Composite 4.5.1 Magnel equations for composite beam 4.5.2 Shrinkage stress calculation 4.5.3 Example of shrinkage stress calculation 4.5.4 Magnel diagrams for a composite beam 4.5.5 Choice of prestress and eccentricity at different sections Cracking
4.7
Thermal
stress
4.7.1
Heating
72
4.7.2
Cooling
73
4.7.3
Calculation of stresses due to thermal
4.7.4
5.
pre-tensioned
design based on engineers' theory of bending 4.2.1 Sign convention 4.2.2 Example of beam designed based on engineer's theory bending Development of SLS design equations 4.3.1 Example of SLS design equations 4.3.2 Magnel diagram 4.3.3
4.4
of
concrete structure
calculation
60 61 64 67 70 71
71
Example
of thermal
4.7.4.1
Thermal stress calculation:
4.7.4.2
Thermal stress calculation:
stress
gradients
calculation
73
74
Heating Cooling
75 78
4.8
Detailing
79
4.9
References to Eurocode 1 and Eurocode 2 clauses
80
Bonded
post-tensioned
5.1
Post-tensioned beams
5.2
Cable 5.2.1
profile in
a
Example 5.2.1.1
5.3
81
structures
81
post-tensioned beam
81
of
82
permitted cable Magnel equations
zone
85
5.2.1.2
Determination of maximum
5.2.1.3
Determination of cable
5.2.1.4
Detailing
of
eccentricity
zone
post-tensioned
tendons
Concept of equivalent loads
87 88 88
5.3.1
General
equation
5.3.2
General
equation for distributed loads for a parabolic
profile
for
85
equivalent
loads
90 91
Contents
ix
5.3.3
6.
Load
Drape of the cable balancing
93
5.4 5.5
Reference to Eurocode 2 clauses
94
94
Statically indeterminate post-tensioned 6.1
95 95
Primary and secondary moments Prestressing of a propped cantilever
Analysis
to
determine the
6.2.1
Equivalent
6.2.2
General
6.2.3
General
loads for
a
97
distribution due
cable
to
prestress
profile of a single parabola
for
equation
equivalent
loads for
a
cable
105 of three
parabolas 111
of two
parabolas 114 116 116
6.3.1
Fixed end
6.3.2
Fixed end moments for of
a
moments
for three-parabola cable profile
four-parabola
cable
profile
continuous beam for moment distribution due
Distribution of shear force
121
Cable profile consisting of linear variation between supports Determination of prestress and cable profile:
Example
118 118
6.4.1 6.6
117
to
prestress 6.5
99
profile
Fixed end moments
Analysis
98
100
of four
parabolic segments 6.2.3.1 Alternative profile consisting instead of four parabolas 6.2.3.2 Alternative profile consisting instead of three parabolas Loss of prestress and equivalent loads
6.2.4
6.4
moment
equation for equivalent loads for a cable profileconsisting of three parabolic segments consisting
6.3
95
6.1.1 6.1.2 6.2
structures
Introduction
of a continuous
bridge
beam
122 122
6.6.1
Analysis
6.6.2
Determination of
126
6.6.3
Refined
128
of the
bridge
123
prestress and eccentricity equivalent loads 6.6.3.1 Fixed end moments for three parabola cable profile 6.6.3.2 Fixed end moments for four parabola cable profile 6.6.3.3 Moments at supports for the cable profile
129
6.6.3.4
129
6.6.3.5
analysis
due to
Choice of prestress at service Stress check at transfer and service
129 129 132
6.7
Concordant cable
6.8
Choice of tendon size and location of tendons
132
6.9
Equivalent loads
134
6.9.1 6.10
Equivalent
132
profile and shift in the centroidal axis
Shift in the centroidal axis in box
girders
loads and variable second moment of
6.11 Thermal stress
analysis
and continuous structures
area
136 136 139
Prestressed Concrete Design
7.
Thermal stress calculation:
6.11.2
Thermal
calculation:
Reduction of moment
6.13
References to Eurocode 2 clauses
Ultimate
support
over
heating
139
cooling
142
in continuous beams
147
bending strength calculations
147
Introduction
7.2
Stress distribution at different stages of
7.3
Stress
7.4
Rectangular
7.5
Stress
-
-
strain strain
149
for concrete
relationship
stress
147
loading
block in bending strength calculations
150
Strain and stress in steel 7.6.1
Prestress and
7.6.2
Strain due to
7.6.3
Total strain and
150
in steel
pre-strain bending in
steel
151
stress in steel
151 151
7.9
bridge
7.10
Ultimate moment calculation of
composite bridge beam
7.11
Use of additional unstressed steel
7.12 7.13
The strain
Stress-strain
Example
149 150
for steel
relationship
compatibility method Properties of a trapezium Ultimate moment calculation of a
7.8
144
145
7.1
7.7
relationship
a
152
for unstressed
beam
152 157
162
reinforcing
steel
163
of ultimate moment calculation with stressed and
unstressed steels
164 167
7.15
Calculation of Mu using tabular values Calculation of Mu for statically indeterminate beams
7.16
Reference to Eurocode 2 clauses
170
7.14
Analysis
171
Introduction
8.2
Cracked section
8.3
Cracked section
8.5
analysis analysis of a double T-beam strain relationship for concrete strain relationship for steel
8.3.1
Stress
8.3.2
Stress
8.3.3
Cracked section
-
-
Partially prestressed Composite 8.5.1
169
171
of cracked sections
8.1
8.4
9.
stress
6.12
7.6
8.
6.11.1
172 174
174 174
analysis
180
beam
183
beam
Magnel diagram
171
for
Ultimate shear and torsional
composite
beam
strength
calculations
184
193
9.1
Introduction
193
9.2
Shear capacity of a section without shear reinforcement and uncracked in flexure
195
Contents
xi
9.2.1
Example
of calculation of shear
capacity
of
a
section
without shear reinforcement and uncracked in bending 9.3
Checking
9.4
Shear capacity of and cracked in 9.4.1
198
section without shear reinforcement
a
bending
198
Example of calculation of shear capacity of a section without shear reinforcement and cracked in
9.5
197
for start of cracked section
Design
199
bending
of shear reinforcement
200
9.5.1
Derivation of
9.5.2
Procedure for shear link
9.5.3
Design
of
a
beam not
9.5.4
Design
of
a
beam
equations (9.7)
and
(9.11)
201
design
203
needing design
shear reinforcement
203 Shear
9.7
Effective web width in the presence of ducts Interface shear between web and flange in T-sections
capacity
9.8.1
of
a
composite
9.10
9.11
204
beam
207
Example of reinforcement calculation between web and
9.9
212 213
for interface shear
flange
214
Interface shear between precast beam and cast in-situ slab Design for torsion
215
9.10.1
227
Spacing
of torsion reinforcement
for combined shear force and torsion
9.12
Design Warping
9.13
References
torsion
219 227 228
to Eurocode 2 clauses
228
Calculation of crack widths
229
10.1
Introduction
10.2
Exposure
10.3
Recommended values of maximum crack width
229
10.4
Minimum steel
231
10.4.1 10.5
229
classes
229
areas
Example
of minimum steel
Calculation of crack
area
calculation
233
10.7
width, wk 10.5.1 Crack spacing, Sr, max 10.5.2 Example of crack width and spacing calculation Example of a partially prestressed beam 10.6.1 Example of minimum steel area calculation 10.6.2 Example of width and spacing of crack Control of cracking without direct calculation
238
10.8
References
239
10.6
11.
shear reinforcement
9.6 9.8
10.
needing design
to
Eurocode 2 clauses
Loss of prestress
234 235
235 236 236 237
241
11.1
Introduction
241
11.2
Immediate loss of prestress
241
Prestressed Concrete Design
11.2.1
Elastic loss in
241
11.2.1.1
pre-tensioned beams Example of elastic loss calculation loss in post-tensioned beams
243
11.2.2
Elastic
244
11.2.3
Loss of prestress due to friction and wobble
Derivation of loss of prestress due to friction 245
11.2.3.2
Example
of calculation of loss of prestress
due to friction and wobble 11.2.3.3
11.2.4
245
11.2.3.1
247
Calculation of 6 for different
profiles
248
Loss due to draw-in of wedges
252
11.2.4.1
254
Example
of loss of prestress due to draw-in
11.3
Loss of prestress due to creep, shrinkage and relaxation 11.3.1 Example of final loss calculation
255
11.4
References
259
Design
to
Eurocode 2 clauses
of slabs
12.1
Introduction
12.2
Typical
257
261 261
beam and slab
261
depths
12.2.1
12.3
Effective span of slabs for different support conditions 262 262 One-way spanning slabs 12.3.1
12.4
12.3.2
Design of a one-way spanning Analysis for applied loading
slab
12.3.3
Choice of prestress
12.3.4
Calculation of losses
12.3.5
Calculation of correct
12.3.6
Calculation of moment distribution
12.3.7
Calculation of stress distribution at service
271
12.3.8
Calculation of stress distribution
272
263 264
266 267
equivalent
loads at service
at transfer
slab
Edge-supported two-way spanning 12.4.1 Design of a two-way spanning
269 270
272 slab
273
12.5
Flat slabs
276
12.6
Methods of
279
12.7 12.8
12.9
analysis of flat slabs Example of the design of flat slab Finite element analysis of flat slab
283
283
12.8.1
Results of anal ys is for dead load
12.8.2
Results of
load pattern 1
288
12.8.3
Results
294
12.8.4
Results
load pattern 2 load pattern 3
12.8.5
Results of
live load pattern 4
303
Finite element
284
analysis for dead plus live of analysis for dead plus live of analysis for dead plus live
analysis
analysis
of
a
for dead
strip
plus
of flat slab
298 307
12.9.1
Results of
12.9.2
Results of
for dead
plus
12.9.3
Results
for dead
plus live
315
12.9.4
Results
for dead
plus
load pattern 2 live load pattern 3
318
12.9.5
Results
for dead
plus
live load pattern 4
321
analysis for dead load
analysis of analysis of analysis of analysis
309 live load partem 1
312
Contents
xiii
12.10
Comparison a
12.11
strip
between the results of
and 324
Eurocode 2 recommendations for
12.12
Grillage analysis
12.13
Example 12.13.1
12.14
analysis of full slab
of slab
of
for
design
Results of
irregular
equivalent frame analysis
column
layout
of flat slab-frame
analysis
327
of slab-f rame model
329
12.13.2 Moment distribution due to prestress 12.13.3 Cable profile
329
Calculation of loss of prestress 12.14.1 Calculation of loss due
331 to
12.14.2 Calculation of loss due to 12.14.3
330
friction and wobble per cable 331
wedge
draw-in
333
Calculation of prestress at service
333
12.14.4 Determination of number of cables 12.15 12.16
Fixed end
moments due
patch
333
loads and concentrated force and
couple Equivalent loads and fixed end moments 12.16.1 Equivalent loads and fixed end moments 12.16.2 Equivalent loads and fixed end moments
334 334 at
transfer
at service
12.16.3
Moment distribution due to
equivalent loads
12.16.4
Moment distribution due to
equivalent
loads at service
12.16.5
Moment distribution due to
equivalent
loads
and external loads 12.16.6
at
at transfer
Moment distribution due to
equivalent
12.6.7 12.16.8
337 340 342
loads and 342
Stress distribution in the slab at transfer and service
Calculation of punching shear stress vEd under the action of a 367
moment
13.7
13.8 13.9
Punching
15.
shear stress under shear force and
moment
acting
369 simultaneously 13.7.1 Special cases of shear force and moment acting together370 371 Punching shear stress checks 373 Example of punching shear capacity design
Torsional constant of thin-walled closed hollow sections411
Example
of analysis of
a
411
beam and slab deck
16.5.1
Bending properties
16.5.2
Section
16.5.3
Section
16.5.4
Torsion
16.5.5
Alternative
expressions
rectangular
cross
412
of precast beam
412
properties of interior composite beam properties of end composite beam
414
416
for composite beam
constant
for
416
approximate value of J
for
sections
16.5.6
Section
16.5.7
Material
16.5.8
Calculation of live loads and
properties
418
of transverse beams
418
properties
419
bending
moment
distribution
in beam elements: SLS 16.6
Stresses due to
16.7
Thermal
stresses
16.7.1
Thermal stresses:
16.7.2
Thermal
shrinkage in the
419
of slab
composite
stresses:
426 beam
426
heating cooling
427 429
16.8
Stress distribution at SLS due to external loads
432
16.9
Magnel diagrams
433
16.10
Calculation of live loads and
16.9.1
16.11
437
bending
moment distribution in
beam elements: ULS
438
Self-weight
443
16.12 Ultimate 16.13
Stress checks
moments
moment
capacity: Mid-span section
443
Ultimate shear force
447
16.13.1 Analysis to determine maximum shear force along the span: Cases 1-4
16.13.2 16.13.3 16.13.4
16.14
Analysis
448
to determine maximum shear force
452
Summary of results
453
Design
of shear reinforcement
454
Design of a post-tensioned box girder bridge 16.14.1
along
the span: Cases 5 -8
Calculation of moments at SLS
459 461
16.14.2 Thermal stresses:
Heating
464
16.14.3 Thermal stresses:
Cooling
466
16.14.4 Determination of prestress and 16.14.5 Stress calculation at SLS
eccentricity
467 471
Prestressed
xvi
Calculation of moments
16.14.6
Design
474
ULS
477
16.14.7 Calculation of moment capacity at ULS 16.14.8 Calculation of shear force at ULS
479
16.14.9 Calculation of twisting
481
16.14.1
ODesign
16.14.11
moment at ULS
483
of shear and torsional reinforcement reinforcement to resist torsion
Longitudinal
16.14.12 Stress
17.
at
Concrete
487 487
analysis of the deck
16.15
Eurocode 2 rules for reinforcement at
489
16.16
External and internal tendons: A
491
16.17
References to Eurocode 2 clauses
Lower bound
anchorages comparison
491
approaches to design
at ultimate
limit state 493
17.1
Introduction
493
17.2
Theory of Plasticity In-plane stresses 17.3.1 Examples of reinforcement calculations Presence of prestressing cables 17.3.2 Designs for a combination of in-plane and flexural forces 17.4.1 Example of design for a combination of in-plane
493
17.3
17.4
494 497 503
504 and
507
flexural forces 17.5
Criterion for cracking
17.6
Out-of-plane
17.7
Strut and tie method of
508
510
shear
17.7.1
B and D
17.7.2
Saint Venant's
17.7.3
An
Design
17.7.5
Types
17.7.6
Correct
511
Regions
example
17.7.4
511
design
512
principle
of strut-tie
of nodal
515
layout
517
zones
17.7.6.1
Correct
17.7.6.2
Correct
layout of struts layout of struts
and ties:
deep
beam
and ties: corbel
520 521
Code recommendation for
design
of corbel
17.7.6.4
Correct layout of struts and ties: half-joint Correct layout of struts and ties: end-block
17.7.6.5
Reinforcement
17.7.6.3
18.
520
of struts and ties
17.7.6.2.1
17.8
514
modelling
of struts
Reference to Eurocode 2 clauses
Design for earthquake resistance
at frame corners
523 525 527 529 531
533
18.1
Introduction
533
18.2
Ductility
535
18.3
Types
of structural systems
536
Contents
18.4
Behaviour
18.5
Ductility
18.6
A brief introduction to structural
540
18.6.1
540
18.6.3 18.6.4
540
system Calculation of eigenvalues
18.6.5 18.6.6
544 545 545
Eigenvectors of [K-o2 M] Properties of eigenvectors
superposition: undamped forced response Mode superposition: damped forced response Mass participation factors and effective mass 18.6.10.1 Mass participation factors: Example
Detailing for local ductility: beams Detailing for local ductility: columns
564
569
18.14
Design shear force in beams and columns Design provisions for ductile walls
18.16
Reference to Eurocode 8 clauses
571
18.12 18.13
19.
542
18.6.7 18.6.9
18.8
538
dynamics Single-degree-of-freedom system Multi-degree-of-freedom system Response to an acceleration of the base Vibration of an undamped free multi-degree-of-freedom
18.6.2
18.7
factor, q
classes
Miscellaneous 19.1
566
topics
573
Introduction
19.2
Unbonded
19.3
Design
of
570
573
design
573
19.3.1
post-tensioned box girder Calculation of live loadings at
