龍騰科技設計工作室

變動負荷

Introduction

A fatigue failure begins with a small crack . The initial is so minute that it can not be detected by the naked eye and is even quite difficult to locate in a Magnaflux or x-ray inspection .The crack will develop at a point of discontinuity in the material, such as a change in cross section , a keyway , or a hole . less obvious points at which fatigue failures are likely to begin are inspection or stamp marks , internal cracks , or even irregularities caused by machining.

疲勞破壞(fatigue failure)-當元件承受循環負荷作用下,元件不連續處會產生裂縫,在高張應力作用下,裂縫會造成擴展,而使斷面積縮減,經過數次循環作用下,斷面淨面積減至無法承受當時之負荷,即造成突然斷裂,此謂之.

疲勞強度(Fatigue Strength)-材料欲達到或超過某一循環壽命N其所能承受的最大完全反覆應力值,即謂之對應於該循環壽命N之疲勞強度.

疲勞限(Fatigue limit or Endurance Limit)-材料(元件)欲達到無窮壽命,其所能承受的最大完全反覆應力值.

The endurance limit

The determinatin of endurance limits by fatigue is now routine , though a lengthy procedure. Generally, Stress testing is preferred to strain testing for endurance limits .

By plotting these as in Fig . it is possible to see whether there is any correlation between the two sets of results . The graph appears to suggest that the endurance limit ranges from about 40 -60 percent of the tensile strength for steel up to about 200 kpsi (1400 Mpa) . Beginning at about Sut = 200 Kpsi , the scatter appears to increase , but the trend seems to level off, as suggested by the dashed horizontal line at S'e= 100 kpsi (700 Mpa) .

Endurance_Limits

Mischke has analyzed a great deal of actual test data from several sources and concluded that endurance limit can. indeed, be related to tensile strength. For steels.

S_e'=	0.506S_ut	S_ut≦200kpsi(1400Mpa)
	100kpsi	S_ut>200kpsi
	700Mpa	S_ut>1400Mpa

The Fatigue Strength

N S_f

₁₀³fS_ut

₁₀^{6
S}_e'

∵S_f=aN^b

^∴log S_f=log a + b log N

As N=10³ _,S_f=fS_ut->log(fS_ut)=log a + 3b-----------------(1)

As N=10⁶ _,S_f=Se' ->log S_e'=log a + 6b---------------------(2)

(1) *2 - (2)

2*log(fS_ut)-logS_e'=log a

a= (fS_ut)²/ S_e'

(1)-(2)

log(fS_ut)--logS_e'=-3b

b=-(1/3) log (fS_ut/ S_e')

Endurance-Limit Modifying Factors

S_e=k_ak_b k_c k_d k_e S_e'

Se= Endurane limit of mechanical element

Se'=Endurance limit of test specimen

k_a=surface factor

k_b=size factor

k_c=load factor

k_d=temperature factor

k_e=miscellaneous-effects factor

surface factor k_a

k_a = a S_ut^b

Surface Finish	Factor a		Exponent b
Surface Finish	kpsi	Mpa	Exponent b
Ground	1.34	1.58	-0.085
Machined or cold-drawn	2.70	4.51	-0.265
Hot-rolled	14.4	57.7	-0.718
As forged	39.9	272	-0.995
Surface Finish Factors

Size Factor k_b

The results for bending and torsion be expressed as

k_b=	(d/0.3)^-0.107	0.11≦d<2 in
	0.859-0.02125d	2<d≦10 in
	(d/7.62)^-0.107	2.79≦d<51 mm
	0.859-0.000837d	51<d≦254 mm

For axial loading there is no size effect.Therefore, use

k_b=1

不回轉樑如所受最大應力0.95以上所具有之體積與回轉樑相同,則有相同之尺寸修正因素.

A_0.95=pi/4 *[d²-(0.95d)²]=0.0766d²

圓形不迴轉樑

A_0.95=∫_0.475d^0.5d (√(d²/4-x² dx))=0.0105d²

0.0105d²=0.0766d_e²

d_e=0.37d

長方形不迴轉樑

A_0.95=0.05bh

0.05bh=0.0766de²

de=0.808(bh)^1/2

These section are shown in Fig together with a channel and a wide-flange or I-beam section. For the channel,

A0.95σ	0.05ab	axis 1-1
A0.95σ	0.052xa+0.1t_f(b-x)	axis2-2

The 95 percent stress area for the wide flange is

A0.95σ	0.10at_f	axis 1-1
A0.95σ	0.05ba (t_f>0.025a)	axis2-2

Load Factor k_c

k_c=α*S_ut^βLN(1,C)

Model of	α		β	C	平均K_c
Model of	kpsi	Mpa	β	C	平均K_c
bending	1	1	0	0	1
axial loading	1.23	1.43	-0.078	0.125	0.85
torsion & shear	0.328	0.258	0.125	0.125	0.59
Marin 負載因數的參數

Temperature Factor k_d

k_d=S_T/S_RT

temperature℃	S_T/S_RT	temperature℉	S_T/S_RT
20	1.0	70	1.0
50	1.010	100	1.008
100	1.02	200	1.02
150	1.025	300	1.024
200	1.02	400	1.018
250	1.0	500	0.995
300	0.975	600	0.963
350	0.943	700	0.927
400	0.9	800	0.872
450	0.843	900	0.797
500	0.768	1000	0.698
550	0.672	1100	0.567
600	0.549
Effect of Operating Temperature on the Tensile Strength of Steel.(ST=Tensile Strength at Operating Temperature;SRT=Tensile Strength at Room Temperature;0.099≦(statistics)σ^≦0.110

kd=[0.975+0.432(10^-3)t_f-0.115(10^-5)t_f²+0.104(10^-8)t_f³+0.595(10^-12)t_f⁴]LN(1,0.11)

NOTCH SENSITIVITY

Notch sensitivity q

q=(K_f-1) /(K_t-1)

K_t為理論(或幾何形狀)應力集中因素,疲勞應力-集中因素K_f.

修正Neuber公式如下:

K_f=(K_t*LN(1,C_kf))/(1+((2/√r)*(K_t-1)*√a/K_t))

K_f:對數常態分佈.

r:缺口半徑.

√a為對應於平均抗拉強度S_ut^¯的函數.

√a & C_kf的Heywood參數
√a
狀況	S_ut^¯,kpsi	S_ut^¯,Mpa	C_kf
穿透孔	5/S_ut^¯	174/S_ut^¯	0.10
肩部	4/S_ut^¯	139/S_ut^¯	0.11
凹部	3/S_ut^¯	104/S_ut^¯	0.15

K_f=1+(K_t-1)/(1+√(a/r))-------------------------(28)

q=1/(1+√(a/r))------------------------------------(29)

√a=0.220353-0.275605(10^-2)S_ut^¯+0.113449(10^-4)(S_ut^¯)²-0.247328(10^-7)(S_ut^¯)³--------(30)

k_t:形狀,負重方式而與材質無關.

k_f=(有缺口試桿之最大應力)/(無缺口試桿之最大應力)

σ_max=kf_*σ₀ σ₀ :公稱應力.

如想將(28)與(29)式應用在低合金鋼扭轉上,可將(30)中的平均極限抗拉強度加大20kpsi然後使用計算√a之值.

註:合金鋼分類下

1.低合金鋼:特別合金元素之和少於8.0%.

2.高合金鋼:特別合金元素之和高於8.0%.

Fluctuating Stresses

Fig 7-13

某些應力時間關係:(a)具有高頻率波紋的波動應力(b)與(c)非正弦波型波動應力(d)正弦波型波動應力(e)覆變應力(f)完全交變正弦波型應力.

Fig 7-14

修正Goodman圖,其中顯示一特定平均應力的所有應力分量所有強度與極限值.

Fig 7-15

拉伸與壓縮區域內的平均應力疲勞損壞圖.利用平均強度與抗拉強度之比S_m/S_ut,平均強度與抗壓強度之比S_m/S_uc及強度振幅與耐久線之比S_a/S_e,即可畫出各種鋼的實驗結果.

Fig 7-16

針對AISI 4340鋼畫出的主疲勞圖.

σ_m=(σ_max+σ_min)/2

σ_a=(σ_max-σ_min)/2

σ_m:平均應力.

σ_a=應力振幅.

當平均應力為壓縮時.只要σ_a=S_e或σ_m=S_yc,就會發生損壞.

Fig 7-17 顯示出各種損壞標準的疲勞圖.對於各標準而言,在個別曲線上或外面的點表示損壞,例如Goodman線上的某一點A顯示σ_m的極限值強度S_m和σ_a(與σ_m成對)的極限值強度Sa的對照關係.

Soderberg方程式如下

S_a/S_e+S_m/S_yt=1--------------------(40)

修正Goodman方程式如下

S_a/S_e+S_m/S_ut=1--------------------(41)

Gerber疲勞曲線方程式如下

S_a/S_e+(S_m/S_ut)²=1--------------------(42)

畸變能橢圓曲線方程式如下

(S_a/S_e)²+(S_m/S_ut)²=1--------------------(43)

Langer 單次-週期-降伏疲勞曲線方程式如下

S_a/S_yt+S_m/S_yt=1--------------------(44)

The stresses and can replace Sa and Sm in Eqs.(40-42) if each strength is divided by a factor of safety n . When this is done , the Soderberg equation becomes