1.1 Introduction

      The design of different reinforced concrete sections of beams will be considered in this chapter.

1.2 Design and Analysis

      The main task of a structural engineer is the analysis and design of structures. The two approaches of design and analysis will be used in this chapter:

            Design of a section. This implies that the external ultimate moment is known, and it is required to compute the dimensions of an adequate concrete section and the amount of steel reinforcement. Concrete strength and yield of steel used are given.

            Analysis of a section. This implies that the dimensions and steel used in the section (in addition to concrete and steel yield strengths) are given, and it is required to calculate the internal ultimate moment capacity of the section so that it can be compared with the applied external ultimate moment.

1.3 Basic Assumptions in Flexure Theory

      Five basic assumptions are made:

1.   Plane sections before bending remain plane after bending.

      2.   Strain in concrete is the same as in reinforcing bars at the same level, provided that the   bond between the steel and concrete is sufficient to keep them acting together under       the different load stages i.e., no slip can occur between the two materials.

      3.   The stress-strain curves for the steel and concrete are known.

      4.   The tensile strength of concrete may be neglected.

      5.   At ultimate strength, the maximum strain at the extreme compression fiber is assumed     equal to 0.003, by the Egyptian Code.

      The assumption of plane sections remaining plane (Bernoulli's principle) means that strains above and below the neutral axis NA are proportional to the distance from the neutral axis, Fig. 1.1. Tests on reinforced concrete members have indicated that this assumption is very nearly correct at all stages of loading up to flexural failure, provided good bond exists between the concrete and steel. This assumption, however, does not hold for deep beams or in regions of high shear.

            FIGURE 1.1. Single reinforced beam section with strain distribution.

1.4 Behavior of a Reinforced Concrete Beam Section Loaded to Failure

            To study the behavior of a reinforced concrete beam section under increasing moment, let us examine how strains and stresses progress at different stages of loading:

1.4.1 Noncracked, Linear Stage

      As illustrated in Fig. 1.2, where moments are small, compressive stresses are very low and the maximum tensile stress of concrete is less than its rupture strength, fctr. In this stage the entire concrete section is effective, with the steel bars at the tension side sustaining a strain equal to that of the surrounding concrete () but the stress in the steel bars is equal to that in the adjacent concrete multiplied by the modular ratio n.  Utilizing the Transformed Area Concept, in which the steel is transformed into an equivalent concrete area , the conventional elastic theory may be used to analyze the "all concrete" area in Fig. 1.2.

            FIGURE 1.2. Transformed section for flexure before cracking.

            This stage should be considered as the basis for calculating the cracking moment Mcr, which produces tensile stresses at the bottom fibers equal to the  modulus of rupture of concrete, Fig. 1.3. The Egyptian Code recommends the flexural formula M/Z to compute the flexural strength of the section:

    (1.1a)

      where  is the moment of inertia of gross concrete section about the centroidal axis, neglecting the reinforcement, yt is the distance from the centroidal axis of cross section, neglecting steel, to extreme fiber tension and fctr is the modulus of rupture of concrete. The Egyptian code (ECCS) suggests an imperical formula relates the modulus of rupture of concrete to its compressive strength:

    N/mm2  (1.1b)

            FIGURE 1.3. Transformed section for flexure just prior to cracking.

1.4.2 Cracked, Linear Stage

      When the moment is increased beyond Mcr, the tensile stresses in concrete at the tension zone increased until they were greater than the modulus of rupture fctr, and cracks will develop. The neutral axis shifts upward, and cracks extend close to the level of the shifted neutral axis. Cracked concrete below the neutral axis is assumed to be not effective and the steel bars resist the entire tensile force. The stress-strain curve for concrete is approximately linear up to 0.40 fcu; hence if the concrete stress does not exceed this value, the elastic (straight line) theory formula M/Z may be used to analyze the "all concrete" area in Fig. 1.4.

            FIGURE 1.4. Transformed section for flexure somewhat after cracking.

1.4.3 Cracked, Nonlinear Stage

For moments greater than these producing stage 2, the maximum compressive stress in concrete exceeds 0.40. However, concrete in compression has not crushed. Although strains are assumed to remain proportional to the distance from the neutral axis, stresses are not and, therefore, the flexural formula M/Z of the conventional elastic theory cannot be used to compute the flexural strength of the section. The Internal Couple Approach, instead, will be used to compute the section strength. This approach allows two equations for equilibrium, for the analysis and design of structural members, that are valid for any load and any section. As Fig. 1.5 indicates, the compressive force C should be equal to the tensile force T, otherwise the section will have a linear displacement plus rotation. Thus,

      

    C = T (1.2a)

                                                               

      The internal moment  is equal to either the tensile force T multiplied by its arm yct    or the compressive force C multiplied by the same lever arm. Thus,

    (1.2b)

  FIGURE 1.5. Transformed section for flexure after cracking.

      The resultant internal tensile force T is given by

   

 (1.3)

      where  is the area of steel and  is the steel stress. The resultant internal compressive force is obtained by integrating the stress block over the area bc. Taking an infinitesimal strip dy of area dA equals b by dy, located at a distance y from the neutral axis and subject to an assumed uniform compressive stress f and strain  X the compressive force C is given by

    (1.4)

            This stage may be considered as the basis for calculating the flexural strength of the section at first yield of the tension steel (known as the yield moment ). When the tension steel first reaches the yield strain (), the strain in the extreme fiber of the concrete may be appreciably less than 0.003. If  the steel reaches the yield strain and the concrete reaches the extreme fiber compression strain of 0.003, simultaneously, the yield moment occurs and equals the ultimate moment Mu. Otherwise, if the concrete crushed before the steel yields, the yield moment will never take place.

1.4.4 Ultimate Strength Stage

      For the given section, when the moment is further increased,  strains increased rapidly until the maximum carrying capacity of the beam was reached at ultimate moment Mu. The section will reach its ultimate flexural strength when the concrete reaches an extreme fiber compression strain Xcu of 0.003 and the tensile steel strain Xs cloud have any value higher or lower than the yield strain .

            As Fig. 1.6 indicates, the compressive forces C1  and C2  are obtained by integrating the parabolic and rectangular stress blocks over the rectangular areas A1 and A2  of  and , respectively.

   
   

                                   FIGURE 1.6. Single reinforced beam section with flexure at ultimate.

      The corresponding lever arms y1 and y2  are given by          

   
   

      The resultant force C is, then, computed from

    (1.5)

      The position of C is at a distance y from the top fiber where y is computed from

   

      The distance between the resultant internal forces, known as the internal lever arm, is

    yct = d - 0.4 c (1.6)

      where d, the distance from the extreme compression fiber to the centroid of the steel area, is known as the effective depth. The ultimate strength Mu is therefore

    (1.7)

1.5 Equivalent Rectangular Compression Stress Block

      As a means of simplification, the Egyptian Code has suggested the replacement of the actual shape of the concrete compressive stress block (a second-degree parabola up to 0.002 and a horizontal branch up to 0.003) by an equivalent rectangular stress block, Fig. 1.7.

                                          FIGURE 1.7. Actual and equivalent stress distribution at failure.

            A concrete stress of  is assumed uniformly distributed over an equivalent compression zone bounded by the edges of the cross section and a line parallel to the neutral axis at a distance  from the fiber of maximum compressive strain, where c is the distance between the top of the compressive section and the neutral axis NA.

            For the resultant compressive forces of the actual and equivalent stress blocks of Fig. 1.7, to have the same magnitude and line of action, the average stress of the equivalent rectangular stress block and its depth are  and  where  and  . These values are as already derived when calculating the ultimate strength Mu in Section 1.4.4.

            The equivalent rectangular stress block applies, as the Egyptian Code permits, to rectangular, T and trapezoidal sections, Fig. 1.8.

      FIGURE 1.8. Applicability of equivalent rectangular stress block to some sections.

            For sections as shown in Fig. 1.9, stress distribution should be based on the actual stress-strain diagram. The above procedure, however, can be implemented to obtain the parameters  and  that correspond to these sections.

            FIGURE 1.9. Inapplicability of equivalent rectangular stress block to some sections.

1.6 Types of Flexural Failure                  

      The types of flexural failure possible (tension, compression and balanced) and  the nominal (ideal) strength Mu of the beam section (a singly reinforced rectangular section) are discussed next.

1.6.1 Tension Failure

      If the steel content of the section is small (an under-reinforced concrete section), the steel will reach its yield strength before the concrete reaches its maximum capacity. The flexural strength of the section is reached when the strain in the extreme compression fiber of the concrete is approximately 0.003, Fig. 1.10. With further increase in strain, the moment of resistance reduces, and crushing commences in the compressed region of the concrete. This type of failure, because it is initiated by yielding of the tension steel, could be referred to as a "primary tension failure," or simply "tension failure." The section then fails in a "ductile" fashion with adequate visible warning before failure.

    FIGURE 1.10. Single reinforced section when the tension failure is reached.

For a tension failure, ; for equilibrium, C = T. Hence from  from

 

    and

      we have  which results in

 

    (1.8)

     

 The nominal strength Mu (which obtained from theory predicting the failure of the section on assumed section geometry and specified materials strengths i.e., = = 1.0), is

 

     (1.9)

 

1.6.2 Compression Failure

      If the steel content of the section is large (an over-reinforced concrete section), the concrete may reach its maximum capacity before the steel yields. Again the flexural strength of the section is reached when the strain in the extreme compression fiber of the concrete is approximately 0.003, Fig. 1.11. The section then fails suddenly in a "brittle" fashion if the concrete is not confined and there may be little visible warning of failure.

                             FIGURE 1.11. Single reinforced section when the compression failure is reached.

            For a compression failure,  as the steel  remains in the elastic range. The steel stress may be determined in terms of the neutral axis depth considering the similar triangles of the strain diagram of Fig. 1.11.

 

   

\

(1.10)

      The steel stress is

      (1.11a)

      or, since Es = 200 kN/mm2,

     (1.11b)

      For equilibrium, , hence

    (1.12)

      The above quadratic equation may be solved to find c and, on substituting a = 0.8c, the nominal strength is

      (1.13)

1.6.3 Balanced Failure

      At a particular steel content, the steel reaches the yield strength and the concrete reaches its extreme fiber compression strain of 0.003, simultaneously, Fig. 1.12. Then,  and from the similar triangles of the strain diagram of Fig. 1.12 we can write

     (1.14)

      where  = neutral axis depth for a balanced failure. Then

    (1.15)

      or, on substituting  = 0.80 , Eq. 1.15 becomes                       

    (1.16)

       FIGURE 1.12. Single reinforced section when the balanced failure is reached.

      For equilibrium, ; hence we have

    (1.17)

      which results in

    (1.18)

      where is the balanced steel ratio.

      The type of failure that occurs will depend on whether the steel ratio m  (where m= ) is less than or greater than . Figure 1.13 shows the strain profiles at a section at the flexural strength for three different steel contents. As Fig. 1.13 indicates, if  for the section m is less than , then c < cb and ; hence a tension failure occurs. Similarly, if m is greater than  , then c > cb and , and  a compression failure occurs.

  FIGURE 1.13. Strain profiles at the flexural strength of a section.

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