Stresses in Beams
TECH 3401
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How the beam “feels” V and M
Shear and moment diagrams look the same regardless of the beam.
The stresses will be different.
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First, the “M”
The stresses in the beam that come from the moment.
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Consider this beam...
The section A-A is only experiencing a bending moment.
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Consider this beam...
The top part of A-A is in compression, the bottom part is in tension.
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Stress and Strain Distribution
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The Flexure Formula
Relates:
Bending Stress in the beam
Bending Moment
Geometric Properties of Cross-Section
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The M turns into this distribution of stress
Maximum stress is on outer layer:
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The Flexure Formula
For maximum stress
For stress at some section
For maximum allowable moment
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Section Modulus
I and c are functions of the size and shape of the beam.
They are properties of the cross-section and do not depend on the material or length of the span.
I/c is called the section modulus and is represented by the symbol S.
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Section Modulus
Listed in design manuals along with other section properties.
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Simpler Flexure Formula
Max bending stress
Allowable moment
Required section modulus
Use section modulus instead of I/c...
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Section Modulus (S)
A measure of the comparative strength of beams.
If all other things are equal, the beam with the higher section modulus will be stronger.
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Flexure Formula Limitations
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Flexure Formula Limitations
6) The beam must have adequate lateral buckling resistance.
7) All component parts must have adequate localized buckling resistance.
8) The beam must be relatively long in proportion to its depth.
9) The cross-section must be not be disproportionately wide.
10 The dimensional change must not be appreciably affected by shear strains.
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Computation of Bending Stresses
Maximum bending stress occurs at the outer layer of the beam where the bending moment “M” is at a maximum.
Watch units, often transition from ft*kips to inch*kips or inch*lbs.
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Example
Calculate the section modulus (S) of the beam:
If the timber were used as a simply supported beam to span 16 ft, with a uniformly distributed load of 400 lb/ft, find the maximum induced bending stress. Neglect the weight of the beam.
Inches
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Example
A W30 x 99 hot rolled shape is used as a simply supported beam on a span of 32 ft. The allowable bending stress for the beam is 24 ksi. The beam supports a uniformly distributed load of 4.0 kips/ft in addition to its own weight. Calculate the maximum bending stress. Is the beam satisfactory?
M =
S =
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Now, the “V”
The stresses that result from shear force.
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Note:
Usually moment is the critical factor. But in short spans, shear can become more important.
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Shear Stresses
This section deals with the shear stresses that result from the shear forces we saw distributed across the beam in the last chapter.
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Shear Stresses
The distribution of shear stress developed over a cross-section is different from the bending stress distribution.
Shear stress is zero at points on the cross-section where bending stress is at a maximum.
Shear stress is almost always at the neutral axis.
In a rectangular cross-section:
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Horizontal Shear Forces
If a shear stress exists vertically, there must be a horizontal force to balance.
Consider glue holding flat planks of a laminated beam
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Horizontal Shear Stress = Vertical Shear Stress
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Calculating the Shear Stress
V will usually come from shear diagram.
Shear will vary up and down the cross-section.
Usually maximum at neutral axis.
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Shear Stresses - Distribution along Cross-Section
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To find the shear stress (𝝉) at an arbitrary plane B-B
Q is statical moment of area
b is breadth of beam
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Q is what?
Q is a quantity known as the statical moment of area.
General Shear Formula:
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Example:
The beam shown is subjected to a shear V of 7,000 lb. Calculate the shear stress at 4” above & below the NA, and then at 2” above and below the NA. Then find the max shear stress at NA.
Plot the stress distribution.
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A convenient shortcut
To find the maximum shear stress in a solid, homogenous, rectangular section:
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Shear Stress Distribution in other Structural Shapes
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Notice this Connection
Comments on the trimmed off flanges? What is affecting the beam the most at this point?
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Average Web Shear
Since the flanges only resist a small portion of the total shear force, the “average web shear” can be used to evaluate shear stress in structural shapes.
d = full web thickness,
tw = web thickness
Note: The average web shear results in a lower value than the theoretically maximum value. Design standards are developed to compensate for this.
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Rewriting General Shear Formula for Shear Capacity
Or, using AVERAGE WEB SHEAR to find shear capacity:
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