 Open access peer-reviewed chapter

# Analytic and Numerical Results of Bending Deflection of Rectangular Composite Plate

Written By

Louay S. Yousuf

Submitted: May 29th, 2020 Reviewed: August 14th, 2020 Published: December 23rd, 2020

DOI: 10.5772/intechopen.93592

From the Edited Volume

## Structural Integrity and Failure

Edited by Resat Oyguc and Faham Tahmasebinia

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## 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

## 2. Equations of motion in terms of displacements

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

σxxσyyσxy=Q11Q120Q12Q22000Q66εxxεyyγxyα1α20ΔTE1

The general bending equation of rectangular plate is as below:

2Mxxx2+22Mxyxy+2Myyy2=0E2

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

MxxMyyMxy=MxxMech.MyyMech.MxyMech.MxxTher.MyyTher.MxyTher.E3

where

MxxMech.MyyMech.MxyMech.=B11B120B12B22000B66uoxvoyuoy+voxD11D120D12D22000D662wox22woy222woxyE4

And,

MxxTher.MyyTher.=k=1NQ11Q12Q12Q22kα1α2kzkzk+1ΔTzdzE5

It can be assumed that all layers have Θ=0, and the same thickness Bij=0, as indicated in Figure 1. Substitute Eq. (4) and into Eq. (3), it can be obtained:

D114wox4+2D12+2D664wox2y2+D224woy4+Nxx2wox2=2MxxTher.x2+2MyyTher.y2E6

## 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 ΔTand in-plane force Nxx. It can be assumed that the temperature is varied linearly through the plate thickness, as in below:

Tmnz=4ab0a0bΔTxyzsinαmxsinβnydxdyE7

where.

T1is the out-off plane uniform temperature when the heat source is applied through the plate thickness.

And,

αm=a,m=1,2,3,,E8a
βn=b,n=1,2,3,,E8b

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

Tmnz=16T1zmnπ2E9

The thermal bending moment is defined as in the following:

MxxTher.MyyTher.=m=1n=1Mmn1Mmn2sinαmxsinβnyE10

where

Mmn1=k=1Nzkzk+1Q11α1+Q12α2TmnzzdzE11a
Mmn2=k=1Nzkzk+1Q12α1+Q22α2TmnzzdzE11b

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

wo=m=1n=1wmnsinαmxsinβnyE12

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

woxy=16T13π2m=1n=1A1αm2+A2βn2sinαmxsinβnymnD11αm4+2D12+2D66αm2βn2+D22βn4Nxxαm2E13

where

A1=k=1NQ11α1+Q12α2zk+13zk3E14a
A2=k=1NQ12α1+Q22α2zk+13zk3E14b

## 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 ΔT, in-plane force Nxx, and bending moment (Mo). As mentioned in the previous section that the temperature is varied linearly through the plate thickness, as below:

Tmz=2a0aΔTxzsinαmxdxE15

where

αm=a,m=1,2,3,,E16

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

Tmz=4T1zE17

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

D114wox4+Nxx2wox2=2MxxTher.x2E18

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

MxxTher.=m=1Mm1sinαmxE19

where

Mm1=k=1Nzkzk+1Q11α1TmzzdzE20

The solution of normal bending deflection is as below:

woxy=woxyH+woxPE21

To find woxP:

woxtp=m=1wmsinαmxE22

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

woxp=4T1A13πm=1αm2sinαmxmD11αm4Nxxαm2E23

where

A1=k=1NQ11α1zk+13zk3E24

To find woxyH, Eq. (6) will be:

D114wox4+2D12+2D664wox2y2+D224woy4+Nxx2wox2=0E25

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

woxyH=m=1YmsinαmxE26

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

woxyH=m=1N1coshαy+N2sinhαy+N3cosβy+N4sinβysinαmxE27

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

woxy=m=1N1coshαy+N2sinhαy+N3cosβy+N4sinβysinαmx+4T1A13πm=1αm2sinαmxmD11αm4Nxxαm2E28

The simply supported boundary conditions from all edges are assumed and the constants N1N2N3and N4are as below:

N1=β2Hα2+β24Moα2+β2D22E29
N2=coshαbβ2Hα2+β2sinhαb+4coshαbMoα2+β2sinhαbD22D12αm2+β2D22Hα2+β2sinhαbD224MoD12αm2Hα2+β2sinhαbD22E30
N3=α2Hα2+β2+4Moα2+β2D22E31
N4=cosβbα2Hα2+β2sinβb4cosβbMoα2+β2sinβbD22α2D22D12αm2Hα2+β2D22sinβb+4MoD12αm2Hα2+β2sinβbD22E32

where

H=4T1A1αm23πmD11αm4Nxxαm2E33

## 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 ΔT), (temperature affect ΔT+ Mo), (temperature affect ΔT+ Nxx), and (temperature affect ΔT+ Mo + Nxx). Multiple values of fiber volume fraction is used such as (25, 40, 50, 60, 70, and 80)%. Table 1 shows the mechanical and thermal properties of the simulated materials.

νf25.07%40%50%60%70%80%
ExcGPa.19.93330.403837.4198844.43551.45258.468
EycGPa.19.93330.403837.4198844.43551.45258.468
EzcGPa.3.08963.817464.533225.57937.2530210.3612
ν120.38350.350980.329150.307320.28550.26366
G12GPa.1.076751.333791.58781.96142.56483.70468
ρckg/m31464.181686.481835.41984.322133.242282.16
αxc1/C25.746 E-621.6044 E-618.5098 E-615.3005 E-612.0234 E-68.70307 E-6
αyc1/C25.746 E-621.6044 E-618.5098 E-615.3005 E-612.0234 E-68.70307 E-6
αzc1/C10.5844 E-67.932 E-66.9852 E-66.3374 E-65.8663 E-65.5082 E-6
kxcW/mC0.45330.6220.7350.8480.9611.074
kycW/mC0.45330.6220.7350.8480.9611.074
kzcW/mC0.21740.26260.300680.35530.434180.55808
CpcJ/kgC768.139780.8944787.7133793.5087798.495802.8304

### Table 1.

Mechanical and thermal properties of the simulated materials.

## 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 60Cand 15Cfor fiber volume fraction (25.076%). The bending deflection value when T=60Cis higher than the value of bending deflection when T=15Cbecause of the expansion and contraction through the plate laminate thickness. Figure 2.Normal bending deflection varying with laminate plate aspect ratio under temperature effect60C∘. Figure 3.Normal bending deflection varying with laminate plate aspect ratio under temperature effect−15C∘.

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 ΔT, bending moment Mo, and in-plane force Nxx. Figure 4.Convergence test of normal deflection and total degrees of freedom under the effect of temperatureΔT, bending momentMo, and in-plane forceNxx.

Table 2 shows the analytic and simulation verification results of bending deflection under combined loadings for fiber volume fraction υf=25.076%and plate aspect ratio (1.8). The value of central deflection of the system with combined loading (ΔT) is higher than the others of combined loading. The deflection of system with combined loading (ΔT+ Mo+ Nxx) and the system with loading (ΔT+ Nxx) is almost the same and in the opposite direction because bending moment has a small effect.

DeflectionLevy method resultsANSYS 18.2 resultsPercentage error (%)
ΔT0.1853e-30.1880e-31.748
ΔT+ Mo−0.1777e-3−0.1882e-35.536
ΔT+ Nxx0.7704e-50.7108 e-57.736
ΔT+ Mo+Nxx−0.9859e-5−0.9365e-55.010

## 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 T=60Cis higher than the value of bending deflection when T=15Cbecause of the expansion and contraction through the plate laminate thickness. The in-plane force (Nxx) has a great effect on the bending deflection value of composite laminate plate, but the bending moment (Mo) has a small effect on the bending deflection value. The normal deflection is decreased with the increasing of fiber volume fraction from 25.07%to 80%under the effect of ΔTand combined loading Mo+ Nxx. Moreover, the normal bending deflection is decreased with the increasing of aspect ratio from 0.8 to 2.4 under the effect of T=60Cand T=15C, respectively.