1. Structures Agenda:
Forces & Architectural Form - review
Material Properties - review
Deflections - not on exam
Allowable Stress Design (ASD)
(member strength + stability)

2. Jurg Conzett – Traversina Bridge
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Add moment diagram
Moment
Jurg Conzett – Traversina Bridge

4. Riccardo Morandi – Santa Barbara Power Station
Add moment diagram
Riccardo Morandi – Santa Barbara Power Station

5. We do all of these crazy shapes and forms to make sure that materials do not reach their capacity, which would cause a failure.

6. Materials Review

7. Stress-Strain curve Fy
This is review of last semester from materials and methods. Modulus of elasticity.
= Modulus of Elasticity = E

8. Stress-Strain curve

9. Comparison of materials
Yield Stress (Fy)
Material
Modulus of Elasticity (E)
bending
compression
tension
Steel
29,000 ksi
36 ksi
36 ksi
36 ksi
Concrete
3100 ksi
0.5 ksi
3 ksi
0.3 ksi
Wood
1700 ksi
1.0 ksi
1.5 ksi
0.7 ksi
Glass
10,000 ksi
24 ksi
145 ksi
24 ksi

10. Comparison of materials
Yield Stress (Fy)
Material
Modulus of Elasticity (E)
bending
compression
tension
Steel 17 36 24 50
Concrete 2
0.5 2
0.5
Wood 1 1 1 1
Concrete uses reinforcing steel for tension
Glass 6 24 97 34

11. Allowable Stress Design
Make sure that materials do not reach their yield stress by providing a factor of safety (FOS).

12. Factor of Safety
Steel: 0.6

13. Allowable flexural stress = factor of safety x yield stress
Steel: 0.6
Allowable flexural stress = factor of safety x yield stress
Fb = 0.6 x Fy

14. Allowable flexural stress limit (Fb)= factor of safety x yield stress
Steel: 0.6
Allowable flexural stress limit (Fb)= factor of safety x yield stress
Fb = 0.6 x Fy
Fb = 0.6 x 36 ksi
Fb = 21.6 ksi

15. Moment = bending stress (fb) x SECTION MODULUS (S)
What is section modulus?

16. Moment = bending stress x SECTION MODULUS (S)
What is section modulus?
Property of the cross sectional shape.
It is what allows us to make the connection between the moment and stress.

17. Moment = bending stress x SECTION MODULUS (S)
What is section modulus?
Property of the cross sectional shape.
Where do you find it?
Look it up in the tables OR calculate it

18. b h2
Section Modulus = S = 6 b b h h
neutral axis

19. Deflection

20. the measured amount a member moves depends upon:
Deflection
the measured amount a member moves depends upon:
Stiffness/Rigidity of the material (E)
Property of the cross sectional shape (I)
Length of beam (L)
Load on beam

21. Deflection Rigidity or stiffness of the material
Modulus of Elasticity (E)
Property of the cross sectional shape
Moment of Inertia (I)

22. Moment of Inertia (I) Property of the cross sectional shape
Where do you find it?
Look it up in tables OR calculate it

23. b h3
Moment of Inertia = I = 12 b b h h
neutral axis

24. 14”
14”
14”
Area = 14 in2
I = 485 in4
Area = 14 in2
I = 229 in4
Area = 14 in2
I = 1.2 in4

25. P L
Bigger S, bigger moment capacity

26. P L P M Rx Ry L
Bigger S, bigger moment capacity

27. P P M Rx Ry P L3 Deflection = 3 E I L L Deflection
Bigger S, bigger moment capacity
P L3
Deflection =
3 E I

28. w w M Rx Ry w L4 Deflection = 8 E I L L Deflection
Bigger S, bigger moment capacity
w L4
Deflection =
8 E I

29. w w Rx Ry Ry 5 w L4 Deflection = 384 E I L L Deflection
Bigger S, bigger moment capacity
5 w L4
Deflection =
384 E I

30. P P Rx Ry Ry P L3 Deflection = 48 E I L L Deflection
Bigger S, bigger moment capacity
P L3
Deflection =
48 E I

31. Moment of Inertia Property of the cross sectional shape
Where do you find it?
Look it up in tables OR calculate it
Bigger Moment of Inertia, smaller deflection

32. STRUCTURAL ANALYSIS:
Determining Strength Capacity

33. From Structural Analysis we have developed an understanding of all :
Actions - Applied forces such as dead load, live load, wind load, seismic load.
Reactions - Forces generated at the boundary conditions (rollers, pins, and fixed ends) that maintain equilibrium.
Internal forces - axial, shear, and moment (P V M) forces inside each element

34. Structural Capacity based upon element’s ability to perform without:
Yielding - permanently deforming (tensile stretching or compression squashing of a squat (short & wide) compression element)
Buckling – slender (tall & thin) compression element loses stability
Deflecting Excessively – movement that may cause damage to attached materials/finishes – floor vibrations or bounce

35. TENSION and ALLOWABLE STRESS:

36. stress
Plastic Range
FY = yield stress
Elastic Range
deformation

37. A = Area P = Force fa = stress
fa = P/A
stress FY fa P1
Force on the spring generates stress and elastic deformation
deformation

38. stress FY
When force is removed, the spring returns to its original shape -- elastic behavior
deformation

39. stress fa FY
larger force (greater than Fy) generates axial stress causing plastic deformation
deformation P2

40. stress fa FY
When the larger force is removed, the plastic deformation remains (permanent offset)
deformation

41. To be sure tension stress does not reach the yield stress, we set an: ALLOWABLE TENSILE STRESS LIMIT (FT) :
FT = 0.60 FY
(capital letter F for limit)
stress FY FT
deformation

42. Using Grade A36 Steel: FY = 36 ksi
Allowable Tensile Stress (FT ):
FT = 0.60 FY = 0.60 (36 ksi) = 21.6 ksi

43. fa stress
A36 Steel : FY = 36 ksi
Allowable Tensile Stress FT:
FT = 0.60 FY = 0.60 (36 ksi) = 21.6 ksi
P = 5,000 lb or 5 kips (get this from P diagram)
fa = P/Area (actual axial stress fa = P/A)
Aarea
P force

44. FT stress
A36 Steel : FY = 36 ksi
Allowable Tensile Stress :
FT = 0.60 FY = 0.60 (36 ksi) = 21.6 ksi
P = 5,000 lb or 5 kips (get this from P diagram)
fA = P/Area
FT = Pmax /AreaRequired
Areq
Pmax

45. FT stress
A36 Steel : FY = 36 ksi
Allowable Tensile Stress :
FT = 0.60 FY = 0.60 (36 ksi) = 21.6 ksi
P = 5,000 lb or 5 kips (get this from P diagram)
fA = P/Area
FT = Pmax /AreaRequired
AreaRequired = Pmax/FT
Areq
Pmax

46. 21.6 ksi
A36 Steel : FY = 36 ksi
Allowable Tensile Stress :
FT = 0.60 FY = 0.60 (36 ksi) = 21.6 ksi
P = 5,000 lb or 5 kips (get this from P diagram)
fA = P/Area
FT = Pmax /AreaRequired
AreaRequired = Pmax/FT = 5k / 21.6 ksi
AreaRequired = .25 in2
Areq 5k

47. FLEXURAL and ALLOWABLE STRESS :

49. stress
Plastic Range
FY = yield stress
Elastic Range
deformation
fb = M/S
S = Section Modulus

50. P1
stress FY
Load on the BEAM generates bending stress (tension and compression) and elastic deformation
(fb = Mmax/S) fb
deformation

51. stress FY
When Force is removed, the BEAM elastically returns to its original shape
deformation

52. P2
stress fb FY
A Larger Force may generate bending stress sufficient to cause plastic deformation
deformation

53. stress fb FY
When the larger force is removed, the plastic deformation remains. (permanent offset)
deformation

54. To be sure bending stress does not reach the yield stress, we set an ALLOWABLE BENDING STRESS LIMIT (Fb):
Fb = 0.60 FY
(capital letter F for limit)
stress FY Fb
deformation

55. A36 Steel : FY = 36 ksi
Allowable Bending Stress (Fb) :
Fb = 0.60 FY = 0.60 (36 ksi) = 21.6 ksi

56. A36 Steel : FY = 36 ksi
Allowable Bending Stress (Fb) :
Fb = 0.60 FY = 0.60 (36 ksi) = 21.6 ksi
Mmax = 316 k-ft (get this from the M diagram)
Mmax = 316 k-ft (12 in / ft) = 3792 k-in

57. A36 Steel : FY = 36 ksi
Allowable Bending Stress (Fb) :
Fb = 0.60 FY = 0.60 (36 ksi) = 21.6 ksi
Mmax = 316 k-ft (get this from the M Diagram)
Mmax = 316 k-ft (12 in / ft) = 3792 k-in
fb = M/S (actual bending stress fb = M/S)

58. A36 Steel : FY = 36 ksi
Allowable Bending Stress (Fb) :
Fb = 0.60 FY = 0.60 (36 ksi) = 21.6 ksi
Mmax = 316 k-ft (get this from the M diagram)
Mmax = 316 k-ft (12 in / ft) = 3792 k-in
fb = M/S
Fb = Mmax / SRequired

59. If using A36 Steel : FY = 36 ksi
Allowable Bending Stress (Fb) :
Fb = 0.60 FY = 0.60 (36 ksi) = 21.6 ksi
Mmax = 316 k-ft (get this from the M diagram)
Mmax = 316 k-ft (12 in / ft) = 3792 k-in
fb = M/S
Fb = Mmax / SRequired
SRequired = Mmax / Fb

60. If using A36 Steel : FY = 36 ksi
Allowable Bending Stress (Fb) :
Fb = 0.60 FY = 0.60 (36 ksi) = 21.6 ksi
Mmax = 316 k-ft (get this from the M Diagram)
Mmax = 316 k-ft (12 in / ft) = 3792 k-in
fb = M/S
Fb = Mmax / SRequired
SRequired = Mmax / Fb = 3792 k-in / 21.6 ksi = 176 in3
Now look up Sxx in Charts

66. If using A36 Steel : FY = 36 ksi
Mmax = 316 k-ft (get this from the M Diagram)
Mmax = 316 k-ft (12 in / ft) = 3792 k-in
Allowable Bending Stress :
Fb = 0.60 FY = 0.60 (36 ksi) = 21.6 ksi
fb = M/S
Fb = Mmax / SRequired
SRequired = Mmax/Fb = 3792 k-in / 21.6 ksi = 176 in3
Use W24x76 : SX-X = 176in3

67. BUCKLING and ALLOWABLE COMPRESSION STRESS :

68. PC
Buckling is a compressive phenomenon that depends on :
‘unbraced length’ of the compression element: (k x l)
cross section geometry: (radius of gyration ryy)
Allowable Material compressive stress: (Fc)

69. Length (l)
‘unbraced length’ (kxl) depends upon the boundary conditions of an element

70. The radius of gyration (ryy) geometric property of member cross section
distance from neutral axis through which member area is considered to act
Iyy = Aryy2
ryy = Iyy/A
(Iyy = moment of inertia, weak axis)

71. Allowable Compression Stress Fc depends on ‘kl/r’
k = 1.0 (from diagram C-2)
l = 15 ft = 180 in (column length)
assume ryy = 3.0” **
kl/r = (1.0)(180”)/3” = 60
Fc = 17.4 ksi (look up on chart C-36)
** we must always come back and verify this assumption **

73. If using A36 Steel : FY = 36 ksi
Pmax = 240 kips (get this from P diagram)
Allowable Compression Stress (Fc) :
FC = 17.4 ksi (get this from Table C-36)
fC = P/Area
FC = Pmax/AreaRequired
AreaRequired = Pmax/FC = 240k / 17.4 ksi = 13.8 in2
Now: verify ryy & look up A in Charts

74. W12x65 A = 19.1 in2
ryy = 3.02 in

77. If using A36 Steel : FY = 36 ksi
Pmax = 240 kips (from P diagram)
Allowable Compression Stress:
FC = 17.4 ksi (from table C-36)
fC = P/Area
FC = Pmax/AreaRequired
AreaRequired = Pmax/FC = 240k / 17.4 ksi = in2
Use W12x65 Area = 19.1 in2
check actual stress: fC = P/A
fC = 240 kips / 19.1 in2 = 12.6 ksi < 17.4 ksi so OK!

79. BUCKLING and ALLOWABLE COMPRESSION STRESS :

80. Allowable Compression Stress depends on slenderness ratio = kl/r

81. Slenderness Ratio = kl/r
k = coefficient which accounts for buckling shape
for our project gravity columns, k=1.0
for moment frames see deformed shape

82. Slenderness Ratio = kl/r
l = unbraced length (inches)

83. Slenderness Ratio = kl/r
r = radius of gyration (inches)
typical use ry (weak direction)
rx > ry

84. Allowable Compression Stress (Fc)
slenderness ratio = kl/r
assume r = 2 in., k = 1.0
lcolumn = 180 in
kl/r = (1.0)(180”)/2”
kl/r = 90
use Table C-36 to
determine Fc = 14.2 ksi

85. AreaRequired = Pmax/FC = 238k / 14.2 ksi = 16.8 in2
Pmax Column 2 = 238 kips
(assume columns are continuous from foundation to roof, total length 30 feet)
FC = 14.2 ksi
AreaRequired = Pmax/FC = 238k / 14.2 ksi = in2

87. AreaRequired = Pmax/FC = 238k / 14.2 ksi = 16.8 in2
Pmax Column 2 = 238 kips
(assume columns are continuous from foundation to roof, total length 30 feet)
FC = 14.2 ksi
AreaRequired = Pmax/FC = 238k / 14.2 ksi = in2
Use W12x65 Area = 19.1 in2
fC = Pmax/Area = 238 k / 19.1 in2 = 12.5 ksi

88. AreaRequired = Pmax/FC = 238k / 14.2 ksi = 16.8 in2
Pmax Column 2 = 238 kips
(assume columns are continuous from foundation to roof, total length 30 feet)
FC = 14.2 ksi
AreaRequired = Pmax/FC = 238k / 14.2 ksi = in2
Use W12x65 Area = 19.1 in2
fC = Pmax/Area = 238 k / 19.1 in2 = 12.5 ksi
check ry for W12x65 and veriFy FC

89. AreaRequired = Pmax/FC = 238k / 14.2 ksi = 16.8 in2
Pmax Column 2 = 238 kips
(assume columns are continuous from foundation to roof, total length 30 feet)
FC = 14.2 ksi
AreaRequired = Pmax/FC = 238k / 14.2 ksi = in2
Use W12x65 Area = 19.1 in2
fC = Pmax/Area = 238 k / 19.1 in2 = 12.5 ksi
check ry for W12x65 and verify FC
ry (W12x65) = 3.02” not 2” as assumed … so must re-check kl/r
kl/r = (1.0)(180 in)/3.02in = 60

90. AreaRequired = Pmax/FC = 238k / 14.2 ksi = 16.8 in2
Pmax Column 2 = 238 kips
(assume columns are continuous from foundation to roof, total length 30 feet)
FC = 14.2 ksi
AreaRequired = Pmax/FC = 238k / 14.2 ksi = in2
Use W12x65 Area = 19.1 in2
fC = Pmax/Area = 238 k / 19.1 in2 = 12.5 ksi
check ry for W12x65 and verify FC
ry (W12x65) = 3.02
kl/r = (1.0)(180 in)/3.02in = 60, using Table C-36
Fc = 17.4 ksi

91. AreaRequired = Pmax/FC = 238k / 14.2 ksi = 16.8 in2
Pmax Column 2 = 238 kips
(assume columns are continuous from foundation to roof, total length 30 feet)
FC = 14.2 ksi
AreaRequired = Pmax/FC = 238k / 14.2 ksi = in2
Use W12x65 Area = 19.1 in2
fC = Pmax/Area = 238 k / 19.1 in2 = 12.5 ksi
check ry for W12x65 and verify FC
ry (W12x65) = 3.02
kl/r = (1.0)(180 in)/3.02in = 60, using Table C-36
Fc = 17.4 ksi > fc , therefore ok

92. Column 2, Efficiency Check: W12x65
fC = 12.5 ksi (actual stress fc = P/A)
FC = 17.4 ksi [allowable stress from chart C-36]
fC/FC < 1.0

93. Column 2, Efficiency Check: W12x65
fC = 12.5 ksi (actual stress fc = P/A)
FC = 17.4 ksi [allowable stress from chart C-36]
fC/FC = 12.5 ksi/17.4 ksi = 0.72 < 1.0

94. Column 2, Efficiency Check: W12x65
fC = 12.5 ksi (actual stress fc = P/A)
FC = 17.4 ksi [allowable stress from chart C-36]
fC/FC = 12.5 ksi/17.4 ksi = 0.72 < 1.0
(72% of capacity is used)

95. ALLOWABLE BENDING + COMPRESSION:

97. 80 kips
40 kips
40 kips
200 kips
200 kips

98. Axial Diagram Moment Diagram 900 k-ft 80 kips 80 kips - compression
+ tension
- compression
fb=M/S
fa=P/Area

104. Combined Stress (fa+fb)=
Axial Stress
(fa)
Bending Stress (fb)
Combined Stress (fa+fb) + =

105. To be certain that the combined stress (bending + axial) never reaches the yield stress, use the INTERACTION EQUATION
fb/Fb + fa/Fa < 1.0 +
Bending Stress (fb)
Axial Stress
(fa) +

106. Mmax = 900 k-ft Pmax = 200 kips
Assume 50% capacity of bending (fb)

107. Mmax = 900 k-ft Pmax = 200 kips
Assume 50% capacity of bending (fb)
50% Fb = (0.5)(21.6 ksi) = 10.8 ksi
SREQ = Mmax/50%Fb = 900k-ft (12in/1ft) / 10.8ksi
SREQ = 1000in3

112. TRY W36x260, Sx-x = 953 in3 A = 76.5 in2 ry-y = 3.78 in

113. TRY W36x260, Sx-x = 953 in3 A = 76.5 in2 ry-y = 3.78 in
fb = Mmax/S = 900 k-ft (12in/1ft) / 953in3 = 11.3 ksi

114. TRY W36x260, Sx-x = 953 in3 A = 76.5 in2 ry-y = 3.78 in
fb = Mmax/S = 900 k-ft (12in/1ft) / 953in3 = 11.3 ksi
Fb = 21.6 ksi
fb/Fb = 11.3ksi/21.6ksi = 0.52

115. TRY W36x260, Sx-x = 953 in3 A = 76.5 in2 ry-y = 3.78 in
fb = Mmax/S = 900 k-ft (12in/1ft) / 953in3 = 11.3 ksi
Fb = 21.6 ksi
fb/Fb = 11.3ksi/21.6ksi = 0.52
fc = Pmax/Area = 200 kips/76.5 in2 = 2.6 ksi
Slenderness ratio: k= l = 180 in
kl/r = (2.0)(180 in)/3.78 in = 95

118. TRY W36x260, Sx-x = 953 in3 A = 76.5 in2 ry-y = 3.78 in
fb = Mmax/S = 900 k-ft (12in/1ft) / 953in3 = 11.3 ksi
Fb = 21.6 ksi
fb/Fb = 11.3ksi/21.6ksi = 0.52
fc = Pmax/Area = 200 kips/76.5 in2 = 2.6 ksi
Slenderness ratio: k= l = 180 in
kl/r = (2.0)(180 in)/3.78 in = 95, using Table C-36
Fc = 13.6 ksi
fc/Fc = 2.6ksi/13.6ksi = 0.19

119. TRY W36x260, Sx-x = 953 in3 A = 76.5 in2 ry-y = 3.78 in
fb = Mmax/S = 900 k-ft (12in/1ft) / 953in3 = 11.3 ksi
Fb = 21.6 ksi
fb/Fb = 11.3ksi/21.6ksi = 0.52
fc = Pmax/Area = 200 kips/76.5 in2 = 2.6 ksi
Slenderness ratio: k= l = 180 in
kl/r = (2.0)(180 in)/3.78 in = 95, using Table C-36
Fc = 13.6 ksi
fc/Fc = 2.6ksi/13.6ksi = 0.19
fb/Fb + fc/Fc = 0.71 < 1.0, therefore ok

120. Assume 70% capacity of bending (fb)

121. Assume 70% capacity of bending (fb)
70% Fb = (0.7)(21.6 ksi) = 15.1 ksi
SREQ = Mmax/70%Fb = 900 k-ft (12in/1ft) / 15.1 ksi
SREQ = 720 in3

123. TRY W33x201, Sx-x = 684 in3 A = 59.1 in2 ry-y = 3.56 in

124. TRY W33x201, Sx-x = 684 in3 A = 59.1 in2 ry-y = 3.56 in
fb = Mmax/S = 900 k-ft (12 in/1 ft) / 684 in3 = 14.2 ksi

125. TRY W33x201, Sx-x = 684 in3 A = 59.1 in2 ry-y = 3.56 in
fb = Mmax/S = 900 k-ft (12 in/1 ft) / 684 in3 = 14.2 ksi
Fb = 21.6 ksi
fb/Fb = 14.2 ksi/21.6 ksi = 0.73

126. TRY W33x201, Sx-x = 684 in3 A = 59.1 in2 ry-y = 3.56 in
fb = Mmax/S = 900 k-ft (12 in/1 ft) / 684 in3 = 14.2 ksi
Fb = 21.6 ksi
fb/Fb = 14.2 ksi/21.6 ksi = 0.73
fc = Pmax/Area = 200 kips/59.1 in2 = 3.4 ksi
Slenderness ratio: k= l = 180 in
kl/r = (2.0)(180 in)/3.56 in = 101

128. TRY W33x201, Sx-x = 684 in3 A = 59.1 in2 ry-y = 3.56 in
fb = Mmax/S = 900 k-ft (12 in/1 ft) / 684 in3 = 14.2 ksi
Fb = 21.6 ksi
fb/Fb = 14.2 ksi/21.6 ksi = 0.73
fc = Pmax/Area = 200 kips/59.1 in2 = 3.4 ksi
Slenderness ratio: k= l = 180 in
kl/r = (2.0)(180 in)/3.56 in = 101, using Table C-36
Fc = ksi
fc/Fc = 3.4 ksi/12.85 ksi = 0.26

129. TRY W33x201, Sx-x = 684 in3 A = 59.1 in2 ry-y = 3.56 in
fb = Mmax/S = 900 k-ft (12 in/1 ft) / 684 in3 = 14.2 ksi
Fb = 21.6 ksi
fb/Fb = 14.2 ksi/21.6 ksi = 0.73
fc = Pmax/Area = 200 kips/59.1 in2 = 3.4 ksi
Slenderness ratio: k= l = 180 in
kl/r = (2.0)(180 in)/3.56 in = 101, using Table C-36
Fc = ksi
fc/Fc = 3.4 ksi/12.85 ksi = 0.26
fb/Fb + fc/Fc = 0.99 < 1.0, therefore ok