 (x) is movement along the reporting axis x of the beam, w(x) is a de-flection. 3. Results Consider an ideal-plastic section of the beam, for which e0 (x) < 0, e0 + h3 κ(x) > 0. Both inequalities will be satisfied if e0 (x) < 0, κ(x) > 0. The state of stress will correspond to figure 2. Then, for the considered part of the beam the longitudinal force N is defined by the equality N = −2σ − 01 +2σ + 03
R h3 h2
R h1 0
b1 (z)dz − 2σ − 02 0
R z1 h1
0
b2 dz + 2σ + 02 0
R h2 z1
0
b2 dz+
+ − + b3 (z)dz = −2b2 σ − 02 + σ 02 z 1 + 2b2 σ 02 h2 + σ 02 h1 −
−2σ − 01
R h1 0
b1 (z)dz + 2σ + 03
The expression for the bending moment M will be
R h3 h2
b3 (z)dz.
R h2 0 0 b2 zdz + 2σ + 02 z 1 b2 zdz+ h 1 R 0 0 2 h3 − + − + 2 2+b + b (z)zdz = −b σ + σ z σ h σ h − +2σ + 3 2 1 2 1 2 03 h2 02 02 02 02 − R h1 + R h3 −2σ 01 0 b1 (z)zdz + 2σ 03 h b3 (z)zdz. 2
M = −2σ − 01
R h1 0
b1 (z)zdz − 2σ − 02
2
R z1
(1)
AMSCM IOP Conf. Series: Journal of Physics: Conf. Series 973 (2018) 1234567890 ‘’“”012014
IOP Publishing doi:10.1088/1742-6596/973/1/012014
y
O b1 (z) 1
h1
2
z1
b 02
h2 3 b3 (z)
h3
z Figure 2. The beam cross section. Numbers 1, 2, 3 refer to the layers of the beam. In the case of transverse bending (N = 0), we have 0
z1 =
+ − b2 σ − 02 h2 + σ 02 h1 − σ 01 0
R h1 0
b1 (z)dz + σ + 03
+ b2 (σ − 02 + σ 02 )
R h3 h2
b3 (z)dz
,
and for z 1 the ratio h1 ≤ z 1 ≤ h2 must be satisfied. Substituting z 1 in (1), we obtain the expression for the limit bending moment +
M0 = 0
0
+ − b2 (σ − 02 h2 +σ 02 h1 )−σ 01 0
2
2
+ +b2 σ − 02 h1 + σ 02 h2 −
R h1 0
b1 (z)dz+σ + 03
+ b2 (σ − 02 +σ 02 ) R h1 2σ − 01 0 b1 (z)zdz
R h3 h2
+ 2σ + 03
b3 (z)dz
R h3 h2
2
+
b3 (z)zdz.
For an ideal-plastic section of the beam with the deformed state
e0 (x) > 0, e0 + h3 κ(x) < 0, e0 (x) > 0, κ(x) < 0. −
Similarly, it is possible to determine the limiting bending moment M 0 − M0
0
=
2
0
− + b2 (σ + 02 h1 +σ 02 h2 )−σ 01 0
2
− −b2 σ + 02 h1 + σ 02 h2 +
R h1 0
b1 (z)dz−σ − 03
− b2 (σ + 02 +σ 02 ) + R h1 2σ 01 0 b1 (z)zdz
R h3 h2
− 2σ − 03
b3 (z)dz
R h3 h2
2
−
b3 (z)zdz.
Thus, in the general case the beam is divided into sections corresponding to the deformed states e0 < 0, κ < 0 or e0 > 0, κ > 0 and e0 = κ = 0. The last correspond to the rigid undeformed state. The number of rigid sections and the plastic hinges separating them will depend on the conditions of fastening of the beams, the type of distributed and concentrated loads and the law of resistance of the supporting environment. Consider a reinforced concrete beam of length l, lying on visco-elastic foundation and cantilever-fixed on the left edge of x = 0.
3
AMSCM IOP Conf. Series: Journal of Physics: Conf. Series 973 (2018) 1234567890 ‘’“”012014
IOP Publishing doi:10.1088/1742-6596/973/1/012014
Q(t) q(x,t) M2(t)
x
l
z Figure 3. A multilayer beam cantilever-fixed on the left edge. The beam is loaded by a distributed transverse load q(x, t), its own weight q s = mg, concentrated load Q(t) and moment M 2 (t) at the end of x = l. Here, m is the mass of length unit of the beam. For a given beam, the mass will be determined by m=
R h1 0
0
(ρa1 µ1 + ρc1 (1 − µ1 )) b1 (z)dz + 2b2 ρc2 (h2 − h1 )+ +
R h3 h2
(ρa3 µ3 + ρc3 (1 − µ3 )) b3 (z)dz
where ρa1 , ρa3 , ρc1 , ρc2 , ρc3 are the density of reinforcement materials and binding concrete, µ1 , µ3 are the coefficients of reinforcement. Represent the current external loads in the form of q(x, t) = q 1 ϕ(x)ψ 1 (t),
Q(t) = Q1 ψ 2 (t),
M 2 (t) = M 12 ψ 3 (t).
A model of an ideal resisting rigid-plastic material will be used for all phase materials of the reinforced concrete beam. Then we have
± σ± 01 = µ1 σ 01
σ± 03
= µ3
a
a σ± 03
+ (1 − µ1 ) σ ± 01 + (1 − µ3 )
c
,
c . σ± 03
For beams of constant cross section, loaded with an uniformly distributed load (ϕ(x) = 1), − if M2 (t) ≥ −M 0 , the movement will be provided, as a rigid rod, with the formation of plastic hinge in the left edge x = 0. Then for the deflection w we will have the expression x w(x, t) = w 0 (t) , l
0≤x≤l
and taking into account that according to the conditions of dynamic equilibrium the main point regarding the support x = 0 of all active forces, including inertia forces must be equal to zero, we obtain the equation + M0
= −M 2 (t) + lQ(t) +
Z lh 0
q(x, t) + qs − q r − mx¨θ xdx, i
(2)
w0 (t) ˙ where θ = ∂w , q r = b0 k 1 w + b0 k2 w, k1 , k 2 are the coefficients of the elastic and ∂x = l viscous resistance of the base, and then we assume that k1 > 0, k 2 > 0, b0 = b3 (h3 ) are the length of the base of the beam cross section. A point denotes a partial derivative in time t.
4
AMSCM IOP Conf. Series: Journal of Physics: Conf. Series 973 (2018) 1234567890 ‘’“”012014
IOP Publishing doi:10.1088/1742-6596/973/1/012014
q q1
q0 t
o
t0 t1
Figure 4. The law of change q(x, t) From (2), we get the equation to determine the deflection ¨ 0 (t) + b0 k 2 w˙ 0 (t) + b0 k1 w0 (t) = γ(t), w m m
(3)
where 3 3g 3 + q 1 ψ1 (t) + (−M 0 + lQ1 ψ2 (t) − M 12 ψ3 (t)). (4) + 2 2m 2 ml ¨ 0 (t), we obtain the equation of quasi-static change of Discarding in (3) the inertial member w the beam overlimited deflection γ(t) =
b0 k 2 ˙ b0 k 1 w 0 (t) + w0 (t) = γ(t), (5) m m where γ(t) is defined by equation (4). Taking the value w0 (t) = 0, we obtain the equation for the first limiting amplitudes of the considered beam 3g 3 3 + q 1 ψ1 (t) + (6) + 2 (−M 0 + lQ1 ψ2 (t) − M 12 ψ3 (t)) = 0. 2m 2 ml From relation (6) we determine the point in time τ 0 when the loads reach the limit value and the plastic hinge occurs at the pinch point of the beam. If at t ≥ τ 0 the loads monotonically quasistatically temporarily increase, then integrating the equation (5) under the initial conditions w0 (τ 0 ) = 0.
(7)
for the deflection we get the expression m − kk1 t w0 (t) = e 2 b0 k 2
Zt
k1 z
γ(z)e k2 dz.
τ0
If we accept that the law of pressure variation on the beam q(x, t) = q 1 ψ 1 (t) changes according to the law as shown in figure 4
5
AMSCM IOP Conf. Series: Journal of Physics: Conf. Series 973 (2018) 1234567890 ‘’“”012014
(
ψ1 (t) =
t if τ 0 τ1 , −(t−τ 1 )
≤ t ≤ τ 1,
(8)
, if τ 1 ≤ t ≤ ∞,
,
e
IOP Publishing doi:10.1088/1742-6596/973/1/012014
and the acting external moment and concentrated force at x = l are absent M 2 (t) = 0,
Q(t) = 0.
Then before reaching the first load limit q 0 at the time 0 ≤ t < τ 0 the beam will be rigid, and for the time moment of formation of the plastic hinge and the value of the first limit load it is true that 2m τ0 = 3q 1
+
3M 0 ml
2
3g − 2
!
2m q0 = 3
τ 1,
+
3M 0 ml
2
3g − 2
!
(9)
.
From relations (7) and assumptions 0 < τ 0 < τ 1 the following inequalities have the form of +
0<
3M 0 ml
2
−
3g < 1. 2
In case of load increase according to the law (8) after reaching the first limit load, for the time interval τ 0 ≤ t ≤ τ 1 the relations (4), (5) take the form of f1 w˙ 0 (t) + f2 w 0 (t) = f4 t + f3
(10)
where f1 =
b0 k 2 , m
f2 =
b0 k 1 , m
f3 =
3g 3 + − M0 , 2 2 ml
f4 =
3 q1 . 2m τ 1
Integrating equation (10) under the initial conditions (7) we obtain an expression for deflection for the time interval τ 0 ≤ t ≤ τ 1 w(t) = g1 eg2 t + g3 t + g4 ,
(11)
where
g1 = −
f2 f3 τ 0 + f4 f3 f1 τ + 2 e f1 0 , f2 f2
g2 = −
f2 , f1
g3 =
f3 , f2
g4 =
f4 f3 f1 − 2 . f2 f2
The deflection rate for the indicated interval will be equal to ˙ w(t) = g1 g2 eg2 t + g3 . We denote the deflection by w 0τ at time t = τ 1 w(τ 1 ) = g1 eg2 τ 1 + g3 τ 1 + g4 = w0τ .
(12)
For time interval τ 1 ≤ t ≤ ∞ the relations (4), (5) take the form ˙ f1 w(t) + f2 w 0 (t) = j4 e−t + j3 , where
6
(13)
AMSCM IOP Conf. Series: Journal of Physics: Conf. Series 973 (2018) 1234567890 ‘’“”012014
j3 =
3g 3 + M0 , − 2 2 ml
IOP Publishing doi:10.1088/1742-6596/973/1/012014
3q 1 τ 1 e . 2m
j4 =
The solution to (13) with conditions (12) has the form w(t) = v1 ev2 t + v3 e−t + v4 ,
(14)
where
v1 = w 0τ −
f2 j3 j4 τ + e−τ1 e f1 1 , f2 f1 − f2
v2 = −
f2 , f1
v3 = −
j4 , f1 − f2
v4 =
j3 . f2
If 0 ≤ t < τ 0 , the beam remains rigid, then the equation to determine the moment has the form ∂2M = q(x, t) + q s , ∂x2 integrating which with the initial conditions
(15)
∂M (l, t) = 0, ∂x we obtain an expression for the moment that is valid when 0 ≤ t < τ 0 M (l, t) = 0,
(16)
2
q t + mgτ 1 2 q 1 tl + mglτ 1 3(q 1 t + mgτ 1 )l . M (x) = 1 x − x− 2τ 1 τ1 2τ 1 The equation of beam deflection in the case of plastic hinge in the place of pinch of a beam has the form ∂2M k1 b0 xw0 (t) k 2 b0 xw˙ 0 (t) = − , q(x, t) + q − s ∂x2 l l adding to which the expression for deflection (10), we obtain ∂2M g2 t = + g x g e t + g 6 5 7 + g8 t + g9 , ∂x2
where g5 = −
(17)
k 1 b0 k2 b0 g1 − g1 g2 , l l
g6 = −
k 1 b0 g3 , l
g7 = −
(18)
k 1 b0 k 2 b0 g4 − g3 , l l
g8 =
q1 , τ1
g9 = mg.
Integrating (18) under the initial conditions (16), we obtain the expression for the moment that is valid for τ 0 ≤ t < τ 1 M = x3
g5 g2 t 6 e
+
g6 6 t
+
g7 6
+ x2 3
g8 2t
+ g53l eg2 t +
+ g6 l 3
g9 2
3
+
−x
g8 2 2 l
g5 l 2
2
eg2 t +
t+
g7 l 3
3
+
g6 l 2
2
+ g8 l t +
g9 2 2l .
g7 l 2
2
+ g9 l +
Substituting in (17) the expression for the deflection (13), we obtain ∂2M v2 t −t −t x v e + v e + v + v9 , = 5 6 7 + v8 e ∂x2 7
(19)
AMSCM IOP Conf. Series: Journal of Physics: Conf. Series 973 (2018) 1234567890 ‘’“”012014
IOP Publishing doi:10.1088/1742-6596/973/1/012014
where v5 = −
k 1 b0 k 2 b0 v1 − v1 v2 , l l
v6 = −
k1 b0 k 2 b0 v3 + v3 , l l
v7 = −
k 1 b0 v4 , l
v8 = q1 eτ1 , v9 = mg.
Integrating (19) under the initial conditions (16), we obtain the expression for the moment that is valid for τ 1 ≤ t < ∞ M = x3 −x
v5 l 2
2
ev2 t +
v6 l 2
2
v5 v2 t 6 e
+ v8 l e−t +
+
v7 l 2
2
v6 −t 6 e
+
v7 6
+ v9 l +
+ x2
v5 l 3
3
v8 −t e 2
ev2 t +
+
v6 l 3
v9 2
3
+
−
v8 2 2 l
e−t +
v7 l 3
3
+
v9 2 2 l .
4. Conclusions Thus, the obtained expressions are determined for the deflections of a beam and the expression of the moments. This solution is true for the entire length of the beam 0 ≤ x ≤ l if the bending − + moment M (x, t) is within −M 0 ≤ M (x, t) ≤ M 0 . This work is carried out with the partial financial support of RFBR grants (projects 17-41210272, 15-01-00825, 16-31-00511). References [1] Gvozdev A To calculation of designs on action of a blast wave 1940 Building industry. 1 18–21 [2] Nemirovski Y and Tikhonov S Dynamics of reinforced concrete beams on the visco-elastic foundation 2017 Vestnik I. Yakovlev Chuvach State Pedagogical University. Series: Mechanics of a limit state. 2 45–64 [3] Nemirovski Y and Tikhonov S Limit conditions of reinforced concrete beams 2016 Vestnik I. Yakovlev Chuvach State Pedagogical University. Series: Mechanics of a limit state. 3 134–158 [4] Erkhov M 1978 Theory of Perfectly Plastic Solids and Structures (Moscow: Nauka) p 352 [5] Shamin V 1989 Calculation of elements of constructions on action of explosive loadings (Moscow: Stroyizdat) p 72 [6] Komarov K and Nemirovski Y 1984 Dynamics of rigid and plastic elements of designs (Novosibirsk: Nauka) p 236 [7] Rzhanicin 1954 Calculation of constructions taking into account plastic properties of materials (Moscow: Stroyizdat) p 288 [8] Savelyev N Calculation of beams of variable section on the elastic base 1956 Calculations on durability. 2 [9] Gorbunov-Posadov M 1949 Beams and plates on the elastic foundation (Moscow: Mashstroyizdat) [10] Nemirovski Y and Boltayev A Nonliner deformation of concrete elements at a longitudinally cross bend 2015 News of higher education institution. Construction. 6 125–129 [11] Cox A and Morland W Dynamic plastic deformation simple supported square plates 1959 J. Mech. Solids. 7 229–241 [12] Johns N, Uran T and Tekin S The dynamic plastic behavior of fully clamped rectangular plates 1970 Int. J. Solid Struct. 12 1499–1512
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