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Warping Factor/Angle in Finite Element Analysis

Warping of an Element:

  • Ideally all the nodes of quadrilateral element should lie on the same plane but at curvatures and complicated geometry profiles it is not possible. "Measure of out of planeness of a Quadrilateral is Warping Factor or Warping Angle".

  • Warping calculation for triangular elements is not applicable, since three points define a plane, this check only applies to quads.

Warping Angle:

  • It is defined as angle between normals of two (triangular) planes formed by splitting the quad element along diagonals.

  • Maximum Angle out of the two possibilities is reported as "Warp Angle".

  • In the case of solid elements, an element's face deviates from being planar. The quad is divided into two trias along its diagonal, and the angle between the trias normals is measured.

  • Ideal Value = 0° ( Acceptable < 5° to 10°)

  • Assume a Quad element is formed with Nodes:1,2,3 &4

  • Split the Quad using the two diagonals (1,3 & 2,4) into triangles as shown below.

  • Plane-1 is formed from nodes: 1,2 &4 and Plane-2 is formed from nodes: 2,3 &4

  • Calculate normals to Plane-1 & Plane-2 as “n1 & n2”. Find the Angle between n1 & n2 and mark it as Θ1.

  • Similarly, for Plane-3 & Plane-4 find the angle between normal n3 & n4 and mark it as Θ2.

  • Maximum of the two Angles Θ1 & Θ2 is Reported as "Warping Angle".

Warping Factor:

  • Some of the Finite Element Softwares calls the Out of Planeness of an element as Warping Factor instead of Warping Angle.

  • Ideal value of Warping Factor=0 (Acceptable: 0 to 1)

  • Warping Factor is calculated by the following procedure:

  • Find the common Normal to the two diagonals (Vectors) AC & BD with vector cross product or by using 3D-Geometry principles.

  • Then generate two planes perpendicular to common Normal and passing through A,C and B,D.

  • These two planes containing AC in one plane BD in other plane are parallel to each other.

  • Construct an Average Plane between these two planes.

  • The projection of Nodes on this average plane are A’B’C’ and D’ respectively and it forms A’B’C’D’ Quadrilateral.

  • The distance of Nodes from Average plane is same i.e., AA’=BB’=CC’=DD’=h

  • Now, find the difference in of heights of nodes of Edge with respect to Average plane.

  • Difference in Nodal height of Edge AB is h-(-h)=2h (Distance between two parallel planes).

  • Find the area ( A ) of the Quadrilateral (A’B’C’D’) formed on the Avg. Plane with the projections of Actual Nodes (A,B,C and D).

  • The Warping Factor is the Ratio between the Distance of Nodes in the direction of Avg. Normal (2h) and Square root of the Area of the Element projected on to the Avg. Normal Plane √A.

So, the Warping Factor=  2h/√A

  • For a flat Quadrilateral, h=0. So, Warping Factor=0.


  1. Finite Element Analysis For Design Engineers by Paul M. Kurowski (Chapter 5.3.1).

  2. Practical Finite Element Analysis by Nitin S Gokhale, Sanjay S Deshpande, Sanjeev V Bedekar and Anand N Thite (Chapter 7.9).

  3. The Finite Element Method : Practical Course by G. R. Liu , S. S. Quek (Chapter 11.4.2).

  4. Ansys theory reference manual.

  5. Hypermesh users guide.

Note (**): Ansys, Hypermesh are Registered trademarks of their respective owners.

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