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How to model bending and bending dominated problems in Abaqus?
As FEA analysts, we often want to use solid elements 3D hexahedral elements, or 2D plane stress / plane strain / axisymmetric quadrilateral elements. Looking into the first order elements, we know these elements are available with two kinds of formulation in ABAQUS, fully-integrated formulation (C3D8, CPS4, CPE4 & CAX4) and reduced integration formulation (C3D8R, CPS4R, CPE4R & CAX4R). The idea of this article is not fully focused on the difference between these two formulations. Nevertheless, to give a quick explanation, reduced integration elements use a lower order integration to form the element stiffness. For example, a first order fully integrated plane stress element will have four integration points as compared to first order reduced integration plane stress element with one integration point at the centroid of the element, taking less time to solve due to the reduced order of integration.
The fundamental issue with these first order elements come up when used with pure bending or bending dominated problems. Fully integrated elements experience overly stiff behavior by energy going into shearing the element rather than bending it. This causes ‘shear locking’. Reduced integration elements on the other hand cannot detect strains at the integration point (center) due to bending, leading to the phenomena called ‘hourglassing’. Hourglassing effect can be significantly reduced by using more elements through the thickness experiencing bending load. A minimum of four (4) reduced integration elements may be required to reduce hourglass effects. When using reduced integration elements, it is possible to evaluate how much “hourglassing” happens with some energy results.
Abaqus also offers ‘incompatible’ mode elements for the quadrilateral and hexahedral elements (C3D8I, CPS4I, CPE4I, CAX4I). These elements enhanced by incompatible model can capture bending more accurately with even one element through the thickness. These elements are fully integrated elements with added internal degrees of freedom (incompatible deformation modes) eliminating ‘parasitic shear stresses’ thereby eliminating the shear locking phenomena. They also remove artificial stiffening due to Poisson’s effect in bending.
Since they are fully integrated, there is no question of hourglass effects. Hence they can capture bending accurately. Incompatible mode elements can be used in almost any scenario however it has to be noted that they tend to be marginally more expensive computational wise than their counterparts. They can however capture bending as accurately as second order elements, with reduced runtime. One of the significant disadvantages of using these elements is their shape. They tend to be less accurate with a parallelogram shape and poor if the elements have trapezoidal shape. So the shape of the element needs to be near perfect.
To summarize, if the problem is bending dominated and 3D Hexahedral or 2D plain strain, plane stress, and axisymmetric elements are to be used, incompatible mode elements are an alternative. When it is possible, the first choice would be to use the reduced integration formulation and use at least four (4) elements in the thickness. This is easier for 2D plane strain / plane stress and axisymmetric models since they are usually small. If it is not possible to use that many elements in the thickness, then the incompatible mode should be used. Note that in Abaqus/Explicit, only C3D8I elements are available.
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