#### Abstract

In this paper both the static and dynamic analyses of the geometrically linear or nonlinear, elastic or elastoplastic nonuniform torsion problems of bars of constant or variable arbitrary cross section are presented together with the corresponding research efforts and the conclusions drawn from examined cases with great practical interest. In the presented analyses, the bar is subjected to arbitrarily distributed or concentrated twisting and warping moments along its length, while its edges are supported by the most general torsional boundary conditions. For the dynamic problems, a distributed mass model system is employed taking into account the warping inertia. The analysis of the aforementioned problems is complete by presenting the evaluation of the torsion and warping constants of the bar, its displacement field, its stress resultants together with the torsional shear stresses and the warping normal and shear stresses at any internal point of the bar. Moreover, the construction of the stiffness matrix and the corresponding nodal load vector of a bar of arbitrary cross section taking into account warping effects are presented for the development of a beam element for static and dynamic analyses. Having in mind the disadvantages of the 3D FEM solutions, the importance of the presented beamlike analyses becomes more evident.

#### 1. Introduction

In engineering practice, we often come across the analysis of members of structures subjected to twisting moments. Curved bridges, ribbed plates subjected to eccentric loading, or columns laid out irregularly in the interior of a plate due to functional requirements are the most common examples.

When the warping of the cross section of a member is not restrained, the applied twisting moment is undertaken from the Saint-Venant [1] shear stresses. In this case the angle of twist per unit length remains constant along the bar. However, in most cases arbitrary torsional boundary conditions are applied either at the edges or at any other interior point of the bar due to construction requirements. This bar under the action of general twisting loading is leaded to nonuniform torsion, while the angle of twist per unit length is no longer constant along it. The consequences of restrained warping were first presented by Marguerre [2].

Although there is extended literature on the Saint-Venant uniform torsion problem for homogeneous isotropic cylindrical bars with simply or multiply connected cross sections [3–8], the extensive use of structural elements subjected to torsional loading necessitates a reliable, accurate, and general analysis of the torsion problem of bars of arbitrary cross section avoiding the restrictions of the Saint-Venant torsion theory.

In the rest of this paper, both the static and dynamic analyses of the geometrically linear or nonlinear, elastic or elastoplastic nonuniform torsion problems of bars of constant or variable arbitrary cross section are presented together with the corresponding research efforts and the conclusions drawn from examined cases with great practical interest.

#### 2. Linear Elastic Nonuniform Torsion of Bars

In the last decades considerable work has been done on the elastic linear problem of nonuniform torsion of bars. Especially because of the mathematical complexity of the problem, the existing analytical solutions are limited to symmetric cross sections of simple geometry, loading, and boundary conditions [9–16]. Moreover, numerical methods such as the finite element method [17] or the boundary element method [18–20] have also been used for the analysis of the nonuniform torsional problem, in the case the geometry of the cross section, the boundary conditions, or the loading are not simple. In all the aforementioned references the analysis of the nonuniform torsional problem is not complete, since the secondary shear stresses due to warping are not evaluated. Finally, Sapountzakis and Mokos in [21, 22] developed a boundary element solution for the general linear elastic nonuniform torsion problem of homogeneous or composite prismatic bars of arbitrary cross section subjected to an arbitrarily distributed or concentrated twisting moment and supported by the most general linear torsional boundary conditions. In these latter research efforts the evaluation of the secondary warping function is accomplished leading to the computation of the secondary shear stresses due to warping, while the developed procedure is a pure BEM [23], since it requires only boundary discretization.

In order to formulate the aforementioned problem, let us consider a prismatic bar of length (Figure 1), of constant arbitrary cross-section of area . The homogeneous isotropic and linearly elastic material of the bar’s cross-section, with modulus of elasticity , shear modulus , and Poisson’s ratio , occupies the two dimensional multiply connected region of the , plane and is bounded by the boundary curves, which are piecewise smooth; that is, they may have a finite number of corners. In Figure 1(b) is the principal coordinate system through the cross section’s centroid , while are its coordinates with respect to system of axes through the cross section’s shear center . The bar is subjected to the arbitrarily distributed or concentrated conservative twisting moment acting in the direction (Figure 1(a)). This torsional loading results in a rotation with respect to the shear center with angle of twist . The shear center coincides with the center of twist of the cross section provided that no other axis or rotation is imposed due to construction requirements.

**(a)**

**(b)**

Under the aforementioned loading the displacement field of the bar with respect to the system of axes is given aswhere the displacement component constitutes the warping of the cross section, denotes the rate of change of the angle of twist regarded as the torsional curvature, and is the warping function with respect to the shear center , which is characterized as the warping function. The index indicates that the warping refers to the rotation axis .

Substituting the aforementioned displacement field to the linearized strain-displacement relations (infinitesimal strain tensor) of the three-dimensional elasticity, the nonvanishing strain components are written aswhile the resulting nonzero stress components (Cauchy stress tensor) in the region employing the stress-strain relations (constitutive relations) and assuming an isotropic and homogeneous material for zero Poisson ratio are derived as

##### 2.1. Equations of Local Equilibrium

Substituting the stress tensor components in the three equilibrium equations of the three-dimensional elasticity theory ignoring body forces yieldsThe first two of these equations, as it can be observed are not satisfied and this constitutes an inconsistency of the nonuniform torsion theory of bars. Only in the case that , that is, the rate of change of is constant (case of uniform *Saint Venant* torsion), and the first two equilibrium equations are satisfied. From (4c) it follows that
Having in mind that the first and third derivatives of the angle of twist are functions of the longitudinal coordinate , it can be observed that the solution of the partial differential equation (5) in the general case should be a function of coordinate as well. This fact contradicts relation (1a), where the warping function is defined as independent of the coordinate . Thus, the left hand side of (5) is a function only of variables , , since , while at the same time the right hand side is a function of variable . To remove this inconsistency it is assumed that the arising shear stresses are decomposed to a *primary *part and a *secondary * one developed due to warping. This decomposition of shear stress is justified by the consideration of equilibrium of normal stresses due to warping in an infinitesimal element of the cross section as shown in Figure 2. Thus, in a bar of arbitrary cross section subjected to nonuniform torsion, normal stresses arise, depending on the developed warping and vary along the longitudinal axis of the bar (Figure 2(a)). Considering the intersection (*A*) in Figure 2(b) it can be observed that the infinitesimal change of normal stresses due to distortion can be equilibrated only by shear stresses along the intersection of (*A*) and which employing Cauchy’s theorem lead to the development of secondary shear stresses (Figure 2(c)) on the plane of the cross section. Thus, according to the previously mentioned primary shear stresses, which express the shear stresses of uniform torsion (*Saint-Venant*) with the difference that is not constant and to the normal stresses due to warping resulting from the deformation (primary ones), secondary shear stresses result so as to equilibrate the aforementioned normal stresses .

Based on the above decomposition of shear stress in primary and secondary components, that isthe primary and secondary components of these stresses and the normal stresses due to warping are defined according to the relations as follows:

where the functions and are called primary and secondary warping function, respectively. Substituting relations (6a)–(9) in the first equilibrium equation of the theory of elasticity ignoring body forces it follows that Equation (10) indicates that the stress state of the bar subjected to torsional loading results from the superposition of primary , secondary shear stress and normal stresses due to warping . In order to satisfy (10) it is required that both the terms originating from the primary shear stresses as well as these from the normal and the secondary shear stresses due to warping to vanish, that is Substituting (7a)–(9) into (11) the following relations are obtained:

To formulate the *boundary conditions* of the primary and secondary warping functions the shear stress components are observed on the boundary of the cross section. By examining in Figure 3 the infinitesimal surface , the following relations are obtained:
where and are the direction cosines of the vector normal to the boundary of the cross section. Substituting relations (6a) and (6b) that express the decomposition of shear stresses in primary and secondary ones into relations (13) yields

Substituting the expressions of primary and secondary shear stresses given by (7a), (7b), (8a), and (8b) in (14a), (14b), (15a), and (15b) the following relations are obtained:Relations (16a), (16c) and (16b), (16d) give the normal and tangential to the boundary of the cross section primary and secondary shear stresses, respectively. Since the lateral surface of the bar is unloaded along the longitudinal direction, the normal shear stresses at the boundary of the cross section should vanish in order to satisfy the equilibrium requirements. To fullfil this boundary condition it is required that both the primary and the secondary shear stress components to vanish, that isSubstituting (16a) and (16c) into (17a) and (17b) and taking into account the fact that in general , the boundary conditions of the primary and the secondary warping functions are obtained as

Summarizing the above mentioned it is concluded that for the determination of the primary and the secondary warping functions the solution of the following boundary value problems is required.

*For the Primary ** Warping Function*. Neumann problem of Laplace differential equation is as follows:*For the Secondary ** Warping Function. *Neumann problem of the Poisson differential equation is as follows:

Finally, based on the above mentioned, the displacement field of the bar ((1a), (1b), and (1c)) is formed as

It is worth here noting that in the case the origin of the coordinate system is a point of the , plane other than the shear center (Figure 1(b)), the warping function with respect to this point is first established from the Neumann problem (19a) and (19b) substituting by . Using the evaluated warping function , is then established using the transformation given by the following equation [2]: where , , , are the coordinates of the shear center with respect to the arbitrary coordinate system (Figure 1(b)) and is an integration constant. The latter is given from the solution of the following linear system of equations:whereis the cross section area, the static moments of inertia relative to the axes , , the bending moments of inertia relative to the axes , , and the product of inertia, respectively, while in relations (23a), (23b), and (23c) the warping moments ,,, are defined asMoreover, the evaluated warping function from the solution of the Neumann problem (20a) and (20b) contains an integration constant (parallel displacement of the cross section along the beam axis), which can be obtained from [24] as follows: and the main secondary warping function is given as

##### 2.2. Equations of Global Equilibrium

The already established shear stresses and yield components of torque, which arise through integration over the cross section. Thus, the resulting twisting moment is obtained as
Introducing the approximation of decomposition of shear stress in primary and secondary components, the twisting moment of the cross section is divided into a *primary* component originating from the *primary* shear stresses due to twisting (as in uniform torsion) and a *secondary* component , originating from the *secondary* shear stresses, that is the restraint of warping (Figure 4). Thus, according to the nonuniform torsion theory in an arbitrary cross section of the bar the following relation is valid:
where after employing (6a) and (6b) and some algebra the aforementioned twisting moment components are given asSubstituting (7a)–(8b) in (30a) and (30b), employing the Gauss-Green theorem, taking into account (17a), (17b), (19a), and (20a) and after some algebra the following relations for the twisting moment components are obtained:whereIn (32a) and (32b) the quantity denotes the *torsional constant* introduced by Saint-Venant, while the quantity denotes the *warping constant*. The quantity denotes the *torsional rigidity*, while the quantity denotes the *warping rigidity* of the cross section. It is noted that the torsional constant is independent of the position of the coordinate system, while the warping constant refers to the center of twist .

**(a)**

**(b)**

To derive the equilibrium equation for the nonuniform torsion problem of a homogeneous isotropic bar, the equilibrium of an infinitesimal part of the bar against twisting moments is examined and observing Figure 5 it follows that or after some algebra Substituting relations (29) and (31a) and (31b) into (34), the following fourth-order differential equation of equilibrium of a homogeneous isotropic bar subjected to nonuniform torsion is obtained: which is independent of the torsional boundary conditions.

In analogy with the bending moments , a new stress resultant is defined, called *warping moment* (or bimoment) and given by
The need for the definition of this new stress resultant stems from the fact that, whereas , there still exist normal stresses acting on the cross section; hence if a new quantity is not considered, the elastic energy due to stresses will be ignored. Substituting the expression (9) of the stress component into (36), the latter is written as
or through (32b)
and therefore relation (9) can be written in the form of
Equation (39) shows that the quantity of warping moment encompasses the generic characteristics of a stress resultant of theory of elasticity. Schardt [25] refers to it as a “*higher order stress resultant*.” Combining relations (31b) and (38), the relation which correlates the secondary twisting moment and the warping moment arises as
Thus, the problem of nonuniform torsion of a homogeneous isotropic bar is reduced to solving the fourth-order differential equation with respect to the angle of twist of the cross section, given by (35). The solution of this equation depends on both the torsional loading of the bar and the torsional support conditions at the ends or inside the bar. The most *general linear torsional boundary conditions* at the ends of the bar are described by the relationsIt is worth noting that all types of conventional boundary conditions (e.g., fixed, forked support, free end, and elastic support) arise from relations (41a) and (41b) after defining appropriately functions , . For example, in the case of a torsionally fixed support the above functions take the values , .

From the examined example problems presented in [18–22] it is concluded that (i)the magnitude of the evaluated normal stresses, due to restrained warping compared with those due to bending, necessitates the consideration of these additional normal stresses near the restrained edges;(ii)the magnitude of the evaluated warping shear stresses due to restrained warping is remarkable and for an “accurate” analysis these additional shear stresses should not be ignored, especially near the restrained edges.

#### 3. Linear Elastic Nonuniform Torsion of Bars of Variable Cross Section

Long span box shaped bridges or concrete slab and beam structures of variable height are the most common examples of structures including members of variable cross section subjected to twisting moments. The extensive use of the aforementioned structural elements necessitates a rigorous analysis. When a bar of variable cross section is subjected to general twisting loading, this member due to this variation and/or due to the arbitrary torsional boundary conditions applied either at the edges or at any other interior point is leaded to nonuniform torsion and its angle of twist per unit length is not constant along its axis.

Several researchers have dealt with beams of variable cross section ignoring the warping effects resulting from the corresponding restraints at the ends of the member [26, 27]. If the aforementioned structures are analyzed or designed for torsion considering only the effect of Saint Venant torsion resistance, the analysis may underestimate the torsion in the members and the design may be unconservative. On the contrary, to the authors’ knowledge relatively little work has been done on the problem of nonuniform torsion of bars of variable cross section with pioneer the work of Cywinski [28] adopting the finite difference method. Wekezer [29] after dividing the bar into segments along its longitudinal axis approximated their shell midsurface by arbitrary triangular shell elements and employed the finite element method to the linear membrane shell theory. This approximation generates inaccuracies, as the warping of the walls of the cross section cannot be taken into account. Moreover, Eisenberger [30] employed FEM upon polynomial approximation of the torsional and warping rigidities using “exact” shape functions to derive the exact stiffness coefficients. This application of shape functions leads also to inaccuracies in stress analysis of beams of variable cross section, as static and kinematic values at nodes and in the element region are computed only approximately and the element may not satisfy local and global equilibrium conditions [31]. In all of the aforementioned procedures the torsion and warping constants have been approximated adopting the thin tube theory. Finally, Sapountzakis and Mokos in [32, 33] developed a boundary element solution for the general linear elastic nonuniform torsion problem of homogeneous or composite bars of arbitrary variable cross section subjected to an arbitrarily distributed or concentrated twisting moment and supported by the most general linear torsional boundary conditions. In these latter research efforts, three boundary value problems with respect to the variable along the beam angle of twist and to the primary and secondary warping functions are formulated and solved employing a pure BEM [23] approach; that is, only boundary discretization is used. Both the variable warping and torsion constants together with the torsional primary shear stresses and the warping normal and secondary shear stresses are computed.

In order to formulate the aforementioned problem, let us consider a bar of length (Figure 6), of an arbitrarily shaped variable along its axis cross section. The homogeneous isotropic and linearly elastic material of the bar’s cross-section, with modulus of elasticity , shear modulus , and Poisson’s ratio , occupies the two dimensional multiply connected region of the , plane and is bounded by the boundary curves, which are piecewise smooth; that is, they may have a finite number of corners. In Figure 6(b) is the principal coordinate system through the cross section’s centroid , while are its coordinates with respect to system of axes through the cross section’s shear center . The bar is subjected to the arbitrarily distributed or concentrated conservative twisting moment acting in the direction (Figure 6(a)).

**(a)**

**(b)**

Adopting the same displacement field (21a), (21b), and (21c) with that of the constant cross section case and following the same procedure presented for the constant cross section bar in Section 2, the following boundary value problem for the angle of twist is derived: where the total twisting moment of the cross section is once again divided into a primary and a secondary component as this is stated in (29), where these components are given fromwith the torsional and the warping constants of the cross section varying along the length of the bar and given from (32a) and (32b), while the resulting total twisting moment of the cross section is given as The warping moment arising from the torsional curvature, similarly with the constant cross section case, is given from (38) and also functions , are specified at the boundary of the beam forming the most general linear torsional boundary conditions for the beam problem including also the elastic support. Finally, as for the constant cross section case the determination of the primary and the secondary warping functions is achieved from the solution of the boundary value problems given from (19a), (19b), (20a), and (20b), respectively, while the primary, the secondary shear stress components, and the normal stresses due to warping are defined according to the relations (7a)–(9).

From the examined example problems presented in [32, 33] it is concluded that (i)the variation of the beam height, as expected, results in increment of the nonuniform beam behavior in torsional loading;(ii)the magnitude of the evaluated normal and warping shear stresses due to restrained warping is remarkable and necessitates the consideration of these additional stresses near the restrained edges, especially for beams with cross section of low torsional rigidity.(iii)the inaccuracy of the thin tube theory in calculating torsional and warping rigidities even for thin walled sections is remarkable;(iv)analyzing a thin-walled cross section in torsional loading by modeling it with shell elements cannot give accurate results, as the warping of its walls cannot be taken into account.

#### 4. 3D Beam Element of Constant or Variable Cross Section Including Warping Effect

As it has been already mentioned, the analysis of rectilinear or curved members of structures of arbitrary constant or variable cross section subjected to twisting moments is often encountered in engineering practice. Nevertheless, accurate analysis of these members is difficult to achieve for two reasons.

According to the first reason, generally commercial programs consider six degrees of freedom at each node of a member of a space frame, ignoring in this way the warping effects due to the corresponding restraint at the ends of the member or due to variable twisting moment along the bar [26, 27, 34]. If the aforementioned structures are analyzed or designed for torsion considering only the effect of Saint Venant torsion resistance, the analysis may underestimate the torsion in the members and the design may be unconservative. Several researchers tried to overcome this inaccuracy only for constant cross section elements by developing a member stiffness matrix including warping degrees of freedom at the ends of a member with open thin-walled section and assuming simple [35–38] or more complicated torsional boundary conditions [39, 40].

According to the second reason, when the variable cross section structures are modeled by space frame elements, they are usually analyzed using Hermite beam elements characterized by shape functions consisting of Hermite interpolation functions (cubic for bending, linear for axial components of displacement and for the angle of twist) [41] or isoparametric beam elements [42]. In these elements the variation of the cross section is generally considered by setting “average” values for the cross section parameters resulting from the corresponding parameters at the start and end nodes of the element. Moreover, application of shape functions results in inaccuracies in stress analysis of beams of variable cross section, as static and kinematic values at nodes and in the element region are computed only approximately and the element may not satisfy local and global equilibrium conditions [31]. This inaccuracy may be reduced by increasing the number of integration points for the stiffness matrix assembly (at least three-point integration) or by refining the mesh of elements, increasing at the same time the effort of preparing the input parameters.

Nevertheless, Sapountzakis and Mokos in [43–45] developed a boundary element solution for the construction of the stiffness matrix and the nodal load vector of a member of arbitrary homogeneous or composite, constant or variable cross section, subjected to an arbitrarily concentrated or distributed twisting moment, taking into account warping effects. The arbitrary low rate variation of the member of variable cross section is continuous so as to assume that its shear center is independent of the member loading and boundary conditions. The developed method can take into account the variable torsional and warping rigidities along the member length. In these latter research efforts boundary value problems with respect to the variable along the bar angle of twist and to the primary warping function are formulated and solved employing a pure BEM [23] approach; that is only boundary discretization is used.

In order to formulate the aforementioned problem, let us consider a prismatic bar of length (Figure 7), of constant or variable arbitrary cross-section. The homogeneous isotropic and linearly elastic material of the bar’s cross-section, with modulus of elasticity , shear modulus , and Poisson’s ratio , occupies the two dimensional multiply connected region of the , plane and is bounded by the boundary curves, which are piecewise smooth; that is, they may have a finite number of corners. In Figure 7(b) is the principal coordinate system through the cross section’s centroid , while are its coordinates with respect to system of axes through the cross section’s shear center .

**(a)**

**(b)**

In order to include the warping behavior in the study of the aforementioned element, in each node at the element ends a seventh degree of freedom is added to the well-known six DOFs of the classical three-dimensional frame element. The additional DOF is the first derivative of the angle of twist denoting the rate of change of the angle of twist , which can be regarded as the torsional curvature (Figure 8) of the cross section. Thus, the nodal displacement vector in the local coordinate system, as shown in Figure 7(a), can be written as and the respective nodal load vector aswhere is the twisting moment at the ends of the element given from (29) and consisting of the primary part defined as the resultant of the primary shear stress distribution given from (31a) or (44a) and the secondary one defined as the resultant of the secondary shear stress distribution due to warping given from (31b) or (44b) for the cases of constant or variable cross section, respectively, that is,Moreover, is the bending moment due to the torsional curvature at the same sections given from (38).

The nodal displacement and load vectors given in (46) and (47) are related with the local stiffness matrix of the spatial beam element written aswhere the coefficients of the stiffness matrix of (49) come from the well-known classical stiffness matrix of the classical three-dimensional frame element. In the special case of a constant cross section element the coefficients are given as [14, 35]whereIn the case of a variable cross section element, the evaluation of the coefficients presumes the solution of the following boundary value problem with respect to the angle of twist (see also (42)): for appropriate values of the , functions. Thereby, for the evaluation of the coefficient it is , at and , at , for the evaluation of the coefficient it is , at and , at , while for the evaluation of the coefficient it is , at and , at . Apparently, for the constant cross section case it is and the governing equation (52) reduces to the one given from (35). Upon the evaluation of the angle of twist , the coefficients are established from its derivatives using relations (38), (48a), and (48b).

According to the nodal load vector, assuming that the span of the bar is subjected to the arbitrarily concentrated or distributed twisting moment (Figure 7(a)), the evaluation of the elements concerning the twisting and the bending moments due to the torsional curvature is accomplished using again relations (38), (48a), and (48b) employing the derivatives of the angle of twist , obtained from the solution of the following boundary value problem:for the constant cross section case: for the variable cross section case:

From the examined example problems presented in [43–45] it is concluded that (i)the discrepancy of the deflections and the internal stress resultants arising from the ignorance of the warping degrees of freedom at the ends of a member and the magnitude of the normal stresses due to warping necessitate the utilization of the member stiffness matrix, especially for beams with open shaped cross sections;(ii)the advantages of a box shaped closed cross section beam subjected to torsional loading compared with that of an open one are verified;(iii)warping is not constant along the thickness of the cross section walls as it is assumed in thin tube theory for thin-walled beams;(iv)comparison of the elements of the resulting stiffness matrix of a variable cross section member taking into account the derivatives of the variable torsional and warping rigidities along the member length with those obtained using a fine mesh of elements having “average” values for the cross section parameters leads to the consideration of these derivatives;(v)the discrepancy between the uncoupled proposed procedure and the formulation that takes into account coupled displacement components is inconsiderable;(vi)having in mind the previous conclusion and that coupled displacement components lead to dependent shear center on the member loading and boundary conditions, and the ignorance of the effect of the transverse displacement components to the longitudinal one is justified.

#### 5. Elastic Linear Torsional Vibrations of Constant or Variable Cross Section Bars

In engineering practice, we often come across the analysis of structures subjected to vibratory twisting loading. The dynamic forces acting on a structure may result from one or more of different causes, such as rotating machinery, wind, asymmetric traffic loading, blast loads, or earthquake forces. The extensive use of the aforementioned structural elements necessitates a rigorous dynamic analysis.

Exact torsional vibration frequencies were presented by Gorman [46] and Belvins [47] for the case of circular cross section shafts subjected to classical boundary conditions, avoiding in this way warping effects. These efforts were extended by Kameswara Rao [48] for elastically restrained edges. Torsional vibration frequencies for beams of open thin-walled sections, subjected to several combinations of classical boundary conditions, taking into account warping effects were first derived by Gere [49]. Since then approximate methods [50] for the calculation of natural frequencies including elastic torsional and warping restraints [51] and employing either discrete [52–55] or distributed mass model systems [56–60] have been presented. In all these references the considered beam is of a constant homogeneous thin-walled cross section, while its torsion and warping constants are evaluated employing the relations of the thin tube theory.

Several researchers have also dealt with beams of variable cross section ignoring the warping effects resulting from the corresponding restraints at the ends of the member [26, 27]. On the contrary, to the author’s knowledge relatively little work has been done on the problem of nonuniform torsion of bars of variable cross section with pioneer the work of Cywinski [28] adopting the finite difference method. Wekezer [29] after dividing the bar into segments along its longitudinal axis approximated their shell midsurface by arbitrary triangular shell elements and employed the finite element method to the linear membrane shell theory. This approximation generates inaccuracies, as the warping of the walls of the cross section cannot be taken into account. Moreover, Eisenberger [30, 61] employed FEM upon polynomial approximation of the torsional and warping rigidities using “exact” shape functions to derive the exact stiffness coefficients. This application of shape functions results also in inaccuracies in stress analysis of beams of variable cross section, as static and kinematic values at nodes and in the element region are computed only approximately and the element may not satisfy local and global equilibrium conditions [31]. In all the aforementioned procedures the torsion and warping constants have been approximated adopting the thin tube theory. Moreover, research efforts have been presented for the corresponding problem of composite beams limited to the formulation of a displacement-based one-dimensional finite element model for the estimation of natural frequencies and corresponding mode shapes of thin-walled composite beams after approximating again the torsion and warping constants with closed form solutions [62, 63].

Only in Ganapathi et al. [64] the warping function for the constant rectangular cross section of sandwich beams is determined by solving the boundary value problem for torsion such that the displacements are continuous at the interfaces of adjacent layers, while the transverse shear stress is continuous at these interfaces and vanishes at the top and bottom surfaces of the beam. Moreover, Sapountzakis in [65, 66] developed a boundary element method for the nonuniform torsional vibration problem of doubly symmetric composite bars of arbitrary constant or variable cross section, respectively. In these latter efforts the beam is subjected to an arbitrarily distributed dynamic twisting moment, while its edges are restrained by the most general linear torsional boundary conditions. A distributed mass model system is employed which leads to the formulation of three boundary value problems with respect to the variable along the beam angle of twist and to the primary and secondary warping functions. The last two problems are solved employing a pure BEM [23] approach that is only boundary discretization is used. Finally, Sapountzakis and Mokos in [67] presented the dynamic analysis of 3D beam elements restrained at their edges by the most general linear torsional, transverse, or longitudinal boundary conditions and subjected in arbitrarily distributed dynamic twisting, bending, transverse, or longitudinal loading. For the solution of this problem, a boundary element method is developed for the construction of the stiffness matrix and the corresponding nodal load vector, of a member of an arbitrarily shaped simply or multiply connected cross section, taking into account both warping and shear deformation effects, which together with the respective mass and damping matrices lead to the formulation of the equation of motion.

In order to formulate the nonuniform torsional vibration problem of doubly symmetric bars of arbitrarily shaped simply or multiply connected constant or variable cross section, let us consider the bar of length of Figure 9. The homogeneous isotropic and linearly elastic material of the bar’s cross-section, with modulus of elasticity , shear modulus , and Poisson’s ratio , occupies the two dimensional multiply connected region of the , plane and is bounded by the boundary curves, which are piecewise smooth; that is, they may have a finite number of corners. In Figure 9(b) is the principal coordinate system through the cross section’s centroid , while , are its coordinates with respect to system of axes through the cross section’s shear center . The bar is subjected to the time dependent arbitrarily distributed or concentrated twisting moment , acting in the direction (Figure 9(a)).

**(a)**

**(b)**

Adopting the same displacement field (21a), (21b), and (21c) with that of the static analysis of the constant cross section case, which for the dynamic problem is written asand following the same procedure presented for the constant cross section bar in Section 2, the following initial boundary value problem for the angle of twist is derived:

where is the first derivative of the angle of twist with respect to time; , are the initial angle of twist and the corresponding initial velocity of the points of the beam axis; is the polar moment of inertia of the cross section about the origin of its two axes of symmetry (see Figure 9) and is the mass density of the cross section material. It is worth here noting that in the constant cross section case the governing equation (57) reduces to In (58a) and (58b) the total twisting moment of the cross section is once again divided into a primary and a secondary component as this is stated in (29), where these components are given from (44a) and (44b), the resulting total twisting moment of the cross section is given from (45), while the warping moment arising from the torsional curvature, similarly with the constant cross section case is given from (38). Also, the functions , are specified at the boundary of the beam forming the most general linear torsional boundary conditions for the beam problem including also the elastic support. Moreover, as for the static constant cross section problem the determination of the primary and the secondary warping functions is achieved from the solution of the boundary value problems given from (19a), (19b), (20a), and (20b), respectively, noting that the secondary warping function in this case is time dependent. Finally, the primary, the secondary shear stress components, and the normal stresses due to warping are defined similarly with relations (7a)–(9), which for the dynamic problem are written as

From the examined example problems presented in [65–67] it is concluded that (i)the discrepancy of the dynamic response of the bar arising from the ignorance of the warping effect necessitates the inclusion of the nonuniform torsional beam behavior;(ii)the discrepancy of the results arising from the ignorance of the warping especially in higher eigenfrequencies is remarkable;(iii)the discrepancy in the analysis of a thin-walled cross section beam employing the BEM after calculating the torsion and warping constants adopting the thin tube theory demonstrates the importance of the proposed procedure even in thin-walled beams, since it approximates better the torsion and warping constants and takes also into account the warping of the walls of the cross section;(iv)the discrepancy of the results arising from the ignorance of the warping degrees of freedom at the ends of a member necessitates the utilization of the member stiffness matrix, especially for beams with open shaped cross section;(v)warping is not constant along the thickness of the cross section walls as it is assumed in thin tube theory for thin-walled beams.

#### 6. Secondary Torsional Moment Deformation Effect in Elastic Linear Torsional Analysis of Bars

As it has already been mentioned in the previous sections in engineering practice we often come across the analysis of rectilinear members of structures subjected to nonuniform torsion. In this case, as it has been analyzed in Section 2, the twisting moment is split into a primary and a secondary part, where the primary and the secondary torsion moments are undertaken from the Saint-Venant shear stresses (primary shear stresses) and the warping shear stresses (secondary shear stresses), respectively, while the warping normal stresses are undertaken from the warping moment (bimoment). Moreover, in engineering practice as well as in the literature very strong torsional warping is assumed to occur only for open shaped cross sections, while for closed shaped ones the warping effect is assumed to be insignificant and therefore negligible. However, this assumption is not always valid especially for bars of closed shaped cross sections and small length. Noting that in the case of direct torsion (equilibrium torsion) the arising normal and additional shear stresses due to warping are equilibrium and not constraint stresses, which means that after a crack these stresses are redistributed, the necessity of the evaluation and inclusion of these additional warping stresses in the analysis is clearly evident.

Despite the wide study from both the analytical [14, 15, 68–71] and numerical [18–22, 24, 72–75] point of view of the nonuniform torsion problem of prismatic bars, relatively little work has been done on the corresponding problem considering secondary torsional moment deformation effect, that is shear deformation due to the nonuniform torsional warping. It is worth here noting that if shear deformations due to shear forces and restrained warping are considered, flexure and torsion of bars of nonsymmetrical cross section are generally coupled, even if the bar is subjected only to twisting loading, while only for bars of doubly symmetrical cross section flexure and torsion are decoupled. According to the research efforts on the analytical solution of the aforementioned nonuniform torsion problem including secondary shear deformation effect the pioneer work of Heilig [76, 77] is mentioned, in which a theoretical formulation of the problem is presented. Later, Roik and Sedlacek [78] presented an analytical solution applying the Force Method (Flexibility Method) and employing the analogy between 2nd order beam theory with tensional axial force and torsion with warping. Moreover, in the work of Schade [79] a theoretical formulation of the corresponding problem is presented including the coupled shear deformation effect due to shear forces and restrained warping. Rubin [80] using the methodology of Roik and Sedlacek [78] presented an analytical solution for continuous prismatic bars by introducing a Three-Moment Equation (similar with the Three-Moment Equation of Clapeyron for the flexure problem). Also, the theory of the shear deformations due to the restrained torsional warping has been validated by a series of torsional experimental tests on fibre reinforced plastics I-beams by Roberts and Al-Ubaidi [81]. Furthermore, from the numerical point of view intensive research works have been made over the last years to develop new beam finite elements which take into account coupled [82] or uncoupled [83] secondary shear deformation. However, in all of the aforementioned research efforts only bars of thin-walled cross section are investigated, since the torsional cross section parameters (warping constant, primary and secondary torsion constant, and shear center) as well as the shear and normal stresses are evaluated employing the Vlasov thin tube theory [68, 69]. Finally, Kraus [84] presented a FEM solution for the calculation of the secondary torsion constant of bars of hot rolled I-sections and Mokos and Sapountzakis in [85] developed a BEM solution for the nonuniform torsion of simply or multiply connected bars of doubly symmetric arbitrary constant cross section (thin and/or thick walled), taking into account secondary torsional moment deformation effect. In this last research effort, to account for secondary shear deformations, the concept of shear deformation coefficient is used leading to a secondary torsion constant, which is computed employing an effective automatic domain integration using the Advancing Front Method (AFM) [86].

In order to formulate the aforementioned problem, let us consider a rectilinear bar of length , of an arbitrary doubly symmetric constant cross section with modulus of elasticity and shear modulus , occupying the two dimensional multiply connected region of the , plane bounded by the curves as shown in Figure 10. The bar is subjected to arbitrarily distributed and/or concentrated twisting and warping moments, while its edges are restrained by the most general torsional boundary conditions.

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**(b)**

In order to account for shear deformation due to restrained torsional warping, the displacements , , along the three axes and the rotation due to twist are spilt into a primary and a secondary part arising from the primary and the secondary torsional moment, respectively, as

where the (total) first derivative of the angle of twist denotes the rate of change of the (total) angle of twist and can be regarded as the (total) torsional curvature. The primary and the secondary part of the first order derivative of the angle of twist represent the primary torsional curvature (Figure 11(a)) arising from the (primary) warping normal stresses and the secondary torsional curvature (Figure 11(b)) arising from the (secondary) warping shear stresses, respectively.

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Assuming small torsional rotation, the displacement field is defined aswhere is the primary warping function with respect to the shear center of the cross section of the bar. Substituting the aforementioned displacement field to the linearized strain-displacement relations (Cauchy strain tensor) and to the stress-strain relations (constitutive relations) of the three-dimensional elasticity the nonzero stress components in the region can be written as

where is the secondary warping function with respect to the shear center of the cross section of the bar. It is worth here mentioning that, for the derivation of the aforementioned relations of shear stresses, the arising terms , have been neglected.

##### 6.1. Equations of Local Equilibrium

The first elasticity equation of equilibrium of the three-dimensional elasticity (equilibrium in the axial direction of the bar) with vanishing body forces is written as Similarly with Section 2, requiring both the primary () and the secondary due to warping () parts of (69) to vanish, as well as the corresponding ones of the traction vector on the free lateral surface of the bar, the Neumann problems of (19a) and (19b) for the primary and of (20a) and (20b) for the secondary warping functions are obtained.

Additionally, applying the stress components given from (67a), (67b), (67c), (68a), and (68b) into the stress resultants given from (29), (30a), (30b), and (36) and taking into account the Neumann problems (19a), (19b), (20a), and (20b), the following relations for the nonzero stress resultants of the bar are obtained: whereare the warping and the primary torsion constants of the cross section, respectively, while and are the warping and the primary torsional rigidities of the cross section, respectively. It is worth noting that the relations (70), (71a), and (71b) are valid inside the bar as well as at the bar ends. Moreover, observing (70) and (71b) equation (40) can be easily obtained, while substituting (71b) into (20a) the partial Poisson type differential equation governing the secondary warping function is obtained as Since the secondary warping is much smaller with regard to the primary one [24], the secondary torsional curvature can be taken into account in the calculation of the angle of twist indirectly using an effective secondary torsion constant. Thus, as in Timoshenko’s beam theory for shear deformable beams [87, 88], the secondary torsional curvature can be approximately evaluated from the following relation [76, 77]: where is the secondary torsional rigidity of the cross section, while is the secondary torsion constant of the cross section which can be written as where the coefficient is called warping shear correction factor and is the warping shear deformation coefficient. Substituting (75) into (74) the secondary torsional curvature can be written as From (76) it is concluded that for the secondary shear deformation effect is neglected, that is and . This assumption is usually employed in the case of open shaped cross sections.

The evaluation of the secondary torsion constant of the cross section can be achieved by applying an energy approach [16, 78]. Thus, taking into account (68a), (68b), and (74) and equating the approximate formula for the evaluation of the secondary shear strain energy per unit length given from with the exact one given from the secondary torsion constant can be obtained as In order to formulate this constant independently from the loading and the material properties of the bar, the values , are employed and the secondary torsion constant is given as where is the unit secondary warping function with respect to the shear center of the cross section of the bar, while taking into account (73) and (20b) the aforementioned function can be established by solving independently the following Neumann problem:Moreover, applying the Green identity for the warping functions and and taking into account the Neumann problems (19a), (19b), (81a), (81b), and (80) the secondary torsion constant is given as where is a domain integral given from the relation while form (75) and (82) the following relation for the warping shear deformation coefficient is obtained:

##### 6.2. Equations of Global Equilibrium

Furthermore, in order to formulate the governing differential equation of moment equilibrium of the bar in the axial direction, the equilibrium of a small segment of the bar (Figure 12) is considered leading to the relation Taking into account (29), (85) can be written as while substituting (40) into (86) the following equilibrium equation is obtained: Having in mind that both the secondary twisting moment and the bimoment (given from relations (71b) and (70), resp.) are expressed in terms of the primary angle of twist , while the primary twisting moment (given from relation (71a)) is expressed in terms of the total angle of twist , an effort is given in the following to formulate also the primary twisting moment with respect to the primary angle of twist .

Introducing an auxiliary geometric constant defined as the relation (76) can be written as while from relation (65b) the total torsional curvature can be written as and according to (71a) and (71b), the primary twisting moment is given with respect to the primary angle of twist as where is a modified warping constant given as It is worth here noting that the auxiliary constant is always smaller or equal to one, that is . Small values of the constant indicate that the secondary torsional moment deformation effect is important and should be considered in the analysis, while in the case of negligible secondary shear deformations . Moreover, according to (29), (71b), and (91) the (total) twisting moment is given with respect to the primary angle of twist from the relation while using (92) the moments and can be written in terms of the modified warping constant as Hence, substituting (91) and (94) into (87) the governing differential moment equation of equilibrium of the bar in the axial direction with respect to the primary angle of twist is obtained as while the boundary conditions are given fromwhere the parameters , are functions specified at the boundary of the bar and the moments and are given form the relations (93) and (94), respectively. It is worth here noting that the boundary conditions (97a) and (97b) are the most general linear torsional boundary conditions including also the elastic support. It is apparent that all types of the conventional torsional boundary conditions (clamped, simply supported, free, or guided edge) can be derived from these equations by specifying appropriately the functions and (e.g., for a clamped edge it is , ).

Furthermore, substituting (95) into (89) and differentiating with respect to , the governing differential equation of equilibrium of the bar in the axial direction with respect to the secondary angle of twist is obtained in terms of the primary angle of twist as while the corresponding boundary conditions are given in terms of the primary angle of twist as where if the primary angle of twist is fixed, that is and , then and , otherwise and .

From the examined example problems presented in [85] it is concluded that (i)the inaccuracy of the thin tube theory in calculating the secondary torsion constant even for thin-walled sections is noteworthy;(ii)as it is also verified by other computational methods [80, 83], for bars with open shaped cross section the secondary torsional moment deformation effect has not significant influence. However, the inclusion of this additional effect leads to more accurate results;(iii)as it is also verified by other computational methods [80, 83], for bars with closed shaped cross sections the secondary torsional moment deformation effect has an important influence and should be considered in the analysis (otherwise the results may be completely wrong especially in the calculation of stresses).

#### 7. Nonlinear Elastic Nonuniform Torsion of Bars

As it has already been mentioned in the previous sections in engineering practice we often come across the analysis of members of structures subjected to twisting moments. Besides, since thin-walled open sections have low torsional stiffness, the torsional deformations can be of such magnitudes that it is not adequate to treat the angles of cross section rotation as small. When finite twist rotation angles are considered, the elastic nonuniform torsion problem becomes nonlinear. Moreover, this problem becomes much more complicated in the case the cross section’s centroid does not coincide with its shear center (asymmetric beams), leading to the formulation of a flexural-torsional coupled problem. The extensive use of the aforementioned structural elements necessitates a reliable and accurate analysis of bars of arbitrary cross section subjected to torsional loading taking into account the geometrical nonlinearity.

Though several researchers have dealt either with the linear nonuniform torsional behaviour of beams [17, 21, 22, 40] or with the nonlinear uniform torsional behaviour of doubly symmetric beams [89, 90], to the author’s knowledge very little work has been done on the corresponding nonlinear nonuniform torsional problem of arbitrary cross section beams. Ghobarah and Tso [91], Attard [92], and Attard and Somervaille [93] have presented a set of displacement relationships for a straight prismatic thin-walled open beam applicable to situations where displacements are finite, the cross section does not distort, strains are small and flexural displacements are small to moderate while cross sectional twist can be large. The presented numerical examples in these studies are concerned only with uniform torsion of either mono- or doubly symmetric cross sections. Finally, Trahair in [94] employing the finite element method and presenting examples of only doubly symmetric cross sections and Mohri et al. in [95] employing similar equations to those established by Attard in [92] and presenting examples of either doubly symmetric cross sections subjected in nonuniform torsion or buckling or postbuckling behavior of arbitrary cross section beams also analyze the nonlinear nonuniform torsional problem. Nevertheless, all of the aforementioned studies, which are the only one considering finite angles of twist in asymmetrical bars (and taking into account all of the arising nonlinear terms) are not general since they are restricted to thin-walled beams. Finally, Sapountzakis and Tsipiras in [96, 97] presented a BE solution for the elastic nonuniform torsion analysis of simply, multiply connected or composite cylindrical bars of arbitrary cross section taking into account the effect of geometric nonlinearity. The torque-rotation relationship is computed based on the finite displacement (finite rotation) theory; that is, the transverse displacement components are expressed so as to be valid for large rotations and the longitudinal normal strain includes the second-order geometrically nonlinear term often described as the “Wagner strain.” These last formulations do not stand on the assumption of a thin-walled structure and therefore the cross section’s torsional rigidity is evaluated exactly without using the so-called Saint-Venant’s torsional constant. The torsional rigidity of the cross section is evaluated directly employing the primary warping function of the cross section [21] depending on its shape. Three boundary value problems with respect to the variable along the beam axis angle of twist, to the primary and to the secondary warping functions are formulated. The first one of these problems is numerically solved employing the Analog Equation Method [98], a BEM based method, leading to a system of nonlinear equations from which the angle of twist is computed by an iterative process. The other two problems are solved using a pure BEM [23] based method. The aforementioned formulation procedure is based on the assumption of no local or lateral torsional buckling or distortion.

In order to formulate the aforementioned problem, let us consider a prismatic bar of length (Figure 13), of constant arbitrary cross-section of area . The homogeneous isotropic and linearly elastic material of the bar’s cross-section, with modulus of elasticity , shear modulus , and Poisson’s ratio , occupies the two dimensional multiply connected region of the , plane and is bounded by the boundary curves, which are piecewise smooth; that is, they may have a finite number of corners. In Figure 13(b) is the principal coordinate system through the cross section’s centroid , while , are its coordinates with respect to system of axes through the cross section’s shear center . The bar is subjected to the combined action of the arbitrarily distributed or concentrated conservative twisting and warping moments acting in the direction (Figure 13(a)).

**(a)**

**(b)**

Under the aforementioned loading the displacement field of the bar with respect to the system of axes for large twisting rotations and small bending ones is given aswhere the transverse displacement components , are valid for large rotations [99]; , are the angles of rotation due to bending of the cross section with respect to its centroid; denotes the rate of change of the angle of twist regarded as the torsional curvature; , are the transverse displacement components of the shear center ; , are the primary and secondary warping functions with respect to the shear center , respectively [21]; is an “average” axial displacement of the cross section of the bar, that will be later explained.

Substituting (100a), (100b), and (100c) in the nonlinear (Green) strain-displacement relations of the nonvanishing strains