S2 module (AD 2000 S2)

1 Equivalent stress per section 5

In this section of the Merkblatt, the equivalent cyclic stress range and the mean equivalent stress are calculated for unidirectional (section 5.1) and multi-axial (section 5.2) cyclic loading.

(5.1) Unidirectional loads

For unidirectional loading the equivalent cyclic stress range 2 σ_va is according to Tresca

(1)	2 σ_a = (σ_MAX − σ_MIN )
(3)	σ_vM = 0.5 (σ_MAX + σ_MIN )

With maximum stress σ_MAX, minimum stress σ_MIN and mean value σ_vM.

Fig. 1: Amplitude σ_va and mean stress σ_vM unidirectional

(5.2) Multi-axial stress state with constant stress directions

From the course of time of the main stresses you can determine the stress diefferences Δσ₁₂, Δσ₂₃, Δσ₃₁;

(4)	Δσ₁₂ = σ₁ − σ₂, Δσ₂₃ = σ₂ − σ₃, Δσ₃₁ = σ₃ − σ₁,

The equivalent cyclic stress range results in:

(5)	2 σ_va = max[Δσ_{12 MAX} − Δσ_{12 MIN} , Δσ_{23 MAX} − Δσ_{23 MIN}, Δσ_{31 MAX} − Δσ_{31 MIN} ]

The index MAX marks the greatest and the index MIN den smallest value. The equivalent mean 2 σ_vM results from the sum of the main stresses Σσ_ij for the principal stresses σ_{i a}nd σ_j at the calculation of σ_va. You have to enter the maximum value and the minimum value of Σσ_ij .

(6a)	Σσ_ij = σ_i + σ_j

(6b)	2 σ_{vM =} Max(Σσ_ij) + Min(Σσ_ij)

2 Equivalent cyclic stress range beyond yield per section 6

(6.1) Mechanical loads

For mechanical loading the equivalent cyclic stress range must be multiplied by an amplification factor. Die pseudo-elastic equivalent cyclic stress range σ^*_vap is

(7)	2 σ^_vap = 2 σ_vapk_e*

The amplification factor can be determined per (8) and (9)

(8)	ke = A1 (σ_vap/ R_{p0.2 T*} −1)^1/2 + 1 ,	für 1 < σ_vap/R_{p0.2 T*} < 1.5

(9)	ke = A2 + A2 σ_vap/ R_{p0.2 T},*	für σ_vap/R_{p0.2 T*} > 1.5

Material group	A1	A2	A3
Ferrite, Rm=800-1000	0,518	0,718	0,432
Austenite, Ferrite Rm<500	0,443	0,823	0,327

(6.2) Thermal loads

(10)	2 σ^_vaw = 2 σ_vawk_v*

(11)	k_v = 0,7 σ_vaw / ( 0,5 σ_vaw + 0,2 R_{p0.2 T} )*

(6.3) Combined loading

For combined loads you have to superpose the stress components from mechanical and thermal loads linearly. The pseudo-elastic stress range is per (12):

(12)	2 σ^_va = 2 σ_vap+wk_e*

3 Allowable stress range per 7.1 (not welded)

The allowable stress for not welded components is:

(13)	2 σ_{a zul} = 2 σ_a f_o f_d f_M f_T*

(7.1.1) Reference stress range for unnotched test bars

The allowable reference stress range 2 σ_a for unnotched, polished test bars made of ferritic and austenitic rolled and forged steels at ambient temperature and mere cyclic loading (average stress = 0)is for R_m < 1000 N/mm² and 10² <= N <= 2. 10⁶

(14)

2 σ_a = 4. 10⁴ / √ N + 0,55 R_m − 10

für (R_m < 1000 N/mm²) und (10² ≤ N ≤ 2.10⁶)

(7.1.2) Correction factor and surface influence

(15)	f_o = F_o^{(0,4343 ln N − 2)/4.301}	für N ≤10⁶
	f_o = F_o ,	für N ≥ 10⁶
	f_o = 1 ,	für R_z < 6 µm

(16)	Fo = 1 − 0,056 (ln R_z)^0,64 ln R_m + 0,289 (ln Rz)^0,53

Guide values for surface roughness:

Surface state	Rolled or extruded	Mechanically processed	Notch-free polished
Rz [µm]	200	50	10

(7.1.3) Correction factor for wall thickness

(17)	f_d = F_d^{(0,4343 ln N − 2)/4.301}	for N ≤ 2. 10⁶
	f_d = F_d,	for N ≥ 10⁶

(18)	F_d = (25 / s)^1/Z , with Z= 10 ,	for s < 150 mm
	F_d =0.84	for s > 150 mm

For forgings, you have to enter the governing heat treatment diameter per DIN 17243 as wall thickness. (s.8 Governing heat treatment diameter per EN 10222-1, Annex B)

(7.1.4) Mean stress correction factor

(7.1.4a) elastic range: 2 σ_va< R_{p0.2 T*}

(19)	fM = [ 1 − M(2+M) σ_vM/ ( σ_a +M σ_a) ]^1/2,	für −R_{p0.2 T} ≤* σ_vM ≤ σ_a /(1+M)
(20)	fM = ( 1+M/3)/(1+M) − M σ_vM /(3 σ_a) ,	für σ_a /(1+M) ≤ σ_vM ≤ R_{p0.2 T}*
(21)	M =0,00035 R_m −0,1

Fig. 2: Stress range as a function of mean stress acc. to Smith

(7.1.4b) Partially plastic

For R_{p0.2 T*} < 2 σ_va< 2 R_{p0.2 T*} or if the principal stress difference is greater than the yield strength.:

(22)	Dσ_v = max[\|Δσ₁₂\| , \|Δσ₂₃\|, \|Δσ₃₁\|] > R_{p0.2 T}*

The reduced equivalent stress is entered in (19) and (20):

(23)	σ_{v red} = R_{p0.2 T*} − σ_va

With the condition |σ_{v red}| <| σ_vM|.

Note: In the fully plastic range, the equivalent stress is reduced acc. section 6.1 and the mean stress is set to zero.

4 Allowable stress range per 7.2 (welded)

The allowable stress range for welded components is:

(26)	2 σ_{a zul} = 2 σ_a f_d f_T*

The reference stress 2 σ_a is

(27)	2 σ_a= ( B1 / N )^1/3

Factor B1 depends on the weld class (K0 to K3 in 7 Table 5 – AD 2000 S2 Table 5).

(7.2.6) For the correction factor fd Z is set to 4 in equation (18) Z = 4 and fd is limited to Fd = 0.64. For the temperature influence factor fT* equations (24) and (25) are valid.

(7.2.7) For stress-relief heat treated components with N ≥ 2. 10⁶ the the fatigue strength increases

(28)	2 σ_{a zul} = 2 σ_a f_d f_T Max{1; 1,3 f_M }*

5 Number of allowable load cycles 8.1 (not welded)

The number of allowable load cycles N_zulis

(29)	N_zul = 16 10⁸ /( 2 σ_a^ −0,55 R_m + 10 )²*

Where the reference stress range σ_a^* results from the stress range σ_vaanalogously per (13)

(30)	2 σ_a^ = 2 σ_va /( f_o f_d f_M f_T)

As the correction factors depend on the number of allowable load cycles N, the solution must be found by iteration.

If the reference stress range is smaller than the endurance strength 2 σ_a^* < 2 σ_va for N > 2 10⁶then there is endurance strength.

6 Number of allowable load cycles 8.2 (welded)

The number of allowable load cycles N_zulis

(31)	N_zul = B1 /( 2 σ_a^ )³*

Where the reference stress range σ_a^* results from the stress range σ_vaanalogously per (13)

(32)	2 σ_a^ = 2 σ_va /( f_d f_T)

Factor B1 depends on the weld class (K0 to K3 in 7 Table 5 – AD 2000 S2 Table 5).

As the correction factors depend on the number of allowable load cycles N, the solution must be found by iteration.

For stress-relief heat treated components it is valid analogously to 7.2.7

(33)	2 σ_a^ = 2 σ_va /( f_d f_Tf_M)*

7 Table 5 – AD 2000 S2 Table 5

Table 5. Examples of construction types and welded connections with the corresponding weld classes (K0, K1, K2 and K3), depending on the type of stress proof (depicted rupture paths exemplary)
No	Figure	Description	Provision	Weld class for stress proof 1 or 2
				1	2
1. Cylindrical and conical shells, dished ends
1.1		Longitudinal or circumferential weld with equal wall thicknesses	Welded on both sides	K1	K1
1.2			Welded on one side with counter weld	K2	K1
1.3			Welded on one side without counter weld	K1	K2
1.4		Longitudinal or circumferential weld with unequal wall thicknesses	Welded on both sides	K2 ¹⁾	K1
1.5				K3	K1
1.6				K1 ¹⁾	K1
1.7			Welded on both sides, Same offset of edges inside and outside	K1	K1
1.8				K2	K1
1.9				-	K1
1.10		Cone with corner joint	Welded on both sides or welded on one side with counter weld	-	K1
1.11			Welded on one side without counter weld	-	K3
1.12		Head to shell joint for dished ends with cylindrical skirt heights acc. to AD-Merkblatt B 3	Stress proof 2: Description of welds, provisions and corresponding weld classes see examples number 1.1 to 1.9

Table 5. Examples of construction types and welded connections with the corresponding weld classes (K0, K1, K2 and K3), depending on the type of stress proof (depicted rupture paths exemplary)
No	Figure	Description		Provision	Weld class for stress proof 1 or 2
No	Figure	Description		Provision	1	2
2. Nozzle welds
2.1		Set-through or set-in nozzle	Welded on both sides or welded on one side with counter weld		-	K1
2.2		Set-through or set-in nozzle	Welded on one side without counter weld		-	K2
2.3		Set-through nozzle (left type in the figure)	Welded on both sides but not continuously welded		-	K2
2.4		Set-in nozzle (right type in the figure)	Welded on both sides but not continuously welded		-	K3
2.5		Set-on nozzle	Welded on one side (no remaining gap), Nozzle drilled or weld root polished		-	K1
2.6			Welded on one side without counter weld or without mechanical processing of the root.		-	K2
2.7		Nozzle with disk type reinforcement. Seam: at outside diameter of the disk			-	K3
2.8		Nozzle with disk type reinforcement. Seam: Nozzle weld	Joint of nozzle pipe with shell and reinforcement disk welded		-	K1

Table 5. Examples of construction types and welded connections with the corresponding weld classes (K0, K1, K2 and K3), depending on the type of stress proof (depicted rupture paths exemplary)
No	Figure	Description		Provision	Weld class for stress proof 1 or 2
No	Figure	Description		Provision	1	2
3. Flanges, block flanges and bolts
3.1		Welding neck flange	Welded on both sides or welded on one side with counter weld		-	K 1
3.2		Welding neck flange	Welded on one side without counter weld		-	K 2
3.3		Slip-on weld flange	Weld according to AD-Merkblatt B 8, Table 1		-	K 2
3.4		Set-in block flange	Welded on both sides or welded on one side with counter weld		-	K 1
3.5		Set-in block flange	Welded on both sides but not continuously welded		-	K 2
3.6		Set-in block flange with welding hub	Welded on both sides or welded on one side with counter weld		-	K 1
3.7		Set-on block flange. Weld at inside diameter (left weld in the figure)			-	K 3
3.7		Set-on block flange. Weld at outside diameter (right weld in the figure)			-	K 2

Table 5. Examples of construction types and welded connections with the corresponding weld classes (K0, K1, K2 and K3), depending on the type of stress proof (depicted rupture paths exemplary)
No	Figure	Description	Provision		Weld class for stress proof 1 or 2
					1	2
4. Jacket – Joint welds
4.1	$S1_3\Image249.gif$	With integrated knuckle: The evaluation is valid for the inner vessel and for the joint weld	Welded on one side	-		K 2
4.2	$S1_3\Image250.gif$	With separate knuckle: The evaluation is valid for the inner vessel wall and for the joint weld of knuckle and inner vessel wall. ( The joint weld of knuckle and outer vessel wall is classified as number 1.3 with K 2)	Welded on both sides or welded on one side with counter weld	-		K 1

Table 5. Examples of construction types and welded connections with the corresponding weld classes (K0, K1, K2 and K3), depending on the type of stress proof (depicted rupture paths exemplary)
No	Figure	Description	Provision	Weld class for stress proof 1 or 2
No	Figure	Description	Provision	1	2
5. Joint weld of flat ends
5.1		Welded end	Welded on both sides	-	K 1
5.2			Welded on both sides but not continuously welded	-	K 2
5.3			Welded on one side without counter weld	-	K 3
5.4		Welded end with relief groove	Welded on one side, Groove dimensions acc. to AD-Merkblatt B 5, Table 1, type e	-	K 2
5.5		End welded to a welding neck flange or slip-on flange	Welded on both sides with fillet welds	-	K 2
5.6		Welded end	Welded on both sides or welded on one side with counter weld	-	K 1
5.7			Welded on both sides but not continuously welded	-	K 2
5.8			Welded on one side	-	K 3
5.9		Partially welded end	Welded on one side	-	K 3
5.10		End with knuckle, forged end or pressed end.	Knuckle radius and skirt height acc. AD-Merkblatt B 5,Table 1, types a and b, Weld and class assignment as number 1.1-1.3

Table 5. Examples of construction types and welded connections with the corresponding weld classes (K0, K1, K2 and K3), depending on the type of stress proof (depicted rupture paths exemplary)
No	Figure		Description	Provision	Weld class for stress proof 1 or 2
					1	2
6. Welded attachments, general Provision: External appearance of the welds acc. EN 25817, classification group B, excluding the characteristics weld reinforcement, lower deviation of the weld and being non-isosceles
6.1		$S1_3\Image251.gif$	Welded attachments without transmitting changing tensile forces and moments 2 The evaluation refers to rib center. For the rib end, the classification is one class worse each.	Welded on both sides	K1	-
6.2				Welded on both sides with filet weld	K2	-
6.3		$S1_3\Image252.gif$		Welded on both sides	K2	K1²
6.4				Welded on both sides with filet weld	K3	K2²
6.5			Reinforcement plate, filler plate with fillet weld joint. No transmitting of changing tensile forces and moments	s₂ ≤ 1,5 · s₁ r ≥ 2 · s2	K2	-
6.6			Welded attachments with transmitting changing tensile forces and moments	Welded on both sides	-	K1
6.7				Welded on both sides but not continuously welded	-	K2

Table 5. Examples of construction types and welded connections with the corresponding weld classes (K0, K1, K2 and K3), depending on the type of stress proof (depicted rupture paths exemplary)
No	Figure		Description	Provision	Weld class for stress proof 1 or 2
No	Figure		Description	Provision	1	2
7. Welded attachments without transmitting of changing tensile forces or moments, examples Provision: External appearance of the welds acc. EN 25817, classification group B, excluding the characteristics weld reinforcement, lower deviation of the weld and being non-isosceles
7.1			Vessel with skirt support	Welded on one side	K2	-
7.2		$S1_3\Image256.gif$	Vessel wall with ring support	Welded on both sides but not continuously welded	K2	-
7.3			Vessel wall with ring stiffener (external pressure)	Welded on both sides but not continuously welded	K2	-
7.4			Vessel wall with ring stiffener (external pressure)	Welded in circumferential direction but not continuously welded	K3	-
7.5			Vessel wall with support bracket (with or without reinforcement)	Welded on one side	K2	-
7.6			Vessel wall with leg support (with or without reinforcement)	Welded on one side	K2	-
7.7			Vessel wall with trunnion (with or without reinforcement)	Welded on one side	K2	-
7.8			Vessel wall with lifting lug (with or without reinforcement	Welded on one side	K2	-
7.9			Vessel wall with lifting lug	Welded on one side	K2	-

8 Governing heat treatment diameter per EN 10222-1, Annex B

DIN EN 10222-1 Table B1; Equivalent thickness t_eq for cross sections differing from governing shape of cross section
Bars		Disks	Rings	Cylindrical hollow profiles with open ends			Cylindrical hollow profiles closed at both ends	Other	Thickness t_R of governing rectangular cross section

t_eq ≈1,5 tR	t_eq ≈ 1,2 t_R	t_eq ≈ t_R	h>t_eq, Di> 200mm t_eq ≈t_R	D1> 200mm t_eq ≈ t_R	200mm >D₁ > 80 mm t_eq ≈ 0,85 t_R	D₁ < 80 mm t_eq ≈ 0,75 t_R	t_eq ≈ 0,6 t_R		b >2 t_R, l > 4 t_R
25	20	16	16	16	15	12	10	In agree-ment. 8.3.2	16
50	40	35	35	35	30	25	20		35
75	60	50	50	50	40	35	30		50
100	80	70	70	70	55	50	40		70
150	120	100	100	100	85	75	60		100
200	160	130	130	130	115	100	80		130
250	200	160	160	160	140	125	100		160
300	250	200	200	200	170	150	120		200
375	300	250	250	250	210	180	150		250
500	400	330	330	330	280	250	200		330
600	480	400	400	400	340	300	240		400
750	600	500	500	500	425	375	300		500
Example, Line 1: The heat treatment diameter of teq = 25 mm with a circular profile is equivalent to a rectangular profile with t_R=16 mm thickness and b>2t_R, l>4t_R.