Dr. Salah Uddin, PhD (Univ of Nottingham, UK)
ME (Structure), BE (Civil)
Associate Professor
Email: Salahuddin@buetk.edu.pk
WhatsApp: +923337950583
Stress & Stress Analysis
Goal of the lecture
• To understand and learn concept of stress and strain
• Stress Analysis
Recap of last Lecture
Explained
• the scope of the subject
• Rigid bodies
• Uniaxial stress and
• Normal Stress
CLO-1 (PLO-1)
Some Important Concepts
• A homogeneous mixture is a solid, liquid or gaseous mixture
that has the same proportions of its components throughout any
given sample. Conversely, a heterogeneous mixture has
components in which proportions vary throughout the sample
Some Important Concepts
• Homogeneous means properties are same on every point.
And isotropic means the property is same in all directions. If
something shows same properties at all points, then it
is homogeneous and if the properties are same in all directions
, then it is isotropic
• ISOTROPIC IS ALWAYS HOMOGENEOUS BUT THE
REVERSE IS NOT TRUE
• Notice that both homogeneity and isotropy are scaledependent quantities: they depend on the spatial scale where
we choose to effectuate our measurements.
To give you a specific example, consider steel: steel is an ironcarbon alloy. At a large enough scale (let's say the mm scale),
steel is homogeneous.
• However, if you look at it close enough (μm scale), this is what you
see
Steel
Granite
Pics Courtesy of: https://physics.stackexchange.com/questions/153008/what-is-differencebetween-homogeneous-and-isotropic-material
A homogeneous but not isotropic pattern on the left and an isotropic but not homogeneous pattern on
the right
Difference between stress
Engineering stress or
Nominal Stress
𝐿𝑜𝑎𝑑
𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝐴𝑟𝑒𝑎
Average stress
𝐿𝑜𝑎𝑑
𝐴𝑟𝑒𝑎
True stress
𝐿𝑜𝑎𝑑
𝐴𝑐𝑡𝑢𝑎𝑙 𝐴𝑟𝑒𝑎
CLO-1 (PLO-1)
Initial Area Vs Actual Area
CLO-1 (PLO-1)
A food for thought for you
What if these requirements are not
satisfied?
I. that the bar be prismatic,
II. the loads act through the centroids of the
cross sections, and
III. the material be homogeneous (that is, the
same throughout all parts of the bar).
CLO-1 (PLO-1)
Engineering/Nominal Strain
• The elongation/Deformation d of the bar is the
cumulative result of the stretching of all elements of the
material throughout the volume of the bar.
• Let us assume that the material is the same everywhere
in the bar. Then, if we consider half of the bar, it will have
an elongation equal to d/2, and if we consider one-fourth
of the bar, it will have an elongation equal to d/4.
CLO-1 (PLO-1)
CLO-1 (PLO-1)
What is Strain
It is the change in length per unit length and is denoted by the Greek letter e
(epsilon)
𝜀=d 𝐿
Tensile force
Compressive force
It’s a dimensionless quantity
Tensile Strain
Compressive Strain
Stress Analysis
1. Equilibrium Analysis
• If necessary, find the external reactions using a free-body diagram (FBD) of the
entire structure.
• Compute the axial force P in the member using the method of sections.
2. Computation of Stress
3. Design Considerations
Stress Analysis
2. Computation of Stress
• After the equilibrium analysis, the
average normal stress in the member
can be obtained from s=P/A, where A is
the cross-sectional area of the member
at the cutting plane.
Solved Problem
The bar ABCD in Fig. (a) consists of three cylindrical steel segments with
different lengths and cross-sectional areas. Axial loads are applied as shown.
Calculate the normal stress in each segment.
Solution