
MECHANICS  THEORY

Beam Deflection, v, and Slope, dv/dx


Previously, beam stresses and strains were investigated and various equations were developed to predict bending and shear stresses. In addition to stresses, deflection and slope are important and need to be calculated. This section (and this chapter) will deal with various methods to calculate beam deflections. 





Deflection Differential Equation

Bending Moment, M, and
Radius of Curvature, ρ are related 

When a moment acts on a beam, the beam rotates and deflects. The relationship
between the radius of curvature, ρ, and the moment, M,
at any given point on a beam was developed in the Bending
Stress and Strain section as
This relationship was used to develop the bending stress equation but
it can also be used to derive the deflection equation.
Recall from calculus, the
radius of curvature for any point of a function, y = f(x), is




Geometry of Beam Bending
and Deflection 

For beams, it is convenient to note the deflection as v (upward is positive)
instead of y. Also, for small deflections, the first derivative or slope,
dy/dx, is small. If it is squared, then it is very small and can be assumed
to be zero. In other words, (dv/dx)^{2} ≈ 0. This simplifies the
radius of curvature equation to
Combining this with the bending stress equation gives standard
momentcurvature equation,
This differential equation is also commonly written as
where the prime, ´ , represents a derivative with respect to x. Since there
are two derivatives, there are two slashes. (It is good to recall from dynamics,
a dot above a letter represents a time derivative, but a slash represents a
spatial derivative.) 





Solving the Deflection Differential
Equation (MomentCurvature Equation)

Each Beam Section Requires its
Own Deflection Equation 

The differential equation EIv´´= M is not useful by itself but
needs to be applied to a beam with specific boundary conditions. Generally, EI
is constant and M is a function of the beam length. Integrating the equation
once gives,
Integration again gives,
The integration constants, C_{1} and C_{2}, are determined from
the boundary conditions. For example, a pinned joint requires the deflection,
v, equals 0. A fixed joint requires both the deflection, v, and slope, v´,
equal 0. Each beam section must have at least two boundary conditions.
Each beam span must be integrated separately, just like when
constructing a moment diagram.
If the moment curve is discontinuous, then a single equation
cannot model the deflection. 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 for Beam Sections 

Boundary Conditions


Determining the boundary conditions is usually the most difficult
part of solving the deflection differential equation. In particular, boundary conditions
for multiple beam sections can be confusing.
For multiple beam sections, many times the boundary between the sections creates
a boundary. These type of conditions are also called "Continuity 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}. These conditions are needed
to solve for the additional integration constants.
The table at the left summarizes most common boundary (and continuity) conditions
for beam deflection and slope. 



