Abstract
Analytical solutions have been developed for nonlinear boundary problems. In this paper, the shifting function method is applied to develop the static deflection of in-plane curved Timoshenko beams with nonlinear boundary conditions. Three coupled governing differential equations are derived via the Hamilton’s principle. The mathematical modeling of the curved beam system can be decomposed into a complete sixth-order ordinary differential characteristic equation and the associated boundary conditions. It is shown that the proposed method is valid and performs well for problems with strong nonlinearity.
1. Introduction
Curved beam structures are widely used in engineering fields, such as mechanical, civil, and aerospace engineering. Reviews of research on such structures have been conducted by Henrych [1], Markus and Nanasi [2], Chidamparam and Leissa [3], and Auciello and De Rosa [4]. The in-plane and out-of-plane problems of plane curved beams have been studied. In most general problems, they are coupled. However, if the cross section of the curved beam is doubly symmetric and the plane is a principal plane of the cross section, then the in-plane and out-of-plane problems are uncoupled.
Many investigators have studied linear problems about the static and free vibration behaviors of curved beams. Washizu [5] obtained the equations for the coupled motions of a curved and pretwisted bar. Rao [6] used Hamilton’s principle to derive the governing equations of motion in consideration of the effects of rotatory inertia and shearing deformation. Fettahlioglu and Mayers [7] studied the static deflection of a ring. Bickford and Maganty [8] used a formulation similar to that of Rao and developed equations of motion for out-of-plane vibrations of symmetrical cross-section thick rings, accounting for curvature variation through the thickness. Their frequency predictions were validated with the experimental data of Kuhl [9]. Based on the moment-displacement relationships presented by Rao, Silva and Urgueira [10] derived the dynamic stiffness matrices for the out-of-plane vibration of curved beams using dynamic equilibrium equations. Lee and Chao [11] developed the exact out-of-plane vibration solutions of curved nonuniform beams. In the book by Cook and Young [12], the exact static analysis of extensional circular curved Timoshenko beams with some special conditions was revealed. Based on the generalized Green function, Lin [13] developed the exact solution of extensible curved Timoshenko beams. Lee and Wu [14] studied the exact in-plane vibration solutions of extensible curved nonuniform Timoshenko beams.
For the beams with time dependent boundary conditions, Lee and Lin [15] generalized the solution method of Mindlin and Goodman [16] and developed the shifting function method to study the problems with general time dependent elastic boundary conditions. Recently, Lee et al. [17, 18] extend the shifting function method to study the exact large deflection of a Bernoulli-Euler beam and Timoshenko beam with nonlinear boundary conditions. From the existing literatures, it shows that exact solutions for curved beam problems with nonlinear boundary conditions are not available.
In the previous studies [17, 18], the governing differential equations are fourth-order differential equations. In the present study, one extends the previous studies and the shifting function method [15] to study the exact large static deflection of in-plane curved Timoshenko beams with nonlinear boundaries. The beam with doubly symmetric cross section is considered. The three coupled governing differential equations for the in-plane curved uniform beams of constant radius are derived via Hamilton’s principle. These three coupled governing differential equations are decoupled and reduced into a sixth-order differential equation with nonlinear boundary conditions. Consequently, the shifting function method is extended and applied to develop the exact solution of the system. It can be shown that the proposed method is valid for problems with strong nonlinearity.
2. Mathematical Modeling of the Curved Beam System
Consider the static response of an extensional curved uniform Timoshenko beam resting on an elastic foundation with nonlinear boundary conditions, subjected to loads , , and , as shown in Figure 1. If the thickness of the curved beam is small in comparison with the radius of the curved beam without considering the warping effects, the displacement fields of the curved beam in cylindrical coordinates arewhere , , and denote the displacement of the curved beam in the , , and directions, respectively, and is the arc length along the neutral axis. is the constant radius of the curved beam and , and are the neutral axis displacements of the curved beam in the and directions, respectively, and is the angle of rotation due to bending in the direction. is measured inward from the neutral axis in the direction.
Substituting (1) into the strain-displacement relations in cylindrical coordinates, only two nonzero strains, namely, and , are obtained:
When is small in comparison with , the two strains reduce to
The strain energy of the curved beam iswhere is the length of the neural axis. , , and denote Young’s modulus, shear modulus, and cross-section area of the curved beam, respectively, and denotes the shear correction factors of the curved beam section about the -axes. Since the cross section of the curved beam considered is doubly symmetric, the integral of the second term in the square brackets vanishes. Equation (4) becomeswhere denotes the second area moments of inertia of the curved beam section about the -axes.
Considering the linear and nonlinear spring and moment on the boundary of the curved beam end and the force and moment loads on the curved beam, the potential energy and work are, respectively,where , , , , , and are the linear spring and rotational stiffness constants in the , , and directions at the left and right ends of the curved beam, respectively. , , , , , and are the nonlinear spring and rotational stiffness constants in the , , and directions at the left and right ends of the curved beam, respectively. , , and are the force and moment loads in the , , and directions, respectively, and , , , , , and are the force and moment loads in the , , and directions at the left and right ends of the curved beam, respectively.
The general form of Hamilton’s principle is
Based on Hamilton’s principle, the governing differential equations and the associated boundary conditions for the curved uniform Timoshenko beam can be derived. The governing differential equations for the in-plane are three coupled differential equations:The associated boundary conditions are as follows:(i)At (ii)At In terms of the following nondimensional quantities,the nondimensional coupled governing characteristic differential equations areThe associated boundary conditions are as follows:(i)At (ii)At
The coupled differential equations (13)–(15) can be decoupled to get two equations, and :where the notation denotes the th-order differentiation with respect to . Consider
Substituting (22)–(24) into (15) and (16)–(21) yields the complete sixth-order ordinary differential characteristic equation and the associated boundary conditions in the direction, respectively. Consider(i)At (ii)At
3. The Shifting Function Method
In this paper, the solution for the sixth-order differential equation (25) with nonlinear boundary conditions (26)-(27) is derived. The shifting function method developed by Lee and Lin [15], given below, is used:whereand , , are the shifting functions to be specified, and is the transformed function. Substituting (28)-(29) into (25)–(27) yields the differential equation for and the associated boundary conditions:(i)At (ii)At
If the shifting functions , , in (28) are chosen to satisfy the differential equationand the following boundary conditions,(i)at (ii)at where is a Kronecker symbol, then the differential equation (30) and the associated boundary conditions (31)-(32) can be reduced to(i)At (ii)At
Once the transformed function and the shifting functions , , , , , and are determined, they are substituted into (28), yielding
Substituting (39) into (22) and (23) yields the solutions and . It can be observed that final solutions include the superposition of the linear and the nonlinear parts. The shifting function method can deal with the nonlinear parts of the boundary conditions very well.
4. Verification and Examples
The previous analysis is illustrated using the following example.
Example. Consider the deflection of a beam subjected to uniform distributed load . The curved beam is clamped at the left end and supported at the right end with linear and nonlinear springs in the direction. The corresponding coefficients arewhere is constant.
Equations (22) and (23) become
The complete sixth-order ordinary differential characteristic equation in (25) and the associated boundary conditions in (26)-(27) become(i)At (ii)At Letwhere
Here is the shifting function to be specified. is the transformed function which satisfies the differential equation (42) and the associated boundary conditions (43)-(44). ConsiderThe boundary conditions are as follows:(i)At (ii)At
It can be found that the function iswhere , , , , , and are given in Appendix.
The shifting function satisfies the following differential equation and boundary conditions:(i)At (ii)At
It can be found that the function iswhere , , , , , and are given in Appendix.
Substituting (45), (49), and (53) into (41) yields the exact solutions of , , and , respectively.
Considering uniform straight Timoshenko beams as and with (41), (45), (49), and (53), one haswhere
With both linear and nonlinear spring stiffness constants being zeros (i.e., ), (54) reduce to
For a Bernoulli-Euler beam without shear deformation (i.e., ), (54) and (55) reduce towhere
It is the exact nondimensional deflection of cantilevered Timoshenko and Bernoulli-Euler curved beams subjected to uniform nondimensional distributed load . Equations (54)–(58) are exactly the same as those given by Lee et al. [18].
Table 1 shows that the exact solutions of the shifting function method are the same as Lin’s results [13]. With increasing distributed load, the amount of deflection in the direction increases at a given location on the curved beam. Table 2 shows the neutral axis displacements in the and directions and the angle of rotation due to bending in the direction. The curved beam with linear and nonlinear boundaries has lower deflection in the direction compared to that of a beam without such boundaries.
Of note, for the curved beam subjected to uniform load in the direction, the deflections at the right end of the curved beam vary, as shown in Figures 2–4. According to (45), (49), and (53), the nonlinear spring stiffness constant is zero, and thus it is reasonable that the deflection at the right end of the curved beam is linearly related to uniform load in the direction. When the nonlinear spring stiffness constant is nonzero value, the relationship can be curvilinear. The spring at the right end of the curved beam is strong enough to reduce the deflection variation, as shown in Figure 2. Figure 3 shows the influence of curvature on the static deflection of the beam in the direction for a given set of boundary conditions. When the center angle is increased, the deflection will decrease. The curved beam itself can be regarded as a spring mechanism. Figure 4 shows that the center angle and boundary conditions significantly affect deflection in the direction. If the load on the curved beam is insufficient and the value of the deflection is less than 1, a linear spring has a greater effect than that of a nonlinear spring. On the contrary, if the value of the deflection is over 1, a nonlinear spring has a greater effect than that of a linear spring.
5. Conclusion
In this paper, the shifting function method is applied to develop the static deflection of in-plane curved Timoshenko beams with nonlinear boundary conditions. Three coupled governing differential equations are derived via Hamilton’s principle and decomposed into a complete sixth-order ordinary differential characteristic equation that depends on the neutral axis displacement in the direction. The explicit relations among the neutral axis displacement in the direction, the angle of rotation due to bending in the direction, and the neutral axis displacement in the direction are revealed. An example was used to illustrate the analysis. It can be found that the center angle and the linear and nonlinear spring stiffness constants will have significant influence on the static deflection of the beam.
Appendix
Coefficients of (49) and (53) are as follows:where
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
It is gratefully acknowledged that this research was supported by the National Science Council of Taiwan, under Grant MOST 103-2218-E-006-016.