Hybrid Post-Processing Procedure for Displacement-Based Plane Elements

In the analysis of high-rise building, traditional 
displacement-based plane elements are often used to get the in-plane internal 
forces of the shear walls by stress integration. Limited by the singular 
problem produced by wall holes and the loss of precision induced by using 
differential method to derive strains, the displacement-based elements cannot 
always present accuracy enough for design. In this paper, the hybrid 
post-processing procedure based on the Hellinger-Reissner variational principle 
is used for improving the stress precision of two quadrilateral plane elements. 
In order to find the best stress field, three different forms are assumed for 
the displacement-based plane elements and with drilling 
DOF. Numerical results show that by using the proposed method, the accuracy of 
stress solutions of these two displacement-based plane elements can be improved.

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For the tional c

Proced lane Ele
where W exp is the work of external forces: exp ( ) Assume the new internal force field as follows: , , where   { } [ ] and 11 j , 12 j , 21 j , 22 j are components of the inver- sion matrix of Jacobian .
After substituting Equation (3) into Equation ( 1), the new form of energy functional can be written as: By the principle of stationary: the parameters of the assumed internal force field can be written as: where Substitute Equation (7) into Equation ( 3), the new internal force can be obtained.
Compared with plate elements, it is much easier to extend this method to plane elements.The according hybrid discrete energy functional of plane elements can be written as [6]: Similar to Equation ( 4), a new form of stress field of displacement-based plane element can be assumed as: Substitute Equation (10) into Equation ( 9), the parameters i  in [ ]  P can be obtained by using the principle of stationary.The according matrix named [ ] for plane elements are as follows: With different matrix of [ ]  P , the stress field will be different too.This new method will be used to try to improve the stress accuracy of plane element named AQ4 and AQ4 [7].

Introduction of AQ4θ and AQ4θλ
AQ4 and AQ4 are two plane elements with drilling DOF formulated using quadrilateral area coordinate methods presented by Long et al. [8,9].They have the merits of high accuracy and robust against mesh distortions.After having been programmed into the software for the analysis of high-rise buildings, the results show that better accuracy is still needed for the stress of shear walls with holes.
The definition of DOF of these two elements is: where and i  is the additional rigid rotation at element node.
The displacements of element are as follows: where   the additional displacement field induced only by the rigid rotation at nodes denoted as i  .
In order to determine the displacement field   0 u , the shape functions in reference [10] is used, they are as follows: where: and Assume the rotational displacement field as polynomials of quadrilateral area coordinates: with the conforming Equations as follows, the rotational displacement can be obtained. where Then the stiffness matrix of element AQ4 can be solved out easily, and the strain matrix is: After substituting Equation ( 21) into the stress-strain relationship, the stress of AQ4 can be obtained.
AQ4 is an improved element based on AQ4 by adding a displacement field which is induced by internal parameters.
The additional displacement fields mentioned above are as follows: where , , ,     are the internal parameters.In order to solve the undetermined parameters, the conforming Equations are taken as: Substitute Equation ( 24) into (25), the shape functions of { }  u can be formulated out, they are: )( ) For element AQ4 , 1 N  is taken as the final internal shape function, then: Through the condense calculation, stiffness matrix of AQ4 can be written as: is the strain matrix of additional displacement field.The according stress fields are as follows: Combined Equation ( 21) with (31), the stress field of AQ4 can be obtained.

Hybrid Post-Processing of AQ4θ and AQ4θλ
In the formulating of stress in elements AQ4 and AQ4 , the strain matrix q     B and    B are the differential results of displacements to coordinates.The differential calculation lowered the accurate order of stress.To avoid the differential, Hybrid Post-processing procedures can be used to improve the stress accuracy.Based on this theory, three forms of stress fields are presented for these two displacement-based elements.

Stress Field I
The first form of stress field is assumed as Equation (32).It is the stress field of a hybrid element developed by Pian and Wu [6]:

Stress Field III
Try to make the three stress components to be independent, the third form of stress field is assumed as follows:  

Strict Patch Test
The constant strain/stress patch test using irregular mesh is shown in Figure 1.Let Young's modulus E = 1000, Poisson's ratio  = 0.25, and thickness of the patch t = 1.
After modified by three different forms of Hybrid Postprocessing, these two elements can produce exact solutions without any problem.

Cook's Skew Beam
This example was proposed by Cook et al. [12].As shown in Figure 2, a skew cantilever beam subjected to distributed shear load along its free edge.The results of  max at point A and  min at point B are listed in Tables 1 and 2.

Conclusion
Using the traditional method to calculate stress of displacement-based elements, the accuracy will descend for the reason of differential when strains are derived from the displacement fields.For improving the stress accuracy, hybrid/mix elements are very effective.However, the formulations of the hybrid elements are more complicated than those displacement-based elements.Based on  the Hellinger-Reissner variational principle, hybrid postprocess procedure can take advantage of the merits of these two kinds of elements to establish the relationship between the displacement and the stress or internal force fields.In this paper, based on this theory, three forms of stress fields are used to improve the stress of plane elements with drilling DOF.Through the numerical results, for element AQ4θ , only the second form of stress field is effective, but for AQ4θ , except for the first form of stress, the other two forms can present better results than the source elements.It is proved that the method of hybrid post-process procedure is workable. where