Ail flexural members must meet the deflection control requirements of AC1 318-95, Section 9.5. Unless the thickness of a beam or one-way slab satisfies the minimum thickness-span ratios given in AC1 3 18-95 Table 9.5(a), long-time deflections must be computed to prove that they are smaller than or equal to the maximum limits given in AC1 318-95 Table 9.5cb). If a member supports or is attached to construction likely to be damaged by large deflections, deflections must be checked even if the requirements of Table 9.5(a) are satisfied. Immediate deflections are to be computed by elastic analysis, using for the moment of inertia the effective moment of inertia fe of the cross section as determined from AC1 3 18-95 Eq.
(9-7). The effective moment of inertia is not to be taken larger than the gross moment of inertia íg.
DEFLECTION 1 through DEFLECTION 5.1 in these design aids have been provided to help in the aigebraic evaluation of the effective moment of inertia as given in DEFLECTION 5.2 combines the several steps involved into one design aid, employing a graphical approach to evaluate the effective moment of inertia for rectangular beams with tension reinforcement only.
DEFLECTION 6.1 provides the moment coefficients of the elastic deflection formulas for the most common cases of loading for simple and continuous spans, and DEFLECTION 6.2 is intended to help in the computation of the immediate deflection by combining the span and the modulus of elasticity of the concrete into one factor.
The evaluation of the immediate deflection for a flexural member is, in spite of all helpful design aids, still a somewhat cumbersome procedure. For this reason, DEFLECTION 7 has been provided to help the designer in obtaining an approximate immediate deflection with relatively little effort. These tables should prove to be of great help, especially during the design stage.
All deflections evaluated with the help of DEFLECTION 1 to DEFLECTION 7 are immediate deflections occurring instantaneously upon each load application.
According to AC1 318-95, Sections 9.5.2.5 and 9.5.3.4, all immediate deflections due to sustained loads shali be multiplied by a factor as given in Section 9.5.2.5 to obtain the additional long-time deflection (creep and shrinkage part). When the creep and shrinkage part is added to the immediate live-load deflection, the total must be below the maximum limits given in Table 9.5(b).
DEFLECTION 8 simplifies the evaluation of the factor given in Section 9.5.2.5 to obtain the additional long-time deflection of a flexural member.
DEFLECTION 9 furnishes the modulus of elasticity as computed for the strength and weight of the concrete used.
While tables giving several significant figures may Eq. (9-7).
suggest a high degree of accuracy is necessary in computation of deflections, such is not the case. The user is reminded that even under laboratory controlled conditions for simply supported beams, "there is approximately a 90 percent chance that the deflections of a particular beam will be within the range of 20 percent less than to 30 percent more than the calculated value, " In' view of this degree of accuracy, interpolation in the use of deflection tables is generally unnecessary.
DEFLECTION 1.1
This chart gives the cracking moments in kip-feet for rectangular sections of various thichesses h but of constant width b = 1 in. for f, ' values from 3000 to 7000 psi.
According to AC1 318-95, Section 9.5.2.3, Eq. (9-8) and (9-9)
For rectangular sections,
Expressed in kip-feet,
Mu = b Ku
where
( - ) h'
6 f ,
= (h*)- t'r: kip - fi per inch of width 9600
AC1 318-95 Section 9.5.2.3(b) specifies that fr shall be multiplied by 0.75 for "all-lightweight" concrete, and 0.85 for "sand-light weight" concrete (for the circumstance where f,, is not specified). This means that IC, from DEFLECTION 1.1 must be multiplied by these values when
"all-lightweight" or "sand-lightweight" concrete is used unless f,, is specified. If f, is specified, multiply II;, by the ratio f,, / (6.7 #) 5 1, in accordance with AC1 318-95 Section 9.5.2.3(a).
' AC1 Committee 435, "Variability of Deflections of Simply Supported Reinforced Concrete Beams," AC1 JOURNAL, Proceedings V. 69, No. 1, Jan. 1972, p.35.
470
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4 b
h
-i______-- - bw I
Fig. DE-1 - Illustration of y, for positive and negative moments
DEFLECTION 1.2 AND 1.3
These charts provide the coefficient K,,,, which relates the cracking moment of a T-section to that of a rectangular section having the same width as the T-section web (b)" ) and the same thickness as the overall thickness of the T-section (h ). K,, can be used in conjunction with coefficient Krr from DEFLECTION 1.1, which is the cracking moment for a l-in. width of rectangular beam.
Multiplying K,, by web width and K,, computes the cracking moment of a T-beam.
DEFLECTION 1.2 provides values of K , for tension at the bottom (positive moment), and DEFLECTION 1.3 provides values for tension at the top (negative moment).
M, = r IE AC1 318-95 Eq. (9-8)
Y,
y, = distance from centroid to tension face, as illustrated in Fig. DE-1.
For a rectangular section
Ig bw h 2
YI 6
For a T- or L-section - - - -
Dividing by 12,000 to change units from pound-inches to
kip-feet , \
f , h 2
or since Kr (from DEFLECTION 1.1) = -
72,000 Mm = bw K, K, , kip-ft
The equations by which K, was evaluated are shown on the Design Aids.
DEFLECTION 2
This chart gives thi: moment of inertia i,, of cracked rectangular sections with tension reinforcement oniy, for various modulus of elasticity ratios n and reinforcement ratios p . Since this condition represents a special case of the derivation for the moment of inertia of a cracked rectangular section with compression and tension reinforcement, refer to the commentary for DEFLECTION 4 where the general case is treated.
B Ei (n-I)A's 1Lbrl nAs
Fig. DE-2 - Internai forces and strains in cracked rectangular section with tension and compression reinforcement Compression force in concrete C, = (bc / 2 - A',) f, = (bc / 2 - p'bd) E, E,
Compression force in compression reinforcement C, = A',f , = p'bda',E, Tension force in tension reinforcement T = A$, = pbdo,E,
Modulus of elasticity of steel E, = 29 X lo6 psi
471
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DEFLECTION 3
This chart provides coefficient Ki, for obtaining the gross moment inertia for an uncracked T-section as
Ig = Ki, (b, h-1f 12)
This formula for Ki, is given on DEFLECTION 3.
DEFLECTION 4
This table furnishes the coefficient Ki2 for calculating the moment of inertia of cracked rectanguiar sections with tension and compression reinforcement, and can also be used for the evaluation of the moment of inertia of cracked T-sections. Because deflection is the response of a structure at service loads, the derivations have been based on the linear relationship between stress and strain.
For a rectangular section with compression reinforcement as illustrated in Fig DE-2, we have from the equation of equilibrium,
T = C , + C , or T - C , - C , = O pbdE$, - p ' b d ~ , ' E , - (bc I 2 - p'bd)~&, = O
Let E, = n E,. The term p'bdEJc represents the reduction in C, caused by displacement of concrete by reinforcement. At the levei of compression reinforcement, E , = E , ' . Substituting E,' for E, in this tem only,
pbâE,nE, - p'bú~,'nE, - (bc I 2 ) ~ & - p'bd~,'E, = O
= o
n - I CEc de, - E d e : ( - -
P 2Pn
I
Let Then
ò, = p ' ( n - 1) I pn
Since E,' = E,(C / d - d' I d) / c Id and E , = ~ ~ ( 1 - c I d) I c Id,
1 - c / d - / d
( c i dc;
c l d òc
From this c / d becomes:
cld = ,/[pn(l + òJ' + 2pn(l + ò, d l I d) - pn(1 + òc)
Assuming that a crack extends to the neutral axis of the section, the moment of inertia of the cracked section about the neutral axis is the sum of the moment of inertia of the uncracked portion of the concrete section (less the area occupied by compression reinforcement) plus the moments of inertia of the compression and tension reinforcement areas, with aíIowance for the relative moduli of elasticity of concrete and steel. Thus
+ (n - l)p'bd(c - d)'
(e I d)' - 2c Id- +
d
Fig. DE-3--Illustration showing how DEFLECTION 4 can be used to determine moment of inertia of cracked T-section by considering capacity of overhanging flange as compression reinforcement and redefining ò, as
h, I d
" P W
P, = [ ; - 1 ) -
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DEFLECTION 4 gives the values of Ki2 for different p, values. To find the moment of inertia of the cracked rectangular cross section, determine pn, d' i d and p, for the section under consideration to find Ki2 which has to be multiplied by b&.
re = [ 3-J3 Ig +
I,, = Ki2 bd'
3
1 - [ 2 ) I-
The most common case of a beam with tension reinforcement only is a special case of this derivation in which p, = O.
IC7 - (c 1 4'
- - - + pn(1-2cid +(c í 4' ) = Ki,
bd' 3
where; cid = {pz n z + 2 p n - pn
The moment of inertia of a cracked rectangular cross section with tension reinforcement only becomes then
I,, = K i , b&
The K,, values can be read from DEFLECTION 2 for ratios of n = E , / E, and p = A, / b d.
DEFLECTION 4 can also be used to determine the moment of inertia of a cracked T-section by considering the capacity of the overhanging flange portion as a kind of compressive reinforcement and by consequently redefining p, as shown in Fig DE-3. Approximating the stress in the flange's overhang by that at its middepth and finding an equivalent amount of reinforcement ple at d' = h, / 2 , we see from the diagram that
Since E,' = E , and E, = n E,, or
Since
(b - bW)+$, = P ' ~ b&,'E, - A,'€$, (b - bw)hpE, = P ' ~ b,,dW, - E,)
( b / k v - l)(hf/ 6) = (n - 1 ) ~ ' ~
we obtain
ò c = [ $ 1 ) - h, ớ d
"P"
This expression is used in DEFLECTION 4 to arrive at a substitute for compression reinforcement for the overhanging flanges.
DEFLECTION 5.1 AND 5.2
According to Section 9.5.2.3 of AC1 318-95, the immediate deflection of a flexural member is to be based on the "effective" moment of inertia of the cross section which is given by Eq. (9-7) as
Dividing both sides by Zg
The values of Ki3 = Z, / lg can be evaluated therefore for various ratios of M,, / Ma and IC, / I,, using DEFLECTION 5.1.
It is noted that the effective moment of inertia is, according to definition, not the same for all loading conditions. Since I , based on M,,/ Md is not the same as I, based on M,, / Md+l (although in many practical cases they are approximately the same), live load deflections can be computed correctly only by an indirect process:
Here ad+l is determined ushg I, based on Mer/ Md+l and a, is determined using I, based on M,, / h í d . Attention
is also drawn to the Commentary on AC1 318-95, Sections 9.5.2.2, 9.5.2.3, and 9.5.2.4.
DEFLECTION 5.2 may be used as an alternate method to determine the effective moment of inertia for cracked rectangular cross sections with tension reinforcement only. The guidelines on the graph (which refer to Deflection Example 7) explain its use.
= - ad
DEFLECTION 6.1 AND 6.2
The diagram and chart were prepared considering the deflection at the center of a beam in terms of the variables. All deflections are stated in terms of the equation
where
M,
1 = spanlength, ft
E, = modulus of elastcity of concrete
= the moment at the center of the beam, or a moment value related to the deflection, kip-ft
-
-
- 33 @, ksi
1000
I, = effective gross moment of inertia, in.4
Kd = a coefficient relating the moment at midspan to the deflection at the center
1728 1' 48 E,
Kai =
DEFLECTIONS 6.1 shows the values of Kd and equations for M, for use in the equation
473
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474
where IKd M, is the sum of all separate Kd M, values for the individual loading conditions.
DEFLECTION 6.2 shows the coefficient Ka, for various spans and values of concrete weight and strength.
The values of Kar depend on E, , but it is not necessary to know E, which is related to the strength and unit weight of the concrete. Ka, can be read directly in terms of span, wc and f,'.
DEFLECTION 7
DEFLECTION 1 through 6.2 have been prepared to aid the designer in arriving at the immediate deflection of a reinforced concrete flexural member as prescribed by Sections 9.5.2.2 and 9.5.2.3 of AC1 318-95. Effective use of these tables can reduce considerable the designer's efforts in the evaluation of the to-be-expected immediate deflection; however, they are still time consuming.
Since deflection must be computed whenever members support or are attached to partitions or other construction likely to be damaged by large deflection, it is highly desirable to have reasonable assurance that, after the selection of the section and reinforcement, the check of the defiection will prove the design satisfactory.
DEFLECTION 7 provides an approximate immediate deflection applicable to uniform loading, based on the moment coefficients for an approximate frame analysis as given in Section 8.3 of AC1 318-95. A deflection computed with DEFLECTION 7 may prove particularly helpful during the design stage.
The approximate immediate deflection at the midspan can be expressed as
ac = ửc K, - W
b
As already shown in relation to DEFLECTION 6.1 and 6.2, the general equation for the deflection a of Point C at midspan can be written as:
If all factors are evaluated for a unit load of 1 kip per ft and a unit width of 1 in., the equation for a, would have to be multiplied by the actual load w and divided by the actual width b , as follows:
Multiplying the a, equation by lg / Zg gives
The first part of the above equation, (Ka, i I&, can be evaluated for various combinations of span and thickness.
These values are called Ka2 and are given in DEFLECTION 7. The second part of the above equation consists of two expressions:
1. The Zg / I, can be evaluated for various reinforcement ratios. The use of only two categories can be considered to be enough for practical purposes:
This division was arbitrarily selected to reflect the behavior of beams so heavily reinforced that the applied moment is considerably larger than the cracking moment.
Therefore, ( M , / M a )3 is negligibly small.
for simple spans:
2. The C(K, M,) can be evaluated as follows:
C(K, M,) = 5 (w 1' / 8)
For continuous spans according to Case 7 of DEFLECTION 6.1 :
E(&Mc) = 5 [ M c - O . I ( M , + M & ) ]
If the computation of the deflection is based on uniformly distributed loadings and on the moment coefficients for an approximate frame analysis as given in Section 8.3 of AC1 3 18-95, then the above expression can be evaluated for each kind of bay with the appropriate moments.
Combining C(K, M,) with the I, / I, values as described under the first expression above gives the 1 3 ~ values in DEFLECTION 7. The final equation for the approximate deflection of a beam at midspan can be written as:
9
where 9 - -
w =
b =
Kd =
bc =
= bc Kd - W
b
deflection at midspan, in.
uniformly distributed load, kip/ft (Note that in a i i deflection calculations the service loads, not the factored loads, are to be used.) width of the member, in.
coefficient from DEFLECTION 7.1
factor depending on type of span and degree of reinforcement
DEFLECTION 8
DEFLECTION 8 is similar to the graph in the Commentary on AC1 318-95, Section 9.5.2.5. It is used to obtain muitipliers for use with computed immediate deflections to estimate the additionai long-the (1 to 60 month) deflections due to the sustained part of the load, sometimes referred to as "creep deflection. "
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DEFLECTION 9
This is a plot of the equation Ec = 33 wC’.’ fi and is
used to obtain modulus of elasticity as provided 111 Section 8.5.1 of AC1 318-95, when concrete strength and unit weight are known.
Values of E, can be obtained from this design aid and used to calculate n and pn for use in DEFLECTION 2 and DEFLECTION 4. The ratio n can also be read directly when E, - 29,000,000 psi.