
MECHANICS  THEORY



In the previous sections,
Integration of the Moment Equation, was shown how
to determine the deflection if the moment equation is known. This section will
extend the integration method so that with additional boundary conditions, the
deflection can be found without first finding the moment equation. 





MomentShearLoad Relationships

Differential Element from Beam 

When
constructing momentshear diagrams, it was noticed that there is a relationship
between the moment and shear (and between the shear and the loading). That relationship
can be derived by applying the basic equations to a typical differential
element from a loaded beam (shown at the left). First, summing the forces in
the vertical direction gives
ΣF_{y} = 0
V  (V + dV)  w(x) dx + 0.5 (dw) dx = 0
Both dw and dx are small, and when multiplied together gives an extremely small
term which can be ignored. Assuming (dw)(dx) = 0, and simplifying gives,
Next, summing moments about the right side (can be anywhere, but an edge is
easier) and ignoring the 3rd order terms gives
ΣM_{right edge} = 0
M + (M + dM)  V dx + [w(x) dx][0.5 dx] = 0
Again, 2nd order terms such as dx^{2}, are assumed extremely small
and can be ignored. This gives
Note, capital "V" is shear and not deflection, which is small "v". 





Extending the Deflection Differential Equation

w, M, V, Slope, and y Relationships
and Sign Conventions
(Note, w(x) is positice downward) 

Recall, the basic
deflection differential equation (MomentCurvature Equation)
was derived as
This can be combined with dM/dx = V to give

EIv´´´ = V(x)

sheardeflection equation 
This equation assumes E and I are constant along the length of the beam section.
They can be combined with dV/dx = w(x) to give

EIv´´´´ = w(x)

loaddeflection equation 
Thus, the deflection can be determined directly from the load function, but
it does require four integrations and four boundary conditions. Where as using
the momentcurvature equation, only two integrations and two boundary conditions
are needed, but the moment equation must first be determined. 





Solving the LoadDeflection Differential
Equation

Each Beam Section Requires its
Own Deflection Equation 

The differential equation EIv´´´´ = w(x) is not useful
by itself but needs to be applied to a beam with specific boundary conditions.
It is assumed that EI is constant and w(x) is a function of the beam length.
The function, w(x), can be equal to 0. In fact, in most situations
it does equal 0. For convenience, w(x) is considered positive when acting downward.
Integrating the equation four times gives,
The integration constants, C_{1}, C_{2}, C_{3} and
C_{4}, are determined from
the boundary conditions. For example, a pinned joint at either end of a beam
requires the deflection, v, equal 0 and the moment, M, equal 0.. A fixed joint
requires both the deflection, v, and slope, v´,
equal 0, but moment and shear are unknown. Each beam section must have at least
four boundary conditions. Details about boundary conditions are given below.
Each beam span must be integrated separately, just like when
constructing a
moment diagram. Thus, each new support or load will start
a new beam section that must be integrated. Examples of beam sections are shown
at the left. 





Boundary Conditions


Determining the boundary conditions is usually the most difficult part of solving the deflection differential equation, especially when integrating four times. In particular, boundary conditions for multiple beam sections can be confusing.
The basic types of boundary conditions are shown below. Those conditions that require two sections are sometimes called continuity conditions instead of boundary conditions. For example, a point force on a beam causes the deflections to be split into two equations. However, the beam's deflection and slope will be continuous at the load location requiring v_{1} = v_{2} and v´_{1} = v´_{2}. Also, the shear difference will equal the applied point load at that location, and the moment will be equal in both beam sections at that point, M_{1} = M_{2}. These conditions are needed to solve for the additional integration constants. 



Typical Boundary Conditions (v, v´, V, M) for
Beam Sections
Using Fourth Order LoadDeflection Equation 





Beam Deflection and Rotation Tool



For complex beams with more than a couple loads, determining rotation and deflection is very difficult. Thus, most structural engineers will use a beam analysis tool to calculate the deflection and rotation curves. A simple one, based on finite element method, can be installed on any mobile device for free. .
For Android mobile devices, "Beam HPC" calculator, can be downloaded at Google Play.
For iOS (Apple) mobile devices, "Beam HPC" calculator, can be downloaded at iTunes. 



