# Serviceability of Reinforced Concrete Beams

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### Introduction

In addition to meeting requirements of flexural strength and ductility, reinforced concrete beams must meet serviceability requirements related to rigidity (such as deflection limits) and durability (such as crack width limits).

Serviceability issues are treated differently from the strength and ductility issues described in the Flexural Design of Reinforced Concrete Beams article in two important ways. First, serviceability limits employ unfactored loadings, which are known as the service loads. Second, behavior is assumed to be within the linear elastic stress range. The following sections summarize and illustrate the ACI 318 serviceability provisions for beams.

### Concepts and Formulas

#### Linear Elastic Behavior

In a properly designed reinforced concrete beam, the steel yields well before the concrete crushes. If the concrete were to crush before the steel yielded, failure would occur suddenly and without warning.

A properly designed beam, then, achieves its moment strength, Mn, by the yielding of its extreme tension steel. When the concrete in the beam crushes (that is, reaches its assumed ultimate strain of 0.003), steel strains are usually in excess of 0.005.

Figure below shows the relationship between moment and midspan deflection for a typical beam loaded to flexural failure.

Initially, the beam is uncracked and the response is essentially linearly elastic, with stresses resisted by the gross section. Cracking is predicted to occur when the maximum tension stress reaches the modulus of rupture, fr. For purposes of serviceability checks, the value of the modulus of rupture used by ACI Sec. 9.5 is

$f_r=7.5\sqrt{f'_c}=\frac{M_{cr}y_t}{I_g}$

$M_{cr}=\frac{7.5\sqrt{f'_c}I_g}{y_t}$

Ig here is the moment of inertia of the gross concrete section about its neutral axis, yt is the distance from the neutral axis to extreme tension fiber prior to cracking, and Mcr is the cracking moment.

After the section cracks, tension is resisted only by the steel, and the neutral axis shifts to a new position. Within the service load range, the member continues to behave linearly under short-term loading, but the moment of inertia is markedly lower than it was for the section before it cracked. To calculate deflection under short-term loading, ACI 318 employs an effective moment of inertia, Ie, that weights the gross and cracked moments of inertia.

$I_e=(\frac{M_{cr}}{M_a})^3 I_g + (1-(\frac{M_{cr}}{M_a})^3)I_{cr} \leq I_g$

Ma is the maximum service load moment ever applied to the beam, and Icr is the moment of inertia of the cracked section.

The section consists of two materials, steel and concrete, that have different properties. To simplify the calculation of elastic stresses and deformations, one material in the section can be replaced with an equivalent area of the other material—that is, the area that would give the same properties—resulting in a fictitious section composed of a single homogeneous material that has the same properties as the actual section. By assuming that a complete bond exists between steel and concrete and that strains must vary linearly through the depth of a section, the steel area can be replaced by an equivalent area, Atr, as shown in Figure above.

$f_sA_s=f^*_sA_{tr}$

$A_{tr}=(\frac{E_s}{E_c})A_s=nA_s$

n is the modular ratio, which is the modulus of elasticity of steel, Es, divided by the modulus of elasticity of concrete, Ec.

In practice, the modular ratio is usually rounded to the nearest integral value; if the calculated value of n is, say, 9.34, it is usually taken as n = 9. Multiplying the actual steel area, As, by the modular ratio replaces the steel with an equivalent strip of concrete smeared across the section at depth d. After this replacement, the usual equations of solid mechanics apply to stresses and deflections. Note, however, that the computed stress, f*s, is a fictitious stress acting over the strip Atr. The actual steel stress is obtained by multiplying the computed stress by n.

$f_s=nf^*_s$

#### Long-Term Behavior

Beams with non-rectangular cross sections and those that contain steel in the compression region are analyzed using basic principles in a manner similar to that illustrated in the previous section. For beams containing steel in the compression region, long-standing practice is to use a modular ratio of 2n to account for the effect of creep deformation in the concrete (in effect, taking the concrete modulus of elasticity as half the instantaneous modulus). The current ACI 318 approach, however, applies a multiplier, λ, to the immediate deflection to account for the long-term deformation.

$\lambda = \frac{\xi}{1+50 \rho '}$

In this formula,

$\rho ' = \frac{A'_s}{bd}$

ξ is an empirical factor to account for the rate of additional deflection = 1.0 for loads sustained 3 mo, 1.2 for loads sustained 6 mo, 1.4 for loads sustained 12 mo, and 2.0 for loads sustained 5 yr or longer.

#### Durability Issues

In many respects, steel and concrete are ideal materials to use in combination. For one, the coefficient of thermal expansion is approximately the same for both, so temperature changes do not induce significant stresses. For another, properly proportioned concrete is practically inert in harsh environments and with adequate cover can protect the steel against corrosion and high temperatures. To ensure this protection, ACI 318 sets limits on certain concrete ingredients and admixtures— especially those containing chloride ions—and in ACI Sec. 7.7 appropriate minimum clear covers are prescribed. These minimums depend on the type of member, the service environment, and the diameter of the reinforcing bars.

To enhance the durability and in some cases the appearance of members, ACI 318 prescribes rules on crack control and distribution of flexural steel. ACI Sec. 10.6 gives the following rules.

1. According to ACI Sec. 10.6.4, the spacing of reinforcement closest to the tension face of a flexural member must not exceed
$s\leq\left\{\begin{matrix} 15(\frac{40,000}{f_s})-2.5c_c\\ 12(\frac{40,000}{f_s}) \end{matrix}\right.$
s is the center-to-center spacing, fs is the calculated service load stress in the steel, and cc is the least distance from surface of reinforcement to the extreme tension face. In lieu of calculating fs using linear elastic theory, ACI 318 permits the stress to be approximated as 2fy/3.

2. Where the flanges of T-beams are in tension, some of the tension reinforcement must extend over a width defined as the lesser of the effective flange width and one-tenth of the span.

3. Where the height of a flexural member, h, exceeds 36 in, longitudinal reinforcement is required on both faces, and must be spread over a distance 1/2 (half the effective depth) from the tension face with a maximum spacing as given above, except that cc in this case is the distance from surface of reinforcing to nearest face of member (see Fig. R10.6.7 in ACI 318).

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