Latest revision as of 16:14, 10 October 2018

Published in Comput. Methods Appl. Mech. Engrg. Vol. 213–216, pp. 362–382, 2012
doi: 10.1016/j.cma.2011.11.023

Abstract

In this work we present a new simple linear two-noded beam element adequate for the analysis of composite laminated and sandwich beams based on the combination of classical Timoshenko beam theory and the refined zigzag kinematics proposed by Tessler et al. [22]. The element has just four kinematic variables per node. Shear locking is eliminated by reduced integration. The accuracy of the new beam element is tested in a number of applications to the analysis of composite laminated beams with simple supported and clamped ends under point loads and uniformly distributed loads. An example showing the capability of the new element for accurately reproducing delamination effects is also presented.

Keywords: Two-noded beam element, zigzag kinematics, Timoshenko theory, composite, sandwich beams

1 INTRODUCTION

It is well known that both the classical Euler-Bernouilli beam theory [1] and the more advanced Timoshenko theory [2] produce inadequate predictions when applied to relatively thick composite laminated beams with material layers that have highly dissimilar stiffness characteristics. Even with a judiciously chosen shear correction factor, Timoshenko theory tends to underestimate the axial stress at the top and bottom outer fibers of a beam. Also, along the layer interfaces of a laminated beam the transverse shear stresses predicted exhibit erroneous discontinuities. These difficulties are due to the higher complexity of the “true” variation of the axial displacement field across a highly heterogeneous beam cross-section.

Indeed to achieve accurate computational results, 3D finite element analyses are often preferred over beam models. For composite laminates with hundred of layers, however, 3D modelling becomes prohibitely expensive, specially for non linear and progressive failure analyses.

Improvements to the classical beam theories have been obtained by the so called equivalent single layer (ESL) theories that assume a priori the behaviour of the displacement and/or the stress through the laminate thickness [3,4]. Despite of being computational efficient, ESL theories often produce innacurate distributions for the stresses and strains (in particular the transverse shear stress) across the thickness.

The need for composite laminated beam theories with better predictive capabilities has led to the development of the so-called higher order theories. In these theories higher-order kinematic terms with respect to the beam depth are added to the expression for the axial displacement and, in some cases, to the expressions for the deflection. A review of these theories can be found in [3,4].

Accurate predictions of the correct shear and axial stresses for thick and highly heterogenous composite laminated and sandwich beams can be obtained by using layer-wise theory. In this theory the thickness coordinate is split into a number of analysis layers that may or not coincide with the number of laminate plies. The kinematics are independently described within each layer and certain physical continuity requirements are enforced [3,4].

A drawback of layer-wise theory is that the number of kinematic variables depends on the number of analysis layers. The layer displacements can be condensed at each section in terms of the axial displacement for the top layer during the equation solution process [5,6]. The displacement condensation processes can be however expensive for problems involving many analysis layers.

Discrete layer theories in which the number of unknowns in the model does not depend on the number of layers in the laminate are described in [7,8,9]. In this class of discrete layerwise theories (called zigzag theories) a piewise linear in-plane displacement function (the zigzag function) is superimposed over a linear displacement field [7,8], a quadratic displacement field [10,11] or a cubic displacement field [12,13,14] through the thickness of the laminate.

Many zigzag theories require ${\textstyle C^{1}}$ continuity for the deflection field, which is a drawback versus simpler ${\textstyle C^{\circ }}$ continuous FEM approximations. Also many zigzag theories run into theoretical difficulties to satisfy equilibrium of forces at a clamped support.

Averill et al. [9,10,16,17] developed linear, quadratic and cubic zigzag beam theories that overcame the need for ${\textstyle C^{1}}$ continuity. The shear strain angle is introduced as a kinematic variable together with the deflection, the rotation and the zigzag function. A ${\textstyle C^{\circ }}$ interpolation can be used for all these variables. The relationship between the shear angle, the deflection and the rotation of each layer is introduced as a constraint via a penalty method. This also ensures the continuity of the transverse shear stress across the laminate depth and the satisfaction of the shear traction boundary conditions. However, Averill theories have difficulties to model correctly clamped boundary conditions. For this reason, analytical and numerical (FEM) studies based on Averill theory have mainly focused on simple supported beams [16,17].

A 2-noded beam element based on Euler-Bernouilli beam theory and an extension of Averill's zigzag theory including a cubic in-plane displacement field within each layer has been recently proposed by Alam and Upadhyay [18]. Good results are reported for cantilever and clamped composite and sandwich beams.

An assessment of different zigzag theories for beam is reported in [19,20]. A review of zigzag theory for plate analysis can be found in [21].

Tessler et al. [22,23] have developed a refined zigzag (RZ) theory starting from the standard Timoshenko kinematic assumptions. This allows one using ${\textstyle C^{\circ }}$ continuous interpolation for all the kinematic variables. Timoshenko beam theory also introduces naturally shear deformation effect for the homogeneous material case, which is advantageous for many problems. The zigzag functions chosen have the property of vanishing on the top and bottom surfaces of a laminate. A particular feature of this zigzag theory is that the transverse shear stresses are not required to be continuous at the layer interfaces. This results in simple piewice-constant functions that approximate the true shear stress distribution. An accurate continuous thickness distribution of the transverse shear stress can be obtained “a posteriori” in terms of the axial stress by integrating the equilibrium equations. This theory also provides good results for clamped supports.

Gherlone et al. [24] have developed two and three-noded ${\textstyle C^{\circ }}$ beam elements based on the RZ theory for analysis of multilayered composite and sandwich beams. Locking-free elements are obtained by using anisoparametric interpolations that are adapted to approximate the four independent kinematic variables that model the beam deformation. A family of beam elements is achieved by imposing different constraints on the original displacement approximation. The constraint conditions requiring a constant variation of the transverse shear force provide an accurate 2-noded beam element [24].

Quite simultaneously to the above work, Oñate et al. [25] proposed a simple 2-noded beam element for composite laminated beams based on the RZ theory. A standard linear displacement field is used to model the four variables of the so called LRZ element. Shear locking is avoided by using reduced integration on selected terms of the shear stiffness matrix.

In this paper we present in detail the formulation of the LRZ beam element originally reported in [25] and explore the capabilities of the new element for multilayered beams and delamination analysis. A study of the locking-free behaviour of the LRZ element for slender beams is presented. The good performance of the element is demonstrated for simply supported and clamped composite laminated beams with different layers under point load and uniformly distributed loads.

Finally, an example showing the capability of the LRZ element to model delamination effects is presented.

2 GENERAL CONCEPTS OF ZIGZAG BEAM THEORY

The kinematic field in zigzag beam theory is generally written as

where

is the zigzag displacement function (Figure 1).

In Eqs.(1) ${\textstyle N}$ is the number of layers, superscript ${\textstyle k}$ indicates quantities within the ${\textstyle k}$th layer with ${\textstyle z_{k}\leq z\leq z_{k+1}}$ and ${\textstyle z_{k}}$ is the vertical coordinate of the ${\textstyle k}$th interface. In Eq.(1a) the uniform axial displacement ${\textstyle u_{0}(x)}$, the rotation ${\textstyle \theta (x)}$ and the transverse deflection ${\textstyle w_{0}(x)}$ are the primary kinematic variables of the underlying equivalent single-layer Timoshenko beam theory. In Eq.(1b) function ${\textstyle \phi ^{k}(z)}$ denotes a piecewise linear zigzag function, yet to be established, and ${\textstyle \Psi (x)}$ is a primary kinematic variable that defines the amplitude of the zigzag function along the beam. Collectively, the interfacial axial displacement field has a zigzag distribution, as shown in Figure 1c.

Figure 1: Thickness distribution of the the zigzag function ${\displaystyle \phi ^{k}}$ (a), the zigzag displacement function ${\displaystyle {\bar {u}}^{k}}$ (b), and the axial displacement (c) in refined zigzag beam theory

The strain-displacement relations are derived by substituting Eq.(1a) into the expressions of classical Timoshenko beam theory, i.e.

In Eq.s (2a) and (2b)

where ${\textstyle {\hat {\boldsymbol {\varepsilon }}}_{p}}$ and ${\textstyle {\hat {\boldsymbol {\varepsilon }}}_{t}}$ are the generalized in-plane and transverse shear strain vectors, respectively. Vector ${\textstyle {\hat {\boldsymbol {\varepsilon }}}_{p}}$ contains the axial elongation ${\textstyle \left({\frac {\partial u_{0}}{\partial x}}\right)}$, the pseudo-curvature ${\textstyle \left({\frac {\partial \theta }{\partial x}}\right)}$ and the derivatives of the amplitude of the zigzag function ${\textstyle \left({\frac {\partial \Psi }{\partial x}}\right)}$. In ${\textstyle {\hat {\boldsymbol {\varepsilon }}}_{t}}$, ${\textstyle \gamma ={\frac {\partial w_{0}}{\partial x}}-\theta }$ is the average transverse shear strain of Timoshenko beam theory and ${\textstyle \beta ^{k}={\frac {\partial \phi ^{k}}{\partial z}}}$. Note that since ${\textstyle \phi ^{k}(z)}$ is piecewise linear, ${\textstyle \beta ^{k}}$ is constant across each layer.

For major principal material axes that are coincident with the beam ${\textstyle x}$ axis, Hooke stress-strain relations for the ${\textstyle k}$th orthotropic layer have the standard form

where ${\textstyle E^{k}}$ and ${\textstyle G^{k}}$ are the axial and shear moduli for the ${\textstyle k}$th layer, respectively.

In the above equations we have distinguished all variables within a layer with superscript ${\textstyle k}$.

3 REFINED ZIGZAG THEORY

3.1 Zigzag kinematics

The key attributes of the refined zigzag (RZ) theory proposed by Tessler et al. [22] are, first, the zigzag function vanishes at the top and bottom surfaces of the beam section and does not require full shear-stress continuity across the laminated-beam depth. Second, all boundary conditions can be modelled adequately. And third, ${\textstyle C^{\circ }}$ continuity is only required for the FEM approximation of the kinematic variables.

Within each layer the zigzag function is expressed as

where ${\textstyle {\bar {\phi }}^{k}}$ and ${\textstyle {\bar {\phi }}^{k-1}}$ are the zigzag function values of the ${\textstyle k}$ and ${\textstyle k-1}$ interface, respectively with ${\textstyle {\bar {\phi }}^{0}={\bar {\phi }}^{N}=0}$ and ${\textstyle \zeta ^{k}={\frac {2(z-z^{k-1})}{h^{k}}}-1}$.

Collectively, function ${\textstyle \phi ^{k}}$ has the zigzag distribution shown in Figure 1a. Due to the dependence between the zigzag displacement function ${\textstyle {\bar {u}}^{k}}$ and ${\textstyle {\bar {\phi }}^{k}}$ (see Eq.(1b)), ${\textstyle {\bar {u}}^{k}}$ also vanishes at the top and bottom layers. The axial displacement field is plotted in Figure 1c.

The above form of ${\textstyle \phi ^{k}}$ gives the constant value of ${\textstyle \beta ^{k}}$ for each layer as

and

The ${\textstyle \beta ^{k}}$ parameter is useful for computing the zigzag function as explained in the next section.

3.2 Computation of the zigzag function

Integrating Eq.(2b) over the cross section and using Eq.(5b) and the fact that ${\textstyle \Psi }$ is independent of ${\textstyle z}$ yields

i.e. ${\textstyle \gamma }$ represents the average shear strain of the cross section, as expected.

The shear strain-shear stress relationship of Eq.(3b) is written as

where ${\textstyle \eta =\gamma -\Psi }$ is a difference function.

Remark 1. Function ${\textstyle \Psi }$ can be interpreted as a weighted-average shear strain angle [22]. The value of ${\textstyle \Psi }$ should be prescribed to zero at a clamped edge and left unprescribed at free and simply supported edges.

Eq.(7) shows that the distribution of ${\textstyle \tau _{xz}^{k}}$ within each layer is constant, as ${\textstyle \eta }$ is independent of the zigzag function and ${\textstyle \beta ^{k}}$ is constant.

The distribution of ${\textstyle \tau _{xz}^{k}}$ is now enforced to be independent of the zigzag function. This can be achieved by constraining the term multiplying ${\textstyle \Psi }$ in Eq.(7) to be constant, i.e.

This is equivalent to enforcing the interfacial continuity of the second term in the r.h.s. of Eq.(7).

Remark 2. We emphasize that this zigzag theory does not enforce the continuity of the transverse shear stresses across the section. This is consistent with the kinematic freedom inherent in the lower order kinematic approximation of the underlying beam theory. An accurate continuous distribution of the transverse shear stress across the thickness of the laminate can be obtained “a posteriori” in terms of axial stresses by integrating the equilibrium equations as explained in Section 7.3.

From Eq.(8) we deduce

Substituting ${\textstyle \beta ^{k}}$ in the integral of Eq.(5b) gives

where ${\textstyle h}$ is the section depth. Substituting Eq.(9) into Eq.(5a) gives the following recursion relation for the zigzag displacement function values at the layer interfaces

Introducing Eq.(11) into (4) gives the expression for the zigzag function as

Recall that superindex ${\textstyle k}$ denotes the number of each material layer.

Remark 3. For homogeneous material ${\textstyle G^{k}=G}$ and ${\textstyle \beta ^{k}=0}$. Hence, the zigzag function ${\textstyle \phi ^{k}}$ vanishes and we recover the kinematics and constitutive expressions of the standard Timoshenko composite laminated beam theory. This is a particular feature of this zigzag theory.

Remark 4. Note that differently from standard Timoshenko beam theory, a shear correction parameter is not needed in the RZ theory.

3.3 Constitutive relationship

The in-plane bending and transverse shear resultant stresses are defined as

In vectors ${\textstyle {\hat {\boldsymbol {\sigma }}}_{p}}$ and ${\textstyle {\hat {\boldsymbol {\sigma }}}_{t}}$, ${\textstyle N,M}$ and ${\textstyle Q}$ are respectively the axial force, the bending moment and the transverse shear force of standard beam theory, whereas ${\textstyle M_{\phi }}$ and ${\textstyle Q_{\phi }}$ are an additional bending moment and an additional shear force which are conjugate to the new generalized strains ${\textstyle {\partial \Psi \over \partial x}}$ and ${\textstyle \Psi }$, respectively.

The generalized constitutive matrices ${\textstyle {\hat {\boldsymbol {D}}}_{p}}$ and ${\textstyle {\hat {\boldsymbol {D}}}_{t}}$ are

with

In the derivation of the expression for ${\textstyle {\hat {\boldsymbol {D}}}_{s}}$ we have used the definition of ${\textstyle \beta ^{k}}$ of Eq.(9).

The generalized constitutive equation can be written as

3.4 Virtual work expression

The virtual work expression for a distributed load ${\textstyle q}$ is

The l.h.s. of Eq.(17) contains the internal virtual work performed by the axial and tangential stresses and the r.h.s. is the external virtual work carried out by the distributed load. ${\textstyle V}$ and ${\textstyle l}$ are the volume and length of the beam, respectively.

Substituting Eqs.(3) into the expression for the virtual internal work and using Eqs.(13) and (14) gives

The virtual work is therefore written as

4 TWO-NODED LRZ BEAM ELEMENT

The four kinematic variables are ${\textstyle u_{0},w_{0},\theta }$ and ${\textstyle \Psi }$. They can be discretized using 2-noded linear ${\textstyle C^{\circ }}$ beam elements of length ${\textstyle l^{e}}$ in the standard form as

with

where ${\textstyle N_{i}={\frac {1}{2}}(1+\xi \xi _{i})}$ with ${\textstyle \xi =1-{\frac {2x}{l^{e}}}}$ are the standard one-dimensional linear shape functions, ${\textstyle {\boldsymbol {a}}_{i}}$ is the vector of nodal kinematic variables and ${\textstyle {\boldsymbol {I}}_{4}}$ is the ${\textstyle 4\times 4}$ unit matrix.

Substituting Eq.(20) into the generalized strain vectors in Eq.(2c) gives

The generalized strain matrices ${\textstyle {\boldsymbol {B}}_{p}}$ and ${\textstyle {\boldsymbol {B}}_{t}}$ are

with

where ${\textstyle {\boldsymbol {B}}_{p_{i}}}$ and ${\textstyle {\boldsymbol {B}}_{t_{i}}}$ are the in-plane and transverse shear strain matrices for node ${\textstyle i}$.

The virtual displacement and generalized strain fields are expressed in terms of the virtual nodal kinematic variables as

The discretized equilibrium equations are obtained by substituting Eqs.(13), (14), (20), (22) and (24) into the virtual work expression (19). After simplification of the virtual nodal kinematic variables, the following standard matrix equation is obtained

where ${\textstyle {\boldsymbol {a}}}$ is the vector of nodal kinematic variables for the whole mesh.

The stiffness matrix ${\textstyle {\boldsymbol {K}}}$ and the equivalent nodal force vector ${\textstyle {\boldsymbol {f}}}$ are obtained by assembling the element contributions ${\textstyle {\boldsymbol {K}}^{e}}$ and ${\textstyle {\boldsymbol {f}}^{e}}$ given by

with

and

Matrix ${\textstyle {\boldsymbol {K}}_{p}^{e}}$ is integrated with a one-point numerical quadrature which is exact in this case. Full integration of matrix ${\textstyle {\boldsymbol {K}}_{t}^{e}}$ requires a two-point Gauss quadrature. This however leads to shear locking for slender composite laminated beams (Section 5).

In order to asses the influence of the reduced integration of matrix ${\textstyle {\boldsymbol {K}}_{t}^{e}}$ for overcoming the shear locking problem we split ${\textstyle {\boldsymbol {K}}_{t}^{e}}$ as follows

with

where ${\textstyle {\boldsymbol {B}}_{s_{i}}}$ and ${\textstyle {\boldsymbol {B}}_{\psi _{i}}}$ are defined in Eq.(23c) and ${\textstyle D_{s}}$ and ${\textstyle g}$ are given in Eq.(15b).

The new linear beam element based on the RZ theory is termed LRZ.

A study of the accuracy of the LRZ beam element for analysis of slender laminated beams using one and two-point quadratures for integrating matrices ${\textstyle {\boldsymbol {K}}_{s}^{e}}$, ${\textstyle {\boldsymbol {K}}_{\psi }^{e}}$ and ${\textstyle {\boldsymbol {K}}_{s\psi }^{e}}$ is presented in the next section.

5 STUDY OF SHEAR LOCKING FOR THE LRZ BEAM ELEMENT

We study the performance of the LRZ beam element for the analysis of a cantilever beam of length ${\textstyle L}$ under an end point load of value ${\textstyle F=1}$ (Figure 2). The beam is formed by a symmetric three-layered material whose properties are listed on Table 1. The analysis is performed for four span-to-thickness ratios: ${\textstyle \lambda =5,10,50,100}$ (${\textstyle \lambda =L/h}$) using a mesh of 100 LRZ beam elements. Results for the LRZ element are labelled “ZZ” in the figures.

The same beam was analized using a mesh of 27000 four-noded plane stress rectangles for comparison purposes (Figure 3). Results for the plane stress analysis are labeled “PS” in the figures.

Figure 2: Cantilever beam under point load

Figure 3: Mesh of 27000 4-noded plane stress rectangular elements for analysis of cantilever and simple supported beams

Figure 4 shows the ratio ${\textstyle r}$ between the end node deflection obtained with the LRZ element (${\textstyle w_{zz}}$) and with the plane stress quadrilateral (${\textstyle w_{ps}}$) (i.e. ${\textstyle r={\frac {w_{zz}}{w_{ps}}}}$) versus the beam span-to-thickness ratio ${\textstyle d={\frac {L}{h}}}$. Results for the LRZ element have been obtained using exact two-point integration for all terms of matrix ${\textstyle {\boldsymbol {K}}_{t}^{e}}$ (Eq.(27)) and a one-point reduced integration for the following three groups of matrices: ${\textstyle {\boldsymbol {K}}_{s}^{e}}$; ${\textstyle {\boldsymbol {K}}_{s}^{e}}$ and ${\textstyle {\boldsymbol {K}}_{s\psi }^{e}}$; and ${\textstyle {\boldsymbol {K}}_{s}}$, ${\textstyle {\boldsymbol {K}}_{s\psi }^{e}}$ and ${\textstyle {\boldsymbol {K}}_{\psi }^{e}}$ (Eqs.(29b)).

Labels ``all', ``S', ``SPsi' and ``Psi' in Figures 4–7 refer to matrices ${\textstyle {\boldsymbol {K}}_{t}^{e}}$, ${\textstyle {\boldsymbol {K}}_{s}^{e}}$, ${\textstyle {\boldsymbol {K}}_{s\psi }^{e}}$ and ${\textstyle {\boldsymbol {K}}_{\psi }^{e}}$ of Eq.(29a), respectively.

Results in Figure 4 clearly show that the exact integration of ${\textstyle {\boldsymbol {K}}_{t}^{e}}$ leads to shear locking as expected. Good (locking-free) results are obtained by one-point reduced integration of the three groups of matrices.

The influence of reduced integration in the distribution of the transverse shear stress was studied next for the three groups of matrices. Figures 5–7 show the thickness distribution of ${\textstyle \tau _{xz}}$ in sections located at distances ${\textstyle {\frac {L}{20}},{\frac {L}{4}},{\frac {L}{2}}}$ and ${\textstyle {\frac {3}{4}}L}$ from the clamped end for span-to-thickness ratios of ${\textstyle \lambda =5,10}$ and 100. Results are compared with the plane stress solution and also with results obtained with a mesh of 300 standard 2-noded elements based on laminated Timoshenko beam theory (labelled TBT in the figures). All TBT results presented in the paper have been used with a simple shear correction factor of ${\textstyle {\frac {5}{6}}}$. Indeed a more accurate value of the shear correction factor in TBT can be used for laminated sections [28].

Figure 4:' ${\displaystyle r}$ ratio ${\displaystyle \left(r={\frac {w_{zz}}{w_{ps}}}\right)}$ versus ${\displaystyle L/h}$ for cantilever under point load analyzed with the LRZ element. Labels ``all', ${\displaystyle S}$, ${\displaystyle SPsi}$ and ${\displaystyle Psi}$ refer to matrices ${\displaystyle {\boldsymbol {K}}_{t}^{e}}$, ${\displaystyle {\boldsymbol {K}}_{s}^{e},{\boldsymbol {K}}_{s\psi }^{e}}$ and ${\displaystyle {\boldsymbol {K}}_{\psi }^{e}}$, respectively

(a) ${\displaystyle \tau _{xz},\lambda =5,L/20}$	(b) ${\displaystyle \tau _{xz},\lambda =5,L/4}$
(c) ${\displaystyle \tau _{xz},\lambda =5,L/2}$	(d) ${\displaystyle \tau _{xz},\lambda =5,3L/4}$
Figure 5: Symmetric 3-layered cantilever thick beam under end point load. Thickness distribution of shear stress for ${\displaystyle \lambda =5}$ at different sections

(a) ${\displaystyle \tau _{xz},\lambda =10,L/20}$	(b) ${\displaystyle \tau _{xz},\lambda =10,L/4}$
(c) ${\displaystyle \tau _{xz},\lambda =10,L/2}$	(d) ${\displaystyle \tau _{xz},\lambda =10,3L/4}$
Figure 6: Symmetric 3-layered cantilever thick beam under end point load. Thickness distribution of shear stress for ${\displaystyle \lambda =10}$ at different sections

(a) ${\displaystyle \tau _{xz},\lambda =100,L/20}$	(b) ${\displaystyle \tau _{xz},\lambda =100,L/4}$
(c) ${\displaystyle \tau _{xz},\lambda =100,L/2}$	(d) ${\displaystyle \tau _{xz},\lambda =100,3L/4}$
Figure 7: Symmetric 3-layered cantilever thick beam under end point load. Thickness distribution of transverse shear stress for ${\displaystyle \lambda =100}$ at different sections

The conclusion is that for small values of ${\textstyle \lambda }$ the reduced or exact reduced integration of matrix ${\textstyle {\boldsymbol {K}}_{t}^{e}}$ leads to similar results. For slender beams, however, results obtained using reduced integration for ${\textstyle {\boldsymbol {K}}_{s}^{e}}$; ${\textstyle {\boldsymbol {K}}_{s}^{e}}$ and ${\textstyle {\boldsymbol {K}}_{s\psi }^{e}}$; and ${\textstyle {\boldsymbol {K}}_{s}^{e}}$, ${\textstyle {\boldsymbol {K}}_{s\psi }^{e}}$ and ${\textstyle {\boldsymbol {K}}_{\psi }^{e}}$ are different. Slightly more accurate results are obtained with the second choice for the section at ${\textstyle x=L/4}$ and ${\textstyle \lambda =100}$ (Figure 7b).

In conclusion, we recommend using a reduced one-point integration for matrices ${\textstyle {\boldsymbol {K}}_{s}^{e}}$ and ${\textstyle {\boldsymbol {K}}_{s\psi }^{e}}$, while matrix ${\textstyle {\boldsymbol {K}}_{\psi }^{e}}$ should be integrated with a 2-point quadrature.

6 CONVERGENCE STUDY

The same three-layered cantilever beam of Figure 2 was studied next for three different set of thickness and material properties for the three layers as listed in Table 2. Material A is the more homogeneous one, while material C is clearly the more heterogeneous.

The problem was studied with six meshes of LRZ elements ranging from 5 to 300 elements. Tables 3–5 show the convergence with the number of elements for the deflection and function ${\textstyle \Psi }$ at the beam end, the maximum axial stress ${\textstyle \sigma _{x}}$ at the end section and the maximum shear stress ${\textstyle \tau _{xz}}$ at the mid section.

Convergence is measured by the relative error defined (in absolute value) as

where ${\textstyle v_{6}}$ and ${\textstyle v_{i}}$ are the values of the magnitude of interest obtained using the finest grid (300 elements) and the ${\textstyle i}$th mesh (${\textstyle i=1,2,\cdots 5}$), respectively.

Results clearly show that convergence is always slower for the heterogeneous material case, as expected.

For a mesh of 25 elements the errors for all the magnitudes considered are less than 1% for materials A and B. For material C the maximum error does not exceed 5% (Table 5). For the 50 element mesh errors of the order of 1% or less were obtained in all cases.

Results for a 10 element mesh are good for material A (errors less than 0.4%), relatively good for material B (errors less than around ${\textstyle 5\%}$) and unacceptable for material C (errors ranging from around 8% to 20%

7 EXAMPLES OF APPLICATION

7.1 Three-layered thick cantilever beam with non symmetric material properties

We present results for a laminated thick cantilever beam under an end point load. The material properties are those of Composite C in Table 2. The span-to-thickness ratio is ${\textstyle \lambda =5}$.

For the laminated sandwich considered the core is eight times thicker than the face sheets. In addition, the core is three orders of magnitude more compliant than the bottom face sheet. Moreover, the top face sheet has the same thickness as the bottom face sheet, but is about three times stiffer. Note that this laminate does not possess material symmetry with respect to the mid-depth reference axis. The high heterogeneity of this stacking sequence is very challenging for the beam theories considered herein to model adequately.

As in previous section, the legend caption PS denotes the reference solution obtained with the structured mesh of 27000 four-noded plane stress quadrilaterals shown in Figure 3. TBT denotes the solution obtained with a mesh of 300 2-noded beam elements based on standard laminated Timoshenko beam theory. LRZ-300, LRZ-50, LRZ-25, LRZ-10 refer to the solution obtained with the LRZ beam element using meshes of 300, 50, 25 and 10 elements, respectively.

Figure 8 shows the deflection values along the beam length. Very good agreement with the plane stress solution is obtained already for the LRZ-50 mesh as expected from the conclusions of the previous section.

TBT results are considerable stiffer. The difference with the reference solution is about six times stiffer for the end deflection value.

Figure 9 shows the distribution of the axial displacements at the upper and lower surfaces of layer 3 (top layer) along the beam length. Excellent results are again obtained with the 50 element mesh. The TBT results are far from the correct ones.

Figure 8: Non symmetric 3-layered cantilever thick beam under end point load (${\displaystyle \lambda =5}$). Distribution of the vertical deflection ${\displaystyle w}$ for different theories and meshes

(a)	(b)
Figure 9: Non symmetric 3-layered cantilever thick beam under end point load (${\displaystyle \lambda =5}$). Axial displacement ${\displaystyle u}$ at the upper and lower surfaces of the top layer (layer 3)