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Chapter 1
Design at Serviceability Limit State (SLS)
1.1. Nomenclature
1.1.1. Convention with the normal vector orientation
The normal vector is chosen to be oriented toward the external part of the considered body. The usual conventions of mechanics of continuous media are chosen, leading to a positive stress for tension and a negative stress for compression.
Figure 1.1. Definition of the normal unit
1.1.2. Vectorial notation
As opposite to the notation used for figures, where vectors are represented with an arrow, in the text, vectors are denoted by bold characters and its components have normal non-bold characters. As an example, we will have “M = Mxi+ Myj + Mzk”.
1.1.3. Part of the conserved reference section
The conserved reference part of the beam used for the calculation of generalized stress in use of the static theorems is the “right” part.
1.1.4. Frame
Figure 1.2. Orientation of the frame for a general section
1.1.5. Compression stress σc,sup in the most compressed fiber
It is admitted that the neutral axis is located inside the cross-section, thus delimiting a tension zone and a compressed zone. This last assumption of a neutral axis inside the cross-section is no more valid when considering additional meaningful normal forces. Typically, under a positive moment (in span), the tension zone is located under the neutral axis, and the compression zone above the neutral axis, as shown in Figure 1.3. Obviously, in the presence of a negative bending moment, the tension zone and the compression zone are permuted with respect to the notation of Figure 1.3. The neutral axis as shown in Figure 1.3 allows the introduction of the concept of extremal compressed fiber, defined from the most distant parallel to the neutral axis belonging to the cross-section. The most compressed fiber in concrete is by definition the fiber associated with the minimal compressive stress in algebraic value, denoted by σc,sup .
Figure 1.3. Concept of extremal compressed fiber
1.2. Bending behavior of reinforced concrete beams – qualitative analysis
1.2.1. Framework of the study
1.2.1.1. Constitutive law of concrete
The constitutive law of concrete is a strong unsymmetrical law in tension and in compression, both from the strength and the postfailure response, which is characterized by its ductility (see Figure 1.4). As a natural choice, the subscript c refers to concrete whereas the subscript s refers to the steel material. We adopt by σc,min the extreme stress at the compression peak.
Figure 1.4. Unsymmetrical response of concrete in uniaxial tension and compression
1.2.1.2. Beam theory in simple bending
In this section, reinforced concrete beams in simple bending are studied (without axial forces), composed of a rectangular cross-section with a total height denoted by h and a width b. This section is reinforced by some steel reinforcement working in tension with a cross-section denoted by As1 and by steel reinforcement working in compression with a cross-section denoted by As2. The center of gravity of the tensile reinforcement is at a distance d of the upper fiber, and one of the compression reinforcements is at a distance d' of the upper fiber of the cross-section.
Figure 1.5. Geometrical parameters of the reinforced cross-section; tensile and compression steel reinforcement
In this case, again, it is implicitly accepted that the bending solicitation corresponds to a positive moment (the lower fibers are in tension with this convention, typically in span). Designing under negative moment (typically at support, for instance) is formally feasible by permuting the behavior of the cross-section. Furthermore, we can introduce the strain εs1 as the strain of the tensile reinforcement with the largest tensile stress σs1 and with the reinforcement area As1.
1.2.2. Classification of cross-sectional behavior
Three kinds of reinforced concrete beam responses can be distinguished, depending on the steel reinforcement density (Figure 1.6). These responses are explicitly detailed in the space of the bending moment with respect to the stress in the most compressed fiber in concrete.
Figure 1.6. Bending behavior of reinforced concrete beams with respect to the steel reinforcement density
F: brittle response, which appears when the beam design does not respect the condition of non-brittleness.
A: beam with low steel reinforcement density, characterized by a global ductile response. Failure is induced by a large drawing of the tensile steel reinforcement. As discussed below, the letter A refers to Pivot A.
B: beam with high steel reinforcement density, characterized by the breaking up of the compressed part of the upper part concrete. The letter B refers to the behavior classified as Pivot B.
In the following, brittle reinforced concrete beams of type F will not be investigated.
1.2.3. Parameterization of the response curves by the stress σs1 of the most stressed tensile reinforcement
In Figure 1.7, the response curves are parameterized by the stress σs1 of the most stressed tensile reinforcement. When reading Figure 1.7 in the...
