# General Information

ARBAT enables you to check the load-bearing capacity and to select the reinforcement for concrete and reinforced concrete structural members, to calculate the deflections in reinforced concrete beams, to check the local strength of reinforced concrete structural members (including inserts) in compliance with requirements of one of the following codes:

• SNiP 2.03.01-84*;
• SNiP 2.03.01-84* with Ukrainian modifications;
• SNiP 52-01-2003 (SP 52-101-03);
• SP 63.13330.2012;
• SP 63.13330.2018.

The analysis is performed taking into account the ultimate and serviceability limit states (strength and crack resistance) for the design combinations of forces (DCF) selected automatically depending on the specified design loads in compliance with SNiP 2.01.07-85*, "Loads and actions" and SNiP 2.03.01-84*, "Concrete and reinforced concrete structures" (SNiP 52-01-2003). If the selected design code is SP 63.13330, then loads will be combined according to the requirements of SP 20.13330.

Design values of loads (forces) are specified in all modes of the application. Characteristic values of loads are used in the analysis of the serviceability limit state. User-specified safety factors for load are used for each load case to convert design values to the characteristic ones.

The selection of reinforcement and the check are performed for beams, columns, and slabs made of heavy-weight, fine-grained and lightweight concrete with the reinforcement steel of the following classes А-I (A240), A-II (A300), A-III (A400), A-IV, A-V and A-VI, and the reinforcing wire of class Bp-I (B500) and reinforcement of classes A400C, A500C (A500), A600C. Moreover, the selection and check of the following types of fiber reinforced polymer (FRP) rebar are implemented in accordance with the requirements of SP 295.1325800: glass, basalt, carbon, aramid, and hybrid.

Moreover, ARBAT plays the role of a manual that provides data on the assortment and properties of reinforcement, characteristic and design values of the concrete strength, service factors for concrete, and allowable deflection limits.

The controls and procedures used to prepare data and to document the results of the analysis, which are implemented in the application, are exactly the same as those in other design and analysis applications included in the SCAD Office® system. The program uses a common multi-tab technique. To switch to a mode, click on its tab or use an appropriate menu item.

## Structural Assessment

### Direct and Inverse Problems

Design codes are developed as a system of checks of the known design, i.e. they solve the problem of structural assessment rather than the problem of its synthesis. ARBAT enables to solve both these problems — assessment task and selection task. However, the latter problem (selection) is solved in a limited formulation as a purposeful search through the list of possible designs.

The described approach to the selection (synthesis) of the cross-section leads to designs that for one reason or another (design considerations, unification, etc.) may not satisfy the designer. He can adjust the design proposed by the program and perform its verification in the check mode.

### Load-bearing Capacity Area of Sections and Structural Assessment

Standard requirements (conditions of strength, general and local stability, limit slenderness, etc.) to a certain design section of a structure can be written in the form of a certain system of inequalities, each of which depends functionally on the values of internal forces $\vec{{S}} = \{ S_{1}, S_{2}, ... , S_{n} \}$ that can arise in the considered section from the action of the design combinations of loadings:

${\rm {\bf \Phi }}\left( {\vec{{S}},\vec{{R}}} \right)\le 1 ;$

or

$\left\{ {{\begin{array}{*{20}c} {f_{1} \left( {\vec{{S}},\vec{{R}}} \right)\le 1;} \\ {f_{2} \left( {\vec{{S}},\vec{{R}}} \right)\le 1;} \\ {...} \\ {f_{j} \left( {\vec{{S}},\vec{{R}}} \right)\le 1;} \\ {...} \\ {f_{m} \left( {\vec{{S}},\vec{{R}}} \right)\le 1;} \\ \end{array} }} \right. ,$

where n is the total number of possible internal forces in the section; m is the number of inequalities that describe the standard requirements; fi is a function of principal variables that implements j-th check; R are generalized resistances.

Based on the values of functions $K_{j} =f_{j} \left( {\vec{{S}},\vec{{R}}} \right)$, we can introduce the concept of a utilization factor of restrictions, and thus represent the analysis criterion as:

$K_{\max } =\max \left\{ {K_{j} \quad \vert \quad j=1,...,m} \right\}\le 1 ,$

where $К_j$ is the left-hand side of the design inequality $f_{j} \left( {\vec{{S}},\vec{{R}}} \right)\le 1$,

where required analyses are included. The value of Кj itself will define a reserve of strength, stability, or another regulated quality parameter available for a particular element (joint, connection, cross-section etc.). If the requirement of the design code is met excessively, then Кj is equal to the relative value of exhaustion of the design requirement (for example, Кj = 0,7 corresponds to the reserve of 30%). If the design requirements are not met, the value of Кj > 1 indicates a violation of a requirement, i.e. it describes a degree of overloading.

Each standard requirement $K_{\max } =\max \left\{ {K_{j} \quad \vert \quad j=1,...,m} \right\}\le 1$ defines a certain area Ωj in the n-dimensional space of internal forces, and the intersection of all areas Ωj forms the load-bearing capacity area of the section Ω in terms of the considered codes (Fig. 1). The maximum utilization factor of restrictions for each point of the load-bearing capacity area of the section is $K_{\max } =\max \left\{ {K_{j} \quad \vert \quad j=1,...,m} \right\}\le 1$.

Figure 1. Formation of the load-bearing capacity area in the two-dimensional space of internal forces

Regulatory restrictions are usually written as follows:

$\varphi_{j} \left( {\vec{{S}}} \right)\le \psi_{j} \left( {\vec{{R}}} \right), (j = 1, …,m),$

where φj, ψj – functions of the main variables which implement the j-th check.

However, not all these requirements can or need to be rewritten in the form:

$f_{j} \left( {\vec{{S}},\vec{{R}}} \right)\le 1, (j = 1,…, m), \text {where}$ $f_{j} \left( {\vec{{S}},\vec{{R}}} \right)=\frac{\varphi_{j} (\vec{{S}})}{\psi_{j} (\vec{{R}})},$

i.e. in the form of the ratio of the left-hand side of the standard inequality to its right-hand side.

To illustrate this let’s consider formula (8.78) of SP 63.13330 as an example:

$T\le T_{0} \sqrt {1-\left( {\frac{M}{M_{0} }} \right)^{2}}.$

If the following ratio was calculated as the value of the utilization factor of this standard check:

$\frac{T}{T_{0} \sqrt {1-\left( {\frac{M}{M_{0} }} \right)^{2}} }\le 1,$

then in the design situation when M < M0, we would obtain a root from a negative number in the denominator, and would not be able to obtain a quantitative estimate of the excess of the bearing capacity, which is avoided by transforming the formula (8.78) as follows:

$\left( {\frac{T}{T_{0} }} \right)^{2}+\left( {\frac{M}{M_{0} }} \right)^{2}\le 1.$

Another example is the formula (8.10) of SP 63.13330

$Ne\leq R_b b x(h_0-x/2)+R_{sc} A_s^\prime(h_0-a^\prime),$

for which the ratio of the left- and right-hand sides can not be used as a factor, because the height of the compressed area x depends on the longitudinal force.

The numerical value of the factor (the value of the utilization factor of restriction) is a measure of how fully the bearing capacity of the structural member is used (or how much it is exceeded), and as a result it allows the designer to make a correct decision about the type of the necessary design modification and only! For example, it hardly makes sense to select another steel grade with a greater design strength in the case when the stability check turned out to be critical.

It should be specially emphasized that there may be no direct proportionality between the value of the factor and the values of the forces appearing in the standard check, as there may be no direct proportionality between the value of the factor and the values of the geometric properties of the cross-sections of the structural elements. This is due to the non-linearity of the standard checks, in particular, the non-linearity of the function of the buckling coefficient, etc.

All values of the Кj factors obtained by analysis are given in the Factors Diagram dialog box or in a full report document. The respective dialog boxes display the value of Кmax – the maximal (i.e., the most dangerous) among the detected Кj values and indicate the type of check (e.g., strength, stability) in which this maximum took place.

The data given in the diagram of factors enable the designer to make a correct decision on the type of necessary structural modifications. For instance, it hardly makes sense to increase the design strength of steel if the stability check turned out to be the critical one.

### Generation of the Load-Bearing Capacity Area and Its Properties

ARBAT enables to generate the load-bearing capacity areas for sections of load-bearing bar elements of reinforced concrete structures.

Sections of bar elements, where six internal forces (longitudinal force, bending moments, transverse forces, and torque) can appear under the load, have a load-bearing capacity area in the form of a six-dimensional geometric object which is very difficult to analyze. The best way to display the load-bearing capacity area of sections is by performing its orthogonal projection onto a certain plane (pair) of internal forces. The generation of a two-dimensional projection of the load-bearing capacity area of a section is performed according to the algorithm given below.

The user selects a pair of internal forces (for example, a pair “longitudinal force N – bending moment My”), in the coordinate system of which an orthogonal projection of the load-bearing area will be generated. The remaining internal forces in the section (Mz, Qy, Qz, Mx) are fixed at a certain level (they are specified by the user or take zero values). At a certain fixed value of the ratio e = My/N, the point closest to the origin is sought, where a certain inequality from the system ${\rm {\bf \Phi }}\left( {\vec{{S}},\vec{{R}}} \right)\le 1$ takes a limit value $f_{j} \left( {\vec{{S}},\vec{{R}}} \right)=1$. This point belongs to the boundary of the two-dimensional orthogonal projection of the load-bearing capacity area of the section.

This method of generating a load-bearing capacity area assumes that the area is starlike, which is a hypothesis, on the one hand, and a restriction of the software implementation, on the other hand.

Moreover, the generated area is an interactive tool for communicating with the user. Using your cursor, you can examine a two-dimensional projection of the area. A certain set of internal forces corresponds to each position of the cursor. Their values are displayed in the respective fields. Depending on the change in the position of the cursor (changes in the respective pair of internal forces), the maximum value of the utilization factor of standard restrictions-inequalities Кmax, corresponding to these forces is output, as well as the type of the inequality for which it is calculated. Clicking the right mouse button on the area enables to see the entire list of performed checks and the values of utilization factors of restrictions $K_{j} (j=\overline {1,m} )$ for the set of internal forces that corresponds to the cursor position. Moreover, the following operations with the mouse can be performed in the program:

• scrolling the wheel changes the scale of the image;
• clicking and holding the middle mouse button (wheel) while dragging the cursor moves the image.

One of the most important properties of the load-bearing capacity area is its convexity. It should be noted that it is the convexity of the load-bearing capacity area of the cross-section that gives us the right to limit ourselves in the linear calculation to the checks of this section for the action of only those combinations of internal forces in the cross-section for which the extreme (minimum or maximum) values are characteristic. The positive result of such checks automatically means that all other conceivable combinations of loads will be acceptable.

The absence of the convexity property of the load-bearing capacity area of the considered section can lead to many unpleasant consequences related to the fact that, traditionally, evaluating unfavorable combinations of internal forces, engineers either do not consider some actions at all (in the case when they have a unloading effect) or take them fully into account. This rule is entirely valid for a convex load-bearing capacity area, while for a nonconvex area a combination with intermediate (not extreme) values of internal forces can turn out to be an unfavorable one.

### Load-bearing Capacity Area of Sections as a Tool for the Analysis of Codes

Several thousand calculations are performed during the computer-aided generation of the load-bearing capacity area of a section, which is apparently the largest check of the considered section. Moreover, the shape of the load-bearing capacity area of the section as well as the character of its boundaries in many cases enables to perform a more detailed analysis of the requirements of the codes than it can be done in other ways. The analysis of the shape of the area enables to check the consistency and completeness of the standard requirements. In this case, it is easy to identify the inconsistency of certain provisions of the codes, in particular the non-smoothness of the transition between the approximations used.

SP 63.13330 Reinforced Concrete Structure

For example, let’s consider the design code for reinforced concrete structures SP 63.13330. Let’s consider a rectangular reinforced concrete section 700 mm × 200 mm, reinforced by 2 rebars with diameters 28 mm. The areas of the load-bearing capacity of this section corresponding to two variants of the standard crack resistance requirements are given in Fig. 2 in the “longitudinal force N – bending moment Мy” axes. The area Ω1 is generated taking into account the regulatory restriction of the crack opening width, and the area Ω2 is generated taking into account the prohibition of crack formation. It should be noted that both methods of the standard check of reinforced concrete sections for crack resistance are semi-empirical. Let’s consider the causes of the non-convexity of the load-bearing capacity area of a reinforced concrete section.

The jump АВ, which is typical for both areas of the load-bearing capacity Ω1 and Ω2, is defined by the postulate of the theory of reinforced concrete, which provides for the immediate exclusion from the operation of the tension area of concrete when tensile stresses appear in it.

The jump ВС is typical only for the load-bearing capacity area of the reinforced concrete section Ω2, generated taking into account the prohibition of crack formation. In this case, the area Ω2 sharply narrows, which is quite expected and, moreover, it acquires a very sophisticated shape, due to a new method of checking reinforced concrete elements for crack resistance, which appeared in the 2004 edition of the codes.

Figure 2. Load-bearing capacity areas of a reinforced concrete section according to SP 63.13330:
Ω1 – restriction of the crack opening width (3-rd category of crack resistance);
Ω2 – prohibition of crack formation (1-st category of crack resistance)

Another instructive example is given in Fig. 3, which shows the load-bearing capacity area of a reinforced concrete section calculated according to a non-linear deformation model without taking into account (variant a) and taking into account (variant b) the random eccentricity. The eccentricity is obviously taken into account “unphysically” in the variant given in the codes, since there is no physical model that would violate the smoothness of the boundary of the load-bearing capacity area. It is specified in the codes that for a statically indeterminate system the value of the eccentricity of the longitudinal force with respect to the center of gravity of the reduced section e0 should be taken equal to the eccentricity value e obtained from the static analysis but not less than a certain fixed value of the random eccentricity ea; and for a statically determinate system, the eccentricity e0 should be taken equal to the value (e + e0). In both cases, when the eccentricity values е are small, an eccentrically compressed element is considered, and the eccentricity e0 is never small.

Figure 3. Effect of random eccentricity on the shape of the projection of the load-bearing capacity area of a reinforced concrete section

The number of examples could be increased, but the ones given here indicate a very real situation when the load-bearing capacity area can turn out to be non-convex. As the analysis shows, in many cases there are some inconsistencies in the formulation of requirements for the bar elements of load-bearing structures, due most likely to insufficient adjustment of the formulations themselves.

The origin of such inconsistencies is related to the fact that the traditional approach based on the manual calculation generated all sorts of “simplifications”, which allowed to skip some checks or to replace the general case with a certain particular case (as at m < 20 for steel elements under bending and compression).

Moreover, the use of “logical switches” that change rules without an exact physical basis leads to an abrupt change in the algorithm, as in the classification of cross-sections in the Eurocode. Modern technologies enable to detect such inaccuracies and determine ways to improve the codes.

Dangers related to the nonconvexity of the load-bearing capacity area point to the necessity of analyzing the closeness of the set of the specified combinations of forces to the section of the boundary of the load-bearing capacity area, where the property of nonconvexity is manifested.

An almost identical analysis can be performed using additional tools provided by SCAD Office. The following buttons are provided for the analysis of problems related to the non-convexity of the load-bearing capacity area:  . They enable to perform the following operations:

– if the forces are specified, clicking this button will draw the entire set of given combinations of forces as points corresponding to the projections of the variants of combinations of forces on the plane of the selected pair of forces (Fig. 4).

– drawing a convex hull of the points specified above, i.e. an entire set of points which may result from a linear combination of specified forces, including their incomplete values.

Despite the fact that these combinations were not subjected to a direct check, in the case when the convex hull of loadings does not leave the load-bearing capacity area of the section, it can be ensured that the various loadings combined from the basic ones are not dangerous.

Figure 4. Given (basic) loadings and their convex hull, combined with the load-bearing capacity area of the section

Figure 5. Illustration of possible design situations

It should be noted that the considered mechanism is a powerful tool for analyzing loading conditions, but it should be used carefully.

The load-bearing capacity area of a section is a body in the six-dimensional space of internal forces (with the coordinates N, My, Qz, Mz, Qy, T). The load-bearing capacity area is a section of the given body by a plane, and the point corresponding to the set of forces is the projection onto this plane. If the point corresponding to a certain set of forces lies within the load-bearing capacity area, it does not follow from this that all the requirements of the codes are satisfied. This is due, for example, to the fact that when generating the load-bearing capacity area (interaction curves), the restrictions of the limit slenderness are ignored (these restrictions do not depend on forces), and a situation may occur when the point belongs to the load-bearing capacity area of the section (lies in the “green area”), but the calculation will show Kmax > 1.

When operating only with two-dimensional orthogonal projections of the load-bearing capacity area, it is possible to “see” the projections of some points (combinations of internal forces) belonging to this area (for which the maximum utilization factor of restrictions does not exceed one Kmax < 1) as displayed outside the projection boundary of the load-bearing capacity area (as, for example, point 1 in Fig. 5). This can happen in the case when the Seismic checkbox is checked for the specified set of combination of internal forces, and as a result the service factor mcr > 1 was used in the calculation. And the generation of the load-bearing capacity area (interaction curves) is performed without taking into account factor mcr. A more complicated situation can arise when analyzing the interaction curves for reinforced concrete structures, since different sets of forces (points) may have different coefficients of duration.

There may also be an erroneous “vision” of a different kind, when the projection of the point lies within the boundaries of the projection of the load-bearing capacity area, and the point itself does not belong to the area (see, for example, point 6 in Fig. 5).

In order to identify such situations, the projections of the points in which the utilization factor of restrictions exceeds one, are displayed in red on the projections of the load-bearing capacity area, otherwise they are displayed in green.

## Interaction Surfaces

In the program the Interaction Curves tab has a button , which enables to generate a surface in 3D where Kmax=1 for the triple of internal forces and moments selected by the user (for example, My-Mz-N). Clicking this button invokes the dialog box, where you can select which force corresponds to a certain coordinate axis (XYZ).

Clicking the Apply button will start the calculation which can be aborted by clicking the Cancel button. Once the calculation is completed, the image of the interaction surface will appear.

Controls in the left side of the window enable to

• select the density of the mesh applied to the surface;
• change the colors used for drawing (as well as the fonts for the names of the axes);
• not to display vertical and/or horizontal meshes on the surface;
• set the transparency and/or lighting mode;
• display geometric perspective and select the distortion factor;
• change the background color and the color of the lighting.

In order to change the color or font, double click on the respective cell and select the parameters.

The Wireframe model checkbox enables to obtain an image of the surface in the form of a wireframe.

Right-clicking invokes the context menu where you can select a projection.

The following operations with the mouse can be performed in the program:

• scrolling the wheel changes the scale of the image;
• clicking and holding the middle mouse button (wheel) while dragging the cursor moves the image;
• clicking and holding the middle mouse button (wheel) together with Shift or Ctrl key while dragging the cursor rotates the surface (it is also possible to use button for switching to the rotation mode).
The Save button enables to save the forces for which Kmax=1 in a text file (this file can later be imported into other programs for further analysis).