TOPIC 1: SIMPLE STRESSES and DEFORMATIONS
Intended Learning Outcomes:
After completing the class, you will:
1. Have reviewed the principles of stress and deformation analysis for several kinds of stresses, including the following:
a. Normal stresses due to direct tension and compression forces
b. Shear stress due to direct shear force
c. Shear stress due to torsional load for both circular and non-circular sections
d. Shear stress in beams due to bending
e. Normal stress in beams due to bending
2. Be able to interpret the nature of the stress at a point by drawing the stress element at any point in a load-carrying
member for a variety of types of loads.
3. Have reviewed the importance of the flexural center of a beam cross section with regard to the alignment of loads on
beams.
4. Have reviewed beam-deflection formulas.
5. Be able to analyze beam-loading patterns that produce abrupt changes in the magnitude of the bending moment in the
beam.
6. Be able to use the principle of superposition to analyze machine elements that are subjected to loading patterns that
produce combined stresses.
7. Be able to properly apply stress concentration factors in stress analyses.
Representing a Stress Element:
One major goal of stress analysis is to determine the point within a load-carrying member that is subjected to the highest
stress level. You should develop the ability to visualize a stress element, a single, infinitesimally small cube from the member in
a highly stressed area, and to show vectors that represent the kind of stresses that exist on that element. The orientation of the
stress element is critical, and it must be aligned with specified axes on the member, typically called x, y, and z.
It is recommended that you visualize the cube form first and then represent a square stress element showing stresses on a
particular plane of interest in a given problem. In some problems with more general states of stress, two- or three-square stress
elements may be required to depict the complete stress condition.
Tensile and compressive stresses, called normal stresses, are shown acting perpendicular to opposite faces of the stress
element. Tensile stresses tend to pull on the element, whereas compressive stresses tend to crush it.
Shear stresses are created by direct shear, vertical shear in beams, or torsion. In each case, the action on an element
subjected to shear is a tendency to cut the element by exerting a stress downward on one face while simultaneously exerting a
stress upward on the opposite, parallel face. This action is that of a simple pair of shears or scissors. But note that if only one
pair of shear stresses acts on a stress element, it will not be in equilibrium. Positive shear stresses tend to rotate the element
in a clockwise direction. Negative shear stresses tend to rotate the element in a counterclockwise direction.
Rather, it will tend to spin because the pair of shear stresses forms a couple. To produce equilibrium, a second pair of shear
stresses on the other two faces of the element must exist, acting in a direction that opposes the first pair. In summary, shear
stresses on an element will always be shown as two pairs of equal stresses acting on (parallel to) the four sides of the element.
1. Normal Stresses and Deformation due to Axial Loads
Stress can be defined as the internal resistance offered by a unit area of a material to an externally applied load. Normal
stresses (s) are either tensile (positive) or compressive (negative). For a load-carrying member in which the external load is
, The following formula computes the stretch due to a direct axial tensile load or the shortening due to a direct axial
compressive load
𝐹𝐿 𝑠𝐿 𝐹̅ 𝑙
𝛿 = 𝐴𝐸 = 𝐸 or from the figure 𝛿 = 𝐴𝐸0
where 𝛿= total deformation of the member carrying the axial load
F = direct axial load
L = original total length of the member
E = modulus of elasticity of the material
A = cross-sectional area of the member
For tensile load:
2. Bearing Stress
A stress develops between two surfaces in contact under load. The bearing area is usually the projected area between the
two contacting surfaces.
𝐹
𝜎𝑏 = 𝐴
𝑏
3. Direct Shear Stress
Direct shear stress occurs when the applied force tends to cut through the member as scissors or shears do or when
a punch and a die are used to punch a slug of material from a sheet. Another important example of direct shear in
machine design is the tendency for a key to be sheared off at the section between the shaft and the hub of a machine
element when transmitting torque.
𝑆ℎ𝑒𝑎𝑟𝑖𝑛𝑔𝑓𝑜𝑟𝑐𝑒 𝑉
𝜏 = 𝑠ℎ𝑒𝑎𝑟𝑖𝑛𝑔𝑎𝑟𝑒𝑎 = 𝐴
𝑠
This stress is more properly called the average shearing stress, but we will make the simplifying assumption that
the stress is uniformly distributed across the shear area.
Intended Learning Outcomes:
After completing the class, you will:
1. Have reviewed the principles of stress and deformation analysis for several kinds of stresses, including the following:
a. Normal stresses due to direct tension and compression forces
b. Shear stress due to direct shear force
c. Shear stress due to torsional load for both circular and non-circular sections
d. Shear stress in beams due to bending
e. Normal stress in beams due to bending
2. Be able to interpret the nature of the stress at a point by drawing the stress element at any point in a load-carrying
member for a variety of types of loads.
3. Have reviewed the importance of the flexural center of a beam cross section with regard to the alignment of loads on
beams.
4. Have reviewed beam-deflection formulas.
5. Be able to analyze beam-loading patterns that produce abrupt changes in the magnitude of the bending moment in the
beam.
6. Be able to use the principle of superposition to analyze machine elements that are subjected to loading patterns that
produce combined stresses.
7. Be able to properly apply stress concentration factors in stress analyses.
Representing a Stress Element:
One major goal of stress analysis is to determine the point within a load-carrying member that is subjected to the highest
stress level. You should develop the ability to visualize a stress element, a single, infinitesimally small cube from the member in
a highly stressed area, and to show vectors that represent the kind of stresses that exist on that element. The orientation of the
stress element is critical, and it must be aligned with specified axes on the member, typically called x, y, and z.
It is recommended that you visualize the cube form first and then represent a square stress element showing stresses on a
particular plane of interest in a given problem. In some problems with more general states of stress, two- or three-square stress
elements may be required to depict the complete stress condition.
Tensile and compressive stresses, called normal stresses, are shown acting perpendicular to opposite faces of the stress
element. Tensile stresses tend to pull on the element, whereas compressive stresses tend to crush it.
Shear stresses are created by direct shear, vertical shear in beams, or torsion. In each case, the action on an element
subjected to shear is a tendency to cut the element by exerting a stress downward on one face while simultaneously exerting a
stress upward on the opposite, parallel face. This action is that of a simple pair of shears or scissors. But note that if only one
pair of shear stresses acts on a stress element, it will not be in equilibrium. Positive shear stresses tend to rotate the element
in a clockwise direction. Negative shear stresses tend to rotate the element in a counterclockwise direction.
Rather, it will tend to spin because the pair of shear stresses forms a couple. To produce equilibrium, a second pair of shear
stresses on the other two faces of the element must exist, acting in a direction that opposes the first pair. In summary, shear
stresses on an element will always be shown as two pairs of equal stresses acting on (parallel to) the four sides of the element.
1. Normal Stresses and Deformation due to Axial Loads
Stress can be defined as the internal resistance offered by a unit area of a material to an externally applied load. Normal
stresses (s) are either tensile (positive) or compressive (negative). For a load-carrying member in which the external load is
, The following formula computes the stretch due to a direct axial tensile load or the shortening due to a direct axial
compressive load
𝐹𝐿 𝑠𝐿 𝐹̅ 𝑙
𝛿 = 𝐴𝐸 = 𝐸 or from the figure 𝛿 = 𝐴𝐸0
where 𝛿= total deformation of the member carrying the axial load
F = direct axial load
L = original total length of the member
E = modulus of elasticity of the material
A = cross-sectional area of the member
For tensile load:
2. Bearing Stress
A stress develops between two surfaces in contact under load. The bearing area is usually the projected area between the
two contacting surfaces.
𝐹
𝜎𝑏 = 𝐴
𝑏
3. Direct Shear Stress
Direct shear stress occurs when the applied force tends to cut through the member as scissors or shears do or when
a punch and a die are used to punch a slug of material from a sheet. Another important example of direct shear in
machine design is the tendency for a key to be sheared off at the section between the shaft and the hub of a machine
element when transmitting torque.
𝑆ℎ𝑒𝑎𝑟𝑖𝑛𝑔𝑓𝑜𝑟𝑐𝑒 𝑉
𝜏 = 𝑠ℎ𝑒𝑎𝑟𝑖𝑛𝑔𝑎𝑟𝑒𝑎 = 𝐴
𝑠
This stress is more properly called the average shearing stress, but we will make the simplifying assumption that
the stress is uniformly distributed across the shear area.
