Module II : Analysis and design of concrete pavements
Lecture 2 : Analysis of Concrete Pavement
 

Axial stress component

It is assumed that the axial stress ($\sigma^A$), which is constant through the thickness of the slab, is generated due to axial temperature component, $T_A$. The axial stress, $\sigma^A$, can be obtained by equating thermal force due to $T^A$ to the total thermal force produced by $T(z)$ as shown in the following (Ioannides and Khazanovich 1998):   
                      \begin{eqnarray*} \int_{-h/2}^{h/2} \sigma^A dz &=& \int_{-h/2}^{h/2}\sigma^{tot... ...\ \Rightarrow T^A &=& \frac{1}{h} \int_{-h/2}^{h/2} T(z) dz \\ \end{eqnarray*}
If $\epsilon^{A,T}$$\sigma^{A,T}$ are axial strain and axial stress respectively, then

                        \begin{eqnarray*} \epsilon_{xx}^{A,T}=\epsilon_{yy}^{A,T} &=& \alpha (T^A - T_0)... ...-\mu} \left( \frac{1}{h} \int_{-h/2}^{h/2} T(z) dz - T_0 \right) \end{eqnarray*}
Most of the time concrete pavement is allowed to expand or contract through various joints. Thus, in such a situation, $\sigma_{xx}^{A,T}$ = $\sigma_{yy}^{A,T}$ = $0$.

 

 

 

 

 

Bending stress component

In a similar way, it is assumed that the bending stress ($\sigma^L$), which varies linearly through the thickness of the slab and assumes a value of zero at the mid-plane of the cross-section, is generated due to bending (linear) temperature component, $T^L$. The bending stress,$\sigma^L$, can be obtained by equating thermal force due to $T^L$ to the total thermal force produced by $T(z)$ as shown in the following (Ioannides and Khazanovich 1998):

\begin{eqnarray*} \int_{-h/2}^{h/2} \sigma^L (z) z dz &=& \int_{-h/2}^{h/2} \sig... ...^{h/2}z^2 dz}= T_0+\frac{12z}{h^3} \int_{-h/2}^{h/2}T(z)z dz \\ \end{eqnarray*}

Accordingly, bending strain ($\epsilon^{L,T}$) and bending stress ($\sigma^{L,T}$) may be expressed as given below:

         \begin{eqnarray*} \epsilon_{xx}^{L,T}(z)=\epsilon_{yy}^{L,T}(z) &=& \alpha \left... ...ht )= - \frac{12Ez\alpha}{(1-\mu)h^3} \int_{-h/2}^{h/2}T(z) z dz \end{eqnarray*}

The maximum values of strain, stress and moment occur at the top ( $z=-h/2$) and bottom ( $z=h/2$) and these values can be calculated as:

                      \begin{eqnarray*} \epsilon_{xx,max}^{L,T}=\epsilon_{yy,max}^{L,T} &=& \frac{6 \a... ...L,T} \\ &=& -\frac{E \alpha}{1-\mu} \int_{-h/2}^{h/2}T(z) z dz \end{eqnarray*}

Most of the time the concrete pavement is restrained against bending. This restraint is primarily due to self weight of the slab. Thus, the strain mentioned above is the restrained strain, and the bending stress does develop in concrete pavement.

 

 

Residual stress component

The shape of  $T(z)$   is arbitrary. Thus, combination of $T^A$ and $T^L$ does not add up to $T(z)$. Thus, $T^{NL}$ is the residual temperature component, obtained after deducting both the axial and linear temperature components from total temperature. The value of $T^{NL}$ can be calculated as follows:  
                  \begin{eqnarray*} \left (T^{NL}(z)-T_0 \right) &=& \left ( T(z)-T_0 \right)- \le... ..._{-h/2}^{h/2}T(z)dz - \frac{12z}{h^3} \int_{-h/2}^{h/2}T(z) z dz \end{eqnarray*}

Thus, the restrained strain and the corresponding stress can be obtained as:
            \begin{eqnarray*} \epsilon_{xx}^{NL,T}(z)=\epsilon_{yy}^{NL,T}(z) &=& \alpha \le... ...h/2}T(z)dz - \frac{12z}{h^3} \int_{-h/2}^{h/2}T(z) z dz \right ) \end{eqnarray*}

1