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7
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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