4. Spring Element Dr. Ahmet Zafer Şenalp Mechanical Engineering Department Gebze Technical University ME 520 Fundamentals of Finite Element Analysis

1-D Line Element (Spring, truss, beam, pipe,...,etc.) 2-D Plane Element (Membrane, plate, shell,...,etc.) Types of Finite Elements ME 520 Dr. Ahmet Zafer Şenalp 2Mechanical Engineering Department, GTU 4. Spring Element 4. Spring Element

3-D Solid Element (3-D fields - temperature, displacement, stress, flow velocity,...,etc.) Types of Finite Elements ME 520 Dr. Ahmet Zafer Şenalp 3Mechanical Engineering Department, GTU 4. Spring Element 4. Spring Element

One Spring Element: Two nodes: i, j Nodal displacements: u i, u j (in, m, mm) Nodal forces: f i, f j (lb, Newton) Spring constant (stiffness): k (lb/in, N/m, N/mm) Spring force-displacement relationship: Spring Element ME 520 Dr. Ahmet Zafer Şenalp 4Mechanical Engineering Department, GTU 4. Spring Element 4. Spring Element linear nonlinear ; is the force needed to produce a unit stretch. We only consider linear problems in this introductory course.

Consider the equilibrium of forces for the spring. At node i, we have and at node j, In matrix form, or, where k = (element) stiffness matrix u = (element nodal) displacement vector f = (element nodal) force vector Note that k is symmetric. Is k singular or nonsingular? That is, can we solve the equation? If not, why? Spring Element (Spring Element) ME 520 Dr. Ahmet Zafer Şenalp 5Mechanical Engineering Department, GTU 4. Spring Element 4. Spring Element

Spring System ME 520 Dr. Ahmet Zafer Şenalp 6Mechanical Engineering Department, GTU 4. Spring Element 4. Spring Element For element 1, element 2, : is the (internal) force acting on local node i of element m (i = 1, 2).

System Stiffness Matrix ME 520 Dr. Ahmet Zafer Şenalp 7Mechanical Engineering Department, GTU 4. Spring Element 4. Spring Element Method 1 – Force Balance: Consider the equilibrium of forces at node 1, consider the equilibrium of forces at node 2, consider the equilibrium of forces at node 3 K : Stiffness matrix (structure matrix) for the spring system. In martix form:

System Stiffness Matrix ME 520 Dr. Ahmet Zafer Şenalp 8Mechanical Engineering Department, GTU 4. Spring Element 4. Spring Element Method 2 – Enlarging the Element Stiffness Matrices : u 1 u 2 u 3 + =

System Stiffness Matrix ME 520 Dr. Ahmet Zafer Şenalp 9Mechanical Engineering Department, GTU 4. Spring Element 4. Spring Element Method 3 – Assembling by Using Row and Column Addresses : K=K= u 2 u 3 u 1 u 2 u1u2u1u2 u1u2u3u1u2u3 u 1 u 2 u 3 u2u3u2u3 K=K=

Example 1 ME 520 Dr. Ahmet Zafer Şenalp 10Mechanical Engineering Department, GTU 4. Spring Element 4. Spring Element Connectivity table: Boundary conditions: a)Displacement boundary conditions: b)Force boundary conditions: E# Element Number N1 Node 1 N2 Node

Example 1 ME 520 Dr. Ahmet Zafer Şenalp 11Mechanical Engineering Department, GTU 4. Spring Element 4. Spring Element Reaction force: Nodal displacement values: Applying boundary conditions; 12

· Deformed shape of the structure · Balance of the external forces · Order of magnitudes of the numbers Checking the Results ME 520 Dr. Ahmet Zafer Şenalp 12Mechanical Engineering Department, GTU 4. Spring Element 4. Spring Element

Notes About the Spring Elements · Suitable for stiffness analysis · Not suitable for stress analysis of the spring itself · Can have spring elements with stiffness in the lateral direction, spring elements for torsion, etc. Notes About the Spring Elements ME 520 Dr. Ahmet Zafer Şenalp 13Mechanical Engineering Department, GTU 4. Spring Element 4. Spring Element

Example 2 ME 520 Dr. Ahmet Zafer Şenalp 14Mechanical Engineering Department, GTU 4. Spring Element 4. Spring Element Given; k 1 =100N/mm, k 2 =200 N/mm, k 3 =100 N/mm, P=500 N Find; (a)the global stiffness matrix (b)displacements of nodes 2 and 3 (c)the reaction forces at nodes 1 and 4 (d)the force in the spring 2 Solution: Connectivity table: 123 E#N1N

Example 2 ME 520 Dr. Ahmet Zafer Şenalp 15Mechanical Engineering Department, GTU 4. Spring Element 4. Spring Element Boundary conditions:  Displacement boundary conditions:  Force boundary conditions: a) Element Stiffness Matrices (N/mm): 123

Example 2 ME 520 Dr. Ahmet Zafer Şenalp 16Mechanical Engineering Department, GTU 4. Spring Element 4. Spring Element Construction of global stiffness matrix : Equilibrium (FE) equation for the whole system is; 123 symmetric and banded.

Example 2 ME 520 Dr. Ahmet Zafer Şenalp 17Mechanical Engineering Department, GTU 4. Spring Element 4. Spring Element b) Applying boundary conditions; 123

Example 2 ME 520 Dr. Ahmet Zafer Şenalp 18Mechanical Engineering Department, GTU 4. Spring Element 4. Spring Element c) From the 1 st and 4 th equations in FE equation for the whole system, we get the reaction forces : d) FE equation for 2. Spring Element: 123 i=2,j=3 Force in the spring 2: F

Example 2 ME 520 Dr. Ahmet Zafer Şenalp 19Mechanical Engineering Department, GTU 4. Spring Element 4. Spring Element c) From the 1 st and 4 th equations in FE equation for the whole system, we get the reaction forces : d) FE equation for 2. Spring Element: 123 i=2,j=3 Force in the spring 2: F

Spring System Example 3 ME 520 Dr. Ahmet Zafer Şenalp 20Mechanical Engineering Department, GTU 4. Spring Element 4. Spring Element Given; Find; the global stiffness matrix Solution : Connectivity table: E#N1N

Spring System Example 3 ME 520 Dr. Ahmet Zafer Şenalp 21Mechanical Engineering Department, GTU 4. Spring Element 4. Spring Element Element Stiffness Matrices : Global stiffness matrix : Symmetric and banded Singular as boundary conditions are not applied; det(K)=0