In Prob. 3.118 , determine the permanent angle of twist of...

Key Concepts

Torsion in Structural Members

Torsion refers to the twisting of a structural member (such as a rod or shaft) when subjected to an applied torque. This concept covers how shear stresses are distributed across the cross-section of the member and how these stresses result in an angular displacement, known as the angle of twist, along its length.

Elastic and Plastic Deformation

Materials typically exhibit elastic behavior up to a yield point, where deformations are fully recoverable. Beyond this point, permanent, plastic deformation occurs. Understanding the distinction between these two phases is crucial when evaluating structural responses, as the elastic component results in recoverable twist whereas the plastic component results in a permanent, residual twist even after the load is removed.

Permanent (Residual) Angle of Twist

The permanent angle of twist refers to the residual rotation remaining in a member after the applied load has been removed, attributable to plastic (non-recoverable) deformations. Calculating this angle involves understanding the material's yield behavior, the distribution of stresses during loading, and the relationship between applied torque and deformation both in the elastic and plastic regimes.

Transcript

00:01 We need to find the length of this value a, given that the angle theta is equal to 45 degrees.

00:09 And to solve this problem, we're going to need to find equations for the different lengths on this diagram, x of a, x of b, y of a, and y of b.

00:20 So i went ahead and wrote out the equation for y of a, and we're going to start by taking a derivative of this y of a.

00:32 So the derivative of y of a with respect to x, that's going to be equal to the hyperbolic sign of x divided by c.

00:54 And we also know that the derivative of y of a with respect to x is equal to the tangent of theta.

01:09 So we're going to take these two expressions and set them equal to each other.

01:17 So we write down this first expression and then set it equal.

01:23 To the second expression.

01:27 And then we can take this, take this equation, and solve for x of a.

01:36 So x of a is equal to c times the inverse hyperbolic sign of the tangent of theta.

01:51 Now we're going to look back at the free body diagram.

01:55 And we notice that the length of the cable s, that's going to be equal to the length from a to c plus the length from c to b.

02:10 So i'm going to write that so we're going to write that s is equal to the length from a to c plus the length from c to b.

02:25 And we can also write that as 10, since you know what s is.

02:33 So you plug in 10 for s, and then ac is equal to c times the hyperbolic sign of x of a divided by c plus c b, which we can write as c times the hyperbic sign of x.

02:55 Of b divided by c and then we can take that equation and solve for x of b so we saw for x of b and that's going to be c times the inverse hyperbolic sign of 10 divided by c minus the hyperbolic sign of x of a divided by c so next i'm going to write out the equations for y of a and y of b so the length y of a is equal to c times the hyperbolic cosine of x of a divided by c and y of b is equal to c times the hyperbolic cosine of x of b divided by c and now we're going to look at the free body diagram and we notice that the tangent of this angle theta is going to be equal to this yf b minus y of a over this length x of a plus x of so we can then write that the tangent of theta is equal to yfb minus y of a divided by x of b plus x of a.

04:43 So to solve this problem, in order to solve this problem, we need to choose a value of c.

04:52 We need to choose some value of c and we plug it into this equation here, which i'll call equation one.

05:02 So we choose the value of c and that will give us a value for x of a.

05:13 So we can then take this value of x of a and plug it into this equation here using the same value of c and this will be equation 2 and then this will give us a value of x of b...

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