Adjacent cross-sections_{}and_{}originally located at distance _{} from each other, as
shown in Figure 1.a, are found after the deformation to be located at distance_{}. The change in the position of the two cross-sections are described
by their displacements _{}and_{}, respectively.

Figure 1 Geometrical Aspects
of the Axial Deformation

(a)
Undeformed Member and (b) Deformed Member

The extensional strain_{} is expressed as:

_{} (1)

Equation (1) is called the **strain-displacement equation**. The cross-section
distribution of the elongation strain _{} is shown in Figure 2. The
elongation strain is a function only of the cross-section position described by
the variable_{}.

Figure 2 Extensional
Strain Distribution

The rest of the generalized
strain tensor components are:** **

_{} (2)

_{} (3)

_{} (4)

From equations (2) through (4) is
evident that a transversal reduction of the cross-section takes place
concomitant with the axial deformation.

The displacement _{}pertinent to a particular cross-section is obtained by
integration from the equation (1):

_{} (5)

where _{}is the displacement at the beginning at the integration
interval.

Consequently, the total elongation
of the member is calculated from equation (5) as:

_{} (6)

where _{}is the total length of the bar.

**5.1.2
****Constitutive
Equation**

The constitutive equation
reflects, as was describe in Lecture 2, the relation between the stress and the
strain. If the **linear elastic material behavior**
is considered, the relation between the normal stress and extension strain _{} for the case of the
axial deformation is written:

_{} (7)

Equation (7) represents the
application of Hook’s Law for the case of axial deformation. The material
constant_{}, the modulus of elasticity, has a value unique to each
specific material and is obtained from tensile tests. The cross-section of the
bar is a small surface and the variation of the modulus of elasticity is
negligible on this surface. In this case the constitutive equation (7) is
expressed as:

_{} (8)

Equation (8) implies that the
normal stress _{} varies only along the
length of the member, but has a constant value on the entire cross-section. The
representation of the normal stress _{} is shown in Figure 3.

The rest of the stress tensor
components are zero:

_{} (9)

_{} (10)

Figure 3
Cross-Section Normal Stress Distribution

**5.1.3
****Cross-Section
Stress Resultants**

Considering the stress
distribution represented by equation (8) through (10) the cross-section stress
resultants are obtained as:

** **_{} (11)

_{} (12)

_{} (13)

The relation between the normal
stress _{} and the cross-section resultants_{}, _{} and _{} is derived using the
notation shown in Figure 4.

Figure 4 Normal Stress and Stress Resultants

(a)
Stress Resultants and (b) Normal Stress

If the axes _{}and _{}of the coordinate system intersect such that the _{}axis passes through the cross-section centroid, the static
moments _{} and _{} are zero and the axial
force_{} remains the only non-zero stress resultant. Then, from
equation (11) the normal stress_{}is calculated as:

_{} (14)

Consequently, it is to be
concluded that a beam made from a linear elastic material undergoes an axial
deformation if the axial force passes through the cross-section centroid.

**5.1.4
****Equilibrium
Equation**

The equilibrium equation pertinent
to the case of axial deformation was derived in Section 4.3 of Lecture 4. The
detailed derivation is not repeated and only the final result expressed as the
differential relation (4.7) is employed. Considering the exterior loading and stress
resultants shown in Figure 5, acting on an infinitesimal volume of length _{} separated from the body
of the beam, the following differential equation is written:

_{} (13)

where _{} is the distributed
loading parallel to the beam longitudinal axis.

Figure
5 Infinitesimal Volume Equilibrium

If the axial stress resultant _{}is orientated as shown in Figure 5 and tends to lengthen the
segment the internal force is called **tension.
**The corresponding normal stress _{} is called **tension stress**. In contrast, if the
stress resultant force _{}is orientated to shorten or compress the segment the
corresponding internal force and normal stress are called **compression** and **compression
stress**, respectively.

Integrating equation (15) the stress
resultant force _{} is calculated:

_{} (16)

where _{}is the value of the axial force at the origin of the
integration interval.

**1.5 Thermal Effects on Axial
Deformation**

The equations obtained in the
previous sections are derived considering only the exterior load action and
neglecting the change in temperature. In this section the effects of the
thermal change are introduced.

During Lecture 2, the thermal
strain along the axis _{}was introduced as:

_{} (17)

where _{} is the thermal
expansion coefficient and _{} is the change in the
member temperature.

The total elongation strain is
the sum of the elongation strain induced by the exterior load action and
thermal effects and is expressed as:

_{} (18)

By substitution of equation (14)
into (18) the elongation strain is obtained as:

_{} (19)

Then, using equation (19) in
equation (6) the total elongation of the member is written as:

_{} (20)

**2 Uniform-Axial
Deformation**

__ __

A special case of axial
deformation frequently encountered in structural engineering is the case of **uniform-axial deformation **shown in
Figure 6. Systems composed of many members subjected to uniform-axial
deformation are also commonly used in structural practice. A typical example is
the plane truss, where each individual member is subjected to uniform-axial
deformation under the action of to tension or compression forces.

The formulation related with the
definition of the uniform-axial member and its application in the investigation
of the statically determinate and indeterminate structures is presented in this
section.

**2.1 Members Subjected to Uniform-Axial
Deformation **

Definition 2 is mathematically
transcribed as:

_{} (21)

_{} (22)

_{} (23)

.

Figure 6 Member
Exhibiting Uniform Axial-Deformation

__Note:__ Equation (23) implies the absence of the distributed load _{} in equation (15)

Rewriting equations (6), (8) and
(14) obtained in the previous section for the case in point of a member with uniform
axial-deformation the following equations are obtained:

_{} (24)

_{} (25)

_{} (26)

In the computer applications
equation (26) takes the following form:

_{} (27)

where _{}is called the **axial
stiffness coefficient**.

The axial stiffness coefficient represents
the force applied to the member ends when the elongation is equal with the unit
length. The unit is [F/L]. The reciprocal value of the axial stiffness is called
the **axial** **flexibility coefficient**. The flexibility coefficient is calculated:

_{} (28)

Using the flexibility coefficient
expression equation (28) is cast in a new format:

_{} (29)

If the change in temperature is
also constant along the entire length of the member the formula (20) are
amended as follows:

_{} (30)

Using the flexibility coefficient
_{} the elongation is
calculated as:

_{} (31)

Rewriting the equation (31) the
force _{} is obtained:

_{} (32)

**5.2.2
****Statically
Determinate Structure **

An example of a statically
determinate structure is shown in Figure 7. A rigid, weightless beam AC is
supported at end A by a column and at end C by a vertical rod CD. The rod is
attached at point D to a “leveling jack”, a component which permits a limited
vertical movement and has the mission to keep the beam AC leveled. The beam AC
is loaded with a vertical force P located at distance _{} from the left end A.
To calculate the axial forces acting on the column and rod the system is
decomposed into three components: the beam, the column and the rod.

Figure 7 Statically
Determinate Structure

The free-body diagram of the beam
is depicted in Figure 8 and is used to calculate the reaction forces _{} and_{}. The connection between the column and the beam requires in
general, two reaction forces. In the absence of any horizontal force the
horizontal component is null and is not shown

Figure 8 Free-Body
Diagram

The reaction forces can be calculated using the following
equilibrium equations:

_{} _{} (33)

_{} _{} (34)

Solving equations (33) and (34)
the reaction forces are obtained:

_{} __Compression in column__ (35)

_{} __Tension in rod__ (36)

__Note:__ The column and the rod are loaded with forces having opposite directions
than the reaction forces calculated in equations (35) and (36). This became
obvious if one attempt to draw the free-body diagram for the rod and column.

From equations (35) and (36) it
is evident that the column and the rod are loaded with compression force _{} and tension force _{}, respectively. These two elements are characterized, using
definition 2, as members with uniform axial-deformation.

Using the formulae (24) through (26)
the following related column values are calculated:

_{} (37)

_{} (38)

_{} (39)

__Note:__ The force _{} is compression and the
column is shortened by the loading action.

Related quantities for the rod CD
are similarly calculated:

_{} (40)

_{} (41)

_{} (42)

__Note:__ The force _{} is tension and the rod
is elongated by the loading action.

For the beam to stay in perfect
horizontal balance the displacement at point A and C should be equal and
manifest in the same direction:

_{} (41)

The expressions of the
displacements at points A and C are calculated using the formula (5).The
displacement at point A is equal with the column elongation, because the
initial displacement at ground connecting point is considered zero:

_{} (42)

The displacement at point C is
calculated as:

_{} (43)

where _{} is the displacement
allowed by the “leveling jack”.

The necessary displacement
allowed by the leveling device is calculated from equation (41):

_{} (44)

__Note:__ The displacement _{} has to be positive for
the device to work properly (the device works only in tension). This means that
the deformation of the column has to be larger than the rod deformation.

The numerical application for
this example is presented in section 6.1.

**5.2.3
****Statically
Indeterminate Structure **

A typical example of a statically
indeterminate structure is shown in Figure 9. This type of structures requires
a more involved methodology for in order to calculate the stress and strain
distributions.

Figure 9 Statically
Indeterminate Structure

The system shown in Figure 9 is
composed of a rigid member AD, pinned into the wall at point A, and two unequal
linear elastic rods, BE and CF. The rods, BE and CF, are attached to the
ceiling at points E and F, respectively. The system is loaded with a
concentrated vertical force _{} at point D. The
free-body diagram used to write the equilibrium equations is shown in Figure 10.

Figure 10 Free-Body
Diagram

The following equilibrium
equations are written:

_{} _{} (47)

_{} _{} (48)

_{} _{} (49)

The equilibrium equations (48)
and (49) contain three unknown reaction forces_{}, _{} and _{}. In order to solve these unknown quantities one additional
equation is necessary. This equation is obtained from the deformation
compatibility condition schematically described in Figure 11. Because the beam
AD is rigid, purely geometric relations between the rod elongations,_{} and_{}, and the rotation angle _{} are written as:

_{} (50)

_{} (51)

Figure 11 Deformation
Notation

Using equation (29) and the
relations (50) and (51) the forces in the rods are expressed as:

_{} (52)

_{} (53)

The stiffness coefficients, _{}and_{}, are calculated from the geometrical and material properties
characteristics of the rods:

_{} (54)

_{} (55)

Substituting equations (52) and (53)
into the equilibrium equations (48) and (49),_{} and _{} are eliminated from
the system leaving only two unknowns, _{} and_{}:

_{} (56)

_{} (57)

Solving the algebraic system, the
two unknowns are found as:

_{} (58)

where _{} is the rotation angle
corresponding to _{}

_{} (59)

Introducing the rotation angle _{} into equations (52)
and (53) the rod forces are calculated:

_{} (60)

_{} (61)

where _{} and _{} are the forces in the
rods in _{}

The rod stresses are also
calculated:

_{} (62)

_{} (63)

The rod elongations are obtained using
equations (50) and (51) as:

_{} (64)

_{} (65)

If the allowable vertical force _{} is required, the
stress in the rods must be compared against the allowable stress_{}:

_{} (66)

_{} (67)

At limit, the relations (66) and (67) are rewritten as:

_{} (68)

_{} (69)

Introducing the rod forces, equations (60) and (61), into
equations (68) and (69) the allowable vertical force admitted by the system is
calculated as:

_{} (70)

The numerical application for
this example is presented in section 6.2.

**3 Nonuniform-Axial
Deformation**

The definition of a member with uniform-axial
deformation is specified in Section 3. If any one of the assumptions contained
in definition 2 is violated the axial deformation is called **nonuniform-axial deformation**. The most
common cases of nonuniform-axial deformation are treated in the following
subsections.

**5.3.1
****Non-homogeneous
Cross-Section Members**

The theory developed in the
previous sections assumed that the cross-section is made from a homogeneous
material described by its modulus of elasticity. In structural engineering practice
it is not uncommon to have a case in which a member constructed from two different
materials bounded together at their interface is forced to undergo an axial deformation.
The members made from different materials but behaving together as a single
member are called **composite sections**.

For the purpose of analysis it is
assumed that the member is made from two materials, each being characterized by
a specific modulus of elasticity (_{} and_{}) and area (_{} and_{}). The strain distribution is, as before, assumed to be a
function of only the variable_{}, the position of the particular cross-section of interest:

_{} (71)

Considering that both materials
are homogeneous linear elastic materials the Hook’s law may be written for each
material as:

_{} (72)

_{} (73)

The total axial force _{}, as expressed in equations (11), is divided into two axial
forces, each acting at the centroid of the corresponding bar cross-section.

_{} (74)

where

_{} (75)

_{} (76)

Substituting the axial stresses
expressed in equations (72) and (73) into equations (75) and (76) the
equilibrium equation (74) is written as:

_{} (77)

Consequently, the elongation
strain is obtained:

_{} (78)

Accordingly, the normal stresses
are calculated employing the equations (72) and (73) as:

_{} (79)

_{} (80)

In general, the stress resultant
moments, as expressed in equations (12) and (13), do not vanish and additional
restrictions regarding the geometrical characteristics of the cross-section
must be imposed. Nullification of the stress resultant moments is obtained by
using symmetric cross-section about one or both centroidal axes.

The frequently encountered
practical case when the areas _{} and_{}are constants along the entire length and the member is
subjected to a constant force _{} is considered below.
It is additionally assumed that the cross-section is symmetrical only about the
vertical axis _{}. Then, the normal stresses developed in each material zone
are calculated, following the equations (79) and (80), as:

_{} (81)

_{} (82)

The assumption regarding the
symmetrical aspect of the cross-section about the vertical axis nullify the
stress resultant moment _{} when the bar local
coordinate system is centroidal. The second stress resultant moment _{} can be made zero by
manipulating the position of the application of the force _{}. It was previously argued that the individual cross-section
resultants _{}and _{}must be located at the centroids of the cross-sections_{} and_{}. This raises the question about the location of the applied force_{} in order that only axial forces are induced in each of the
individual bars composing the cross-section. This situation is solved by
replacing the axial forces _{}and _{} with a resultant force
and a null moment written about a horizontal axis. The stress resultant moment about the vertical
axis has a zero value due to the imposed symmetry of the cross-section about
that axis. The moment equation, written about an horizontal axis passing
through the application point of the force _{}, is:

_{} (83)

where _{} is the distance
between the centroids of cross-sections_{} and_{}, while _{} is the distance
between the application point of the force _{} and the centroid of
the area _{}.

The distance _{} is then calculated:

_{} (84)

__Note__: In general the application point of the force _{} does not correspond
with the centroid of the composite cross-section.

**3.2 Non-homogeneous Cross-Section Members Subjected to Thermal Changes**

Consider the composite section
described in Section 3.1 subjected to change in temperature. For generality, in
this discussion, let be assumed that each individual bar undergoes a different
change in temperature _{}and_{}, respectively. The notation employed in the Section 3.1 is
maintained. In the absence of the exterior forces, the equation of equilibrium
is:

_{} (85)

The axial forces pertinent to
each one of the bars is constant along their respective lengths and thus, the
equation (85) becomes:

_{} (86)

The equilibrium equation contains
two unknown cross-sectional forces,_{}and _{}, and consequently the system is statically indeterminate. An
additional equation is necessary to obtain these forces. This equation is derived
from the condition of equality of the elongations for the two individual bars
imposed by the existence of the rigid members attached at their ends. This
equation is written as:

_{} (87)

The elongations may be express
using the formulation of equation (31):

_{} (88)

_{} (89)

Substituting equations (88) and (89)
into equation (87) the second necessary equation is obtained:

_{} (90)

Solving the algebraic equation
system (86) and (90) the cross-section forces are calculated:

_{} (91)

The axial stress in the bars is:

_{} (92)

_{} (93)

**3.3 Heated Member with a Linear Temperature
Variation**

The slender beam shown in Figure 12.a
is heated by a heating coil capable of producing a linearly varying temperature
as shown in Figure 12.b. The beam has a constant cross-section area _{}and is made from a linear elastic material. The beam is
attached to rigid supports at ends A and B.

Figure 12 Heated
Uniform Member

(a) Geometry and (b)
Temperature Variation

In the absence of any applied
forces between A and B, the equilibrium equation is written as:

_{} (94)

The system is statically
indeterminate containing two unknown reaction forces _{}and_{}. An additional equation is necessary. This equation is
obtained by observing that the total elongation of the beam is null:

_{} (95)

Following the equations (20) and
(31) the total elongation is calculated:

_{} (96)

where _{}

Accordingly, with the temperature
variation shown in Figure 12.b the thermal variation in a particular
cross-section is:

_{} (97)

The integral contained in the equation
(96) is calculated using the expression (97):

_{} (98)

Substituting (98) into (96) the
total elongation is expressed as:

_{} (99)

Imposing the condition (95) the
force _{} is found:

_{} (100)

Using the equation (94) results that:

_{} (101)

The normal stress is then calculated as:

_{} (102)

**4.1 Normal Stress in the Vicinity
of the Load Application**

A vertical prismatic bar
characterized by a constant cross-section along its entire length_{} is loaded at one end by a concentrated force_{}and supported at the other end as illustrated in Figure 13.
Based on Newton’s
Law the constraint at the support generates a uniform distributed reaction _{} opposed to the action.
The free-body diagram is shown in Figure 13.a. The example considered falls in
the category of members with uniform axial-deformation studied in Section 2.
The exact determination of the distribution of normal stress along the length
of the beam requires advanced methodologies employed in the Theory of
Elasticity and for this reason only the results are presented. Analyzing the
results illustrated in Figure 13 it is found that the formulae obtained in Section
2 are valid in the majority of the cross-sections except of those located in
the vicinity of the ends. The perturbation zone has a length _{}roughly equal to the width of the cross-section.

Figure 13 Normal Stress Distribution

__Note:__ It can be concluded that for practical purposes the formulae
obtained in Section 2 based on the assumptions contained in definition 1 are
valid especially if the ratio_{}. Special attention has to be given to the areas located near
the point of load application or near abrupt changes in the cross-section. The
application of Saint Venant’s Principle is valid for the case of the beam.

**4.2 Stress Concentrations**

In the theoretical development
pertinent to the axial deformation of linear members, the area _{}of the cross-section is considered as a smoothly varying
function of the position _{}. If discontinuities appear in the definition of the
cross-sectional area, the formulae obtained in the preceding sections are
invalid and the concept of **stress
concentration** must be introduced.

Figure 14
Concentration of Stress

A typical case is shown in Figure
14 where a prismatic type linear member having a circular hole of diameter _{} is subjected to a
constant tension force_{} . The member cross-section is described by the height _{} and thickness_{}.

The normal stress _{}around the hole can be significantly greater than _{} and varies as a function
of the ratio_{}. For practical applications the coefficient_{}, called the **stress-concentration
factor **is introduced. This factor is
defined as the ratio of the maximum normal stress _{}around the hole to the normal stress calculated in the
absence of the hole_{}, called **nominal
stress**.

_{} (103)

_{}. (104)

The average normal stress
calculated with the formula (14) is:

_{}. (105)

The variation of the
stress-concentration factor _{}, calculated using the Theory of Elasticity methods is
illustrated in Figure 1 The stress concentration factor for the configuration
under consideration varies from 2.3 to 3.0.

Figure 15 Variation
of the Stress-Concentration Factor

Analyzing the variation, it is
concluded that if the diameter_{}decreases the concentration factor _{}also increases. This is somewhat misleading and is due to the
way the chart in Figure 15 is constructed. Using the average normal stress_{}in the definition of the stress concentration factor_{}, equation (103) the following formula is obtained:

_{} (106)

Considering the extreme cases
shown in Figure 15 and applying these within equation (106) it can be concluded
that the maximum normal stress _{}increases with the increase in the diameter of the hole and
varies as:

_{} (107)

For small diameter holes the
stress concentration disappears at relatively small distance from the hole.
This is an example of the application of Saint Venant’s Principle.

**4.3 Limits of Poisson’s
Ratio**

The volumetric strain_{}was calculated in Section 2.9.3 for the general case when the
normal stresses_{}, _{} and _{}are applied simultaneously. For the case of axial deformation
only the normal stress _{} has a non-zero value
and, consequently, the equation (2.111) is written as:

_{} (108)

Because the volume can not
decrease during the tensioning of the axially deformed member the volumetric
strain is a positive value. Mathematically this condition is enforced as:

_{} (109)

Consequently, from physical
reality and equation (108) it is seen that:

_{} (110)

The limits established for Poisson’s ratio _{} by the expression (110)
are generally valid for all materials used in structural engineering. The cases
representing the limiting values, _{} and _{}, are pertinent to cork and water, respectively. Structural
steel has a Poisson’s ratio of 0.33. There are some cases when the material has
a negative Poisson’s ratio. These materials are called **swollen solids** and this unusual behavior is characteristic of certain
materials subjected to radiation.

The transversal contraction of the cross-section is
similarly obtained as:

_{} (111)

**5 Design of
Members Subjected to Axial Deformation **

In the design of the members
subjected to axial deformation two important factors, the load _{} and the resistance _{}, are considered. The load _{} represents the maximum
axial force that occurs in the specific member when subjected to the action of
exterior forces or change in temperature. The most common definition for the resistance
_{} is the force which is
developed in the member when the normal stress reaches the yielding value_{}.

The design of the member
subjected to axial deformation is conducted under the condition that the
capacity of the member, represented by its resistance force_{}, must always be greater or equal to the demand force _{}. Mathematically, this assumption is expressed as:

_{} (112)

In the calculation of the load _{} and resistance _{} forces there are
typically a number of factors (load magnitudes and directions, material
characterization, manufacturing tolerance, etc.) which are not known with
absolute certainty. The degree of uncertainty may be treated using the concepts
proper of Probability Theory. In order to circumvent the difficulties inherent
in the rigorous application of probabilistic methods, a global factor _{}, called the **safety
factor**, encompassing all possible uncertainties is introduced as:

_{} (113)

The allowable resistance _{} is then used in place
of the resistance _{} in equation (112).

_{} (114)

The safety factor is always greater
than or equal to unity:

_{} (115)

This is the approach used by the
method called **ultimate strength design
method** and was adopted by American Concrete Institute (ACI) and American
Institute of Steel Construction (AISC).

If the relation between the
stress and strain is linear, than a similar safety factor may be defined by
limiting the value of the normal stress in the axially deformed member.

_{} (116)

The design formula (114) is modified
using the relationship between maximum normal stress _{} and the allowable normal
stress_{}:

_{} (117)

The formula (117) was used for a
long period of time in a procedure known as the **allowable-stress design. D**ue to the simplicity of application, this
method is still commonly used in United States for the design of
steel structures.

The application of the
theoretical formulae developed in this lecture is illustrated in the following
examples.

**5.6.1
****Statically
Determinate Structure **

The structure illustrated in
Figure 7 was investigated in Section 2. The following numerical values are
considered in the numerical application of the generic formulation:

_{}_{}

_{}_{} _{}_{} _{}_{} _{}

_{}_{} _{}_{}

_{}_{} _{}_{}

The reaction forces are:

_{} _{} __Compression in column__

_{}_{} __Tension in rod__

The column related values are:

_{}_{}

_{}

_{}_{}

The rod related values are:

_{}_{}

_{}

_{}_{}

The displacement _{} of the leveling device
is calculated as:

_{}_{}

**5.6.2
****Statically
Indeterminate Structure **

The statically indeterminate
structure illustrated in Figure 9 is solved in Section 2.3. For the numerical
application of the formulation developed the following data are employed:

_{}_{} _{}_{} _{}_{}

_{}_{} _{}_{}

_{}_{}

_{}_{} _{}_{}

_{}_{}

The stiffness coefficients are:

_{}_{}

_{}_{}

The rotation angle _{} is:

_{}_{}

The rod forces _{} and _{} are calculated as:

_{}_{}

_{}_{}

The allowable vertical force _{} for the system is:

_{}_{}