Understanding the Direction of Rotation for Stress State Equations

  • Thread starter Master1022
  • Start date
  • #1
231
35

Summary:

If we are given a set of stress state equations and we want to 'rotate' them, which direction shall I use?

Main Question or Discussion Point

Hi,

The context for my question is: A thin plate, which lies in the [itex] x-y [/itex] plane, contains a small hole of radius [itex] a [/itex]. Consider a polar co-ordinate system r,􏰆 with its origin at the centre of the hole and 􏰆defined as the angle that a radial line makes with the x-axis. A uniform uniaxial tensile stress 􏰋[itex] \sigma^{app}_x [/itex] is applied to the plate in the x-direction. The stress state in the plate is:
Screen Shot 2020-01-08 at 6.04.27 PM.png


Determine the state of stress in a plate that is subjected to a uniform uniaxial tensile stress 􏰋[itex] \sigma^{app}_y [/itex] which is aligned with the y-direction.

My Method:
I defined some new angle [itex] \phi = \theta + 90 [/itex] (CCW) (with the main point being that I chose to define it with a +). Following on, I got that: [itex] \sin(2(\theta + 90)) = -\sin(\theta) [/itex] and that [itex] \cos(2(\theta + 90)) = -\cos(\theta) [/itex]. Then we just substitute those in to the three equations and change [itex] \sigma^{app}_x [/itex] to [itex] \sigma^{app}_y [/itex] to get the answer. The answers come out the same in this instance regardless of the direction of definition of the new angle. However, in later parts of the question, we basically follow the same method for rotating the state by 45 degrees, and the direction makes a difference there (some signs are different).

My question: Why has the solution chosen to define the 'new' [itex] \theta_{new} = \theta_{old} - 90 [/itex] (CW)?

(Note: I only posted the earlier question as the same exact concept applies to the later one, but there are more parts between now and then which would just unnecessarily add to the reading time)
 
Last edited:

Answers and Replies

  • #3
231
35
Are you familiar with dyadic tensor notation?
Thank you for your reply. I am not familiar with it. I have just done a quick search to see what it is, but I haven't seen it in my courses before.
 

Related Threads on Understanding the Direction of Rotation for Stress State Equations

Replies
1
Views
774
  • Last Post
Replies
1
Views
5K
Replies
4
Views
3K
  • Last Post
Replies
14
Views
386
  • Last Post
Replies
1
Views
7K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
1
Views
22K
  • Last Post
Replies
8
Views
8K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
6
Views
9K
Top