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Russian Journal of Building Construction and Architecture

Based on the equilibrium equation of the moments of internal and external forces acting in section I––I in relation to the axis perpendicular to this section and passing through the point of application of the resultant forces in the compressed zone (Tb,I = 0), we get:

Here Т is the shear stress of torsion in the compressed concrete determined by means of projecting the diagram σі – εі onto the plane – and onto the plane I––I and distributed proportionally to the ratio Q : T.

Based on this equation we determine the unknown T (for the first group of limit states) and if T is specified by the Rsup:T ratio and considering the second group of limit states, this equation is used to identify T .

Meanwhile the following condition is tested:

Here T ,pl is the tangential torsional stress in compressed concrete (taking into account the

Q : T ratio) which corresponds to the maximum in the diagram σі – εі. If the condition (4) is not met, τT is assumed to equal τT,pl and the parameter ypl is determined based on the transformed equation (3) (Fig. 1):

Here 0 are prestresses in the stressed reinforcement at the moment of decreasing the prestress in concrete to zero by external forces acting on the structure considering prestress losses in the stressed reinforcement that correspond to the investigated construction stage.

If the condition (7) is not met, s,i is assumed to equal Rs.

The second supporting block. It is separated from the reinforced concrete element by a spatial section formed by a spiral-shaped crack and a vertical section passing through the compressed zone of concrete through the end of the front of the spatial crack.

The equilibrium of this block is ensured if the following conditions are met.

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The sum of the moments of all the internal and external forces acting in a vertical longitudinal plane in relation to the z axis passing through the point of application of the resultant forces in the compressed zone is zero (∑Mb = 0, block II):

Here аm,b is the horizontal distance from the support to the center of gravity of the compressed zone of concrete in section k.

It should be noted that in this equation the moments qsw,T c2 occurring on the lateral faces as

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a result of the longitudinal forces in the transverse reinforcement are mutually balanced in relation to the point B. The same should be noted about the moments caused by the “arched” components in the longitudinal reinforcement [11]. Based on the equation (8), the unknown σs is determined.

Here, it is also necessary to point out the parameter σb (the stresses in compressed concrete of section k, Fig. 1). This parameter is determined using the equation of equilibrium of the moments of the internal and external forces in the spatial section in relation to the z axis in relation to the point ОК of application of the resultant forces (section k, Fig. 1) in compressed concrete (∑MО,К =0), we get:

Here аm,S am is the horizontal distance from the support to the center ofgravity of the entire longitudinalreinforcementinthesectionk.Usingtheequation(9),theunknownσb isdetermined. The sum of the projections of all the forces acting in the spatial section on the axis x is zero (∑X = 0, block II):

Here c is the parameter equal to the numerical coefficient to the parameter ; q2sw is the linear “in-line” force in the clamps [11] occurring on the lateral faces of the reinforced concrete element (the condition is not shown in Fig. 1). Using the equation (10), the unknown xb is determined. The sum of the projections of all the forces acting in the spatial section on the axis y is zero ((∑Y = 0, block II):

Here Q is the tangential stress in the compressed concrete determined by means of projecting the diagram σі – εі onto the plane – (considering the distribution in proportion to the Q : T

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Russian Journal of Building Construction and Architecture

ratio) and projecting the component stresses of the plane k onto the plane perpendicular to the longitudinal axis of the reinforced concrete element; qsw,Q is the linear force in the clamps occurring on the lateral faces of the reinforced concrete element from the transverse force Q (Fig. 1); Qs are “indented” forces in the longitudinal reinforcement [11] (not shown in Fig. 1). The unknown qsw,Q is determined using the equation (11).

The sum of the moments of the internal and external forces in the vertical transverse plane relative to the axis x passing through the point of application of the resultant forces in the compressed zone is zero (∑Tb=0, block II):

qsw, c2 b2 (h0 0.5xb ) 2qsw,T b /2 (h0 xb )2 c2 2 T b /2 xb T 0. (12) Here T is the shear stress caused by torsion in the compressed concrete determined by means of projecting the diagram σі – εі onto the plane – (considering the distribution proportional to the Q: T ratio) and projecting the component stresses of the plane k onto the plane perpendicular to the longitudinal axis of the reinforced concrete element; is the filling factor of the plot of shear torsional stresses in compressed concrete; qsw,Т is the linear force in the clamps occurring on the lateral faces of the reinforced concrete element from the torque T (Fig. 1, a); qsw, is the linear force in the clamps occurring on the lower edge of the reinforced concrete element from the torque T (Fig. 1). The unknown qsw,T is determined based on the equation (12).

The sum of the projections of all the forces acting in the spatial section on the axis z is zero (∑Z = 0, block II):

onto the plane parallel to the longitudinal axis of the reinforced concrete element and equal to the parameter to the numerical coefficient .

Using the (13) the unknown qsw,σ is determined. Therefore the considered method for calculating the resistance of reinforced concrete structures under the combined action of bending moment, torque and shear force for the second stage of the stress-strain (case 1) can be employed when spatial cracks of the first type occur on the lower face of the structure.

The second scheme is implemented when the resistance of reinforced concrete elements subjected to the combined effects of torques and transverse forces.

The calculating diagram of the resistance of a reinforced concrete structure under the combined action of bending moment, torque, and shear force (case 2) is shown in Fig. 2.

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In order to design the calculation equations, two blocks will be separated from the reinforced concrete element using the section method (Fig. 2).

The first block is separated by a cross section I––I passing at the end of a spatial crack. This block is in equilibrium under the action of external forces applied to the block from the support side and internal forces occurring in the section.

Based on the equation of equilibrium of the moments of internal and external forces in this section I––I in relation to the point passing through the point of application of the resultant forces in the tensile reinforcement (∑MO, I = 0), we get:

Here am ,S is the horizontal distance from the support in the direction of the axis y to the center of gravity of the operating longitudinal reinforcement in the section I––I (pointО1 ). In this case, it is necessary to emphasize that the moment created by Rsup am,S will be twisting in relation to the axis x and to t; moment M will be bending with respect to the y axis and relative to the pointO1; the moment created by b,I Ab h0 y,2 xb ,x x will be bending in relation to the axis z and to the pointO1. Here y,2 xb,x is the static and geometric parameter

considering the location of the center of gravity of the compressed concrete zone in section I––I (in the хb section the diagram of compressive stresses is rectangular in the section х – хb it is triangular); Rsup is the support reaction in the first block (Fig. 2), for the second group of limit states this parameter is known.

Based on this equation, the unknown b,I is identified.

Based on the equilibrium equation of the projections of all the forces acting in the section I––I on the axis x, we determine the height of the compressed zone of concrete x in this section. The equation takes the following form

Here пр і, і is the parameter that takes into account the projection of the diagram σі – εі on the direction perpendicular to the plane k (Fig. 2); is the parameter that considers the projection of the component stresses in the plane k onto the plane I––I perpendicular to the longitudinal axis of the reinforced concrete element; y,2 xb ,x is the parameter equal to the numerical coefficient to the parameter y xb ,x .

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Russian Journal of Building Construction and Architecture

Fig. 2. The calculating diagram of the resistance of the reinforced concrete structure under the combined action of bending moment, torque and shear force (case 2):

is the compressed zone of the spatial section; is the compressed zone of the section I––I

Based on the equilibrium equation of the moments of the internal and external forces acting in the section I––I in relation to the axis perpendicular to this section and passing through the point of application of the resultant forces in the compressed zone (Tb,I = 0), we get:

Based on this equation, we determine the unknown T (for the first group of limit states) and if T is specified by the Rsup:T ratio and when considering the second group of limit states, this

If the condition (17) is not met, τT is assumed to equal τT,pl and based on the transformed equation (16) the parameter ypl is identified (see Equation (18) and Fig. 2.

2 [ T,pl ypl b /2 0.5ypl 0,5 T ,pl (b /2 ypl )2/3 b /2 ypl ] x T 0. (18) It should be noted that for the scheme II of the diagram τT is normally similar to the rectangular. Based on the hypothesis of proportionality of longitudinal strains (the equation similar to (19)s,I is identified.

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The second supporting block is separated from the reinforced concrete element by a spatial section formed by a spiral crack and a vertical section passing through the compressed zone of concrete through the end of the front of the spatial crack.

The equilibrium of this block is ensured if the following conditions are met.

The sum of the moments of all the internal and external forces acting in the vertical longitudinal plane in relation to the axis z to the point of application of the resultant forces in the compressed zone is zero (∑Mb = 0, block II).

Here am ,b is the horizontal distance from the support in the direction of the axis y to the center of gravity of the compressed zone of concrete in the section k (the points b). At the same time, it is necessary to point out that the moment created byRsup am,b will be twisting in relation to the axis x and to т. b; the moment M will be bending in relation to the axis y and to the point b; the moment created by smAs (h0 0.5xb )will be bending in relation to the axis z and to the point b.

from the longitudinal forces in the transverse reinforcement are mutually balanced in relation to the point B.

The same should be noted about the moments caused by the “arched” components in the longitudinal reinforcement.

Using the equation (21) the unknown σs is identified.

Here it is also necessary to point out the parameter σb (the stresses in compressed concrete of the section k, Fig. 2). This parameter is determined using the equation of equilibrium of the moments of the internal and external forces in the spatial section in relation to the axis z in relation to the point ОК of application of the resultant forces (section k, Fig. 2) in compressed concrete (∑MО,К = 0), we get:

Here аm,S is the horizontal distance from the support to the center of gravity of the entire longitudinal reinforcement in the section k. Using the equation (22), the unknown b is determined.

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Russian Journal of Building Construction and Architecture

The sum of the projections of all the forces acting in the spatial section on the axis x is zero (∑X = 0, block II).

Using the equation (23), the unknown xb is identified.

The sum of the moments of the internal and external forces in the vertical transverse plane in relation to the axis x passing through the point of application of the resultant forces in the compressed zone is zero (∑Tb = 0, block II):

Using the equation (24) the unknown qsw,T is identified.

The sum of the projections of all the forces acting in the spatial section in the axis z is zero (∑Z = 0, block II):

Using the equation (25) the unknown qsw,Q is identified.

The unknown qsw, is identified based on the following considerations. This linear force occurs on the lateral face as a result of Т + Rsup eQ as well as the linear force qsw,Т occurring on the upper and lower faces. Thus its difference from the latter will be that the b2 : h2 ratio and the characteristics of the reinforcement employed will have to be taken into consideration. Then

“Indentation forces” in the longitudinal Qs and the transverse reinforcement qsw,2 are determined using the special model of the “indentation effect” discussed in [11].

Thus the investigated method for calculating the resistance of reinforced concrete structures under the combined action of bending moment, torque and shear force for the second stage of the stress-strain state (case 2) can be employed when spatial cracks of the first type occur on the side face of the structure.

3. Calculation of the distance between spatial cracks and the width of their opening in reinforced concrete structures during torsion with bending (case 1). When calculating reinforced concrete structures for the action of transverse forces, bending and torques, it becomes necessary to evaluate a complex stress-strain, which is even more complicated when there are spatial cracks.

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After cracks have occurred, the continuity of concrete is disrupted and the application of the formulas of the mechanics of a solid deformed body is no longer valid. However, in order to determine the actual stress-strain of reinforced concrete structures, it becomes necessary to identify the full picture of crack formation during loading. It is important to have not only different levels of crack formation of spatial cracks, but also to have their full picture.

First, the entire fan of spatial cracks of all types needs to be applied. After determining a dangerous spatial crack using the criterion for the formation or the largest width of their opening, it is necessary to apply the whole picture of spatial cracks.

Moreover, as shown by the practice of calculations and design of reinforced concrete structures, the distance between the first type of spatial cracks for the first level of crack formation located along the transverse or longitudinal reinforcement can be determined using the

following ratio (Fig. 3, Fig. 4):

Fig. 3. The design scheme for determining the distance between cracks of the first type (case 1):

а is the scheme of efforts and the choice of the coordinate system to the formation of the first spatial crack

Hence

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Russian Journal of Building Construction and Architecture

In order to determine the distance between spatial cracks of the second level of their formation, the ratio between the stresses in the reinforcement in section I––I and in the section with a dangerous spatial crack identified using the criterion of the maximum width of their opening is employed.

Fig. 4. Location of an adjacent crack of the next level between two cracks of the previous level: а is along the axis of the transverse reinforcement; b is along the axis of the longitudinal reinforcement

Moreover, a new level of crack formation corresponds to the load level at which the following inequality is met

(32) where along the transverse reinforcement from a dangerous inclined crack is determined using the following ratios (Fig. 4, b):

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Moving to the right along the longitudinal reinforcement, such ratios will take the following form (Fig. 4, b):

Moving to the left of the dangerous spatial crack, we compare the functional lcrc and l* (see Fig. 4), and if necessary, the same ratio is used.

Moreover, we do not go outside the area where bt y bt ,u .

in the longitudinal reinforcement in the section of spatial inclined cracks, the ratios (35) and (37) are somewhat modified, i. e., in addition to the stress ratio in the reinforcement, the ratio of the longitudinal reinforcement areas (before and after the break) is also taken into account. As a result, these formulas will take the form, respectively, moving to the right:

Thus, cracking continues until failure occurs. At the same time, not one (as is customary in a number of the well-known methods), but several levels of crack formation are distinguished:

Comparing the functional and level value lcrc , the possible realization of the appearance of subsequent levels of cracking is analyzed. With the levels of crack formation along the longitudinal and transverse reinforcement of the reinforced concrete structure available, a

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