A horizontal elastic beam is supported at each end and is subjected to forces at points 1, 2, and 3, as shown in Figure 1. Let f in R^3 list the forces at these points, and let y in R^3 list the amounts of deflection (that is, movement) of the beam at the three points. Using Hooke’s law from

Question

A horizontal elastic beam is supported at each end and is subjected to forces at points 1, 2, and 3, as shown in Figure 1. Let f in $\mathbb{R} ^{3}$ list the forces at these points, and let y in $\mathbb{R} ^{3}$ list the amounts of deflection (that is, movement) of the beam at the three points. Using Hooke’s law from physics, it can be shown that

y = Df

where D is a flexibility matrix. Its inverse is called the stiffness matrix. Describe the physical significance of the columns of D and $D^{-1}$ .

y = Df

where D is a flexibility matrix. Its inverse is called the stiffness matrix. Describe the physical significance of the columns of D and [latex] D^{-1} [/latex].

Accepted Answer

Write [latex] I_{3} [/latex] = [ [latex] e_{1} e_{2} e_{3} [/latex] ] and observe that

D = D[latex] I_{3} [/latex] = [ [latex] De_{1} De_{2} De_{3} [/latex] ]

Interpret the vector [latex] e_{1} [/latex] = (1, 0, 0) as a unit force applied downward at point 1 on the beam (with zero force at the other two points). Then [latex] De_{1} [/latex], the first column of D, lists the beam deflections due to a unit force at point 1. Similar descriptions apply to the second and third columns of D .

To study the stiffness matrix [latex] D^{-1} [/latex], observe that the equation f = [latex] D^{-1} [/latex]y computes a force vector f when a deflection vector y is given. Write

[latex] D^{-1} = D^{-1}I_{3} = [ D^{-1}e_{1} D^{-1}e_{2} D^{-1}e_{3} ] [/latex]

Now interpret [latex] e_{1} [/latex] as a deflection vector. Then [latex] D^{-1}e_{1} [/latex] lists the forces that create the deflection. That is, the first column of [latex] D^{-1} [/latex] lists the forces that must be applied at the three points to produce a unit deflection at point 1 and zero deflections at the other points. Similarly, columns 2 and 3 of [latex] D^{-1} [/latex] list the forces required to produce unit deflections at points 2 and 3, respectively. In each column, one or two of the forces must be negative (point upward) to produce a unit deflection at the desired point and zero deflections at the other two points. If the flexibility is measured, for example, in inches of deflection per pound of load, then the stiffness matrix entries are given in pounds of load per inch of deflection.

Question 2.2.3: A horizontal elastic beam is supported at each end and is su...

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