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Catégorie :Category: mViewer GX Creator App HP-Prime
Auteur Author: matheusalmeida
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07Ch07.qxd 9/27/08 1:18 PM Page 571

7
Analysis of Stress
and Strain

Plane Stress
1200 psi
Problem 7.2-1 An element in plane stress is subjected to stresses sx ⫽ 4750 psi,
sy ⫽ 1200 psi, and txy ⫽ 950 psi, as shown in the figure.
Determine the stresses acting on an element oriented at an angle u ⫽ 60˚ from the
x axis, where the angle u is positive when counterclockwise. Show these stresses on a 950 psi
sketch of an element oriented at the angle u.
4750 psi

Solution 7.2-1
sx ⫽ 4750 psi sy ⫽ 1200 psi txy ⫽ 950 psi sx ⫺ sy
tx1y1 ⫽ ⫺ sin(2u) + txy cos(2u)
u ⫽ 60° 2
sx + sy sx ⫺ sy tx1y1 ⫽ ⫺2012 psi ;
sx1 ⫽ + cos(2u) + txy sin(2u)
2 2 sy1 ⫽ sx + sy ⫺ sx1
sx1 ⫽ 2910 psi ; sy1 ⫽ 3040 psi ;

Problem 7.2-2 Solve the preceding problem for an element in plane stress 80 MPa
subjected to stresses sx ⫽ 100 MPa, sy ⫽ 80 MPa, and txy ⫽ 28 MPa, as shown in
the figure.
Determine the stresses acting on an element oriented at an angle u ⫽ 30˚ from 28 MPa
the x axis, where the angle u is positive when counterclockwise. Show these stresses
on a sketch of an element oriented at the angle u. 100 MPa

571
07Ch07.qxd 9/27/08 1:18 PM Page 572

572 CHAPTER 7 Analysis of Stress and Strain

Solution 7.2-2
sx ⫽ 100 MPa sy ⫽ 80 MPa txy ⫽ 28 MPa sx ⫺ sy
tx1y1 ⫽ ⫺ sin(2u) + txy cos(2u)
u ⫽ 30° 2
sx + sy sx ⫺ sy tx1y1 ⫽ 5.30 MPa ;
sx1 ⫽ + cos (2u) + txy sin (2u) sy1 ⫽ sx + sy ⫺ sx1
2 2
sx1 ⫽ 119.2 MPa ; sy1 ⫽ 60.8 MPa ;

Problem 7.2-3 Solve Problem 7.2-1 for an element in plane stress subjected to 2300 psi
stresses sx ⫽ ⫺5700 psi, sy ⫽ ⫺2300 psi, and txy ⫽ 2500 psi, as shown in the figure.
Determine the stresses acting on an element oriented at an angle u ⫽ 50˚ from the x
axis, where the angle u is positive when counterclockwise. Show these stresses on a sketch 2500 psi
of an element oriented at the angle u.
5700 psi

Solution 7.2-3
sx ⫽ ⫺5700 psi sy ⫽ ⫺2300 psi txy ⫽ 2500 psi sx ⫺ sy
tx1y1 ⫽ ⫺ sin (2u) + txy cos (2u)
u ⫽ 50° 2
sx + sy sx ⫺ sy tx1y1 ⫽ 1240 psi ;
sx1 ⫽ + cos (2u) + txy sin (2u) sy1 ⫽ sx + sy ⫺ sx1
2 2
sx1 ⫽ ⫺1243 psi ; sy1 ⫽ ⫺6757 psi ;

Problem 7.2-4 The stresses acting on element A in the web of 160 MPa
a train rail are found to be 40 MPa tension in the horizontal
direction and 160 MPa compression in the vertical direction
(see figure). Also, shear stresses of magnitude 54 MPa act in the
directions shown. 40 MPa
A
Determine the stresses acting on an element oriented at a
counterclockwise angle of 52˚ from the horizontal. Show these A
54 MPa
stresses on a sketch of an element oriented at this angle.
Side Cross
View Section
07Ch07.qxd 9/27/08 1:18 PM Page 573

SECTION 7.2 Plane Stress 573

Solution 7.2-4
sx ⫽ 40 MPa sy ⫽ ⫺160 MPa txy ⫽ ⫺54 MPa sx ⫺ sy
tx1y1 ⫽ ⫺ sin (2u) + txy cos (2u)
u ⫽ 52° 2
sx + sy sx ⫺ sy tx1y1 ⫽ ⫺84.0 MPa ;
sx1 ⫽ + cos(2u) + txy sin(2u) sy1 ⫽ sx + sy ⫺ sx1
2 2
sx1 ⫽ ⫺136.6 MPa ; sy1 ⫽ 16.6 MPa ;

Problem 7.2-5 Solve the preceding problem if the normal 18,500 psi
and shear stresses acting on element A are 6500 psi, 18,500 psi,
and 3800 psi (in the directions shown in the figure).
Determine the stresses acting on an element oriented at a
counterclockwise angle of 30˚ from the horizontal. Show these 6500 psi
stresses on a sketch of an element oriented at this angle. A
A
3800 psi
Side Cross
View Section

Solution 7.2-5
sx ⫽ 6500 psi sy ⫽⫺18500 psi txy ⫽ ⫺3800 psi sx ⫺ sy
tx1y1 ⫽ ⫺ sin (2u) + txy cos (2u)
u ⫽ 30° 2
sx + sy sx ⫺ sy tx1y1 ⫽ ⫺12725 psi ;
sx1 ⫽ + cos (2u) + txy sin (2u) sy1 ⫽ sx + sy ⫺ sx1
2 2
sx1 ⫽ ⫺3041 psi ; sy1 ⫽ ⫺8959 psi ;

Problem 7.2-6 An element in plane stress from the fuselage of an airplane is subjected 5.5 MPa
to compressive stresses of magnitude 27 MPa in the horizontal direction and tensile
stresses of magnitude 5.5 MPa in the vertical direction (see figure). Also, shear stresses
of magnitude 10.5 MPa act in the directions shown.
Determine the stresses acting on an element oriented at a clockwise angle of 35˚ 27 MPa
from the horizontal. Show these stresses on a sketch of an element oriented at this
angle.
10.5 MPa
07Ch07.qxd 9/27/08 1:18 PM Page 574

574 CHAPTER 7 Analysis of Stress and Strain

Solution 7.2-6
sx ⫽ ⫺27 MPa sy ⫽ 5.5 MPa txy ⫽ ⫺10.5 MPa sx ⫺ sy
...

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