The displacement functions can be expressed as Fourier series of the form:
$$\begin{aligned}&w(x)=\sum _{m=1}^{\infty } A_m \sin \alpha _m x, \quad 0 \leqslant x \leqslant L \end{aligned}$$
(15)
$$\begin{aligned}&u_1(x)=B_0+\sum _{m=1}^{\infty } B_m \cos \alpha _m x, \quad 0 \leqslant x \leqslant L \end{aligned}$$
(16)
$$\begin{aligned}&u_2(x)=C_0+\sum _{m=1}^{\infty } C_m \cos \alpha _m x, \quad 0 \leqslant x \leqslant L \end{aligned}$$
(17)
where
\(\alpha _m=m\pi /L\) and
\(A_m\),
\(B_m\) and
\(C_m\) are the Fourier coefficients. In general, the applied load can be given by
$$\begin{aligned} q(x)=\sum _{m=1}^{\infty } Q_m \sin \alpha _m x, \quad 0 \leqslant x \leqslant L \end{aligned}$$
(18)
where
$$\begin{aligned} Q_m=\frac{2}{L}\int _{0}^{L } q(x)\sin \alpha _m x \mathrm{d}x \end{aligned}$$
(19)
With a given external force
q(
x), these Fourier coefficients can be determined by substituting the displacement functions [Eqs. (
15), (
16) and (
17)] in the governing differential equations of Eqs. (
6), (
9) and (
11). The derivation process is presented as follows.
Based on Stokes’ transformation, the first derivative of
w(
x) can be expressed as:
$$\begin{aligned} w'(x)=\sum _{m=1}^{\infty }\alpha _mA_m \cos \alpha _m x, \quad 0 \leqslant x \leqslant L \end{aligned}$$
(20)
However, with regard to the first derivative of
\(u_1\) and
\(u_2\), term-by-term differentiation is only valid on open interval of
\(0<x<L\). They can be expressed as:
$$\begin{aligned}&u'_1=-\sum _{m=1}^{\infty }\alpha _mB_m\sin \alpha _m x \end{aligned}$$
(21)
$$\begin{aligned}&u'_2=-\sum _{m=1}^{\infty }\alpha _mC_m\sin \alpha _m x \end{aligned}$$
(22)
Substituting Eqs. (
21) and (
22) in Eq. (
6), the first governing equation can be expressed as:
$$\begin{aligned} \sum _{m=1}^{\infty }\alpha _m(E_1A_1B_m+E_2A_2C_m)\sin \alpha _m x=0 \end{aligned}$$
(23)
Since Eq. (
23) is valid for any
x (or Cantor’s theorem [
46]), it can be concluded that
$$\begin{aligned} E_1A_1B_m+E_2A_2C_m=0 \end{aligned}$$
(24)
The second derivative of
\(u_1\) on interval of
\(0 \leqslant x \leqslant L\) can be obtained as:
$$\begin{aligned} u_1''=\frac{u'_1(L)-u'_1(0)}{L}+\sum _{m=1}^{\infty }\left\{ \frac{2}{L}\big [(-1)^m u'_1(L)-u'_1(0)\big ]-\alpha _m^2B_m\right\} \cos \alpha _m x \end{aligned}$$
(25)
Substituting Eqs. (
16), (
17), (
20) and (
25) in the second governing equation of Eq. (
9), it gives
$$\begin{aligned} \begin{aligned}&\frac{E_1A_1}{k}\big [u'_1(L)-u'_1(0)\big ]+k(C_0-B_0) +\sum _{m=1}^{\infty }\Big \{\frac{2 E_1A_1}{L}\big [(-1)^mu'_1(L)-u'_1(0)\big ]-E_1A_1\alpha _m^2B_m\\&\quad +\,k(C_m-B_m)+kh\alpha _mA_m\Big \}\cos \alpha _m x=0 \end{aligned} \end{aligned}$$
(26)
Based on Cantor’s theorem [
46], coefficients of the cosine series in Eq. (
26) equal zero. Applying Eq. (
24), it arrives
$$\begin{aligned} \frac{k}{E_1A_1}h\alpha _m A_m+\frac{2\big [(-1)^mu'_1(L)-u'_1(0)\big ]}{L}-\left( \alpha _m^2+\frac{k}{EA}\right) B_m=0 \end{aligned}$$
(27)
in which
\(\frac{1}{EA}=\frac{1}{E_1A_1}+\frac{1}{E_2A_2}\) . Further simplification can be made by substituting the boundary conditions of Eq. (
14c), which results in
$$\begin{aligned} \frac{k}{E_1A_1}h\alpha _m A_m+2\sum _{i=1}^{M}D_{\mathrm{im}}\alpha _i^2B_i-\left( \alpha _m^2+\frac{k}{EA}\right) B_m=0 \end{aligned}$$
(28)
The derivation of
\(D_{\mathrm{im}}\) is shown in
6.
At last, the fourth derivative of
w and the third derivative of
\(u_1\) are given by
$$\begin{aligned}&w''''(x)=-\sum _{m=1}^{\infty }\alpha _m\left\{ \frac{2}{L}\big [(-1)^mw''(L)-w''(0)\big ]-\alpha _m^3A_m\right\} \sin \alpha _m x \end{aligned}$$
(29)
$$\begin{aligned}&u_1'''(x)=-\sum _{m=1}^{\infty }\alpha _m\left\{ \frac{2}{L}\big [(-1)^m u_1'(L)-u_1'(0)\big ]-\alpha _m^2B_m\right\} \sin \alpha _m x \end{aligned}$$
(30)
in which
\(0<x<L\). After substituting the load function of Eq. (
18) and displacement derivatives of Eqs. (
29) and (
30) in Eq. (
11), the first governing equation can be expressed as:
$$\begin{aligned} \begin{aligned}&\sum _{m=1}^{\infty }\Big \{-EI_o\alpha _m\frac{2}{L}\big [(-1)^mw''(L)-w''(0)\big ] -E_1A_1h\alpha _m\frac{2}{L}\big [(-1)^m u_1'(L)-u_1'(0)\big ]\\&\quad +\,EI_o\alpha _m^4A_m+E_1A_1h\alpha _m^3B_m-Q_m\Big \}\sin \alpha _m x=0 \end{aligned} \end{aligned}$$
(31)
Based on Cantor’s theorem [
46], coefficients of the trigonometric series in Eq. (
31) equal zero, which arrives
$$\begin{aligned} \begin{aligned}&-EI_o\alpha _m\frac{2}{L}\big [(-1)^mw''(L)-w''(0)\big ] -E_1A_1h\alpha _m\frac{2}{L}\big [(-1)^m u_1'(L)-u_1'(0)\big ] \\&\quad +EI_o\alpha _m^4A_m+E_1A_1h\alpha _m^3B_m=Q_m \end{aligned} \end{aligned}$$
(32)
Applying Eq. (
14b), Eq. (
32) can be rearranged as:
$$\begin{aligned} \alpha _m\frac{2}{L}\big [(-1)^{m}R_\mathrm{L}w'(L)+R_0w'(0)\big ]+EI_o\alpha _m^4A_m+E_1A_1h\alpha _m^3B_m=Q_m \end{aligned}$$
(33)
Using Eq. (
20), it can be obtained that
$$\begin{aligned} w'(L)=\sum _{m=1}^{\infty }(-1)^m\alpha _mA_m, \quad w'(0)=\sum _{m=1}^{\infty }\alpha _mA_m \end{aligned}$$
(34)
Substituting Eq. (
34) into Eq. (
33), it gives
$$\begin{aligned} EI_o\alpha _m^4A_m+\frac{2}{L}\alpha _m\sum _{i=1}^{\infty }\left[ (-1)^{i+m}R_\mathrm{L}+R_0\right] \alpha _iA_i+E_1A_1h\alpha _m^3B_m=Q_m \end{aligned}$$
(35)
For convenience, a non-dimensional parameter,
r, was introduced herein to replace the rotational restraint stiffness,
R, as
$$\begin{aligned} r=\frac{1}{1+\frac{3EI_o}{RL}} \end{aligned}$$
(36)
in which
r is so-called end fixity factors, which is widely used in semi-rigid steel frame connections [
38,
47]. In particular,
\(r=0\) represents the simply supported condition and
\(r=1\) is used for the fully clamped. Accordingly, Eq. (
35) can be expressed as:
$$\begin{aligned} EI_o\alpha _m^4A_m+\frac{6EI_o\alpha _m}{L^2}\sum _{i=1}^{\infty }\left[ (-1)^{i+m}\frac{r_\mathrm{L}}{1-r_\mathrm{L}}+\frac{r_0}{1-r_0}\right] \alpha _iA_i+E_1A_1h\alpha _m^3B_m=Q_m \end{aligned}$$
(37)
Equations (
28) and (
37) are two infinite systems of linear equations with respect to coefficients
\(A_m\) and
\(B_m\). (Here, the subscripts were set as the same for convenience.) The infinite number of equations can be truncated to finite terms, say
M, for each coefficients. (The truncated terms for
\(A_m\) and
\(B_m\) can be different as well.) Then, it will produce 2
M equations with 2
M unknown coefficients, which can be easily solved. Once coefficients
\(B_m\) have been determined,
\(C_m\) can also be calculated based on Eq. (
24). As a result, all exact displacements,
w(
x),
\(u_1(x)\) and
\(u_2(x)\), can be determined. In addition, the slip can be obtained by using Eq. (
1). With all these displacements known, other moments and inner forces can easily be derived, which are not presented here.