Prove that if we apply a unit force to node i in a structure...

Prove that if we apply a unit force to node $i$ in a structure and measure the displacement of node $j$ in the direction of the force, then we obtain the same value if we apply the force to node $j$ and measure the displacement at node $i$ in the same direction. Hint: First, solve Exercise 6.1.6.

Key Concepts

Reciprocity Theorem

The reciprocity theorem in structural mechanics states that in a linear elastic system, the response (displacement) at one point due to a unit load applied at another point is equal to the response at the latter point when the same load is applied at the former. This concept underpins many fundamental analyses in structural engineering and helps to explain how forces are distributed symmetrically across a structure.

Symmetry of the Stiffness Matrix

In linear elastic systems, the stiffness matrix, which relates nodal displacements and applied forces, is symmetric. This symmetry is a mathematical expression of the physical fact that the influence between two nodes is reciprocal, meaning that the displacement at node j from a unit force at node i is the same as the displacement at node i from the same unit force applied at node j.

Principle of Virtual Work

The principle of virtual work is an energy-based method used in the analysis of mechanical systems, which states that the work done by external forces during any virtual displacement is equal to the internal work done in the system. This concept is crucial in proving symmetry and reciprocity properties in structures, as it provides a fundamental link between forces and displacements in equilibrium.

Transcript

00:01 Okay, so in this problem, we have a support and we have a force f acting on it.

00:06 And we're told that the x, y component, or the force in the xy plane of this force f, is 7 kilonutons.

00:15 So you can imagine that as this little blue shadow that i drew underneath f.

00:20 And we're told that the shadow force, right, or the xy component of the force, forms a 40 degree angle with the x axis and a 30 degree.

00:30 Angle with the force f itself.

00:33 So that's the red and green angles that i've drawn in this little diagram here.

00:37 And so we have to use those to find the magnitude and the coordinate direction angles of the force f.

00:44 So to do that, you can actually first imagine turning this 3d plane or 3d environment into a 2d plane.

00:52 Right.

00:52 So we're actually going to draw a second little diagram of the x and y plane.

00:59 Right.

00:59 So now we're 2d again.

01:03 And our triangle is going to be somewhere in, or our forest line is going to be somewhere here.

01:09 Right? and so we have an angle here, and we actually can determine two values from this.

01:18 We can determine this value here and this value here.

01:21 So we can find fx and fy.

01:25 So to find fx, we just know that it's equal to the magnitude.

01:32 In this case, our magnitude, is 7 kilonutons, right? because it's the shadow, and it actually gives us the magnitude of the little shadow line.

01:44 So 7 kilonutons times for the y value, it's actually going to be cosine of 40 degrees.

01:55 For the x value, it's going to be cosine of 40 degrees.

02:00 And if you plug this into a calculator, you should get about 5, 3, 6...

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