chapter 6: bending - university of illinois urbana-champaign
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Chapter 6: Bending
Chapter Objectives
Determine the internal moment at a section of a beam
Determine the stress in a beam member caused by bending
Determine the stresses in composite beams
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Bending analysisThis wood ruler is held flat against the table at the left, and fingers are poised to press against it. When the fingers apply forces, the ruler deflects, primarily up or down. Whenever a part deforms in this way, we say that it acts like a βbeam.β In this chapter, we learn to determine the stresses produced by the forces and how they depend on the beam cross-section, length, and material properties.
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Support types:
Load types:
Concentrated loads Distributed loads Concentrated moments
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Sign conventions:
How do we define whether the internal shear force and bending moment are positive or negative?
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ππ
Given:ππ = 2 kNπΏπΏ = 1 m
πΏπΏ πΏπΏ
Shear and moment diagrams Find:ππ π₯π₯ ,ππ π₯π₯ , Shear-Bending moment diagram along the beam axis
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ππ π€π€
Given:ππ = 4 kN
π€π€ = 1.5 kN/mπΏπΏ = 1 m
πΏπΏ πΏπΏ 2πΏπΏ
Example: cantilever beam Find:ππ π₯π₯ ,ππ π₯π₯ , Shear-Bending moment diagram along the beam axis
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Relationship between load and shear:
Relationship between shear and bending moment:
Relations Among Load, Shear and Bending Moments
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Wherever there is an external concentrated force, or a concentrated moment, there will be a change (jump) in shear or moment respectively.
Concentrated force F acting upwards:
Concentrated Moment π΄π΄ππacting clockwise
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ππ
Given:ππ = 2 kNπΏπΏ = 1 m
πΏπΏ πΏπΏ
Shear and moment diagrams Use the graphical method the sketch diagrams for V(x) and M(x)
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ππ π€π€
Given:ππ = 4 kN
π€π€ = 1.5 kN/mπΏπΏ = 1 m
πΏπΏ πΏπΏ 2πΏπΏ
Example: cantilever beam Use the graphical method the sketch diagrams for V(x) and M(x)
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Pure bendingTake a flexible strip, such as a thin ruler, and apply equal forces with your fingers as shown. Each hand applies a couple or moment (equal and opposite forces a distance apart). The couples of the two hands must be equal and opposite. Between the thumbs, the strip has deformed into a circular arc. For the loading shown here, just as the deformation is uniform, so the internal bending moment is uniform, equal to the moment applied by each hand.
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We assume that "plane sections remain plane" βAll faces of βgrid elementsβ remain at 90ππ to each other, hence
Therefore,
No external loads on y or z surfaces:
Thus, at any point of a slender member in pure bending, we have a state of uniaxial stress, since πππ₯π₯ is the only non-zero stress component
For positive moment, M > 0 (as shown in diagram):Segment AB decreases in lengthSegment AβBβ increases in length
Hence there must exist a surface parallel to the upper and lower where
This surface is called NEUTRAL AXIS
Geometry of deformation
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Geometry of deformationπ΅π΅πΎπΎ
π΅π΅π΅πΈπΈ
π΄π΄π½π½
π΄π΄π΅π·π·
π¦π¦
π¦π¦
π₯π₯
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Constitutive and Force Equilibriumπππ₯π₯ = πΈπΈπππ₯π₯ = β
πΈπΈπ¦π¦ππ
Force equilibrium:
Constitutive relationship:
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Moment Equilibrium
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Centroid of an area
The centroid of the area A is defined as the point C of coordinates οΏ½Μ οΏ½π₯ and οΏ½π¦π¦, which satisfies the relation
In the case of a composite area, we divide the area A into parts π΄π΄1, π΄π΄2, π΄π΄3
π΄π΄π‘π‘πππ‘π‘π‘π‘π‘π‘ οΏ½π¦π¦ = οΏ½ππ
π΄π΄ππ οΏ½π¦π¦πππ΄π΄π‘π‘πππ‘π‘π‘π‘π‘π‘οΏ½Μ οΏ½π₯ = οΏ½ππ
π΄π΄πποΏ½Μ οΏ½π₯ππ
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Example: Find the centroid position in the π¦π¦π¦π¦ coordinate system shown for π‘π‘ = 20cm
2 π‘π‘ π‘π‘π‘π‘
π‘π‘
3 π‘π‘
π¦π¦
π¦π¦
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Second moment of area The 2nd moment of the area A with respect to the x-axis is given by
The 2nd moment of the area A with respect to the y-axis is given by
Example: 2nd moment of area for a rectangular cross section:
ππ
πππ¦π¦
π¦π¦
π¦π¦
β2
β2
π¦π¦
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Centroids and area moments of area: Formula sheet
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Parallel-axis theorem: the 2nd moment of area about an axis through C parallel to the axis through the centroid Cβ is given by
Example: Find the 2nd moment of area about the horizontal axis passing through the centroid assuming t = 20 cm
2 π‘π‘ π‘π‘π‘π‘
π‘π‘
3 π‘π‘
π¦π¦
π¦π¦
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The maximum magnitude occurs the furthest distance away from the neutral axis. If we denote this maximum distance βcβ, consistent with the diagram below, then we can write
Bending stress formula
ππππ =ππ πππΌπΌπ§π§
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Bending stress sign
ππππ
ππππ
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Example: Find the maximum tensile and compressive stresses in this beam subjected to moment πππ§π§ = 100 N-m with the moment vector pointing in the direction of the z-axis. Again take t = 20 cm.
2 π‘π‘ π‘π‘π‘π‘
π‘π‘
3 π‘π‘
π¦π¦
π¦π¦
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Why I-beams?
http://studio-tm.com/constructionblog/wp-content/uploads/2011/12/steel-i-beam-cantilevered-over-concrete-wall.jpg
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β’ Maximum stress due to bending
ππ =πππππΌπΌ
β’ Bending stress is zero at the neutral axis and ramps up linearly with distance away from the neutral axis
β’ πΌπΌ is the 2nd moment of area about the neutral axis of the cross section
β’ Be sure to find the cross-sectionβs centroid and evaluate I about an axis passing through the centroid, using the parallel axis theorem if needed
β’ To determine stress sign, look at the internal bending moment direction:
β’ Side that moment curls towards is in compressionβ’ Side that moment curls away from is in tension
Summary of bending in beams
π₯π₯