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Shear Calculation

The document discusses shear stresses in beams. It explains that the maximum shear stress occurs at the neutral axis and is 50% larger than the average shear stress. It also notes that the shear stress formula is only valid for beams made of linearly elastic materials with small deflections and rectangular cross-sections. The document further discusses shear flow in built-up composite beams, defining shear flow as the force per unit length along the beam axis. Shear flow is used to determine shear forces in fasteners that hold segments of a composite beam together.

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John Patric Ranjo Sia
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116 views12 pages

Shear Calculation

The document discusses shear stresses in beams. It explains that the maximum shear stress occurs at the neutral axis and is 50% larger than the average shear stress. It also notes that the shear stress formula is only valid for beams made of linearly elastic materials with small deflections and rectangular cross-sections. The document further discusses shear flow in built-up composite beams, defining shear flow as the force per unit length along the beam axis. Shear flow is used to determine shear forces in fasteners that hold segments of a composite beam together.

Uploaded by

John Patric Ranjo Sia
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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SHEAR STRESSES IN BEAMS

Source: Gere & Goodno

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CAGAYAN STATE UNIVERSITY – CARIG CAMPUS AGGonzales


; ;

The quantity dM/dx is equal to the shear force V (see Eq.


4-6), and therefore the preceding expression becomes

The integral in this equation is evaluated over the shaded


part of the cross section (Fig. 5-28d), as already explained.
Thus, the integral is the first moment of the shaded area with
respect to the neutral axis (the z axis). In other words, the
integral is the first moment of the cross-sectional area above
the level at which the shear stress t is being evaluated. This
first moment is usually denoted by the symbol Q:
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CAGAYAN STATE UNIVERSITY – CARIG CAMPUS AGGonzales


Distribution of Shear Stresses in a Rectangular Beam

The maximum value of the shear stress occurs at the neutral


axis (y1 = 0) where the first moment Q has its maximum value.
Substituting y1 = 0

in which A = bh is the cross-sectional area. Thus, the maximum


shear stress in a beam of rectangular cross section is 50% larger
than the average shear stress V/A.

Note: they are valid only for beams of linearly elastic materials
with small deflections. A common error is to apply the shear
formula to crosssectional shapes for which it is not applicable.
For instance, it is not applicable to sections of triangular or
semicircular shape.

To avoid misusing the formula, we must keep in mind the following


assumptions that underlie the derivation: (1) The edges of the
cross section must be parallel to the y axis (so that the shear
stresses act parallel to the y axis), and (2) the shear stresses
must be uniform across the width of the cross section. These
assumptions are fulfilled only in certain cases, such as those
discussed in this and the next two sections.
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CAGAYAN STATE UNIVERSITY – CARIG CAMPUS AGGonzales


Shear Flow in Built-Up Members

The integral represents Q, that is, the moment of the


segment’s area A’ about the neutral axis for the entire cross
section. Since the segment has a length dx, the shear flow, or
force per unit length along the beam, is q = dF/dx. Hence
dividing both sides by dx and noting that V = dM/dx; thus

Shear flow is a measure of the force per unit length along


the axis of a beam.This value is found from the shear formula
and is used to determine the shear force developed in fasteners
72

and glue that holds the various segments of a composite beam


Page

together.

CAGAYAN STATE UNIVERSITY – CARIG CAMPUS AGGonzales


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