OPEN ACCESS
In this work, the SaintVenant torsion problem of prismatic bars with rectangular crosssections was presented as a boundary value problem (BVP) of the theory of elasticity. The governing partial differential equation was formulated and shown to be a Poisson equation in terms of the Prandtl stress functions. The Poisson equation governing the SaintVenant torsion problem was expressed in variational form using Galerkin variational method. The trial function that apriori satisfies the boundary conditions was chosen as a trigonometric cosine series of infinite terms; and in terms of unknown undetermined coefficients or parameters. The unknown parameters were determined by solving the Galerkin variational integral; thus fully determining the Prandtl stress function. The shear stresses were then determined. The maximum shear stress was also obtained. The moment of the crosssection was determined and found to depend on non – dimensional torsional parameters F1(a/b). The maximum shear stress was also found to depend on dimensionless torsional parameters F2(a/b) which were determined and tabulated. It was found that the solutions obtained using the Galerkin method were mathematically closed form solutions because the exact shape functions were used to approximate the trial solution. Expressions obtained for the Prandtl stress function, shear stresses and moment of crosssection were exact and agreed with solutions in the technical literature.
galleria variational method, poison equation, Prandtl stress function, Saint Venant torsional problem
When a torque is applied to a beam with noncircular crosssection, the crosssection rotates about the longitudinal axis of the beam and simultaneously undergoes a significant distortion. The crosssection thus undergoes both twisting and warping deformations [18]. Such problems are formulated using theory of elasticity principles. The foundational assumptions of the formulation are the straindisplacement (kinematic) relations of infinitesimal/small deformation assumptions, the stressstrain laws, the differential equations of equilibrium and compatibility requirements. SaintVenant formulated the problem using theory of elasticity and Prandtl solved the problem in terms of Prandtl’s stress functions.
Prandtl’s formulation of the SaintVenant torsion problem leads to a Poisson type partial differential equation (PDE); which can be solved using analytical or numerical methods. Available analytical techniques include the method of separation of variables, eigenfunction expansion methods, integral transform methods and Green’s function methods. The numerical methods that can be used to solve the torsion problem are the numerical methods available for solving the boundary value problems (BVP) in engineering. Some of the numerical techniques for solving BVP which are applicable to the Poisson type PDE are Galerkin’s variational method, Extended Galerkin’s variational method, Ritz’s method, Finite Element method (FEM), Finite difference method (FDM); and Boundary element methods (BEM) [1–13].
In this work, the SaintVenant problem of torsion of prismatic bars with rectangular crosssection will be formulated in variational form, and solved using the Galerkin variational method.
Research aim and objectives
The research aim is to use the Galerkin variational method to solve the SaintVenant torsion problem for prismatic bars with rectangular crosssections. The specific objectives are:
(1) to formulate the SaintVenant torsional problem for prismatic bars with rectangular crosssections using theory of elasticity principles and techniques
(2) to show that the formulated SaintVenant torsional problem is governed by a partial differential equation called the Poisson equation when expressed in terms of the Airy’s and Prandtl’s stress functions of elasticity
(3) to express the boundary value problem (BVP) in variational form using the Galerkin variational method
(4) to solve the Galerkin variational equation for the problem and thus obtain solutions for the unknown parameters in the assumed (trial) solutions for the Airy’s or Prandtl’s stress function
(5) to obtain analytical expressions for the torque, shear stresses and torsional parameters
(6) to show that Galerkin’s variational method can be used to obtain closed form mathematical solutions to the BVP of SaintVenant torsion for prismatic bars with rectangular crosssections.
The study considered an isotropic, homogeneous long bar with prismatic crosssection denoted by R^{2} on the yz coordinate plane. The longitudinal axis is coincident with the x – Cartesian coordinate axis. The bar is fixed at x = 0. The end at x = l is subject to a torsional moment which twists it by an angle ${l\theta}'$ where ${\theta }'$ is the twist rate and l is the length of the bar [14–21]. Other relevant literature can be found in Roohi et al. [22] and Heydari et al. [23].
The assumptions of the formulation are as follows:
(1) the crosssections in the yz coordinate plane undergo rotation as a rigid body. For noncircular crosssections, the crosssection will experience twisting. It is deflected in the x – coordinate direction
(2) the deflection and twist rate are constant along the longitudinal axis of the bar. This renders the problem a twodimensional (2D) problem in the theory of elasticity
(3) the material of the bar is isotropic and homogeneous.
2.1 Displacement field
The three dimensional (3D) Cartesian components of the displacement field, following SaintVenant hypothesis [14–21] are given by:
$u(x, y, z)=\theta^{\prime} \varphi(y, z)=\beta(y, z)$ (1)
$v(x, y, z)=\theta^{\prime} x z=\beta x z$ (2)
$w(x, y, z)=\theta^{\prime} x y=\beta x y$ (3)
where j(y, z) is an unknown function which is used to define the deflection and is a function of the y and z Cartesian coordinate variables. u, v, and w are the components of displacement in the x, y and z Cartesian coordinate directions, respectively. b = θ' is the twist rate.
2.2 Strain field
Using the straindisplacement equations for infinitesimally small deformation, the strain fields are obtained as follows:
$\varepsilon_{x x}=\frac{\partial u}{\partial x}=0$ (4)
${{\varepsilon }_{yy}}=\frac{\partial v}{\partial y}=0$ (5)
${{\varepsilon }_{zz}}=\frac{\partial w}{\partial z}=0$ (6)
${{\gamma }_{xy}}=2{{\varepsilon }_{xy}}=\left( \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x} \right)=\beta \left( \frac{\partial \varphi }{\partial y}z \right)$ (7)
${{\gamma }_{xz}}=2{{\varepsilon }_{xz}}=\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}=\beta \left( \frac{\partial \varphi }{\partial z}+y \right)$ (8)
${{\gamma }_{yz}}=2{{\varepsilon }_{yz}}=\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}=\beta x+\beta x=0$ (9)
ε_{xx}, ε_{yy}, ε_{zz} are the normal strains while γ_{xy}, γ_{yz} and γ_{xz} are the shear strains. Thus, the strain – compatibility equation becomes:
$\frac{\partial {{\gamma }_{xz}}}{\partial y}\frac{\partial {{\gamma }_{xy}}}{\partial z}=\frac{\partial }{\partial y}\beta \left( \frac{\partial \varphi }{\partial z}+y \right)\frac{\partial }{\partial z}\beta \left( \frac{\partial \varphi }{\partial y}z \right)=\beta \left( \frac{{{\partial }^{2}}\varphi }{\partial y\partial z}+1\frac{{{\partial }^{2}}\varphi }{\partial z\partial y}1 \right)=2\beta $ (10)
Provided $\frac{{{\partial }^{2}}\varphi }{\partial y\partial z}=\frac{{{\partial }^{2}}\varphi }{\partial z\partial y}$ (11)
Eq.(11) implies that φ(y, z) is required to be a continuous function of y and z.
2.3 Stress fields
The generalized Hooke’s law of linear isotropic elasticity is given generally by
${{\tau }_{ij}}=\lambda {{\partial }_{ij}}({{\varepsilon }_{xx}}+{{\varepsilon }_{yy}}+{{\varepsilon }_{zz}})+2G{{\varepsilon }_{ij}}=\lambda {{\partial }_{ij}}{{\varepsilon }_{v}}+2G{{\varepsilon }_{ij}}$ (12)
where ${{\partial }_{ij}}=1$ for i = j; ${{\partial }_{ij}}=0$ for i≠j
λ and G are the Lamé’s constants G is the shear modulus. ε_{v} is the volumetric strain.
${{\varepsilon }_{v}}={{\varepsilon }_{xx}}+{{\varepsilon }_{yy}}+{{\varepsilon }_{zz}}=0$ (13)
The stress fields are given by
${{\sigma }_{xx}}={{\sigma }_{yy}}={{\sigma }_{zz}}=0$ (14)
${{\tau }_{xy}}=G{{\gamma }_{xy}}=2G{{\varepsilon }_{xy}}=\beta G\left( \frac{\partial \varphi }{\partial y}z \right)$ (15)
${{\tau }_{xz}}=G{{\gamma }_{xz}}=2G{{\varepsilon }_{xz}}=\beta G\left( \frac{\partial \varphi }{\partial z}+y \right)$ (16)
${{\tau }_{yz}}=G{{\gamma }_{yz}}=0$ (17)
where σ_{xx}, σ_{yy}, σ_{zz} are normal stresses τ_{xy}, τ_{yz}, τ_{xz} are shear stresses.
2.4 Differential equations of equilibrium
The differential equations of equilibrium in the absence of body forces f_{i} given in general by Eq. (18).
${{\sum }_{j}}{{\partial }_{j}}{{\tau }_{ij}}={{f}_{i}}=0$ (18)
Simplify to become Equations (19) – (21).
$\frac{\partial {{\tau }_{xy}}}{\partial y}+\frac{\partial {{\tau }_{xz}}}{\partial z}=0$ (19)
$\frac{\partial {{\tau }_{xy}}}{\partial x}=0$ (20)
$\frac{\partial {{\tau }_{xz}}}{\partial x}=0$ (21)
2.5 Prandtl’s stress function ϕ(y, z)
Prandtl defined stress functions ϕ(y, z) which are functions of the y and z coordinate variables of the crosssection, and independent of x such that the differential equations of equilibrium are satisfied by the nonvanishing stress components τ_{xy} and τ_{xz} as follows:
${{\tau }_{xy}}=G\beta \frac{\partial \phi }{\partial z}(y,z)$ (22)
${{\tau }_{xz}}=G\beta \frac{\partial \phi }{\partial y}(y,z)$ (23)
It is observed that for Prandtl’s stress functions Equations (22) and (23), Equation (19) becomes:
$\frac{\partial }{\partial y}\left( G\beta \frac{\partial \phi }{\partial z} \right)+\frac{\partial }{\partial z}\left( G\beta \frac{\partial \phi }{\partial y} \right)=G\beta \frac{{{\partial }^{2}}\phi }{\partial y\partial z}G\beta \frac{{{\partial }^{2}}\phi }{\partial z\partial y}=0$ (24)
if
$\frac{{{\partial }^{2}}\phi }{\partial y\partial z}=\frac{{{\partial }^{2}}\phi }{\partial z\partial y}$ (25)
Prandtl's stress functions are solutions of the differential equation of equilibrium if the functions are continuous. The strain components are given in terms of the Prandtl stress function ϕ(x, z) as:
${{\gamma }_{xy}}=\frac{{{\tau }_{xy}}}{G}=\beta \frac{\partial \phi }{\partial z}$ (26)
${{\gamma }_{xz}}=\frac{{{\tau }_{xz}}}{G}=\beta \frac{\partial \phi }{\partial y}$ (27)
Then the strain compatibility equation is
$\frac{\partial {{\gamma }_{xz}}}{\partial y}\frac{\partial {{\gamma }_{xy}}}{\partial z}=\beta \frac{{{\partial }^{2}}\phi }{\partial {{y}^{2}}}\beta \frac{{{\partial }^{2}}\phi }{\partial {{z}^{2}}}=\beta \left( \frac{{{\partial }^{2}}\phi }{\partial {{y}^{2}}}+\frac{{{\partial }^{2}}\phi }{\partial {{z}^{2}}} \right)$ (28)
From Equation (10), Equation (25) can be expressed as:
$\beta \left( \frac{{{\partial }^{2}}\phi }{\partial {{y}^{2}}}+\frac{{{\partial }^{2}}\phi }{\partial {{z}^{2}}} \right)=2\beta $ (29)
Simplifying,
$\Delta \phi =\frac{{{\partial }^{2}}\phi (y,z)}{\partial {{y}^{2}}}+\frac{{{\partial }^{2}}\phi (y,z)}{\partial {{z}^{2}}}={{\nabla }^{2}}\phi (y,z)=2$ (30)
where
$\Delta ={{\nabla }^{2}}=\frac{{{\partial }^{2}}}{\partial {{y}^{2}}}+\frac{{{\partial }^{2}}}{\partial {{z}^{2}}}$ (31)
△ or ▽^{2} is the Laplace differential operator.
2.6 Boundary condition
The boundary condition for the Prandtl stress function for a crosssectional profile with no holes is
$\phi (y,z)=0$ (32)
on the boundary Γ.
2.7 Torque, section moment and shear stresses
The torque or torsional moment, M is computed as the double integral over the crosssection
$M=\iint\limits_{{{R}^{2}}}{({{\tau }_{xy}}z+{{\tau }_{xz}}y)}dydz$ (33)
where R^{2} is the crosssection of the bar.
$M=\iint\limits_{{{R}^{2}}}{\left( G\beta \frac{\partial \phi }{\partial z}zG\beta \frac{\partial \phi }{\partial y}y \right)}dydz$ (34)
$M=G\beta \iint\limits_{{{R}^{2}}}{\left( \frac{\partial \phi }{\partial z}z+\frac{\partial \phi }{\partial y}y \right)}dydz$ (35)
Using the method of integration by parts,
$\iint\limits_{{{R}^{2}}}{\frac{\partial \phi }{\partial z}z}\,dydz=\int\limits_{\Gamma }{\phi z\,{{n}_{z}}}\,ds\iint\limits_{{{R}^{2}}}{\phi (y,z)}\,dydz=\iint\limits_{{{R}^{2}}}{\phi (y,z)}\,dydz$ (36)
$\iint\limits_{{{R}^{2}}}{\frac{\partial \phi }{\partial y}y}\,dydz=\iint\limits_{{{R}^{2}}}{\phi (y,z)}\,dydz$ (37)
$M=2G\beta \iint\limits_{{{R}^{2}}}{\phi (y,z)}\,dydz=G\beta J$ (38)
where
$J=2\iint\limits_{{{R}^{2}}}{\phi (y,z)}dydz$ (39)
J is the moment of the crosssection, or torsional constant.
The modulus of the shear stress is
$\left T \right={{\left( \tau _{xy}^{2}+\tau _{xz}^{2} \right)}^{1/2}}$ (40)
For a rectangular crosssection on the yz Cartesian coordinate plane defined by
$\frac{a}{2}\le y\le \frac{a}{2};\,\,\frac{b}{2}\le z\le \frac{b}{2}$
where a ≥b > 0 the Prandtl stress function that satisfies the boundary condition Eq. (32) is assumed in terms of the unknown parameters C_{mn} as the infinite series:
$\phi (y,z)=\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{{{C}_{mn}}\cos \frac{m\pi y}{a}\cos \frac{n\pi z}{b}}}$ (41)
m = 1, 3, 5, 7, 9, …; n = 1, 3, 5, 7, 9, …
Since
$\phi \left( y=\pm \frac{a}{2},z \right)=\phi \left( y,z=\pm \frac{b}{2} \right)=0$ (42)
The Galerkin variational integral becomes:
$\int\limits_{b/2}^{b/2}{\int\limits_{a/2}^{a/2}{({{\nabla }^{2}}\phi +2)\cos \frac{{m}'\pi y}{a}\cos \frac{{n}'\pi z}{b}}}\,dydz=0$ (43)
Expanding,
$\int\limits_{b/2}^{b/2}{\int\limits_{a/2}^{a/2}{\left\{ \left( \frac{{{\partial }^{2}}}{\partial {{y}^{2}}}+\frac{{{\partial }^{2}}}{\partial {{z}^{2}}} \right)\left( \sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{{{C}_{mn}}\cos \frac{m\pi y}{a}\cos \frac{n\pi z}{b}}} \right)+2 \right\}\cos \frac{{m}'\pi y}{a}\cos \frac{{n}'\pi z}{b}dydz=0}}\,$ (44)
The Galerkin variational integral is
$\int\limits_{b/2}^{b/2}{\int\limits_{a/2}^{a/2}{\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\left( \frac{{{\partial }^{2}}}{\partial {{y}^{2}}}+\frac{{{\partial }^{2}}}{\partial {{z}^{2}}} \right)}}\left( {{c}_{mn}}\cos \frac{m\pi y}{a}\cos \frac{n\pi z}{b} \right)}}$
$\cos \frac{{m}'\pi y}{a}\cos \frac{{n}'\pi z}{b}\,dydz$
$=\int\limits_{b/2}^{b/2}{\int\limits_{a/2}^{a/2}{2\cos \frac{{m}'\pi y}{a}}}\cos \frac{{n}'\pi z}{b}dydz$ (45)
$\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{{{C}_{mn}}\int\limits_{b/2}^{b/2}{\int\limits_{a/2}^{a/2}{\left( {{\left( \frac{m\pi }{a} \right)}^{2}}+{{\left( \frac{n\pi }{b} \right)}^{2}} \right)\cos \frac{m\pi y}{a}\cos \frac{n\pi z}{b}\cos \frac{{m}'\pi y}{a}\cos \frac{{n}'\pi z}{b}}}dydz}}$
$=\int\limits_{b/2}^{b/2}{\int\limits_{a/2}^{a/2}{2\cos \frac{{n}'\pi z}{b}\cos \frac{{m}'\pi y}{a}}}dydz$ (46)
2 is expanded in Fourier cosine series as:
$2=\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{{{a}_{mn}}\cos \frac{m\pi y}{a}\cos \frac{n\pi z}{b}}}$ (47)
where
${{a}_{mn}}=\frac{4}{ab}\int\limits_{b/2}^{b/2}{\int\limits_{a/2}^{a/2}{(2)\cos \frac{m\pi y}{a}\cos \frac{n\pi z}{b}}}dydz$ (48)
${{a}_{mn}}=\frac{4}{ab}\mathsf{\cdot }2\mathsf{\cdot }\frac{2a}{m\pi }{{(1)}^{\frac{m1}{2}}}\frac{2b}{n\pi }{{(1)}^{\frac{n1}{2}}}$ (49)
${{a}_{mn}}=\frac{(2){{4}^{2}}}{mn{{\pi }^{2}}}{{(1)}^{\frac{m+n2}{2}}}=\frac{(2){{4}^{2}}{{(1)}^{\frac{m+n}{2}1}}}{mn{{\pi }^{2}}}$ (50)
So,
$2=\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{{{(1)}^{\frac{m+n}{2}1}}\frac{(2){{4}^{2}}}{mn{{\pi }^{2}}}}}\cos \frac{m\pi y}{a}\cos \frac{n\pi z}{b}$ (51)
So,
$\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{{{C}_{mn}}\int\limits_{b/2}^{b/2}{\int\limits_{a/2}^{a/2}{\left( {{\left( \frac{m\pi }{a} \right)}^{2}}+{{\left( \frac{n\pi }{b} \right)}^{2}} \right)\cos \frac{m\pi y}{a}\cos \frac{n\pi z}{b}}}}}\cos \frac{{m}'\pi y}{a}\cos \frac{{n}'\pi z}{b}dydz$
$=\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\int\limits_{b/2}^{b/2}{\int\limits_{a/2}^{a/2}{\frac{{{(1)}^{\frac{m+n}{2}1}}(2){{(4)}^{2}}}{mn{{\pi }^{2}}}\cos \frac{m\pi y}{a}\cos \frac{n\pi z}{b}}}}}\cos \frac{{m}'\pi y}{a}\cos \frac{{n}'\pi z}{b}dydz$ (52)
${{C}_{mn}}=\frac{(2){{4}^{2}}{{(1)}^{\frac{m+n2}{2}}}}{mn{{\pi }^{2}}}{{\left( {{\left( \frac{m\pi }{a} \right)}^{2}}+{{\left( \frac{n\pi }{b} \right)}^{2}} \right)}^{1}}=\frac{{{2}^{5}}{{(1)}^{\frac{m+n}{2}1}}}{mn{{\pi }^{2}}}\left( \frac{1}{{{\pi }^{2}}\left( {{\left( \frac{m}{a} \right)}^{2}}+{{\left( \frac{n}{b} \right)}^{2}} \right)} \right)$
$=\frac{{{2}^{5}}{{(1)}^{\frac{m+n}{2}1}}}{mn{{\pi }^{4}}}\left( \frac{1}{\left( \frac{{{b}^{2}}{{m}^{2}}+{{n}^{2}}{{a}^{2}}}{{{a}^{2}}{{b}^{2}}} \right)} \right)$ (53)
${{C}_{mn}}=\frac{{{2}^{5}}{{(1)}^{\frac{m+n}{2}1}}{{a}^{2}}{{b}^{2}}}{{{\pi }^{4}}}\frac{1}{mn({{m}^{2}}{{b}^{2}}+{{n}^{2}}{{a}^{2}})}$ (54)
The Prandtl stress function is then:
$\phi (y,z)=\frac{{{2}^{5}}{{a}^{2}}{{b}^{2}}}{{{\pi }^{4}}}\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\frac{{{(1)}^{\frac{m+n}{2}1}}\cos \frac{m\pi y}{a}\cos \frac{n\pi z}{b}}{mn({{m}^{2}}{{b}^{2}}+{{n}^{2}}{{a}^{2}})}}}$ (55)
ϕ(y, z) is obtained as a trigonometric cosine series of infinite terms. The series is a convergent series since
$\left {{C}_{mn}}\cos \frac{m\pi y}{a}\cos \frac{n\pi z}{b} \right\le \left {{C}_{mn}} \right\le \text{constant}\left( \frac{1}{{{m}^{2}}{{n}^{2}}} \right)$
and, $\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\frac{1}{{{m}^{2}}{{n}^{2}}}}}=\left( \sum\limits_{m}^{\infty }{\frac{1}{{{m}^{2}}}} \right)\left( \sum\limits_{n}^{\infty }{\frac{1}{{{n}^{2}}}} \right)$
4.1 Moment of the crosssection (torsional constant, j)
From Eq. (39),
$J=2\int\limits_{b/2}^{b/2}{\int\limits_{a/2}^{a/2}{\frac{{{2}^{5}}{{a}^{2}}{{b}^{2}}}{{{\pi }^{4}}}}}\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\frac{{{(1)}^{\frac{m+n}{2}1}}\cos \frac{m\pi y}{a}\cos \frac{n\pi z}{b}}{mn({{m}^{2}}{{b}^{2}}+{{n}^{2}}{{a}^{2}})}}}\,dydz$ (56)
$J=\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\frac{{{2}^{6}}{{a}^{2}}{{b}^{2}}}{{{\pi }^{4}}}}}\frac{{{(1)}^{\frac{m+n}{2}1}}}{mn({{m}^{2}}{{b}^{2}}+{{n}^{2}}{{a}^{2}})}\int\limits_{b/2}^{b/2}{\int\limits_{a/2}^{a/2}{\cos \frac{m\pi y}{a}\cos \frac{n\pi z}{b}}}dydz$ (57)
$J=\frac{{{2}^{8}}{{a}^{3}}{{b}^{3}}}{{{\pi }^{6}}}\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\frac{1}{{{m}^{2}}{{n}^{2}}({{m}^{2}}{{b}^{2}}+{{n}^{2}}{{a}^{2}})}}}$ (58)
Let
$\frac{a}{b}=r$ (59)
or
a=br (60)
then,
$J=\frac{{{2}^{8}}{{b}^{6}}{{r}^{3}}}{{{\pi }^{6}}}\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\frac{1}{{{m}^{2}}{{n}^{2}}({{m}^{2}}{{b}^{2}}+{{n}^{2}}{{b}^{2}}{{r}^{2}})}}}$ (61)
$J=\frac{{{2}^{8}}{{b}^{6}}{{r}^{3}}}{{{\pi }^{6}}}\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\frac{1}{{{m}^{2}}{{n}^{2}}{{b}^{2}}({{m}^{2}}+{{n}^{2}}{{r}^{2}})}}}$ (62)
$J=\frac{{{2}^{8}}{{b}^{4}}{{r}^{3}}}{{{\pi }^{6}}}\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\frac{1}{{{m}^{2}}{{n}^{2}}({{m}^{2}}+{{n}^{2}}{{r}^{2}})}}}$ (63)
Alternatively,
$J=\frac{{{2}^{8}}{{a}^{6}}}{{{r}^{3}}{{\pi }^{6}}}\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\frac{1}{{{m}^{2}}{{n}^{2}}\left( \frac{{{m}^{2}}{{a}^{2}}}{{{r}^{2}}}+{{n}^{2}}{{a}^{2}} \right)}}}$ (64)
$J=\frac{{{2}^{8}}{{a}^{6}}}{{{r}^{3}}{{\pi }^{6}}}\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\frac{1}{{{m}^{2}}{{n}^{2}}\left( \frac{{{m}^{2}}{{a}^{2}}+{{n}^{2}}{{a}^{2}}{{r}^{2}}}{{{r}^{2}}} \right)}}}$ (65)
$J=\frac{{{2}^{8}}{{a}^{6}}}{{{r}^{3}}{{\pi }^{6}}}\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\frac{{{r}^{2}}}{{{m}^{2}}{{n}^{2}}{{a}^{2}}({{m}^{2}}+{{n}^{2}}{{r}^{2}})}}}$ (66)
$J=\frac{{{2}^{8}}{{a}^{6}}}{r{{\pi }^{6}}}\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\frac{1}{{{m}^{2}}{{n}^{2}}{{a}^{2}}({{m}^{2}}+{{n}^{2}}{{r}^{2}})}}}$ (67)
$J=\frac{{{2}^{8}}{{a}^{4}}}{r{{\pi }^{6}}}\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\frac{1}{{{m}^{2}}{{n}^{2}}({{m}^{2}}+{{n}^{2}}{{r}^{2}})}}}$ (68)
or
$J=\frac{{{2}^{8}}{{a}^{3}}{{b}^{3}}}{{{\pi }^{6}}}\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\frac{1}{{{b}^{2}}{{m}^{2}}{{n}^{2}}\left( \frac{{{m}^{2}}{{b}^{2}}+{{n}^{2}}{{a}^{2}}}{{{b}^{2}}} \right)}}}$ (69)
$J=\frac{{{2}^{8}}{{a}^{3}}b}{{{\pi }^{6}}}\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\frac{1}{{{m}^{2}}{{n}^{2}}\left( {{m}^{2}}+\frac{{{n}^{2}}{{a}^{2}}}{{{b}^{2}}} \right)}}}$ (70)
$J=\frac{{{2}^{8}}}{{{\pi }^{6}}}\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\frac{{{(a\text{/}b)}^{2}}}{{{m}^{2}}{{n}^{2}}\left( {{m}^{2}}+\frac{{{n}^{2}}{{a}^{2}}}{{{b}^{2}}} \right)}}}\mathsf{\cdot }a{{b}^{3}}$ (71)
$J={{F}_{1}}\left( \frac{a}{b} \right)a{{b}^{3}}$ (72)
where
${{F}_{1}}\left( \frac{a}{b} \right)=\frac{{{2}^{8}}}{{{\pi }^{6}}}\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\frac{{{(a\text{/}b)}^{2}}}{{{m}^{2}}{{n}^{2}}\left( {{m}^{2}}+\frac{{{n}^{2}}{{a}^{2}}}{{{b}^{2}}} \right)}}}$ (73)
4.2 Shear stress tensors
The shear stresses are found from the Prandtl stress function as:
${{\tau }_{xy}}(y,z)=G\beta \frac{\partial \phi }{\partial z}(y,z)=G\beta \frac{{{2}^{5}}{{a}^{2}}{{b}^{2}}}{{{\pi }^{4}}}\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\frac{{{(1)}^{\frac{m+n}{2}1}}}{mn({{m}^{2}}{{b}^{2}}+{{n}^{2}}{{a}^{2}})}}}\cos \frac{m\pi y}{a}\frac{\partial }{\partial z}\cos \frac{n\pi z}{b}$ (74)
${{\tau }_{xz}}(y,z)=G\beta \frac{\partial \phi }{\partial y}(y,z)\,=\,G\beta \frac{{{2}^{5}}{{a}^{2}}{{b}^{2}}}{{{\pi }^{4}}}\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\frac{{{(1)}^{\frac{m+n}{2}1}}}{mn({{m}^{2}}{{b}^{2}}+\,{{n}^{2}}{{a}^{2}})}}}\cos \frac{n\pi z}{b}\frac{\partial }{\partial y}\left( \cos \frac{m\pi y}{a} \right)$ (75)
${{\tau }_{xy}}(y,z)=\frac{\beta G{{2}^{5}}{{a}^{2}}b}{{{\pi }^{3}}}\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\frac{{{(1)}^{\frac{m+n}{2}}}}{m({{m}^{2}}{{b}^{2}}+{{n}^{2}}{{a}^{2}})}}}\cos \frac{m\pi y}{a}\sin \frac{n\pi z}{b}$ (76)
m = 1, 3, 5, 7, 9, …; n = 1, 3, 5, 7, 9, …
${{\tau }_{xz}}(y,z)=G\beta \frac{{{2}^{5}}a{{b}^{2}}}{{{\pi }^{3}}}\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\frac{{{(1)}^{\frac{m+n}{2}}}}{n({{m}^{2}}{{b}^{2}}+{{n}^{2}}{{a}^{2}})}}}\sin \frac{m\pi y}{a}\cos \frac{n\pi z}{b}$ (77)
m = 1, 3, 5, 7, 9, …; n = 1, 3, 5, 7, 9, …
The maximum shear stress τ_{max} is found as:
${{\tau }_{\max }}=\frac{G\beta {{2}^{5}}{{a}^{2}}}{{{\pi }^{3}}b}\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\left\{ \frac{{{(1)}^{\frac{m1}{2}}}}{m\left( {{m}^{2}}+{{n}^{2}}\frac{{{a}^{2}}}{{{b}^{2}}} \right)} \right\}}}$ (78)
${{\tau }_{\max }}=G\beta a{{b}^{2}}{{F}_{2}}\left( \frac{a}{b} \right)$ (79)
where τ
${{F}_{2}}\left( \frac{a}{b} \right)={{F}_{1}}\left( \frac{a}{b} \right)\frac{{{\pi }^{3}}}{{{2}^{5}}{{\left( \frac{a}{b} \right)}^{2}}}\left\{ \sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\frac{{{(1)}^{\frac{m1}{2}}}}{m\left( {{m}^{2}}+\frac{{{n}^{2}}{{a}^{2}}}{{{b}^{2}}} \right)}}} \right\}$ (80)
The nondimensional torsional parameters ${{F}_{1}}\left( \frac{a}{b} \right)$ and ${{F}_{2}}\left( \frac{a}{b} \right)$ for the Saint Venant torsion of prismatic bars with rectangular crosssections are tabulated as Tables 1 and 2 for various values of the ratio a/b for the present study and ${{\bar{F}}_{1}},{{\bar{F}}_{2}}$ are for results from Jan Francu et al. [3].
Table 1. Variation of torsional parameter F_{1} with a/b
$r=a\text{/}b$ 
${{F}_{1}}(a\text{/}b)$ 
${{\bar{F}}_{1}}(a\text{/}b)$ [3] 
1 
0.1406 
0.141 
1.2 
0.1661 

1.5 
0.1958 
0.196 
2 
0.2287 
0.229 
2.5 
0.2494 

3 
0.2633 
0.263 
4 
0.2808 
0.281 
5 
0.2913 
0.291 
6 
0.298 
0.298 
8 
0.307 
0.307 
10 
0.3123 
0.312 
∞ 
1/3 
1/3 
$r=a\text{/}b$ 
${{F}_{2}}(a\text{/}b)$ 
${{\bar{F}}_{2}}(a\text{/}b)$ [3] 
1 
0.208 
0.208 
1.5 
0.231 
0.231 
2 
0.246 
0.246 
3 
0.267 
0.267 
4 
0.282 
0.282 
5 
0.292 
0.292 
6 
0.299 
0.299 
8 
0.307 
0.307 
10 
0.313 
0.313 
∞ 
1/3 
1/3 
$\frac{\partial \varphi }{\partial y}z=\frac{\partial \phi }{\partial z}$ (81)
and
$\frac{\partial \varphi }{\partial z}+y=\frac{\partial \phi }{\partial y}$ (82)
Thus,
$\frac{\partial \varphi }{\partial y}=\frac{\partial \phi }{\partial z}+z$ (83)
or,
$\frac{\partial \varphi }{\partial z}=\frac{\partial \phi }{\partial y}y$ (84)
By integration of Equation (83) we obtain Equation (85) as φ(y, z)
$\varphi (y,z)=\frac{{{2}^{5}}{{a}^{3}}b}{{{\pi }^{4}}}\sum\limits_{m}^{\infty }{\sum\limits_{n}^{\infty }{\frac{{{(1)}^{\frac{m+n}{2}}}\sin \frac{m\pi y}{a}\sin \frac{n\pi y}{b}}{{{m}^{2}}({{m}^{2}}{{b}^{2}}+{{n}^{2}}{{a}^{2}})}}}+yz$ (85)
4.3 Numerical problem
A numerical problem to illustrate the validity of the results obtained in this study considers the calculation of torsional constant J, given by the expressions in Eqns. (72) and (73) where Eq. (73) is presented as Table 1 in terms of a/b. The torsional contant is important in torsion problems since it determines the torsional rigidity D_{t} as follows:
${{D}_{t}}=GJ$ (86)
We compare our results with results from Jan Francu et al. [3] who presented a Navier series solution of the torsion problem leading to results that are identical with the results from the present study which employed the Galerkin method.
Numerical solutions are presented for various values of a and b as follows and compared with results from Jan Francu et al. [3].
Table 3. Results for torsional stiffness for various crosssections, and comparison with results from Jan Francu et al. [3]
Crosssection 
a/b 
F_{1} 
J=ab^{3}F_{1} Present study 
F_{1} 
$J=a{{b}^{3}}{{\bar{F}}_{1}}$ Jan Francu et al [3] 

a (cm) 
b (cm) 


(cm^{4}) 

(cm^{4}) 
2 
2 
1 
0.1406 
2.2496 
0.141 
2.256 
4 
2 
2 
0.2287 
7.3184 
0.229 
7.328 
6 
2 
3 
0.2633 
12.6384 
0.263 
12.624 
8 
2 
4 
0.2808 
17.9712 
0.281 
17.984 
10 
2 
5 
0.2913 
23.304 
0.292 
23.28 
12 
2 
6 
0.298 
28.604 
0.298 
28.604 
16 
2 
8 
0.307 
39.296 
0.307 
39.296 
20 
2 
10 
0.3123 
49.968 
0.312 
49.92 
∞ 
2 
∞ 
1/3 
∞ 
1/3 
∞ 
This work has successfully presented the SaintVenant torsion problem of prismatic bars with rectangular crosssection as a boundary value problem (BVP) of the theory of elasticity using Prandtl’s stress function $\phi$(y, z). The resulting BVP was observed to be a Poisson type partial differential equation in terms of the Prandtl’s stress functions. The basis (shape) functions that satisfies the boundary conditions given in terms of trigonometric (cosine) functions; and the assumed (trial) Prandtl stress function used was given as Eq. (45) – a double trigonometric cosine series of infinite terms.
The Galerkin variational formulation of the SaintVenant torsion equation was obtained as Eq. (45). The unknown parameters of the Galerkin formulation was obtained by solving the Galerkin variational statement of the Poisson equation as Eq. (54). The Prandtl stress function was thus completely determined as Eq. (55), which was found to be a rapidly convergent double trigonometric cosine series with infinite terms.
The moment of the crosssection was obtained in terms of the ratio of the crosssectional dimensions (a/b) as Eqns. (71), and (72) where Eq. (72) is expressed in terms of the nondimensional torsion parameter, F_{1}(a/b) F_{1}(a/b) is found to depend on the ratio (a/b) as Eq. (73). Values of F_{1}(a/b) for various values of a/b were calculated and shown in Table 1.
The nonvanishing shear stress fields were found as Eqns. (76) and (77). The maximum shear stress was obtained as Eq. (78) and presented in terms of the dimensionless torsion parameter F_{2}(a/b) as Eq. (79). The dimensionless torsion parameter F_{2}(a/b) was calculated for various values of (a/b) and presented as Table 2.
The numerical results obtained for the torsional constant J for various values of the crosssectional dimensions were identical with the results obtained by Jan Francu et al. [3] who used Navier series method.
The conclusions of this study are as follows:
x, y, z Cartesian coordinates in three dimensions
$\left. \begin{align} & u(x,y,z) \\ & v(x,y,z) \\ & w(x,y,z) \\\end{align} \right\}$ Displacement field components in the x,
y, and z Cartesian coordinate directions
θ’=β twist rate
φ(y, z) unknown function related to deflection and used to define the deflection
εxx, εyy, εzz normal strains
γxy, γxz, γyz shear strains
λ Lamé’s content
G shear modulus or modulus of rigidity
Dt torsional rigidity
εv volumetric strain
∂ij Kronecker’s delta
τij stress using indicial notation
εij strain using indicial notation
σxx, σyy, σzz normal stresses
τxy, τxz, τyz shear stresses
$\phi$(y, z) Prandtl’s stress function
R2 crosssection of the bar
M torque, torsional moment
G boundary of the crosssection
J moment of the crosssection, St Venant torsional constant
m, n, m', n' integers
a,b inplane dimensions (length and width)
Cmn unknown parameter of the Prandtl’s stress function
amn cosine series parameter
r aspect ratio
$\left. \begin{align} & {{F}_{1}}\left( a\text{/}b \right) \\ & {{F}_{2}}\left( a\text{/}b \right) \\\end{align} \right\}$ dimensionless torsion parameters
$\left. \begin{align} & {{{\bar{F}}}_{1}}\left( a\text{/}b \right) \\ & {{{\bar{F}}}_{2}}\left( a\text{/}b \right) \\\end{align} \right\}$ dimensionless torsion parameters obtained by Jan Francu et al. [3] using Navier series method
2D two dimensional
3D three dimensional
BEM boundary element method
BVP boundary value problem
PDE partial differential equation
FEM finite element method
FDM finite difference method
MATHEMATICAL SYMBOLS
$\sum$ summation
$\sum\sum$ double summation
$\int$ integration (integral)
$\int\int$ double integration (double integral)
$\frac{\partial }{\partial x}$ partial derivative with respect to x
$\frac{\partial }{\partial y}$ partial derivative with respect to y
$\frac{{{\partial }^{2}}}{\partial x\partial y}$ mixed partial derivative
△ =▽^{2} Laplacian
[1] Abdelkader, K., Toufik, Z., Mohamed, B.J. (2015). Torsional stress in noncircular crosssections by the finite element method. Advances in Mechanical Engineering, 7(5): 120. https://doi.org/10.1177/1687814015581979
[2] Torsion Introduction. https//www.public.iastate.edu/~_m.424/Torsion%20intro%20Plus. pdf.
[3] Francu, J., Novackova, P., Janicek, P. (2012). Torsion of a Non – circular bar. Engineering Mechanics, 19(1): 4560.
[4] Barber, J.R. (2002). Elasticity. Springer, Netherlands. http://doi.org/10.1007/0306483955
[5] Sadd, M.H. (2014). Elasticity: Theory, Applications and Numerics. 3rd Edition. Academic Press, USA.
[6] Srinath, L.S. (2009). Advanced Mechanics of Solids. Third Edition. Tata McGraw Hill Publishing Company Limited, New Delhi.
[7] Shalpari, S., Hematiyan, M.R. (2013). Closed form solutions for torsion analysis of structural beams considering webflange junction fillets. Journal of Theoretical and Applied Mechanics, Warsaw, 51(2): 393407.
[8] Ecsedi, I., Baksa, A. (2010). Prandtl’s formulation for the SaintVenant’s torsion of homogeneous piezoelectric beams. International Journal of Solids and Structures, 47(22): 30763083. https://doi.org/10.1016/j.ijsolstr.2010.07.007
[9] Epele, L.N., Fanchiotti, H., Garcia Canal, C.A. (2012). General solution of laplace and poisson equations in a multiply connected circular domain: Applications to torsion. SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), 72(3): 919934. https://doi.org/10.1137/110852231
[10] Abbasi Nasser, M. (2006). Review of FEM solution for the torsion problem of a rectangular crosssection. https/12000.org/my_courses/UCI_COURSES/CREDIT …/index.htm. Nov. 27 2006. Accessed on 10 Aug. 2018.
[11] Selvadurai, A.P.S. (2000). Partial Differential Equation in Mechanics 2. The Biharmonic Equation, Poisson’s Equation. Springer, New York.
[12] Liu, C.S. (2006). New methods for elastic torsion of bar with arbitrary shape of crosssection. Proceedings of Symposium on Advances of Mechanics in Honour of President Robert R. Hwang, May 12, 2006. Keelung, Taiwan, pp. 17.
[13] Timoshenko, S.P., Goodier, J.M. (1969). Theory of Elasticity. 3rd Edition. McGraw Hill, New York.
[14] Little, R.W. (1973). Elasticity. PrenticeHill, New Jersey.
[15] Ockendon, J.R., Howison, S.D., Lacey, A.A., Movchan, A.B. (2003). Applied Partial Differential Equations Revised Edition. Oxford University Press, Oxford, Great Britain.
[16] Eslami, M.R. (2014). Finite elements methods in mechanics solid mechanics and its applications. Springer International Publishing Switzerland. https://doi.org/10.1007/9783319080376
[17] Teimoni, H., Faal, R.T., Das, R. (2016). SaintVenant torsion analysis of bars with rectangular crosssection and effective coating layers. Applied Mathematics and Mechanics, 37(2): 237253. https://doi.org/10.1007/s1048301620288
[18] Jog, C.S., Mokashi, I.S. (2014). A finite element method for the SaintVenant torsion and bending problems for prismatic beams. Computers and Structures, Pergamon Press, New York, USA, 135: 6272. https://doi.org/10.1016/j.compstruc.2014.01.010
[19] Karihaloo, B.L., Xiao, Q. (2016). St Venant torsion and bending of prismatic composite shafts. Proceedings Indian National Science Academy, 82(2): 183200.
[20] Ecsedi, I. (2009). Some analytical solutions for SaintVenant torsion of nonhomogeneous cylindrical bars. European Journal of Mechanics –A/Solids. Elsevier 2009, 28(5): 985. https://doi.org/10.1016/j.euromechsol.2009.03.010
[21] Romano, G., Barretta, A., Barretta, R. (2012). On torsion and shear of SaintVenant beams. European Journal of Mechanics A/Solids, 35(2012): 4760. https://doi.org/10.1016/j.euromechsol.2012.01.007
[22] Roohi, R., Heydari, M.H., Aslami, M., Mahmoudi, M.R. (2018). A comprehensive numerical study of spacetime fractional bioheat equation using fractional – order legendre functions. The European Physical Journal Plus, 133(10): 412. https://doi.org/10.1140/epjp/i201812204x
[23] Heydari, M.H., Mahmoudi, M.R., Shakibia, A., Avazzadeh, Z. (2018). Chebyshev cardinal wavelets and their application in solving nonlinear stokchastic differential equations with fractional Brownian motion. Communications in Nonlinear Science and Numerical Simulation, Elsevier, 64: 98121. https://doi.org/10.1016/j.cnsns.2018.04.018