# An Intuitive Derivation of Beam Models of Arbitrary Order

## Abstract

**:**

## 1. Introduction

## 2. Form of the Present Beam Model

#### 2.1. General Definitions

#### 2.2. Kinematics

#### 2.3. Simplifying Assumptions

- 1.
- All externally applied loads act parallel to the z-axis.
- 2.
- All externally applied concentrated moments act about axes that are parallel to the y-axis.
- 3.
- All imposed linear displacements occur parallel to the z-axis.
- 4.
- All imposed angular displacements (rotations) occur about axes that are parallel to the y-axis.
- 5.
- The beam is configured such that none of the applied loadings result in the development of any non-zero values of ${v}_{n}$, ${\varphi}_{{\scriptscriptstyle v}}$, or ${\theta}_{{\scriptscriptstyle v}}$.
- 6.
- The beam is configured such that none of the applied loadings result in the development of any torsional deformations of the beam.

#### 2.4. Governing Equations

## 3. Section Constants and Section Functions for the Present Beam Model

#### 3.1. General

- 1.
- The beam comprises a multi-layered laminate, wherein each lamina is composed of a linear elastic orthotropic material that has three orthogonal symmetry planes, and wherein the surface normal of each of the aforementioned symmetry planes is parallel to one of the x, y, and z coordinate axes of the beam.
- 2.
- The undeformed beam exhibits a constant rectangular sectional geometry and a constant material composition along its length.
- 3.
- All stresses act parallel to the x-z plane.
- 4.
- All normal strains and normal stresses act parallel to the x-axis of the beam; therefore, the effects of Poisson’s ratio are ignored.

#### 3.2. Elasticity Relationships Pertaining to the First Term

#### 3.3. Elasticity Relationships Pertaining to Higher-Order Terms

#### 3.4. Truncation of the Infinite Series

#### 3.5. Calculation of Section Constants

#### 3.5.1. Conceptual Description of Procedure

#### 3.5.2. Assumption of Sinusoidal Loading to Simplify Procedure

## 4. Boundary Conditions for the Present Beam Model

## 5. Practical Implementation of the Present Beam Model

- 1.
- In order to utilize the present beam model, it is first necessary to calculate the various section constants that are included in its governing equations. The procedure that is described in Section 3.5.2 can be employed in order to calculate ${C}_{\left\{1\right\}}\cdots {C}_{\left\{{N}_{}\right\}}$. Accordingly, ${C}_{\left\{1\right\}}\cdots {C}_{\left\{{N}_{}\right\}}$ can be determined using Equation (95), provided that the value of ℵ is set approximately equal to zero. Once ${C}_{\left\{1\right\}}\cdots {C}_{\left\{{N}_{}\right\}}$ has been determined, ${A}_{\left\{1\right\}}\cdots {A}_{\left\{{N}_{}\right\}}$ can be calculated by substituting each value of ${C}_{\left\{i\right\}}$ into Equation (58), for each integer value of i from $i=1$ to $i={N}_{}-1$, inclusively.
- 2.
- Once ${C}_{\left\{1\right\}}\cdots {C}_{\left\{{N}_{}\right\}}$ has been determined, the equations and relationships that are presented in Section 3.2 and Section 3.3 can be employed in order to calculate the ${R}_{\left\{j\right\}}$ and ${Q}_{\left\{j\right\}}$ section functions that correspond to each integer value of j from $j=1$ to $j={N}_{}$, inclusively. In the case of a multi-layered laminated composite beam, each section function shall be expressed as a piecewise polynomial.
- 3.
- Once the ${C}_{\left\{1\right\}}\cdots {C}_{\left\{{N}_{}\right\}}$ and ${A}_{\left\{1\right\}}\cdots {A}_{\left\{{N}_{}\right\}}$ section constants have been determined, these section constants can be substituted into Equations (14), (16), (18) and (23), and the resulting expressions can be combined with Equations (1), (13), (15), (17) and (22) in order to define the general form of the governing equations of the present beam model. For any particular beam configuration, it is possible to establish at least one loading function that describes the loading that is imposed upon the beam; each loading function may be an expression of ${M}_{yy}$, ${V}_{z}$, or ${g}_{z}$. The specific governing equations that pertain to the beam can then be defined by substituting the aforementioned at least one loading function into the aforementioned general form of the governing equations. While adhering to the relevant boundary conditions in accordance with the provisions that are set out in Section 4, the governing equations can then be solved using one of numerous possible techniques, such as analytical methods, the finite difference method, the finite element method, or any other method that is applicable to the solution of ordinary differential equations. The solution of the governing equations facilitates the calculation of the values of ${w}_{n}$, ${\varphi}_{{\scriptscriptstyle w}}$, ${\theta}_{{\scriptscriptstyle w}}$, and ${\theta}_{\left\{j\right\}}$ that correspond to each x-coordinate, for each integer value of j from $j=1$ to $j={N}_{}$, inclusively.
- 4.
- Once the section functions have been calculated and the governing equations have been solved, the equations and relationships that are presented in Section 2.4, Section 3.2, and Section 3.3 can be employed in order to recover the sectional fields of ${u}_{{}_{N}}$, ${\epsilon}_{x}$, ${\gamma}_{xz}$, ${\sigma}_{x}$, and ${\tau}_{xz}$ that correspond to each section of the beam.

## 6. Validation of the Present Beam Model

## 7. Concluding Remarks

## 8. Consent for Publication

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Supplementary Definitions

## References

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**Figure 1.**Deformed shape of a transversely loaded beam, illustrating some of the geometric dimensions and kinematic variables that are relevant to the present beam model.

**Figure 2.**Loading configuration of the example beams used to validate the present beam model [5]. The reader is directed to the Consent for Publication section.

Physical Constraint | Boundary Conditions at Position of Constraint |
---|---|

Roller (no transverse deflection) | ${w}_{n}=0$ |

Guided (no rotation) | $\frac{d{w}_{n}}{dx}=0$ |

Fixed (clamped) | ${w}_{n}=0$, ${\varphi}_{{\scriptscriptstyle w}}=0$, and $\frac{{d}^{(2j-2)}{\varphi}_{{\scriptscriptstyle w}}}{d{x}^{(2j-2)}}=0$ |

Free End | $\frac{{d}^{(2j-1)}{\varphi}_{{\scriptscriptstyle w}}}{{dx}^{(2j-1)}}=0$ |

where $j$ is an integer value that satisfies $0<j\le {N}_{}$. |

**Table 2.**Laminates employed in the example beams used to validate the present beam model [5]. The reader is directed to the Consent for Publication section.

Laminate A | Laminate B | Laminate C | |
---|---|---|---|

Lamina 3 | Carbon ${0}^{\circ}$ | Carbon ${0}^{\circ}$ | Carbon ${0}^{\circ}$ |

${E}_{}=$ 119,000 $\mathrm{N}/{\mathrm{mm}}^{2}$ | ${E}_{}=$ 119,000 $\mathrm{N}/{\mathrm{mm}}^{2}$ | ${E}_{}=$ 119,000 $\mathrm{N}/{\mathrm{mm}}^{2}$ | |

${G}_{}=4230\mathrm{N}/{\mathrm{mm}}^{2}$ | ${G}_{}=4230\mathrm{N}/{\mathrm{mm}}^{2}$ | ${G}_{}=4230\mathrm{N}/{\mathrm{mm}}^{2}$ | |

$\mathrm{Thickness}=1\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ | $\mathrm{Thickness}=1\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ | $\mathrm{Thickness}=1\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ | |

Lamina 2 | Carbon ${0}^{\circ}$ | Carbon ${90}^{\circ}$ | Modified Carbon ${90}^{\circ}$ |

${E}_{}=$ 119,000 $\mathrm{N}/{\mathrm{mm}}^{2}$ | ${E}_{}=9270\mathrm{N}/{\mathrm{mm}}^{2}$ | ${E}_{}=9270\mathrm{N}/{\mathrm{mm}}^{2}$ | |

${G}_{}=4230\mathrm{N}/{\mathrm{mm}}^{2}$ | ${G}_{}=3228\mathrm{N}/{\mathrm{mm}}^{2}$ | ${G}_{}=323\mathrm{N}/{\mathrm{mm}}^{2}$ | |

$\mathrm{Thickness}=1\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ | $\mathrm{Thickness}=1\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ | $\mathrm{Thickness}=1\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ | |

Lamina 1 | Carbon ${0}^{\circ}$ | Aluminium | Carbon ${0}^{\circ}$ |

${E}_{}=$ 119,000 $\mathrm{N}/{\mathrm{mm}}^{2}$ | ${E}_{}=$ 68,900 $\mathrm{N}/{\mathrm{mm}}^{2}$ | ${E}_{}=$ 119,000 $\mathrm{N}/{\mathrm{mm}}^{2}$ | |

${G}_{}=4230\mathrm{N}/{\mathrm{mm}}^{2}$ | ${G}_{}=$ 26,200 $\mathrm{N}/{\mathrm{mm}}^{2}$ | ${G}_{}=4230\mathrm{N}/{\mathrm{mm}}^{2}$ | |

$\mathrm{Thickness}=1\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ | $\mathrm{Thickness}=1\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ | $\mathrm{Thickness}=1\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ |

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**MDPI and ACS Style**

Honickman, H.
An Intuitive Derivation of Beam Models of Arbitrary Order. *Appl. Mech.* **2023**, *4*, 109-140.
https://doi.org/10.3390/applmech4010008

**AMA Style**

Honickman H.
An Intuitive Derivation of Beam Models of Arbitrary Order. *Applied Mechanics*. 2023; 4(1):109-140.
https://doi.org/10.3390/applmech4010008

**Chicago/Turabian Style**

Honickman, Hart.
2023. "An Intuitive Derivation of Beam Models of Arbitrary Order" *Applied Mechanics* 4, no. 1: 109-140.
https://doi.org/10.3390/applmech4010008