Appendix 1 51 conversion table

Imperial SI Metric

1 in. 25.4mm 1 ft 304.8 mm Length 3.28 ft lm

1 in.' 645.2 mm2

1 ft 2 0.093 m2 Area 10.7 ft 2 1m2

llb 0.454 kg 2.2lb 1 kg Mass 1 ton 1016.05 kg (weight) 0.982 tons 1 t

1 ton/ft' 107.2 kN/m2

1 tonfin2 15 480 kN/m2 Live loads 15.48 N/mm2

20.88 lb/ft2 1.0 kN/m2

llbf 4.448 N 0.225lb f IN Force 1 ton f 9964N moments

9.9 kN 1 ton xin. 0.253 kNxm

llbfin.' 0.0069 N/mm2

1 ton/in. 2 15.45 N/mm2

144.93lb/in.' l Nfmm2 Stress 1000 lb/in.' 6.9 Nfmm2

0.065 ton/in.' 1 Nfmm2

135

Appendix 2 Calculation of a roof truss

~ 1360 kg Elevation

t-~r~ E // C) ~12o C) C)

c-, ~--;~"

~// / /_/ r: // ~// ~2 ILL

11 87S

Plan

136

680 kg

680 kg

1360 kg i'

~T/71 V//~ ///

kgjm21 ~//I 1// ~(! ~"" / 'il v/'lJ, ~~4 m I

H

1160

K

A

8

G

= = M

• G

Forces scale : 1 mm = 40 kg

137

Appendix 3 Calculation of timber purlin on truss shown in Appendix 2

w = 680 kg

~11111111111111111111111111111111111111111111111111111111~ L = 3,000 m

680 X 3 M = 8 255 kg m = 2 500 000 N mm = 2 500 kN m

f=7N/mm 2

Z = 2 500 OOO 357 142.8 mm3 7

Assume B = 75 mm

75D 2 Z = - 6- = 357 142.8 mm3

D = J6 x 357 142.8 = "\128 500 = 168 mm 75

Select 75 X 175 mm

Check for deflection

5 WL 3 d=---

384 E/

Win newtons = 9.806 X 680 = 6700 N

E = 10 500 N/mm2

I= 34 200 000 mm4

138

d= 5x6700x30003

384 X 10 500 X 34 200 000

5x6.7x33 103 x109

3.84 X 1.05 X 3.42 102 X 104 X 107

65.6 6 5 =w= . 6mm

Permissible deflection 3~5 of span

6.56 1 3QOo = 46o span

/= 75 X 1753 = 34.2 X 106 mm4 12

139

Appendix 4 Calculation of a timber box girder

Shear diagram

Loading 120 kg/m2 W = 1800 kg M = W X L = 1800 X 10

req. 8 8

= 2250 kg m

or 22.0 kNm

Check M possible for f = 6.3 N /mm 2

Cross-section

I= 112 (BD3 - bd3 ) = 1

12 (200 X 5003 -160 X 4003 )

= _!__ (2 X 53 -1.6 X 4 3 ) X 108 = 12.3 X 108 12

z =!.."' 12·3 x 108 = 0 4925 x 108 = 4925 000 mm3 y 250 .

Mposs. = 4 925 000 X 6.3 = 31 000 000 mN X m

= 3150 kg x m

adequate

140

Check for shear

s - 1800 k f max --2- g

s Shear stress = ---

A flanges

1 I l I l

e

~I J/~~~/ //,// ~/ /0/ ,0-~///-:, ~/

'V/ /~ ~// // /'/ ///////////,

I I

10 000 m

Plan

1800 = 900 2x20x400 16000

= 0.056 kg/mm2 = 0.55 N/mm2

Adequatefspenn = 1 N/mm2

Ends to be strengthened against buckling.

141

Appendix 5 Calculation of a slender timber column

142

E = = -_,

f= 5.5 N/mm2

H= 4.800 m

Effective height %X 4.800 = 3.600 m

Slenderness ratio !:_ r

r= JI= y8.33= 28.8 ~ = 125

/perm: (Table Fig. 151) interpolated between 120 and 130 factor= 0.38

/perm= 0.38 x 5.5 N/mm2 = 2.09 N/mm2

Ppermissible = 2.09 X 1002 = 20 900 N = 2130 kgf

~{~ 1 1oo I

1004 I=--= 8 333 000 mm4

12

From Euler:

P = L; E/ E (assumed for grade S2 = 10 300 N/mm2)

1r P = 36002 X 10 300 X 8 333 000 N

= ~ X 1 03 X 8 3 104 X l 06 = 20 800 N 3.6 . . 10

= 2120 kgf

143

Appendix 6 "Kern" dimensions for hollow shapes

1. Squares

144

8

""la 3

z e=-A

IB - /b - 1 B4 - b4 z- B/2 - 12_B_ x 2

A =B2 -b2

Z 1 B 4 - b4

A= 6B B2 -b2

= .1_ (B2+b2) 6B

For very thin walls

b approaches B

which is 2 X eccentricity of solid

2. Tubes

0

lo 2

z =In- /d = ..!!._ D4- d4

D/2 64 D/2

A= 1T(D2- d2) 4

For very thin walls

d approaches D

which is 2 x eccentricity of the solid.

145

Appendix 7 Calculation of a continuous reinforced concrete floor slab

6.000

Live load 5 kNfm2 [100 lb/ft2 )

Dead load 3.6 kNfm2 [72lb/ft2 ) assuming a slab 150 mm thick

8.6 kNfm2

W= 6 X 8.6 = 51.6kN

M = 51.6 x 6 = 31 0 kN m 10 .

D= ~ =y'0.0177 \I l750 X 1 = 0.130m

Steel

M = A 5 X fs X lever

M = P~B BD2 [load factor method]

PcB = 7 N/mm2 = 7000 kN/m2

M= 1750BD2

Lever arm= 1v = 1 x 0.130 = 0.097 m 4 4

fs = 210 000 kN/m2

31.0 = A 8 X 210 000 X 0.097

A = ___1!:Q_ = 0.00 147m2 = 2290 mm2 s 21 000

rrD 2 Assume 10 bars 147 mm2 each 4 = 147

Use 10 bars 14 mm<fl or nearest D = 14 mm.

146

)L.:.,..__~ • _.l:_ver • ____,~• • \ d 0 = 130 mm

cover 30 mm

160 mm

147

Index

stresses (see Allowable Stress)

shear walls

62 62, 72, 74-7 94,96

also

American pavilion, Brussels 23, 24 Arch 9, 16, 39

'middle third'line of 94 Arch action (hyperbolic para-

boloid) 16, 19, 20, 22 Arches Norman 16

parabolic 16, 39 reinforced concrete 69 Roman 16 semi-circular 16 stone 16, 25

Architecture, Gothic modern 1, 4

Arresting forces 5, 10, 27

Balcony, structural action in 2 7, 28

Basic equilibrium 5, 7 Beam 30,31

action 61 ff. in planks 70, 71, 104 in shells 68, 69 in slabs 66, 68

calculations 60, App_ 3 continuous 113-15 forces in 31 precast concrete I 00 rectangular 56, 57, 62 reinforced concrete 65 steel, calculation of 59, 60 theory 61 ff. timber App_ 3

Bending, elastic theory of 53 examples of 54, 55 stress 50

Bending moments 30, 31, 53, 105 in columns 90 in continuous beams 113-

15 initial Ill in portal frames 39-45 in pre-stressed structures

97-101 maximum 31, 32, 38

Bicycle in

Bond

wheel, 23, 24 65

structural action

Box beam, plywood 109, App. 4 Bridge, masonry arch 23

parabolic arch 16, 18, 20 suspension 8, 16, 17

Brunelleschi 19, 21 Brussels, American pavilion 23, 24 Buckling 8, 81 Buckminster Fuller 25 Buttress I, 6, 7, 13, 14, 16, 18

in hyperbolic paraboloid 20,22

Cable 8, 9, 14, 16, 39, 98-9, 100 Calculations, of basic beam 60

of reinforced concrete beam 65

of timber box girder App. 4 Calculus, use of, in considering

deflection 103 California, Market Hall 9, 20, 22 Camber, in pre-stressed beams 99 Cantilever 27, Ill Cantilevered hoops 20 Carbon fibre 132 Castellated beam 62, 74, 109 Catenary 16, 19, 20, 22, 23, 43,

79, 131 circular 23 in portal frame 42, 43 within thickness of arch 94

Chain, structural action in suspended 16

Centroid of section 54 Characteristic loads 129 Circle ('middle quarter') 92 Collapse moment 123, 124 Collapse, theory of 128 Column 8, 109, 110

load-bearin~; capacity of 82, 109

mushroom pre-stressed slender 82,

98,99

117 98, 99 App. 5 (p. 142)

149

Components, horizontal and Dome-cont. vertical 5, 6, 12, 14

resultant of 11 Compression 81

circular 23 in concrete 65, 66 radial 23

Compressive and tensile stresses, in beams 65

in pre-stressing 97, 98 Concrete, beam 65 ff.

'E' values 51 reinforced 60, 65 slab (see also Slabs) 109 shells 69, 7 5

Continuity in structure 110, 111, 113, 129, 132

Continuous beam 113-16 Contraflexure 33, 43, 44

in portal frames 43, 44, 45, 111

Counter-rotation 27 Couple 14, 65, 69, 70, 131 Curvature, due to deflection 101,

103

Deflection (see also Deforma-tion) 66, 103, 104, 106, 113, 128, 129

and continuity 113 and span-load relationship

106-7, 108 in concrete beams 65, 66 in long spans 104 slope of 112

Deformation 48, 49, 53, 66, 103, 123

elastic shear

53,54 74, 75

Detours, caused by bending moments 131

structural 9, 27, 30, 105 Diagonal members (see also Trussed

joist) 6, 62 Diagrids 11 7 Diesel engine Direct transmission of forces 8, 14,

39 Distributed

Loads) Dome 23

load (see 9,37, 105,106

and rings 23

also

by Buckminster Fuller 25 circular 18 masonry 18

150

of Sta. Maria della Fiore, Florence 19

of St. Paul's, London 18

'E' values for

131, 132 steel,

timber 51-2 in deflection 104

concrete,

in pre-stressing 100 in reinforced concrete design

65,66 relative (in plastic theory)

127, 128 Eccentricity, on foundations 44,

95 Eccentric loading 90, 93, 98, 129 Eccentric pressure 92, 93, 94

in pre-stressing 98 tensile and compressive

stresses 94 Economy, in use of materials Edge beam, use of, in shell

construction 76 Effective length 86

in columns 84, 86 in walls 88

Effective depth, in reinforced concrete beam 65

Efficiency of sections 55 Elastic behaviour 49, 65, 123 Elastic deformation 49 Elastic theory (see also Beam

action) 53, 59, 129 Elasticity 49, 123 Equilibriant 11, 27, 29, 31, 53, 96 Equilibrium 5, 7, 9, 12, 13, 14, 27,

29, 47, 53, 96, 131 basic 96 demonstrated on

board 28-9 direct 14 static 27

Essence of structure 131 Euler theory 82, 85 Euler values 82

Fibres, stresses in 103 Fixing moment 112, 113

pulley

Fixity, condition of 112, 113 Flanges, value of, in structural

sections 56 Flutter, tendency of parabolic roofs

to 20 Flying buttress Flywheel analogy 83

Folded slabs 67, 119 used in space frames 119

Forces, and levers 28 defined 47 direct transmission of 5, 8 direction of 10 equilibrating 11 polygon of 10, 12 triangle of 11, 12 tensile and compressive 6, 8

Form, meaningless 1 Forth Bridge 16, 17 Foundation design 44, 92 Fulcrum 28 Funicular action 23, 24, 25 Funicular, in shells 69 Funicular line 14, 16, 19, 23, 25,

39,42,69,131

Geometry, of action of forces 7 of structural layout 105,

129 Glass fibre materials 131 Gothic arches 19 Gothic piers 94 Gothic pinnacles 95 Gravity, action of 5 Grip strength, in reinforced concrete

65 Gyration, radius of 83, 84

Half-timber construction 1 High-tensile materials 9, 19, 131 Hinge, in portal frame 41, 123

plastic 124 structural 35, 43, 129

Hollow shapes 93 limits of eccentricity of 9 3

Hollow tiles, use of, in reinforced concrete construction 67, 109

Hooke's Law 47, 49, 50, 51, 54, 104

Hoop action 23, 24, 25 Hoop arches 20 Hoop stresses 2 3, 2 5 Horizontal components 12 Hyperbolic curves 20 Hyperbolic paraboloid 22

'I' -beams, plywood 7 7 sections 106-7 span-load relationship of 1 06

Inertia, moment of 55-8, 66, 82, 83,128,131

Joists, 'ideal' shape of 3 trussed 62

Kern of section 93 'Knees', in portal frames 40

Lamp posts, structural action in 98 Lateral thrust 9 4-9 Lever arm 28, 54, 66

in bridging members 131 in concrete beam 65, 66

Limit state design 128-9 Line of action 7 Load 48

centrally-placed point 37 characteristic 129 on catenary 16 distributed 9, 37, 38, 128,

129 factor 124, 125, 128, 129

in reinforced concrete design 127

parallel 23 point 37,38 radial 23 symmetrical 32

Load-bearing walls 88-90 Long spans (see also Span)

deflection due to 103, 104 in space frames 119-21

Maillart 16 Mass 5 Maximum stress (see also

Stress) 123 'Middle third' rule 92, 94, 96 'Middle quarter' (in circular sections)

92 Modulus, plastic 126-8

sectional 59, 60 Young's 51

Moments 27, 28, 30 distribution of, in beams

32-3 negative 30, 33, 111

and positive 30, 33, 111

of inertia 54, 55-8, 66, 82, 83, 106, 109, 128, 131

of resistance 58, 125 Mushroom columns 117, 119

Natural thrust line 94 (see also Catenary)

Negative moment 30, 33, 98, 111 Nervi 16 Neutral axis 54, 58, 61, 67, 85 Newton (unit of measurement) 5 Newton's Second Law of Motion 5

151

No-moment line 39, 40, 42, 45, 79 (see also Catenary)

No-shear line 79 No-tension limit 94 Norman arch 16 Notre Dame, Cathedral of, Paris

Olympic Swimming Pool 20, 22 Open section (see also Truss) 105,

107, 109

Parabolic arch 16, 20, 39 Parabolic curve 23 Parabolic ribs 19 Parabolic shell 16 Paraboloid, hyperbolic 20 Permissible loads (reduction factors

in concrete columns) 85 Permissible stress, shear, for steel,

concrete, timber 79 Planks, precast concrete beams

and 100 Plastic method of reinforced concrete

design 127 Plastic modulus 125 Plastic stress distribution 126 Plastic stage, stress distribu-

tion 124 Plastic theory, of structural

design 1231/., 127, 129 Plywood, use of, in structural

members 22, 67 Point loads 37, 38, 131 Polygon, of forces 10, 12

vector 12 Portal frames 39 If., 129

"ideal" shape for 42 and deviation from

42,43 load-factor method applied to

design of 123, 124 moments in 39, 40, 125 one-pin (hinge) 35 two-pin (hinge) 43 three-pin (hinge) 42

Positive moments 30, 33, Ill Post-and-beam construction 131 Precast concrete beams, and

planks 100, 101, 109 Pre-stress cables 99 Pre-stress conditions, in rings and

domes 23 Pre-stressing 23, 97, 98, 99 Pre-tensioned and pre-stressed

concrete members 100

152

Pulley systems, illustrating moments 28, 29

Purlin calculations App. 3

'Queen-post' truss 73

Radialloads 23 Radial tension 23 Radius of gyration 83, 84 Rafters, couples in 14

thrust of 6, 30 Rankine 82 Reactions, and reaction forces 4 7 Rectangularbeam 105, 107 Reinforced concrete beams 65 If.,

and slabs 66 If. Reinforcement, in concrete struc-

tures, bars II 7 shear 75, 76 steel 65 two-way 117

Renaissance period and its structural expression 18, 19

Resistance, moment in concrete beam 65

moment of 58-9, 65, 121 plastic 125 to shear forces 74, 75

Resultant 94 Ribbed slab 67 Rigid frames 44 Rigidity 44, 113 Ring action 23 Rings, and domes 23 If. Rolled steel joist, (R.SJ-) 57, 125 Roof 20-2

collar beam 30 flat 119-21

Roman arch 15, 16 Rotation 27, 29, 72

and counter-rotation 27

'Saddle shape' 20 Sta. Maria della Fiore, Florence,

dome of 19, 21 St. Paul's, London, dome of 18,

19, 20 Sectional modulus 59, 60 Shape factor 54, 69, 125, 126, 129 Shear action 70, 71, 72 Shear couples 70 Shear force 6, 62, 99

at supports 72 in diagonal members 7 3, 7 4,

99

Shear force-cont. in pre-stressed members 97,

100 maximum 72 vertical, and horizontal 71,

77 Shear stress 73, 74, 75

distribution of, over cross section 74, 78

in pre-stressed members I 00 in timber 77 maximum 72 permissible 79 related to span 109

Shell construction, in reinforced concrete 16, 22, 75, 76

Short columns II 0 Short spans 109 S.I. (Systeme International)

notation 5, App. I Skin structures 131 Slabs, continuous 116

folded 67 reinforced concrete 66, 109 ribbed 67 shear in 109 stresses in 113, 116, 117 two-way 117

Span-load relationship 103, 104, 106, 107

considered as to depth of members 104-5

due to bending 104 due to deflection I 06

Statically determinate frames 42 Statically indeterminate frames 12 Steel, reinforcement of, for

concrete 65, 108, 117 Steel stresses 50

'E' value 51 high tensile 9, 18, 131 relative price of 132

Stiffness, due to'/' value 55-7 in columns 81-3 in portal frames 41,43 of a section 55, 62, 82

Stonehenge 132 Strain 48, 49

in concrete 69 ff. Stress, characteristic 4 7, 48, 129

compressive 53, 58 definition of 4 7 distribution of 58-9 patterns 65, 91 tensile 53, 58 values 50

Stress-cont. working 50, 51

Structural behaviour 7, 8 Structural detours 9, 27, 30, 105 Structural form 105, 106 Structure, essence of 131 Strut 39 Suspended roofs 131 Suspension bridges 17, 61, 131 Systeme International 5, App. I

Tensile cracks 94 Tension, 5, 6, 8, 14

radial 23 T-beams 66, 67, 117 Tide forces 10, 11 Ties, action of 6, 14 Timber, 'E' values of 52 Triangle, of forces 11, 12

vector 11, 13, 14, 15 Triangulation 14 Three-hinged frames (portals) 42,

123 Trusses 62, 73, 105, 119

shear in 73-4 Turin, Exhibition Hall 16 Turning effect 112 Two-pin frames 43

Vectors 10, 13 Vector polygon 12 Vector triangle 11, Vertical components Vertical shear 70 Vierendeel girder 74

13, 14, 15, 27 12

Walls 88-90 effective height 88 effective slenderness 88 effective thickness 88 11 inch cavity 88 stress, permissible (allowable)

in 88 Weather resistance 94

due to pre-stressing 98 Weight, related to gravity 5

increase of 62 Wind forces 10 Wind pressure 94, 95 Working stress 50, 51

in concrete 51 in steel 51,66 timber 51

153

Yield point 50, 51 Young 49 Young's Modulus 49, 51, 104

154

Zero moment 43, Ill Zero line 54 Zurich, 1939 Exhibition shell, by

Maillart 16