**Plane Stress** **6**

**Transformations**

ASEN 3112 Lecture 6 – Slide 1

**Plane Stress State**

**Recall that in a body in plane stress, the general 3D stress state with**
**9 components (6 independent) reduces to 4 components (3 independent): **

σ σ

σ τ τ

ττ* ^{xx}* τ

^{xy}*τ*

_{yy}

^{xz}*zz*
*yz*
*zy*
*zx*

*yx*

*σ**xx* *τ**x y*

*τ**x y*

*τ**yx*

*τ**yx*

*σ**yy*

0 0

0 0 0

**plane stress**

with =

**Plane stress occurs in thin plates and shells (e.g. aircraft & rocket skins, **
**parachutes, balloon walls, boat sails, ...) as well as thin wall structural **
**members in torsion.**

**In this Lecture we will focus on thin flat plates and associated**
**two-dimensional stress transformations**

ASEN 3112 Lecture 6 – Slide 2

**Flat Plate in Plane Stress**

*x*

*y* *z*

**Inplane dimensions: in x,y plane****Thickness dimension** **or transverse dimension**

**Top surface**

ASEN 3112 Lecture 6 – Slide 3

**Mathematical Idealization as ** **a Two Dimensional Problem**

*x*
*y*

### Midplane

### Plate

ASEN 3112 Lecture 6 – Slide 4

*x*

*y*
*z*

*x*

*x* *x*

*x*
*y*
**In-plane stresses**

σ σ

*xx*

τ = τ*xy* *yx* *yy*

*y*
**In-plane strains**
*h* *h*

*h*

ε ε

*xx*

γ = γ_{xy }_{yx}*yy*

*y*
*y*

**In-plane displacements**

*u* *u*

*x*

*y*

*h*

*y*

*y*
*x*

*x*
**In-plane internal forces**

**Thin plate in plane stress**

*p*_{xx}*p*_{xy}*p*_{yy}

**+ sign conventions**
**for internal forces,**
**stresses and strains**

*dx*

*dx*

*dy*

*dy*

*dx dy*

*dx dy* *dx dy*

*h*

**In-plane body forces**

*b* *b*

*x*

*y*

*dx dy*
*dx*
*dy*

**Internal Forces, Stresses, Strains**

ASEN 3112 Lecture 6 – Slide 5

** Stress Transformation in 2D**

*x*
*y*

*z*

*x*
*y*

*t*

*n*
*z*

*P* σ*xx*

τ*xy*

τ*yx*

σ*yy*

σ*nn*

τ*nt*

τ*tn*

σ*tt*

θ Global axes

*x,y stay fixed *

*Local axes n,t *
rotate by θ with
*respect to x,y *

*P*

**(a)** **(b)**

ASEN 3112 Lecture 6 – Slide 6

**Problem Statement**

**This transformation has two major uses:**

**Find stresses along a given skew direction**

** Here angle θ is given as data**

**Find max/min normal stresses, max in-plane shear **
**and overall max shear**

** Here finding angle θ is part of the problem**

**Plane stress transformation problem: **

**given **σ , σ , τ **and angle **θ

**express **σ , σ and τ** in terms of the data **

*xx* *yy* *xy*

*nn* *tt* *nt*

*x*
*y*
*t*

*n*
*z*

σ*nn*

τ*nt*

τ*tn*

σ*tt*

θ

*P*

ASEN 3112 Lecture 6 – Slide 7

**Analytical Solution**

2 2

2 2

2 2

σ = σ cos θ + σ sin θ + 2 τ sin θ cos θ σ = σ sin θ + σ cos θ − 2 τ sin θ cos θ τ = −(σ − σ )sin θ cos θ + τ (cos θ − sin θ)

*xx*
*xx*

*xx*

*yy*

*yy*
*yy*

*xy*
*xy*
*xy*
*nn*

*tt*
*nt*

σ + σ = σ + σ *nn* *xx* *yy*

For quick checks when θ is 0 or 90 , see Notes. The sum of the two transformed normal stresses

*is independent of the angle* θ**: it is called a stress invariant**

(mathematically, this is the trace of the stress tensor). A geometric interpretation using the Mohr's circle is immediate.

ο ο

*tt*

This is also called method of equations in Mechanics of Materials books. A derivation using the wedge method gives

ASEN 3112 Lecture 6 – Slide 8

**Double Angle Version**

σ = cos 2θ + τ sin 2θ τ = − sin 2θ + τ cos 2θ

*xy*

*xy*
*nn*

*nt*

2 2

σ + σ σ − σ *xx* *yy* *xx* *yy*

+
2
σ − σ *xx* *yy*

Using double-angle trig relations such as cos 2θ = cos θ - sin θ and sin 2θ = 2 sin θ cos θ, the transformation equations may be rewritten as

2 2

Here σ*tt* is omitted since it may be easily recovered as σ + σ − σ *xx* *yy* *nn*

ASEN 3112 Lecture 6 – Slide 9

**Principal Stresses: Terminology**

**The max and min values taken by the in-plane normal stress **σ
when viewed as a function of the angle θ are called principal stresses
(more precisely, principal in-plane normal stresses, but qualifiers

"in-plane" and "normal" are often omitted).

**The planes on which those stresses act are the principal planes.**

**The normals to the principal planes are contained in the x,y plane.**

They are called the principal directions.

The θ* angles formed by the principal directions and the x axis are*
called the principal angles.

*nn*

ASEN 3112 Lecture 6 – Slide 10

**Principal Angles**

To find the principal angles, set the derivative of σ with respect to θ to zero. Using the double-angle version,

*nn*

*nn*

*xx*

*xy*
*xx*
*yy*

*p* *yy*

*p1*

*p1*
*p2*

*p2*

*p*

*d *σ

*d *θ *= * (σ − σ ) sin 2θ + 2τ cos 2θ = 0 * *
*This is satisfied for *θ = θ if

tan 2θ = 2 τ σ − σ

It can be shown that (*) provides two principal double angles, 2θ and 2θ , within the range of interest, which is [0, 360 ] or [−180 ,180 ] (range conventions vary between textbooks).

The two values differ by 180 . On dividing by 2 we get the principal
angles θ and θ **that differ by 90 ** . Consequently the

**two principal directions are orthogonal. **

o o o

**o **

**o **

(*)

ASEN 3112 Lecture 6 – Slide 11

**Principal Stress Values**

*nn*

Replacing the principal angles given by (*) of the previous slide into the expression for σ and using trig identities, we get

2

σ _{1,2} = + τ 2 _{xy}

1,2

2
σ + σ *xx* *yy*

2

σ − σ *xx* *yy*

in which σ denote the principal normal stresses. Subscripts 1 and 2 correspond to taking the + and − signs, respectively, of the square root.

A staged procedure to compute these values is described in the next slide.

ASEN 3112 Lecture 6 – Slide 12

**Staged Procedure To Get Principal Stresses**

1. Compute

Meaning: σ is the average normal stress (recall that σ + σ is an
invariant and so is σ* ), whereas R is the radius of Mohr's circle *
*described later. This R also represents the maximum in-plane shear *
value, as discussed in the Lecture notes.

2. The principal stresses are

σ = σ + *R , *σ = σ − *R*

3. The above procedure bypasses the computation of principal angles.

Should these be required to find principal directions, use equation (*) of
**the Principal Angles slide.**

2 2

σ av* = , R = + * + τ *xy*

1 av 2 av

av

av

2
σ + σ *xx*

*xx*
*yy*

*yy*

2

σ − σ *xx* *yy*

ASEN 3112 Lecture 6 – Slide 13

**Additional Properties**

1. The in-plane shear stresses on the principal planes vanish

2. The maximum and minimum in-plane shears are +R and −*R,*
respectively

3. The max/min in-plane shears act on planes located at +45 and -45 from the principal planes. These are the principal shear planes

4. A principal stress element (used in some textbooks) is obtained by drawing a triangle with two sides parallel to the principal planes and one side parallel to a principal shear plane

For further details, see Lecture notes. Some of these properties can be
visualized more easily using the Mohr's circle, which provides a
**graphical solution to the plane stress transformation problem **

ASEN 3112 Lecture 6 – Slide 14

**Numeric Example**

*x*

*x*
*y*

*x*
*t* *y*

*n*
*P* σ * _{xx}*=100 psi

τ *xy*= τ * _{yx}*=30 psi
σ

*= 20 psi*

_{yy}θ

θ 1=18.44 θ 2= 108.44

*x*

|τ |_{max} *= R=50 psi*
18.44 +45
= 63.44

principal planes principal directions

principal planes

principal planes

**(a)** **(b)** **(c)**

*P*

**(d)**
σ_{1} =110 psi
σ2 =10 psi

*P*

plane of max inplane shear

principal stress element

*P* ^{45}

45

For computation details see Lecture notes

ASEN 3112 Lecture 6 – Slide 15

**Graphical Solution of Example Using Mohr's Circle**

*x*
*y*

*P* σ * _{xx}*=100 psi
τ

*xy*= τ

*=30 psi σ*

_{yx}*= 20 psi*

_{yy}**(a)**

σ = normal
** stress**
τ = shear

** stress**

0 20 40 60 80 100 50

40 30 20 10 0

−10

−20

−30

−40

−50

H

V C

σ = 110_{1}
σ = 10

2

coordinates of blue points are H: (20,30), V:(100,-30), C:(60,0)

τ = min −50
τ = 50_{max}

**Radius R = 50**

2θ = 36.881

2θ _{2}= 36.88 +180 = 216.88

**(b) Mohr's circle**
**(a) Point in plane stress**

ASEN 3112 Lecture 6 – Slide 16

**What Happens in 3D?**

**This topic be briefly covered in class if time allows,** **using the following slides.**

**If not enough time, ask students to read Lecture notes** **(Sec 7.3), with particular emphasis on the computation ** **of the overall maximum shear**

ASEN 3112 Lecture 6 – Slide 17

**General 3D Stress State**

### σ

### σ , σ , σ

### σ σ

### τ τ

### τ τ

^{xx}### τ

^{xy}

_{yy}### τ

^{xz}*zz*
*yz*
*zx* *zy*

*yx*

**There are three (3) principal stresses, identified as**

1 2 3

ASEN 3112 Lecture 6 – Slide 18

**Principal Stresses in 3D (2)**

**The ** σ ** turn out to be the eigenvalues of the stress matrix.**

**They are the roots of a cubic polynomial (the so-called** **characteristic polynomial)**

**The principal directions are given by the eigenvectors ** **of the stress matrix.**

**Both eigenvalues and eigenvectors can be numerically ** **computed by the Matlab function eig(.)**

### σ −σ

### σ −σ

### σ −σ τ τ

### τ

### τ

^{xx}### τ τ

*xy* *xz*

*yy*

*zz*
*yz*
*zy*

*zx*

### C (σ) = det

*yx*

*i*

### = −σ +

^{3}

* I *

_{1}

### σ −

^{2}

* I* σ +

_{2}

*I*

_{3}

### = 0

ASEN 3112 Lecture 6 – Slide 19

**3D Mohr Circles **

**(Yes, There Is More Than One)**

**All possible stress states **
**at the material point lie**
**on the grey shaded**
**area between the outer**
**and inner circles**

σ **= normal **
** stress**
τ** = shear **

** stress**

**Principal stress** σ _{1}
**Outer Mohr's **

**circle**
**Inner Mohr's **

**circles**

σ
**Principal stress** σ 2

3

**Overall**
**+ max shear**

**The overall maximum shear, which is the radius of the**
**outer Mohr's circle, is important for assessing strength **
**safety of ductile materials**

ASEN 3112 Lecture 6 – Slide 20

**The Overall Maximum Shear is** **the Radius of the Outer Mohr's Circle**

2

**If the principal stresses are algebraically ordered as** **then **

**Note that the intermediate principal stress ** σ ** does not appear.**

**If they are not ordered it is necessary to use the max function ** **in a more complicated formula that picks up the largest **

**of the three radii: **

2 3

### σ σ σ

1 1 3### σ − σ τ

^{overall}

_{max}*= R = *

_{outer}### 2

2

### σ − σ

1### 2

3

### σ − σ

2### 2

3 1

### σ − σ τ

^{overall}

_{max}*= max , , * 2

ASEN 3112 Lecture 6 – Slide 21

### σ , σ , σ

_{1}

_{2}

### =0

**where** σ **and ** σ **are the inplane principal stresses obtained** **as described earlier in Lecture 6.**

1 2

3

**Consider plane stress but now account for the third dimension. **

**One of the principal stresses, call it for the moment ** σ ,

_{3}

** is zero:**

**Plane Stress in 3D: The 3rd Principal Stress**

**The zero principal stress ** σ ** is aligned with the z axis (the ** **thickness direction) while ** σ ** and **

**is aligned with the z axis (the**

_{1}

### σ

_{2}

** act in the x,y plane:**

**act in the x,y plane:**

3

*x*

*y*
*z*

### σ

1### σ

^{2}

### σ

3Thin plate

ASEN 3112 Lecture 6 – Slide 22

**Plane Stress in 3D: Overall Max Shear**

**Let us now (re)order the principal stresses by algebraic value as**

**(A) Inplane principal stresses have opposite signs. Then the** ** zero stress is the intermediate one: ** σ ** , and **

_{2}

**To compute the overall maximum shear 2 cases are considered:**

1 3

### σ − σ 2

1 2 3

### σ

1### 2

**(B) Inplane principal stresses have the same sign. Then** **If ** σ ** ** σ ** 0 and ** σ ** = 0, ** τ

^{overall}

_{max}### =

2

### σ

3**If ** σ ** **

_{3}

### σ ** 0 and ** σ ** = 0, **

_{1}

### τ

^{overall}

_{max}### = − 2 τ

^{overall}

_{max}### = τ

^{inplane}

_{max}### =

2 3

### σ σ σ

1ASEN 3112 Lecture 6 – Slide 23

**Plane Stress in 3D: Example**

σ = normal
** stress**
τ ** = shear **

** stress**

0 20 40 60 80 100 50

40 30 20 10 0

−10

−20

−30

−40

−50

σ = 110_{1}
τ = _{max} 50

σ = 10_{2}
σ = 0

3

inplane

τ = _{max}^{overall} 55

**Yellow-filled circle is the in-plane Mohr's circle**

*x*
*y*

*P* σ * _{xx}*=100 psi
τ

*xy*= τ

*=30 psi σ*

_{yx}*= 20 psi*

_{yy}**Plane stress example **
**treated earlier:**

ASEN 3112 Lecture 6 – Slide 24