chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/ch-3-review-load-and-stress-analysis.pdf ·...
TRANSCRIPT
![Page 1: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/1.jpg)
Chapter 3
Load and
Stress Analysis
![Page 2: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/2.jpg)
Shear Force and Bending
Moments in Beams
Internal shear force V & bending moment M
must ensure equilibrium
Saturday, September 26,
2015
Fig. 3−2 Mohammad Suliman Abuhaiba, Ph.D., P.E.
2
![Page 3: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/3.jpg)
Sign Conventions for Bending
and Shear
Mohammad Suliman Abuhaiba, Ph.D., P.E.Fig. 3−3
![Page 4: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/4.jpg)
Distributed Load on Beam
Distributed load q(x) = load intensity
Units of force per unit length
Saturday, September 26,
2015
Fig. 3−4 Mohammad Suliman Abuhaiba, Ph.D., P.E.
4
![Page 5: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/5.jpg)
Relationships between Load, Shear,
and Bending
Saturday, September 26,
2015
Mohammad Suliman Abuhaiba, Ph.D., P.E.
5
![Page 6: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/6.jpg)
Plane stress occurs = stresses on one surface
are zero
Saturday, September 26,
2015
Fig. 3−8
Cartesian Stress Components
Mohammad Suliman Abuhaiba, Ph.D., P.E.
6
![Page 7: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/7.jpg)
Plane-Stress Transformation Equations
Saturday, September 26,
2015
Fig. 3−9Mohammad Suliman Abuhaiba, Ph.D., P.E.
7
![Page 8: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/8.jpg)
Principal Stresses for Plane Stress
principal directions
principal stresses
Zero shear stresses at principal surfaces
Third principal stress = zero for plane stress
Saturday, September 26,
2015
Mohammad Suliman Abuhaiba, Ph.D., P.E.
8
![Page 9: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/9.jpg)
Extreme-value Shear Stresses for
Plane Stress
Max shear stresses: on surfaces that are
±45º from principal directions
Two extreme-value shear stresses:
Saturday, September 26,
2015
Mohammad Suliman Abuhaiba, Ph.D., P.E.
9
![Page 10: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/10.jpg)
Mohr’s Circle Diagram
Relation between x-y stresses and principal
stresses
Relationship is a circle with center at
C = (s, t) = [(sx + sy)/2, 0]
Saturday, September 26,
2015
2
2
2
x y
xyRs s
t
Mohammad Suliman Abuhaiba, Ph.D., P.E.
10
![Page 11: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/11.jpg)
Mohr’s
Circle
Diagram
Saturday, September 26,
2015
Fig. 3−10
Mohammad Suliman Abuhaiba, Ph.D., P.E.
11
![Page 12: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/12.jpg)
For a stress element undergoing sx, sy, and
sz, simultaneously,
Saturday, September 26,
2015
Elastic Strain
Mohammad Suliman Abuhaiba, Ph.D., P.E.
12
![Page 13: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/13.jpg)
Hooke’s law for shear:
Shear strain g = change in a right angle of astress element when subjected to pure
shear stress.
G = shear modulus of elasticity
For a linear, isotropic, homogeneous
material,
Saturday, September 26,
2015
Elastic Strain
Mohammad Suliman Abuhaiba, Ph.D., P.E.
13
![Page 14: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/14.jpg)
For tension and compression,
For direct shear (no bending present),
Uniformly Distributed Stresses
Saturday, September 26,
2015
Mohammad Suliman Abuhaiba, Ph.D., P.E.
14
![Page 15: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/15.jpg)
Normal Stresses for Beams in Bending
Straight beam in positive bending
x axis = neutral axis
xz plane = neutral plane
Neutral axis is coincident with centroidal
axis of the cross section
Saturday, September 26,
2015
Fig. 3−13
Mohammad Suliman Abuhaiba, Ph.D., P.E.
15
![Page 16: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/16.jpg)
Bending stress varies linearly with distance
from neutral axis, y
Saturday, September 26,
2015
Fig. 3−14
Normal Stresses for Beams in
Bending
Mohammad Suliman Abuhaiba, Ph.D., P.E.
16
![Page 17: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/17.jpg)
Transverse Shear Stress (TSS)
TSS is always accompanied
with bending stress
Saturday, September 26,
2015
Fig. 3−18
Mohammad Suliman Abuhaiba, Ph.D., P.E.
17
![Page 18: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/18.jpg)
Transverse Shear Stress in a
Rectangular Beam
Saturday, September 26,
2015
Mohammad Suliman Abuhaiba, Ph.D., P.E.
18
![Page 19: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/19.jpg)
Torsion
Angle of twist for a solid round bar
Saturday, September 26,
2015
Fig. 3−21
Mohammad Suliman Abuhaiba, Ph.D., P.E.
19
![Page 20: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/20.jpg)
Stress Concentration
Localized increase
of stress near
discontinuities
Kt = Theoretical
(Geometric) Stress
Concentration
Factor
Saturday, September 26,
2015
Mohammad Suliman Abuhaiba, Ph.D., P.E.
20
![Page 21: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/21.jpg)
Theoretical Stress
Concentration Factor
A-15 and A-16
Peterson’s Stress-Concentration
Factors
Saturday, September 26,
2015
Mohammad Suliman Abuhaiba, Ph.D., P.E.
21
![Page 22: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/22.jpg)
Stress Concentration for Static
and Ductile Conditions
With static loads and ductile materials
Highest stressed fibers yield (cold work)
Load is shared with next fibers
Cold working is localized
Overall part does not see damage unless
ultimate strength is exceeded
Stress concentration effect is commonly
ignored for static loads on ductile
materials
Saturday, September 26,
2015
Mohammad Suliman Abuhaiba, Ph.D., P.E.
22
![Page 23: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/23.jpg)
Stresses in Pressurized Cylinders
Tangential and radial stresses
Saturday, September 26,
2015
Fig. 3−31
Mohammad Suliman Abuhaiba, Ph.D., P.E.
23
![Page 24: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/24.jpg)
Special case of zero outside pressure, po = 0
Saturday, September 26,
2015
Stresses in Pressurized Cylinders
Mohammad Suliman Abuhaiba, Ph.D., P.E.
24
![Page 25: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/25.jpg)
If ends are closed, then longitudinal stresses
also exist
Saturday, September 26,
2015
Stresses in Pressurized Cylinders
Mohammad Suliman Abuhaiba, Ph.D., P.E.
25
![Page 26: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/26.jpg)
Thin-Walled Vessels
Cylindrical pressure vessel with wall
thickness 1/10 or less of the radius
Radial stress is quite small compared to
tangential stress
Average tangential stress
Saturday, September 26,
2015
Mohammad Suliman Abuhaiba, Ph.D., P.E.
26
![Page 27: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/27.jpg)
Thin-Walled Vessels
Maximum tangential stress
Longitudinal stress (if ends are closed)
Saturday, September 26,
2015
Mohammad Suliman Abuhaiba, Ph.D., P.E.
27
![Page 28: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/28.jpg)
Curved Beams in Bending
Saturday, September 26,
2015
Fig. 3−34
Mohammad Suliman Abuhaiba, Ph.D., P.E.
28
![Page 29: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/29.jpg)
Location of neutral axis
Stress distribution
Saturday, September 26,
2015
Curved Beams in Bending
Mohammad Suliman Abuhaiba, Ph.D., P.E.
29
![Page 30: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/30.jpg)
Stress at inner and outer surfaces
Saturday, September 26,
2015
Curved Beams in Bending
Mohammad Suliman Abuhaiba, Ph.D., P.E.
30
![Page 31: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/31.jpg)
Example 3-15
Plot the distribution of stresses across
section A–A of the crane hook shown in Fig.
3–35a. The cross section is rectangular, with
b = 0.75 in and h = 4 in, and the load is F =
5000 lbf.
Saturday, September 26,
2015
Mohammad Suliman Abuhaiba, Ph.D., P.E.
31
![Page 32: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/32.jpg)
Example 3-15
Fig. 3−35
Saturday, September 26,
2015
Mohammad Suliman Abuhaiba, Ph.D., P.E.
32
![Page 33: Chapter 3 - site.iugaza.edu.pssite.iugaza.edu.ps/.../09/Ch-3-Review-Load-and-Stress-Analysis.pdf · Shear Force and Bending Moments in Beams Internal shear force V & bending moment](https://reader031.vdocuments.net/reader031/viewer/2022022513/5aeb83827f8b9ac3618f5055/html5/thumbnails/33.jpg)
Formulas for Sections of
Curved Beams (Table 3-4)
Saturday, September 26,
2015
Mohammad Suliman Abuhaiba, Ph.D., P.E.
33