长安大学结构设计原理课程设计(英文版).doc

Design of the PRC SSB

Name: 何书进
Number: 201421080218
Class: 2014210802

Catalogue
1.The data of design 3
2.Main beam size 3
3.Geometric features of all cross section 4
4.The main beam inner force of calculation 5
5..Estimation of reinforcement area and layout steel cable. 5
6.Calculation of geometric features of main beam cross-section. 9
7.Calculation of carrying capacity ultimate limit state 10
8.Estimation of pre-stress losses 12
9.Stress check 16
10.Check of cracking 18
11.Deformation calculate 19
12.The layout of pre-arch value 21

1. The data of design
(1) Span of SSB: 28.18m, computed span L=26.84m
(2) Design load: vehicular load-Ι, pedestrian load is 3.0kN/m2, coefficient of importance γ0=1.0
(3) Environment: Bridge locates in a common field, environment class-Ι, relative humidity(average of per year) is 75%.
(4) Materials: Tendon: low slack strands (1×7): standard value of tension strength fpk=1860MPa, Design value of tension strength fpd=1260MPa, nominal area = 140mm2, nominal diameter is 15.2mm, elastic modulus Ep=1.95×105Mpa, strand tapered of group anchors. Nonprestressed reinforcement: HRB 400, fsk=100MPa, fsd=330MPa; HRB 335: d<12mm, normal value of tension strength fsk=335MPa, design value of tension strength fsd=280MPa,; elastic modulus Es=2.0×105MPa; concrete C50: Ec=2.45×104MPa; normal value of compress strength fck=32.4MPa. Design value of compress strengthfcd=22.4MPa; normal value of tension strength ftk=2.65MPa; design value of tension strength ftd=1.83MPa.
(5) Design requirements: JTG D62-2004, total pre-stress concrete.
(6) Construction method: post-tension: pre-cost girder; TD Jack; cast-in-site wet connection joint with width of 40m; asphalt lager with 80mm thickness.
2. Main beam size
Just like the picture.

3. Geometric features of all cross section
(1) Calculation of effective width bf’
According to the high way bridge gauge, the minimum value of the follow three value can be got.
① 1/3 simple supported beam span, l/3=26850/3=8950mm.
② The average distance between two adjacent beams. To the middle beams is 2200mm.
③ (b+2bh+12hf'), bh=0, so let hf is equivalent with average of web thickness about middle span section
So b+2bh+12hf'=200+6×0+12×228=2936mm.
Above all, the bf'=2200mm.
(2) Calculating geometric features of all cross-section
Generally, geometric features of main beam often use block numerical summation method. Area of all cross-section: A=Ai
Distance between gravity center line of the all cross-section and the top of the beam:
yu=AiyiA
Ai—the area of separated section
yi—the distance between the center of grarity
of separated section and edge of beam

Geometry features of all cross-section forΙ-Ιsection
Number of block
Area of block
yi
(mm)
Si=Aiyi
(mm)
yu-yi
(mm)
Ix=Aiyu-yi2
(mm4)
Ii
(mm4)
Figure of block section
①
360000
90
32400×103
454
74.202×109
0.972×109

②
96000
220
21120×103
324
10.078×109
0.077×109
③
320000
800
256000×103
-256
20.972×109
68.267×109
④
20000
1533
30660×103
-989
19.562×109
0.044×109
⑤
80000
1700
136000×103
-1156
106.907×109
0.267×109
Summary
876000
yu=544
yb=1256
476180×103

231.721×109
69.627×109
I=IX+Ii=301.348×109

4. The main beam inner force of calculation, caused by permanent action and variable action.

跨中截面
L/4截面
变化点截面
支点截面
Mmax
V
Vmax
M
Mmax
V
Vmax
M
Mmax
Vmax
Vmax
一期标准值
①
2124.9
0
0
2124.9
1593.6
141.7
141.7
1593.6
1051.7
201.4
283.3
二期标准值
现浇湿接缝
②
196.7
0
0
196.7
147.4
13.2
13.2
147.4
97.3
18.6
26.2
桥面与栏杆
③
458.7
0
0
458．7
344.1
30.5
30.5
344.1
227
43.5
61.2
人群荷载
④
61.8
0
2.1
30.9
42.5
4.3
4.6
34.8
24.4
6
7.7，
公路一级汽车荷载标准值（不计冲击系数）
⑤
1431.8
60.5
95.5
1074.1
1104.2
270
270
1104.2
821.3
207
306.6
公路以及汽车荷载标准值（计冲击系数）
⑥
1550.6
65.5
103.4
1163.3
1195.8
292.3
292.3
1195.8
889.4
223.2
332.1
持久状态的应力计算的可变作用标准值组合
⑦
1612.5
65.5
105.4
1194.2
1238.3
296.7
296.9
1230.6
913.8
230.2
339.8
承载能力极限状态计算的基本组合
⑧
5576.5
91.7
147.1
5000
4223.9
636.6
636.9
4215.3
2923.8
636.8
918.4
正常使用极限状态按作用短期效应组合计算
⑨
1064.1
42.2
68.9
782.8
815.4
193.3
193.6
807.7
599.3
150.9
222.3
正常使用极限状态按作用长期效用组合计算
⑩
597.4
24.2
39
442
458.7
109.8
109.8
455.6
338.3
85.2
125.7
5. Estimation of reinforcement area and layout steel cable.
(1) According to the requirement of resistance crack from normal section to estimate number pre-stressed rebar. Full pre-stressed concrete structure of the effective pre-stress in middle span cross-section is:
Npe=Msw-0.7ftk1A+epw
Ms=MG1+MG2+MQs=3844.4 kN∙m
Assume ap=100mm, the length from the point of resultant force to the cross gravity axis is: ep=yb-ap=1156mm.
According to the features of all cross section to estimate rebar, the area of middle span cross-section is A=876000mm2.
The elastic resistance moment of the all cross-section to the resistance cracking edge: W=Iyb=301.348×1091256=923.927×106mm3
So the resultant force of effective pre=applied force is:
Npe≥Msw-0.7ftk1A+EPW=2.3773×106N
The control stress of pre-stressed rebar is:
σcon=0.75fpk=0.75×1860=1395MPa
The pre-stresses losses are estimated as 20% σcon
So get the area of pre-stressed reinforcement is:
Ap=Npe(1-0.2)σcon=2130mm2
So use 3 bundle 6фs15.24 strand, the area of pre-stressed reinforcement is:
Ap=3×6×140=2520mm2
And use the strand tapered group anchors ф70 metal corrugated pipe into the hole.
(2) Lay out pre-stressed reinforcement.
① Arrangement of requirement of the Road Bridge Gauge, arrangement of pre-stressed reinforcement in mid-span section shown in the figure.
② Lay out steel cable of anchor cross-section.

③ Location and angle of steel cable from other sections.
1. Bending shape, angle and radius of bending.
Obtain interpolate curve between straight lines. To make the pre-applied fore be perpendicular to anchor bearing plate, bending angle of N1, N2, N3 are 8°.
RN1=45000mm
RN2=30000mm
RN3=15000mm

2. Calculating horizontal distance between load point and anchoring point.
According to the formula:
Ld=c∙cotθ0
Ld=400×cot8°=2846mm

Calculating horizontal distance between load point and bending point.
According to the formula:
Lb2=R∙tanθ2=1049mm
Lw=Ld+Lb2=3895mm
xk=268502+312-Lw=9842mm
According to the same theory, we can get location of control point from N1 and N2 as shown in the following table.
钢束号
升高值
弯起角
弯起半径
支点至锚固点水平距离
弯起点至跨中截面水平距离
弯止点至跨中截面水平距离
N1
1610
8
45000
156
545
6374
N2
900
8
30000
256
6224
10198
N3
500
8
15000
312
9842
11930

3. Calculating location and angle of steel cable from all kinds of sections.
According to the figure, the distance point I to bottom of beam
ai=a+ci, and angle θ, a=100mm
When xi-xk≤0, ci=0, ai=a=100mm, θi=0.
When 0<xi-xk≤(Lb1+Lb2), ci=R-R2-(xi-xk)2, θi=sin-1xi-xkR.
When (xi-xk)>(Lb1+Lb2), θi=θ0=9°,
ci=xi-xk-Lb2tanθ0.

计算截面
钢束编号
xk
Lb1+Lb2
xi-xk
θ0
ci
ai=a+ci
跨中截面
xi=0
N1
545
5829
为负值钢束尚未弯起
0
0
100
N2
6224
3974
N3
9842
2088
L/4截面
xi=6712.5
N1
545
5829
xi-xk≤(Lb1+Lb2)
8
425
525
N2
6224
3974
0<xi-xk<3974
0.933
4
104
N3
9842
2088
为负值
0
0
100
变化点截面
xi=10178
N1
545
5829
xi-xk≤(Lb1+Lb2)
8
1043
1143
N2
6224
3974
0<xi-xk<3974
8.647
262
362
N3
9842
2088
为负值
0
0
100
变点截面
xi=13878
N1
545
5829
xi-xk≤(Lb1+Lb2)
8
2020
2120
N2
6224
3974
xi-xk≤(Lb1+Lb2)
8
993
1093
N3
9842
2088
xi-xk≤(Lb1+Lb2)
9
553
653

④ The position and angle in flat bend zone.
N1, N2, N3, are in the same plane at the middle span, but at anchor terminal they are all in the middle lane, to get this result, N2, N3 must be bend from both sides to the middle line in the main beam lab. N1, N3 are taking the same to bend up and the position in flat bend as figure shows. There are two curve arc each angle is :
θ=6388000×180π=4.569°

⑤ Requirement for non-prestressed reinforcement.
To be satisfied with the ultimate limited state, the number of non-prestressed reinforcement are: after deciding the reinforcement number, non-prestressed reinforcement is decided according to the normal cross-section’s ultimate limited state.
Assume the distance from prestressed and non-prestressed reinforcement’s resultant fore point to the cross-section bottom is a=80mm.
h0=h-a=1720mm
First, assume first kind of T shape, from γ0Md≤fcdbf'x(h0-x2), so we can get x=71.5mm <hf'=228mm
So the As=fcdbf'x-fpdApfsd=1055.5mm2
Select 5 ∅ 18mm, HRB 400, As=1272.5mm2>0.003bh=1032mm2, the space is 75mm, as=45mm.

6. Calculation of geometric features of main beam cross-section.
Choose the first stage mid-span cross-section as example.
分块名称
分块面积
重心至梁顶距离
对梁顶地面积矩
自身惯性矩
yu-yi
Ix=Ai(yu-yi)2
I=Ii+Ix
混凝土全截面
804×103
584.2
469.697×106
285.204×109
-7.2

非预应力钢筋换算面积
6.104×103
1755
10.713×106
0
-1178
8.47×109

预留管道面积
-11.545×103
1700
-19.627×106
0
-1123
-14.56×109

净截面面积
798.599×103
577
460.783×106
285.204×109

-6.048×109
278.976×109

受力阶段
计算截面
A
yu
yb
rp
I
wu=Iyu
wb=Iyb
wp=Iep
阶段一：孔道压浆前
跨中截面
798.5×103
577
1223
1123
278.9×109
4.8×108
2.3×108
2.484×108
L/4截面
798.5×103
578.3
1220.7
959.7
282.8×109
4.9×108
×2.3108
2.95×108
变化点截面
798.5×103
582.9
1217.1
712.4
287.5×109
5.2×108
2.99×108
4.036×108
支点截面
1050.5×103
657.1
1142
173.6
341.7×109
5.1×108
2.58×108
19.685×108
阶段二：管道结硬后至湿接缝结硬前
跨中截面
821.8×103
608.8
9
1091.2
307.5×109
5×108
2.54×108
2.82×108
L/4截面
821.8×103
606.5
1193.5
932.5
303.7×109
5×108
2.5×108
3.26×108
变化点截面
821.8×103
603
1197
692.3
299×109
5.2×108
3×108
4.32×108
支点截面
1073.8×103
660.8
1139.2
169.9
342.4×109
5.7×108
2.6×108
20.15×108
阶段三：湿接缝结硬后
跨中截面
893.8×103
567.4
1232.6
1132.6
325.4×109
5.7×108
2.64×108
2.87×108
L/4截面
893.8×103
565.3
1234.7
973.7
321.5×109
5.6×108
2.6×108
3.3×108
变化点截面
893.8×103
562.1
1237.9
733.2
316.6×109
5.6×108
2.558×108
4.318×108
支点截面
893.9×103
625
1175
205.7
364.4×109
5.7×108
3.1×108
17.715×108
7. Calculation of carrying capacity ultimate limit state
(1) Normal section capacity
① Compressive height x:
z=fpdAp+fsdAsfcdbf'=73mm<hf'=180mm
The neutral axis is in the compressive plate calculation of bearing capacity of rectangular cross-section. So it is type I cross-section.
② Normal section capacity
a=fpdApap+fsdAsasfpdAp+fsdAs=93.6mm
h0=h-a=1706.4mm
Md=5576.5kN∙m
Mu=fcdbf'xh0-x2=6007.365kN∙m
Mu>γoMd=5576.5kN∙m
So the normal section capacity meet the requirement.
(2) Inclined section capacity
① Shear capacity calculation
0.5×10-3α2ftdbh0≤γ0vd≤0.51×10-3fcu,kbh0
vd=636.8kN, fcu,k=50MPa, b=200mm
a=fpdApap+fsdAsasfpdAp+fsdAs=451mm
So h0=1800-451=1349mm, α2=1.25.
γ0vd=1.0×636.8=636.8kN
0.5×10-3α2ftdbh0=308.584kN≤γ0vd
0.51×10-3fcu,kbh0=972.965kN≤γ0vd
So the size of cross section meets the requirement, but it need to exist
resistant shear steel.
γ0vd≤vcs+vpb
P=100p=100×Ap+Apb+Asbh0=1.4
So choose the R=10mm, double wire HRB 335 steel.
fsv=280MPa, sv=200, Asv=2×78.54=157.08mm2
psv=Asvsvb=0.00393
sinθp is the average of three pre-stressed steel value.
sinθp=0.089
So vcs=α1α2α30.45×10-3bh02+0.6pfcu,kpsvfsv
=763.542kN
vpd=211.945kN
vcs+vpd=975.487kN>γ0vd=636.8kN
So it meets the requirement.
8. Estimation of pre-stress losses
(1) The control stress
σcon=0.75fpk=1395MPa
(2) Stress loss in cable
① Friction in pre-stressed reinforcement and pipeline.
σc1=σcon[1-e-(Nθ+kx)]
For the mid-span cross section: x=l/2+d, d is the distance from anchor point to bearing central line; N is the friction coefficient of reinforcement and pipe; k is the local deviation of pipe per meter length’s influence on the friction coefficient. Knowing N=0.25, k=0.0015, θ is the pipe rolls angle from tensioned place to the mid span. Only N1 is a siphon standpipe, so θN1=θ0=8°, N2, N3 is not only a siphon but also a flat bend pipe, the siphon angle θv=8°, θH=9.138°, θN2=θN3=θH2+θv2=12.145°
Mid span (I-I) cross section friction loss σl1:
钢束编号
θ

Nθ

x

kx

β=1-e-(Nθ+kx)

σcon

σc1
（°）
（rad）
N1
8
0.1396
0.0349
13.986
0.02098
0.0543
1395
75.75
N2
12.145
0.2120
0.0530
14.086
0.02113
0.0714
1395
99.60
N3
12.145
0.2120
0.0530
14.142
0.02121
0.0715
1395
99.78
平均值
91.71

Also, we can calculate other cross-section σl1. The average friction loss of cross section can be list in the following table.
截面
跨中
L/4
变化点
支点
91.71
53.44
53.44
24.75
0.49
② Anchorage deformation、steel retraction causes stress losses
Post tensioned structure should consider the influence of friction, its influence length is :

The influence length of friction shows in the following Table.
截面
钢束编号
x
lf
△σ
σl2
各截面σl2平均值
跨中截面
N1
13486
11854
131.5969032
x>lf截面不受反摩擦影响
0
N2
13586
10361
150.5637822
N3
13642
10377
150.3373309
L/4截面
N1
8321
11854
131.5969032
39.224
31.64640158
N2
8421
10361
150.5637822
28.192
N3
8477
10377
150.3373309
27.522
变化点截面
N1
3856
11854
131.5969032
88.791
91.35958522
N2
3956
10361
150.5637822
93.076
N3
4012
10377
150.3373309
92.211
支点截面
N1
156
11854
131.5969032
129.87
140.8419546
N2
256
10361
150.5637822
146.84
N3
312
10377
150.3373309
145.82
(3) Concrete elastic compression caused prestress losses
For simple supported beam, calculate l/4 section, and take its value as the whole beam cross-section’s prestressed reinforcement losses’ average value:

m---Tension batch number, m=3;
---versus of prestressed reinforcement elastic modulus to concrete elastic modulus, calculate as the concrete true strength when tensioned, is assumed as 90% design value, , we can obtain

---Concrete normal stress that all prestressed reinforcements’ resultant force at its application point caused, , section features use the first stage in the upper table;

So Np=3282.023kN, σl4=28.712MPa
(4) Steel relaxation causes pre-stress losses:
The pre-stress losses caused by steel relaxation is :

---Tension coefficient, use super tension, so =0.9;
---Steel loose coefficient, to low relaxation strand, =0.3;
---Reinforcement stress at force anchor ,
So σl5=33.04MPa.
(5) Stress losses caused by concrete creep and shrinkage：
Stress losses caused by concrete creep and shrinkage could calculate by use the following formula:

、——load age is , final value of concrete shrinkage strain, and final value of concrete creep.
——Load age，the load age till the 90% of the design strength， , ；To the second stage dead load , the load age , assume =90d；
The bridge is at the common area of field. The relative humidity is 75%，thickness of the member (I－I section), get ，check the table, get the final value of creep