AAE 3521 AAE 352 Lecture 08 Matrix methods - Part 1 Matrix methods for structural analysis Reading Chapter 4.1 through 4.5

AAE 3522 Objectives Introduce the matrix stiffness method used to analyze aerospace structures Provide definitions & concepts –Matrix operations –Node points –Stiffness matrix

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AAE 3524 We need accurate static and dynamic analysis of complex structures to estimate stresses, deflections and natural frequencies After we have applied loads, we need to create analytical models that provide input/output relationships between applied loads and internal forces, moments and stresses

AAE 3525 Nodes and degrees of freedom - definitions A node is a point on a structural model where we can apply forces and also measure or compute displacements. A degree of freedom is a displacement on or within the structure or on a structural boundary.

AAE 3526 Statically indeterminate structure Computing internal forces requires knowledge of structural stiffness-here is an example F1F1 F2F2 PoPo PoPo F1F1 F2F2 F2F2 F1F1

AAE 3527 Displacement compatibility between elements must be preserved – some points must move together u1u1 u1u1 u2u2 u2u2 u3u3 u3u3 u4u4 u4u4 u 1 =0 u 4 =0 u 2 =u 3

AAE 3528 Use element displacement constraints to get the additional equation we need to solve for the internal forces u1u1 u1u1 u2u2 u2u2 u3u3 u3u3 u4u4 u4u4 u 1 =0 u 4 =0 u 2 =u 3

AAE Internal forces depend on the stiffness of elements and structural displacements Displacement Compatibility

AAE Final results - there are many forces that satisfy equilibrium, but only two that allow the common points to move together F1F1 F2F2 F 2 is compressive F 1 is tensile This approach is hard to apply if there are many different elements

AAE Here is the matrix method approach bar elements (also called rods) extend or contract like springs F = kx F F x k = EA/L

AAE The element stiffness matrix is at the heart of an equation that relates nodal forces (F’s) to nodal displacements (  ’s) k = EA/L This matrix equation accounts for all possible combinations of applied forces and displacements for the bar/rod. Notice that there is a naming pattern (11, 12, ij, element number, node number)  11  12 L F 11 F

AAE The element stiffness matrix relates internal nodal forces to internal nodal displacements  11  12 L1L1 F 11 F 12  21  22 L2L2 F 21 F 22

AAE The system stiffness matrix relates external nodal forces to external nodal displacements- global systems are related to the local system  11  12 L1L1 F 11 F 12  21  22 L2L2 F 21 F 22 P1P1 P2P2 P3P3 u1u1 u2u2 u3u3

Write nodal(joint) equilibrium equations between applied forces (the P’s) and internal forces (the F’s) AAE F 11 F 12 F 21 F 22 P1P1 P2P2 P3P3 P1P1 F 11 F 12 F 21 P2P2 F 22 P3P3

Write nodal force equilibrium equations in terms of system displacements AAE P1P1 F 11 F 12 F 21 P2P2 F 22 P3P3

AAE Two element result – the “global stiffness matrix” k1k1 k2k2 In Lecture 8 we will show in more detail how to assemble this matrix from the elemental matrices

AAE Putting in the structural boundary constraints k1k1 k2k2

AAE Now solve for displacements and reactions k1k1 k2k

AAE This requires that we know how to take the matrix inverse k1k1 k2k This determinant is singular if we don’t include constraints

AAE Compute the reaction force if the two elements are fixed on the right k1k1 k2k

AAE Compute the displacements k1k1 k2k

Summary A procedure based on the modeling of a structure as a discrete set of elements – such as a truss- leads to construction of matrix equations of equilibrium These matrix equations contain all of the required displacement constraints Solution of the matrix equations gives us all the internal load information required to compute stresses and design the structure AAE 35223