Design size of a dowel bar at expansion joint for following data:
Thickness of slab = 25 cm
Width of joint = 2.5 cm
Allowable flexural stress in bars = 1400 kg/cm²
Allowable shear stress in bars = 1000 kg/cm²
Allowable bearing stress in concrete = 100 kg/cm²

To find the length of the dowel bar the sum L_d + δ is used, where L_d is the embedded length and ? is the width of the joint. The equation to find the embedded length is given by:
\(L_d=5d\left(\frac{F_f}{F_b} ×\frac{L_d+1.5?}{L_d+8.8?}\right)^{1/2}\)
Where, F_f is the allowable flexural stress = 1400 kg/cm²
F_b is the allowable bearing stress in concrete = 100 kg/cm²
d is the diameter of the dowel bar and it can be assumed as 2.5 cm
δ is the width of the joint = 2.5 cm
After substituting all the values in the equation,
\(L_d=5×2.5\left(\frac{1400}{100}×\frac{L_d+1.5×2.5}{L_d+8.8×2.5}\right)^{1/2}\)
\(L_d=12.5\left(14×\frac{L_d+3.75}{L_d+22}\right)^{1/2}\)
The next step is to find the value of Lsub>d using trial and error method. This is in accordance with the method prescribed by IRC 58:1988.
Assuming L_d = 45 cm for the first trial,
\(L_d=12.5\left(14×\frac{45+3.75}{45+22}\right)^{1/2}\)=39.89 cm
39.89 is less than 45 cm, now assume 39.5 cm
\(L_d=12.5\left(14×\frac{39.5+3.75}{39.5+22}\right)^\frac{1}{2}\)=39.22 cm
39.22 cm is less than 39.5 cm, now assume 39.2 cm,
\(L_d=12.5\left(14×\frac{39.2+3.75}{39.2+22}\right)^{1/2}=39.18 ?39.2\) cm
Therefore, the L_d is obtained as 39.2 cm.
The length of dowel bar would be = L_d + δ = 39.2 + 2.5 = 41.7 cm