How is the 𝐂𝐨𝐧𝐜𝐞𝐩𝐭 𝐨𝐟 𝐃𝐞𝐠𝐫𝐞𝐞𝐬 𝐨𝐟 𝐅𝐫𝐞𝐞𝐝𝐨𝐦 (𝐃𝐎𝐅) defined in Finite Element Analysis (𝐅𝐄𝐀)

🎯 How is the 𝐂𝐨𝐧𝐜𝐞𝐩𝐭 𝐨𝐟 𝐃𝐞𝐠𝐫𝐞𝐞𝐬 𝐨𝐟 𝐅𝐫𝐞𝐞𝐝𝐨𝐦 (𝐃𝐎𝐅) defined in Finite Element Analysis (𝐅𝐄𝐀)?

In FEA, Degrees of Freedom refers to the number of independent ways a node or element can move in a system. The DOF determines how the structure deforms under applied loads and boundary conditions.

✔️ DOF in Structural Analysis ⦿ In structural mechanics, DOFs are typically associated with displacements and rotations at each node: ⦿Translational DOF: Movement in the x, y, z directions. ⦿ Rotational DOF: Rotation about the x, y, z axes.

For example: ⦿ 1D elements (Beam, Truss): Can have 1 to 6 DOFs per node depending on the type. ⦿ 2D elements (Shells, Membranes): Typically have 3 to 6 DOFs per node. ⦿ 3D elements (Solid elements): Usually have 3 DOFs per node (translations in x,y,z).

✔️ DOF in Different FEA Elements ⦿ Truss Element - 3 (Translation in x,y,z) ⦿ Beam Element - 6 (Translation + Rotation in x,y,z) ⦿ Shell Element - 6 (3 Translation + 3 Rotation) ⦿ Solid Element (3D Brick/Tetrahedral) - 3 (Translation in x,y,z).

✔️ Importance of DOF in FEA ⦿ Determines the stiffness matrix size. ⦿ Affects computational complexity. ⦿ Governs the accuracy of results (more DOF = higher accuracy but increased computation time).

✔️ How DOF Affects Stiffness Matrix Size ⦿ The total number of DOFs in the system (n) is determined by: n = (Number of nodes)×(DOFs per node). Since the stiffness matrix is always square, its size is: (n×n)

✔️ Why 3𝐃 𝐒𝐨𝐥𝐢𝐝 Elements Have 𝐓𝐡𝐫𝐞𝐞 𝐓𝐫𝐚𝐧𝐬𝐥𝐚𝐭𝐢𝐨𝐧 DOF ?

💡 3D solid elements, such as tetrahedra and hexahedra, have three translational DOFs per node (X, Y, Z) and no rotational DOFs. They capture material deformation using displacement gradients, making explicit rotational DOFs unnecessary. Rotational effects emerge naturally within the strain tensor. Unlike beams or shells, solid elements model bulk materials where deformations like stretching, compression, and shear are fully described by translations.

💡 The stress-strain relationship is computed through nodal displacements, ensuring accurate structural behavior representation. Since solid elements lack thin surfaces or rotational axes, bending and twisting are indirectly captured as internal strain variations within the material.

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