Seismic Behavior of Reinforced Concrete Buildings_Sozen

13 Seismic Behavior of Reinforced Concrete Buildings 13.1 Int 13.1 ntrrod odu uct ctio ion n 13.2 13 .2 Es Esti tima mati ting ng Per Perio iod d Approximate Solution for the Period of a Reinforced Concrete Approximate Frame • Approximate Solution for the Period of a Building with a Dominant Reinforced Concrete Wall

13.33 Est 13. Estima imating ting Bas Basee She Shear ar Str Streng ength th Response of a Section Subjected to Axial Load and Bending Moment • Limit Analysis

13.4 13 .4 Es Esti tima mati ting ng Dr Drif iftt Linear and Nonlinear Response • Drift • Drift Determination for a Seven-Story Frame

13.5 Estimating Drift Capacity  An Example of Hysteresis for Reinforced Concrete • Sources Contributing to Capacity for Drift

13.66 Tran 13. ransv svers ersee Rein Reinfor forcem cement ent Axial Load • Combined Bending and Shear

Mete A. Sozen

13.7 Reinf 13.7 Reinforc orced ed Con Concr crete ete Wall allss 13.8 13 .8 Fu Futu turre Chal Challe leng nges es 13.9 13 .9 Co Conc nclu ludi ding ng Rem Remar arks ks

13.1 Introduction Reinforced concrete has had a less than perfect record in the earthquake environment. Nevertheless, major earthquakes of the past two decades have not revealed any surprises for reinforced concrete construction. It would appear that the major vulnerabilities have already been identified. We are not in a position to be able to say with confidence that “we have seen everything twice” with respect to behavior of reinforced concrete in earthquakes; however, it is not overly optimistic to believe that our current theory and experience should suffice to avoid the mistakes of the past. We understand the reasons for the vulnerabilities that we have observed in the field and in the laboratory, and we are able to avoid them by design. This chapter summarizes a few basic concepts for understanding the response of reinforced concrete to strong ground motion and the principles behind the rules used for proportioning and detailing.1 1

“Hands-on” examples in Imperial and SI units are provided in appendices to this chapter, posted on the Web site of this book.

© 2004 by CRC Press LLC

Because the concern is with design but the focus is on behavior, a note of caution is in order. Current building codes are prescriptive. It is important for the engineer to understand how the structure will respond to strong ground motion, but the engineer’s first responsibility is to satisfy the code of practice. The response of reinforced concrete structures to strong ground motion can be controlled through  judicious balancing b alancing of three ratios: 1. Ratio Ratio of of mass mass to to stiffn stiffness ess 2. Ratio Ratio of weigh weightt to stre strengt ngth h 3. Ratio of lateral lateral displac displacemen ementt to height height The stated ratios defy precise definitions. In earthquake-resistant design that is almost universally the case, starting with the definition of the ground motion. Earthquake-resistant design of reinforced concrete is closer to art than to science. One must expect the unexpected. The three ratios cited are simply vehicles for understanding and projecting experience. In this chapter, we will think of structures as if they were fixed at their bases and as if they existed in only one vertical plane. Neither notion is correct but they do serve to help us understand behavior. Our notional two-dimensional structure sitting on rigid ground is an approximation of the actual structure. No matter how exact our analysis is of the notional structure, the results are, at best, an approximation. The most convenient definition of the mass-to-stiffness ratio is the translational vibration period of  the notional structure corresponding to its lowest natural frequency. That is the quantity we will refer to whenever we invoke “period” unless we specify that it refers to a higher mode. For preliminary  proportioning, the period is the most important characteristic of a structure threatened by strong ground motion. The ratio of weight to strength is usually expressed in terms of the “base shear strength strength”” of the structure, the maximum base shear strength calculated for an arbitrary distribution of lateral forces applied at floor levels. The ratio of tributary weight to strength (usually called the “base shear strength coefficient”) is an approximate measure of how strong a structural system is in reference to its tributary load. It is not an important property of the structure as long as it is above a threshold value related to the groundmotion demand. The third ratio is more properly called “drift capacity” or the “limiting-drift ratio.” It can be defined as the capacity for the “mean drift ratio,” the roof drift divided by the height to roof above base, or the capacity for the “story drift ratio,” the lateral displacement in one story divided by the height of that story. It is a measure of the ability of the structure to distort without losing its integrity. For a given ground-motion demand, the period and the drift capacity are the two critical criti cal considerations for proper proportioning and detailing of a structure. The base shear strength coefficient is not likely to be important unless the engineer has made unreasonable choices (such as assigning an entire parking structure to two lateral-force resisting frames) or unless the peak ground velocity turns out to be high (more than one m/sec). The earthquake-resistant design of a reinforced concrete structure can be accomplished entirely by  use of appropriate software — as it should be. The material in the following sections is not intended for detailed design. Rather, it is intended for preliminary design, which ought to be accomplished without automatic devices, and for ensuring that the final design, obtained with the help of software, is reasonable. reasonable. The old adage — that an engineer should not accept a result from a computational tool he or she could not have guessed at to within ±20% — is most relevant for proportioning and detailing for earthquake resistance.

13.2 Estimating Period The period of a notional structure is an important index that identifies vulnerability to excessive drift. Even within its constrained domain defined above (lowest mode of a two-dimensional building based on unyielding ground), it may have different meanings depending on how it has been obtained.

© 2004 by CRC Press LLC

Because the concern is with design but the focus is on behavior, a note of caution is in order. Current building codes are prescriptive. It is important for the engineer to understand how the structure will respond to strong ground motion, but the engineer’s first responsibility is to satisfy the code of practice. The response of reinforced concrete structures to strong ground motion can be controlled through  judicious balancing b alancing of three ratios: 1. Ratio Ratio of of mass mass to to stiffn stiffness ess 2. Ratio Ratio of weigh weightt to stre strengt ngth h 3. Ratio of lateral lateral displac displacemen ementt to height height The stated ratios defy precise definitions. In earthquake-resistant design that is almost universally the case, starting with the definition of the ground motion. Earthquake-resistant design of reinforced concrete is closer to art than to science. One must expect the unexpected. The three ratios cited are simply vehicles for understanding and projecting experience. In this chapter, we will think of structures as if they were fixed at their bases and as if they existed in only one vertical plane. Neither notion is correct but they do serve to help us understand behavior. Our notional two-dimensional structure sitting on rigid ground is an approximation of the actual structure. No matter how exact our analysis is of the notional structure, the results are, at best, an approximation. The most convenient definition of the mass-to-stiffness ratio is the translational vibration period of  the notional structure corresponding to its lowest natural frequency. That is the quantity we will refer to whenever we invoke “period” unless we specify that it refers to a higher mode. For preliminary  proportioning, the period is the most important characteristic of a structure threatened by strong ground motion. The ratio of weight to strength is usually expressed in terms of the “base shear strength strength”” of the structure, the maximum base shear strength calculated for an arbitrary distribution of lateral forces applied at floor levels. The ratio of tributary weight to strength (usually called the “base shear strength coefficient”) is an approximate measure of how strong a structural system is in reference to its tributary load. It is not an important property of the structure as long as it is above a threshold value related to the groundmotion demand. The third ratio is more properly called “drift capacity” or the “limiting-drift ratio.” It can be defined as the capacity for the “mean drift ratio,” the roof drift divided by the height to roof above base, or the capacity for the “story drift ratio,” the lateral displacement in one story divided by the height of that story. It is a measure of the ability of the structure to distort without losing its integrity. For a given ground-motion demand, the period and the drift capacity are the two critical criti cal considerations for proper proportioning and detailing of a structure. The base shear strength coefficient is not likely to be important unless the engineer has made unreasonable choices (such as assigning an entire parking structure to two lateral-force resisting frames) or unless the peak ground velocity turns out to be high (more than one m/sec). The earthquake-resistant design of a reinforced concrete structure can be accomplished entirely by  use of appropriate software — as it should be. The material in the following sections is not intended for detailed design. Rather, it is intended for preliminary design, which ought to be accomplished without automatic devices, and for ensuring that the final design, obtained with the help of software, is reasonable. reasonable. The old adage — that an engineer should not accept a result from a computational tool he or she could not have guessed at to within ±20% — is most relevant for proportioning and detailing for earthquake resistance.

13.2 Estimating Period The period of a notional structure is an important index that identifies vulnerability to excessive drift. Even within its constrained domain defined above (lowest mode of a two-dimensional building based on unyielding ground), it may have different meanings depending on how it has been obtained.

© 2004 by CRC Press LLC

A few of the possible definitions of period for a reinforced concrete building structure may refer to the: 1. Measured Measured or estimated estimated (based on measured data of similar buildings) period period of the entire building including the effects of nonstructural elements 2. Calculated Calculated period period based on gross gross sections sections of the structural structural elements elements 3. Calculated Calculated period period based on cracked cracked sections sections of the structural structural elements elements 4. Calculated Calculated period period based on gross or cracked cracked sections sections but consideri considering ng also the effects effects of the nonstructural elements One must be discriminating when a period is mentioned in relation to design. In this chapter, we will use the second definition and caution the reader whenever we deviate from it. A reinforced concrete structure does not have a unique period — neither before an earthquake because of time-dependent changes nor during an earthquake because of amplitude-dependent changes in its stiffness. We do not pretend that the period we calculate is an actual attribute of the building. Its importance is due to the fact that an experienced engineer can tell from the calculated period of the structure (for a given/assumed earthquake demand) whether there will be problems with drift. It is simply an index to the mass/stiffness properties of the building structure.

13.2.1 13.2.1 Approxima Approximate te Soluti Solution on for for the the Period Period of a Reinforce Reinforced d Concret Concretee Frame Frame A preliminary estimate of the period of a building with reinforced concrete frames acting to resist earthquake demand can be obtained using the time-honored expression T  =

 N  10

(13.1)

where T is the period in seconds and N is the number of stories. Equation 13.1 was used in the Uniform Building Code for many years until it was replaced (ICBO, 1997) by  T

=

3 0.03h 4

(13.2)

where h is the height of the reinforced concrete frame above its base in feet. Both of the above expressions are based on data from and analyses of buildings in California. They  are applicable as long as the dimensions d imensions of the frame elements are comparable to those used in California. It is not reasonable to assume that either expression will provide an acceptable approximation to the period of a structure in, say, Paducah, Kentucky. If it is desired to examine the effect on period of changes in values and distributions of the member stiffnesses and story masses, Equations 13.1 and 13.2 do not provide help. Use of a particular software system is intellectually the least demanding option. However, However, there are occasions when the software does not produce a value that appears reasonable. At those times, the method to use is the Rayleigh Principle (1877). The Rayleigh Principle states that the energy of a vibrating system is conserved. In reference to the single-degree-of-freedom (SDOF) oscillator with stiffness k in Figure 13.1, it can be stated that at maximum displacement, X max, where the mass, M, is momentarily at rest, the potential energy is 0.5 k*X max 2.

(13.3)

When the mass is at its initial position, the velocity is at maximum and can be expressed as w*X max , for simple harmonic motion. The corresponding kinetic energy is 0.5 M*(wX max )2.

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(13.4)

Hatched area = Potential Energy X max  X max 

Displacement

Mass

X max 

Stiffness, k 

kX max 

Displacement

Velocity 2

Force

X max 

X max 

Acceleration FIGURE 13.1

Response of an SDOF oscillator in harmonic motion. Shear spring representing story stiffness Story mass

7 @ 3.65 = 25.5 m

3 @ 9.15 = 27.45 m FIGURE 13.2

Seven-story frame and its “shear beam” representation. representation.

Equating the kinetic to the potential energy maxima, the circular frequency is obtained as

w=

k  M 

(13.5)

in radians per second and the period T in seconds is obtained as 2p/v. Rayleigh suggested that the method would also work for systems with many degrees of freedom as long as the assumed deflected shape was a good approximation of the correct modal shape. To apply the Rayleigh principle to a frame, we will use the numerical procedure by Newmark (1962). Consider a single two-dimensional frame of a seven-story building (Figure 13.2). The overall dimensions are indicated in the figure. Other relevant properties are: Young’s modulus Column dimensions Width Dept Depth h (in (in plan planee of of fra frame me)) Girder dimensions Width Depth Tributary mass at each level

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25,000 MPa 0.60 m 0. 0.660 m 0.45 m 0.90 m 145 ton-metric

To simplify the calculations, we will ignore the increase in stiffness at the joints (wire frame). We approximate the frame by a “shear beam” or a string of concentrated masses connected by shear springs illustrated in Figure 13.2 by concentrating the tributary weight at the floor levels and defining story  stiffnesses. Whereas the story stiffness can be determined easily if the girders are assumed to be rigid, in frames with long spans this choice may lead to underestimating the period. To include the effect of girder flexibility, we use an old expression dating from the early design methods for wind (Schultz, 1992). For intermediate stories, the story stiffness k typ is defined by 

ktyp

= 24 ◊

E c  2

H 

1

◊

Ê  Á Á ÁË 

ˆ  + + n n n ˜  ˜  kc  k gb k ga ˜  ¯  i =1 i =1 i =1 2

c

1

gb

1

(13.6)

ga

  Â

k typ = stiffness of a story with flexible girders above and below the story  Ec = Young’s modulus for concrete H = story height nc = number of columns k c =

I c 

H  Ic = moment of inertia of prismatic column ngb = number of girders below  k gb =

I  gb

L Igb = moment of inertia of prismatic girder below  L = span length nga = number of girders above Iga = moment of inertia of prismatic girders above i = number identifying column or girder To determine the stiffness for the first story, the stiffnesses of the girders below, k gb, are assumed to be infinitely large, with belief in the notion that the footings are inert. Accordingly, the second term in the denominator drops out. The moments of inertia for the columns and the girders as well as the story stiffnesses are determined in Calculation Sheet A in Appendix A_SI2. Axial and shear deformations are neglected. The spreadsheet format (Table A1) offers a very convenient platform for implementing the calculations. The procedure is simple. We assume a deflected shape. The story deflections can be determined arbitrarily but it is helpful if an approximation to the first-mode is selected. In Table A1 the assumed initial story drifts are assumed to vary linearly with height (column D). Mass and stiffness (columns B and C) are entered as coefficients of values listed in cells D17 and D18. The steps in the arithmetic are described in the glossary  that is included in Appendix A. The period from the first iteration is 0.7 sec. (It is usually sufficient to determine a structural period to one tenth of a second.) In iteration 2, we start with the deflected shape resulting from the first iteration. We note that the second iteration does not lead to a change in period, at least not in terms of one tenth of a second. We should note that this is not necessarily always the case. If the masses and the story  stiffnesses vary, satisfactory convergence may require a few iterations.

2

The material in Appendix A is repeated in Imperial and SI unit systems.

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7 @ 12 ft = 84 ft

FIGURE 13.3

Frame with dominant wall.

For the frame analyzed, Equations 13.1 and 13.2 lead to periods of 0.7 and 0.8 sec, respectively. For a “regular” frame with members having appropriate sizes and reasonably uniform distribution of typical story heights, the two simple expressions provide satisfactory results and we could have used their results as a frame of reference. The advantage of the calculation we made is that it allows us to get a sense of  the effects of changes in stiffnesses and it provides us with a satisfactory deflected shape for the first mode. Both are very useful in preliminary proportioning of reinforced concrete frames. Tables A2 and A3 illustrate changes in period and modal shape for different stiffness distributions. A reduction in the first-story stiffness, which may be attributed to foundation flexibility and cracking in the column, has a strong effect on the period and the mode shape. Table A3 suggests that increasing girder stiffness is an effective way to reduce the period. The tabular form enables quick investigations of  effects of changes in stiffness in any and all members on period.

13.2.2 Approximate Solution for the Period of a Building with a Dominant Reinforced Concrete Wall In this section, we consider the period of a structure with a wall (Figure 13.3) that dominates the response such that we can neglect the contribution of the frame to its stiffness. If the wall is prismatic and the story masses are reasonably uniform over the height of the building, a simple and blunt approximation is provided by  T W  =

 N  20

(13.7)

where Tw  is the period of the building and N is the number of stories. Although Equation 13.7 provides a good target for the period, it is likely to underestimate the period unless the wall fills the profile of the building and has a tributary area defined by a width not exceeding approximately 40 ft. Another simple and direct procedure would be to assume the building to be represented as a uniform cantilever wall with the mass distributed uniformly over its length. For this condition, the period is 2◊ p

T w  = 3.5 ◊

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E c ◊ I w  m ◊ H 4

(13.8)

Ec Iw  m H

= Young’s modulus for concrete = moment of inertia of wall = unit mass assumed to be the total tributary mass divided by height = height of wall

Consider a seven-story structure with its lateral stiffness provided by a prismatic wall that is slender enough to permit defining its stiffness based on its bending flexibility. The relevant properties are assumed as follows: Young’s modulus Tributary mass for each level Unit mass Moment of inertia Total height (uniform story height)

25,000 MPa 145 ton-metric 40 ton/m 15.5 m4 25.55 m

We obtain preliminary estimates, correct to one tenth of a second, from Equation 13.7 Tw  = 0.4 sec and from Equation 13.8 Tw =0.4 sec The result from Equation 13.8 suggests that the assumed parameters for the wall are within the experience on which Equation 13.7 is based. Next, we use the Rayleigh principle as illustrated in Table A4, Appendix A_SI. The arithmetic involved is described in the annotations. The calculated period, to one tenth of a second, is the same as that from the one determined using the simple equations. The only additional information we have is a new  approximation to the deflected shape. The spreadsheet solution provides us with a convenient tool to understand the effects of changes in structural properties. We know that if the wall is cracked uniformly over its full height to have a stiffness of, say, 20% of its initial stiffness, we can estimate its period using Equation 13.5 as a guide for estimating the effect of a reduction in stiffness of 5 Twcracked = Tw  *

5

or approximately 0.9 sec. But what happens if the stiffness is reduced only at the lowest three levels? Table A5 in Appendix_SI gives us the approximate answer and the understanding that changes in the stiffness of the lower levels, by design or by accident, are critical for the period. What about such changes in the top three levels? Table A6 provides an answer. The effect on period is negligible. These exercises prepare us for making quick estimates of the relative drifts of competing framing systems.

13.3 Estimating Base Shear Strength 13.3.1 Response of a Section Subjected to Axial Load and Bending Moment The assumptions and procedures for determining the moment curvature relationship for reinforced concrete have been presented in many textbooks and notably in Chapter 5 of the book (see Blume et al. 1961). The reader is referred to that document for detailed background information.

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60 N0.4

2 in. Conc. Comp. Strength 4070 psi Yield Stress 53.3 ksi 10 in.

12 in. N0.6 6 in.

3 ft

3 ft

3 ft

50   s   p    i    k   n    i    d   a   o    L    d   e    i    l   p   p    A    l   a    t   o    T

40

30

20

10

0

0

1

2

3

4

5

Mid-Span Deflection in in.

FIGURE 13.4

Measured load-deflection relationship for a reinforced concrete beam loaded at third-points of its span.

For sections with proportions appropriate for earthquake-resistant structures, the flexural strength can be approximated closely using no more than the principles of equilibrium and a rudimentary  knowledge of the strengths of the materials involved. Determination of the unit curvature for a specified strain is also easy. However, establishing the limiting unit curvature is difficult because it is sensitive to many parameters. To boot, using this limit to determine rotation is even more difficult. That is why it is preferable to handle the problem of toughness in design by controlling parameters such as the mean unit stress on the section and the amounts of longitudinal and transverse reinforcement. An explicit and accurate calculation of the limiting rotation that is universally applicable under cycling loading is not yet within reach. Before we start with the details of the computational process, it is instructive to examine the observed flexural response of reinforced concrete members under two different conditions: (1) in a span with no moment gradient and (2) in a span with moment gradient. The measured moment–deflection response of a reinforced concrete member subjected to bending moment only is shown in Figure 13.4 (Gaston, 1952). Ideally, the moment–deflection curve exhibits three stages identified by different slopes or stiffnesses: (1) Stage 1, before flexural cracking, (2) Stage 2, after flexural cracking and before yielding and (3) Stage 3, after yielding. We must remember that the section considered has specific properties. It is reinforced moderately to develop yielding of tensile reinforcement before the compressed concrete reaches its limiting strain in compression. The reinforcement is concentrated at one layer in the tension and compression flanges. The properties cited are representative of  those of members used in earthquake-resistant structures. However, we are considering a case with monotonically increased moment. That is not typical loading for a structure subjected to strong ground motion. The three stages are convenient for describing the general force–displacement properties of a reinforced concrete section but we must not conclude that they would all appear in every cycle of response of a member subjected to moment reversals. For the case of monotonically increased load, we note that the slope in Stage 3 is close to “flat.” If we ignore Stage 1, which we can do justifiably in all but very lightly reinforced members, we can think of  the moment–deflection curve as elasto-plastic. Furthermore, we can notice the reflection of the nearly  elasto-plastic stress–strain curve for the reinforcement in the moment–deflection curve. The measured moment–deflection response of a reinforced concrete member subjected to combined bending and shear is shown in Figure 13.5 (Wight, 1973). For this element the result of one complete cycle of load is shown. The striking feature is that the moment–deflection curve does not have a flat slope in Stage 3 of the first loading to a maximum and the shape of the curve for the reversal does not lend itself well to idealizing it as a straight line. © 2004 by CRC Press LLC

100 80

2 in.

60 40

10 in.

12 in. No. 6

   N    k  ,   r   a   e    h    S

6 in.

20 0 20 40 60 80

3 ft 4.5 in. Conc.Comp. Strength Yield Stress

FIGURE 13.5

100

3 ft 4.5 in.

120 80

4850 psi 72 ksi

60

40

20

0

20

40

60

80

Drift, mm

Load–deflection curve for a reinforced concrete beam with moment constantly changing along its span

10   n    i   a   r    t 8    S    d    l   e    i    Y    / 6   n    i   a   r    t    S 4    d   e   r   u   s 2   a   e    M

0

0

1

2

3

4

Measured Deflection/Yield Deflection

Measured moment–deflection relationship showing the “jump” phenomenon in reinforcement strain as the applied moment exceeds the yield moment in a span with changing moment. FIGURE 13.6

Why is the “plastic” portion of the load–displacement relationship lost even in the first loading to a. maximum? The data in Figure 13.6 reveal the reason. Measured tensile-reinforcement strains, as a coefficient of the yield strain, are plotted against the measured deflection at load point (as a coefficient of the deflection at yield). At the yield deflection, the tensile strain in the reinforcement increases with hardly any increase in deflection. In effect, the reinforcement strain goes abruptly from the yield strain to the value at which strain hardening begins. The flat portion of the stress–strain curve for the reinforcement is thus not perceptible in the load–deflection relationship. Increase in deflection beyond yield requires an increase in moment. Members resisting earthquake effects are typically subjected to moment gradients. We can and will use the elasto-plastic response to help us simplify certain concepts of analysis, but we should not assume that it is correct. And we should remember that the elasto-plastic response is likely to be less correct for cyclic loading. In this section we will first consider the moment–curvature relationship as a vehicle for understanding some of the design rules and not as a direct tool for design. For a moderately reinforced section subjected to constant axial load and monotonically increasing moment, a representative moment curvature relationship is shown in Figure 13.7. Engineering literature contains considerable material on the construction of the moment–curvature diagrams for reinforced concrete sections. It is almost impossible not to think that the topic is at once © 2004 by CRC Press LLC

1.2

   d    l   e    i    Y    t   a    t 0.8   n   e   m   o    M    /    t 0.4   n   e   m   o    M

0.0 0.0000

0.0004

0.0008

0.0012

Curvature, 1/in FIGURE 13.7

Calculated relationship between bending moment and unit curvature.

5000    i   s   p  ,   s   s   e   r    t    S   e   v    i   s   s   e   r   p   m   o    C

4000 3000 2000 1000 0 0.000

0.001

0.002

0.003

0.004

Unit Strain FIGURE 13.8

The Hognestad stress–strain relationship for unconfined concrete in compression.

very well understood and very important. Actually, the relationship between moment and curvature under reversals into the nonlinear range of response is well beyond our ability to determine to the degree of exactness often implied. And fortunately, safe design does not require exact calculation of the relationship between moment and curvature. The critical factor is the detailing of reinforcement. However, understanding the moment–curvature relationship under monotonically increasing moment is helpful to understanding behavior. We start with the assumption that the reader is familiar with Euclidean geometry and the conditions of  equilibrium, two theoretical concepts necessary for relating internal stresses to external forces at a section. We ignore the effects of the tensile strength of concrete. It is indeed negligible unless the amount of  reinforcement is negligible. The concept for determining the moment–curvature relationship for a reinforced concrete section is simple and direct. For a given section subjected to moment about one axis, two sets of operations are involved. The first set includes those that define the section, the materials and the axes about which moments are traditionally defined. (The moment can be defined about any axis, provided it is consistent with the definition of the external moment.) 1. Assume a relationship between unit stress and unit strain for compressed concrete. An example for unconfined concrete is shown in Figure 13.8 (Hognestad, 1951). 2. Assume a relationship between unit stress and unit strain for the reinforcement. An example is shown in Figure 13.9. 3. Determine the location of the plastic centroid. The plastic centroid is defined as the centroid of  the maximum forces in the materials constituting the section. An example is provided in Figure © 2004 by CRC Press LLC

800   a    P 600    M  ,   s   s   e   r 400    t   s    t    i   n    U 200

0 0.00

0.02

0.04

0.06

0.08

0.10

Unit Strain FIGURE 13.9

Representative idealized stress–strain relationship for steel reinforcement.

d c 

b 

As 1 y  d t 

h 

As 2

dc (As1 fy) + dt (As2 fy) + bhf c y=

h 2

(As1 fy + As2 fy + bh)

fy: yield stress of reinforcement f c: compressive strength of concrete in the section The plastic centroid has been determined using “gross section.” FIGURE 13.10

Definition of plastic centroid for a rectangular reinforced concrete section.

13.10. For a symmetrical section symmetrically reinforced about the bending axis, the plastic centroid coincides with the geometric centroid. The second set of operations are the ones that are repeated until the correct solution is obtained. 1. Assign a strain Abs(ec)