Lecture 38:

• Select a beam that has adequate strength to carry a given load in bending.
  
  
• Determine the bending stresses in a given beam caused by a specific loading condition.
  
  
• Determine the greatest moment a given beam will resist (or the greatest load it will carry)
  

S is the Section Modulus in in^3. This beam property can either be calculated or read from tables for most beams.

M is the moment in the beam in inch-pounds (or inch-kips). In the selection of a beam it is taken from the moment diagram.

F is the bending stress (psi or ksi) (Fb is usually used to denote the allowable bending stress; fb is used to indicate the actual bending stress).

• The beam has a constant, prismatic cross-section and is constructed of a flexible, homogenous material that has the same Modulus of Elasticity in both tension and compression (shortens or elongates equally for same stress).
  
• The material is lineraly elastic; the relaitonship between the stress and strain ar directly proportional.
• The beam material is not stressed past its proportional limit.
  
• A plane section within the beam before bending remains a plane after bending (see AB & CD in the image below).
  
• The neutral plane of a beam is a plane whose length is unchanged by the beam's deformation. This plane passes throught the centroid of the cross-section.
  In order to visualize this, think of a black rubber beam with three lines drawn on its side. The dashed lines in the diagram below represent this beam, and the neutral axis and lines AB and CD are drawn parallel to their respective sides. Lines AB and C D are separated by some distance that is determined, but is not of consequence for the following discussion. These two lines were parallel before bending. As the beam bends, these lines remain perpendicular to the neutral axis.
  
  
  
  
  
  Thus, as the beam bends, developing a curve in the neutral plane that reflects the bending, the line CD does not remain parallel to line AD. There is a distinct shortening at the top face of the beam and elongation at the bottom face. CC' is equal to d (delta), or the shortening of the top fiber. Similarly, DD' is equal to (delta) or the elongation of the bottom fiber (tension). If these were measured, they would be found to be equal, but opposite. Knowing that the Modulus of Elasticity (E) describes a linear relationship between the strain and the amount of stress in a material and that this material is homogeneous and has a certain value for E, we can determine the stress. The magnitude of the strain at any point along C'D' can be found by using similar triangles. Therefore, the stress is also proportional to the distance from the neutral plane, as illustrated in the following diargram.
  
  
  
  
  
  The loads and reactions acting on this beam segment cause a tendency for clockwise rotation (a clockwise moment). An internal moment counteracts this tendency so that the beam segment remains in equilibrium. Thus, the magnitude of the internal moment is exactly equal to the moment due to the external loads and reactions; but in the opposite direction. There is a state of equilibrium:
  
  Internal Moment = External Moment
  The illustration above shows the stress prisms developed due to the straining of the material. These prisms (compression and tension) can each be resolved into a single force that acts through the centroid of each prism. These two resulting forces (one is a compressive force (C) and the other a tensile force (T) ) are found to have equal but opposite magnitudes. They are separated by a distance that is approximatly 2/3 of the depth of the beam (because of their triangular geometry) and create an internal couple(!) which causes the internal moment resistance of the beam. The exact location of the resultant forces of the stress prisms depends upon the locaton of the centroid of the entire stress prism. The location of the centroid is in turn dependent upon the exact distribution of the stresses across the section which this is determined by the geometry of the cross-section.
  

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  GENERAL CASE
  If one accepts the assumptions given in the discussion of the bending theory above, the formula for determining the bending stress at any given fiber in a section can be derived as follows.
  
  
  
  
  Given the beam as shown above loaded so that it has an internal bending moment. The stress at the extreme face (the distance C in this case) from the Neutral Axis (NA) is equal to Sigma or fb. The stress at any distance "y"; from the NA can be found using similar triangles. Thus,
  
  fy = (fb/c)(y)
  
  If "a" is a very small area at a distance "y" from the NA, then magnitude of the stress on area "a" equals (fb/c)(y). Remembering that stress is equal to a force distributed over an area, the force on the area "a" is equal to the stress times that area, or (fb/c)(y)(a). And, the moment of the force on area "a" about the NA is equal to (fb/c)(y)(a)(y).
  
  Now, if all of the moments of all of the forces on all of the tiny "a" areas across the face of the beam are added together we arrive at an expression of the internal moment resistance of a beam;
  
  M =Sum (f/c)ay^2 or M = (f/c) Sum ay^2
  
  In this case, the value of (f/c) is a constant for any beam and Sum ay^2 is the second moment of the area, better known as the Moment of Inertia. Therefore, substituting items that we know, the equation becomes:
  
  M = (fb/c) I
  
  or rearranging
  
  M = (I/c) fb
  or
  M = S*fb
  (I/c) is known as the Section Modulus (S) and is also tabulated.
  
  

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  BENDING STRESSES REVIEW
  The flexure formula: M = (f/c) I or M = (I/c) f
  

  M is the internal moment in a beam created by the loads and reactions. It is numerically equal to the external moment at any section. Its value can be taken directly from the moments diagram. M is in inch-pounds (or inch-kips) .
  
  f is the stress at any distance "c" from the NA. If "c" is taken as the distance from the NA to the most remote fiber, then the stress "f" is the greatest stress. c is in inches; f is psi (or ksi).
  
  I is the Moment of inertia of the beam. I is in inches^4.
  
  S is the Section Modulus. It is (I/c.) and has the units of inches^3.

M = (Fb/c) I to find the resisting moment of a beam

fb = M c /I to find the stress in a beam

I / c = Fb to select a beam

Sreq = M / Fb to select a beam