Module Content:

Module Reading, Problems, and Demo:

MAE 2310 Str. of Materials © E. J. Berger, 2010 7- 1

Module 7: Axial LoadingFebruary 3, 2010

1. Axial loading results in normal stress and normal strain, which may be functions of position on the structure.2. Several common loading scenarios simplify the axial load analysis into an easy-to-remember formula relating load and deformation.

Reading: Sections 4.1, 4.2Problems: Prob. 4-5, 4-8Demo: noneTechnology: http://pages.shanti.virginia.edu/som2010

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Concept: Axial Loading• axial loading refers to a structural loading scenario in which the external applied load acts in the direction of the

long axis of the structure

• Ex.: a material tensile test, a massless rope or cable

• this external applied load results in a normal stress (like in Ch. 1) and normal strain (like in Ch. 2) which can be calculated using the “average” equations we have already seen

2

undeformed

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Concept: Axial Loading• axial loading refers to a structural loading scenario in which the external applied load acts in the direction of the

long axis of the structure

• Ex.: a material tensile test, a massless rope or cable

• this external applied load results in a normal stress (like in Ch. 1) and normal strain (like in Ch. 2) which can be calculated using the “average” equations we have already seen

2

undeformed deformed

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Concept: Axial Loading• axial loading refers to a structural loading scenario in which the external applied load acts in the direction of the

long axis of the structure

• Ex.: a material tensile test, a massless rope or cable

• this external applied load results in a normal stress (like in Ch. 1) and normal strain (like in Ch. 2) which can be calculated using the “average” equations we have already seen

2

undeformed deformed

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Concept: Axial Loading• axial loading refers to a structural loading scenario in which the external applied load acts in the direction of the

long axis of the structure

• Ex.: a material tensile test, a massless rope or cable

• this external applied load results in a normal stress (like in Ch. 1) and normal strain (like in Ch. 2) which can be calculated using the “average” equations we have already seen

2

undeformed deformed

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Concept: Axial Loading• axial loading refers to a structural loading scenario in which the external applied load acts in the direction of the

long axis of the structure

• Ex.: a material tensile test, a massless rope or cable

• this external applied load results in a normal stress (like in Ch. 1) and normal strain (like in Ch. 2) which can be calculated using the “average” equations we have already seen

2

undeformed deformed

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Concept: Saint-Venant’s Principle

3

http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Saint-Venant.html

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Concept: Saint-Venant’s Principle

3

http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Saint-Venant.html

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Concept: Saint-Venant’s Principle

3

http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Saint-Venant.html

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Concept: Saint-Venant’s Principle

3

http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Saint-Venant.html

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Theory: Axial (Elastic) Deformation• consider a bar in equilibrium, acted upon by two end loads P1 and P2, and a distributed axial load along its

length

• the cross sectional area A may be a function of the spatial coordinate x, i.e., A=A(x)

• upon loading, the total deformation is δ

4

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Theory: Axial (Elastic) Deformation• consider a bar in equilibrium, acted upon by two end loads P1 and P2, and a distributed axial load along its

length

• the cross sectional area A may be a function of the spatial coordinate x, i.e., A=A(x)

• upon loading, the total deformation is δ

4

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Theory: Axial (Elastic) Deformation• consider a bar in equilibrium, acted upon by two end loads P1 and P2, and a distributed axial load along its

length

• the cross sectional area A may be a function of the spatial coordinate x, i.e., A=A(x)

• upon loading, the total deformation is δ

4

P (x)

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Theory: Axial (Elastic) Deformation• consider a bar in equilibrium, acted upon by two end loads P1 and P2, and a distributed axial load along its

length

• the cross sectional area A may be a function of the spatial coordinate x, i.e., A=A(x)

• upon loading, the total deformation is δ

4

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Theory: Axial (Elastic) Deformation• consider a bar in equilibrium, acted upon by two end loads P1 and P2, and a distributed axial load along its

length

• the cross sectional area A may be a function of the spatial coordinate x, i.e., A=A(x)

• upon loading, the total deformation is δ

4

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Theory: Axial (Elastic) Deformation• consider a bar in equilibrium, acted upon by two end loads P1 and P2, and a distributed axial load along its

length

• the cross sectional area A may be a function of the spatial coordinate x, i.e., A=A(x)

• upon loading, the total deformation is δ

4

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Theory: Axial (Elastic) Deformation• consider a bar in equilibrium, acted upon by two end loads P1 and P2, and a distributed axial load along its

length

• the cross sectional area A may be a function of the spatial coordinate x, i.e., A=A(x)

• upon loading, the total deformation is δ

4

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Theory: Axial (Elastic) Deformation• consider a bar in equilibrium, acted upon by two end loads P1 and P2, and a distributed axial load along its

length

• the cross sectional area A may be a function of the spatial coordinate x, i.e., A=A(x)

• upon loading, the total deformation is δ

4

!(x) =P (x)

A(x)stress-load:

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Theory: Axial (Elastic) Deformation• consider a bar in equilibrium, acted upon by two end loads P1 and P2, and a distributed axial load along its

length

• the cross sectional area A may be a function of the spatial coordinate x, i.e., A=A(x)

• upon loading, the total deformation is δ

4

!(x) =P (x)

A(x)stress-load:

!(x) =d"

dxstrain-displacement:

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Theory: Axial (Elastic) Deformation• consider a bar in equilibrium, acted upon by two end loads P1 and P2, and a distributed axial load along its

length

• the cross sectional area A may be a function of the spatial coordinate x, i.e., A=A(x)

• upon loading, the total deformation is δ

4

!(x) =P (x)

A(x)stress-load:

!(x) =d"

dxstrain-displacement:

!(x) = E"(x)stress-strain:

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Theory: Axial (Elastic) Deformation• consider a bar in equilibrium, acted upon by two end loads P1 and P2, and a distributed axial load along its

length

• the cross sectional area A may be a function of the spatial coordinate x, i.e., A=A(x)

• upon loading, the total deformation is δ

4

!(x) =P (x)

A(x)stress-load:

!(x) =d"

dxstrain-displacement:

!(x) = E"(x)stress-strain:

d! =P (x)dx

EA(x)

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Theory: Axial (Elastic) Deformation• consider a bar in equilibrium, acted upon by two end loads P1 and P2, and a distributed axial load along its

length

• the cross sectional area A may be a function of the spatial coordinate x, i.e., A=A(x)

• upon loading, the total deformation is δ

4

!(x) =P (x)

A(x)stress-load:

!(x) =d"

dxstrain-displacement:

!(x) = E"(x)stress-strain:

d! =P (x)dx

EA(x)

! =

! L

0

P (x)

EA(x)dx

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Theory: Another Perspective• the deformation of this thin segment of material is only a function of the increment in loading over this

section (that is, of )

• this equilibrium analysis is designed to “prove” that the FBD of Fig. 4-2 from the book is actually valid (since it fails to illustrate the distributed load)

5

P P + dP

P (x)

dx!Fx = (P + dP ) ! P + P (x)dx = 0 " P (x) =

dP

dx

A = A(x)

d!

dP/dx

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Theory: Another Perspective• the deformation of this thin segment of material is only a function of the increment in loading over this

section (that is, of )

• this equilibrium analysis is designed to “prove” that the FBD of Fig. 4-2 from the book is actually valid (since it fails to illustrate the distributed load)

5

P P + dP

P (x)

dx!Fx = (P + dP ) ! P + P (x)dx = 0 " P (x) =

dP

dx

A = A(x)

d!

dP/dx

stress at a point:

analysis proceeds from here...

!(x) =P (x)

A(x)

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Remarks on Axial Loading• in many practical cases, the distributed load is absent and only end loads act on the structure; in this case the

internal force is constant:

• clearly, then, if the cross section is also constant:

• and if the structure has step changes in cross section, then we simply sum the deformations over each section:

6

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Remarks on Axial Loading• in many practical cases, the distributed load is absent and only end loads act on the structure; in this case the

internal force is constant:

• clearly, then, if the cross section is also constant:

• and if the structure has step changes in cross section, then we simply sum the deformations over each section:

6

! =P

E

! L

0

dx

A(x)e.g., Prob. 4-25

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Remarks on Axial Loading• in many practical cases, the distributed load is absent and only end loads act on the structure; in this case the

internal force is constant:

• clearly, then, if the cross section is also constant:

• and if the structure has step changes in cross section, then we simply sum the deformations over each section:

6

! =P

E

! L

0

dx

A(x)e.g., Prob. 4-25

! =

P

EA

! L

0

dx =

PL

EAe.g., lecture module 6

MAE 2310 Str. of Materials © E. J. Berger, 2010 7-

Remarks on Axial Loading• in many practical cases, the distributed load is absent and only end loads act on the structure; in this case the

internal force is constant:

• clearly, then, if the cross section is also constant:

• and if the structure has step changes in cross section, then we simply sum the deformations over each section:

6

! =P

E

! L

0

dx

A(x)e.g., Prob. 4-25

! =

! PL

EAe.g., Prob. 4-15

! =

P

EA

! L

0

dx =

PL

EAe.g., lecture module 6