Mechanics of Materials Chap 05-02 PDF
Mechanics of Materials Chap 05-02 PDF
Mechanics of Materials Chap 05-02 PDF
CHAPTER 5
Stresses in Beams
Design of Beams
P
s2
s2
Steel rail
Wood
tie
d
b
Steel
girder
(b)
s1
(a)
P
Steel rail
Wood
tie
d
b
s1
(b)
(a)
3P(s1 s2 )
bsallow
5b
A
2b
D
P
I
b 35 mm
d 4
64
MINIMUM DIAMETER
(96)(36 N)(35 mm)
96Pb
d3
sallow
(30 MPa)
1,283.4 mm3
Mmax P(3b)
5d 2
dmin 10.9 mm
C
2b
SECTION 5.6
Design of Beams
P 2500 lb
q 200 lb/ft
L = 6 ft
Solution 5.6-3
Cantilever beam
L 6 ft
TRIAL SECTION W 8 21
S 18.2 in.3
M0
q0 L
378 lb-ft 4536 lb-in.
2
Beam is satisfactory.
W 8 21
P = 4000 lb
Solution 5.6-4
q0 21 lb/ft
7.5 ft
q = 400 lb/ft
L = 15 ft
Simple beam
TRIAL SECTION W 8 28
q0 28 lb/ft
S 24.3 in.3
2
M0
q0 L
787.5 lb-ft 9450 lb-in.
8
Beam is satisfactory.
305
306
CHAPTER 5
Stresses in Beams
Solution 5.6-5
Simple beam
L 24 ft
q0 25.4 lb/ft
M0
PL qL2
12,000 lb-ft 7,200 lb-ft
4
32
19,200 lb-ft 230,400 lb-in.
Mmax
S
q0 L
1829 lb-ft 21,950 lb-in.
8
Beam is satisfactory.
Use S 10 25.4
Chess
Pontoon
Balk
Pontoon bridge
Chess
LR 2.0 m
Pontoon
Lb 3.0 m
2.0 m
Lb length of balks
3.0 m
SECTION 5.6
Section modulus S
KN
q 8.0 m
b
Mmax
b
Lb 3.0 m
S
W total load
wLb Lc
Design of Beams
307
b3
6
Mmax 9,000 N m
562.5 106 m3
sallow
16 MPa
b3
562.5 106 m3andb3 3375 106 m3
6
wLc
W
q
2Lb
2
(8.0 kPa)(2.0 m)
2
8.0 kN/m
Solution 5.6-7
Planks
s
s
L
Joists
Floor joists
q
Mmax
qL2 1
(13.333 lbin.)(126 in.) 2 26,460 lb-in.
8
8
Required S
L 10.5 ft
308
CHAPTER 5
Stresses in Beams
Problem 5.6-8 The wood joists supporting a plank floor (see figure) are
40 mm 180 mm in cross section (actual dimensions) and have a span
length L 4.0 m. The floor load is 3.6 kPa, which includes the weight
of the joists and the floor.
Calculate the maximum permissible spacing s of the joists if the
allowable bending stress is 15 MPa. (Assume that each joist may be
represented as a simple beam carrying a uniform load.)
Solution 5.6-8
Planks
h = 180 mm
s
s
L
Joists
b = 40 mm
L 4.0 m
w floor load 3.6 kPa
s spacing of joists
allow 15 MPa
L 4.0 m
q ws
S
bh2
6
Mmax
S
SPACING OF JOISTS
qL2 wsL2
8
8
Mmax
wsL2
bh2
sallow 8sallow
6
smax
4 bh2sallow
3wL2
SECTION 5.6
Design of Beams
309
q
A
L
3.033
in.
B
L
2.384 in.
0.649 in.
10.0 in.
c1 2.384 in.
Mt
c2 0.649 in.
q load on overhang
Mc
L length of overhang
3.0 ft = 36 in.
ALLOWABLE STRESSES
t 18 ksi
c 12 ksi
Tension governs.
Mmax
(q q0 )L2
2
Tension on top; compression on bottom.
(q q0 )L2
2Mallow
qallow q0
2
L2
qallow
2Mallow
2(29,750 lb-in.)
q0
2.5 lbin.
2
L
(36 in.) 2
310
CHAPTER 5
Stresses in Beams
Solution 5.6-10
L
2
b
L
P 1.2 kN
L 2.1 m
IC 1.85948(0.41421h)4 0.054738h4
Section modulus
b 0.41421h
PL (1.2 kN)(2.1 m)
630 N m
4
4
S
IC
0.054738h4
0.109476h3
h2
h2
Minimum height h
M
M
S
s
S
630N
m
0.109476h3
3.15 106 m3
200 MPa
h3 28.7735 106 m3 h 0.030643 m
s
b
C
2
h
2
360 360
45
n
8
b b
tan (from triangle)
b
2 h
2
b h
h cot
2 b
2
b
hmin 30.6 mm
Alternative solution (n 8)
M
b
b
PL
b 45tan 2 1cot 2 1
4
2
2
b ( 2 1)hh ( 2 1)b
For 45:
b
45
tan
0.41421
h
2
h
45
cot
2.41421
b
2
Moment of inertia
IC
b
b
nb4
cot 3 cot2 1
192
2
2
IC
8b4
(2.41421) [3(2.41421) 2 1] 1.85948b4
192
IC
S
11 82 4
42 5 4
b
h
12
12
42 5 3
3PL
h h3
6
2(42 5)sallow
4000 lb
5 ft
2000 lb
B
16 ft
SECTION 5.6
Solution 5.6-11
311
Moving carriage
P2
P1
M RA x 125(43x 3x2)
B
L
L
L
x
x
5
(4000 lb) 1 (2000 lb) 1
16
16 16
125(43 3x)
(x ft; M lb-ft)
RA
(x ft; RA lb)
43
43 2
3 R
6
6
Select S 8 23
(S 16.2 in.3)
A
B
d
P
L
Minimum diameter
Mmax allow S
PL
gd 2L2
d 3
sallow
8
32
sallow d 3 4g L2 d 2
Design of Beams
qL2
gd 3L2
PL
2
8
Section modulus S
d
32
32
(400 N)(0.45 m) 0
dmin 31.61 mm
312
CHAPTER 5
Stresses in Beams
Solution 5.6-13
4a
11qa
RA 8
RB
4a
4a
45qa
8
qmax
D
2a
W 16 57
RD 2qa
2qa2
11a
8
2sallow S
5a2
S 92.2 in.3
121 qa2
128
B
qmax
5q a2
sallow S
2
Data: a 6 ft 72 in.
4a
922 lb/ft
qallow 922 lb/ft 57 lb/ft 865 lb/ft
2a
5qa2
2
Pin
Mmax
Pin
Compound beam
q
4b
h=
3
L2
L1
L2
2.5 m
(5.5 kPa)
6875 Nm
2
2
L1 2.1 m L 2 2.5 m
Floor dimensions: L 1 L 2
Design load w 5.5 kPa
5.5 kN/m3 (weight density of wood beam)
allow 15 MPa
Weight of beam
q0 gbh
4gb2 4
(5.5 kNm2 )b2
3
3
7333b2 (N/m)
(b meters)
SECTION 5.6
(q
1
(6875 Nm 7333b2 )(2.1 m) 2
2
2
15,159 16,170b2 (N m)
q0 )L21
Mmax
Design of Beams
bh2 8b3
6
27
Mmax allow S
S
h
4b
0.2023 m
3
Required dimensions
8b3
15,159 16,170b2 (15 106 Nm2 )
27
b 152 mm
h 202 mm
y
b
1.5 in.
1.25 in.
z
12 in.
1.5 in.
16 in.
Solution 5.6-15
b
C1
1.25 in.
C2
A3
1.5 in.
A1
12 in.
B
1.5 in.
16 in.
A A1 A2 A3 39 1.5b (in.2)
First moment of the cross-sectional area about the
lower edge B-B
QBB a yi Ai (14.25)(1.5b) (7.5)(15) (0.75)(24)
130.5 21.375b (in.3)
Distance c2 from line B-B to the centroid C
QBB 130.5 21.375b 45
in.
A
39 1.5b
7
c1 4
sbottom c2 3
c2
4
60
c1 h 8.57143 in.
7
7
Solve for b
stop
3
45
c2 h 6.42857 in.
7
7
A2 (12)(1.25) 15 in.2
A3 (16)(1.5) 24 in.2
C
A2
A1 1.5b
313
314
CHAPTER 5
Stresses in Beams
y
t
50 mm
120 mm
Solution 5.6-16
Channel beam
Areas of the cross section (mm 2)
A1 ht 50t A2 b1 t 120t 2t 2
A 2A1 A2 220t 2t 2 2t(110t)
A1
A2
c1
c2
A1
t
B
B
t
h 50 mm
b1
b 120 mm
t thickness (constant)
(t is in millimeters)
b1 b 2t 120 mm 2t
c2
Q BB t(2500 60t t 2 )
A
2t(110 t)
2500 60t t 2
15 mm
2(110 t)
7
h 35 mm
10
c2
3
h 15 mm
10
Solve for t
2(110 t)(15) 2500 60t t 2
t 2 90t 800 0
t 10 mm
Problem 5.6-17 Determine the ratios of the weights of three beams that
have the same length, are made of the same material, are subjected to the
same maximum bending moment, and have the same maximum bending
stress if their cross sections are (1) a rectangle with height equal to twice
the width, (2) a square, and (3) a circle (see figures).
h = 2b
SECTION 5.6
315
Design of Beams
a3
a (6S) 13
6
A2 a2 (6S)2/3 3.3019S 2/3
(2) Square: S
d 3
32S 13
d
32
d 2 32S 23
A3
3.6905 S 23
4
4
(3) Circle: S
t
A
B
L
(a)
q
A
D
B
C
L
(b)
Solution 5.6-18
D
B
C
L
(a)
L
( 12 1)
2
Substitute x into the equation for either M1 or M2 :
M1
Mmax
Solve for x: x
M2
M2
(b)
L 900 mm
b 300 mm
t 20 mm
allow 5.0 MPa
(L 4x)
8
2
qL2
(3 212)
8
Eq. (1)
bt 2
Eq. (2)
Equate Mmax from Eqs. (1) and (2) and solve for q:
qmax
4bt 2sallow
3L2 (3 212)
316
CHAPTER 5
Stresses in Beams
Problem 5.6-19 A steel plate (called a cover plate) having crosssectional dimensions 4.0 in. 0.5 in. is welded along the full length of
the top flange of a W 12 35 wide-flange beam (see figure, which shows
the beam cross section).
What is the percent increase in section modulus (as compared to the
wide-flange beam alone)?
Solution 5.6-19
W 12 35
c2
6.25
L
6.25
q
C
2b
B
2L
SECTION 5.6
Solution 5.6-20
Design of Beams
317
2b
b
2L
L
9qL
RB
4
3qL
RA
4
9qL2
32
C
0
A
qL2
2
Mmax
L 150 mm
q 3.5 kN/m
allow 60 MPa
77.0 kN/m3
qL2
bh2 2b3
S
2
6
3
b 0.00995 m 9.95 mm
qL2
2b3
sallow
2
3
3qL2
4sallow
2b3
3
(q q0 )L2 1
(q 2g b2 )L2
2
2
Mmax allow S
1
2b3
(q 2g b2 ) L2 sallow
2
3
Rearrange the equation:
4allow b 3 6L2 b2 3qL2 0
Substitute numerical values:
(240 106)b3 10,395b2 236.25 0
(b meters)
Solve the equation:
b 0.00996 m 9.96 mm
3 in.
p1 = 100 lb/ft2
12 in.
diam.
12 in.
diam.
5 ft
3 in.
Top view
p2 = 400 lb/ft2
Side view
318
CHAPTER 5
Stresses in Beams
Solution 5.6-21
Retaining wall
q1
t
qs2 p2 bs2
8
8
Mmax allow S
or
S
bt 2
6
p2 bs2
bt 2
sallow
8
6
Solve for s:
s
4 sallow t 2
72.0 in.
B 3p2
q2
6
32
Solve for s:
3 sallow d 3
s
81.4 in.
16h2 (2p1 p2 )
Plank governs
a
a
z
C
a
a
SECTION 5.6
Solution 5.6-22
removed
am m
1
319
Design of Beams
1.10
h1
(S)
S0 max 1.0535
Eq. (1)
a P P1
a
Eq. (1)
1.00
0
0.1
.50
0.2
19
0.3
I0 a3 12
a4
a
c0
S0
c0
12
12
12
(1 b) 4a4
12
(1 b)a
12
I 12 a3
S
(1 3b)(1 b) 2
c
12
9
d
b
(a)
9
(b)
320
CHAPTER 5
Solution 5.6-23
Stresses in Beams
Graph of
d
(a)
(b)
S2 b[8h (h 2d) ]
h
S1
2d
9(h 2d)(bh2 )
9 1
h
3
8 1
d
0
h
d
h
S2
S1
0
0.25
0.50
0.75
1.00
1.000
0.8426
0.8889
1.0500
1.2963
S2
S1
1.0
0.5
0.2937
0
0.6861
0.5
1.0
d
h
S2
0.8399
S1 min