**6.6. Creep deformation**

The forces acting on the slab shell in the secondary cooling zone can be modeled as the bending of a rectangular thin plate under loading (i.e., static pressure of molten steel). One segment of the slab along the strand direction is taken to build the model. In the model, the slab is a rectangular thin plate fixedly supported along two sides and simply supported along the other two sides. In addition, the thin plate is subjected to lateral loads, and the temperature field linearly changes in the thickness direction of the slab. Because the width of the slab is much greater than the gap between the rollers, the effects of the slab boundary on the internal side of the slab can be ignored according to the Saint-Venant principle. According to plate theory, the slab shell is viscoelastic at high temperature, and the stress and deformation satisfy the Maxwell creep law. As shown in Figure 13.

**Figure 13.** Model of slab shell and the force model of slab creep. (Sun J et al,1996)

In the secondary cooling zone, the total stress equals the sum of elastic strain and creep strain, when the slab shell creepily bends under the static pressure of molten steel. The elastic strain changes little with time. The elastic deflection is expressed as: (Sun J et al,1996):

$$\begin{split} W\_c(\mathbf{x}, y) &= W\_{qc} + W\_{Te} = \frac{4q\alpha^4}{\pi^5 D} \sum\_{i=1,3,\dots}^{\infty} \frac{1}{m^5} \times \\ \left(1 - \frac{\alpha\_m ch\alpha\_m + sh\alpha\_m}{\alpha\_m + sh\alpha\_m ch\alpha\_m} ch\frac{2\alpha\_m}{b} y + \frac{\sigma\_m sh\alpha\_m}{\alpha\_m + sh\alpha\_m ch\alpha\_m} \cdot \frac{2y}{b} sh\frac{2\alpha}{b} y\right) \\ &\times \sin\frac{m\pi}{\alpha} \mathbf{x} + \frac{48\alpha aT\_0}{\pi^3 h^3 (1 - 2\gamma)(1 - \gamma)} \times \sum\_{i=1,3,\dots}^{\infty} (1 - \frac{ch\frac{2\alpha\_m}{b} y}{ch\alpha\_m}) \sin\frac{m\pi}{a} \mathbf{x}. \end{split} \tag{18}$$
  $\int\_{\gamma}^{b/2} T(\gamma)\gamma \,d\pi$ .

where: /2 <sup>0</sup> /2 () d *<sup>h</sup> h T Tz z z* .

The expression of elastic deflection has the series of hyperbolic function, and converges rapidly, thus setting m=1 is sufficiently accurate for calculation. Using equation (18) and the Cauchy equation, the creep deformation of slab shell on the narrow side can be derived as:

Numerical Simulation of Slab Broadening in Continuous Casting of Steel 575

**Figure 14.** Comparion of calculated and measured side creep results. ( FU JianXun et al.2011(a))

574 Numerical Simulation – From Theory to Industry

**6.6. Creep deformation** 

speed.

the cast-rolling segment. The degree of extension and broadening increases with casting

The forces acting on the slab shell in the secondary cooling zone can be modeled as the bending of a rectangular thin plate under loading (i.e., static pressure of molten steel). One segment of the slab along the strand direction is taken to build the model. In the model, the slab is a rectangular thin plate fixedly supported along two sides and simply supported along the other two sides. In addition, the thin plate is subjected to lateral loads, and the temperature field linearly changes in the thickness direction of the slab. Because the width of the slab is much greater than the gap between the rollers, the effects of the slab boundary on the internal side of the slab can be ignored according to the Saint-Venant principle. According to plate theory, the slab shell is viscoelastic at high temperature, and the stress

and deformation satisfy the Maxwell creep law. As shown in Figure 13.

**Figure 13.** Model of slab shell and the force model of slab creep. (Sun J et al,1996)

<sup>4</sup> <sup>1</sup> (,)

*<sup>q</sup> W xy W W D m*

 

48

 

In the secondary cooling zone, the total stress equals the sum of elastic strain and creep strain, when the slab shell creepily bends under the static pressure of molten steel. The elastic strain changes little with time. The elastic deflection is expressed as: (Sun J et al,1996):

> 5 5 1,3

*ch sh sh <sup>y</sup> ch y sh y sh ch b sh ch b b*

 

*ch y m m aT <sup>b</sup> x x*

sin (1 )sin . (1 2 )(1 )

The expression of elastic deflection has the series of hyperbolic function, and converges rapidly, thus setting m=1 is sufficiently accurate for calculation. Using equation (18) and the Cauchy equation, the creep deformation of slab shell on the narrow side can be derived as:

 

1,3

*h ch a*

 

*i m*

2

 

*m*

(18)

4

<sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>1</sup>

*i*

 

0

*mm m m mm m mm m mm*

 

3 3

*e qe Te*

 

<sup>0</sup> /2 () d *<sup>h</sup> h T Tz z z* .

where: /2

The amount of creep deformation for the narrow side of the slab was calculated using Matlab software; the results are shown in Figure 14. The figure also shows the measured results from the experiments of a stagnant slab. The agreement between the calculated results from the Maxwell model and the measured results illustrates that the Maxwell model is able to reveal deformation behavior at high temperature.
