Currently, finite element numerical analysis has been widely used in various industries. However, during numerical analysis, sometimes there are situations where the software calculation results are inconsistent with theoretical analysis or the calculation results are not within a reasonable range. Although we have chosen reasonable element types and material constitutive models during the process, the final result is inexplicable. This may involve the four common defects in finite element analysis: shear locking, volume locking, hourglass mode, and zero-energy mode.
This article mainly introduces the basic concept of shear locking, taking the assumption of the cross-section in common material mechanics as an example, as shown below.
In the above figure, the first model is the assumption of the planar section in material mechanics. When the section undergoes pure bending deformation, the upper surface is elongated by the horizontal tensile stress, and the lower surface is compressed by the horizontal compressive stress. The surface perpendicular to the middle section before deformation is still perpendicular to the middle section after deformation, and the vertical straight line before and after deformation remains the same length and is always perpendicular to the upper and lower surfaces, that is, there is no shear deformation, and the shear stress is 0. At this time, the section deformation is the pure bending deformation in theoretical calculation.
The second model is the before-and-after comparison of the model section in finite element analysis. During numerical analysis, we need to discretize the section. When we use linear elements for analysis, since each side is composed of two nodes, the element still remains straight before and after deformation, and the upper surface is stretched while the lower surface is compressed, so the middle line no longer remains straight and has increased or decreased, that is, shear deformation occurs. Due to the existence of shear deformation, part of the deformation energy is consumed, which makes the element too stiff, and the bending deformation decreases, resulting in certain differences with the theoretical calculation, which is called shear locking phenomenon.
In short, shear locking refers to the occurrence of shear deformation in elements that theoretically have no shear deformation, also known as associated shear deformation. It generally occurs when linear elements with full integration are used in a pure bending state. The main phenomenon is that the model’s bending stiffness increases, which leads to smaller deformation. The solution methods are as follows:
- Use reduced integration;
- Refine the mesh;
- Use nonconforming elements;
- Assume a shear strain method.
Taking a simply supported beam as an example, the shear locking phenomenon in finite element analysis will be briefly verified, and the calculation software ANSYS will be used.
As shown below, the simply supported beam has a length of 2 meters and a section size of 0.2m x 0.2m, and the material E = 30Gpa, U = 0.2. The midpoint is subjected to a uniformly distributed load F = 20kN/m. The maximum displacement at the midpoint position is calculated theoretically as follows: W=5ql^4/(384EI)=5202000^412/(38430e4*200^4)=1.042mm
During the calculation, Plane182 considering the thickness of the plane stress element is used, and full integration is used for the integration method. The models controlled by different meshes are respectively expressed as Model 1 to Model 5 according to the degree of mesh density.
The ANSYS calculation results of each model are as follows:
The models one to four have significant errors compared to the theoretical calculations, with model one having the highest error at approximately 30.36%. This is unacceptable in engineering. However, model five is much closer to the theoretical calculations. The main reason for this is that models one to four experienced different degrees of shear lock-up. It can be found that shear lock-up is mainly related to two situations:
Element shape After dividing the elements, the narrower and longer the element, the more likely it is to experience shear lock-up. For example, models two to four have a large aspect ratio, while the dimensions of the elements in model five are more similar.
Element force situation Even if the element size meets the requirements, if the element deformation is mainly due to bending, the probability of shear lock-up occurring will be high. This is the case in model one.
As mentioned earlier, there are four methods to eliminate the shear lock-up phenomenon. For ANSYS, taking the above example as an example, the following feasible methods can be used:
1. Reduced integration method With conventional linear solid elements.
The software defaults to full integration format, with two Gaussian integration points in each direction. This means that two points determine a straight line, and after the element deformation, each edge is still a straight line, which may cause shear force locking. If only one integration point is used for the same element, can shear force locking be avoided? Perhaps this is possible, but will one Gaussian integration point reduce the calculation accuracy? Through calculation analysis, using the above analysis model and the same calculation grid, the calculation results are as follows after setting the element keyword 1 of Plane182 to use reduced integration:
From the table, although reduced integration is used, there are still significant errors, especially in model one, where abnormal results even appear. Therefore, sometimes, even if reduced integration is used, if the grid size is not appropriate, the result error will still be large. This is mainly related to another defect of finite element, the Hourglass defect, which will be discussed in the next article.
2. Using non-conforming elements
Although reduced integration can eliminate shear lock-up to some extent, its effect is not always good, and sometimes Hourglass phenomenon occurs, as in the above example. Here, another method is introduced, which is to use the Enhanced Strain Mode (ESM), also known as non-conforming elements. The main idea is to make the strain linearly change in a certain direction in some way, and to make the internal strain pattern of the element linearly change by adding some virtual additional degrees of freedom. Since the increase in deformation gradient is completely internal to the element and independent of the element nodes, it neither increases the overall degrees of freedom of the structural solution nor affects the continuity of displacement on the boundary. Using the above analysis model and the same calculation grid, the calculation results are as follows after setting the element keyword 1 of Plane182 to use the enhanced strain algorithm:
It can be seen from the results that the fundamental changes have been made to models one to four after using this method, and the results of the finite element numerical analysis using the enhanced strain mode are very close to the theoretical solution. This indicates that the enhanced strain elements are very effective. However, enhanced strain elements also have their limitations and weaknesses. For example, when the element shape is very irregular, the calculation result will be very poor if the angle between each adjacent line of the element is too large or too small.
3. Using Higher Order Elements
The essential reason for shear locking is that structures with mainly bending deformation are discretized using linear elements in finite element analysis, resulting in linear deformation at the boundaries. To alleviate shear locking to some extent, increasing the order of element shape functions can be employed, but the computation workload increases significantly due to the increase in element nodes.
In summary, shear locking may occur due to various factors such as structural loading, element shape, and element mode selection. When shear locking is unavoidable, methods such as reduced integration, enhanced strain elements, and higher order elements can be used to eliminate this phenomenon as much as possible. However, it is also necessary to understand the applicability, advantages, and disadvantages of each method and select the appropriate and correct method for analysis to obtain reasonable computation results.