2.1. Shell Subjected to Concentrated Force at Joining Interface and Functional Boundary Conditions
Kirchhoff–Love shell theory is utilized to establish the deformation model of single curvature skin, regarded as a thin cylindrical shell. The curvilinear coordinate system of the undeformed shell is denoted as
Oαβγ. Curve edges of the panel are located on fixturing boards, which induce the position variations on boundary conditions, as shown in
Figure 1.
The equilibrium equations of cylindrical shell [
11] along axes of local coordinate system
Oτ1τ2τ3 are established by
where
,
. Moreover, the third terms of second equation in Equation (1) are small enough quantities to be eliminated for thin shell [
19]. The stress components are expressed in terms of the middle surface stain components, as follows:
where
is a unit matrix, elements of matrix [
QI] are
,
,
, other elements of [
QI] are zeroes, strains are
,
and
, changes in curvature and twist are
,
and
. The stress resultants and stress couples of a thin cylindrical shell are given by:
where
.
Substituting Equation (3) into Equation (1), governing equations of the deformation of a thin cylindrical shell can be derived as
Differential operator matrix of the above equation is denoted by:
Since the symmetric matrix
is a linear differential operator matrix, Equation (5) can be transformed into the combination of several differential equations of potential functions. The introduced potential functions
have the following relations with displacements:
where
is the cofactor of determinant
. Substituting Equation (6) into Equation (4) yields:
The Galerkin approach is employed to derive solutions of the governing equations. According to the expressions satisfying the simply supported boundary conditions prescribed on the two straight edges of shell in [
20], displacement and rotation components are assumed as:
where
,
. The displacements and stress acting on the straight edges of shell,
and
, have the following relationships:
which are also the boundary conditions on straight edges of panel.
Therefore, the variables in potential functions and the loads are decoupled by expanding the functions into the form of Fourier series as Equation (8):
and
where
Substituting Equations (8) and (11) into Equation (4), the displacements and loads with
can be rewritten as:
where
,
is the coefficient to be determined. When
, substituting the expanded Equations (10)–(12) into Equation (7) and eliminating the trigonometric terms, the following equations can be obtained:
where constant
. The solution
of Equation (14) must be derived to calculate
, which is the functions in the displacement Equation (6), i.e., the solutions of shell deformation equation.
The complete solution
is the sum of the homogeneous solution and the particular solution. Firstly, the solutions of the homogeneous equation corresponding to Equation (14) are calculated by solving the characteristic equation:
All roots of Equation (15) are obtained by:
where real and imaginary parts
of complex roots
are listed as Equation (A1) in
Appendix A. Solutions of the homogeneous equation corresponding to Equation (14) can be expressed by:
where elements of
are
,
,
,
and other elements of
are zeroes,
is a set of real coefficients in displacement expressions to be determined.
Secondly, the particular solutions of Equation (14) are derived by using the Wronskian determinant. The Wronskian determinant of the functions
, a fundamental set of solutions to the associated homogeneous equation to Equation (14), is defined by:
The determinant is denoted by replacing the j-th column of the Wronskian with the column (0,0,…,0,1)T.
Particular solutions of Equation (14) can be derived from the following expression:
The load functions of concentrated force and uniform load encountered in the practical joining process of aircraft panel assembly are introduced to derive the explicit expressions of Eqaution (19). When the concentrated force
f is imposed on the joining interface of shell at point
, the associated distributed load
q introduced into the deformation equation is expressed as the Dirac delta function
. Using Equation (12), the following expression can be derived:
Then, the particular solution of Equation (14) with concentrated force applied on shell when
is transformed as:
When the uniform load is imposed on the shell, the particular solution of Equation (14) with
can be easily obtained by:
where
qim is a constant or linear function of
α.
Since the distributed load is a combination of concentrated force and uniform load, the complete solution of Equation (14) with should be the superposition of Equations (17), (21) and (22) as . Subsequently, substituting into Equations (10) and (6), the displacements expressions with coefficients can be obtained as Equation (A2).
Moreover, the presented displacement functions should satisfy the boundary conditions prescribed on shell edges. The initial variations on the skin curve edges are regarded as the non-zero kinematic boundary conditions imposed on edges
, as follows:
where
is the rotation around the axis
β. Expanding
,
,
and
into Fourier series as the form of displacements expressions in Equations (A3) and (A4), a set of linear algebraic equations specific to coefficients are generated. Then, coefficients
can be solved. Deformation of arbitrary point (
α,β,γ) on the shell is determined by
,
and
. Eventually, the semi-analytical expression of shell deformation with concentrated force, uniform load and arbitrary functional boundary conditions applied are achieved.
2.2. Fundamental Equation of Spatial Stiffener with Arbitrary Boundaries
Deformation of the stiffener is analyzed by the modified EBB model. Modified EBB approach introduces the transformation relations of displacements and rotations between arbitrary points on the same cross-section of the stringer entity. Cartesian coordinate system
Obxyz of beam is shown in
Figure 1.
,
and
are the rotations around three coordinate axes, respectively.
With the assumptions imposed in EBB theory, the stress resultants can be described as [
21,
22]:
where elements of matrix
[B] are
,
,
,
,
, other elements of
[B] are zeroes,
is the axial strain,
,
and
denote components of the curvature vector of axis
x.
The governing differential equations of EBB are derived based on the principle of minimum total potential energy, which implies that the total potential energy of the system must be stationary and the stationary value is always a minimum. The total potential energy of the system is defined by:
where
is a functional,
is total strain energy,
is the work done by external forces. To achieve the stationary total potential energy, the variation of Equation (25) should satisfy the following relations:
The strain energy of a beam is defined in terms of six components: one axial force, two bending moments, two shear forces and one torsional moment [
23]. Since the shear deformation is neglected in the EBB theory, the total strain energy is expressed by the sum of the strain energy components due to tension or compression, bending and torsion force. The variation of the strain energy is described as:
The virtual work of external forces is given by:
where the notation
means the minus of the function values with selected independent variable values
x2 and
x1. Using the Green’s theorem and method of integration by parts, Equation (26) is rewritten in the following form:
Simplifying Equation (29), the governing equations and the static boundary conditions are derived as:
The governing equation of EBB is expressed by substituting stress resultants–strain relation Equation (24) into Equation (30), as follows:
The position error and clamping variations on the edges of stiffeners are regarded as the following kinematic boundary conditions:
For each joining point on the cross section of the beam
, eight supplementary boundary conditions should be added into the EBB deformation equation as:
Thus, displacements ub, vb and wb, the piecewise functions with unknown coefficient Cj, could be expressed by the polynomial functions of independent variable x since the derived governing equation is a set of the linear partial differential equation.
In particular, the modified EBB method introduces the transformation relations between the displacements of the centroid and arbitrary joining point on the cross-section of the stringer, the following expression [
24] is presented by:
where the coordinate of arbitrary joining point on the cross-section
with respect to Cartesian coordinate system
Obxyz is
, rotation matrix
.
Consequently, the displacements and rotations of arbitrary points on the deformed stringer with positioning and clamping variations and the mechanical joining interaction are derived as Equations (32) and (35).