# Wiki source code of Concrete Design

1.1 1 {{box cssClass="floatinginfobox" title="**Contents**"}}
63.1 2 {{toc/}}
1.1 3 {{/box}}
4
3.1 5 (% style="text-align: justify;" %)
63.1 6 FEM-Design performs design calculations for reinforced concrete-, steel- and timber structures according to Eurocode. The following design considers EC1992-1-1 (standard) with the following National Annexes (NA) for:
3.1 7
63.1 8 * Denmark
9 * Estonia
10 * Finland
11 * Germany
12 * Hungary
65.1 13 * Latvia
63.1 14 * Norway
15 * Poland
16 * Romania
17 * Sweden
18 * United Kingdom.
3.1 19
25.1 20 = {{id name="Design forces"/}}{{id name="Design forces"/}}Design forces =
1.1 21
22
8.1 23 The design forces are the forces that the reinforcements should be designed for in the reinforcement directions. The term design forces have meaning only in surface structures like plate, wall or 3D plate. In beam structures the design forces are equivalent to the internal forces. The necessary reinforcement calculations are based on the design forces.
24
25 (% style="text-align: justify;" %)
26 The way of calculating the design forces is common in all modules and in all standards.
27
28 (% style="text-align: justify;" %)
29 In FEM-Design the design forces calculation is based on the mechanism of optimal reinforcement calculation for skew reinforcements made by M.P. Niel- sen, Wood-Armer and Dr. Ferenc Németh, see [2]. The following description will show the way of calculation for moments but the way of the calculation is the same for normal forces too. Just substitute the m signs with n and you will have the calculation for normal forces.
30
1.1 31 (((
8.1 32 (% style="text-align: justify;" %)
33 For the calculation of the design forces we have given:
34
35 * ξ, η reinforcement directions,
36 * α, β angle of global x direction and the ξ, η reinforcement directions,
37 * mx, my, mxy internal forces.
38 )))
39
40 The results will be the design moments:
41
42 * [[image:1543503447758-943.png||height="25" width="79"]]
43
44 (% style="text-align: justify;" %)
45 In the first step we are taking a ξ-ϑ coordinate system and transform the internal forces into this system:
46
47 [[image:1543503517958-895.png||height="311" width="399"]]
48
15.1 49 (% style="text-align: justify;" %)
50 Now the design forces will be chosen from four basic cases called a), b), ξ) and η). The possible design moment pairs of the cases:
8.1 51
15.1 52 a) case:
8.1 53
15.1 54 [[image:1543503652914-266.png||height="91" width="351"]]
8.1 55
15.1 56 b) case:
8.1 57
15.1 58 [[image:1543503750827-594.png||height="100" width="350"]]
8.1 59
15.1 60 ξ) case:
8.1 61
15.1 62 [[image:1543503838340-661.png||height="87" width="165"]]
63
64 η) case:
65
66 [[image:1543503816725-369.png||height="88" width="336"]]
67
68 From the four cases the one is invalid where:
69
70 * the signs are different: **m,,ξ,,*m,,η,, < 0**
71 * the crack tensor invariant is less than the internal forces invariant:
72
73 [[image:1543503891591-532.png||height="75" width="316"]]
74
75 (% style="text-align: justify;" %)
76 The valid positive pair will be the design moment for bottom reinforcement; the valid negative pair will be the design moments for the top reinforcement (positive means positive and zero values; negative means negative and zero values).
77
78 (% style="text-align: justify;" %)
79 So the result will be four values in a certain point: two moment values for each reinforcement directions. It can sound strange that the reinforcements are used for both positive and negative moment in one direction at the same time, but if we are looking at a plate where the **m,,x,, **is positive and the **m,,y,,** is negative and the reinforcements have an angle of 45 degree to the x direction we could imagine that the bottom reinforcement bars make equilibrium to the **m,,x,, **and the top reinforcement bars make equilibrium to the **m,,y,,**. So a certain reinforcement direction takes positive and negative loads at the same time.
80
8.1 81 (((
1.1 82 ----
83 )))
84
2.1 85 = {{id name="Shrinkage as load action"/}}Shrinkage as load action =
1.1 86
19.1 87 (((
88 (% style="text-align: justify;" %)
64.1 89 In the Plate and 3D Structure modules the shrinkage behaviour of reinforced concrete slabs can be taken into consideration as load action. The program add this movement effect (specific rotation) calculated from the formulas written below to the structure as invisible load (one load case must be defined as [[+Shrinkage>>doc:Manuals.User Manual.Loads.Shrinkage (Load).WebHome]] type).
1.1 90
19.1 91 (% class="box warningmessage" style="text-align: justify;" %)
1.1 92 (((
19.1 93 Note: The shrinkage effect has to be used together with applied reinforcement.
94 )))
95 )))
96
97 The effect of the shrinkage for the surface reinforcement bars in one direction (here X) (it is also valid in other bar directions):
98
99 (% style="text-align: justify;" %)
100 [[image:1543504152702-119.png||height="413" width="739"]]
101
102
103 (% style="text-align: justify;" %)
104 The specific normal force causing the given shrinkage value (ε,,cs,, [‰] at concrete materials) in the concrete zone of the section is (here in X direction):
105
106 (% class="mark" %)N,,X,, = E,,c ,,A,,c,, ε,,cs,, [k&Nu;/m]
107
108
109 (% style="text-align: justify;" %)
110 The position change of centre of gravity considering reinforcement bars (here X-direction; see dashed line):
111
112 (% style="text-align: justify;" %)
113 [[image:1543504296967-109.png||height="60" width="122"]]
114
115 (% style="text-align: justify;" %)
63.1 116 where:
19.1 117
118 (% style="text-align: justify;" %)
119 n = E,,s,, / E,,c ,,and S,,s,, is the statical moment of (here) X-directional bars around the Y axis of the calculation plane.
120
121
122 (% style="text-align: justify;" %)
123 The moment around the Y axis of the calculation plane from N,,X,, because of the position change of centre of gravity:
124
125 (% style="text-align: justify;" %)
126 (% class="mark" %)M,,Y,, = N,,X,, z,,s,,
127
128
129 (% style="text-align: justify;" %)
130 The specific rotation (curvature) from M,,Y,, for 1 meter wide section:
131
132 [[image:1543504426046-612.png||height="117" width="222"]]
133
134
135 (((
1.1 136 ----
137 )))
138
2.1 139 = {{id name="Design calculations for surface structures"/}}Design calculations for surface structures =
1.1 140
141
20.1 142 == Ultimate limit state ==
143
22.2 144 (% style="text-align: justify;" %)
25.1 145 The design of the slab is performed with respect to the design moments described in [[Design forces>>doc:||anchor="Design forces"]].
22.2 146
147 (% style="text-align: justify;" %)
148 In order to minimize cracking in the slab a good way is to reinforce according to the elastic moments which normally also leads to good reinforcement economy. The required bending reinforcement is designed according to EC2 3.1.7, where a rectangular stress distribution as shown below has been assumed.
149
150
151 (% class="MsoBodyText" style="margin-top:0cm; margin-right:5.25pt; margin-bottom:.0001pt; margin-left:5.5pt; text-align:justify; margin:0cm 0cm 0.0001pt 5.3pt" %)
152 [[image:1543932761308-583.png||height="175" width="392"]]
153
154
155 λ = 0,8 for f,,ck,, ≤ 50 MPa
156
157 λ = 0,8 - (f,,ck,, - 50)/400 for 50 < f,,ck,, ≤ 90 MPa
158
159 and:
160
161 η = 1,0 for f,,ck,, ≤ 50 MPa
162
163 η = 1,0 - (f,,ck ,,- 50)/200 for 50 < f,,ck,, ≤ 90 MPa
164
22.3 165 (((
22.2 166
167
22.3 168 (% style="text-align: justify;" %)
169 If the current moment is larger than the moment representing balanced design, compression reinforcement will be provided if allowed by the user otherwise an error message will be displayed. If the spacing regulations for the reinforcement are exceeded before adequate moment capacity can be reached a warning mes- sage will be displayed.
170 )))
171
172 (% style="text-align: justify;" %)
173 Note, that the required bending reinforcement is at design level primary not affected of the presence of user defined reinforcement. However, when user defined applied reinforcement is selected the stiffness will be effected, which in most cases will influence the moment distribution and thus secondary the required bending reinforcement.
174
20.1 175 == Shear capacity ==
176
28.1 177
178 (% style="text-align: justify;" %)
30.1 179 The shear capacity is calculated according to EC2 6.2.2 and 6.2.3 considering applied bending reinforcement when the option //Checking //has been selected. Otherwise, the required bending reinforcement according to [[this chapter>>doc:||anchor="Checking"]]. The design criteria for the shear capacity is:
28.1 180
181 (% class="mark" %)VSd < VRd1(%%)
182
183
63.1 184 where:
28.1 185
186 V,,Sd,, is the design shear force;
187
188 V,,Sd ,,= Q, which is calculated as [[image:1543933319172-750.png||height="30" width="121"]]
189
190 V,,Rd,, is the shear capacity.
191
192
193 If the section in which the shear force is acting has an angle with respect to the reinforcement directions the shear capacity is calculated as: [[image:1543933343993-360.png||height="31" width="190"]]
194
20.1 195 == Punching ==
196
32.1 197
198 The punching capacity is calculated according to EC2 6.4.3 - 6.4.5.
199
200
29.1 201 === {{id name="Checking"/}}Checking ===
202
32.1 203 (% style="text-align: justify;" %)
204 **Punching without shear reinforcement**
205
206 (% style="text-align: justify;" %)
207 A concrete compression check on u0 is made according to 6.4.5 (6.53). A concrete shear check on **u,,1,,** is made for a capacity calculated according to 6.4.4 (6.47).
208
209
210 (% style="text-align: justify;" %)
211 **Punching with shear reinforcement**
212
213 (% style="text-align: justify;" %)
63.1 214 A concrete compression check on **u,,0,,** is made according to 6.4.5 (6.53). Reinforcement is calculated with regard to critical perimeters u,,1,,, u,,2,,, ... u,,nReinf  ,,according to 6.4.5 (6.52 ).
32.1 215
216 (% style="text-align: justify;" %)
217 (u,,i,, are control perimeters above the reinforced region, distance between them is ”Perimeter distance”, defined in the calculation parameter).
218
219
220 (% style="text-align: justify;" %)
221 A concrete shear check on **u,,out,,** is made for a capacity calculated according to 6.4.4 (6.47)
222
223 (% style="text-align: justify;" %)
63.1 224 (**u,,out,,** is either the first perimeter that does not need reinforcement, or if it is not found, the perimeter that is** k d,,eff,,** distance from the outer perimeter of the reinforcement).
32.1 225
226
227 (% style="text-align: justify;" %)
228 **Warnings**
229
230 (% style="text-align: justify;" %)
231 A warning message is shown, if reinforcement does not comply with the detailing rules in 9.4.3.
232
233
29.1 234 === Design ===
235
42.2 236 1. Check, if reinforcement is needed at all,
237 1. If reinforcement is needed, it is designed to satisfy the detailing rules in (% style="color:#e74c3c" %)9.4.3(%%), if possible,
238 1. Design fails and a warning message is displayed if, **u,,out,,** is not found within 6 **d,,eff,,** distance from the column perimeter.
239
241
242 (% style="text-align: justify;" %)
243 - openings are not considered when control perimeters are generated.
244 - the position of the column relative to the plate is considered only in the generation of control perimeter. It means, (user or program defined) reinforcement may be partly out of the plate, but it won't affect the calculation.
245 - If ”Calculate β automatically” is set in the calculation parameter, β is calculated according to equation (% style="color:#e74c3c" %)6.4.3 (6.39)(%%).
246
247
20.1 248 == Serviceability limit state ==
249
250
42.2 251 **Method of solution**
20.1 252
42.2 253 (% style="text-align: justify;" %)
254 The program performs crack- and deflection control for all load combinations according to EC2 7.3 and 7.4. Two limiting conditions are assumed to exist for the calculations: **Stadium I **(the uncracked condition) and **Stadium II **(the fully cracked condition).
20.1 255
42.2 256
258
259 If the user does not activate the option **Cracked section analysis**, the calculation will be performed with respect to the total stiffness of the slab.
260
261
262 **Stadium II Fully cracked condition**
263
264 (% style="text-align: justify;" %)
265 If the option **Cracked section analysis**, is activated the program will consider the decrease in slab stiffness on behalf of cracking. This means an iterative calculation where the slab in the beginning is assumed to be uncracked when the section forces are calculated. Sections which are not loaded above the level which would cause the tensile strength of the concrete to be exceeded will be considered to be uncracked (**Stadium I**). Sections which are expected to crack will behave in a manner intermediate between the uncracked and fully cracked conditions and an adequate prediction of behavior used in the program is shown below.
266
267 (% style="text-align: justify;" %)
268 The stiffness calculation is performed considering the required or the applied reinforcement depending on what option has been selected. If applied reinforcement has been selected this is used in all load combinations. If applied reinforcement is not present or not selected the required reinforcement is used instead. In the latter case the required reinforcement in every element is calculated as the maximum value from all load combinations, which means that all calculations of serviceability limit values are performed with the same reinforcement.
269
270 (% style="text-align: justify;" %)
271 In the next step a new calculation based on the new stiffness distribution is performed and so on. When the deflection values resulting from two calculations does not differ more than a defined percentage of the first one or the maximal number of allowed calculations has been reached the calculation is stopped.
272
273
274 **Crack width**
275
276 Crack width is according to EC2 7.3.4 calculated as:
277
278 (% class="mark" %)w,,k,, = S,,r,max,, (ε,,sm,, - ε,,cm,,)
279
280 where:
281
282 (% style="text-align: justify;" %)
283 **S,,r,max,,** is the maximum crack spacing,
284
285 (% style="text-align: justify;" %)
286 **ε,,sm,,** is the mean strain in the reinforcement under the relevant com- bination of loads, including the effect of imposed deformations and taking into account the effects of tension stiffening. Only the additional tensile strain beyond the state of zero strain of the concrete at the same level is considered,
287
288 (% style="text-align: justify;" %)
289 **ε,,cm,,** is the mean strain in the concrete between cracks.
290
291
292 ε,,sm,, - ε,,cm,, may be calculated from the expression:
293
294 [[image:1557391677004-144.png||height="67" width="337"]]
295
296 where:
297
298 **σ,,s,, **is the stress in the tension reinforcement assuming a cracked section. For pretensioned members, σ,,s,, may be replaced by &Delta;σ,,p,, the stress variation in prestressing tendons from the state of zero strain of the concrete at the same level,
299
63.1 300 **α,,e,,**= is the ratio** Es / Ecm**
42.2 301
302 **A,,p,,** and **A,,c,eff,,** are as defined in (% style="color:#e74c3c" %)7.3.2 (3)(%%),
303
304 **ξ,,1,,** according to Expression (% style="color:#e74c3c" %)(7.5)(%%),
305
306 **k,,t,,** is a factor dependent on the duration of the load.
307
308
310
312
313 **Ap´ = 0,0** (pre or post-tensioned tendons)
314
315 **Ac,eff:**
316
317 [[image:1557392170606-753.png||height="170" width="449"]]
318
319 [[image:1557392181561-817.png||height="172" width="541"]]
320
321
322 (% class="mark" %)h,,cef ,,= min (2,5 * (h - d), (h - x) / 3, h / 2 )
323
324 (% class="mark" %)s,,r,max,, = k,,3,,c + k,,1,, k,,2,, k,,4,,φ /ρ,,p,eff,,
325
326
327 where:
328
329 (% style="text-align: justify;" %)
330 **φ** is the bar diameter. Where a mixture of bar diameters is used in a
331 section, an equivalent diameter, **φ,,eq,,**, should be used. For a section with **n,,1,,** bars of diameter **φ,,1,,** and **n,,2,,** bars of diameter **φ,,2,,**, the following expression should be used,
332
333 [[image:1557392495138-639.png||height="66" width="164"]]
334
335 **c **is the cover to the longitudinal reinforcement,
336
337 **k,,1,,** is a coefficient which takes account of the bond properties of the bonded reinforcement:
338
339
63.1 340 **k,,1,,** =** 0,8 **for high bond bars,
42.2 341
342 **k,,1,,** **= 1,6** for bars with an effectively plain surface (e.g. prestressing tendons),
343
344 **k(% style="font-size:10.5px" %)2(%%)** is a coefficient which takes account of the distribution of strain:
345
346 **k(% style="font-size:10.5px" %)2(%%)** **= 0,5 **for bending,
347
348 **k(% style="font-size:10.5px" %)2(%%)** **= 1,0 **for pure tension,
349
350 (% class="mark" %)k,,2,, = (ε1 + ε,,2,,) / 2ε,,1,,
351
352 where ε,,1,, is the greater and ε,,2,, is the lesser tensile strain at the boundaries of the section considered, assessed on the basis of a cracked section.
353 Recommended values of **k,,3,, = 3,4** and **k,,4,, = 0,425** are used.
354
355
356 **Maximum crack spacing:**
357
358 s,,r,max,, = 1,3 (h - x)
359
360 **Equivalent quantities perpendicular to crack direction:**
361
362 * **Reinforcement area:**
363
364 [[image:1557393319443-687.png||height="31" width="274"]]
365
366 * **Number of bars:**
367
368 [[image:1557393330997-858.png||height="107" width="168"]]
369
370 * **Diameter:**
371
372 [[image:1557393343977-995.png||height="47" width="301"]]
373
374
375 **Deflections**
376
377 The calculations is performed according to EC2 7.4.3.
378
379
381
382 Load depended curvature is calculated as:
383
384 (% class="mark" %)1 / r,,f,, = M / E,,c,ef,, I,,1,,
385
386
387 where:
388
389 **M** is current moment,
390
391 **I,,1,,** is Moment of Inertia in **Stadium I**,
392
393 **E,,c,ef,,** is the modulus of elasticity with respect to creep.
394
395
396 The modulus of elasticity is calculated as:
397
398 (% class="mark" %)E,,c,eff,, = E,,cm,, / (1 + φ)
399
400 where φ is the creep coefficient.
401
402 Curvature with respect to shrinkage is considered according to (% style="color:#e74c3c" %)2.2.2(%%) above.
403
404
405 **Stadium II Fully cracked condition**
406
407 Load depended curvature is calculated as:
408
409 (% class="mark" %)1 / r,,f,, = M / E,,c,ef,, I,,2,,
410
411
412 where:
413
414 **E,,c,ef,,** is the modulus of elasticity as shown above,
415
416 **I,,2,,** is the moment of Inertia in Stadium II,
417
418 **M** is current moment.
419
420
63.1 421 Curvature with respect to shrinkage is considered according to(% style="color:#e74c3c" %) 2.2.2(%%) above.
42.2 422
423
424 (% style="text-align: justify;" %)
425 Sections which are expected to crack will behave in a manner intermediate between the uncracked and fully cracked conditions and an adequate prediction of this behavior is given by:
426
427 (% style="text-align: justify;" %)
428 α = ζ α,,II,, + (1 - ζ) α,,I,,
429
44.2 430
42.2 431 (% style="text-align: justify;" %)
44.2 432 where:
42.2 433
44.2 434 (% style="text-align: justify;" %)
435 α is in this case the curvature calculated for the uncracked and fully cracked conditions,
42.2 436
44.2 437 ζ is a distribution coefficient given by ζ = 1 - β (σ,,sr,, / σ,,s,,)^^2^^
42.2 438
44.2 439 ζ is zero for uncracked sections,
42.2 440
44.2 441 β is a coefficient taking account of the influence of the duration of the loading or of repeated loading on the average strain,
442
443 σ,,s,, is the stress in the tension steel calculated on the basis of a crack- ed section,
444
445 σ,,sr,, is the stress in the tension steel calculated on the basis of a cracked section under the loading which will just cause cracking at the section being considered.
446
447
448 (% class="box warningmessage" style="text-align: justify;" %)
1.1 449 (((
44.2 450 Note that stresses and moment of inertia are calculated with applied reinforcement if it is selected, otherwise with required reinforcement.
451 )))
452
453 (((
1.1 454 ----
455 )))
456
2.1 457 = {{id name="Design calculations for bar structures"/}}Design calculations for bar structures =
21.1 458
459
460 == Material properties ==
461
44.2 462
463 **Concrete**
464
465
466 [[image:1557394141236-821.png||height="358" width="397"]]
467
468
469 * **Ultimate limit states:**
470 Continuous line is used.
471
60.2 472 * **Servicibility limit states:**
44.2 473 Stage II is used (dashed line, without horizontal section).
474
475 **Steel**
476
477
48.2 478 [[image:1557394271242-563.png||height="311" width="452"]]
44.2 479
480
481 * **Ultimate limit states:**
482 **B** graph with horizontal line is used.
483
60.2 484 * **Servicibility limit states:**
44.2 485 The same as ultimate but without safety factor.
486
21.1 487 == Longitudinal reinforcement ==
488
25.1 489 (% lang="EN-US" style="font-family:~"Times New Roman~",~"serif~"; font-size:12pt; letter-spacing:-0.05pt" %)**Analysis of second order effects with axial load**
21.1 490
44.2 491 According to EC2 5.8.
21.1 492
48.2 493 * For calculation of 2nd order effect **Nominal curvature** method ((% style="color:#e74c3c" %)5.8.8(%%)) is used.
494 * If there is no compression force in the section the eccentricity is equal to **0,0**.
495 * Buckling lengths **l,,0x,,** and **l,,0y,,** are specified by the user.
496 * Curvature:
44.2 497
48.2 498 1 / r = kr kϕ 1 / r0
44.2 499
48.2 500 where:
44.2 501
48.2 502 **k,,r,,** is a correction factor depending on axial load,
503
504 **κ,,ϕ,,** is a factor for taking account of creep,
505
506 1 / r0 = ε,,yd,, / (0,45 d),
507
508 **d** is the effective depth,
509
510 d = (h / 2) + is
511
512 where **i,,s,,** is the radius of gyration of the total reinforcement area.
513
514 k,,r,, = (n,,u,, - n) / (n,,u,, - n,,bal,,) ≤ 1
515
516 where:
517
518 n = N,,Ed,, / (A,,c,, f,,cd,,), relative axial force,
519
520 **N,,Ed,,** is the design value of axial force, n,,u,, = 1 + ω,
521
522 **n,,bal,,** is the value of **n** at maximum moment resistance; the value 0,4 is used,
523
524 ω = A,,s,, f,,yd,, / (A,,c,, f,,cd,,),
525
526 **A,,s,,** is the total area of reinforcement,
527
528 **A,,c,,** is the area of concrete cross section,
529
530 k,,ϕ,, = 1 + β ϕ,,ef,, ≥ 1
531
532 where:
533
534 ϕ,,ef,, is effective creep ratio, defined by the user,
535
536 β = 0,35 + fck /200 - λ / 150,
537
538 λ is the slenderness ratio.
539
540
541 * 2nd order effect is ignored, if:
542
543 λ ≤ λ,,lim,,
544
545 λ,,lim ,,= 20 A B C / √n
546
547 where: A = 1 / (1 + 0,2 ϕ,,ef,,),
548
549 B = √1 + 2 ω,
550
551 C = 0,7
552
553 ϕ,,ef,, is effective creep ratio,
554
555 ω = A,,s,, f,,yd,, / (A,,c,, f,,cd,,), mechanical reinforcement ratio,
556
557 **A,,s,,** is the total area of longitudinal reinforcement,
558
559 n = N,,Ed,, / A,,c,, f,,cd,,), relative normal force,
560
561 r,,m,, = M,,01,, / M,,02,,, moment ratio,
562
563 **M,,01,,, M,,02,,** are the first order en moments |M,,02,,| ≥ |M,,01,,|.
564
565
566 * Geometric imperfection (5.2 (7) a):
567
568 e,,i,, = l,,0,, / 400
569
570 * The minimum of all eccentricities (1st order + imperfection + 2nd order effect): max (20,0; h / 30,0).
571 * Imperfection and 2nd order effect considered in both directions.
572 * The eccentricity is calculated in four possible positions:
573 ** Stiff direction+, weak direction+
574 ** Stiff -, weak+
575 ** Stiff+, weak-
576 ** Stiff-, weak-
577
578 **Torsion**
579
580 * Necessary longitudinal reinforcement area (**A,,sl,,**):
581 **T,,Ed,,** is the applied design torsion (see Figure 6.11):
582
583 [[image:1557398781245-600.png||height="193" width="441"]]
584
585 (% style="text-align: justify;" %)
63.1 586 The required cross-sectional area of the longitudinal reinforcement for torsion** &Sigma;A,,sl,,** may be calculated from:
48.2 587
588 [[image:1557398836470-691.png||height="46" width="203"]]
589
590 where:
591
592 **u,,k,, **is the perimeter of the area **A,,k,,**,
593
594 **f,,yd,,** is the design yield stress of the longitudinal reinforcement **A,,sl,,**,
595
596 **θ** is the angle of compression struts, θ = 45 deg.
597
598
599 (% style="text-align: justify;" %)
600 Considering torsion in calculation of longitudinal bars:
601
602 (% style="text-align: justify;" %)
603 Calculation of torsional capacity by edges, considering all bars placed in tef strip. The minimum of capacities gives the torsional capacity of the section. Utilization for torsion calculated for all bars placed in the strip one by one.
604
605 (% style="text-align: justify;" %)
606 Area of these bars decreased in the calculation of axial effects (**N, My, Mz**) in proportion of utilization (see formula below):
607
608 [[image:1557399352673-816.png||height="34" width="114"]]
609
610 where:
611
612 **A** is area of the bar,
613
614 **A’** is decreased area used in calculation
615
616
617 **ULS checking**
618
619 [[image:1557399421450-202.png||height="263" width="441"]]
620
621
622 **SLS checking**
623
624 Crack width calculated according to EC2 7.3.
625
626 • Crack width calculated as:
627
628 w,,k,, = s,,r,max,, (ε,,sm,, - ε,,cm,,)
629
630 where:
631
632 **s,,r,max,,** is the maximum crack spacing,
633
634 (% style="text-align: justify;" %)
635 **ε,,sm,,** is the mean strain in the reinforcement under the relevant combination of loads, including the effect of imposed derforma- tions and taking into account the effects of tension stiffening. Only the additional tensile strain beyond the state of zero strain of the concrete at the same level is considered,
636
637 (% style="text-align: justify;" %)
638 **ε,,cm,,** is the mean strain in the concrete between cracks.
639
640 (% style="text-align: justify;" %)
641 ε,,sm,, - ε,,cm,, may be calculated from the expression:
642
50.2 643 [[image:1557399634681-893.png||height="66" width="360"]]
48.2 644
50.2 645 where:
48.2 646
50.2 647 **σ,,s,,** is the tension reinforcement assuming a cracked section,
648
649 **α,,e,,** is the ratio E,,s,, / E,,cm,,
650
651 **ρ,,p,eff,,**,, ,,= A,,s,, / A,,c,eff,,,
652
653 **A,,c,eff,,** is calculated as below,
654
655 **k,,t,,** is a factor dependent on the duration of the load,
656
658
659 **k,,t,, **= 0,4 for long term loading (always supposed by the program),
660
661 **A,,c,eff,,**:
662
663 [[image:1557400792364-457.png||height="154" width="522"]]
664
665
60.2 666 h,,c,ef,, = min (2,5 (h - d), (h - x) / 3, h / 2)
50.2 667
60.2 668 s,,r,max,, = k,,3,, c + k,,1,, k,,2,, k,,4,, φ / ρ,,p,eff,,
50.2 669
60.2 670 where:
50.2 671
60.2 672 (% style="text-align: justify;" %)
673 **φ** is the bar diameter. Where a mixture of bar diameters is used in a section, an equivalent diameter, φ,,eq,,, should be used. For a section with n,,1,, bars of diameter φ,,1,, and n,,2,, bars of diameter φ,,2,,, the following expression should be used,
50.2 674
60.2 675 [[image:1557402498643-546.png||height="45" width="141"]]
676
677 **c** is the cover to the longitudinal reinforcement,
678
679 **k,,1,,** is a coefficient which takes account of the bond properties of the bonded reinforcement:
680
681 **k,,1,, = 0,8** for high bond bars,
682
683 **k,,1,, = 1,6** for bars with an effectively plain surface (e.g. prestressing tendons),
684
685 **k,,2,,** is a coefficient which takes account of the distribution of strain:
686
687 **k,,2,, = 0,5** for bending,
688
689 **k,,2,, = 1,0** for pure tension
690
691 For cases of eccentric tension or for local areas, intermediate values of k2 should be used which may be calculated from the relation:
692
693 k,,2,, = (ε,,1,, + ε,,2,,) / 2 ε,,1,,,
694
695 where: ε1 is the greater and ε,,2,, is the lesser tensile strain at the boundaries of the section considered, assessed on the basis of a cracked section.
696
697 Recommended values of **k,,3,, = 3,4** and **k,,4,, = 0,425** are used.
698
699 • Maximum crack spacing:
700
701 s,,r,max,, = 1,3 (h - x)
702
703
704 **Space between bars**
705
706 * **Minimum distance:**
707 The clear distance (horizontal and vertical) between individual parallel bars or horizontal layers of parallel bars should be not less than the maximum of k,,1,, bar diameter, (d,,g,, + k,,2,, mm) or 20 mm where d,,g,, is the maximum size of aggregate.
708
709 * **Maximum distance:**
710 The longitudinal bars should be so arranged that there is at least one bar at each corner, the others being distributed uniformly around the inner periphery of the links, with a spacing not greater than 350 mm.
711
712 **Lengthening and anchorage**
713
714 * **Because of shear effect (//shift rule//):**
715
716 a,,i,, = 0,9 max (h, b)
717
718 The code prescribes **d** instead of **h**, but the difference can be ignored.
719
720 * **Anchorage:**
721
722 f,,bd,, = 2,25 η,,1,, η,,2,, f,,ctd ,,
723
724 where:
725
726 (% style="text-align: justify;" %)
727 **f,,ctd,,** is design value of concrete tensile strength. Due to the increasing brittleness of higher strength concrete, **f,,ctk,0,05,,** should be limited here to the value for **C60/75**, unless it can be verified that the average bond strength increases above this limit
728
729 (% style="text-align: justify;" %)
730 η,,1,, is a coefficient related to the quality of the bond condition and the position of the bar during concreting:
731
732 (% style="text-align: justify;" %)
733 η,,1,, = 0,7
734
735 (% style="text-align: justify;" %)
736 η,,2,, is related to the bar diameter:
737
738 (% style="text-align: justify;" %)
739 η,,2,, = 1,0 for φ ≤ 32 mm,
740
741 (% style="text-align: justify;" %)
742 η,,2,, = (132 - φ) / 100 for φ > 32 mm
743
744 (% style="text-align: justify;" %)
745 l,,b,rqd,, = (φ / 4) (σ,,sd,, / f,,bd,,)
746
747 (% style="text-align: justify;" %)
748 where:
749
750 (% style="text-align: justify;" %)
751 σ,,sd,, = f,,yd,, (fully utilized bar supposed),
752
753 (% style="text-align: justify;" %)
754 l,,bd,, = α,,1,, α,,2,, α,,3,, α,,4,, α,,5,, l,,b,rqd,, ≥ l,,b,min,,,
755
756 (% style="text-align: justify;" %)
757 α,,i,, = 1,0
758
759 (% style="text-align: justify;" %)
760 **l,,b,min,,** is the minimum anchorage length if no other limitation is applied:
761
762 * for anchorages in tension:
763 l,,b,min,, > max (0,3 l,,b,rqd,,; 10 φ; 100 mm),
764
765 * for anchorage in compression:
766 l,,b,min,, > max (0,6 l,,b,rqd,,; 10 φ; 100 mm),
767 Rule given for compression is used.
768
21.1 769 == Stirrups ==
770
771 === Shear ===
772
60.2 773 In Figure 6.5 below the following notations are shown:
774
775 **α** is the angle between shear reinforcement and beam axis perpendicular to the shear force (measured positive as shown in Figure 6.5),
776
777 **θ** is the angle between the concrete compression strut and the beam axis per- pendicular to the shear force,
778
779 **F,,td,,** is the dessign value or the tensile force in the longitudinal reinforcement,
780
781 **F,,cd,,** is the design value of the concrete compression force in the direction of the longitudinal member axis,
782
783 **b,,w,,** is the minimum width between tension and compression chords,
784
785 **z** is the inner lever arm, for a member with constant depth, corresponding to the bending moment in the element under consideration. In the shear analy- sis of reinforced concrete without axial force, the approximate value **z = 0,9 d** may normally be used.
786
787 [[image:1557404124068-157.png||height="311" width="496"]]
788
789
790 * **Member do not require shear reinforcement, if:**
791
792 The design value for the shear resistance **V,,Rd,c,,** is given by:
793
794 (% class="mark" %)V,,Rd,c,, = [C,,Rd,c,, k (100 ρ,,l,, f,,ck,,)1/3 + k,,1,, σ,,cp,,] b,,w,, d
795
796 with a minimum of:
797
798 (% class="mark" %)V,,Rd,c,, = (v,,min,, + k,,1 ,,σ,,cp,,) b,,w,, d
799
800 where: **f,,ck,,** is in MPa
801
802 [[image:1557404308626-814.png||height="80" width="280"]]
803
804 (% style="text-align: justify;" %)
63.1 805 **A,,sl,,** is the area of the tensile reinforcement, which extends: ≥ (**l,,bd,,** +** d**) beyond the section aonsidered (see Figure 6.3), **b,,w,,** is the smallest width of the cross-section in the tensile area [mm],
60.2 806
807 σ,,cp,, = N,,Ed,, / A,,c,, < 0,2 f,,cd,, [MPa],
808
809 **N,,Ed,,** is the axial force in the cross-section due to loading or prestressing [in N] (**N,,Ed ,,> 0** for compression). The influence of imposed deformations on **N,,E,,** may be ignored,
810
811 **Ac** is the area of concrete cross section [mm2],
812
813 **V,,Rd,c,,**,, ,,is [N]
814
815 (% style="text-align: justify;" %)
63.1 816 The recommended value for **C,,Rd,c,,** is **0,18 / γ,,c,,**, that for** v,,min,,** is given by the expression below and that for **k,,1,,** is **0,15**.
60.2 817
818 [[image:1557404927126-189.png||height="24" width="136"]]
819
820 [[image:1557404943122-538.png||height="192" width="607"]]
821
822 * **Upper limit of shear:**
823
824 (% class="mark" %)V,,Rd,max,, = α,,cw,, b,,w,, z ν,,1,, f,,cd,, / (cotθ + tanθ)
825
826 where:
827
828 **A,,sw,,** is the cross-sectional area of the shear reinfocement,
829
830 **s** is the spacing of the stirrups,
831
832 **f,,ywd,,** is the design yield strength of the shear reinforcement,
833
834 **ν,,1,,** is a strength reduction factor for concrete cracked in shear,
835
836 **α,,cw,, **is a coefficient taking account of the state of the stress in the compression chord.
837
838 (% style="text-align: justify;" %)
839 The recommended value of ν,,1,, is ν (see expression below). The recommended value of α,,cw,, is as follows:**1** for non-prestressed structures,
840
841 [[image:1557405487679-357.png||height="82" width="120"]]
842
843 Capacity of stirrups:
844
845 [[image:1557405540899-916.png||height="37" width="170"]]
846
847 where:
848
849 **A,,sw,,** is the cross-sectional area of the shear reinforcement,
850
851 **s** is the spacing of the stirrups,
852
853 **f,,ywd,, **is the design yield strength of the shear reinforcement.
854
855
21.1 856 === Torsion ===
857
60.2 858 **T,,Ed,,** is the applied design torsion (see Figure 6.11)
859
860 [[image:1557405627044-695.png||height="210" width="480"]]
861
862
863 **A,,k,,** is the area enclosed by the centre-lines of the connecting walls, including inner hollow areas,
864
865 **τ,,t,i,, **is the torsional shear stress in wall i,
866
867 **t,,ef,i,, **is the effective wall thickness. It may be taken as A/u, but should not be taken as less than twice the distance between edge and center of the longitudinal reinforcement. For hollow sections the real thickness is an upper limit,
868
869 **A** is the total area of the cross-section within the outer circumference, including inner hollow areas,
870
871 **u** is the outer circumference of the cross-section,
872
873 **z,,i,,** is the side length of wall i defined by the distance between the intersection points with the adjacent walls,
874
875 θ = 45 deg, in all calculations.
876
877
878 * **Member do not require torsional reinforcement, if:**
879
880 (% class="mark" %)T,,Rd,c,, = f,,cd,, t,,ef,, 2 A,,k,, ≤ T,,Ed,,
881
882 * **Upper limit of torsion:**
883
884 T,,Rd,max,, = 2 ν α,,cw,, f,,cd,, A,,k,, t,,ef,i,, sinθ cosθ
885
886 where ν and α,,cw,, are as above.
887
888 * **Force in stirrups:**
889
890 (% class="mark" %)T,,Rd,max,,=2να,,cw,,f,,cd,,A,,k,,t,,ef,i,,sinθcosθ
891
892 The shear force **V,,Ed,i,,** in a wall **i** due to torsion is given by:
893
894 (% class="mark" %)V,,Ed,i,, = τ,,t,i ,,t,,ef,i,, z,,i,,
895
896 **z,,i,,** is section height used to be able to sum with shear.
897
898 * **Capacity of stirrups:**
899
900 See **Shear**.
901
21.1 902 === Shear and torsion ===
903
61.1 904 * **Forces in stirrups:**
905
906 V,,Ed,, = V,,Ed(shear) ,,/ 2 + V,,Ed(torsion),,
907
908 * **No stirrup required:**
909
910 T,,Ed,, / T,,Rd,c,, + V,,Ed,, / V,,Rd,c,, ≤ 1,0
911
912 * **Upper limit of the effects:**
913
914 T,,Ed,, / T,,Rd,max,, + V,,Ed,, / V,,Rd,max,, ≤ 1,0
915
916 * **Calculation is done in two directions y' and z' independently.**
917
21.1 918