Final report on

Normal Stresses

in Straight One-Dimensional Structural Element

by

Ahmed Ismail Ahmed Gouda

Ahmed Khaled Ali Abdel-Moaty

Ahmed Alaaeldin Fouad Shehata

Bassem Hassan Abdel-Baset

construction department

pre-master technical writing course

Faculty of Engineering, Cairo University

Giza, Egypt.

January 2013

i

Abstract

This report covers how to calculate the normal stress at any point, in any section of a

straight one-dimensional structural element subjected to external loads, but the

straining actions on the "section to be analyzed" must be known, we won't discuss

how to get the straining actions in this report

ii

Table of contents

Pages

Abstract .................................................................................................. i

Table of content ..................................................................................... ii

List of figures ........................................................................................ iii

List of symbols ....................................................................................... iv

Chapter 1: Introduction .......................................................................... 1

1.1 General ................................................................................... 1

1.2 Definitions .............................................................................. 2

1.3 Assumptions ........................................................................... 4

Chapter 2: Longitudinal strains and normal stresses in

straight one dimensional element ............................................................ 5

2.1 Internal strain equations in S.O.D.S.E fibers .......................... 5

2.2 Internal stress equations in S.O.D.S.E fibers .......................... 14

2.3 Equilibrium between straining actions and internal stresses.. 15

2.4 Getting the stress equation for any point in the section ........ 18

Chapter 3: conclusion ............................................................................ 19

References ............................................................................................. 20

Appendix I ........................................................................................... 1-I

Appendix II ........................................................................................... 1-II

iii

List of Figures

Figure (1.1) .S.O.D.S.E's section with straining actions at certain centroidal axes.....1

Figure (1.2) .S.O.D.S.E showing its section and its longitudinal lines........................2

Figure (1.3) .Radius of curvature of a part of elastic line after deformation...............3

Figure (2.1) .Infinitesimal part of S.O.D.S.E. and straining action acting on it..........5

Figure (2.2) .The general movement of a section of S.O.D.S.E. w.r.t. its other

section..........................................................................................................................6

Figure (2.3) .effect of the movement x on the infinitesimal part dz........................7

Figure (2.4) .Effect of the movement y on the infinitesimal part dz.......................8

Figure (2.5) .Effect of the movement Cz on the infinitesimal part dz........................8

Figure (2.6) .Showing general point "p" in the section with its X,Y coordinates.......9

Figure (2.7) .Deformation in general line "p" due to the rotation of the section about

x-axis. .......................................................................................................................10

Figure (2.8) .Deformation in general line "p" due to the rotation of the section about

y-axis. .......................................................................................................................11

Figure (2.9) .Deformation in general line "p" due to axial movement of the

section in z-axis direction.........................................................................................12

iv

List of symbols

S.O.D.S.E: straight one-dimensional structural element

dz : length of infinitesimal part of one dimensional element “ S.O.D.S.E.”.

: Radius of curvature about x-axis.

: Radius of curvature about y-axis.

: Angle of rotation of S.O.D.S.E's section due to rotation of the section about x-

axis.

: Angle of rotation of S.O.D.S.E's section due to rotation of the section about y-

axis.

: Axial Deformation of S.O.D.S.E's section due to the axial movement of the

section.

: Deformation of general Point “P” on Cross section due to rotation of the section

about x-axis.

: Deformation of general Point “P” on Cross section due to rotation of the section

about y-axis.

: Deformation of general Point “P” on Cross section due axial movement of the

section in z-axis direction.

: Strain of general point “P” due to rotation of the section about x-axis

: Strain of general point “P” due to rotation of the section about y-axis

: Strain of general point “P” due to transition of the section in the longitudinal

direction.

: Total strain at general point “P” due to section movement.

σ: Normal Stress at general point “P” due to section movement.

N: Normal Force acting on cross section.

: Bending Moment acting on cross section about x-axis.

: Bending Moment acting on cross section about y-axis.

E: Modules of Elasticity of Material of S.O.D.S.E .

: Moment of Inertia (or second moment of area) of the section about the centroidal

x-axis.

: Moment of Inertia (or second moment of area) about of the section about the

centroidal y-axis.

: Product Moment of Inertia about centroidal axes x,y.

X: coordinate of general point “P” on x-axis.

Y: coordinate of general point “P” on y-axis.

1

Chapter (1)

Introduction

1.1 General

Our aim in this report is to get the normal stress at any point, in any section of a

straight one-dimensional structural element subjected to external loads.

In order to get the normal stress, straining actions (N, Mx, My) about centroid of the

section, see figure (1.1) should be known.

Analysis of the element to get the straining actions is not our concern in this report.

Under some assumptions in the report, we will be able to get the distribution of

normal stress but first some definitions and assumptions should be known.

Figure (1.1) represents straining actions on a S.O.D.S.E's section, around two

perpendicular axes x and y, and the two axes x,y having their point of intersection at

the centroid of the section.

Figure (1.1) .S.O.D.S.E's section with straining actions at certain centroidal axes.

2

1.2 Definitions

1- Straight One-dimensional structural element

It is a 3 dimensional element, which has one dimension long with respect

to the other two dimensions.

The long dimension must be straight.

The 2 small dimensions are perpendicular to the long one.

The 2 small dimensions form the one-dimensional element's cross

sectional area, while the long ones are called the longitudinal lines.

For better understanding see figure (1.2)

In the rest of the report we will call the Straight One-dimensional

structural element (S.O.D.S.E.) .

2- Centroid or Center of area of the section (C.A.)(see Appendix I)

It is the point at which the section area can be concentrated.

If the element is tensioned from this point no moment is generated on the

element sections, and if the element is tensioned from any other point

moments will be generated on the element.

To determine the location of this point on a section see Appendix I.

Figure (1.2) .S.O.D.S.E showing its section and its longitudinal lines.

3

3- Elastic line of the O.D.S.E

It is the longitudinal line passing through the Centroid of the S.O.D.S.E.'s

sections.

4-Raduis of curvature (ρ) of a part of the elastic line

Any infinitesimal part of the elastic line can be considered as a part of a

circle (after loading) with a certain radius "radius of curvature". See

figure (1.3)

The radius of curvature of a part along the elastic line of the beam is

dependent on the value of straining actions (bending moments) at this

point.

In figure (1.3), the straight black line between the supports represents the

elastic line of the beam before deformation, while the curved green line

represents the elastic line of the beam after deformation.

Figure (1.3) .Radius of curvature of a part of elastic line after deformation.

4

1.3 Assumptions

1-The material forming the S.O.D.S.E. is homogenous and isotropic.

2-The material forming the S.O.D.S.E. is perfectly elastic and obeys Hooks law.

3-The material forming the S.O.D.S.E. behavior is the same in tension and

compression.

4-Plane section, normal to the elastic line of the S.O.D.S.E. before deformation

remain plane and normal to the elastic line after deformation.

5

Chapter 2

Longitudinal strains and normal stresses in one

dimensional element

2.1 Internal strain equations in S.O.D.S.E. fibers

Taking an infinitesimal part of the S.O.D.S.E. with straining actions on it. see

figure (2.1)

Figure (2.1) .Infinitesimal part of S.O.D.S.E. and straining action acting on it.

6

For compensating straining actions on this part of the S.O.D.S.E., one of the faces

(sections) of the infinitesimal part will move, while the other won't. see figure (2.2)

This movement will generate longitudinal strains in the whole parts of the

infinitesimal part.

Since this S.O.D.S.E. material is perfectly elastic, strains will generate stresses.

The stresses generated in the infinitesimal part will be in equilibrium with the

straining actions on the infinitesimal part.

Figure (2.2) .The general movement of a section of S.O.D.S.E. w.r.t. its other

section.

7

General movement of the section can be divided into three component of movement.

1-Rotation of the section about x-axis by an angle . see figure (2.3)

This movement will make the infinitesimal part a part of a circle with a radius x

.

Figure (2.3) .effect of the movement x on the infinitesimal part dz

8

2-Rotation of the section about y-axis by an angle . see figure (2.4)

This movement will make the infinitesimal part a part of a circle with a radius y

.

Figure (2.4) .Effect of the movement y on the infinitesimal part dz

3-Translation of the origin (point of the centroid) by a movement . see figure ( 2.5).

Figure (2.5) .Effect of the movement Cz on the infinitesimal part dz

9

Any line in the infinitesimal part will change its length, due to the movement of

one face (section) of the infinitesimal part w.r.t. the other face.

Let us take a general line "p"(having coordinates X,Y (see figure 2.6) in the same

system of coordinates at which the straining actions are calculated), and calculate its

strain due to the 3 component of movement each at a time.

Figure (2.6) .Showing general point "p" in the section with its X,Y coordinates.

10

1- Due to 1st component or x ,"p" will be elongated by the value . see figure

(2.7)

In figure (2.7) the straight blue line represents the length of point "p" before

deformation (=dz), while the curved blue line represents the length of point "p" after

deformation ( = dz+ ).But the length of elastic line is the same before and after

deformation.

Figure (2.7) .Deformation in general line "p" due to the rotation of the section about

x-axis.

x

x

(1)

= dz *

(2)

Strain that happened in "P" is by definition: the deformation( ) in the line over the

original length(dz)

=

(3)

=

x

=

x (4)

11

2- Due to 2nd

component or y ,"p" will be elongated by the value . see

figure (2.8).

In figure (2.8) the straight blue line represents the length of point "p" before

deformation (=dz), while the curved blue line represents the length of point "p" after

deformation ( = dz+ ). But the length of elastic line is the same before and after

deformation.

Figure (2.8) .Deformation in general line "p" due to the rotation of the section

about y-axis.

y

y

(5)

= dz *

(6)

Strain that happened in “P” is by definition the deformation in the line ( ) over its

original length (dz).

=

(7)

=

y

=

y (8)

12

3- Due to 3rd

component Cz ,"P" will be elongated by the value . see figure (2.9)

Figure (2.9) .Deformation in general line "p" due to axial movement of the section in

z-axis direction.

= Cz (9)

Strain that happened in "P" is by definition the deformation in the line over its

original length

=

(10)

13

Conclusion:

The strain of the general line "p", due to the general movement, is equal to the

summation of the strains produced by the three movements.

= (11)

Substitute in equation (11) by equations (4), (8) and (10) .

=

x

y

(12)

Equation (12) represents the strain of a general line "p" in the section having

coordinates X,Y in the defined x,y coordinate system.

The strain of any other point in the section can be determined by substitution of the

coordinates of the point in equation (12).

14

2.2 Internal stress equations in S.O.D.S.E fibers

According to stress-strain relation of elastic materials

σ = E * (13)

substitute in equation (13) by the value of the strain in equation (12) .

σ = E *(

x

y

) (14)

The equation (14) is considered the equation of stress for any point in the section

having coordinates X,Y w.r.t. the centroidal coordinate system

σ = (

x

y

)

But the values of x , y ,and Cz/dz must be known for the application of the

equation (14).

These values( that represents the movement that happened in the infinitesimal part

dz ) are determined by applying the equation of equilibrium between straining

actions and internal stresses.

15

2.3 Equilibrium between straining actions and internal stresses

Straining actions N, Mx, My are defined about the same axes used for the deduction

of the equation of internal stresses

For equilibrium:

1-Total normal force is the summation of the normal stresses over all the section

N=

(15)

Substitute in equation (15) by the value of stress in equation (14)

N=

x

y

(16)

N=

x

-

y

+

The result of integrations

, and

about centroidal axes are equal

to zero. see Appendix I

The result of integration

is equal A(area)

N=

(17)

=

(18)

The value of the 3rd movement (Cz/dz) is now known.

16

2-The value of the moment about X axis “ must be equal the summation of the

stress *dA*y

=

(19)

Substitute the value of stress in equation (19) by the stress in equation (14) .

=

x

y

(20)

=

x

-

y

+

The result of the integration

is named the second moment of area

(moment of inertia) about the x-axis ( ).To be able calculate it for any section see

Appendix I.

The result of the integration

is named the product moment of area

(product moment of inertia) about the x and y axes ( ).To be able calculate it for

any section see Appendix I.

While the result of the integration

about the centroidal axis is equal to zero.

=

x -

y (21)

In equation (21) there is only 2 unknowns x and y .

17

3-The value of the moment about y axis “ must be equal the summation of the

stress *dA*X

=

(22)

Substitute the value of stress in equation (22) by the stress in equation (14) .

=

x

y

(23)

=

x

-

y

+

(24)

The result of the integration

is named the second moment of area

(moment of inertia) about the y-axis ( ).To be able calculate it for any section see

Appendix I.

The result of the integration

is named the product moment of area

(product moment of inertia) about the x and y axes ( ).To be able calculate it for

any section see Appendix I.

While the result of the integration

about the centroidal axis is equal to zero.

=

x -

y (25)

In equation (25) there is only 2 unknowns x and y

,they are the same unknowns

of equation (21) .

18

2.4 Getting the stress equation for any point in the section

Solving the equations (21) and (25) together, we get the values of the two unknowns

x and y

.

For detailed solution of the 2 equations see Appendix II.

We get the expressions of x and y .

=

(26)

=

(27)

Now substitute by the values of x , y ,and Cz/dz from equations (18),(26), and

(27) in the stress equation (14)

=

(28)

19

Chapter (3)

Conclusion

At any section of a S.O.D.S.E., if the location of the section centroid is known, and

the straining actions about a certain axes(x,y), and the moments of inertia about

these axes (x,y) are known, then the normal stress at any point (having coordinates

X,Y) can be determined by the following equation :

=

For positive values of straining action.

N is positive if it is tension.

Mx is positive if it follows the right hand rule in x-axis direction.

My is positive if it follows the right hand rule in y-axis direction.

For the values of section properties,

Ix, Iy , Ixy are the moments of inertia about centroidal axes (must be same axes used

in calculating Mx, My).

20

References

1. Bakhoum, M. “structural Mechanics (volume – one )” pg.(71, 72, 77, 90, 91, 99,

100)

1

Appendix I

Centroid or center of gravity of Area:

It is the point at which the resultant of all elementary areas passes. i.e. the point at

which the whole area could be conceived as being concentrated[1] .

First Moment of Area:

The statical moment, or first moment of an area about an axis is the sum of the

product of each elementary area by the normal distance from that axis.

By definition the statical moment of an area about an axis y, is given by

=

Similarly, the statical moment about the x axis , is

=

The statical moment may be positive or negative, according to the location of the

reference axes with respect to the area

If the centroid of an area is at a certain point C, whose coordinates referred to the X

and Y axes are and then, from the definition of the centroid , the sum of

moments of all elementary areas about the y- axis should be equal to the total area

multiplied by the normal distance of the centroid from y-axis , i.e. one may write

[1] .

=

= A

=

= A

=

and =

2

Moment of Inertia:

By definition. The moment of inertia of an area A about an axis y is given by

=

dA

Similarly the moment of inertia about the x-axis is

=

dA

Obviously, the moment of inertia should always have appositive sign, regardless the

position of the reference axis [1] .

Moment of Inertia about parallel axes

If the Moment of inertia about an axis y is known, then, the moment of inertia about

another axis , parallel to y and at a normal distance d from it is given by

=

dA =

dA +

dA +

dA

= + A + 2 d

If the y-axis passes through the centroid of the area, then, =0 and one gets [1] .

= + A

Product of inertia

By definition, the product of inertia of an area about any two orthogonal axes, X and

Y is given by

=

dA

Obviously, the product of inertia may be either positive or negative according to the

position of axes with respect to the section.

If either axis x or y is an axis of symmetry the product of inertia will vanish [1] .

3

Moment of inertia about inclined axes

For any section if the moments of inertia and product of inertia with respect to any

pair of orthogonal axes y and x through any point C are known, then, the moment of

inertia and product of inertia, referred to any other set of orthogonal axes

inclined at an angle θ to the first set, may be obtained in terms of the values referred

to the x and y axes .

= y cos θ + x sin θ

= - y sin θ + x cos θ

=

dA =

) dA

dA +

dA +

dA

=

+ +

Similarly one may prove that

=

+ -

1

Appendix II

=

x -

y (21)

=

x -

y (25)

Multiply eq.(21) by

Multiply eq. (25) by

-

= - +

= -

By superposition of above 2 eq. we can get:

(

= –

= (

–

=

By substitute in eq. (21), we can get

=