General case of reinforced concrete rod elements calculation using the diagram method

In ВС 63.13330 the general calculation case for the diagram method is oblique off-center compression, which takes into account only three components of internal force factors in the cross section: the longitudinal force - Nz – and two bending moments relative to the corresponding axes – Mx and My. The other three components-the QX and Qy transfer forces and the MZ torque – are left out of consideration. In addition, for this case, the search in the available literature, including the founders of the diagram method, for the output of calculation formulas, was not successful – in all sources they are given in ready-made form without evidence. This article is intended to try filling in these gaps. For this purpose, based on the expressions for rod displacements that are generally accepted in mechanics, in particular on the Mora integral of displacements, the resolving expressions of the diagram method are obtained in the most general form.


Introduction
BC 63.13330 also includes the diagrammatic method as an alternative to the ultimate force method for the reinforced concrete rods' design. Its distinctive feature lies in the possibility of calculating for both groups of limiting states according to the same formulas, as well as obtaining a stress-strain state (SSS) of the considered elements at all loading stages: from zero to failure. In this set of rules, the general design case for this method is oblique offcenter compression, which takes into account only three components of internal force factors in the section: longitudinal force -Nz -and two bending moments about the corresponding axes -Mx and My. The other three components are shearing forces Qx and Qy and torque Mzremain out of consideration. In addition, for oblique eccentric compression, the search in the available literature, including the founders of the diagram method [1] - [3], the calculation formulas' derivation was not crowned with success -in all sources they are already presented in a finished form without proof. This article is intended to try filling in these gaps. (1) (2)

Models and Methods
In the special case of oblique eccentric compression, the governing equations for the rod shown in Fig. 1, a, are known. For example, in [4] such expressions are given: where for the section shown in Fig. 1b, the components of the stiffness matrix are:  Comparing the obtained expressions with the formulas (8.39) -(8.47) BC 63.13330, we find a lot of similarities. The difference between the set of rules is, firstly, in the rotation of the coordinate axes Оху about the axis Oz on 90 0 clockwise, and secondly, in the complete absence of minus signs in the stiffness matrix, which is apparently connected with the rotation of the axes. The third difference lies in the indexing of bending moments: in this article, the generally accepted in the resistance of materials is used -the subscript at the moment denotes the coordinate axis relative to which rotation occurs (for example, the momentМх causes rotation about the axisОх); in BC 63.13330otherwise: the subscript at the moment indicates the coordinate axis along which the moment acts. At the same time, although the plane in which a particular moment acts is concretized in BC, without this specification, rotation relative to any of the two remaining axes can occur, which causes confusion. Therefore, in our works we use the designations generally accepted in the strength of materials and structural mechanics.
To derive the general case of calculating the rod ( Fig. 1, a) by the diagrammatic method, we write the Mohr displacement integral [4]: The formulas are valid if at the pole О the rod is set to the right coordinate system Оxyz: axis zalong the rod, axis хfrom drawing plane, axis y-up. Moreover, the axes are the main central ones. The positive direction for the moments and angles of rotation is counterclockwise as viewed from the positive end of that axis. For compactness, the moments and angles of rotation are designated as spin vectors.
Let us consider each of the six movements separately. So, in the action 1 1 z N  and z N (the rest of the force factors are equal to zero), using the rule of multiplying the diagrams, we obtain a displacement along the axis Oz: . Pole rotation Оrod relative to the axis Оxat the corner x  possible both from the action of the moments , and from the action of cutting forces which create a moment equal y Q z and the corresponding movement The rest of the displacement components are obtained by analogy: where: z D is the axial stiffness of the rod, x D , y Dflexural, Q Dshear, k Dtwisting; l -rod length; x k , y k are the shear shape factors (for a rectangular section 1, 2 x y k k   ).More details about the stiffness will be given at the end of the article. To find the shape coefficients during shear, the following expressions are used (reinforcement is neglected): where b A shows the concrete section area; b is a concrete section width; bx I , by I are the moments of inertia of a concrete section relative to the axesОх and Оу; отс bx S , отс by S define the static moments of the cut-off part of the concrete section relative to the corresponding axes.
The vector of forces applied at the rod pole О, we write it in the form: the corresponding displacement vector:

Research results and their analysis
In the expressions (6), we differentiate along the coordinate z linear displacements z u once:  is a relative angle of twist about the axisОz. Now we get: In these formulas, we make the replacement: y , then we will have: The equations 2, 3 and, accordingly, 5,6 in the formulas (11) are identical -they determine one and the same quantity, therefore we discard, for example, 2 and 3. We obtain the following system of equations: Or in a matrix form: The expressions obtained do not contain shearing forces and the shifts are taken into account implicitly. To take them into account explicitly, it is necessary to discard the last two expressions in the formulas (11) and put z=l, 3 5 2 ... 8 8 . Then (11) will appear in the form: Or in matrix form: It should be noted that, although shear forces are taken into account in the expressions (14) and (15), in the deformation vector, shear deformations γare absent, they are taken into account indirectly through shear stiffness Q D , which depends on τandγ. We also note that in the structure of the formulas for curvatures, the first term is responsible for the pure shear strains effect on the curvature, the second term is for bending deformations caused by a shearing force, and the third term is for bending deformations caused by the moment action.
In addition, the proposed approach carries the development prospects due to taking into account, in addition to longitudinal deformations z  also transverse deformations x  and y  . Possible options for such clarifications are given in the works [5], [6], [7].
Prior to this, all formulas were derived with respect to the main central axes of the rod. With an arbitrary choice of the coordinate system O'x'y'z' it is necessary to make adjustments to the stiffness characteristics of the section based on the following reasoning ( Fig. 1, b). The longitudinal force has no eccentricity relative to the main central axes, ax=ay=0, and the centrifugal moment of inertia and the corresponding centrifugal stiffness of the rod section are equal to zero: When specifying the coordinate system arbitrarily, it is necessary to take into account additional deformations of the section rotation caused by the eccentrically applied longitudinal force z N , as well as additional turns from Mxabout the axis Oyand fromMyabout the axis Ox. This is achieved by including the formulas (12) or (14) to the first expression, the relative deformations caused by bending: And, in addition, it is necessary to replace the bending stiffnesses with the following values: where is the stroke «'»means that this value is calculated with respect to arbitrary axes It is also necessary to replace bending moments and shear forces with: