# Composite Beams ```Composite Beams
examples
plastic coated steel pipe
bimetallic
wood reinforced with steel plate
sandwich - corrugated core
sandwich - plastic core
sandwich - honeycomb core
reinforced concrete
stress and strain
Moment-curvature relationship for a composite beam
Flexure formulas for a composite beam
Transformed-Section Method
This method is an alternative to the preceding section. It provides a convenient way to visualize the
calculations.
steps
1. Transform the cross section of a composite beam into an equivalent cross
section (called the transformed section) of an imaginary beam composed of only one
material.
2. Analyze the transformed section as customary for a beam of one material.
3. Convert the stresses back to the original beam.
modular ratio
To transform the beam into material 1:
The dimensions of area 1 remain unchanged, and the width of area 2 is multiplied by n. (all
dimensions perpendicular to the neutral axis remain the same)
A similar procedure can be used to transform the beam into material 2 or a completely different
material. One can also extend this technique to cover beams of more than two materials.
flexure formulas
MECHANICS - THEORY
Introduction
Composite beams are constructed from more than one
material to increase stiffness or strength (or to reduce
cost). Common composite-type beams include I-beams
where the web is plywood and the flanges are solid
wood members (sometimes referred to as "engineered
I-beams"). Pipe beams sometimes have an outer liner
made from another type of material.
Various Examples of Two-Material
Composite Beams
In this section, two-material composite beams will be
examined. Of course, two materials can be arranged in
multi-sections 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.
Two-material Composite Beams
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
ε1 = ε = σ1/E1
ε2 = ε = σ2/E2
Eliminating ε gives,
σ1/E1 = σ2/E2
The total load P must equal the stresses times their
respective areas, or
P = A1σ1 + A2σ2
Strain and Stress in Two-material
Composite Beam undergoing
Combining the previous two equations gives
Two-material Composite Beams
Similar to axially loaded two-material beams, when a
beam is subjected to a moment, the strain is still
continuous, but the stress is discontinuous. Where the
bending strain and stress is a linear function through
the thickness for each material section as shown at the
left.
Strain and Stress in Two-material
Composite Beam undergoing
The bending stress equations require the location of the
neutral axis. For non-composite 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 non-composite 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.
The distance h can be determined by recalling that the
stresses through the cross section must be in
equilibrium. Summing forces in the x-direction gives,
Recall, the bending stress in any beam is related to
the radius of curvature, ρ, as σ = -Ey/ρ,
Neutral Axis Location for
Composite Beam
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 = E1Q1 + E2Q2
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 = E1 (y1 A1) + E2 (y2 A2)
where y1 and y2 are the distance from the NA to the
centroid of the material area. Notice, "h" is not in this
equation, but both y1 and y2 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
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
Neutral Axis Location for
Composite Beam
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 two-material composite
beams only.
Alternative Method - Equivalent Area
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,
Equivalent Area Method Cross Section
n = E2 / E1
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.
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