Mechanical and thermal properties of the simulated materials.

## Abstract

In this chapter, the derivation of analytic formulation of bending deflection has been done using the theory of classical laminate plate. The method of Navier and Levy solutions are used in the calculation. The composite laminate plate is exposed to out-off plane temperatures and combined loading. The temperature gradient of thermal shock is varied between 60C∘ and −15C∘. The combined loading are the bending moment (Mo) in the y-direction and in-plane force (Nxx) in the x-direction. The in-plane force (Nxx) has a great effect on the bending deflection value within a 95.842%, but the bending moment (Mo) has a small effect on the bending deflection value in the rate of 4.101%. The results are compared and verified for central normal deflection.

### Keywords

- classical plate theory
- composite laminate plate
- temperature affect
- combined loading

## 1. Introduction

The effect of temperature and combined loading on composite plate is one of the primary life limiting factors of a bridge engineering application. This chapter will consider the structural evaluation of the localized effect in the bridge engineering. The application of bridge engineering can be found in a structural bridge deck panel. Ray studied the fiber-matrix debonding by applying the thermal shock of thermal fatigue, taking into account the conditioning time. He performed a three-point bending test on glass fiber reinforced with unsaturated polyester and epoxy resin composites in which it exposed to 75

## 2. Equations of motion in terms of displacements

The stress and strain relationship is varied through the laminate thickness, as indicated in Eq. (1):

The general bending equation of rectangular plate is as below:

By taking into account the temperature effect, the mechanical and thermal bending moments are:

where

And,

It can be assumed that all layers have

## 3. Formulation of bending deflection distribution using Navier solution

The normal deflection distribution is derived based on the solution of classical laminate plate theory using Navier equation. Navier solution assumed that the boundary condition is simply supported from all edges under the effect of temperature

where.

And,

By integrating Eq. (7) with respect to (x) and (y), the temperature distribution through the plate thickness is:

The thermal bending moment is defined as in the following:

where

The general solution of normal deflection for simply supported boundary condition from all edges is:

Substitute Eq. (12) and Eq. (10) into Eq. (6), the solution of bending deflection is illustrated in the following equation:

where

## 4. Formulation of bending deflection distribution using levy solution

The theory of classical laminate plate of Levy solution is used to derive the solution of normal deflection. The Levy solution assumed that the variation of the bending deflection should be along the x-axis. Levy solution can be used on any type of boundary condition which gives flexibility on any type of loading such as

where

By integrating Eq. (15) with respect to (x), the temperature distribution through the plate thickness is:

Ignore the variation of thermal bending moment and normal deflection along y-axis, Eq. (6) will be:

As mentioned earlier, the thermal bending moment is varied along x-axis, as below:

where

The solution of normal bending deflection is as below:

To find

Substitute Eq. (19) and Eq. (22) into Eq. (18) to obtain the particular solution of bending deflection along x-axis,

where

To find

The solution of Eq. (25) is as below:

Substitute Eq. (26) into Eq. (25), to obtain the homogeneous solution of Eq. (25) along x- and y-directions:

Substitute Eq. (27) and Eq. (23) into Eq. (21) to obtain the general solution of normal bending deflection, as indicated below:

The simply supported boundary conditions from all edges are assumed and the constants

where

## 5. Numerical simulation procedure

In this chapter, the finite element discretization is carried out by using ANSYS Ver. 18.2. (SHELL 132) element is used to mesh the composite laminate plate. SHELL 132 is defined by eight nodes having six degrees of freedom at each node to calculate the central normal deflection. In the simulation analysis, the central point of laminate plate is used to calculate the normal deflection. Always the convergence test is needed to determine the size of elements in which the value of normal bending deflection settles down. Finite element analysis of convergence curve defines the relationship between the grid interval and the analysis accuracy. Four types of combined loading is used such as: (temperature affect only

25.07 | 40 | 50 | 60 | 70 | 80 | |
---|---|---|---|---|---|---|

19.933 | 30.4038 | 37.41988 | 44.435 | 51.452 | 58.468 | |

19.933 | 30.4038 | 37.41988 | 44.435 | 51.452 | 58.468 | |

3.0896 | 3.81746 | 4.53322 | 5.5793 | 7.25302 | 10.3612 | |

0.3835 | 0.35098 | 0.32915 | 0.30732 | 0.2855 | 0.26366 | |

1.07675 | 1.33379 | 1.5878 | 1.9614 | 2.5648 | 3.70468 | |

1464.18 | 1686.48 | 1835.4 | 1984.32 | 2133.24 | 2282.16 | |

25.746 E-6 | 21.6044 E-6 | 18.5098 E-6 | 15.3005 E-6 | 12.0234 E-6 | 8.70307 E-6 | |

25.746 E-6 | 21.6044 E-6 | 18.5098 E-6 | 15.3005 E-6 | 12.0234 E-6 | 8.70307 E-6 | |

10.5844 E-6 | 7.932 E-6 | 6.9852 E-6 | 6.3374 E-6 | 5.8663 E-6 | 5.5082 E-6 | |

0.4533 | 0.622 | 0.735 | 0.848 | 0.961 | 1.074 | |

0.4533 | 0.622 | 0.735 | 0.848 | 0.961 | 1.074 | |

0.2174 | 0.2626 | 0.30068 | 0.3553 | 0.43418 | 0.55808 | |

768.139 | 780.8944 | 787.7133 | 793.5087 | 798.495 | 802.8304 |

## 6. Results and discussions

Figures 2 and 3 show the verification test of normal bending deflection using Levy and Navier solutions, taking into consideration ANSYS 18.2 results. The normal bending deflection decreased with the increasing of plate aspect ratio because of the increasing in plate bending stiffness under the temperature effect

Figure 4 shows the convergence test of normal bending deflection with total degrees of freedom for different fiber volume fractions using ANSYS software. The normal central deflection decrease with the increasing of fiber volume fraction under the effect of temperature

Table 2 shows the analytic and simulation verification results of bending deflection under combined loadings for fiber volume fraction

Deflection | Levy method results | ANSYS 18.2 results | Percentage error ( |
---|---|---|---|

0.1853e-3 | 0.1880e-3 | 1.748 | |

−0.1777e-3 | −0.1882e-3 | 5.536 | |

0.7704e-5 | 0.7108 e-5 | 7.736 | |

−0.9859e-5 | −0.9365e-5 | 5.010 |

## 7. Conclusions

As mentioned in Introduction section, Levy and Navier solutions are used to describe the theory of bending deflection by taking into consideration the use of simply supported boundary condition from all edges. ANSYS software is used in the convergence test. The bending deflection value when

## Nomenclature

α1, α2 | thermal expansion coefficient in longitudinal and lateral directions, 1/C∘. |

ΔT | gradient uniform temperature, C∘. |

A1, A2 | bending moment due to temperature, N.m/C∘. |

Mxx,Myy, and Mxy | bending and twist moments, N.m. |

Qij | reduced stiffness elements, N/m2. |

w0 | midplane deflection along z-direction. |

zk, zk+1 | upper and lower lamina surface coordinates along z-direction, m. |

a, b | length of large and small spans of rectangular plate (m). |

m, n | double trigonometric of Furrier series. |

N | total number of layers. |