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An analysis of the loads and stresses in a pre-stressed roof beam, including dead loads, live loads, and ultimate internal forces. It also includes calculations for the moment of inertia, cross-section area, and stresses at various points in the beam. intended for engineering students or professionals in the field of civil or structural engineering.
What you will learn
Typology: Thesis
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Faculty of Engineering
Civil Engineering
Graduation Project 2
Mentor: Prepared By:
Assist. Prof. Dr. Jelena Ristic
Email:
Skopje, 2019
1.Project Conditions
a) Span of Prestressed main girder L=26,4m
b) Span of Prestressed roof beam L 1
=8,10m
c) Total Height of the hall H=6,8m
d) Foundation on soil with bearing capacity σ
z
doz
=0,26 MPa
2.Geometric Characteristics
d p
λ- half with of flange (initial value)
d ≥
=0,405 m =40,5 cm
d -Total Height of roof beam
Adopted: d=50cm
λ - Distance between webs axis of roof beam
λ =( 40 − 75 )
λ = 50 cm 2 λ = 100 cm
b-Width of the web
b=(8-10cm)
Adopted: b= 10cm
b 0
=b+2cm
Adopted: b 0
= 12cm
2b 0
=24cm Adopted: 2b
0
= 25cm
b 1
=2b 0
+2=27cm
d =
l
+( 0,2 ÷ 0,4) m
d ≈
+0.3=1,32+ 0,3=1,62 m
l=span =26,40m
Adopted Value: d=1,7m
d 1 =( 0,1−0,15) d =( 0,1−0,15) ∙ 1,7=0,17−0,
Adopted : d 1 =0,20 m
b =( 0,3−0,4) d =( 0,3−0,4 ) ∙ 1,7=0,51−0,
Adopted : b =0,6 m
b
0
=0,25 ∙ b ≥ 12
b
0
= 20 cm =0,2 m
b
0
− thickness of web
3)Reinforced concrete longitudinal beams
Fig.Disposition of the structure in longitudinal direction
Calculation of beam dimensiong:
Height of beam:
d =
(
)
∙ l
0
(
)
∙ 8,10=(1,01−0,675) m
-Width of beam
Accepted: d=0.8m
b =
(
)
d
0
(
)
=( 0,4−0,27) m
b=0,3m
4)Reinforced concrete columns
Lateral Longitudinal
Direcrion Direction
l
i
= 13,6m
l
i
= 4,5m
j=0, m
4
j=0, m
4
m
2
m
2
i=0,2309 i=0,
λ=58,90 λ=25,
λ =
l
i
i
No problem with slenderness 25< λ <
5)General and seismic dilatation
δ
min
= 3 cm
δ > 3 +
h − 5
=3,60 cm
Adopted δ = 5 cm
Statical Calculation
Prefabricated Pre-Stressed roof beam
Analysis of loads:
Calculation Analysis of loads for roof beam dead load and permanent loads :
-Load from roof sheet d 1
=1.0mm 1,0∙0,36[kN/m]=0,36kN/m`
(weight of steel sheet 0,36kN/m
2
-Load from termoisolation 1,0∙0,1=0,1kN/m`
(weight of 0,10[kN/m
2
-Self load of beam 0,167∙25,0=4,715 kN/m`
(weight of RC 25[kN/m
3
Σ total g=5,18 kN/m`
g=5,18 KN/m`
Live loads on Roof Beam
-Load by snow (H<500 mNv) s=0,75kN/m
2
(from Standards)
Load from wind:
-Exposed to wind
-H<10m (h=6,8m) (from Standards)
q' w
=0,7 kN/m
2
nominal value
Analysis of loads for
Main pre-stressed girder
1)Calculation of dead load and other permanent loads:
-Self weight of main girder
0,525[m
2
]∙25,0[kN/m
3
]= 13,13 kN/m`
-Reaction from roof beams
2x20,98 kN/m` (2x= 2 beams supported on main girder)
Total g=55,09 kN/m`
g
max
g l
2
=4799,4 kNm
A
g
B
g
=721,2 kN
Live Loads of main girder
-Reaction from roof beam from snow 3,04 kN/m`
p=3,04kN/m`
p
max
p ∙ l
2
=264,84 kNm
A
p
B
p
p ∙ l
=40,1 3 kN
Ultimate in termal forces for main gider:
u
g
p
= 1,6∙4799,4+1,8∙264,84=8155,75 kNm
u
g
p
Centric loaded foundation – calculation of cross sectional area of the foundation
σ real
Z,allowed
F= cross-section area of foundation (AxB)
σ
Z ,allowed ❑
=0,26 MPa = 260 kN / m
2
σ
Z , allowed ❑
σ
Z ,allowed
❑
= Allowed tension stress ∈ soil
0
o
2
2
=3,90 m
B=1,97m minimal value for side B
Adopted: B=2,15m
Adopted: A=1,65m
Determining height of foundation:
a)From condition of column not punching through:
h
t
0
ͳ
a ∙
( m ) 0-Peripheral circumference of column
ͳ a
=allowed shear stress
0-letter0=2∙0,8+2∙0,6=2,80m
a
=0,6 MPa=600 KPa
ht =
=0,453 m
45,3+5=50,3cm
Adopted: ht=60,0cm
b)From condition of tension due to bending of foundation
Pressure from soil p u
0
=214,61 kPa
F= Area of the foundation (AxB)
u
u
∙(0,675∙1,65)=239,0 kN – bending force from soil
Bending moment from soil :
u
u
.S=239,0∙0,3375=80,66 kNm
Static width of the cross section:
b st
h m
=h st
+a
h m
=kh
Mu
bst
=18,97 cm =0,1897 m
h a
u
z
∙ ͳr ∙ b
st
=0,285 m
Reinforcement rebar RA400/500 ; f y
=400 MPa yield strength
f u
= 500 MPa ultimate tensile strength
a
= 10% elongation in steel reinforcement
b
=2,4% elengation on concrete
From table: for grade 30 K h
For grade 30 K z
*We adopt the largest value ht=0,6m
h a
= needed height from soil tension condition
b st
=statical width of foundation
h m
=h st
+a
u
=ultimate moment
h
=height coefficient (depends on reinforcement and concrete)
y d
=y c
=30,19cm
y=50-30,19=19,81cm
2.Analysis of Loads
2.1.Dead loads
2.1.1 Self weight of the beam g=0,167∙25,0=4,18 kN/m`
2.1.2 Other dead loads
=1,0mm
=10 cm
Total: g 1
=0,46kN/m`
2.2 Live loads
p=0,96 kN/m`
3.Determining internal forces.
3.1. Real Span of the beam,l
l 1
=8,10m
b=0,60m support length
l=l 1
-b+2∙0,1=8,10-0,60+2∙0,1=7,7m
-Static system: Simply supported beam
3.2.Internal forces from:
3.2.1-Self weight G
3.2.2-Other dead loads g 1
3.2.3-live loads
Bending
moment
[M] 0,1 l
0,77m
0.2l
1,54m
0,3l
2,31m
0,4l
3,08m
0,5l
3,85m
g
[kNm] 11,2 19,8 26,0 29,8 31,
g
[kNm] 1,2 2,2 2,9 3,3 3,
p
[kNm] 2,6 4,6 6,0 6,8 7,
q
[kNm] 15,0 26,6 34,9 39,9 41,
4.Calculating prestressing and needed wires
4.1.Determining final prestressing force
g
− σ
bz ∙W
d
e
k
g
σ bz
= 0 –tension stress in concrete is zero
ɡ
= Total moment
k
ɡ
e
k + k
g
a k
= 0,15d= 0,15∙50,0=7,5cm
e k=
y d
-a k
=30,19-7,5=22,69cm
k
k
=134,43 kN
4.2 Determining the needed initial pre-stressed force
p
k
=192,04 kN
4.3 Determining bearing capacity of one wire
σ m
=1850 MPa –(tensile strength in state of tearing of the wire )
-Cross sectional areas of the wires
ϕ=3,00mm A p
m
2
ϕ=5,00mm A p
m
2
d g
= Diameter of the largest aggregare in the concrete
Φ= Diameter of the wires
a=4,0+0,6+
=5,35 cm
Adopted: a=5,5cm
e p
= e
k
=30,19-5,5=24,69cm –eccentricity
4.6. Control of the stresses in the upper and lower edge of the cross section
4.6.1. Control of the stresses in cross-section in the middle of the beam (0.5l)
4.6.1.1 Control of stresses in the pre-stressing phase t=
σ g
p
f
b
p ∙ e
p
w
g
g
g
∙ 10 ≤ σ
bz
(σ bz
=allowed tension stress in concrete)
σ g=
1,185-2,347+1,489=0,327MPa≤σσ bz
σ d
p
f
b
p ∙e p
w
g
g
g
∙ 10 ≤ σ
bpr
(σ bpr
=allowed compressing stress in concrete)
σ d
=1,185+3,577-2,270=2,492MPa≤σ σ bpr
4.6.1.2. Control of stresses in serviceability state t= ∞
σ g
k
b
k∙ e k
w
g
g
g
σ g=
0,829-1,643+1,994=1,180MPa≤σ σ bpr
σ g=
p
f
b
p ∙e
p
w
g
g
g
σ g=
0,829+2,504-3,039=0,294MPa≤σ σ bpz
Stress Diagrams
Cross section in mid-span of the beam
σ g
p
b
p ∙ e
p
w
g
∙ 10 ≤ σ
bz
–Check for tension –Top edge
σ g
=1,185-2,347=-1,162MPa ≤σσ bz
=-2,4MPa
σ d=
p
f
b
p ∙e
p
w
d
∙ 10 ≤ σ
bz
-Check for compression -Bottom edge
σ g
=1,185+3,577=4,762MPa ≤σσ bz =
σ pr
=15,0MPa
4.6.2.2 Control of Stress Inserviceability State t=∞
σ g
k
b
k∙ e k
w
g
g
g
∙ 10 ≤ σ
bz
σ g
=0,829-1,643=-0,814MPa ≤σσ bz
=-1,5MPa -Tension
σ d=
k
f
b
k ∙ e k
w
d
∙ 10 ≤ σ
bpR
(0.7=during exploitation
compression)
σ d
=0,829+2,504=3,333MPa≤σσ bpR
=12MPa –Compression
Adopted concrete grade M30/30MPa compressive strength