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PROBLEM 11.93
Knowing that the couple shown acts in a vertical plane,
determine the stress at (a) point A, (b) point B.
PROBLEM 11.94
(a) Using an allowable stress of 120 MPa, determine the largest couple
M that can be applied to a beam of the cross section shown. (b) Solve
part a, assuming that the cross section of the beam is an 80-mm square.
PROBLEM 11.95
A steel bar (Es=210 GPa) and an aluminum bar (Ea=70 GPa) are
bonded together to form the composite bar shown. Determine the
maximum stress in (a) the aluminum, (b) the steel, when the bar is bent
about a horizontal axis, with M = 60N ≅ m
consent of McGraw-Hill Education.
PROBLEM 11.96
A single vertical force P is applied to a short steel post as shown.
Gages located at A, B, and C indicate the following strains:
500= −
A
εµ
1000= −
B
εµ
200
= −
C
εµ
Knowing that
6
29 10 psi,E= ×
determine (a) the magnitude of
P, (b) the line of action of P, (c) the corresponding strain at the
hidden edge of the post, where
2.5 in.x= −
and
1.5 in.z= −
SOLUTION
34 34 2
66
11
(5)(3) 11.25 in (3)(5) 31.25 in (5)(3) 15 in
12 12
2.5 in., 2.5 in., 2.5 in., 2.5 in.
1.5 in., 1.5 in., 1.5 in., 1.5 in.
(29 10 )( 500 10 ) 14,500 psi 14.5 ksi
xz
xz
A BCD
ABC D
IIA
M Pz M Px
x xxx
zzz z
E
sε
−
= = = = = =
= = −
=−= = =−
== =−=−
= = × −× =− =−
PROBLEM 11.97
Two vertical forces are applied to a beam of
the cross section shown. Determine the
maximum tensile and compressive stresses
in portion BC of the beam.
PROBLEM 11.98
In order to increase corrosion resistance, a 2-mm-thick cladding of
aluminum has been added to a steel bar as shown. The modulus
of elasticity is 200 GPa for steel and 70 GPa for aluminum. For a
bending moment of 300 N ⋅ m, determine (a) the maximum stress
in the steel, (b) the maximum stress in the aluminum, (c) the radius
of curvature of the bar.
PROBLEM 11.99
A 6 × 10-in. timber beam has been strengthened by bolting to it
the steel straps shown. The modulus of elasticity is
E=1.5 × 106 psi for the wood and E=30 × 106 psi for the steel.
Knowing that the beam is bent about a horizontal axis by a couple
of moment 200 kip ≅ in., determine the maximum stress in (a) the
wood, and (b) the steel.
PROBLEM 11.100
The four forces shown are applied to a rigid plate supported by a solid steel
post of radius a. Knowing that P = 24 kips and a = 1.6 in., determine the
maximum stress in the post when (a) the force at D is removed, (b) the forces
at C and D are removed.
SOLUTION
consent of McGraw-Hill Education.
PROBLEM 11.101
A couple M of moment 8 kN ⋅ m acting in a vertical plane is
applied to a W200 × 19.3 rolled-steel beam as shown. Determine
(a) the angle that the neutral axis forms with the horizontal plane,
(b) the maximum stress in the beam.
SOLUTION
For
W200 19.3×
rolled steel sector,
6 4 64
6 4 64
33
33
16.6 10 mm 16.6 10 m
1.15 10 mm 1.15 10 m
203 101.5 mm
2
102 51 mm
2
(8 10 )cos5 7.9696 10 N m
(8 10 )sin5 0.6972 10 N m
z
y
ABDE
A B DE
z
y
I
I
yy y y
zzzz
M
M
−
′
−
′
=×=×
=×=×
= =−=−= =
=−=−= = =
= × °= × ⋅
=− × °=− × ⋅
(a)
6
6
16.6 10
tan tan tan( 5 ) 1.2629
1.15 10
z
y
I
I
ϕθ
−
−
×
= = −° =−
×
51.6
ϕ
=−°
51.6 5
α
= °− °
46.6
α
= °
(b) Maximum tensile stress occurs at point D.
3 3 33
66
(7.9696 10 )( 101.5 10 ) ( 0.6972 10 )( 51 10 )
16.6 10 1.15 10
yD
zD
Dzy
Mz
My
II
σ
−−
−−
×− × − ×−×
=−+ =− +
××
6
79.6 10 Pa= ×
79.6 MPa
D
σ
=
PROBLEM 11.94
(a) Using an allowable stress of 120 MPa, determine the largest couple
M that can be applied to a beam of the cross section shown. (b) Solve
part a, assuming that the cross section of the beam is an 80-mm square.
PROBLEM 11.95
A steel bar (Es=210 GPa) and an aluminum bar (Ea=70 GPa) are
bonded together to form the composite bar shown. Determine the
maximum stress in (a) the aluminum, (b) the steel, when the bar is bent
about a horizontal axis, with M = 60N ≅ m
consent of McGraw-Hill Education.
PROBLEM 11.96
A single vertical force P is applied to a short steel post as shown.
Gages located at A, B, and C indicate the following strains:
500= −
A
εµ
1000= −
B
εµ
200
= −
C
εµ
Knowing that
6
29 10 psi,E= ×
determine (a) the magnitude of
P, (b) the line of action of P, (c) the corresponding strain at the
hidden edge of the post, where
2.5 in.x= −
and
1.5 in.z= −
SOLUTION
34 34 2
66
11
(5)(3) 11.25 in (3)(5) 31.25 in (5)(3) 15 in
12 12
2.5 in., 2.5 in., 2.5 in., 2.5 in.
1.5 in., 1.5 in., 1.5 in., 1.5 in.
(29 10 )( 500 10 ) 14,500 psi 14.5 ksi
xz
xz
A BCD
ABC D
IIA
M Pz M Px
x xxx
zzz z
E
sε
−
= = = = = =
= = −
=−= = =−
== =−=−
= = × −× =− =−
PROBLEM 11.97
Two vertical forces are applied to a beam of
the cross section shown. Determine the
maximum tensile and compressive stresses
in portion BC of the beam.
PROBLEM 11.98
In order to increase corrosion resistance, a 2-mm-thick cladding of
aluminum has been added to a steel bar as shown. The modulus
of elasticity is 200 GPa for steel and 70 GPa for aluminum. For a
bending moment of 300 N ⋅ m, determine (a) the maximum stress
in the steel, (b) the maximum stress in the aluminum, (c) the radius
of curvature of the bar.
PROBLEM 11.99
A 6 × 10-in. timber beam has been strengthened by bolting to it
the steel straps shown. The modulus of elasticity is
E=1.5 × 106 psi for the wood and E=30 × 106 psi for the steel.
Knowing that the beam is bent about a horizontal axis by a couple
of moment 200 kip ≅ in., determine the maximum stress in (a) the
wood, and (b) the steel.
PROBLEM 11.100
The four forces shown are applied to a rigid plate supported by a solid steel
post of radius a. Knowing that P = 24 kips and a = 1.6 in., determine the
maximum stress in the post when (a) the force at D is removed, (b) the forces
at C and D are removed.
SOLUTION
consent of McGraw-Hill Education.
PROBLEM 11.101
A couple M of moment 8 kN ⋅ m acting in a vertical plane is
applied to a W200 × 19.3 rolled-steel beam as shown. Determine
(a) the angle that the neutral axis forms with the horizontal plane,
(b) the maximum stress in the beam.
SOLUTION
For
W200 19.3×
rolled steel sector,
6 4 64
6 4 64
33
33
16.6 10 mm 16.6 10 m
1.15 10 mm 1.15 10 m
203 101.5 mm
2
102 51 mm
2
(8 10 )cos5 7.9696 10 N m
(8 10 )sin5 0.6972 10 N m
z
y
ABDE
A B DE
z
y
I
I
yy y y
zzzz
M
M
−
′
−
′
=×=×
=×=×
= =−=−= =
=−=−= = =
= × °= × ⋅
=− × °=− × ⋅
(a)
6
6
16.6 10
tan tan tan( 5 ) 1.2629
1.15 10
z
y
I
I
ϕθ
−
−
×
= = −° =−
×
51.6
ϕ
=−°
51.6 5
α
= °− °
46.6
α
= °
(b) Maximum tensile stress occurs at point D.
3 3 33
66
(7.9696 10 )( 101.5 10 ) ( 0.6972 10 )( 51 10 )
16.6 10 1.15 10
yD
zD
Dzy
Mz
My
II
σ
−−
−−
×− × − ×−×
=−+ =− +
××
6
79.6 10 Pa= ×
79.6 MPa
D
σ
=
