Microsoft Word tn292 floor deflection 032109. doc

                                                                            

                   

Technical Note

 

 

 

 

15

 

 

FIGURE EX2-3  DEFLECTED SHAPE WITH ALLOWANCE FOR CRACKING, 

USING SIMPLIFIED METHOD. 

 

 

 

Using  Equivalent Moment of Inertia (Ie) Combined with Numerical Integration 

 

The next step in increased accuracy of deflection calculation is (i) the use of equivalent moment 

of inertia Ie, (ii) the strip method as outlined in the preceding example, and (iii) numerical 

integration. In this scheme each span will be subdivided in a number of segments, typically 10 

to 20 divisions. The equivalent moment of inertia for each division will be calculated separately, 

and a solution obtained with recognition of a variable moment of inertia along the length of each 

span. This procedure along with a detailed numerical example is described in ADAPT TN294 

 

Figure 3 is an example showing the variation of moment along the first span of a two-span 

member, subdivision of the span into smaller segments, and the equivalent moment of inertia 

for each segment due to cracking. 

 

Using the computer program ADAPT-RC, the above procedure is employed to determine the 

deflection of the design strip shown in Fig, 4, with due consideration for cracking. The calculated 

deflection by the program is 0.235 in (5.97 mm), compared to 0.264 in (6.71 mm) where the 

simplified averaging of effective moment of inertia was used in calculation of cracked deflection. 

 

Using this method, the total deflection is estimated as: 

 

 

Total deflection = 2 * 0.235 = 0.470 in.  (11.94 mm) 

 

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Technical Note

 

 

 

 

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FIGURE 3  VARIABLE MOMENT INERTIA ALONG A  

MEMBER  DUE TO CRACKING 

 

 

 

FIUGRE 4 DEFLECTED PROFILE OF THE DESIGN STRIP WITH ALLOWANCE 

 FOR CRACKING USING NUMERICAL INTEGRATION 

 

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Technical Note

 

 

 

 

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Using  Finite Element Method With No Allowance for Cracking 

 

Using finite element method (FEM), the salient features of the geometry and loading that are 

idealized in other previously explained options can be faithfully modeled.  This leads to a more 

valid  estimate of slab deflection. Figure 5 shows the Discretization of the floor system used in 

the previous examples into finite element cells. 

 

 

FIGURE 5  DISCRETIZATION OF THE TYPICAL FLOOR SLAB FOR FINITE 

ELEMENT ANALYSIS (FLOOR-PRO) 

 

 

 

EXAMPLE 

For the same geometry and parameters of examples 1 and 2, using a finite element program 

determine the deflection at center of the panel identified in Fig. EX1-1. 

 

ADAPT-FLOOR Pro

4

 program was used to model the slab and obtain a solution. The 

distribution of deflection for the given load is shown in Fig. EX3-1.  The maximum deflection at 

the center of the panel under consideration is reported as  0.54 in. 

 

 

 

 

                                                 

4

  ADAPT-Floor Pro is a finite element program for analysis and design of conventionally reinforced or post-

tensioned floor systems. www.adaptsoft.com 

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Technical Note

 

 

 

 

18

 

FIGURE EX3-1 DEFLECTION CONTOUR OF THE FLOOR SYSTEM UNDER THE COMBINED 

ACTION OF DEAD AND LIVE LOADS 

 

 

 

Using  Finite Element Method With Due Allowance for Cracking 

Formulation of finite elements with allowance for cracking is somewhat complex. The complexity 

arises from the fact that cracking and reduction in stiffness depend on the presence, amount 

and orientation of reinforcement.  Before a solution is obtained, the reinforcement detailing of a 

floor system must be fully known, since the loss of stiffness in each finite element cell depends 

on the availability and exact location of the reinforcement in that cell. 

 

The following briefly describes the steps for a finite element deflection calculation, with 

allowance for cracking. 

 

1.  Using the geometry, boundary conditions, material properties, and the load combination for 

which the deflection is sought, the program discretizes the structure, sets up the system 

stiffness matrix of the structure based on gross moment of inertia (Ig), and obtains the 

distribution of moments (Ma) over the entire structure. 

 

2.  If required the program performs a design check, using a building code, and adds the 

required reinforcement to the floor system for the specified loads. 

 

3.  The Program scans the entire floor system to detect the reinforcement available in the 

beams and the slab regions. The available reinforcement is either determined by the 

Program prior to the initiation of deflection calculation, or is a combination of program calculated 

and user defined/edited reinforcement.  

 

Once the deflection calculation is initiated, the available reinforcement remains unchanged. The 

reinforcement can be in one or more of the following forms, with no restriction on the orientation, 

length, or the position of each reinforcement within the floor system. 

 

a -  User defined one or more top and bottom reinforcement mesh; 

b -. User defined grouped or distributed reinforcement bars at top and/or bottom of 

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