# Egr 315

Topics: Force, Shear stress, Second moment of area Pages: 7 (2105 words) Published: March 25, 2013
EGR 315
Final Paper
Joe Hyde
There are 5 student learning outcomes covered in this semester. These leaning outcomes where the concentreation of time was spent in the semester and were deemed important by the powers that be. By grasping the concepets involved in these leaning outcomes they are what should be taken from this course and are to be applied in the workplace in the future. Not all of the topics can be defined in 10 pages. So the topics that follow are ones that I found importance in and were used very frequently to solve the problems in this semester. The topics to come are Shear Stresses for Beams in bending, Castigliano’s Theorem, Distortion Energy Theorem for Ductile Materials, Mechanics of Power screws, and Fatigue loading of helical compression springs. The Two learning objectives not covered where number 5 which is “design welded and brazed joints for mechanical components using correct welding symbols”. The welding chapters were not covered in the course of this semester, and number 7 which is “analyze stress and deflection using ANSYS Finite Element Analysis software for design optimization. Being that it may be a future class in the mechanical engineering program. Student learning outcome:

Demonstrate proficiency in analysis of stresses in different mechanical components such as beams and pressure vessels. Topic:
Shear Stresses for Beams in Bending
In most cases involving beams there are two things present, being shear forces and bending moments. Very infrequently do we have a beam in pure bending, having zero shear force. The flexure formula is developed on this assumption of pure bending. Shear force will be denoted as V and bending moment will be denoted as M at x. external loading causes the shear force and bending moment depend on the value of x. This shear force leads to shear stress, and if shear stress is assumed to be uniform. Thus

The term dM/I can be removed from inside the integral being that it is constant. Also b,dx can be moved to the right side of the equation. And from V=dM/dx. Yielding Equation 1
In this equation, the integral is the first moment of area with respect to the neutral axis. This integral is usually represented by Q. yielding Equation 2
Were for the isolated area y1 to c, y’ is the distance in the y direction from the neutral plane to the centroid of the area A’. thus yielding Equation 3
This stress from equation 3 is known as the transverse shear stress, and is always accompanied with bending stress. Defining the variables in this equation, b is the width of section at y=y1, and I is the second moment of area of the entire section about the neutral axis. Being that cross shears are equal, and area A’ being finite, the shear stress can be calculated with equation 3. The shear stress distributing in a beam depends on how Q/b varies as a function of y1. For a beam with a rectangular cross sectional area, subjected to a shear force V and a bending moment M. as a result of the bending moment a normal stress is developed on a cross section, which is compression above the neutral axis and it is tension below the neutral axis. To investigate the shear stress at a distance y1 above the neutral axis. Then dA=bdy, so equation 2 becomes Equation 4

Subbing this value in for Q into equation 3 gives
Equation 5
Equation 5 is known as the general equation for shear stress in a rectangular beam. The second moment of area for a rectangular section from appendix A-18, is

Subbing in ,
This yield s

If this value is known for I, equation 5 becomes
Equation 6
Also being that shear stress is at a maximum when y1=0, which is at the bending neutral axis gives us Equation 7

Student learning outcome:
Define and apply the concepts of deflection of beams due to bending and strain energy, Castigliano’s method, Euler column. Topic:
Castigliano’s Theorem
One of the simplest ways to approach deflection analysis with the energy method is to use Castigliano’s theorem....

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