1 Introduction

Recently, a method of reverberation ray matrix (MRRM) has been proposed for analysis of dynamic response of 2D and 3D isotropic framed structures^[1-3]. In dynamic analysis, the arriv ing and departing wave amplitudes for joints are chosen as unknown quantities and calculated by determination on a reverberation-ray matrix reflecting the properties of wave propagation in beam members. The MRRM has been used to analyze the propagation of elastic waves in lay ered solid and liquid^[4-6]. More recently, the MRRM has also been used as an effective method for evaluation on damage detection in various beam structures and soils^[7-9]. In contrast, for static analysis of space structures composed of bar elements, displacements and rotational angles (displacement vector) at the ends of each bar member can be directly considered as unknown quantities^[10]. The corresponding formulation of reverberation matrix incorporating the effect of carry-over and distribution matrix and source vector for joints and structure for arbitrary connection and constraint conditions is given for isotropic structure^[11].

This paper aims to develop a computational algorithm for static analysis of planar framed structure made of anisotropic materials under the frame of MRRM. The expressions for stiffness matrices for anisotropic beam members are presented. The carry-over and distribution matrix and source vector are obtained using the equilibrium equations and compatibility conditions. The phase matrix and permutation matrix of the structure are obtained based on the relation between the displacements of two ends of each member under the prescribed dual coordinates. A reverberation matrix is derived analytically for united exact determination on the displacements and internal forces at both the ends of each member and arbitrary cross-sectional locations. Numerical examples are given and compared with finite element method (FEM) results toward demonstrating the accuracy of the present model. The analysis of some parameters, i.e., height span ratio and material constants (anisotropic and isotropic), is carried out to examine the accuracy of Timoshenko (T) beam theory and the error of use of classical Euler-Bernoulli (EB) beam theory. The resulting reverberation matrix can be used for exact calculation of anisotropic framed structures as well as for parameter analysis of geometrical and material properties of the framed structures.

2 Formulation of reverberation ray matrix

As shown in Fig. 1, the joints in structure members are identified by the numerals 1, 2, 3, … or by the letters J, K, L, …, and the structural members by two numerals 12, 23, … or letters of both end joints JK, KL, …. The positions of the joints are measured in a fixed Cartesian coordinate (X, Y), called global coordinate. For any member JK, two right-handed Cartesian coordinate systems (x, y)^JK originated at the joint J and (x, y)^KJ at the joint K are introduced, and they thus form dual local coordinate systems for each member. The direction of the axis, x^JK, is always opposite to that of x^KJ, and that of y^KJ is opposite to y^JK by choice. According to the well-known sign convention for displacement and internal force components in each local system, every component at the same point in one member under dual local coordinate systems has the following relations:

(1)

The corresponding displacement, rotation, internal force components in the global coordinate can be derived using the transform relation as follows:

(2)

where θ_nN is the angle between the n- and N-axes in the local and global coordinates, respec tively.

2.1 Equilibrium equation and compatibility relation for joint displacements

The internal force F^JK= (F_F^JK F_M^JK)^T = (F_x^JK(0) F_y^JK(0) F_M^JK(0))^T at the end of member JK can be written as

(3)

where is the resulting internal force at the end J from the external loading acting in the member. a^JK = (u^JK(0) v^JK(0) ϕ^JK(0))^T and d^JK=(u^JK(l) v^JK(l) ϕ^JK(l))^T represent the near- and far-end displacement vectors for the joint J, respectively. A^JKand D^JKare two 3 × 3 square matrices characterizing the relation between the internal forces and displacements, which will be determined in Section 3.

For a completely hinged joint J, F_M^JK ≡ 0, the equilibrium equation for the joint in the global coordinate is written as

(4)

where , and f_F^J = (f_x^J f_y^J)^T is the load vector at the joint J. Substituting (3) into (4) yields

(5)

where

In addition, the translational displacements at the joint J with which members (J1, J2, …, Jm^J) are connected are the same, and the corresponding bending moments are all zeros, i.e,

(6)

where u_F^J = (u_x^J u_y^J)^T is the translational displacement vector for the joint J in the global coordinate. Using (6), assembling all equations for members connected at the joint J, we can obtain a compatibility relation in the form of matrix for the hinged joint J as follows:

(7)

in which

For other type of joints (e.g., complete rigid joints), analysis of compatibility equation is de scribed in Ref. [11].

2.2 Carry-over and distribution matrix and source vector for joints and structure

Combination of (5) and (7) leads to the relation between near- and far-end displacement vectors as follows:

(8)

where S^Jand s^Jare the carry-over and distribution matrix and the source vector for joints, respectively.

With d^J = 0 and (8), combination of (5) and (7) gives the result for source vector s^J

(9)

Similarly, with F₁^J= 0 and , the result for carry-over and distribution matrix S^Jis derived as follows:

(10)

Continuing the processes of stacking the joint-wave amplitudes d^Jand a^Jin (8) for all joints of the structure (J = 1, 2, …, n), we obtain the global departing wave vector d and global arriving wave vector aof entire structure. Similarly, all joint carry-over and distribution matrix and source vectors are assembled to form the two corresponding global matrices, S and s. The relation between a and d is thus written from (8) by

(11)

2.3 Reverberation matrix for structure

For the member JK, by using (1), we derive the following relation for the displacements in dual local coordinate systems:

(12)

where p= diag(-1, -1, 1) is called phase matrix for the member JK. For the entire structure, there is

(13)

in which P= kron(eye(2∗m), p) is a global phase matrix. Here, the "kron" means the Kronecker product of eye(2 ∗ m) and p. Two vectors and acontain the same elements but in different permutations. The sequential orders of the same element in two column matrices may be related by , where U is a permutation matrix consisting of a single non-vanishing element of unity in each and every row and column. Then, (11) becomes

or

(14)

in which R= SPU is called the reverberation matrix for the structure. Assembling all F^JK for members to a global matrix F for the structure, the internal forces are determined by

(15)

3 Expressions for stiffness matrices A^JKand D^JKfor member JK

The displacement components incorporating the effect of transverse shear deformation of a beam are assumed to take the form of

(16)

The geometrical equations are

(17)

The constitutive equation is

(18)

The internal resultant forces thus can be expressed by

(19)

where A is the area of cross section, I_z=∬_Ay²dA is the inertial moment of cross section with respect to the z axis. is the reduced compliance matrix, and = (a)^-1 is the stiffness matrix. κ is the modified coefficient of transverse shear deformation.

The equilibrium equations for the internal forces are written by

(20)

Substituting (19) into (20) yields

(21)

which results in

(22)

where 2b₂ = c₁ and 3b₃ = c₂. The near- and far-end displacements for a member are given by (u(0) v(0) ϕ(0))^T and (u(1) v(1) ϕ(1))^T with

(23)

All coefficients in (22) can be expressed by displacements at the ends of a beam. Then, using (19), one gets the relation between the internal forces and displacements for the member JK as

(24)

where

(25)

(26)

By using the above resulting stiffness matrices for arbitrary member, the displacement and internal forces at ends of every member can be calculated by (14) and (15), while the displace ment and internal forces at arbitrary sectional location of a member are determined by (22) and (19). Especially, if the effect of transverse shear is neglected by setting Aκc₆₆ →∞ in (25) and (26), the solution will be simplified to that useful for analysis of the anisotropic framed structures composed of classical EB beam members.

4 Numerical examples and comparison

As shown in Fig. 2, a planar framed structure is chosen as the computational model. The horizontal member AB is hinged with the vertical member BC, where joints A and C are fixed to the foundation. The uniform distributed loading is along the negative direction of the axis Y . All members have the same length of l=350 m and rectangular cross section with width of b=1 m and height of h. In computation, two height-span ratios, i.e., h/l = 1/7 and h/l = 1/4 are considered. The modified coefficient is κ = π²/12. The materials of members are graphite epoxy, and material constants are E₁=156GPa, E₂=13GPa, G₁₂=7 GPa, and ν₁₂ = 0.23.

To verify the present model, an FEM analysis is done using the ANSYS code. The BEAM3 element in the code is used to construct a numerical model with total 20 elements and 22 nodes by discretizing every member to 10 elements. Based on the present MRRM using the T beam theory and classical EB beam theory, the solutions for the displacements and internal forces for the case of height-span ratio h/l = 1/7 are compared with corresponding FEM results, as listed in Tables 1 and 2, respectively. The units for the displacement and rotational angle are "m" and "rad", respectively, while those for the internal forces and moments are "m" and "Nm", respectively.

The results show that the MRRM using the T beam theory agrees well with the FEM simulation. The use of EB beam theory leads to larger errors, especially for rotational angle and bending moment. The calculation indicates that a larger height-span ratio can cause much error when the EB theory is used. To evaluate the effect of material property on deformation, results for displacements of the same geometry of the structure from anisotropic and isotropic materials are shown in Table 3. It is observed that for the small height-span ratio, compared with T theory, the classical EB theory is applicable when the material is isotropic. However, when the material is anisotropic, a large error is evident. The anisotropy of material will increase the transverse shear deformation and thus results in larger translational and rotational deformations.

The displacements and internal forces for the anisotropic structure for different values of height-span ratio are listed in Tables 4 and 5. It is seen that a larger ratio will cause a larger error between the T and EB theories for displacements and internal forces as expected, especially for rotational and bending deformations. The classical EB theory cannot be adopted for the case of anisotropic materials, especially for a lager value of height-span ratio.

5 Conclusions

Following the procedure of the MRRM for dynamic analysis, a corresponding reverbera tion matrix for planar framed structures composed of anisotropic T beam members containing completely hinged joints is developed for static analysis of such structures. Translational and rotational displacements at the ends of each beam member are directly considered as unknown quantities. The expressions for stiffness matrices for anisotropic beam members are deduced. The reverberation matrix is derived analytically for exact and united determination on the displacements and internal forces at both the ends of each member and arbitrary cross sec tional locations in the structure. Numerical examples are given and compared with the FEM results. The resulting reverberation matrix can be used for exact calculation of anisotropic framed structures as well as for parameter analysis of geometrical and material properties of the anisotropic framed structures.