Nomenclature

b,	width of the cross-section;	F,	concentrated force;
h,	height of the cross-section;	q,	distributed load;
κ0,	initial curvature of the beam;	E,	elastic modulus of the material;
κ1,	curvature increment;	n,	material constant;
s0,	initial arc length;	L,	reference length;
s0*,	arc length after deformation;	L0,	initial length of the beam;
u,	horizontal displacement;	σ,	stress;
w,	vertical displacement;	ε,	strain;
φ0,	initial slope;	N,	resultant axial force;
φ,	slope after deformation;	M,	resultant bending moment;
φ1,	slope increment;	H,	resultant force component along the x-axis.
V,	resultant force component along the y-axis;

1 Introduction

As one of the basic structural elements, beams are widely used in practical engineering. Flexible beams are often associated with large deflection, and the pioneer Euler has established the governing equations for the flexible beams involving large deflection. This kind of problems is named as "elastica", and has been investigated for over 200 years. Bisshopp and Drucker^[1] obtained the closed-form solution in terms of elliptical integrals for the large deflection of cantilever beams loaded by a vertical concentrated force at the free end. Holden^[2] investigated the large deflection of linearly elastic cantilever beams subjected to a distributed load with the Runge-Kutta method. Li^[3] put forward a new integral method for the large deflection of cantilever beams, and solved the elastica problem for the cantilever beams under complex loads through simple numerical techniques. However, all these investigations are restricted to linearly elastic materials, i.e., only geometric nonlinearity is considered.

In fact, many commonly-used materials exhibit nonlinear stress-strain relationships^[4-5]. Thus, in some practical cases, both material nonlinearity and geometric nonlinearity are involved. Since the late 1960's, many researchers have made their efforts to reveal the mechanism of the large deflection of flexible beams with the consideration of both material nonlinearity and geometric nonlinearity. Lewis and Monasa^[6] and Prathap and Varadan^[7] obtained the solution for the large deflection of cantilever beams made of the Ludwick type and Ramberg-Osgood type materials subjected to an end moment and a vertical tip load, respectively. Lee^[8] derived the governing equations of the large deflection of cantilever beams made of the Ludwick type material by using the shearing force formulation instead of the bending moment formulation, and the obtained numerical results were consistent with those given by Lewis and Monasa^[9]. Baykara et al.^[10] investigated the large deflection of cantilever beams made of nonlinear bi-modulus materials to reflect the mechanical property difference when being pulled and being compressed. Borboni and Santis^[11] solved the large deflection of cantilever beams loaded by not only bending couple but also horizontal and vertical forces at the free end. Brojan et al ^[4] obtained the numerical solution for the large deflection of non-prismatic beams made of generalized Ludwick type materials. Elastica problems have been extended to the non-homogenous materials such as nonlinearly elastic functionally graded composite beams in recent years^[12].

It should be stressed that the above mentioned investigations are mostly restricted to straight beams. In fact, curved beams are also extensively employed in engineering and of great interest. For curved beams, Lau^[13] gained the closed-form solutions for the large deflection of a cantilever beam, where the initial curvature radius of the beam was constant. Based on the assumption that the axis of the beam was extensible, Li et al.^[14] established a mechanical model for the large deflection of extensible curved beams. However, these investigations only contain geometric nonlinearity. When both geometrical and mechanical nonlinearities are incorporated simultaneously, the elastica problem of curved beams is much more complicated compared with their straight counterparts. It seems that there is still no research to deal with the elastica problems of curved beams involving both mechanical and geometric nonlinearities in the open literature.

The Ludwick relation is a generalization of Hooke's law. It is a well-known empirical relation to represent the experimental nonlinear stress-strain relation of the metals, such as annealed copper and N.P.8 aluminum alloy^[15-16]. Due to the fact that the formula of the Ludwick relation is simple and can express the known properties of a wide range of polycrystalline materials, a new theoretical approach is proposed in this paper to describe the large deflection of curved beams made of the Ludwick type material, and the given results not only reveal the nonlinear coupling characteristics for such problems, but also help to make better use of the Ludwick type material.

2 Theoretical models for the large deflection of curved beams made of the Ludwick type material

Consider a curved beam of small curvature, which has a uniform rectangular cross-section with the height h and the width b. An arbitrary point C(x, y) on the central axis of the undeformed beam moves to the point C': (X, Y)= (x+u, y+w) after deformation (see Fig. 1). The horizontal and vertical displacements are u and w, respectively. The arc lengths of the points C and C', which are measured from the left end of the beam, are s₀ and s₀^*, respectively, and

(1)

where

The normal strain ε₀ of the axis is

(2)

The elastic beam is assumed to satisfy the Euler-Bernoulli postulation. Before deformation, the length of the arbitrary longitudinal segment ds at the distance η from the central axis of the beam can be written as follows:

(3)

Similarly, the corresponding length of the segment ds after deformation becomes

(4)

where φ₀ and φ are the angles from the positive x-axis to the tangent line of the beam axis before and after deformation, respectively. φ₁=φ -φ₀ is the slope increment. Thus, the normal strain of the point at the distance η from the beam axis is

(5)

where κ₀ and κ₁ are defined by

(6)

After deformation, the curvature of the central axis becomes

(7)

Using the geometrical relationship, we have

(8)

The resultant axial force N and the bending moment M can be written as follows:

(9)

(10)

where A is the area of the cross-section. When the material is linearly elastic, the normal stress σ is proportional to ε, and σ = Eε, where E is Young's modulus of the material. Substituting this constitutive equation and Eq.(5) into Eqs.(9) and (10), we have^[14]

(11)

(12)

where C₁, C₂, and C₃ are the parameters reflecting the geometrical properties of the cross-section and the initial curvature κ₀, and are defined by

(13)

Especially, C₂ reflects the coupling between elongation and bending. Denote

From Eqs.(11) and (12), we can rewrite ε₀ and κ₁ in terms of N and M as follows:

(14)

(15)

Obviously, for linearly elastic materials, an explicit expression for κ₁ and ε₀ can be given with respect to N and M. As a consequence, the governing equations for the large deflection of curved beams can be established easily. However, for the Ludwick type material, it is rather difficult to get such an explicit expression for κ₁ and ε₀. The constitutive equation for the Ludwick type material is a nonlinear function, i.e.,

(16)

where n is the material constant. If n=1, the material is linearly elastic. For beams with small curvature, the curvature radius of the curved beam is much larger than the height of the cross-section, i.e., . In such a case, (1+ηκ₀)^-1/n can be approximately expressed as follows:

(17)

More polynomials on the right side of Eq.(17) are retained, and a higher precision can be obtained at the cost of more complicated computation. As a matter of fact, it is precise enough to retain the first two polynomials in Eq.(17). Substitute Eqs.(16) and (17) into Eqs.(9) and (10). Then, N and M can be derived.

(ⅰ) When κ₁=0,

(18)

(19)

(ⅱ) When κ₁ ≠ 0, set ε₀= lκ₁. Then,

(20)

(21)

If -h/2 ≤ l ≤ h/2,

(22)

(23)

If l>h/2,

(24)

(25)

If l < -h/2,

(26)

(27)

Set

Substitute them into Eqs.(20)-(27). Then, we have

(28)

where

(29)

As shown in Eq.(28), β is a single-valued function of α, i.e., a unique β can be determined as long as α is specified. To illustrate the single-valued characteristics, set n=2.16 and κ₀h=0.1. The relation between β and α is depicted in Fig. 2. Obviously, the curve, where n=2.16 and κ₀h=0.1, appears as a single-valued function, and the case for κ₁=0 is included in Eq.(28) when α approaches ±∞.

Therefore, once β is determined, we can get the value of l by solving Eq.(28). Substituting the value of l into Eqs.(20) and (21), we have

(30)

(31)

Unlike linearly elastic materials, the coupling between elongation and bending cannot be expressed in an explicit form similar to C₂ in Eq.(13).

Consider a micro-segment of the curved beam loaded by the distributed force q = (q_x, q_y) and the bending couple m (see Fig. 3). The equilibrium equations can be written as follows:

(32)

where H and V are the resultant force components along the x-and y-axes, respectively. Equation (32) can be described in a differential form with respect to s₀, i.e.,

(33)

To make the governing equations more concise, the following non-dimensional parameters are introduced:

(34)

where L is a reference length, which is equal to the initial length of an undeformed straight beam or the radius of a curved beam. Taking s₀^*, u, w, φ₁, H, V, and M as seven unknowns in terms of s₀, we can get the following closed-form governing equations:

(35)

where .

The governing equations in Eq.(35) are nonlinear with the consideration of both geometric nonlinearity and material nonlinearity, and some variables are coupled with each other in a complex manner. Therefore, numerical methods rather than analytical solutions are adopted to solve this kind of problems. The shooting method is a convenient numerical method to compute the initial missing values. The detail of the shooting method can be found in Ref.[17]. To solve the governing equation with the shooting method, it is necessary to specify the boundary conditions. For a cantilever loaded by a concentrated force P_F at the free end, the boundary conditions are

(36)

(37)

For a simply supported beam loaded by a concentrated forceP_F at the midpoint, the boundary conditions can be written as follows:

(38)

(39)

For a fixed-fixed beam loaded by a concentrated forceP_F at the midpoint, the boundary conditions are specified as follows:

(40)

(41)

3 Numerical examples and discussion

To inspect the validity of the theoretical model of curved beams in Section 2, we degenerate the curved beam into a straight beam by setting κ₀= 0. The cantilever beam with an initial length L₀ carries a distributed load q_y and a vertical concentrated force F at its free end (see Fig. 4). The material constant n is taken as 2.16, and h/L₀ is set to be 0.1, which are the same as those in Ref.[8]. The vertical and horizontal displacements at the free end are denoted as δ_v and δ_h, respectively.

Since the beam is straight, the initial length L₀ of the beam is taken as the reference length, i.e., L =L₀. Two examples are carried out, and the obtained results are compared with those in Ref.[8]. The first example is a beam loaded by a concentrated force F, and the second is loaded by a combined load consisting of a distributed load q_y satisfying q_yL = F, where F is a concentrated force (see Table 1 and Table 2). For the sake of comparison, the non-dimensional force L₀ⁿ⁺¹/K_n defined in Ref.[8], which is different from the non-dimensional force P_F in our study, is retained and also listed in Table 1 and Table 2. The non-dimensional force L₀ⁿ⁺¹/K_n in Ref.[8] can be expressed in terms of the variables in the present paper, as shown in Eq.(42), and the relationship between K_n and P_F can be written as follows:

(42)

(43)

As shown in Table 1 and Table 2, the predictions in our study are in good agreement with those in Ref.[8] for both concentrated force and combined load. Therefore, some numerical examples are further given for curved beams. The initial curvature of the beams is set to be constant, and the deflections of the cantilever and simply supported beams are calculated. For both of the beams, the radius angle is 60º, and the radius of the curvature of the central axis is taken as the reference length, i.e., L = R. The parameters of the beams are specified as n=2.16 and h/L=0.1. The deformed configuration of the simply supported beam and the cantilever beam are illustrated in Figs. 5 and 6 and Figs. 7 and 8, respectively, whereas Figs. 5 and 7 are shown for the beams loaded by a concentrated force F, and Figs. 6 and 8 are shown for a combined load consisting of a distributed load q_y with πRq_y/3 =F and a concentrated force F. The concentrated force is exerted on the midpoint for the simply supported beam and on the free end for the cantilever beam. From Figs. 5-8, one can easily find that, when the non-dimensional load increases, the deformation of the curved beams becomes more significant. Under the same non-dimensional load, the deflections of the cantilever beam are much greater than those of the simply supported beam.

The thickness-length ratio also has an effect on the beam displacement. For a given non-dimensional force defined in the present paper, the non-dimensional displacement changes. As shown in Fig. 9, the vertical displacement at the center of a simply supported curved beam subjected to a concentrated force at its midpoint is plotted as a function of the thickness-length ratio h/L₀. The radius angle of the beam is fixed at 60º, and the non-dimensional force P_F is set to be 0.4. It is clear that the non-dimensional vertical displacement δ_v/R decreases when h/L₀ increases. It should be noted that a constant P_F does not mean a fixed loading. The beam thickness and beam length have been incorporated in the expression of P_F in Eq.(34). A much smaller load is actually acting on the slenderer beam. Although only a simply supported beam is studied, the effects of the thickness-length ratio on the beams with other boundary conditions under various kinds of loads can be analyzed in a similar way.

4 Conclusions

The large deflection of curved beams made of the Ludwick type material is investigated, wherein the geometric nonlinearity and material nonlinearity are both incorporated. Through the piecewise-integration method, the geometric equations and governing equations are established. The numerical solutions are presented through the shooting method. Particularly, a degenerated case for the straight beam is first given, and the corresponding predictions by the present paper agree well with those in the existing literature, which indicates that the theoretical model and the numerical method in this paper are effective. For curved beams, the deformed configurations of simply supported and cantilever beams are given. One can observe that the deformation of the curved beams increases with the increase in the applied load. The present formulation and solving method can be extended to the non-conservative load case. Although the detailed forms of the governing ordinary differential equations and the boundary conditions under the conservative load and non-conservative load are different, both of them are boundary value problems of the ordinary differential equations with strong nonlinearity, and can be solved numerically with the shooting method, etc. The results also show that only beams with small initial curvature are applicable.