Lateral instability of sandwich beams under uniform bending

Abstract

This paper investigates the lateral-torsional instability of soft-core sandwich beams. This physical phenomenon, which has received limited attention in the framework of sandwich structures, is investigated here in the context of fully nonlinear analysis for the first time. The research addresses, quantifies, and explores the lateral-torsional instability with emphasis on the unique features of the sandwich structure and its geometrically nonlinear response. The research questions relate to the nature of the instability and to the role of the deformability of the core layer in the unique mechanism. The results of the investigation include a new, high-order, nonlinear model for the analysis of the torsional-lateral response as well as new insight into the evolution of the phenomenon in sandwich beams.

Keywords

Lateral-torsional instability lateral buckling sandwich beams soft core sandwich beams high-order sandwich theory geometrically nonlinear analysis

Introduction

Modern sandwich structures are comprised of thin and stiff face sheets placed on the outer surfaces of a flexible and lightweight core. This configuration provides the composite structure with improved flexural stiffness and strength. Nevertheless, the response of a sandwich beam may not be limited to flexure about the main axis but instabilities that involve additional and more complex structural mechanisms may also evolve. The lateral-torsional instability is such. The lateral instability occurs in beams subjected to bending about the main axis and involves torsion and bi-axial flexure. While global instabilities and wrinkling of sandwich structures are well documented in the literature (e.g., in Carlsson and Kardomateas¹), lateral instability of such sandwich beams is not. The nature of lateral instability in soft core sandwich beams and the mechanisms involved in such instability are open questions. This paper aims to answer these questions.

Lateral torsional instability is a geometrically nonlinear phenomenon where flexure about the major axis suddenly transforms into a bi-axial flexure and a torsional rotation about the longitudinal axis (Trahair²). The lateral instability is triggered by bending while the axial stress resultant is zero. The cross-sections of an I-section beam after reaching lateral instability and its soft-core sandwich beam counterpart are illustrated in Figure 1. In the I-shaped beam, the cross-section does not distort the displaced state. In soft-core sandwich beams, distortion of the cross-section is anticipated. These unique features of the soft-core beam go way beyond any lateral instability theory developed for I-shaped beams. As such, they require a special theory that takes the deformability of the core into account. Such theory was not found in the literature.

The anticipated cross-sectional distortion requires the extension of the soft-core sandwich theory beyond the fundamental beam, plate, or shell theories. Approaches that are based on high-order deformation theories face this challenge. The displacement field in such theories consists of high-order polynomials (hence their name) or other rich functional forms. Examples of such theories are the High-order Sandwich Panel Theory (HSAPT) in Frostig et al.,³ and the Extended High-order Sandwich Panel Theory (EHSAPT) in Phan et al.⁴ The first neglects the longitudinal stiffness of the core and provides a closed-form solution for the core’s displacement fields. The second theory uses kinematic assumption of the same polynomial order as the HSAPT but takes the longitudinal stiffness of the core into account. Rabinovitch⁵ and later Odessa et al.⁶ extended the theory to account for delaminations between the layers utilizing cohesive laws. These works lay the foundation for the analysis of soft-core sandwich beams, but they do not address the case of torsion or lateral instability.

Teimoori et al.⁷ developed a model for Saint Venant’s torsion in coated rectangular-shaped beams, which can also be applied to torsion in sandwich beams of similar geometries. The model is solved using the Finite Fourier Cosine Transform but it only refers to Saint Venant’s torsional mechanism. Warping, the features of the soft core, or the typical localized effects are left out of the equation. Also, the torsional phenomenon is addressed in the geometrically linear regime. An extension of this model to the nonlinear regime was not found in the literature. Pozorski⁸ developed an analytical model that employs classical theories (e.g., sectoral coordinates) for the two torsional mechanisms (Saint-Venant, warping). The torsional differential equation is solved globally without specific reference to the layered configuration. Considering the torsional mechanisms clarifies their interaction but the model is also limited to the linear regime.

Wurf et al.⁹ developed a generalized EHSAPT model for the geometrically linear analysis of sandwich beams subjected to torsion and bending. The model uses an enriched displacement field of the core and the face sheets with kinematics that introduces the Saint-Venant and the warping torsional mechanisms. The generalized torsional form of the EHSAPT kinematics reduces the 3D elasticity-based formulation to a set of ODEs. Nevertheless, this work is also limited to geometrically linear regime and, therefore, does not account for the lateral instability.

Challamel and Girhammar¹⁰ investigated the lateral instability of layered sandwich beams with interlayer slip. Only the outer layers (face sheets) were explicitly modeled, and the core layer was referred to as the shear connection between the face sheets linking their interlayer slip through the respective core modulus. This simplifies the direct contribution of the core but it tends to overlook the unique stress fields that evolve in the core. This particularly refers to the warping mechanism, which tends to trigger significant localized stresses, see Wurf et al.⁹ The bifurcation analysis implemented in this model yields the critical lateral buckling load, but the post-buckling behavior and the complete nonlinear behavior still have to be explored.

Several models for the geometrically nonlinear bending analysis of sandwich beams are available in the literature (see the review in Carlsson and Kardomateas¹) but they mainly deal with the global buckling of the sandwich beam or with localized wrinkling of the compressed face sheet, but not with the torsional instability. Phan et al.¹¹ explored the global and wrinkling nonlinearities in sandwich beams subjected to compression using HSAPT and EHSAPT. This work showed a notable difference between the two high-order sandwich theories and indicated that the EHSAPT is better suited for the description of such instabilities. Yuan and Kardomateas^12,13 further explored the geometrically nonlinear response of sandwich beams using the EHSAPT and analytically investigated the buckling and the post-buckling behavior of such beams. Yuan and Kardomateas¹⁴ applied this concept to the dynamic stability of soft-core sandwich beams. These works explore and quantify a spectrum of features of the nonlinear response of sandwich beams, but they do not deal with the lateral-torsional nonlinearity. In fact, the survey of the literature does not reveal any work that addresses this challenge. The lateral-torsional instability in soft core sandwich beams is still an open question.

The goal of this study is to explore, quantify, and explain the lateral-torsional instability phenomenon in soft core sandwich beams. This particularly refers to the unique features of the soft core and their role in the geometrically nonlinear response. To face this challenge, a general analytical model for the combined flexural and torsional instability is developed. Through this model, the investigation aims to contribute to the understanding of the structural behavior of the soft core sandwich structure and its failure modes.

Mathematical formulation

The mathematical formulation refers to the geometrically nonlinear regime. The geometrically linear case is addressed in Wurf et al.⁹ The model assumes large deformations, moderate rotations, and small strains in the face sheets and linear-elastic material behavior of all layers. Each layer is modeled independently, and they are joined by compatibility conditions at the interfaces. The geometry, coordinates, and notations appear in Figure 2.

The nonlinear model is derived using the principle of minimum potential energy:

δ (U + V) = 0

(1)

where

U

is the strain energy,

V

is the potential of the external loads, and

δ

is the variational operator. The first variation of the strain energy is:

δ U = {δ U}^{t} + {δ U}^{c} + {δ U}^{b}

(2)

where the subscripts

t, c, b

designate the top face sheet, core, and bottom face sheet, respectively.

The first variation of the strain energy in the $i$ -th layer is:

{δ U}^{i} = \int_{V_{i}} \begin{array}{l} [σ_{x x}^{i} (x, y_{i}, z_{i}) \cdot {δ ε}_{x x}^{i} (x, y_{i}, z_{i}) + σ_{y y}^{i} (x, y_{i}, z_{i}) \cdot {δ ε}_{y y}^{i} (x, y_{i}, z_{i}) \\ + σ_{z z}^{i} (x, y_{i}, z_{i}) \cdot {δ ε}_{z z}^{i} (x, y_{i}, z_{i}) + τ_{x y}^{i} (x, y_{i}, z_{i}) \cdot {δ γ}_{x y}^{i} (x, y_{i}, z_{i}) \\ + τ_{x z}^{i} (x, y_{i}, z_{i}) \cdot {δ γ}_{x z}^{i} (x, y_{i}, z_{i}) + τ_{y z}^{i} (x, y_{i}, z_{i}) \cdot {δ γ}_{y z}^{i} (x, y_{i}, z_{i})] d V_{i} \end{array}

(3)

where

V_{i}

is the volume of the

i

-th layer (

i = t, c, b

σ

and

τ

denote normal and shear stresses, respectively,

ε

and

γ

denote normal strain and shear angles, respectively, and

x

y_{i}

, and

z_{i}

are the coordinates of the

i

-th layer.

Kinematic relations for the face sheets

The face sheets are modeled as beams that deform in all three directions $x$ , $y$ , and $z$ . A generalized beam theory that includes the rotation about the longitudinal axis (torsional rotation) and the lateral deformations (in $y$ direction) is separately attributed to each face sheet:

u^{i} (x, y_{i}, z_{i}) = u_{0}^{i} (x) - y_{i} \cdot φ_{z}^{i} (x) - z_{i} \cdot φ_{y}^{i} (x)

(4)

v^{i} (x, y_{i}, z_{i}) = v_{0}^{i} (x) - z_{i} \cdot θ_{x}^{i} (x)

w^{i} (x, y_{i}, z_{i}) = w_{0}^{i} (x) + y_{i} \cdot θ_{x}^{i} (x)

where

u^{i} (x, y_{i}, z_{i})

v^{i} (x, y_{i}, z_{i})

, and

w^{i} (x, y_{i}, z_{i})

are the axial, lateral, and vertical components of the displacement field, respectively, of the

i

-th (

i = t, b

) face sheet,

u_{0}^{i} (x)

v_{0}^{i} (x)

, and

w_{0}^{i} (x)

are the displacements, respectively of the reference axis,

φ_{y}^{i} (x)

and

φ_{z}^{i} (x)

are the bending rotations about the lateral and vertical axes, respectively, and

θ_{x}^{i} (x)

is the torsional rotation about the longitudinal axis.

The kinematic relations assume small strains, moderate rotations, and large displacements:

ε_{x x}^{i} (x, y_{i}, z_{i}) = u_{, x}^{i} (x, y_{i}, z_{i}) + \frac{1}{2} \cdot {(v_{, x}^{i} (x, y_{i}, z_{i}))}^{2} + \frac{1}{2} \cdot {(w_{, x}^{i} (x, y_{i}, z_{i}))}^{2} = u_{, x}^{i} (x, y_{i}, z_{i})

+ \frac{1}{2} \cdot {(F_{v} \cdot v_{0, x}^{i} (x) - F_{θ} \cdot z_{i} \cdot θ_{x, x}^{i} (x))}^{2} + \frac{1}{2} \cdot {(F_{w} \cdot w_{0, x}^{i} (x) + F_{θ} \cdot y_{i} \cdot θ_{x, x}^{i} (x))}^{2}

(5)

\begin{array}{l} ε_{y y}^{i} (x, y_{i}, z_{i}) = v_{, y}^{i} (x, y_{i}, z_{i}) & γ_{x y}^{i} (x, y_{i}, z_{i}) = u_{, y}^{i} (x, y_{i}, z_{i}) + v_{, x}^{i} (x, y_{i}, z_{i}) \\ ε_{z z}^{i} (x, y_{i}, z_{i}) = w_{, z}^{i} (x, y_{i}, z_{i}) & γ_{x z}^{i} (x, y_{i}, z_{i}) = u_{, z}^{i} (x, y_{i}, z_{i}) + w_{, x}^{i} (x, y_{i}, z_{i}) \\ γ_{y z}^{i} (x, y_{i}, z_{i}) = v_{, z}^{i} (x, y_{i}, z_{i}) + w_{, y}^{i} (x, y_{i}, z_{i}) \end{array}

where the comma designates a partial derivative and

F_{v}

F_{w}

, and

F_{θ}

are flags (

= 0

1

) that allow to control the contribution of the nonlinear terms.

F_{v} = F_{w} = F_{θ} = 0

reduce the formulation to the linear model of Wurf et al.⁹

The stress resultants at the $i$ -th face sheet ( $i = t, b$ ) are as follows (also see Wurf et al.⁹):

\begin{array}{l} N_{x}^{i} (x) = \int_{A_{i}} σ_{x x}^{i} (x, y_{i}, z_{i}) {d A}_{i} & V_{y}^{i} (x) = \int_{A_{i}} τ_{x y}^{i} (x, y_{i}, z_{i}) {d A}_{i} \\ M_{z}^{i} (x) = \int_{A_{i}} y_{i} \cdot σ_{x x}^{i} (x, y_{i}, z_{i}) {d A}_{i} & V_{z}^{i} (x) = \int_{A_{i}} τ_{x z}^{i} (x, y_{i}, z_{i}) {d A}_{i} \\ M_{y}^{i} (x) = \int_{A_{i}} z_{i} \cdot σ_{x x}^{i} (x, y_{i}, z_{i}) {d A}_{i} & M_{R}^{i} (x) = \int_{A_{i}} (z_{i}^{2} + y_{i}^{2}) \cdot σ_{x x}^{i} (x, y_{i}, z_{i}) {d A}_{i} \end{array}

(6)

M_{x}^{i} (x) = \int_{A_{i}} [τ_{x z}^{i} (x, y_{i}, z_{i}) \cdot y_{i} - τ_{x y}^{i} (x, y_{i}, z_{i}) \cdot z_{i}] {d A}_{i}

where

A_{i}

is the cross-sectional area,

N_{x}^{i} (x)

is the axial resultant,

V_{y}^{i} (x)

and

V_{z}^{i} (x)

are the lateral and vertical shear resultants, respectively,

M_{z}^{i} (x)

and

M_{y}^{i} (x)

are the bending moments about the vertical and lateral axes, respectively,

M_{x}^{i}

is the torque about the longitudinal axis, and

M_{R}^{i}

is the high-order Wagner’s moment.¹⁵

The first variation of the potential of the external loads, which exerted on the face sheets only, is:

\begin{array}{l} δ V = {δ V}^{t} + {δ V}^{b} & \begin{array}{l} {δ V}^{i} = - \int [n_{x}^{i} (x) \cdot {δ u}_{0}^{i} (x) + m_{x}^{i} (x) \cdot {δ θ}_{x}^{i} (x) \\ + q_{y}^{i} (x) \cdot {δ v}_{0}^{i} (x) + q_{z}^{i} (x) \cdot {δ w}_{0}^{i} (x)] d x \end{array} \end{array}

(7)

where

n_{x}^{i} (x)

q_{y}^{i} (x)

, and

q_{z}^{i} (x)

are distributed loads in the axial, lateral, and vertical directions, respectively, and

m_{x}^{i} (x)

is a distributed torque about the longitudinal axis.

Kinematic relations for the core

The core layer is modeled as a 3D elastic medium and follows the high-order kinematic assumption developed in Wurf et al.⁹ for the torsional analysis of soft-core sandwich panels:

u^{c} (x, y_{c}, z_{c}) = u_{0}^{c} (x) - z_{c} \cdot u_{1}^{c} (x) + z_{c}^{2} \cdot u_{2}^{c} (x) + z_{c}^{3} \cdot u_{3}^{c} (x) + z_{c} \cdot y_{c} \cdot u_{4}^{c} (x) - y_{c} \cdot u_{5}^{c} (x)

(8)

v^{c} (x, y_{c}, z_{c}) = v_{0}^{c} (x) - z_{c} \cdot θ_{x}^{c} (x) + z_{c}^{2} \cdot v_{2}^{c} (x) + z_{c}^{3} \cdot v_{3}^{c} (x)

w^{c} (x, y_{c}, z_{c}) = w_{0}^{c} (x) + z_{c} \cdot w_{1}^{c} (x) + z_{c}^{2} \cdot w_{2}^{c} (x) + y_{c} \cdot θ_{x}^{c} (x)

where

u_{0}^{c} (x)

v_{0}^{c} (x)

, and

w_{0}^{c} (x)

are the axial, lateral, and vertical displacements of the core’s reference axis, respectively,

u_{1}^{c} (x)

and

u_{5}^{c} (x)

are the bending rotations about the lateral and vertical axes, respectively,

θ_{x}^{c} (x)

is the torsional rotation,

u_{4}^{c} (x)

is the warping displacement, and

u_{2}^{c} (x)

u_{3}^{c} (x)

v_{2}^{c} (x)

v_{3}^{c} (x)

w_{1}^{c} (x)

, and

w_{2}^{c} (x)

are the high-order displacement components.

Frostig et al.¹⁶ showed that the nonlinear part of the kinematic assumption for the core does not play an important role in the geometrically nonlinear response of the panel. The comparison of such nonlinear analysis with a geometrically linear behavior of small deformations shows that the linear kinematic relations for the core are sufficient while the nonlinear kinematics of the face sheets plays the major role in the response. The kinematic relations of the core layer follow this observation and reduce to the linear form as follows:

\begin{array}{l} ε_{x x}^{c} (x, y_{c}, z_{c}) = u_{, x}^{c} (x, y_{c}, z_{c}) & γ_{x y}^{c} (x, y_{c}, z_{c}) = u_{, y}^{c} (x, y_{c}, z_{c}) + v_{, x}^{c} (x, y_{c}, z_{c}) \\ ε_{y y}^{c} (x, y_{c}, z_{c}) = v_{, y}^{c} (x, y_{c}, z_{c}) & γ_{x z}^{c} (x, y_{c}, z_{c}) = u_{, z}^{c} (x, y_{c}, z_{c}) + w_{, x}^{c} (x, y_{c}, z_{c}) \\ ε_{z z}^{c} (x, y_{c}, z_{c}) = w_{, z}^{c} (x, y_{c}, z_{c}) & γ_{y z}^{c} (x, y_{c}, z_{c}) = v_{, z}^{c} (x, y_{c}, z_{c}) + w_{, y}^{c} (x, y_{c}, z_{c}) \end{array}

(9)

The compatibility conditions at the interfaces of the face-sheets and the core are:

\begin{array}{l} u^{c} (x, y_{c}, z_{c} = - \frac{h_{c}}{2}) = u^{t} (x, y_{t}, z_{t} = \frac{h_{t}}{2}) & u^{c} (x, y_{c}, z_{c} = \frac{h_{c}}{2}) = u^{b} (x, y_{b}, z_{b} = - \frac{h_{b}}{2}) \\ v^{c} (x, y_{c}, z_{c} = - \frac{h_{c}}{2}) = v^{t} (x, y_{t}, z_{t} = \frac{h_{t}}{2}) & v^{c} (x, y_{c}, z_{c} = \frac{h_{c}}{2}) = v^{b} (x, y_{b}, z_{b} = - \frac{h_{b}}{2}) \\ w^{c} (x, y_{c}, z_{c} = - \frac{h_{c}}{2}) = w^{t} (x, y_{t}, z_{t} = \frac{h_{t}}{2}) & w^{c} (x, y_{c}, z_{c} = \frac{h_{c}}{2}) = w^{b} (x, y_{b}, z_{b} = - \frac{h_{b}}{2}) \end{array}

(10)

The compatibility conditions (Equation (10)) define the core’s high-order components ( $u_{2}^{c} (x)$ , $u_{3}^{c} (x)$ , $v_{2}^{c} (x)$ , $v_{3}^{c} (x)$ , $w_{1}^{c} (x)$ , $w_{2}^{c} (x)$ ) in terms of the remaining displacements and rotations. This procedure affects the formulation of the boundary conditions and couples the stress resultants at the free edge. As a result, the stress resultants of the core are no longer the variational counterparts of the displacement components.

The stress resultants of core are:

M_{z^{k} y^{l}}^{c} (x) = \int_{A_{c}} z_{c}^{k} \cdot y_{c}^{l} \cdot σ_{x x}^{c} (x, y_{c}, z_{c}) {d A}_{c} \begin{array}{l} k = 0 \dots 3 \\ l = 0 \dots 1 \end{array}

(11)

R_{z^{k} y^{l}}^{c} (x) = \int_{A_{c}} z_{c}^{k} \cdot y_{c}^{l} \cdot σ_{z z}^{c} (x, y_{c}, z_{c}) {d A}_{c} \begin{array}{l} k = 0 \dots 2 \\ l = 0 \dots 1 \end{array}

M_{x}^{c} (x) = \int_{A_{c}} [τ_{x z}^{c} (x, y_{c}, z_{c}) \cdot y_{c} - τ_{x y}^{c} (x, y_{c}, z_{c}) \cdot z_{c}] {d A}_{c}

(12)

{Q x y}_{z^{k} y^{l}}^{c} (x) = \int_{A_{c}} z_{c}^{k} \cdot y_{c}^{l} \cdot τ_{x y}^{c} (x, y_{c}, z_{c}) {d A}_{c} \begin{array}{l} k = 0 \dots 3 \\ l = 0 \dots 1 \end{array}

{Q x z}_{z^{k} y^{l}}^{c} (x) = \int_{A_{c}} z_{c}^{k} \cdot y_{c}^{l} \cdot τ_{x z}^{c} (x, y_{c}, z_{c}) {d A}_{c} \begin{array}{l} k = 0 \dots 3 \\ l = 0 \dots 1 \end{array}

{Q y z}_{z^{k} y^{l}}^{c} (x) = \int_{A_{c}} z_{c}^{k} \cdot y_{c}^{l} \cdot τ_{y z}^{c} (x, y_{c}, z_{c}) {d A}_{c} \begin{array}{l} k = 0 \dots 3 \\ l = 0 \dots 1 \end{array}

where l and k are powers of the coordinates and

M_{x}^{c} (x)

is the torque about the longitudinal axis.

Change of variables

To simplify the application of the boundary conditions, a change of variables is implemented. This procedure transforms the “Physical” stress resultants of equations (11) and (12) into new “Natural” ones:

N_{x, N}^{i} (x) = N_{x}^{i} (x) + \frac{2 \cdot M_{z^{2} y^{0}}^{c} (x)}{h_{c}^{2}} - β^{i} \cdot \frac{4 \cdot M_{z^{3} y^{0}}^{c} (x)}{h_{c}^{3}}

(13)

V_{y, N}^{i} (x) = v_{y}^{i} (x) + \frac{2 \cdot {Q x y}_{z^{2} y^{0}}^{c} (x)}{h_{c}^{2}} - β^{i} \cdot \frac{4 \cdot {Q x y}_{z^{3} y^{0}}^{c} (x)}{h_{c}^{3}} + F_{v} \cdot N_{x}^{i} (x) \cdot v_{0, x}^{i} (x)

- F_{v} \cdot F_{θ} \cdot M_{y}^{i} (x) \cdot θ_{x, x}^{i} (x)

V_{z, N}^{i} (x) = v_{Z}^{i} (x) - β^{i} \cdot \frac{{Q x z}_{z^{1} y^{0}}^{c} (x)}{h_{c}} + \frac{2 \cdot {Q x z}_{z^{2} y^{0}}^{c} (x)}{h_{c}^{2}} + F_{w} \cdot N_{x}^{i} (x) \cdot w_{0, x}^{i} (x)

+ F_{w} \cdot F_{θ} \cdot M_{z}^{i} (x) \cdot θ_{x, x}^{i} (x)

M_{z, N}^{i} (x) = M_{z}^{i} (x) + \frac{2 \cdot M_{z^{2} y^{1}}^{c} (x)}{h_{c}^{2}} - β^{i} \cdot \frac{4 \cdot M_{z^{3} y^{1}}^{c} (x)}{h_{c}^{3}}

M_{y, N}^{i} (x) = M_{y}^{i} (x) + β^{i} \cdot \frac{M_{z^{2} y^{0}}^{c} (x) \cdot h_{i}}{h_{c}^{2}} - \frac{2 \cdot M_{z^{3} y^{0}}^{c} (x) \cdot h_{i}}{h_{c}^{3}}

M_{x, N}^{i} (x) = M_{x}^{i} (x) - β^{i} \cdot \frac{{Q x z}_{z^{1} y^{1}}^{c} (x)}{h_{c}} - β^{i} \cdot \frac{{Q x y}_{z^{2} y^{0}}^{c} (x) \cdot h_{i}}{h_{c}^{2}} + \frac{2 \cdot {Q x z}_{z^{2} y^{1}}^{c} (x)}{h_{c}^{2}}

+ \frac{2 \cdot {Q x y}_{z^{3} y^{0}}^{c} (x) \cdot h_{i}}{h_{c}^{3}} + F_{w} \cdot F_{θ} \cdot M_{z}^{i} (x) \cdot w_{0, x}^{i} (x)

- F_{v} \cdot F_{θ} \cdot M_{y}^{i} (x) \cdot v_{0, x}^{i} (x) + F_{θ} \cdot M_{R}^{i} (x) \cdot θ_{x, x}^{i} (x)

\begin{array}{l} N_{x, N}^{c} (x) = M_{z^{0} y^{0}}^{c} (x) - \frac{4 \cdot M_{z^{2} y^{0}}^{c} (x)}{h_{c}^{2}} & V_{y, N}^{c} (x) = {Q x y}_{z^{0} y^{0}}^{c} (x) - \frac{4 \cdot {Q x y}_{z^{2} y^{0}}^{c} (x)}{h_{c}^{2}} \\ M_{z, N}^{c} (x) = M_{z^{0} y^{1}}^{c} (x) - \frac{4 \cdot M_{z^{2} y^{1}}^{c} (x)}{h_{c}^{2}} & V_{z, N}^{c} (x) = {Q x z}_{z^{0} y^{0}}^{c} (x) - \frac{4 \cdot {Q x z}_{z^{2} y^{0}}^{c} (x)}{h_{c}^{2}} \\ M_{y, N}^{c} (x) = M_{z^{1} y^{0}}^{c} (x) - \frac{4 \cdot M_{z^{3} y^{0}}^{c} (x)}{h_{c}^{2}} & M_{o, N}^{c} (x) = M_{z^{1} y^{1}}^{c} (x) - \frac{4 \cdot M_{z^{3} y^{1}}^{c} (x)}{h_{c}^{2}} \end{array}

(14)

M_{x, N}^{c} (x) = M_{x}^{c} (x) + \frac{4 \cdot {Q x y}_{z^{3} y^{0}}^{c} (x)}{h_{c}^{2}} - \frac{4 \cdot {Q x z}_{z^{2} y^{1}}^{c} (x)}{h_{c}^{2}}

where

β^{t} = 1

and

β^{b} = - 1

N_{x, N}^{i} (x)

is the “Natural” axial stress resultant,

V_{y, N}^{i} (x)

and

V_{z, N}^{i} (x)

are the “Natural” lateral and vertical shear stress resultants, respectively,

M_{x, N}^{i} (x)

M_{z, N}^{i} (x)

, and

M_{y, N}^{i} (x)

are the “Natural” torque and bending moment resultants about the respective axis, and

M_{o, N}^{c} (x)

is the “Natural” high-order torque stress resultant (Bi-moment) in the core.

Equilibrium equations

The equilibrium equations read:

N_{x, N, x}^{i} (x) = {- n}_{x}^{i} (x) + \frac{4 \cdot {Q x z}_{z^{1} y^{0}}^{c} (x)}{h_{c}^{2}} - β^{i} \cdot \frac{12 \cdot {Q x z}_{z^{2} y^{0}}^{c} (x)}{h_{c}^{3}}

V_{y, N, x}^{i} (x) = {- q}_{y}^{i} (x) + \frac{4 \cdot {Q y z}_{z^{1} y^{0}}^{c} (x)}{h_{c}^{2}} - β^{i} \cdot \frac{12 \cdot {Q y z}_{z^{2} y^{0}}^{c} (x)}{h_{c}^{3}}

V_{z, N, x}^{i} (x) = {- q}_{z}^{i} (x) - β^{i} \cdot \frac{{R z}_{z^{0} y^{0}}^{c} (x)}{h_{c}} + \frac{4 \cdot {R z}_{z^{1} y^{0}}^{c} (x)}{h_{c}^{2}}

M_{z, N, x}^{i} (x) = V_{y, N}^{i} (x) + \frac{4 \cdot {Q x z}_{z^{1} y^{1}}^{c} (x)}{h_{c}^{2}} - \frac{2 \cdot {Q x y}_{z^{2} y^{0}}^{c} (x)}{h_{c}^{2}} + β^{i} \cdot \frac{4 \cdot {Q x y}_{z^{3} y^{0}}^{c} (x)}{h_{c}^{3}}

- β^{i} \cdot \frac{12 \cdot {Q x z}_{z^{2} y^{1}}^{c} (x)}{h_{c}^{3}} + F_{v} \cdot F_{θ} \cdot M_{y}^{i} (x) \cdot θ_{x, x}^{i} (x) - F_{v} \cdot N_{x}^{i} (x) \cdot v_{0, x}^{i} (x)

(15)

M_{y, N, x}^{i} (x) = V_{z, N}^{i} (x) + β^{i} \cdot \frac{{Q x z}_{z^{1} y^{0}}^{c} (x)}{h_{c}} + β^{i} \cdot \frac{2 \cdot {Q x z}_{z^{1} y^{0}}^{c} (x) \cdot h_{i}}{h_{c}^{2}}

- \frac{2 \cdot {Q x z}_{z^{2} y^{0}}^{c} (x)}{h_{c}^{2}} - \frac{6 \cdot {Q x z}_{z^{2} y^{0}}^{c} (x) \cdot h_{i}}{h_{c}^{3}} - F_{w} \cdot F_{θ} \cdot M_{z}^{i} (x) \cdot θ_{x, x}^{i} (x) - F_{w} \cdot N_{x}^{i} (x) \cdot w_{0, x}^{i} (x)

M_{x, N, x}^{i} (x) = - β^{i} \cdot \frac{{R z}_{z^{0} y^{1}}^{c} (x)}{h_{c}} + \frac{4 \cdot {R z}_{z^{1} y^{1}}^{c} (x)}{h_{c}^{2}} - β^{i} \cdot \frac{2 \cdot {Q y z}_{z^{1} y^{0}}^{c} (x) \cdot h_{i}}{h_{c}^{2}} + \frac{6 \cdot {Q y z}_{z^{2} y^{0}}^{c} (x) \cdot h_{i}}{h_{c}^{3}}

N_{x, N, x}^{c} (x) = - \frac{8 \cdot {Q x z}_{z^{1} y^{0}}^{c} (x)}{h_{c}^{2}} M_{z, N, x}^{c} (x) = V_{y, N}^{c} (x) + \frac{4 \cdot {Q x y}_{z^{2} y^{0}}^{c} (x)}{h_{c}^{2}} - \frac{8 \cdot {Q x z}_{z^{1} y^{1}}^{c} (x)}{h_{c}^{2}}

(16)

V_{y, N, x}^{c} (x) = - \frac{8 \cdot {Q y z}_{z^{1} y^{0}}^{c} (x)}{h_{c}^{2}} M_{y, N, x}^{c} (x) = V_{z, N}^{c} (x) - \frac{8 \cdot {Q x z}_{z^{2} y^{0}}^{c} (x)}{h_{c}^{2}}

V_{z, N, x}^{c} (x) = - \frac{8 \cdot {R z}_{z^{1} y^{0}}^{c} (x)}{h_{c}^{2}} M_{x, N, x}^{c} (x) = - \frac{8 \cdot {R z}_{z^{1} y^{1}}^{c} (x)}{h_{c}^{2}} + \frac{12 \cdot {Q y z}_{z^{2} y^{0}}^{c} (x)}{h_{c}^{2}}

M_{o, N, x}^{c} (x) = M_{x, N}^{c} (x) + 2 \cdot {Q x y}_{z^{1} y^{0}}^{c} (x) - \frac{4 \cdot {Q x y}_{z^{3} y^{0}}^{c} (x)}{h_{c}^{2}} - \frac{8 \cdot {Q x z}_{z^{2} y^{1}}^{c} (x)}{h_{c}^{2}}

The equilibrium equations consists of two types of nonlinear components. The first one reflects the interaction of the axial load and the derivative of the respective displacement. The second includes the bending moment multiplied by the derivative of the torsional rotation. These terms stem from the interaction of the bending about the two transversal directions through the torsional rotation. The “Natural” stress resultant of the torque consists of three different nonlinear components. One is associated with the second-order Wagner effects and the other two link the bending moments through the bending angle to the torque. This link reflects the nonlinear interaction between the bending moments and the face sheets torque.

Constitutive relations

The constitutive relations for the core are derived using the three-dimensional Hooke’s law of isotropic materials:

{\begin{cases} \begin{array}{l} σ_{x x}^{c} (x, y_{c}, z_{c}) \\ σ_{y y}^{c} (x, y_{c}, z_{c}) \end{array} \\ \begin{array}{l} σ_{z z}^{c} (x, y_{c}, z_{c}) \\ τ_{x y}^{c} (x, y_{c}, z_{c}) \end{array} \\ \begin{array}{l} τ_{x z}^{c} (x, y_{c}, z_{c}) \\ τ_{y z}^{c} (x, y_{c}, z_{c}) \end{array} \end{cases}} = [\begin{array}{l} \begin{array}{l} \begin{array}{l} \begin{array}{l} a_{1} \cdot E_{c} \\ a_{2} \cdot E_{c} \end{array} \\ \begin{array}{l} a_{2} \cdot E_{c} \\ 0 \end{array} \\ \begin{array}{l} 0 \\ 0 \end{array} \end{array} & \begin{array}{l} \begin{array}{l} a_{2} \cdot E_{c} \\ a_{1} \cdot E_{c} \end{array} \\ \begin{array}{l} a_{2} \cdot E_{c} \\ 0 \end{array} \\ \begin{array}{l} 0 \\ 0 \end{array} \end{array} \end{array} & \begin{array}{l} \begin{array}{l} \begin{array}{l} a_{2} \cdot E_{c} \\ a_{2} \cdot E_{c} \end{array} \\ \begin{array}{l} a_{1} \cdot E_{c} \\ 0 \end{array} \\ \begin{array}{l} 0 \\ 0 \end{array} \end{array} & \begin{array}{l} \begin{array}{l} 0 \\ 0 \end{array} \\ \begin{array}{l} 0 \\ G_{c} \end{array} \\ \begin{array}{l} 0 \\ 0 \end{array} \end{array} \end{array} & \begin{array}{l} \begin{array}{l} \begin{array}{l} 0 \\ 0 \end{array} \\ \begin{array}{l} 0 \\ 0 \end{array} \\ \begin{array}{l} G_{c} \\ 0 \end{array} \end{array} & \begin{array}{l} \begin{array}{l} 0 \\ 0 \end{array} \\ \begin{array}{l} 0 \\ 0 \end{array} \\ \begin{array}{l} 0 \\ G_{c} \end{array} \end{array} \end{array} \end{array}] \cdot {\begin{cases} \begin{array}{l} ε_{x x}^{c} (x, y_{c}, z_{c}) \\ ε_{y y}^{c} (x, y_{c}, z_{c}) \end{array} \\ \begin{array}{l} ε_{z z}^{c} (x, y_{c}, z_{c}) \\ γ_{x y}^{c} (x, y_{c}, z_{c}) \end{array} \\ \begin{array}{l} γ_{x z}^{c} (x, y_{c}, z_{c}) \\ γ_{y z}^{c} (x, y_{c}, z_{c}) \end{array} \end{cases}}

(17)

where

a 1 = \frac{η_{c} - 1}{2 \cdot η_{c}^{2} + η_{c} - 1}

and

a 2 = \frac{- η_{c}}{2 \cdot η_{c}^{2} + η_{c} - 1}

E_{c}

G_{c}

, and

η_{c}

are the elastic and shear moduli of the core and its Poisson’s ratio.

The constitutive relations for face sheets follow the Euler-Bernoulli beam theory, and the notation:

v_{0, x}^{i} (x) = φ_{z}^{i} (x) w_{0, x}^{i} (x) = φ_{y}^{i} (x)

(18)

The relations then read:

\begin{array}{l} N_{x}^{i} (x) = E_{i} \cdot h_{i} \cdot a \cdot (u_{0, x}^{i} (x) + F_{v} \cdot \frac{1}{2} \cdot {(v_{0, x}^{i} (x))}^{2} + F_{w} \cdot \frac{1}{2} \cdot {(w_{0, x}^{i} (x))}^{2}) \\ + F_{θ} \cdot \frac{E_{i} \cdot h_{i}^{3} \cdot a^{3}}{6 \cdot (h_{i}^{2} + a^{2})} \cdot {(θ_{x, x}^{i} (x))}^{2} \end{array}

(19)

M_{z}^{i} (x) = - E_{i} \cdot \frac{h_{i} \cdot a^{3}}{12} \cdot (φ_{z, x}^{i} (x) - F_{w} \cdot F_{θ} \cdot w_{0, x}^{i} (x) \cdot θ_{x, x}^{i} (x))

M_{y}^{i} (x) = - E_{i} \cdot \frac{h_{i}^{3} \cdot a}{12} \cdot (φ_{y, x}^{i} (x) + F_{v} \cdot F_{θ} \cdot v_{0, x}^{i} (x) \cdot θ_{x, x}^{i} (x))

M_{x}^{i} (x) = G_{i} \cdot \frac{h_{i}^{3} \cdot a^{3}}{3 \cdot (h_{i}^{2} + a^{2})} \cdot θ_{x, x}^{i} (x)

\begin{array}{l} M_{R}^{i} (x) = \frac{E_{i} \cdot h_{i}^{3} \cdot a^{3}}{3 \cdot (h_{i}^{2} + a^{2})} \cdot (u_{0, x}^{i} (x) + F_{v} \cdot \frac{1}{2} \cdot {(v_{0, x}^{i} (x))}^{2} + F_{w} \cdot \frac{1}{2} \cdot {(w_{0, x}^{i} (x))}^{2}) \\ + F_{θ} \cdot \frac{E_{i} \cdot h_{i} \cdot a}{16} \cdot (\frac{a^{4}}{10} + \frac{h_{i}^{2} \cdot a^{2}}{9} + \frac{h_{i}^{4}}{10}) \cdot {(θ_{x, x}^{i} (x))}^{2} \end{array}

where

E_{i}

and

G_{i}

are the elastic and shear moduli of the

i

-th face sheet, respectively.

By utilizing the relations presented in equations (17)–(19), nonlinear expressions for the “Physical” stress resultants in terms of the “Natural” stress resultants and the unknown displacements are derived. Those expressions are later used to derive a set of 19 ODEs in terms of the derivatives of the unknown displacements (6 in each face sheet and seven in the core). This procedure takes the following sequential steps:

1. The strain fields of the core (Equation (9)) are written in terms of the unknown displacements.

2. Using (Step 1), the 3D Hooke’s law (Equation (17)) and the stress fields are written in terms of the unknown displacements.

3. The “Physical” stress resultants (Equations (11) and (12)) of the core are expressed as functions of the unknown displacements by integration over the cross-section area.

4. The “Physical” stress resultants are transformed into the “Natural” ones (Equations (13) and (14)) and then introduced into the equilibrium equations written in terms of equations (15) and (16).

5. The “Physical” stress resultants of the core (Step 3) and the constitutive relations of the face sheets (Equations (18) and (19)) are introduced into the “Natural” stress resultants (Equations (13) and (14)) and equilibrium equations (Equations (15) and (16)).

6. The equations resulting from Step 5 are written as a set of 38 nonlinear differential equations in terms of the unknown displacements and the “Natural” stress resultants.

The resulting 38 nonlinear equations are long and complex and are not explicitly presented here. The equations file will be provided by the corresponding author upon request.

Boundary conditions

The boundary conditions at the edges of the face sheets ( $i = t, b$ ) and the core are:

α (x_{0}) \cdot N_{x, N}^{i} (x = x_{0}) = N_{x, e x t}^{i} (x = x_{0}) o r u_{0}^{i} (x = x_{0}) = u_{0, e x t}^{i} (x = x_{0})

(20)

α (x_{0}) \cdot V_{y, N}^{i} (x = x_{0}) = V_{y, e x t}^{i} (x = x_{0}) o r v_{0}^{i} (x = x_{0}) = v_{0, e x t}^{i} (x = x_{0})

α (x_{0}) \cdot V_{z, N}^{i} (x = x_{0}) = V_{z, e x t}^{i} (x = x_{0}) o r w_{0}^{i} (x = x_{0}) = w_{0, e x t}^{i} (x = x_{0})

- α (x_{0}) \cdot M_{z, N}^{i} (x = x_{0}) = M_{z, e x t}^{i} (x = x_{0}) o r φ_{z}^{i} (x = x_{0}) = φ_{z, e x t}^{i} (x = x_{0})

- α (x_{0}) \cdot M_{y, N}^{i} (x = x_{0}) = M_{y, e x t}^{i} (x = x_{0}) o r φ_{y}^{i} (x = x_{0}) = φ_{y, e x t}^{i} (x = x_{0})

α (x_{0}) \cdot M_{x, N}^{i} (x = x_{0}) = M_{x, e x t}^{i} (x = x_{0}) o r θ_{x}^{i} (x = x_{0}) = θ_{x, e x t}^{i} (x = x_{0})

α (x_{0}) \cdot N_{x, N}^{c} (x = x_{0}) = N_{x, e x t}^{c} (x = x_{0}) o r u_{0}^{c} (x = x_{0}) = u_{0, e x t}^{c} (x = x_{0})

(21)

α (x_{0}) \cdot V_{y, N}^{c} (x = x_{0}) = V_{y, e x t}^{c} (x = x_{0}) o r v_{0}^{c} (x = x_{0}) = v_{0, e x t}^{c} (x = x_{0})

α (x_{0}) \cdot V_{z, N}^{c} (x = x_{0}) = V_{z, e x t}^{c} (x = x_{0}) o r w_{0}^{c} (x = x_{0}) = w_{0, e x t}^{c} (x = x_{0})

- α (x_{0}) \cdot M_{z, N}^{c} (x = x_{0}) = M_{z, e x t}^{c} (x = x_{0}) o r u_{1}^{c} (x = x_{0}) = u_{1, e x t}^{c} (x = x_{0})

- α (x_{0}) \cdot M_{y, N}^{c} (x = x_{0}) = M_{y, e x t}^{c} (x = x_{0}) o r u_{5}^{c} (x = x_{0}) = u_{5, e x t}^{c} (x = x_{0})

α (x_{0}) \cdot M_{o, N}^{c} (x = x_{0}) = M_{o, e x t}^{c} (x = x_{0}) o r u_{4}^{c} (x = x_{0}) = u_{4, e x t}^{c} (x = x_{0})

α (x_{0}) \cdot M_{x, N}^{c} (x = x_{0}) = M_{x, e x t}^{c} (x = x_{0}) o r θ_{x}^{c} (x = x_{0}) = θ_{x, e x t}^{c} (x = x_{0})

where

x_{0} = 0

L

α (x_{0} = 0) = - 1

α (x_{0} = L) = 1

, and the subscript

e x t

denotes quantities (load or displacement) prescribed at the boundaries.

Solution method

The governing equation of the model takes the form of a coupled set of nonlinear differential equations. This set is not subjected to a linearization process, but it is directly solved in its nonlinear form. This extends the analysis beyond the determination of the linearized buckling load to the full nonlinear response path. In that sense, the present analysis goes way beyond what is found in the literature in the context of lateral-torsional instability of soft core sandwich structures.

The physical nature of the nonlinear behavior of sandwich beams involves instabilities and folds of the equilibrium path. A solution algorithm that involves a continuation procedure is therefore needed. For that purpose, the continuation procedure of Odessa et al.⁶ is adopted. The external load is generally written as $λ \cdot g (x)$ , where $λ$ is a constant and $g (x)$ is a known distribution function. To determine $λ$ for a given value of the continuation parameter, an additional differential equation is added. Since $λ$ is constant this equation reads:

λ_{, x} = 0

(22)

The set of equations now requires an additional boundary condition that reads:

y (x_{0}) = ψ

(23)

where

y (x_{0})

stands for any unknown function and

ψ

is the continuation parameter (e.g., the controlled displacement). With this prescribed parameter, the unknown external load

λ

, is solved for. When

ψ

is monotonic, the set can be solved using the Newton-Raphson method.

To allow for the lateral instability (and control its direction), a small level of distributed load is applied to the compressed face sheet in the lateral direction. The magnitude of the load is determined so that the displacement that develops due to that load is at least two orders of magnitude smaller than the peak lateral displacements observed along the nonlinear response. The model is solved numerically for each $ψ$ increment using the ODEs solver for BVP problems implemented in the software package Maple.

Numerical study

The numerical results are presented for a case where $F_{v} = F_{w} = 1$ , and $F_{θ} = 0$ and are compared with the results of a reference model (Challamel and Girhammar¹⁰) and a Finite-Element (FE) analysis. This selection of flags is in conjunction with the uniform bending load investigated here. Further investigation of the case where $F_{θ} = 1$ as well is left for future study.

The sandwich beam and its boundary conditions are described in Figure 3. The beam is subjected to uniform bending introduced through concentrated axial loads at the edges.

The face sheets in the numerical study are constrained at the vertical and lateral directions on both edges. The upper face sheet is compressed at both edges by external axial loads of magnitude $M_{e n d} / l$ where $l = h_{t} / 2 + h_{c} + h_{b} / 2$ . The core is free of loads and constraints. Due to the sandwich configuration, the cross-sectional bending moment combines localized bending moments in the face sheets, $M_{y}^{t} (x)$ and $M_{y}^{b} (x)$ , and a couple formed by the axial forces in the face sheets. The distribution of the global bending moment among the three components is a function of the elasto-geometric properties. In the investigated beam, where the face sheets are significantly thinner than the entire cross section, the first two components are expected to be three to four orders of magnitude smaller than the third. Considering that, the external loading is limited to the third component. This assumption, and its deviation from the way the load is applied in the reference model¹⁰ are not expected to affect the results. This also applies to the fact that the axial forces in the face sheets do not vanish at the edges. By virtue of Saint Venant’s principle, and due to the minor expected contribution of the localized bending moments in the face sheets, the representation of pure bending load by means of the two axial loads is expected to be sufficient.

The material properties for both face sheets ( $i = t, b$ ) and the core are $E_{i} = 27, 400 M P a$ , $G_{i} = 10, 538 M P a$ , $η_{i} = 0.3$ , $E_{c} = 75 M P a$ , $G_{c} = 30 M P a$ , and $η_{c} = 0.25$ .

The additional boundary condition chosen for the continuation procedure is the vertical displacement of the upper face sheet at midspan $w_{0}^{t} (L / 2)$ . This measure monotonically increases up to a max value of $10 m m$ . To introduce this condition, the longitudinal symmetry is exploited (and thus forcing all lateral instability modes to be symmetric about $x = L / 2$ ).

The imperfection load is $q_{y}^{t} (x) = 10^{- 3} N / m m$ . The lateral displacement due to this load is two orders of magnitude smaller than the one observed under the lateral instability.

The FE analysis of the sandwich beam uses ANSYS APDL 2022 R1 software. The FE model is illustrated in Figure 4 and it uses eight-node 3D solid-shell elements (SOLSH190). The mesh uses 208,593 nodes and 180,000 elements, 120,000 for the core and 30,000 for each face sheet. The size of the element in the core and face sheets is $1 m m x 1 m m x 0.476 m m$ and $1 m m x 1 m m x 0.1 m m$ , respectively, see Figure 4(b). The FE model is solved using a static, large displacements nonlinear incremental solver that uses the arc-length method. A hard stop condition is applied when the lateral displacement of the upper faces sheet exceeds 15 mm.

The imperfection load used in the FE analysis is a $0.5 N$ concentrated load acting at midspan. The peak lateral displacement due to this load is similar to the one detected in the present model.

The investigation looks into several aspects of the response. It begins with the comparison of the model’s nonlinear curves shown in Figure 5 with the corresponding curves of the FE analysis. The critical lateral buckling load determined by the three models is also investigated. Then the deformed shapes right before reaching the critical load and at a point well within the nonlinear regime are explored. Finally, the evolution of the torque resisting mechanisms and the distribution of the stresses in the core are studied. In conjunction with the assumption of small strains, moderate rotations, and large displacements, the results are shown only to the point where they are still valid. The results shown in Figure 5 clearly reveal the lateral instability. Figure 5(a) reveals a characteristic bi-linear behavior of the external moment when plotted against the vertical displacement. The slope of the curve significantly changes when the point of instability is reached. The critical load in the present model is determined based on that change of slope as $M_{c r, m o d e l} = 12.650 N m$ . After reaching the point of instability, a rapid development of lateral displacement of the upper face sheet is observed in Figure 5(b). This mechanism forces the sandwich cross-section to twist, distort, and laterally displace.

Figure 5 reveals that the model developed here, and the FE reference analysis are in good agreement, both in terms of the point of instability ( $M_{c r, m o d e l} = 12.650 N m$ and $M_{c r, F E} = 13.300 N m$ , respectively) and in terms of the overall behavior. This correlation also supports the high-order, yet geometrically linear, kinematic relations attributed to the core. The reference lateral buckling model follows equation (30 in Challamel and Girhammar¹⁰) and refers to the double symmetric case with two identical sub-elements that model the face sheets. This model reveals a lower level of critical load, $M_{c r [10]} = 7.707 N m$ . The main assumption in the reference analysis relates to the modeling of the soft core by means of vertical and horizontal slip layers that link the tractions in the core’s interfaces with the relative displacements of the face sheets. This assumption is introduced into the model through two parameters, K_tran and K_lat that integrate the core’s geometry and material properties. The reference analysis is also focused on linearized buckling that determines the critical load. Figure 5 shows that this tends to underestimate the critical moment. The high-order kinematic assumptions used for the core in the present analysis reflect the distortion of the core’s cross-section and yields a better agreement with the FE reference. In addition, it extends beyond the determination of the critical load and detects the entire nonlinear response curve.

In the context of the above comparison, it is mentioned that the derivation of a simple approximation of the behavior based on the theory derived here is rather difficult, mainly due to the complexity of the physical response and the integration of different length scales. The characteristic length scales range from localized effects spread over lengths of the order of the thickness of the face sheets, transition zones between torsional resisting mechanisms spread over lengths of the order of the thickness of the core, to global effects that span the entire length of the beam. A simplified, Euler type of approximation tends to capture the latter but reflect the former two. To further look into this feature, the results of the current analysis are compared with such Bernoulli-Euler beam theory-based Euler buckling load given by Timoshenko and Gere:¹⁷

{(M_{0})}_{c r} = \frac{π}{l} \sqrt{{E I}_{η} \cdot C \cdot (1 + \frac{C_{1}}{C} \cdot \frac{π^{2}}{l^{2}})}

(24)

where

{(M_{0})}_{c r}

is the critical uniform bending moment,

l

is the length, and

{E I}_{η}, C,

and

C_{1}

are bending stiffness about the minor axis, Saint-Venant’s torsional stiffness, and the warping cross-sectional stiffness, respectively.

The comparison of the current analysis with the benchmark given in equation (24) heavily depends on the assessment of the torsional stiffnesses of the sandwich cross-section. Assessing the stiffness based on classical methods yields a buckling load that equals $19.067 N m$ . This value is 1.5 times larger than the one determined by the current model. When the torsional stiffnesses are estimated based on the current model by means of the procedure described in Wurf et al.,⁹ the classical torsional-flexural buckling load reads $12.832 N m$ . This value is only $1.44 %$ higher than the one determined by the current nonlinear analysis. This observation further highlights the unique nature of the sandwich structure, the role played by the high order theory in the assessment of its stability and nonlinear behavior, and the one played by the localized effects.

The deformed shapes of the beam for $M_{e n d} = 11.368 N m$ ( $\approx 0.90 M_{c r, m o d e l}$ ) and $M_{e n d} = 13.342 N m$ ( $\approx 1.05 M_{c r, m o d e l}$ ) are displayed in Figures 6 and 7, respectively, for the symmetric half of the beam.

Figures 6 and 7 reveal how the beam shifts from uniform bending about the major axis to a mixed bending about both major and minor axes and torsional rotation about the longitudinal one. Figure 6 reveals a typical uniform bending behavior but Figure 7 shows how the compressed upper face sheet moves in the lateral direction and “pulls” the sandwich beam to this complex response. The results also reveal the warping of the cross-section at $x = 0$ and the corresponding out-of-plane displacements of both face sheets. Finally, the deformed shapes reveal the cross-section distortion as the two face sheets are no longer parallel, and the edges of the core cross-section deviate from their original plane configuration. On the global scale, these figures highlight the evolution of the instability and show how a relatively small increase in the applied moment causes a sudden and significant increase in lateral displacement which is a prominent result of the lateral instability.

Figure 8 reveals the evolution of the two torque resisting mechanisms at different response states of the beam. Figure 8(a) reveals the initial response due to the imperfection load. This response is mostly based on Saint-Venant mechanism whereas warping mainly evolving near the supports. In Figure 8(b), which depicts the response still before the critical point of instability, the level of Saint-Venant mechanism along the span is not significantly changed, but the localized evolution of warping increases. Figure 8(c)–(d) which reflect the response after the evolution of instability and into the nonlinear range. Here, the two mechanisms have an opposite direction and similar magnitudes. Also, the sign of the Saint-Venant component changes and the role of warping significantly increases. It is also observed that the magnitude of each mechanism slowly increases toward the critical load. When the critical load is reached, the torsional torque increases significantly and rapidly, in conjunction with the complex behavior introduced by the lateral instability.

Figures 9–11 highlight the evolution of the stresses, $σ_{x x}^{c}$ , $τ_{x z}^{c}$ , and $σ_{z z}^{c}$ , in the core through the states of the response: the initial linear state, right before the instability, and deep within the nonlinear state. Two cross sections are examined: one at midspan (for $σ_{x x}^{c}$ , and $σ_{z z}^{c}$ ) and at x = 50 mm (for $τ_{x z}^{c}$ ) and one at x = 20 mm, where the localized stresses concentrations associated with the free edge of the core are expected to decay.

At the initial stage ( $M_{e n d} = 6.918 N m$ ), Figure 9 reveals a clear pattern of $σ_{x x}$ that is typical to uniform bending with only slight impact of the imperfection load. The increase of the external end moment to $M_{e n d} = 11.368 N m$ does not significantly affects the pattern of the cross-sectional $σ_{x x}$ stress distribution. Deep into the nonlinear state, the stress distribution and the magnitude of the stresses take a sharp turn. A relatively small increase of the external end moment to $M_{e n d} = 13.342 N m$ yields a significant increase in the stress magnitudes, particularly at x = 20 mm. Also, the cross-sectional stress distribution is no longer typical to pure bending about the major axis but it reflects skewed bending about the two axes and warping. The latter is reflected by the curved contours of $σ_{x x}$ . Such pattern deviates from the linear contours typical to skewed bending. Another observation is that when the response is driven into deep nonlinear regime, the stress evolution is by no means proportional, and a unique stress state evolves along the nonlinear path.

Figure 10 shows that as expected, in the initial pre-buckling state, $τ_{x z}^{c}$ almost vanishes. The selected $τ_{x z}^{c}$ cross-sections are not the same as the $σ_{x x}^{c}$ due to applied symmetry resulting in zero $τ_{x z}^{c}$ stresses at midspan. Before reaching the critical end moment, the $τ_{x z}^{c}$ stresses due to imperfection load are not significant. Deep into the nonlinear state, $τ_{x z}^{c}$ significantly increases due to lateral flexure and torsional rotation.

Conclusions

An extended geometrically nonlinear High-order Sandwich Panel Theory for the lateral instability of soft-core sandwich beams has been developed. One innovative step taken in this paper lies in the development of a high-order model that aims at describing the lateral instability of such beams while addressing their unique features. The consideration of these features is essential for the analysis and quantification of the evolution of the nonlinear response of such soft-core sandwich beams undergoing lateral-torsional instability. This challenge has not been addressed in the literature. The quantitative exploration of these effects is an innovative and original contribution of the present work.

A comparison of the numerical results of the current model with a fully nonlinear FE analysis has revealed a good agreement, both in terms of the critical load of instability and the overall response. Compared with a reference model for the critical buckling load, the presented model has indicated on a better agreement with the reference FE analysis. It has also been emphasized and demonstrated how the current model extends beyond the buckling load to the quantification of the complete nonlinear analysis. Exploration of the distortion of the cross-section of the soft core has revealed its evolution with the nonlinear lateral-torsional response as well as the evolution of the cross-sectional stresses. These observations highlight the role played by the soft core layer and the need for its refined modeling. The torsional mechanisms and the cross-sectional stresses that evolve at the different response states have revealed how and when the different response mechanisms take place in the nonlinear structural response.

The model introduced here enables a thorough nonlinear investigation of the sandwich beam response and the unique phenomena that characterizes them. It reveals how the global bending drives the beam into a new equilibrium state where it develops significant lateral and rotational deformations and the distinctive core stresses it induces. The lateral instability torque resisting mechanisms gradually shift to a unique case where they have an opposite direction and similar magnitudes. All these extend far beyond what is available in the literature and allow a deeper understanding of the lateral instability of sandwich beams.

Footnotes

Acknowledgements

Oded Rabinovitch gratefully acknowledges the support of the Abel Wolman Chair in Civil Engineering.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Avi Wurf

Oded Rabinovitch

Data Availability Statement

The equations file containing the final set of 38 differential equations of the model will be provided by the corresponding author upon request.

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