PROBLEM 11.30
Wooden beams and steel plates are securely bolted together to form the composite
member shown. Using the data given below, determine the largest permissible bending
moment when the member is bent about a horizontal axis.
Wood Steel
Modulus of elasticity:
6
2 10 psi×
6
29 10 psi×
Allowable stress: 2000 psi 22 ksi
SOLUTION
PROBLEM 11.31
A steel bar and an aluminum bar are bonded together to form the composite
beam shown. The modulus of elasticity for aluminum is 70 GPa and for steel is
200 GPa. Knowing that the beam is bent about a horizontal axis by a couple of
moment
1500 N m,M= ⋅
determine the maximum stress in (a) the aluminum,
(b) the steel.
SOLUTION
PROBLEM 11.32
A steel bar and an aluminum bar are bonded together to form the composite
beam shown. The modulus of elasticity for aluminum is 70 GPa and for steel is
200 GPa. Knowing that the beam is bent about a horizontal axis by a couple of
moment
1500 N m,
M= ⋅
determine the maximum stress in (a) the aluminum,
(b) the steel.
SOLUTION
PROBLEM 11.33
The
6 12-in.×
timber beam has been strengthened by bolting to it the steel
reinforcement shown. The modulus of elasticity for wood is
6
1.8 10 psi×
and for
steel,
6
29 10 psi.×
Knowing that the beam is bent about a horizontal axis by a
couple of moment
450 kip in.,M= ⋅
determine the maximum stress in (a) the
wood, (b) the steel.
SOLUTION
PROBLEM 11.34
The
6 12-in.
×
timber beam has been strengthened by bolting to it the steel
reinforcement shown. The modulus of elasticity for wood is
6
1.8 10 psi×
and for
steel,
6
29 10 psi.×
Knowing that the beam is bent about a horizontal axis by a
couple of moment
450 kip in.,M= ⋅
determine the maximum stress in (a) the
wood, (b) the steel.
SOLUTION
PROBLEM 11.35
For the composite beam indicated, determine the radius of curvature caused by
the couple of moment 1500 N ⋅ m.
Beam of Prob. 11.31.
SOLUTION
See solution to Prob. 11.31 for the calculation of I.
94 9
1
99
852.44 10 m 70 10 Pa
1 1500 0.02513 m
(70 10 )(852.44 10 )
a
IE
M
EI
ρ
−
−
−
=×=×
= = =
××
39.8 m
ρ
=
consent of McGraw–Hill Education.
PROBLEM 11.36
For the composite bar indicated, determine the radius of curvature caused by the
couple of moment 1500 N ⋅ m.
Beam of Prob. 11.32.
SOLUTION
See solution to Prob. 11.32 for calculation of I.
94 9
1
99
936.71 10 m 70 10 Pa
1 1500 0.02288 m
(70 10 )(936.71 10 )
a
IE
M
EI
ρ
−
−
−
=×=×
= = =
××
43.7 m
ρ
=
consent of McGraw–Hill Education.
PROBLEM 11.37
For the composite beam indicated, determine the radius of curvature caused by the
couple of moment 450 kip ⋅ in.
Beam of Prob. 11.33.
SOLUTION
46
3
361
6
1873.77 in 1.8 10 psi
450 kip in 450 10 lb in.
1 450 10 133.421 10 in.
(1.8 10 )(1873.77)
−−
= = ×
= ⋅= × ⋅
×
= = = ×
×
w
IE
M
M
EI
ρ
7495 in. 625 ft
ρ
= =
consent of McGraw–Hill Education.
PROBLEM 11.38
For the composite beam indicated, determine the radius of curvature caused by the
couple of moment 450 kip ⋅ in.
Beam of Prob. 11.34.
SOLUTION
See solution to Prob. 11.34 for calculation of I.
46
3
361
6
1874.69 in 1.8 10 psi
450 kip in. 450 10 lb in.
1 450 10 133.355 10 in.
(1.8 10 )(1874.69)
ρ
−−
= = ×
= ⋅= × ⋅
×
= = = ×
×
w
IE
M
M
EI
7499 in. 625 ft
ρ
= =
PROBLEM 11.39
The reinforced concrete beam shown is subjected to a positive bending
moment of 175 kN ⋅ m. Knowing that the modulus of elasticity is 25 GPa
for the concrete and 200 GPa for the steel, determine (a) the stress in the
steel, (b) the maximum stress in the concrete.
SOLUTION
2 2 32
32
200 GPa 8.0
25 GPa
4 (4) (25) 1.9635 10 mm
44
15.708 10 mm
s
c
s
s
E
nE
Ad
nA
ππ
= = =
=⋅= = ×
= ×
Locate the neutral axis.
3
23 6
300 (15.708 10 )(480 ) 0
2
150 15.708 10 7.5398 10 0
− × −=
+ × − ×=
x
xx
xx
Solve for x.
3 32 6
15.708 10 (15.708 10 ) (4)(150)(7.5398 10 )
(2)(150)
177.87 mm, 480 302.13 mm
x
xx
− ×+ × + ×
=
= −=
3 32
3 32
9 4 34
1(300) (15.708 10 )(480 )
3
1(300)(177.87) (15.708 10 )(302.13)
3
1.9966 10 mm 1.9966 10 m
Ix x
nMy
I
σ
−
= +× −
= +×
=×=×
= −
(a) Steel:
302.45 mm 0.30245 my=−=−
36
3
(8.0)(175 10 )( 0.30245) 212 10 Pa
1.9966 10
σ
−
×−
=−=×
×
212 MPa
σ
=
(b) Concrete:
177.87 mm 0.17787 my= =
36
3
(1.0)(175 10 )(0.17787) 15.59 10 Pa
1.9966 10
σ
−
×
=− =−×
×
15.59 MPa
σ
= −
PROBLEM 11.31
A steel bar and an aluminum bar are bonded together to form the composite
beam shown. The modulus of elasticity for aluminum is 70 GPa and for steel is
200 GPa. Knowing that the beam is bent about a horizontal axis by a couple of
moment
1500 N m,M= ⋅
determine the maximum stress in (a) the aluminum,
(b) the steel.
SOLUTION
PROBLEM 11.32
A steel bar and an aluminum bar are bonded together to form the composite
beam shown. The modulus of elasticity for aluminum is 70 GPa and for steel is
200 GPa. Knowing that the beam is bent about a horizontal axis by a couple of
moment
1500 N m,
M= ⋅
determine the maximum stress in (a) the aluminum,
(b) the steel.
SOLUTION
PROBLEM 11.33
The
6 12-in.×
timber beam has been strengthened by bolting to it the steel
reinforcement shown. The modulus of elasticity for wood is
6
1.8 10 psi×
and for
steel,
6
29 10 psi.×
Knowing that the beam is bent about a horizontal axis by a
couple of moment
450 kip in.,M= ⋅
determine the maximum stress in (a) the
wood, (b) the steel.
SOLUTION
PROBLEM 11.34
The
6 12-in.
×
timber beam has been strengthened by bolting to it the steel
reinforcement shown. The modulus of elasticity for wood is
6
1.8 10 psi×
and for
steel,
6
29 10 psi.×
Knowing that the beam is bent about a horizontal axis by a
couple of moment
450 kip in.,M= ⋅
determine the maximum stress in (a) the
wood, (b) the steel.
SOLUTION
PROBLEM 11.35
For the composite beam indicated, determine the radius of curvature caused by
the couple of moment 1500 N ⋅ m.
Beam of Prob. 11.31.
SOLUTION
See solution to Prob. 11.31 for the calculation of I.
94 9
1
99
852.44 10 m 70 10 Pa
1 1500 0.02513 m
(70 10 )(852.44 10 )
a
IE
M
EI
ρ
−
−
−
=×=×
= = =
××
39.8 m
ρ
=
consent of McGraw–Hill Education.
PROBLEM 11.36
For the composite bar indicated, determine the radius of curvature caused by the
couple of moment 1500 N ⋅ m.
Beam of Prob. 11.32.
SOLUTION
See solution to Prob. 11.32 for calculation of I.
94 9
1
99
936.71 10 m 70 10 Pa
1 1500 0.02288 m
(70 10 )(936.71 10 )
a
IE
M
EI
ρ
−
−
−
=×=×
= = =
××
43.7 m
ρ
=
consent of McGraw–Hill Education.
PROBLEM 11.37
For the composite beam indicated, determine the radius of curvature caused by the
couple of moment 450 kip ⋅ in.
Beam of Prob. 11.33.
SOLUTION
46
3
361
6
1873.77 in 1.8 10 psi
450 kip in 450 10 lb in.
1 450 10 133.421 10 in.
(1.8 10 )(1873.77)
−−
= = ×
= ⋅= × ⋅
×
= = = ×
×
w
IE
M
M
EI
ρ
7495 in. 625 ft
ρ
= =
consent of McGraw–Hill Education.
PROBLEM 11.38
For the composite beam indicated, determine the radius of curvature caused by the
couple of moment 450 kip ⋅ in.
Beam of Prob. 11.34.
SOLUTION
See solution to Prob. 11.34 for calculation of I.
46
3
361
6
1874.69 in 1.8 10 psi
450 kip in. 450 10 lb in.
1 450 10 133.355 10 in.
(1.8 10 )(1874.69)
ρ
−−
= = ×
= ⋅= × ⋅
×
= = = ×
×
w
IE
M
M
EI
7499 in. 625 ft
ρ
= =
PROBLEM 11.39
The reinforced concrete beam shown is subjected to a positive bending
moment of 175 kN ⋅ m. Knowing that the modulus of elasticity is 25 GPa
for the concrete and 200 GPa for the steel, determine (a) the stress in the
steel, (b) the maximum stress in the concrete.
SOLUTION
2 2 32
32
200 GPa 8.0
25 GPa
4 (4) (25) 1.9635 10 mm
44
15.708 10 mm
s
c
s
s
E
nE
Ad
nA
ππ
= = =
=⋅= = ×
= ×
Locate the neutral axis.
3
23 6
300 (15.708 10 )(480 ) 0
2
150 15.708 10 7.5398 10 0
− × −=
+ × − ×=
x
xx
xx
Solve for x.
3 32 6
15.708 10 (15.708 10 ) (4)(150)(7.5398 10 )
(2)(150)
177.87 mm, 480 302.13 mm
x
xx
− ×+ × + ×
=
= −=
3 32
3 32
9 4 34
1(300) (15.708 10 )(480 )
3
1(300)(177.87) (15.708 10 )(302.13)
3
1.9966 10 mm 1.9966 10 m
Ix x
nMy
I
σ
−
= +× −
= +×
=×=×
= −
(a) Steel:
302.45 mm 0.30245 my=−=−
36
3
(8.0)(175 10 )( 0.30245) 212 10 Pa
1.9966 10
σ
−
×−
=−=×
×
212 MPa
σ
=
(b) Concrete:
177.87 mm 0.17787 my= =
36
3
(1.0)(175 10 )(0.17787) 15.59 10 Pa
1.9966 10
σ
−
×
=− =−×
×
15.59 MPa
σ
= −