Module Content:

Module Reading, Problems, and Demo:

MAE 2310 Str. of Materials © E. J. Berger, 2010 18- 1

Module 18: Combined StressApril 7, 2010

1. Pressure vessels are structurally important components which are capable (through clever design enabled by an understanding of the stresses) of containing very highly pressurized gases and liquids, or preventing pressure-related collapse of a structure.2. Pressure vessels offer us our first opportunity to consider multiple stress components simultaneously, and this will lead to our study of combined stresses.

Reading: Sections 8.1, 8.2Problems: Ex. 8.2, Prob. 8-15, 8-17Demo: noneTechnology: http://pages.shanti.virginia.edu/som2010

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Concept: Stress at a Point• all of our stress analysis so far has focused on the idea of stress at a point being composed of a total of six

components (a “general” state of stress)

• these stress components all act at a point, although we usually visualize the stresses on a “cube” of material which has differential volume dV (i.e., is a point in space in the limit)

• these stresses are a bit like vectors in the sense that we cannot simply add up different flavors of stress

2

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Concept: Stress at a Point• all of our stress analysis so far has focused on the idea of stress at a point being composed of a total of six

components (a “general” state of stress)

• these stress components all act at a point, although we usually visualize the stresses on a “cube” of material which has differential volume dV (i.e., is a point in space in the limit)

• these stresses are a bit like vectors in the sense that we cannot simply add up different flavors of stress

2

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Concept: Stress at a Point• all of our stress analysis so far has focused on the idea of stress at a point being composed of a total of six

components (a “general” state of stress)

• these stress components all act at a point, although we usually visualize the stresses on a “cube” of material which has differential volume dV (i.e., is a point in space in the limit)

• these stresses are a bit like vectors in the sense that we cannot simply add up different flavors of stress

2

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Concept: Stress at a Point• all of our stress analysis so far has focused on the idea of stress at a point being composed of a total of six

components (a “general” state of stress)

• these stress components all act at a point, although we usually visualize the stresses on a “cube” of material which has differential volume dV (i.e., is a point in space in the limit)

• these stresses are a bit like vectors in the sense that we cannot simply add up different flavors of stress

2

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Concept: Stress at a Point• all of our stress analysis so far has focused on the idea of stress at a point being composed of a total of six

components (a “general” state of stress)

• these stress components all act at a point, although we usually visualize the stresses on a “cube” of material which has differential volume dV (i.e., is a point in space in the limit)

• these stresses are a bit like vectors in the sense that we cannot simply add up different flavors of stress

2

Question: how do we know if one set of stresses is more “severe” than another set?

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Concept: Stress at a Point• all of our stress analysis so far has focused on the idea of stress at a point being composed of a total of six

components (a “general” state of stress)

• these stress components all act at a point, although we usually visualize the stresses on a “cube” of material which has differential volume dV (i.e., is a point in space in the limit)

• these stresses are a bit like vectors in the sense that we cannot simply add up different flavors of stress

2

Question: how do we know if one set of stresses is more “severe” than another set?

Answer: we develop a way of combining all these stresses into a simple characterization that allows us to compare different stress states.

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Concept: Pressure Vessel• a thin walled pressure vessel is often used to contain pressurized fluid or gas and is subject to stresses in all

directions simultaneously

• thin walled: the wall thickness is small compared to the radius of the vessel

• pressurized fluid or gas: internal pressure is different from ambient

• stresses in all directions: the wall of the pressure vessel is “stretched” in all directions

3

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Concept: Pressure Vessel• a thin walled pressure vessel is often used to contain pressurized fluid or gas and is subject to stresses in all

directions simultaneously

• thin walled: the wall thickness is small compared to the radius of the vessel

• pressurized fluid or gas: internal pressure is different from ambient

• stresses in all directions: the wall of the pressure vessel is “stretched” in all directions

3

t << r

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Concept: Pressure Vessel• a thin walled pressure vessel is often used to contain pressurized fluid or gas and is subject to stresses in all

directions simultaneously

• thin walled: the wall thickness is small compared to the radius of the vessel

• pressurized fluid or gas: internal pressure is different from ambient

• stresses in all directions: the wall of the pressure vessel is “stretched” in all directions

3

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Concept: Pressure Vessel• a thin walled pressure vessel is often used to contain pressurized fluid or gas and is subject to stresses in all

directions simultaneously

• thin walled: the wall thickness is small compared to the radius of the vessel

• pressurized fluid or gas: internal pressure is different from ambient

• stresses in all directions: the wall of the pressure vessel is “stretched” in all directions

3

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Cylindrical Pressure Vessels• first, we make the case that we can consider two normal stresses in orthogonal directions

• the “hoop” direction

• the “axial” direction:

4

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Cylindrical Pressure Vessels• first, we make the case that we can consider two normal stresses in orthogonal directions

• the “hoop” direction

• the “axial” direction:

4

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Cylindrical Pressure Vessels• first, we make the case that we can consider two normal stresses in orthogonal directions

• the “hoop” direction

• the “axial” direction:

4

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Cylindrical Pressure Vessels• first, we make the case that we can consider two normal stresses in orthogonal directions

• the “hoop” direction

• the “axial” direction:

4

2 [!1(tdy)]

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Cylindrical Pressure Vessels• first, we make the case that we can consider two normal stresses in orthogonal directions

• the “hoop” direction

• the “axial” direction:

4

2 [!1(tdy)]!p(2r)dy

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Cylindrical Pressure Vessels• first, we make the case that we can consider two normal stresses in orthogonal directions

• the “hoop” direction

• the “axial” direction:

4

2 [!1(tdy)]!p(2r)dy = 0

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Cylindrical Pressure Vessels• first, we make the case that we can consider two normal stresses in orthogonal directions

• the “hoop” direction

• the “axial” direction:

4

2 [!1(tdy)]!p(2r)dy = 0

!1 =

pr

t

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Cylindrical Pressure Vessels• first, we make the case that we can consider two normal stresses in orthogonal directions

• the “hoop” direction

• the “axial” direction:

4

!1 =

pr

t

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Cylindrical Pressure Vessels• first, we make the case that we can consider two normal stresses in orthogonal directions

• the “hoop” direction

• the “axial” direction:

4

!1 =

pr

t

!2(2"rt)

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Cylindrical Pressure Vessels• first, we make the case that we can consider two normal stresses in orthogonal directions

• the “hoop” direction

• the “axial” direction:

4

!1 =

pr

t

!2(2"rt) !p!r2

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Cylindrical Pressure Vessels• first, we make the case that we can consider two normal stresses in orthogonal directions

• the “hoop” direction

• the “axial” direction:

4

!1 =

pr

t

!2(2"rt) !p!r2= 0

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Cylindrical Pressure Vessels• first, we make the case that we can consider two normal stresses in orthogonal directions

• the “hoop” direction

• the “axial” direction:

4

!1 =

pr

t

!2(2"rt) !p!r2= 0

!2 =pr

2t

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Spherical Pressure Vessels• because of complete axisymmetry, we have only one normal stress component

5

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Spherical Pressure Vessels• because of complete axisymmetry, we have only one normal stress component

5

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Spherical Pressure Vessels• because of complete axisymmetry, we have only one normal stress component

5

!2(2"rt)

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Spherical Pressure Vessels• because of complete axisymmetry, we have only one normal stress component

5

!2(2"rt)!p!r2

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Spherical Pressure Vessels• because of complete axisymmetry, we have only one normal stress component

5

!2(2"rt)!p!r2= 0

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Spherical Pressure Vessels• because of complete axisymmetry, we have only one normal stress component

5

!2(2"rt)!p!r2= 0

!2 �pr

�t

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Remarks• we have assumed that these pressure vessels are thin, and the implication is that the radial stress through the

thickness of the wall is negligible (in fact, it is substantially smaller than either the hoop or axial stress)

• this is the first time we have seen a problem with two normal stresses in two different directions

• clearly we can calculate the two separately

• we can apply this idea to any problem with multiple loading types (axial, bending, torsion, etc.)

• in this section, we are neglecting the possibility of external pressure higher than internal pressure, in which case the stresses are compressive and for which we must consider potential buckling (Ch. 13)

• failure mechanisms for pressure vessels include over-pressurization and rupture...how does this look?

6

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Combined Loading by Example• Example 8.2: determine the state of stress at locations B and C

7

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Combined Loading by Example• Example 8.2: determine the state of stress at locations B and C

7

① FBD/method of sections at B-C

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Combined Loading by Example• Example 8.2: determine the state of stress at locations B and C

7

① FBD/method of sections at B-C

② internal reactions at B-C

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Combined Loading by Example• Example 8.2: determine the state of stress at locations B and C

7

① FBD/method of sections at B-C

② internal reactions at B-C

③ Q: what types of stress?

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Combined Loading by Example• Example 8.2: determine the state of stress at locations B and C

7

① FBD/method of sections at B-C

② internal reactions at B-C

③ Q: what types of stress?

④ stress calculations

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Combined Loading by Example• Example 8.2: determine the state of stress at locations B and C

7

① FBD/method of sections at B-C

② internal reactions at B-C

③ Q: what types of stress?

④ stress calculations

(a) axial (normal, P/A)

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Combined Loading by Example• Example 8.2: determine the state of stress at locations B and C

7

① FBD/method of sections at B-C

② internal reactions at B-C

③ Q: what types of stress?

④ stress calculations

(a) axial (normal, P/A)(b) bending (normal, My/I)

MAE 2310 Str. of Materials © E. J. Berger, 2010 18-

Theory: Combined Loading by Example• Example 8.2: determine the state of stress at locations B and C

7

① FBD/method of sections at B-C

② internal reactions at B-C

③ Q: what types of stress?

④ stress calculations

(a) axial (normal, P/A)(b) bending (normal, My/I)