report on normal stresses in straight one-dimensional structural element
DESCRIPTION
This report provides you the method of calculating the normal stresses at any point inside the straight structural elementTRANSCRIPT
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
=