Lesson #30

Reading: 12:1-2

 

 

Deflection in Beams

 

If the bending moment or distributed load cannot be written as a single function over the entire length of the beam, the beam must be broken into sections in the same way we would for a shear and moment diagram.  Then a differential equation could be written for each section.  If we use the second order equation, we have two constants of integration for each section.  If the fourth order equation is used, there are four constants of integration for each section.  Now, the boundary conditions alone are not sufficient to determine all the constants.  We must use matching conditions.

 

Matching conditions are conditions based on beam continuity and equilibrium at the point where two sections meet.  Using the second order equation, our matching conditions are v₁ = v₂ and v₁’ = v₂’.  These are called continuity conditions because the beam would be broken if the deflection and slope from the two sections did not match at the meeting point.  If there is a pin or roller at the matching point, we get three matching conditions instead of two: v₁ = 0 , v₂ = 0 and v₁’ = v₂’.  Using the fourth order equation, we also consider equilibrium of a small piece of beam at the matching point.  Assuming that there might be a force and moment at the point,

 

SF_y = 0:          V₁ = V₂ + P_o                  or          E I v₁’’’ = E I v₂’’’ + P_o

 

SM = 0:          M₁ + M_o = M₂               or          E I v₁’’ + M_o = E I v₂’’

 

 

 

 

If there is a roller or pin support at the matching point, we would have a reaction force on the free body diagram.  Since the reaction is not known, we can ignore the force equation and just use the moment equation.  We will still have sufficient conditions because of the extra continuity condition.  There will always be four matching conditions at each matching point when using the fourth order differential equation.

 

When a beam has multiple sections, the number of constants of integration and the number of boundary and matching conditions makes for a large system of equations that needs to be solved algebraically.  This can take considerable time and is not particularly instructive.  Therefore, when we have a beam with multiple sections, we will just set up the problem, that is, write the differential equation for each section and then list the necessary boundary and matching conditions.  We will assume the problem can be solved given sufficient time.

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