4-7 BEAMS WITH COMPRESSION REINFORCEMENT
3. Change of mode of failure from compression to tension. When a beam fails in a brittle manner through crushing of the compression zone before the steel
yields. A moment–curvature diagram for such a beam is shown in Fig. 4-32
When enough compression steel is added to such a beam, the compression zone is strengthened sufficiently to allow the tension steel to yield before the concrete crushes.
The beam then displays a ductile mode of failure. For earthquake-resistant design, all beam sections are required to have
4. Fabrication ease.When assembling the reinforcing cage for a beam, it is cus- tomary to provide small bars in the corners of the stirrups to hold the stirrups in place in the form and also to help anchor the stirrups. If developed properly, these bars in effect are compression reinforcement, although they generally are disregarded in design, because they have only a small affect on the moment strength.
rœ Ú 0.5r.
1rœ = 02. r 7 rb,
Curvature Fig. 4-31
Effect of compression rein- forcement on strength and ductility of under-reinforced beams. (From [4-16].)
Analysis of Nominal Moment Strength,
The flexural analysis procedure used for doubly reinforced sections, as illustrated in Fig. 4-33, essentially will be the same as that used for singly reinforced sections. The analysis is done for a rectangular section, but other section shapes will be included in the following sections. The area of compression reinforcement is referred to as the depth to the centroid of the compression reinforcement from the extreme compression fiber of the section is the strain in the compression reinforcement is and the stress in the compression reinforcement is
A linear strain distribution is assumed, as shown in Fig. 4-33b, and for the evaluation of the nominal moment capacity, the compression strain in the extreme concrete compres- sion fiber is set equal to the maximum useable concrete compressive strain, As was done for singly reinforced sections, the section is assumed to be under-reinforced, so the strain in the tension reinforcement is assumed to be larger than the yield strain. The exact magnitude of that strain is not known, and thus, the depth to the neutral axis,c, also is un- known. An additional unknown for a doubly reinforced section is the strain in the com- pression reinforcement, Unlike the tension-reinforcement strain, it is not reasonable to assume that this strain exceeds the yield strain when analyzing the nominal moment strength of a beam section. The following relationship can be established from similar tri- angles in the strain diagram:
or
(4-30) The assumed distribution of stresses is shown in Fig. 4-33c. As before, the real concrete compression stress distribution is replaced by Whitney’s stress block. The stress in the compression reinforcement, is not known and cannot be determined until the depth to the neutral axis has been determined. As was done in the analysis of a singly reinforced section, the stress in the tension reinforcement is set equal to the yield stress,fy.
fsœ, esœ = ac - d¿
c becu esœ
c - d¿ = ecu c esœ.
ecu. fsœ.
esœ, d¿,
Aœs, Mn
b
d d
d
c a b1c
Neutral axis
a/2
h
As
As
ecu
es ey
e es
F T fs fy f
fs
Cc Cs 0.85fc
(a) Doubly reinforced section. (b) Strain distribution. (c) Stress distribution. (d) Internal forces.
Fig. 4-33
Steps in analysis of Mnin doubly reinforced rectangular sections.
The internal section forces (stress resultants) are shown symbolically in Fig. 4-33d.
The concrete compression force, is assumed to be the same as that calculated for a singly reinforced section.
(4-13b) This expression contains a slight error, because part of the compression zone is occupied by the compression reinforcement. Some designers elect to ignore this error, but in this presentation, the error will be corrected in the calculation of the force in the com- pression steel by subtracting the height of the compression stress block from the stress in the compression reinforcement, By correcting this error at the level of the compression reinforcement, the locations of the section forces are established easily. So, the force in the compression reinforcement is expressed as
(4-31) The stress in the compression reinforcement is not known, but can be expressed as (4-32) The tension force is simply the area of tension reinforcement multiplied by the yield stress. Thus, establishing section equilibrium results in the following:
or
(4-33) In this expression, there are two unknowns: the neutral axis depth,c, and the stress in the compression reinforcement, The compression steel stress can be assumed to be linearly related to the compression steel strain, as expressed in the first part of Eq. (4-32).
Also, the compression steel strain is linearly related to the neutral axis depth given in Eq. (4-30). Thus, the section equilibrium expressed in Eq. (4-33) could be converted to a quadratic equation in terms of one unknown,c.
However, the solution of such a quadratic equation has two potential problems. First, after a value has been found for the neutral axis depth,c, a check will be required to con- firm the assumption that the compression steel is not yielding in Eq. (4-32). If the com- pression steel is yielding, Eq. (4-33) would need to be solved a second time (linear solution) starting with the assumption that The second, and more important potential problem, is that the engineer does not develop any “feel” for the correct answer.
What is a reasonable value for c? What should be done if cis less that the depth to the com- pression reinforcement,
To develop some “feel” for the correct solution, the author prefers an iterative solution for the neutral axis depth, c. With some experience this process converges quickly and allows for modifications during the solution. The recommended steps are listed below and described in a flowchart in Fig. 4-34.
1. Assume the tension steel is yielding,
2. Select a value for the neutral axis depth, c(start with a value between d/4 andd/3).
3. Calculate the compression steel strain,esœ,Eq. (4-30).
es Ú ey. dœ?
fsœ = fy. esœ, fsœ.
Asfy = 10.852fœcbb1c + Asœ1fsœ - 0.85fcœ2 T = Cc + Cs
fsœ = Esesœ … fy Cs = Asœ1fsœ - 0.85fcœ2 fsœ.
Cc = 10.852fcœbb1c = 10.852fcœba Cc,
4. Calculate the compression steel stress, Eq. (4-32).
5. Calculate the compression steel force, Eq. (4-31).
6. Calculate the concrete compression force, Eq. (4-13b).
7. Calculate the tension steel force,
8. Check section equilibrium, Eq. (4-33). If (difference less than 5 percent of T), then go to step 9.
(a) If increase cand return to step 3.
(b) If decrease c and return to step 3.
9. Confirm that tension steel is yielding in Eq. (4-18 to find ).
10. Calculate nominal moment strength,Mn,as given next.
es T 6 Cc + Cs,
T 7 Cc + Cs,
T Cc + Cs T = Asfy.
Cc, Cs,
fsœ,
1. Assume esey
2. Select value for c
4. fs Eses fy
5. Cs As(fs 0.85fc)
6. Cc 0.85 fcbb1c
7. T Asfy
8. Is TCc Cs No, TCc Cs
Yes
No, T Cc Cs
Decreasec Increasec
3. es ecu
cd c
9. es ecu ey
dc c
10. Mn Cc da Cs(dd) 2
Fig. 4-34
Flowchart for analysis of doubly reinforced beam sections.
As stated previously, this process quickly converges and gives the engineer control of the section analysis process. To answer the one question raised previously, if during this process it is found that cis less that the author recommends removing from the cal- culation because the compression steel is not working in compression. Thus, the section should be analyzed as if it is singly reinforced, following the procedure given in Section 4-4.
This will often happen when a beam section includes a compression flange, as will be dis- cussed in the next section of the text.
Once the process has converged and section equilibrium is established (step 8), and it has been confirmed that the tension steel is yielding (step 9), the section nominal moment strength can be calculated by multiplying the section forces times their moment arms about a convenient point in the section. For the analysis presented here, that point is taken at the level of the tension reinforcement. Thus,Tis eliminated from the calculation and the resulting expression for is
(4-34) where with defined previoulsly in Eq. (4-14).
Analysis of Strength-Reduction Factor,
The next step in the flexural analysis of a doubly reinforced beam section is to determine a value for the strength-reduction factor, so the value of can be compared with the factored design moment, that must be resisted by the section. The general procedure is the same for all beam sections. The value of the tension strain in the extreme layer of tension reinforcement, can be determined from a strain compatibility expression simi- lar to Eq. (4-18) with the distance to the extreme layer of tension reinforcement, used in place of d.
(4-35) If the value of is greater than or equal to 0.005, the section is tension-controlled, and If is less than or equal to 0.002, the section is compression controlled, and If is between these two limits, the section is in the transition zone, and Eq. (4-28a) can be used to determine the value for For tension-controlled sections, this process can be shortened if the value of calculated in step 9 of the section analysis process described previously, is found to be greater than or equal to 0.005. Because the value of is always greater than or equal to d, then will always equal or exceed and thus would be greater than 0.005.
Minimum Tension Reinforcement and Ties for Compression Reinforcement
Minimum tension reinforcement, which is seldom an issue for doubly reinforced beam sec- tions, is the same as that for singly reinforced rectangular sections, as given in Eq. (4-11).
As a beam section reaches in maximum moment capacity, the compression steel in the beam may buckle outward, causing the surface layer of concrete to spall off. For this reason, ACI Code Section 7.11 requires compression reinforcement to be enclosed within stirrups or ties over the length that the bars are needed in compression. The spacing and size of the ties is similar to that required for columns ties, as will be discussed in Chapter 11.
es et
dt
es, f. et
f = 0.65.
et f = 0.90.
et
et = ¢dt - c c ≤ecu
dt, et,
Mu,
fMn f,
f b1
a = b1c,
Mn = Ccad - a
2b + Cs1d - d¿2 Mn
Cs d¿,
Frequently, longitudinal reinforcement, which has been detailed to satisfy bar cutoff rules in Chapter 8, is stressed in compression near points of maximum moment. These bars normally are not enclosed in ties if the compression in them is not included in the calculation of the section nominal moment strength. Ties are required throughout the portion of the beam where the compression steel is used in compression when determining the nominal moment strength of a beam section. If the compression steel will be subjected to stress reversals, or if this steel is used to resist torsion, closed stirrups must be used to confine these bars. Details for closed stirrups will be discussed in Chapters 6 and 7 on design to resist shear and torsion.
EXAMPLE 4-4 Analysis of Doubly Reinforced Rectangular Beam Section
Compute the nominal moment strength, and the strength-reduction factor, for the doubly reinforced rectangular beam shown in Fig. 4-35. This beam section is very simi- lar to the section for Beam 3 of Example 4-3. For the beam section in Fig. 4-35, three No. 9 bars have been used as compression reinforcement, and a closed No. 3 stirrup-tie is used to help hold the top bars in place during casting. This example will demonstrate how beam section behavior can be changed by adding compression reinforcement. Assuming that the beam has a clear cover, we will assume the distance from the compression edge to the centroid of the compression reinforcement, is 2.5 in. The values for dand are the same as used for Beam 3 of Example 4-3. Assume the material properties are
and Recall that for the given concrete compressive strength and that the steel modulus
1. Use the iterative procedure discussed in the prior paragraphs to establish