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8 Figure 4. Effective span length ratios. predates modern computer analysis. Assumptions need to be Step 2: Define the centroidal location of the total compres- made in order to make the closed-form analytical problem sive force in the slab tractable. With modern day approaches such as the Finite Determine the vertical location of the resultant compres- Element Method, however, many of those classical assump- sive force using statics. The distance from the top of the slab tions do not need to be made. Nor do the same classical def- to this centroidal location is defined as zo (see Figure 1). initions of effective flange width need to be used. In order to enforce both assumptions, both Cslab and zo must remain unchanged. Step 3: Determination of maximum longitudinal stress in 2.2 NEW DEFINITION FOR EFFECTIVE WIDTH the slab The review of literature revealed that the classical defini- Extract the maximum longitudinal stress (max) in the slab tion of effective width was more suited to a stiffened plate directly from the finite element analysis results. For instance, than to a composite girder-deck system undergoing flexure. the maximum slab longitudinal stress is located at the extreme In the latter, the deck (plate) is sufficiently thick for stress compression fiber in the elastic response (see Figure 1). variation through the thickness to be an important considera- tion. Thus, a new definition of effective width was developed. Step 4: Calculate minimum longitudinal stress of the slab The new definition (Chiewanichakorn et al., 2004) enforced The term "minimum longitudinal stress (min)" of the slab two conditions that the traditional definition does not: in Figure 1 can also be described as an equivalent longitudi- nal stress at the bottom of the slab. Because of a linear vari- · Enforce the same moment in the idealized Bernoulli- ation in the strain profile, simple beam theory assumes a lin- Euler line girder as in the 3-D FEM slab-girder system ear variation in the stress profile for the elastic response, that (the classical definition requires only the same force), and is, a trapezoidal shape. · Enforce moment equilibrium as well as force equilibrium. In order to satisfy the two assumptions, the centroidal location of the resultant compressive force must be the same This new definition is applied according to the following for both finite element analysis and simple beam theory. procedure in a positive moment section. With the pre-determined values of max and zo, compute the minimum longitudinal stress (min) of the slab such that the Step 1: Calculate total compressive force in the slab conditions of total force and resultant location are similar to Compute the total or resultant compressive force in the those obtained from finite element analysis. slab by summing up all element forces in the slab using Equation 1. Step 5: Computation of "effective slab width" After the value of min is obtained from Step 4, calculate n Cslab = i Areai Equation 1 the equivalent compressive block (area of the trapezoid) and i =1 determine the effective slab width using Equation 2: where Cslab Cslab beff = = Equation 2 Cslab = total or resultant compressive force in the slab A 0.5 tslab ( max + min ) = element longitudinal stress where Area = element cross-sectional area i = element number beff = effective slab width