The inner temperature of concrete when reinforced concrete construction is suffering from fire, can be calculated theoretically ; and its investigated results concur with experimental consequence considerably.
From its results, the concrete covering to prevent temperatural lowering of strength of steel can be inferred.
The theory of thermometric conduction is based on fundamental linear or secondary differential equation, that is
(∂θ)/(∂t)=a(∂²θ)/(∂x²) (1)   0<x<+∞     a = diffusivity
or thermometric conductivity of concrete.
(∂θ)/(∂t)=a((∂²θ)/(∂x²)+(∂²θ)/(∂y²)) (2) 0<x<+∞
               0<y<+∞
at (1) eq. boundary conditions are
θ=Θ   at x=0
θ=θ₀  at  t=0
Then its solution is
θ=(Θ-θ₀){1-Φ(x/2√at)}+θ₀ (3)
in this eq. Φ(x)=2/√π ∫^x₀ e ^-α2dα (4)
It is known as probability function that is Gauss’ theory of errors.
In (3) eq. when x=1(cm), Θ=1120°C, θ₀=20°C, a=0.0012m²/hr ; θ is approximately coincide with A. S. T. M. fire standard curve, and when x=2(cm), θ is equal to surface temperature of concrete that is suffering from A. S. T. M. standard fire. And then inner temperature of concrete is expressed as follows.
θ=1100 [1-Φ((x+2)/(2√at))] -40   (5)
     θ≧100°C  a=0.0012
θ=1100 [1-Φ((x+2)/(2√at))]+20   (6)
     θ≦100°C  a=0.0024
x=depth from heated concrete surface (cm)
But at the corner of concrete, fire frame heats at the two surfaces that are x-axis and y-axis surface. In this case, (4) eq. is fundamental equation. Boundary conditions are
θ=Θ   at x=0, y=0
θ=Θ₀  at t=0
Its solution is
θ=Θ+(θ₀-Θ)Φ((x/2√at), (y/2√at))   (7)
Φ((x/2√at,) (y/2√at))=4/π ∫ ^x/2√at ₀ ∫ ^y/2√at ₀
        e ^-(α2+β2)dαdβ  (8)