Analysis of dowel bar

Analysis

Recall the formulation developed as Equation 13 , in the lecture on 'analysis of concrete pavement' , representing beam on elastic foundation. By putting appropriate boundary conditions to that equation, for a semi-infinite beam with a moment, M_o, and a point load, P, the following solution emerges for Winker's foundation (Porter 2001, Timoshenko and Lessels 1925 ):

(25)

where,
y  =  the deflection along the x direction, , and is called as the relative stiffness of the beam on the elastic foundation, k = spring constant, E = elastic modulus of the beam and I = moment of inertia of the beam section.

From Equation 1, the deflection of the dowel bar at the face of the joint, y_o, can be obtained as (Friberg 1940):

(26)

where, relative stiffness of the dowel bar resting on concrete (assumed as elastic foundation), k_d  = modulus of dowel support, b_d = dowel bar width (i.e. diameter), E_d = elastic modulus of the dowel bar, I_d = moment of inertia of the dowel bar, P_d = load transferred through the dowel bar and z =joint width.

Though the above equation assumes dowel bar to be semi-infinite, Porter (2001) showed that Equation (26) gives a reasonably good estimate of deflection obtained from more rigorous analysis.

Thus, the bearing stress developed, σ_b^d, can be expressed as the product of modulus of dowel support ( k_d) and the deflection at the face of joint ( y_o) (Porter 2001), i.e.

(27)

For a successful dowel bar design, the value of   σ_b^d needs to be kept lower than the allowable bearing stress of concrete, σ_b^d, specified as (ACI 1956, Porter 2001, IRC:58 2002):

(28)

where, σ_b^d  is expressed in MPa, f_ck = characteristic compressive strength of concrete in MPa and b_d  is the diameter (i.e. width) of the dowel bar in mm,

Joint load transfer efficiency

Ideally, the dowel bar system should be able to transfer the whole load applied to it. However, voids developed due to repetitive loading reduce the joint load transfer efficiency ( JLTE ) of the dowel bar. The JLTE (expressed in percentage) of a dowel bar can be defined as (Porter 2001, Ioannides and Korovesis 1992 )

where, P_a  is the load applied to the dowel bar. The force transmitted being somewhat difficult to measure,    joint load transfer efficiency is sometimes measured in terms ratio of deflections. If impact loading (generally, falling weight deflectometer ( FWD ) is used for this purpose) is applied near the transverse dowelled joint of a slab, the JLTE can be defined as the percentage of the deflection of the unloaded slab with reference to the deflection of the loaded slab. Figure 1 schematically shows two extreme situations as JLTE = 0% and JLTE = 100%. Various researches and documents (Porter 2001, Yoder and Witczak 1975, ACPA 1991 ) suggest that the load transfer efficiency varies between 35 to 50%.

                                             
Figue-21 Schematic diagram explaining the deflection based load transfer efficiency (Chakroborty and Das 2003)

Distribution of load

A part of the load applied is shared by the dowel bar system. Essentially, this load is not shared by only one dowel bar, rather, it is shared by a group of dowel bars. (These dowel bars are placed at some designed interval). Thus, it is important to know (i) how many dowel bars participate in load transfer, and (ii) how is this load shared across the various participating dowel bars.

Fridberg (1940) suggested that a length of up to 1.8 × radius of relative stiffness (refer to Equation (11) of the lecture 'analysis of concrete pavement' for definition) participate in the load transfer. It is also suggested that the load may be taken as linearly varying with maximum share taken by the dowel bar which is just vertically below the wheel. Tabatabaie et al. (Porter 2001) suggested that instead of taking the factor as 1.8 it should be taken as 1.0 .
From design point of view, the wheel can be placed in two ways over the transverse edge of the slab, viz. case (i) the wheel at one edge or, case (ii) the wheel is at the middle. Obviously, the maximum load shared by a dowel bar in case (i) will be more than case (ii) . Hence case (i) would govern the design of dowel bar.