Maximum Tensile Stress In Relation To Temperature Drop In A Steel Bar

Strings (SiPjAjk) = S7P3A31     Base Sequence = 12735     String Sequence = 12735 - 3 - 31

Expressions Of Pj Problems
Maximum Tensile Stress In Relation To Temperature Drop In A Steel Bar
Math A steel bar in the form of a frustum (truncated cone) is rigidly fixed at both of its ends (figure 125.2). Its dimensions are:
length = 24 inches; diameter of circular section at one end = 1 inch.
diameter of circular section at the other end = 3 inches.
Modulus of elasticity, E = 30,000,000 lb/in2.
Coefficient of thermal expansion = 0.0000065/oF.

Suppose bar is subjected to a drop in temperature of 50oF. Determine maximum tensile stress in bar.

The strings: S7P3A31 (Force - Pull).

The math:
Pj Problem of Interest is of type force (pull). Assumptions: bar is rigidly supported; bar is free to contract; proportional limit of steel bar ≥ maximum stress; superposition applicable.

Temperature drop caused a shortening of bar. Tensile Stress of interest is the maximum required to produce an elongation equal to the shortening (i.e stretch bar back to its original length).

shortening, e due to temperature drop = 50(0.0000065)24 = 0.00780 in.
Consider a circular cross-sectional area, A with diameter d, and distance x inches from the end with the smaller diameter (figure 125.2). Then by symmetry:
(d -1)/x = (3-1)/24. So, d = 1 + x/12.
So, Area A = (π/4)(1 + x/12)2
So, e = elongation = ∫024 P/[(πE/4)(1 + x/12)2] = 0.00000034P = 0.00780.
So, P = 22,900 lb

Maximum stress occurs where cross-sectional area is minimum, that is, cross-sectional area with diameter 1.
So, maximum stress, σmax = P/[(π/4)(1)2] = 22,900/0.786 = 29.135 lb/ in2. The point . is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.
Single Variable Functions
Conics
Ordinary Differential Equations (ODEs)
Vector Spaces
Real Numbers
Separation Of Variables As Solution Method For Homogeneous Heat Flow Equation
Newton And Fourier Cooling Laws Applied To Heat Flow Boundary Conditions
Fourier Series
Derivation Of Heat Equation For A One-Dimensional Heat Flow The Universe is composed of matter and radiant energy. Matter is any kind of mass-energy that moves with velocities less than the velocity of light. Radiant energy is any kind of mass-energy that moves with the velocity of light.
Periodic Table
Composition And Structure Of Matter
How Matter Gets Composed
How Matter Gets Composed (2)
Molecular Structure Of Matter
Molecular Shapes: Bond Length, Bond Angle
Molecular Shapes: Valence Shell Electron Pair Repulsion
Molecular Shapes: Orbital Hybridization
Molecular Shapes: Sigma Bonds Pi Bonds
Molecular Shapes: Non ABn Molecules
Molecular Orbital Theory
More Pj Problem Strings

Blessed are they that have not seen, and yet have believed. John 20:29

TECTechnic Logo, Kimberlee J. Benart | © 2018 | All rights reserved | Founder and Site Programmer, Peter O. Sagay.