The Material is considered as continuous and cohesive
Stress is the quotient of the force and area. It describes the intensity
of the internal force on a specific plane (area) passing through a
Normal stress: The intensity of force, or force per unit area, acting
normal to ΔA is defined as the normal stress, σ (sigma), since ΔFz is
normal to the area then:
If the normal force or stress “pulls” on the area element ΔA, it is
referred to as tensile stress, whereas if it “pushes” on ΔA it is called
Shear stress: The intensity of force, or force per unit area, acting
tangent to ΔA is called shear, .Here we have shear stress
In the International Standard or SI system, the magnitudes of both
normal and shear stress are specified in the basic units of Newtons
per square meter (N/m2). This unit, called a Pascal (1Pa=1N/m2) rather
small, and in engineering work prefixes such as kilo (103), symbolized
by k, mega (106), symbolized by M, or Giga (109), symbolized by G, are
used to represent larger, more realistic values of stress.
Average normal stress distribution
Provided the bar is subjected to a constant uniform deformation as noted, then
this deformation is the result of a constant normal stress σ.
As a result, each area ΔA on the cross section is subjected to a force ΔF=σ
ΔA, and the sum of these forces acting over the entire cross-sectional area
must be equivalent to the internal resultant force P at the section.
dAA and thereforeIf we let dFF
Then, recognizing σ is constant, we have
dAdF . ; P=σ.A,
σ: average normal stress at any point on the
P: Internal resultant normal force, which is
applied through the centroid of the cross
sectional area. P is determined using the method
of sections and the equations of equilibrium.
A: Cross sectional area of the bar.
It should be apparent that only a normal stress
exists on any volume element of material located
at each point on the cross section of an axially
loaded bar. If we consider vertical equilibrium of
the element, then applying the equation of force
0zF , σΔA-σ’ΔA=0, σ=σ’
In the other words, the two normal stress components on the element
must be equal in magnitude but opposite in direction. This is referred
to as uniaxial stress.
Average shear stress
In order to show how this stress can develop, we will consider the effect of
applying a force F to the bar.
If the supports are considered rigid,
and F is large enough, it will cause
the material of the bar to deform and
fail along the planes identified by AB
A free body diagram of the unsupported
center segment of the bar indicates that the
shear force V=F/2 must be applied at each
section to hold the segment in equilibrium.
The average shear stress distributed over each
sectioned area that develops this shear force is
avg : average shear stress at the section, which is assumed to be the
same at each point located on the section.
V: internal resultant shear force at the section
determined from the equations of equilibrium.
A: Area at the section.
Notice that average shear is in the same direction as
V, since the shear stress must create associated forces
all of which contribute to the internal resultant force V
at the section.
The steel and wood joints are examples of single-shear connections and are
often referred to as lap joints.
An engineer in charge of the design of a structural member or
mechanical element must restrict the stress in the material to a level
that will be safe.
To ensure safety, it is necessary to choose an allowable stress that
restricts the applied load to one that is less than the load the member
can fully support.
One method of specifying the allowable load for the design or analysis of a
member is to use a number called the factor of safety.
The factor of safety (F.S.) is a ratio of the failure load Ffail divided by the
allowable load, Fallow.
If the load applied to the member is linearly related to the stress
developed within the member, as in the case of using σ = P/A and τavg
=V/A, then we can express the factor of safety as a ratio of the failure
stress σfail (or τfail) to the allowable stress σallow (or τallow);
In any of these equations, the factor of safety is chosen to be
greater than 1 in order to avoid the potential for failure.