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7. Ананьина Е.В., Коцарь Ю.Н., Мордовии Б.М. Машины брошюровочно-переплетного производства. Часть1 и 2. М.: Книга, 1974.
DIRECT DEFINITION OF CROSS-SECTIONAL FORCE IN SLOPING SECTIONSOF REINFORCED CONCRETE STRUCTURES Morozov A.N. Email: [email protected]
Morozov Aleksei Nikolaevich — Doctor of engineering, independent Researcher, TALLINN, ESTONIA
Abstract: in order to calculate the shear force in oblique sections it is 7drawn the formula Q = bz T (1), the direct solution to which is still in the process of search, and various approximate methods are used in practice. Thus, an empiric method is used for calculation in Russia, while beam models (options of the truss similarity) are applied in Europe and the USA. In [1], [2] it is shown that the scheme of calculation on the basis of the compression stress diagram in the normal cross-section crossing the top of the inclined crack reflects the value of the shear force well. However, derivation of the formula (1) and estimation of the strength of concrete criterion under the state ofplane stress [3] is performed based on the triangular compression normal stress diagram. Actually, the above diagram possesses a notch, which is oriented at the top of the inclined crack [1], [3] that exactly determined the choice of the scheme of calculation according to fig.1.
Keywords: shear force, obligue section, normal gross-section grossing the top of the inclined crack stress, tangent lines.
ПРЯМОЕ ОПРЕДЕЛЕНИЕ ПОПЕРЕЧНОЙ СИЛЫ В НАКЛОННЫХ СЕЧЕНИЯХ ЖЕЛЕЗОБЕТОННЫХ КОНСТРУКЦИЙ Морозов А.Н.
Морозов Алексей Николаевич — кандидат технических наук, cамостоятелъный исследователь,
г. Таллин, Эстония
Аннотация: для расчета поперечной силы в наклонных сечениях выведена формула Q=bzx (1), прямое решение которой находится до сих пор в поиске и на практике применяются различные приближенные методы. Так в России используется эмпирический метод расчета, а в Европе и Америке применяются стержневые модели (варианты ферменной аналогии). В [1], [2] было показано, что расчетная схема, основанная на эпюре напряжений сжатия в нормальном сечении, проходящем через вершину наклонной трещины, хорошо отражает величину поперечной силы. Однако вывод формулы (1) и оценка критерия прочности бетона при сложном напряженном состоянии [3] производится по треугольной эпюре нормальных напряжений. В реальности в этой эпюре имеется вырез, ориентированный на вершину наклонной трещины [1], [3], что и определило выбор расчетной схемы согласно fig. 1. Ключевые слова: поперечная сила, наклонное сечение, нормальное сечение, проходящеe через вершину наклонной трещины, касательные напряжения.
UDC 62407221012
1. Introduction
Failure of the reinforced concrete elements might occur along the inclined cracks as the result of a simultaneous effect of bending moments and shear forces that was the reason for performance of a large-scale research of this kind of failure of the reinforced concrete structures. The beginning of the research concerned can be marked up with the research papers by R. Saliger [4], who suggested considering the beams with inclined cracks as arcs with the tie-beams; E. Morsch [5], who was one of the first to draw the formula (1); and A. Talbot [6], who suggested the scheme of the cross-tie system. In the USSR the problem concerned was investigated by A. A. Gvozdev [7] and M, S. Borishansky [8], who suggested the formula of the shear force taken by the concrete; and A. S. Zalesov [9], who considered the normal cross-section crossing the top of the inclined crack that formed the basis for develop-ment of the given scheme of calculation. The research papers by A. S. Silantjev [10], Yu. V. Krasnoshchekov [11], V. I. Zharnitsky [12], and I. N. Starishko [13], related to various issues of investigation of the strength of oblique sections should have been noted in Russia. However, the direct determination (1) is not found in the references, and only its particular values are provided. Thus in the regulations, which are currently in effect, Qbmax = 2.5i Rbtbh0 at r0 max = 2 . 5 Rm. The scheme of
calculation for the direct determination of the shear force representing the problem to be solved in this research paper is based on three concepts:
1. According to [1][2] the shear force is well determined upon the normal section crossing the top of the inclined crack.
2. At failures of the oblique sections resulting from compression of the concrete (as it is typical for most cases of failures) the height of the compressed zone is determined within the design section diagram based on the joint solution to the equations of equilibrium for the moments in normal and oblique sections (the moments of axial and shear forces) bx 0a>Rbz = bzmza (2) [ 1] [2] [ 14] [18]
3. The stress tangent lines in (2) are determined on the basis of the diagram for the normal stresses with the notch (see fig. 1) according to (3).
+l 2 - IY1 " V
a A x,
t =
Q
bz
1
2m
1 x2 1--2
v
x
0
0 y
Q
bz
m2(3)
Derivation of the formula (3) is provided below.
Fig. 1. Design section diagram
2. Methods
The tests were performed for the autoclaved aerated concrete with y = 600 — 700kg/ 3 and
m
Rb = 2.2 — 4.&MPa.
There were tested the beams having the cross-section of 15x23 cm; L= 130 - 190 cm and the test beams were loaded according to Figure 2
Fig. 2. Diagram showing application of loads to the test beams
a=40-60 cm; a = 20 - 3 1, fJ°% = 0.18 — 0.98.Deformations of the concrete were measured
ho
by the strain gauges installed in the quantity of 100...200 pieces per beam. The strain gauges with l = 5 mm were used in the gauge rosettes, while in the rest of the cases the gauges with l = 20 mm were used. Digital bridges with further computer processing of the measurements were used for measurement recording. Having no large-size filler the autoclaved aerated concrete allows applying the strain gauges with a small base and its high elasticity (y > 0,9) provides for a more accurate estimation of the stresses. The beams made of heavy concrete with Rb = 18.5 — 29MPa were tested
in the applicable references [10], [16].
The first concept was tested on the basis of the experimental data of [2], [14] upon (1) with
, YS , where V S is the sum of static moments of the deformation bands in relation to the
z=h0— YF
fiber, while Y F is the area of the deformations of compression (the elasticity in the autoclaved
aerated concrete is y > 0.9). Finally, the average T = 0.97Rbtand Q = L02,( = 0.096
Qfact
[1][2]. Based on the data of [15] in [2] it was determined the strength criterion for the autoclaved
1 — 0 95 (mc
r (4) that for the neutral axis at
aerated concrete in the state of plane stress
Rb,
6.35 Rbt
R
Tmax = amt = ~amc Provides for T
= 0.185Rb (triangular compression diagram). For the scheme of calculation (3) at X2 = 0 the expression in square brackets is m0 = 4(a 1'
2m
.(5). At the
average value of completeness of the compression diagram T = 0.37 m0 = 0,65 and the average
R
-bt- = 0.116 Tn Rb
T = 0.97 R
= 1.59Rbt and T0 = m0Tmax = 1.03R&t that matches with the experimental
bt ■
The second concept on the basis of (2) has two options - 1. based on the rectangular compression diagram accepted by the regulations
<^R = ¿;0m [1][2] and different values of z and 2. when the equations of equilibrium for the
moments of the axial and shear forces form a match (2) and equal values of z [1][3]. The stress tangent lines in the scheme of calculation are determined based on (3) and drawn from the condition of equilibrium of the difference between normal stresses acting in two parallel cross-sections with the distance between them, which is equal to dl . While considering the above conditions we can put down for the point with the ordinate X^ that
0
Tbdl = J bdx2 dab
From fig. 1 it follows that ®xqRb — ^b + Rb X, + , therefore a = R
2
2
X2-1 + 2®
(6)
at Xi — Xq X2
At r = M taking into consideration (6) ah =
M
®x„bz
®x0bz
\
■-1+2®
v xq
and at
dM ~df
da, —
Qdl
f
®XQbz
X2-1 + 2®
therefore bdl — ^ P I ^ -1 +
r/TX Z I X
\
-1 + 2®
dX2 Finally, we have
t —
Ql
bz
2®
A-2
V xo
+
2--
®
1 - fl
V xq J
= Q
— m2 (3). bz 2
As it is shown above, at x=0 m, turns into m0 upon (5) and at 0 = 0.5 - into the basic value
Q
1
a X
of t = — . From (6) it follows that ®=-(1 + —b---2-)(7). — ^ = — — ^(8)
X m t a
bz
2
Rb xq
hQ ® Rb hQ
3. Results and Discussion
Based on the data from [17] table 1 shows comparison of the measured and calculated upon (7) values of the completeness coefficients of the compression diagram.
Table 1. The comparision of the measured and calculated upon (7) values of the completeness coefficients of the
compression diagram
\xq
X
2
v x0
1
Beam No. 1 2 3 4 5 6 7 8
x2/ / xq 0.64 0.56 0.68 0.44 0.16 0.72 0.70 0.55
ab/ /Rb 0.31 0.23 0.42 0.19 0.17 0.30 0.46 0.25
® fact 0.32 0.33 0.32 0.38 0.54 0.29 0.28 0.35
® (7) 0.33 0.33 0.37 0.38 0.51 0.29 0.38 0.35
®/ / ® fact 1.03 1.00 1.15 1.00 0.94 1.00 1.35 1.00
It is obtained the average value of = 1.06 , ( = 0.13 and it can be considered that the
/ ®fact
basic parameter of ® for the method of direct determination of the shear force based on the given scheme of calculation, corresponds to the real compression diagram in the design section.
Table 2 represents the results of calculations of m2 and mQ depending upon the values of
A = a and B — ^ Rb xq
\A Values of m2 Values of m0
b\ 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9
0.1 0.99 1.12 1.22 1.29 1.35 1.00 1.17 1.28 1.37 1.44
0.3 0.79 0.91 0.99 1.05 1.09 0.75 1.00 1.17 1.28 1.37
0.5 0.58 0.69 0.75 0.79 0.82 0.33 0.75 1.00 1.17 1.28
0.7 0.38 0.45 0.49 0.51 0.53 - 0.33 0.75 1.00 1.17
0.9 0.15 0.17 0.18 0.185 0.19 - - 0.33 0.75 1.00
It is evident from table 2 that at the same values of A and B in some cases m2 > m0 that indicates at the position of the maximum stress tangent lines in the top of the inclined crack related to the cut off of the concrete (x = X2 ). Thus, at A=0.1 and B=0.5 m2 = 0.58, and m0 = 0.33 .
At the equality of the values of m the boundary condition is B=2A; and at the excess of 2A m2 > m0, while at CO < 0.25 one can apply the generalized formula m = 2.94(0- 0.15)
(9). At z = (l)h0 (10) P = 1 + 6C when CO < 0.5 and P = 1 + 2C at CO > 0.5
12 6
As far as the elements of heavy concrete are concerned then according to the formulas (141) (142)
t
1 + -
of SNiP 2.03.01 - 84 for the neutral axis where T = a t = —a T
mt mc -
R,.
Rb at a=0.2+0.01B but R,.
not less than 0.5, and B is the class of compressive strength of the concrete. Therefore,
Rbt R,
(11) (triangular compression diagram). For the scheme of calculation t0 = m0tand T2 = m2T
with calculation of z according to the formulas provided above.
In the practical sense, the basic parameter in this scheme of calculation is represented by assigning the value of completeness coefficient of the compression diagram C . Preliminarily, it can be accepted
that 0 = 0.33,m0 — 0.5, P = 0.25. Pursuant to table 2 it can be seen that in most of the cases
m0 > m2, but there occur also the cases when tmax correspond to the top of the inclined crack
related to the cut off of the concrete. In [10][16] A. S. Silantiev provided the experimental data with respect of the failure of oblique sections depending upon the cut off of the concrete with fixation of the
shift deformations equal to their limit values. The same papers provide also the values of x0 in the
design section so that only one unknown value of C remains in the formula Qb = bzmT while using (9) (10)(11). The results of calculations under the formulas concerned are provided in table 3.
t =
Beam No. 1 4 5 6 7 8 9 10
Qfac kN 23.11 21.5 48.4 50.69 31.44 42.66 74.85 85.00
Rb MPa 30.5 18.5 29.0 29.0 22.0 22.0 20.5 20.5
Rbt Mpa 2.22 1.6 2.19 2.19 1.82 1.82 1.72 1.72
& 0.147 0.26 0.27 0.21 0.124 0.091 0.42 0.355
T MPa 5.17 3.86 5.26 5.26 4.37 4.37 3.94 3.94
( 0.245 0.278 0.331 0.342 0.276 0.311 0.600 0.536
m 0.280 0.376 0.532 0.564 0.371 0.474 1.321 1.135
ß 0.206 0.222 0.249 0.250 0.221 0.239 0.366 0.345
QbkN 22.92 21.89 47.93 53.95 33.19 45.49 83.18 82.37
Qb Qfac 0.991 1.018 0.990 1.065 1.050 1.075 1.110 0.969
It is evident from table 3 that based on the value of the experimental height of the compression zone, the values of the shear force calculated on the basis of the given techniques form a good match
with their experimental values - on average = 1033 c = 0.049. Therefore, the considered
Qfac
scheme of calculation is also applicable at failure of the oblique sections related to the cut off of the concrete. However, the equation (2) is not applicable to determining the height of the compression zone, because the shear force is determined by stresses of the cut off of the concrete but not its compression. Considering an insufficient quantity of the experimental data in the first approximation,
= 0.49c — 0.044(12). Setting the value of C , one can determine the height of the
compression zone. It was shown in [1] that the strength of the oblique sections depending upon the cut off of the concrete upon [10][16] could be also calculated on the basis of (2) accepting that ю = 0.33. 4. Conclusions
1 The scheme of calculation provided for the option of the direct determination of the shear force in the oblique sections (1) really reflects the deformation stress state of this cross-section.
2. At failure of oblique sections depending upon the compression of the concrete, the height of the compressed zone in the design section is determined by a common solution to the equation of equilibrium for the axial and shear forces (normal and oblique sections) upon (2); whereas the said height is dependent upon the value of relative span of the cut off — = —
h0 Qh0
3. Calculated values of the stress tangent lines are determined on the basis of the diagram for the normal stresses with a notch (3).
4. The calculated height of the compressed zone at failure of the oblique sections due to cut off of the concrete is preliminarily recommended to be determined on the basis of (12) at C = 0.33 .
5. The basic parameter for calculation under this scheme of calculation is represented by assigning the value of the coefficient C
References / Список литературы
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Morozov A.N. Raschet prochnosti gazobetonnykh kon-struktsii na deistvie poperechnykh sil [Calculation of strength of the autoclaved aerated concrete structures against shear force effects] // Concrete and reinforced concrete, 1991. № 5. P. 13-14.
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