Stresses & Strains
TECH 3401
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Strength of Materials
This section deals with the relationships between forces acting on bodies and the internal stresses and strains that are generated by these forces.
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Tensile Force
Stress is force per unit area
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Compressive Forces
Stress is still force per unit area
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Stress
This gives the average computed stress. Is is known as the direct stress formula.
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Stress Units
Hint: if you remember that psi can be stress units, it will give you the formula.
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Allowable Stress
A level of stress that is predetermined (by building codes, design guides, etc.) to be acceptable.
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Allowable Stress
Max load:
Min cross section:
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Bearing Stress
Compressive stress exerted on an external surface of a body (such as the ground).
Pressure from gases or fluids, or pressure on the soil (as in a footing) are bearing stress.
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Example
Calculate the tensile stress developed in a steel bar 2 inches by 2 inches in cross section if it is subjected to an axial tensile load of 95 kips.
Determine the tensile stress if the bar is replaced with a section of W8X31.
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Example
Steel hangers are used to support pipes in a manufacturing facility. The rods are ½ inch in diameter and have an allowable tensile stress of 24,000 psi. Calculate the allowable load in the rods.
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Example
Calculate the required size of a short, square dressed (S4S) timber post subjected to a compressive load of 24,000 lb if the allowable compressive stress is 800 psi.
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Example
A flat steel bar .5 inch thick and 4.0 inches wide is subjected to a 20 kip load. Two .75 diameter holes are located as shown. Determine the average tensile stress at A-A and B-B.
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Shear Stresses
Stresses are tangential to the force applied.
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Shear Stresses
Notice the forces are parallel to the surface bearing the load.
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Punching
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Design Example
The rod shown in red is to support a load of 20 kips.
The bar has an allowable stress of 7500 psi.
What is the minimum diameter of the bar?
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Tensile and Compressive Strain
and Deformation
Strain:
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Example
Compute the total elongation of a 60 foot long steel wire if the strain is .00067 in/in.
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Keep in mind...
If a rod is suspended from its upper end and supports only its own weight, the strain varies uniformly from zero at the lower end to a maximum value at the upper end.
The average strain is
Since the strain is linear, the strain at the top is 2X the average, or
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The Stress-Strain Relationship
For elastic region:
For steel,
E is 29,000 ksi
(manual of steel construction, 9th ed, AISC)
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Shear Strain
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Stress and Strain in Shear
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PL/AE
By substituting some of the above equations, we get a convenient way to describe total deformation in terms of PLA&E.
Works for an axially loaded homogeneous prismatic shape.
𝛅 = total axial deformation
P = Load in lb, kips, (N)
L = Length in in, (mm)
A = Cross-sectional area
E = Modulus of elasticity in psi, ksi (Pa, MPa)
IF and ONLY IF the stress is below the proportional limit.
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Example
A tensile member is subjected to an axial load of 5,000 lb. It has a length of 30 inches and is made from a steel tube having an OD of .75 inches and an ID of .5 inches. Calculate the tensile stress in the tube and the total axial deformation. Verify that the stress is below the proportional limit.
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Example
The steel bar shown has a cross-sectional area of one square inch. What is the total elongation of the bar?
Use E = 30,000,000 psi
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