MECHANICS  THEORY



Introduction

Various Examples of TwoMaterial Composite Beams 

Composite beams are constructed from more than one material to increase stiffness
or strength (or to reduce cost). Common compositetype beams include Ibeams
where the web is plywood and the flanges are solid wood members (sometimes referred
to as "engineered Ibeams"). Pipe beams sometimes have an outer liner
made from another type of material.
In this section, twomaterial composite beams will be examined. Of course, two materials
can be arranged in multisections but only two different type of materials will
be used. Beams with three or more materials are possible, but are rare and increase
the complexity of the equations. 





Twomaterial Composite Beams
 Axial Load

Strain and Stress in Twomaterial Composite Beam undergoing
Axial Loading 

The simplest loading in a composite beam is
axial loading. The strain is continuous across the beam
cross section but the stress is discontinuous as shown in the diagram at the
left. When axially loaded, the normal strains are equal since the two materials
are rigidly attached. From Hooke's law, this gives
ε_{1} = ε = σ_{1}/E_{1}
ε_{2} = ε = σ_{2}/E_{2}
Eliminating ε gives,
σ_{1}/E_{1} = σ_{2}/E_{2}
The total load P must equal the stresses times their respective areas, or
P = A_{1}σ_{1} +
A_{2}σ_{2}
Combining the previous two equations gives






Twomaterial Composite Beams
 Moment Load

Strain and Stress in Twomaterial Composite Beam undergoing
Moment Loading 

Similar to axially loaded twomaterial beams, when a beam is subjected to a
moment, the strain is still continuous, but the stress is discontinuous. Where
the stress and strain in axial loading is constant, the bending strain and stress
is a linear function through the thickness for each material section as shown
at the left.
The bending stress equations require the location of the neutral axis. For noncomposite
beams, the neutral axis (NA) is the centroid of the cross section. This is not
the case for composite beams and is one of the main difficulties in solving for
the bending stress. Thus, the first step in calculating bending stress is locating
the NA. Then the
bending stress equation, My/I, can be used to find the
stress in each material. There will be a separate equation for the bending stress
in each material section. 





Neutral Axis (NA) Location



As with noncomposite beams, the neutral axis (NA) is the location where the
bending stress is zero. The location of the NA depends on the relative stiffness
and size of each of the material sections.
Generally, the NA location is determined relative to the bottom surface of the
beam. However, this is not mandatory, and the location can be relative to any location.
If the bottom is used, then the NA axis is a distance "h" from the
bottom as shown in the diagram at the left. 



Neutral Axis Location for
Composite Beam 

The distance h can be determined by recalling that the stresses through
the cross section must be in equilibrium. Summing forces in the xdirection gives,
Recall, the bending stress in any beam is related to the radius
of curvature, ρ, as σ =
Ey/ρ,
Since the curvature is the same at all locations of a given cross section, this equation simplifies to
The two integrals are the first moment of each material area which is commonly
noted as simply Q, giving

0 = E_{1}Q_{1} + E_{2}Q_{2}


Generally, Q is not solved using the integral form since the centroid of each
material area will be known (or found in the
Sections appendix). Thus the equation
can also be written as

0 = E_{1} (y_{1} A_{1}) + E_{2} (y_{2} A_{2}) 

where y_{1} and y_{2} are the distance from the NA to the centroid
of the material area. Notice, "h" is not in this equation, but both y_{1} and
y_{2} depend
on h. Thus, the only unknown will be h and can be determined. Note, y will be negative if the centroid of the material area is below the NA. 





Bending Stresses

Neutral Axis Location for
Composite Beam 

The bending stress in a composite beam can be found by using the moment equilibrium
equation at any internal location. Summing the moments give,
Using the relationship between the bending stress and the radius
of curvature, σ =
Ey/ρ, gives,
Notice that the integral is the second moment of the area which is also the
area moment of inertia, I. This simplifies to
Rearranging gives
The bending stress in each material section is related to the beam curvature
as
Substituting the curvature into the above equations gives the final bending
stress for each material section.
Each equation is only valid for its material area. Also, these two equations
are for twomaterial composite beams only. 





Alternative Method  Equivalent Area

Equivalent Area Method Cross Section 

Another way to analyze composite beams is to use an equivalent area to represent the increased (or decreased) stiffness of the second material. The new equivalent cross section is assumed to be made completely from material 1. The area of material 2 is simply scaled to account for the stiffness difference using the scaling factor, n,
n = E_{2} / E_{1}
Note, the area scaling must only be done in the horizontal direction. The vertical dimension of either material cannot be changed.
The neutral axis can be found by finding the centroid of the full cross section, as was done with single material beams. Also, the bending stresses can be determined from the basic beam bending equation,
where I is the moment of inertia of the full equivalent cross section, and y is the distance from the neutral axis (down is negative).
While this method simplifies the equations, it is still basically the same calculations. It is important to be make sure the scaling factor, n, is correctly determined and applied to the area of the second material.



