The classical shell theory (CST) without considering the shear deformation has been commonly used in the calculation of shells structures recently. However, the impact of theory of plates and shells subjected to the shear deformation on the calculation is increasingly pronounced along with the wide use of composite laminated structures. In this paper, based on firstorder shear deformation theory (FSDT) of cylindrical shells, the displacement control differential equation of moderately thick cylindrical shells has been obtained, so has been the edge force at longitudinal of the shells. Meanwhile, a group of unit force is introduced to deduce the displacement of edge beam under the action of edge force. A join condition of moderately thick cylindrical ribbed shells is established according to the continuity of displacement as well. Most notably, the displacement analytical solution of bending problems of moderately thick cylindrical ribbed shells is obtained, which has profound theoretical significance for further improving the analytical solution of moderately thick cylindrical shells.
With the advent of new materials, various shell structures have experienced unprecedented development. In comparison with the beam and plate structural and the spherical shell structure, the cylindrical shell is of more reasonable structure, superior mechanical performance, simpler construction, and manufacturing. Therefore, the cylindrical shells have been widely used in tunnel, subway, and pipe jacking and other municipal engineering and machinery and aviation industries [
Since Sophie Germain (1813) gained correct expression of bending of thin plate and Aaron (1874) and Love (1888) established theory of thin shells; the theory of plates and shells have been developed with the shells structure gradually application in engineering [
Highorder shear deformation theory (HSDT) and FSDT are adopted to analyse the effect of transverse shearing deformation in the moderately thick plates and shells theory [
Take coordinate system of cylindrical shells as shown in Figure
Coordinate system of cylindrical shells.
Then the corresponding Lamé coefficients are
The radiuses of curvature are
Hence, the geometrical equations can be written as
And the constitutive equations are
Also, the equilibrium equations can be written as
Substituting (
Bending problems of moderately thick cylindrical shells are summarized as mathematical problems that solve the displacementbased differential equation (
In the equation
Meanwhile, we can get
Substituting (
By the CauchyRiemann conditions, it does not lose its generality [
Substituting (
Equations (
Then (
To make the data easily analyzed, we introduce the nondimensional coordinate
Equations (
The displacement equations can be obtained by (
And the angles of rotation can be expressed as
According to (
According to (
By opening up the displacement function as trigonometric series in satisfying the boundary conditions, the control equation can be simplified to a solvable eighthorder ordinary differential equation and a secondorder ordinary differential equation. After solving the solution of the control equation sets, displacement, angle, and internal force can be solved too, and then set ten undetermined coefficients that it contains by using boundary conditions or join conditions to solve the solution of moderately thick cylindrical shells.
In order to study the coworking situation of shells and edge beam, the shells and edge beam are cut and split along the edge of the shells, this section will have the internal forces
Force at the place where the shells link up with the edge beam.
Now let us discuss the displacement state of bearing load of edge beams
Coordinate graph of edge beam.
In order to calculate the deformation state of edge beam, the edge beam bears the edge load as shown in Figure
Rules of a set of displacement and stress on shear center
According to the displacement differential equation of given beam and the SaintVenant torsional equation [
The displacements
Defining integral to (
The displacement on the point
Displacement on the point









0  0 

0 

0 

0 

0 




0 


0  0  0 

0 

0 

0  0 

In the table, the first column of the first row represents the displacement
Displacement rules of point
Generally, the edge force of shells often acts on the edge of beam, instead of acting on point
From the equilibrium condition [
Defining differential and integral to (
By comparing (
Equivalent load.




 


1  0  0 

0 

0  1  0 

0 




0  0 

0  0  0  1  0 

0  0  0  0  1 
In Table
By using the combination of the displacement of point
Displacement of point












0 














0  0 





0 

0 



0 


0 

For the convenience of data analysis, record
Displacement of point












0 














0  0 





0 

0 



0 


0 

Recently, open cylindrical shells, especially the reinforced concrete open cylindrical shells which are usually used to be largespan roofs, have been applied successfully in many projects. The curvy edges of cylindrical shells are braced by the reinforced concrete circulararc thin plate or thin arch circle, which can be regarded as simply supported edges with large stiffness. Thus, we can introduce
Substituting (
Then, substituting (
The particular solution of (
To obtain the solution of (
Equation (
The homogeneous solution of (
The general solution of (
And the solution of (
To satisfy boundary conditions or join conditions, displacement components, stress components, and bending moment components are expanded according to trigonometric series.
According to (
According to (
According to (
According to (
According to (
There are five edge forces at the place where the shells is connected with the edge beam, namely,
on the edge
on the edge
The edge forces are as shown in Figure
Edge forces distribution at the place where the shells are connected with the edge beam.
In order to use the results in Table
Decompose
Decompose
According to Table
Thus, the subdisplacement and angle of rotation of edge beam I in
Decompose
Decompose
According to Table
Thus, the subdisplacement and angle of rotation of edge beam II in
The expression for shell’s displacements is
Then, shell’s displacement on the edge is as follows.
Edge I is
Edge II is
By using the continuous conditions of displacements to compare (
Edge I is
Edge II is
According to the join condition, the ten undetermined coefficients can be solved, and trigonometric series expressions of displacements, angles of rotation, internal forces, and moments can be further obtained, whose changing curves can be drawn with MATLAB program.
With the cylindrical shells roof that has been widely used in practical engineering as an example, assume the shells and edge beams are steel reinforcement and concrete materials. The calculating parameters are as follows: elastic modulus is
It can be seen from Figure
Comparison of the displacement and internal force under three boundary conditions.
With the same calculating parameters (including dimensions of shells and edge beam and the applied loads) of Example
Generally speaking, as seen in Figures
Comparison of the displacement with different values of
Comparison of the displacement with different values of
Based on FSDT considering the effect of transverse shear, the displacementbased differential equation set for general problems of moderately thick cylindrical shells is obtained; the problem is simplified into a solvable eighthorder ordinary differential equation and a secondorder ordinary differential equation by introducing four displacement functions.
In consideration of the effect of edge beams, the displacement of edge beams under edge force is obtained by using the given displacement differential equation of beams, and the displacement of edge beams is converted into the shells coordinate system. The join condition of moderately thick cylindrical ribbed shells is established according to the continuity of displacement. By using this join condition, the ten undetermined constants the displacement function contains are solved, and thus the solution of bending problem of moderately thick cylindrical ribbed shells is obtained. In addition, by comparing the displacement, internal force of ribbed shells with the displacement, internal force of fixed edges, and simply supported edges, the numerical results show that the displacement and internal force of ribbed shells are between these of fixed edges and simply supported edges, which explains that the solving methods proposed and the join condition established in this paper are correct.
For ease of the calculation, the firstorder model considering the effect of transverse shear is adopted in this paper. The basic equations derived by using the above model are simpler, but they cannot fully satisfy the edge condition that the shear stress is 0 on the free surface, which is inconsistent with the actual situation and is also the defectexisting in the firstorder model itself. To remedy this defect, the higherorder model can be adopted for calculation in the followup study, so that the calculation results can get closer to the actual.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is partially supported by National Natural Science Foundations of China (Grant no. 51478044). The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.