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## Static Analysis: Load Factor Analysis

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**Static Analysis:**Load Factor Analysis**Objectives**Section II - Static Analysis Module 8 - Load Factor Analysis Page 2 The objective of this module is to introduce the equations and numerical methods used to determine if a system is stable. • Buckling is a failure mode, due to compressive components becoming unstable. • The type of analysis presented in this module is frequently called load factor, linear buckling, or eigenbuckling analysis.**Load Deflection Curve**Section II - Static Analysis Module 8 - Load Factor Analysis Page 3 • The figure shows the force versus deflection curve of a system. • As the load is increased, the slope (tangent stiffness) progresses to a point at which it is zero. • When the stiffness is zero, the application of an infinitesimally small load will cause a very large displacement. • This type of behavior is typical of a system subjected to compressive loading. F u**Governing Equations**Section II - Static Analysis Module 8 - Load Factor Analysis Page 4 • The governing equations for a static finite element analysis can be written as • The tangent stiffness matrix, , has three components where are the linear, displacement, and stress dependent contributions. are the displacement increments.**Governing Equations**Section II - Static Analysis Module 8 - Load Factor Analysis Page 5 is the unbalanced load array. It is the difference between two arrays. is an array of external forces acting on the nodes. This array is obtained from the external virtual work term. is an array of node forces associated with the stresses inside the body. This array is obtained from the internal virtual work term.**Zero Tangent Stiffness Matrix**Section II - Static Analysis Module 8 - Load Factor Analysis Page 6 • When the system is in equilibrium, the unbalanced load vector is equal to zero and the equations reduce to • There is the possibility that the system is unstable and that another equilibrium configuration exists that the system could move to without an increase in the external loads • The stress levels at which unstable configurations exist can be determined by introducing a Load Factor (l) into the above equation**Eigenvalue Problem**Section II - Static Analysis Module 8 - Load Factor Analysis Page 7 • This equation is a homogeneous set of equations that has a solution only when the determinant of the coefficient matrix is zero. • The determinant is a polynomial that will be equal to zero only for specific values of l. The order of the polynomial is equal to the number of equations. det Characteristic Polynomial l1 l3 l l2**Load Factor Interpretation**Section II - Static Analysis Module 8 - Load Factor Analysis Page 8 • If l is equal to 1, then the system is unstable as loaded. • If l is between 0 and 1, then the system is unstable at a load level less than the applied load. • If l is greater than 1, then the load can be increased by a multiple of l before it becomes unstable. • If l is equal to -1, the system is unstable if the load is reversed. • If l is between 0 and -1 then the system is unstable at a load level less than the applied load if the load is reversed. • If l is greater than, -1 then the system becomes unstable at a multiple of l when the load is reversed. Only the lowest value of l is of practical interest.**Example**Section II - Static Analysis Module 8 - Load Factor Analysis Page 9 • Simulation is used to compute the load factor and buckled shape of the cantilevered beam that has an axial load of 5 lbs. located at its free end. 1 inch wide x 12 inch long x 1/8 inch thick. Material - 6061-T6 aluminum. Brick elements with mid-side nodes are used to improve bending accuracy through the thin section. 0.0625 inch element size. Fixed end boundary condition 5 lbs.**Example – Analysis Parameters**Section II - Static Analysis Module 8 - Load Factor Analysis Page 10 Use Sparse matrix solver Use all threads/cores during the solution (fastest) Compute first five load factors (lowest is only one of practical interest) Use BCS-LIB parallel sparse solver**Example - Results**Section II - Static Analysis Module 8 - Load Factor Analysis Page 11 The buckling load is 5 lbs. x 5.6 load factor = 28 lbs. Buckled shape for lowest load factor. Beam will buckle at a load that is 5.6 times that applied. Caution – initial distortions in the shape of the beam could make the buckling load be smaller.**Summary**Section II - Static Analysis Module 8 - Load Factor Analysis Page 12 • A method for determining the stability of a system based on the tangent stiffness matrix was introduced. • A load factor was introduced that acts like a safety factor to tell how far from being unstable the system is at an applied load level. • An example showing results for a simple system were presented.