AAE 352 Lecture 08 Matrix methods Part 1

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AAE 352 Lecture 08 Matrix methods - Part 1 Reading Chapter 4. 1 through

AAE 352 Lecture 08 Matrix methods - Part 1 Reading Chapter 4. 1 through 4. 5 Matrix methods for structural analysis AAE 352 1

Objectives • Introduce the matrix stiffness method used to analyze aerospace structures • Provide

Objectives • Introduce the matrix stiffness method used to analyze aerospace structures • Provide definitions & concepts – Matrix operations – Node points – Stiffness matrix AAE 352 2

Product development and operation • • • The “fuzzy front” – A need –

Product development and operation • • • The “fuzzy front” – A need – New discovery – New capability Generation of high level “mission” or operational details and “paper napkin” concepts Development of “systems level requirements” Creation of viable concept Permission to proceed Design is heavily involved in the front end

We need accurate static and dynamic analysis of complex structures to estimate stresses, deflections

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 352 4

Nodes and degrees of freedom definitions A node is a point on a structural

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 352 5

Statically indeterminate structure Computing internal forces requires knowledge of structural stiffness-here is an example

Statically indeterminate structure Computing internal forces requires knowledge of structural stiffness-here is an example Po Po F 1 F 2 F 2 F 1 AAE 352 6

Displacement compatibility between elements must be preserved – some points must move together u

Displacement compatibility between elements must be preserved – some points must move together u 1=0 u 1 u 3 u 2=u 3 u 1 u 4=0 u 4 u 2 AAE 352 7

Use element displacement constraints to get the additional equation we need to solve for

Use element displacement constraints to get the additional equation we need to solve for the internal forces u 1=0 u 1 u 3 u 2=u 3 u 1 u 4=0 u 4 u 2 AAE 352 8

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

Internal forces depend on the stiffness of elements and structural displacements Displacement Compatibility AAE 352 10

Final results - there are many forces that satisfy equilibrium, but only two that

Final results - there are many forces that satisfy equilibrium, but only two that allow the common points to move together F 1 F 2 is compressive F 1 is tensile This approach is hard to apply if there are many different elements AAE 352 10

Here is the matrix method approach bar elements (also called rods) extend or contract

Here is the matrix method approach bar elements (also called rods) extend or contract like springs F = kx k = EA/L F F x AAE 352 11

The element stiffness matrix is at the heart of an equation that relates nodal

The element stiffness matrix is at the heart of an equation that relates nodal forces (F’s) to nodal displacements (d’s) d 11 F 11 d 12 L 2 1 F 12 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) AAE 352 12

The element stiffness matrix relates internal nodal forces to internal nodal displacements d 11

The element stiffness matrix relates internal nodal forces to internal nodal displacements d 11 F 11 L 1 d 12 d 21 F 12 F 21 AAE 352 L 2 d 22 F 22 13

The system stiffness matrix relates external nodal forces to external nodal displacementsglobal systems are

The system stiffness matrix relates external nodal forces to external nodal displacementsglobal systems are related to the local system d 11 L 1 d 12 d 21 F 12 F 11 L 2 d 22 F 21 P 1 u 3 u 2 P 2 AAE 352 P 3 14

P 1 Write nodal(joint) equilibrium equations between applied forces (the P’s) and internal forces

P 1 Write nodal(joint) equilibrium equations between applied forces (the P’s) and internal forces (the F’s) P 3 P 2 F 11 F 22 F 21 P 2 F 11 F 12 F 21 AAE 352 P 3 F 22 15

Write nodal force equilibrium equations in terms of system displacements P 1 P 2

Write nodal force equilibrium equations in terms of system displacements P 1 P 2 F 11 F 12 F 21 AAE 352 P 3 F 22 16

Two element result – the “global stiffness matrix” k 1 k 2 In Lecture

Two element result – the “global stiffness matrix” k 1 k 2 In Lecture 8 we will show in more detail how to assemble this matrix from the elemental matrices AAE 352 17

Putting in the structural boundary constraints k 1 k 2 AAE 352 18

Putting in the structural boundary constraints k 1 k 2 AAE 352 18

Now solve for displacements and reactions 1 k 1 2 AAE 352 k 2

Now solve for displacements and reactions 1 k 1 2 AAE 352 k 2 3 19

This requires that we know how to take the matrix inverse 1 k 1

This requires that we know how to take the matrix inverse 1 k 1 2 k 2 3 This determinant is singular if we don’t include constraints AAE 352 20

Compute the reaction force if the two elements are fixed on the right 1

Compute the reaction force if the two elements are fixed on the right 1 k 1 2 AAE 352 k 2 3 21

Compute the displacements 1 k 1 2 AAE 352 k 2 3 22

Compute the displacements 1 k 1 2 AAE 352 k 2 3 22

Summary • A procedure based on the modeling of a structure as a discrete

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 352 23