Mecanica de materiales beer
Analysis of Stress and Strain
Plane Stress
Problem 7.2-1 An element in plane stress is subjected to stresses 6500 psi, y 1700 psi, and xy 2750 psi, as shown in the x figure. Determine the stresses acting on an element oriented at an angle 60° from the x axis, where the angle is positive when counterclockwise. Show these stresses on a sketch of an element oriented at the angle .
y
y
=1700 psi
O
= 6500 psi x xy = 2750 psi
x
Solution 7.2-1
Plane stress (angle )
y
x
2920 psi 5280 psi 60 O x 3450 psi
sx1
6500 psi 60 sx sy 2 5280 psi sx sy
y
1700 psi sy 2
xy
2750 psi
sx
cos 2u
txy sin 2u
tx1y1 sy1
sin 2u txy cos 2u 2 3450 psi sx sy sx1 2920 psi
Problem 7.2-2 52 MPa, y
Solve the preceding problem for x 80 MPa, 48 MPa, and 25°(see figure). xy
52 MPa 48 MPa 80 MPa
425
426
CHAPTER 7
Analysis of Stress and Strain
Solution 7.2-2
Plane stress (angle )
y
x
20.2 MPa 111.8 MPa 25 O x 20.1 MPa
sx1
80 MPa 25 sx sy 2 111.8 MPa sx sy
y
52 MPa sy 2
xy
48 MPa
sx
cos 2u
txy sin 2u
tx1y1 sy1
sin 2u txy cos 2u 2 20.1 MPa sx sy sx1 20.2 MPa
Problem 7.2-3 Solve Problem 7.2-1for x 9,900 psi, 3,400 psi, xy 3,600 psi, and 50° (see figure). y
3,400 psi 3,600 psi 9,900 psi
Solution 7.2-3
y 10,760 psi
Plane stress (angle )
x
2,540 psi 50 O x 2,580 psi
sx1
9900 psi 50 sx sy sx 2 2540 psi sx sy
y
3400 psi sy
xy
3600 psi
2
cos 2u
txy sin 2u
tx1y1 sy1
sin 2u txy cos 2u 2 2580 psi sx sy sx1 10,760 psi
Problem 7.2-4 The stressesacting on element A in the web of a train rail are found to be 42 MPa tension in the horizontal direction and 140 MPa compression in the vertical direction (see figure). Also, shear stresses of magnitude 60 MPa act in the directions shown. Determine the stresses acting on an element oriented at a counterclockwise angle of 48° from the horizontal. Show these stresses on a sketch of an elementoriented at this angle.
140 MPa
A A Side View Cross Section
42 MPa 60 MPa
SECTION 7.2
Plane Stress
427
Solution 7.2-4
Plane stress (angle )
y
x
20.2 MPa 118.2 MPa 48 O x 84.2 MPa
sx1
42 MPa 48 sx sy
y
140 MPa sy
xy
60 MPa
tx1y1 sy1
cos 2u txy sin 2u 2 2 118.2 MPa sx sy sin 2u txy cos 2u 2 84.2 MPa sx sy sx1 20.2 MPa
20,500 psi
sx
Problem7.2-5 Solve the preceding problem if the normal and shear stresses acting on element A are 7,500 psi, 20,500 psi, and 4,800 psi (in the directions shown in the figure) and the angle is 30° (counterclockwise).
A
7,500 psi 4,800 psi
Solution 7.2-5
Plane stress (angle )
y
x xy
9,340 psi 3,660 psi
sx1
30 O x 14,520 psi
7,500 psi y 4,800 psi 30 sx sy sx 2 3,660 psi sx sy 2
20,500psi
sy
cos 2u
txy sin 2u
tx1y1 sy1
sin 2u txy cos 2u 2 14, 520 psi sx sy sx1 9,340 psi
Problem 7.2-6 An element in plane stress from the fuselage of an airplane is subjected to compressive stresses of magnitude 25.5 MPa in the horizontal direction and tensile stresses of magnitude 6.5 MPa in the vertical direction (see figure). Also, shear stresses of magnitude 12.0 MPa act inthe directions shown. Determine the stresses acting on an element oriented at a clockwise angle of 40° from the horizontal. Show these stresses on a sketch of an element oriented at this angle.
6.5 MPa
25.5 MPa 12.0 MPa
428
CHAPTER 7
Analysis of Stress and Strain
Solution 7.2-6
Plane stress (angle )
y
x xy
17.8 MPa 18.5 MPa O
sx1
25.5 MPa 12.0 MPa 40 sx sy sx 2 0.5MPa sx sy
y
6.5 MPa
sy 2
cos 2u
txy sin 2u
x 40 0.5 MPa
tx1y1 sy1
sin 2u txy cos 2u 2 17.8 MPa sx sy sx1 18.5 MPa
Problem 7.2-7 The stresses acting on element B in the web of a wide-flange beam are found to be 11,000 psi compression in the horizontal direction and 3,000 psi compression in the vertical direction (see figure). Also, shear stresses of magnitude 4,200 psi...
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