Microsoft Word tn292 floor deflection 032109. doc
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- Bu səhifədəki naviqasiya:
- Technical Note
- Source Immediate deflection Creep
- Total Long-Term Displacement From Removal of Forms
- Load Combination for Code Checks
Technical Note
19 slab and beams; c - Reinforcement calculated and reported by the program for minimum requirements of the code, strength check, initial condition, or other code related criteria; and d - Post-tensioning tendons defined by the user, each with its own location and force.
4. The program matches the calculated moment (Ma) of each finite element cell with the existing reinforcement in that cell. Using the parameters given below, the Program calculates the effective moment of inertia for each cell:
a. Finite element cell thickness (to obtain uncracked second moment of area); b. Available reinforcement associated with each cell (nonprestressed and prestressed), with recognition of orientation and height of each individual reinforcement; c. Cracking moment of inertia associated with each cell in each direction (lcr); d. Cracking moment associated with each cell (Mcr); and e. Applied moment (Ma calculated in 2).
5. Having determined the effective second moment of area of each finite element cell in each of the principal directions, the Program re-constructs the stiffness matrix of each cell.
6. The Program re-assembles the system stiffness matrix and solves for deflections. At this stage the solution given from the first iteration is a conservative estimate for the floor deflection with cracking. For practical engineering design, it is recommended to stop the computations at this stage.
For a more accurate solution the newly calculated deflection can be compared with that of a previous iteration. If the change in maximum deflection is more than a pre-defined tolerance, the Program will go into another iteration starting from step (2). In this scenario, the iterations are continued, until the solution converges to within the pre-defined tolerance.
The Program accounts for loss of bending stiffness in beams and slab regions and its combinations..
EXAMPLE Using finite elements determine the deflection of the panel identified in Fig. EX1 for the loads and conditions described in Example 1.
Calculate the deflection for the combination of dead and live loads. Use ACI318-08 to determine the reinforcement necessary for both the in-service and strength requirements of the code. Use the calculated minimum reinforcement of the code to determine the cracked deflection. No other reinforcement is added.
illustrated as a contour image in Fig. EX4-1. The deflection at the center of the panel under consideration is 0.70 in. (17.78 mm) compared to 0.54 in. (13.72 mm) for the uncracked slab. The cracked deflection can be reduced by adding reinforcement at the locations of crack formation in addition to the minimum requirements of the code already included in the analysis. Technical Note
20
FIGURE EX4-1 DEFLECTION CONTOUR OF SLAB WITH CRACKING
EX4-3. At each location, the reduction in effective moment of inertia is based on the calculated moment at that location and the amount, position and orientation of reinforcement at the same location. The largest loss of stiffness occurs over the columns and the support lines joining the columns. The maximum loss of stiffness is 69% reducing the effective moment of inertia to 31% of its uncracked value.
Technical Note
21
FIGURE EX4-2 EXTENT OF CRACKING SHOWN THROUGH REDUCTION IN EFFECTIVE MOMENT OF INERTIA Ie ABOUT Y-Y AXIS
FIGURE EX4-3 EXTENT OF CRACKING SHOWN THROUGH REDUCTION IN EFFECTIVE MOMENT OF INERTIA Ie ABOUT X-X AXIS Technical Note
22 Deflection of Post-Tensioned Floor Systems
Two-way post-tensioned floor systems designed to ACI-318 provisions and the PTI- recommended slab-to-depth ratios (Table 4), either do not crack under service condition, or crack to an extent that does not invalidate calculations based on gross cross-sectional geometry and linear elastic theory. This is because, unlike other major non-US codes, the allowable tensile stresses in ACI are relatively low.
The preceding observation does not hold true for post-tensioned one-way slab and beams, where ACI-318 permits designs based on post-cracking regime. For such post-tensioned floor systems, designers must include allowance for cracking in their designs.
As illustrated in the design examples, the calculated deflections design engineers use from the ??? currently available procedures for conventionally reinforced concrete vary greatly. Table 6 lists the outcome of the various methods. Note that for the typical floor system selected, the difference between the various methods can be as much as three times. Finite Element Method with due allowance for crack formation gives the largest deflection. The strip method with no allowance for cracking produces the smallest value.
TABLE 6 DEFLECTION VALUES AT CENTER OF PANEL OF THE NUMERICAL
1 Closed form formulas 0.480(12.19) 69 %
2 ACI318 – Simplified method 0.528(13.42) 75 %
3 Strip method (uncracked) 0.462(11.74)
66 % 4 Strip method with cracking and numerical integration 0.470(11.94) 67 % 5
No allowance for cracking 0.540(13.72) 77 % 6
With allowance for cracking 0.700(17.78) 100 %
Technical Note
23 LONG-TERM DEFLECTIONS A concrete member’s deformation changes with time due to shrinkage and creep. Shrinkage of concrete is due to loss of moisture. Creep is increase in displacement under stress. Under constant loading, such as selfweight, the effect of creep diminishes with time. Likewise, under normal conditions, with loss of moisture, the effect of deformation due to shrinkage diminishes. Restraint of supports to free shortening of a slab due to shrinkage or creep can lead to cracking of slabs and thereby an increase in deflection due to gravity loads.
While it is practical to determine the increase in instantaneous deflection of a floor system due to creep and shrinkage at different time intervals, the common practice for residential and commercial buildings is to estimate the long-term deflection due to ultimate effects of creep and shrinkage.
It is the long-term shrinkage due to loss of moisture through the entire volume of concrete that impacts a slab’s deformation. Plastic shrinkage that takes place within the first few hours of placing of concrete does not play a significant role in slab’s deflection and its impact in long- term deflection is not considered.
Long-term shrinkage results in shortening of a member. On its own, long-term shrinkage does not result in vertical displacement of a floor system. It is the presence of non-symmetrical reinforcement within the depth of a slab that curls it (warping) toward the face with less or no reinforcement. The slab curling is affine to its deflection due to selfweight, and hence results in a magnification of slab’s natural deflection.
It is important to note that, deflection due to shrinkage alone is independent of the natural deflection of slab. It neither depends on the direction of deflection due to applied loads, nor the magnitude. The shrinkage deflection depends primarily on the amount and position of reinforcement in slab.
A corollary impact of shrinkage is crack formation due to restraint of the supports. This is further discussed in connection with the restraint of supports. It is the crack formation due to shrinkage that increases deflection under gravity loads.
Shrinkage takes place over a time period extending beyond a year. While the amount of shrinkage and its impact on deflection can be calculated at shorter intervals, the common practice is to estimate the long-term deflection due to the ultimate shrinkage value.
Shrinkage values can vary from zero, when concrete is fully immersed in water to 800 micro strain. Typical ultimate shrinkage values are between 400 to 500 micro strain.
Creep is stress related. It is a continued magnification of the spontaneous displacement of a member with reduced rate of creep with time. Values of creep vary from 1.5 to 4. Typical ultimate creep values for commercial and building structures are between 2 to 3.
Restraint of supports, such as walls and columns to free movement of a slab due to shrinkage can lead to tensile stresses in the slab and early cracking under applied loads. Early cracking will reduce the stiffness of the slab and increase its deflection.
Technical Note
24 Multiplier Factors for Long-Term Deflections For design purposes, the long-term deflection of a floor system due to creep and shrinkage can be expressed as a multiplier to its instantaneous deflection.
Long-term deflection due to sustained load:
Δ l
= C * Δ i
(8)
Where Δ l = long-term deflection; Δ i
= instantaneous deflection; and C = multiplier.
ACI-318 suggests the multiplier factor shown in Fig. 5 to estimate long term deflections due to sustained loads
FIGURE 5 MULTIPLIER FOR LONG-TERM DEFLECTION The multiplier can be reduced, if compression reinforcement is present. The factor ( λ) for the reduction of the multiplier is given by:
ρ’ )
(9) Where
ρ’ is the value of percentage of compression rebar at mid-span for simple and continuous members and at support for cantilevers.
ACI’s recommended multipliers account for the cracking of slab. Hence, they are intended to be applied to cracked deflections. Several Investigators recommend long--term multiplier coefficients for deflections based on gross cross-sectional area. These coefficients are higher Technical Note
25 than the ACI-318 multiplier (Fig. 5). Table 7 lists the recommended values of multipliers for non- prestressed slabs. Δ l
λ c + λ sh ) * Δ i
(10) Where λ c = creep multiplier; λ sh = shrinkage multiplier; Or, simply Δ l
Δ I
, where C = ( 1 + λ c + λ sh )
TABLE 7 MULTIPLIERS FOR LONG-TERM DEFLECTIONS Source Immediate deflection Creep λ c Shrinkage λ sh Total C Sbarounis(1984) 1.0 2.8 1.2 5.0 Branson(1977) 1.0 2.0 1.0 4.0 Graham and Scanlon (1986b) 1.0 2.0 2.0 5.0 ACI-318 1.0 2.0 3.0
Based on the author’s observation and experience, it is recommended that structures built in California use the following values:
For conventionally reinforced floor systems C = 4
For post-tensioned floor systems
= 3
LOAD COMBINATIONS
The load combination proposed for evaluating the deflection of a floor system depends on the objective of the floors evaluation. The following describe several common scenarios.
(1.0*SW + 1.0*SDL + 1.0*PT + 0.3*LL)* C Where
SW = selfweight;
SDL = superimposed dead load, (floor cover and partitions); PT
= post-tensioning; and
LL = design live load.
The above load combination is conservative as it assumes the application of superimposed loads as well as the application of sustained live load of the structure to take place at the time of removal of the supports below the cast floort. The factor 0.3 suggested for live load is for Technical Note
26 “sustained” load combination. The significance of the above load combination is that it provides a measure for the total deflection from the position of the forms at the time of concrete casting. Its magnitude must be evaluated for aesthetics and drainage of surface water, if applicable. It is used for checking the deflection of parking structure decks or roofs, where the floor is placed in service in its as-cast condition.
Load Combination for Code Checks
For the acceptability of a floor deflection in connection with the code specified maximum values listed in Table 1, the following two load combinations apply.
1.0*LL
C1*C*(SW + SDL + PT) + 0.3*C2*C*LL + 0.7*LL
Where C1 is the fraction of long-term deflection coefficient related to the balance of long-term deflection subsequent to construction installation likely to be damaged by deflection of slab. Partitions and other fixtures are generally installed when more than one-half of long-term deflection has taken place. As a result, C1 is generally less than 50% of the long-term deflection multiplier, assuming that the superimposed dead load and partitions are not installed before 40 days from date of casting the floor (Fig. 6). C2 relates to the time, when construction is complete and in-service live load applied. This is generally less than 20% of the long-term multiplier. Figure 6 can be used as a guideline for values of C1 and C2. For example, if the in-service live load of a structure is put in place six months subsequent to casting the floor, the value of C2 will be approximately 0.25 (value associated with 180 days in Fig. 6).
FIGURE 6 LONG-TERM SHORTENING OF CONCRETE MEMBERS DUE TO CREEP AND SHRINKAGE WITH TIME
LIVE LOAD DEFLECTION Technical Note
27 Even when using linear elastic theory to calculate a floor system’s deflection, cracking will result in a non-linear response. For the same load, the deflection of a slab depends on the extent of cracking prior to the application of the load. Therefore, when calculating the deflection due to the instantaneous application of live load, one must use the following procedure:
Deflection due to LL = (deflection due to DL+LL) – (deflection due to DL)
The above accounts for loss of stiffness due to dead load prior to the application of live load.
APPENDIX A
CHARACTERISTICS OF PARISSA APARTMENTS TYPICAL FLOOR
Geometry Slab thickness and support dimensions (see plan)
Concrete f’c (28 day cylinder strength)
= 5000 psi (34.47 MPa) Wc (unit weight)
= 150 pcf (2403 kg/m 3 ) Ec (modulus of elasticity at 28 days) = 4,287 ksi (29558 MPa)
Non-Prestressed Reinforcement Yield stress
= 60 ksi (400 MPa)
Aalami, B. O. and Bommer, A. (1999) “Design Fundamentals of Post-Tensioned Concrete Floors,” Post-Tensioning Institute, Phoenix, AZ, pp. 184.
ACI318-08 Bares, R., (1971), “Tables for the Analysis of Plates, Slabs and Diaphragms Based on the Elastic Theory,” Bauverlag GmbH, Wiesbaden und Berlin, 1971, pp. 626
Branson, D. E., (1977), “ Deformation of Concrete Structures ,” McGraw-Hill Book Company, New York, pp. 546.
Graham, C. J., and Scanlon, A., (1986), “ Long Time Multipliers for Estimating Two-Way Slab Deflections ,” ACI Journal, Proceedings V. 83, No.5, pp. 899-908.
PTI (1990). Post-Tensioning Manual, 5 th Edition, Post-Tensioning Institute, Phoenix, AZ, pp. 406 Technical Note
28 Sarbounis, J. A., (1994) “ Multistory Flat Plate Buildings: Measured and Computed One-Year Deflections, ” Concrete International, Vol. 6, No. 8, August, pp. 31-35. Yüklə 228 Kb. Dostları ilə paylaş:
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