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Analysis of Loads and Stresses in a Pre-stressed Roof Beam, Thesis of Interface between Computer Science and Economics

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

  • What is the allowed compressing stress in the concrete for the foundation?
  • What are the ultimate internal forces for the roof beam?
  • What are the live loads on the main girder?
  • What is the allowed tension stress in the soil for the foundation?
  • What are the dead loads on the main girder?

Typology: Thesis

2019/2020

Uploaded on 01/26/2020

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Faculty of Engineering
Civil Engineering
“Design of Pre-Stressed Concrete T-Beam”
Graduation Project 2
Mentor: Prepared By:
Assist. Prof. Dr. Jelena Ristic
Email:
Skopje, 2019
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Download Analysis of Loads and Stresses in a Pre-stressed Roof Beam and more Thesis Interface between Computer Science and Economics in PDF only on Docsity!

Faculty of Engineering

Civil Engineering

“Design of Pre-Stressed Concrete T-Beam”

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

  • thickness of beam flange

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

  1. Determining geometrical characteristics of the main girder

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

A=0,

m

2

A=0,

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`

M

g

max

g l

2

=4799,4 kNm

R

A

g

=R

B

g

=721,2 kN

Live Loads of main girder

-Reaction from roof beam from snow 3,04 kN/m`

p=3,04kN/m`

M

p

max

p ∙ l

2

=264,84 kNm

R

A

p

=R

B

p

p ∙ l

=40,1 3 kN

Ultimate in termal forces for main gider:

M

u

=1,6M

g

+1,8M

p

= 1,6∙4799,4+1,8∙264,84=8155,75 kNm

R

u

=1,6R

g

+1,8R

p

6)REINFORCED CONCRETE FOUNDATION

Centric loaded foundation – calculation of cross sectional area of the foundation

σ real

N
F

Z,allowed

F= cross-section area of foundation (AxB)

σ

Z ,allowed

=0,26 MPa = 260 kN / m

2

A≥
N

σ

Z , allowed

σ

Z ,allowed

= Allowed tension stresssoil

N

0

=721,20+40,13= 761,33 KN
N

o

  • axial force acting on the foundation
0,75 B

2

B

2

=3,90 m

B=1,97m minimal value for side B

Adopted: B=2,15m

A=0,75B

Adopted: A=1,65m

Determining height of foundation:

a)From condition of column not punching through:

h

t

N

0

ͳ

a ∙

( m ) 0-Peripheral circumference of column

ͳ a

=allowed shear stress

0-letter0=2∙0,8+2∙0,6=2,80m

T

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

N

0

F

=214,61 kPa

F= Area of the foundation (AxB)

Q

u

=P

u

∙(0,675∙1,65)=239,0 kN – bending force from soil

Bending moment from soil :

M

u

=Q

u

.S=239,0∙0,3375=80,66 kNm

Static width of the cross section:

b st

A

=0,825 m

h m

=h st

+a

h m

=kh

Mu

bst

=18,97 cm =0,1897 m

h a

Q

u

K

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

M

u

=ultimate moment

K

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

  • Steel sheet with thickness d 1

=1,0mm

  • Termoinsulation with thickness d 2

=10 cm

Total: g 1

=0,46kN/m`

2.2 Live loads

  • Snow loads s=0,75∙1,0=0,75 kN/m`
  • Wind loads w=0,21∙1,0=0,21 kN/m` -acting only from side

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

CROSS-SECTION

[M] 0,1 l

0,77m

0.2l

1,54m

0,3l

2,31m

0,4l

3,08m

0,5l

3,85m

M

g

[kNm] 11,2 19,8 26,0 29,8 31,

M

g

[kNm] 1,2 2,2 2,9 3,3 3,

M

p

[kNm] 2,6 4,6 6,0 6,8 7,

M

q

[kNm] 15,0 26,6 34,9 39,9 41,

4.Calculating prestressing and needed wires

4.1.Determining final prestressing force

  • Final pre-stressing force N k
M

g

σ

bz ∙W

d

e

k

  • k

g

  • For completely pre-stressed elements:

σ bz

= 0 –tension stress in concrete is zero

M

ɡ

= Total moment

N

k

M

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

N

k

N

k

=134,43 kN

4.2 Determining the needed initial pre-stressed force

N

p

N

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

N

p

f

b

N

p ∙ e

p

w

g

M

g

W

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

N

p

f

b

N

p ∙e p

w

g

M

g

W

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

N

k

F

b

N

k∙ e k

w

g

M

g

W

g

σ g=

0,829-1,643+1,994=1,180MPa≤σ σ bpr

σ g=

N

p

f

b

N

p ∙e

p

w

g

M

g

W

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

N

p

F

b

N

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=

N

p

f

b

N

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

N

k

F

b

N

k∙ e k

w

g

M

g

W

g

10 ≤ σ

bz

σ g

=0,829-1,643=-0,814MPa ≤σσ bz

=-1,5MPa -Tension

σ d=

N

k

f

b

N

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