Deflection of Beams - Mechanics of Materials

Elastic Curve: showing the slope and displacement for a beam when loading is applied. Often called the deflection diagram, showing the elastic curve. This curve is fairly simple to sketch just knowing simple restrictions for the slope and displacement.
Supports such as a pin or a roller will resist a force and therefore will restrict displacement.
Supports such as a fixed wall will resist a moment and will restrict displacement and rotation(slope)

Drawing the moment diagram also helps when sketching the elastic curve. When there is a positive internal moment the elastic curve will be concave upward, and concave downward when the internal moment is negative. When the moment is zero, this is the inflection point on the elastic curve.

Looking at the elastic curve where the slope is zero, this point may be a maximum deflection point. This depends on the magnitude and spacing of the force(s) applied to the beam and the spacing of supports.


Three coordinates will help create a relationship regarding the internal moment and the radius of curvature.

• X-axis - locates the element along the initially straight longitudinal axis
• V-axis - measures the displacement from the x-axis to the centroid
• Y-coordinate - measured from the neutral axis and locates the position of a fiber in the element of the beam

1 / ρ = M / E I

where...

• ρ is the radius of curvature
• M is the internal moment at the point where ρ is
• E is the modulus of elasticity for the material
• I is the moment of inertia about the neutral axis of the cross section of the beam

E I is commonly know as the flexural rigidity. This is always a positive number.
1 / ρ is known as the curvature.



Slope and Displacement by Integration

The equation for the elastic curve is a second-order differential, called the elastica. This equation uses the curvature in terms of v and x. But this equation only gives the exact shape for beam defections that have bending.

Skipping some mathematical steps we are left with 3 equations which are...

• EI (d'''' v / dx'''') = -w(x) - Distributed Load
  
• EI (d''' v / dx''') = V(x) - Shear Force
  
• EI (d'' v / dx'') = M(x) - Bending Moment
  

When integrating the above equations remember to solve for the constant of integration. With each equation the integration yields a constant which will be solved for to get a unique solution. Boundary conditions will allow the evaluation of the constant from integration. These constants are determinded by evaluating the function at a point.


Boundary Conditions

• Supports(pin or roller) located at the end of a beam: displacement and moment are zero
• Beam supported by a support(not located at the end): displacement are
• The fixed end of a beam: slope and displacement are zero
• The free end of a beam: shear stress and moment are zero
• Internal pin/hinge: moment will be zero

When the above conditions can't be used to solve for the slope or the deflection continuity conditions will be used.

Continuity Condition

Take 2 different sections of a beam with one common point. The function for the slope or deflection of section 1 of the beam must be equal to the function for the slope or deflection of section 2 at the common point. Both functions must produce the same answer at that common point so the elastic curve will be continuous.
Example: For displacement v1(a) = v2(a) and for slope θ1(a) = θ2(a)


Superposition: analizing each load on a beam then algebraically adding the various components of the loads together to get the original beam and loadings. This is used to determine the slope and displacement for different points on the beam.


Statically Indeterminate Beams (Method of Superposition)

An indeterminate beam is when there are more unknown reactions than equations needed to solve the reactions.

Force method: Identify the redundant support reaction.

• Redundant support: The extra support on the beam which is not needed to keep the beam in a stable state of equilibrium.

Remove the redundant(s) from the beam which yields the primary beam and this will be statically determinate, stable, and subjected to external load. This primary beam will either be a cantilever beam or a simply supported beam.

By principle of superposition we obtain the original beam and loadings from the addition of a succession of supported beams, each loaded with a separate redundant.

Where each of the redundants act, a condition of compatibility must be written. The redundant will have a zero change in displacement in the actual beam, so the sum of the displacements from the different superimposed beams will be zero. By using this condition of compatibility, the redundant support reaction can be solved. Tables are used to look up information for the slopes and deflections of beams.

Once the redundant(s) have been solved for, the other reactions can be found through equations of equilibrium.


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