statics of building structures i., erasmus

Statics of Building Structures I., ERASMUS

Transversally loaded frame and grid

• Basic properties of a transversally loaded frame

Department of Structural MechanicsFaculty of Civil Engineering, VŠB-Technical University of Ostrava

• Basic properties of a transversally loaded frame• Simple transversally loaded open frame• Basic properties of a grid• Approximate solution of regular ceiling grids• Utilization of the symmetry of a grid

Types of transversally loaded frames

2 / 32Basic properties of a transversally loaded frame

Transversally loaded frame in a vertical and horizontal plane

Types of transversally loaded frames

3 / 32Basic properties of a transversally loaded frame

Closed transversally loaded frame in a horizontal plane

Simple transversally loaded open frame

4 / 32Simple transversally loaded open frame

Simple transversally loaded frame, decomposition into partial loading states

Example 8.1


Problem definition of example 8.1,formulation of a basic structure and marking of indeterminate reaction

Example 8.1 Loading states Bending moments Torsional moments


Solution of example 8.1,diagrams of bending

moments and torsional moments in partial

loading states

Example 8.1


External forces and diagrams of

internal forces for example 8.1

Example 8.2

1,3

⋅=⋅=

==43-

t43-

Sss

m101,733I ,m101,067I

(b) picture see , nsymmetry utilizing when n :Solution

(a) picture toaccording loading and dimensions has beamBalcony :Problem

01 =+⋅⋅=⋅=

1011

t

:condition nalDeformatio

m101,733I ,m101,067I

δδ X


Problem definition of example 8.2,formulation of the basic static scheme and marking of indeterminate interactions

Example 8.2

EEE

EEE

881456,1)68,7(

10733,0

4,22

3

6,1)68,7(

10067,1

12

13477116,1

10733,0

4,2212,31

10067,1

1

3310

3311

−=⋅−⋅⋅⋅

⋅+⋅−⋅⋅⋅

⋅=

=⋅⋅⋅⋅⋅

⋅+⋅⋅⋅⋅⋅

=

−−

−−

δ

δ

:tscoefficien naldeformatio ofn Calculatio

kNmX

EEE

540,613477

88145

6,1)68,7(10733,0

2310067,1

2

11

101

3310

=−−=−=

=⋅−⋅⋅⋅

⋅+⋅⋅⋅

⋅= −−

δδ

δ

Loading states Bending moments Torsional moments


Solution of example 8.2, diagrams of bending and torsional moments in the partial loading states

Example 8.2

1,140 kNm 1,140 kNm

6.540 kNm

1,14

0

1,14

0

6.54


External forces, interactions and internal forces diagrams for example 8.2

Grid

11 / 32Basic properties of a grid

Samples of right-angled, oblique and circular grids

Bridge and ceiling grid

12 / 32Basic properties of a grid

Samples of bridge and ceiling grids

Approximate solution of regular ceiling grids

axis), with x parallel barsMM moments bending b)

forcesshear a)

:grids theof bars in the forces internal of components 3generally are There

y (

,

≡

≡≡ zVV

solution). eapproximatin neglected are(they moments torsionalc)

axis),y with parallel barsMM moments bending x (≡

link-swingshort or very joint -ballimaginary with

replaced are bars theof joints Monolithic

13 / 32Approximate solution of regular ceiling grids

Replacement of internal link by interaction in the joint of the grid

solution. eapproximatin

Approximate solution of regular ceiling grids


Simplified computational model of the grid and formulation of partial loading states

Example 8.3

0,002mI is m 6llenght ofbar for the

0,004mI is m 9llenght ofbar for the b)

below picture see - dimensions and supports loading, a)

:Problem

4

4

====

0,002mI is m 6llenght ofbar for the 4==


Problem definition of example 8.3 and basic statically determinate scheme

Example 8.3, solution

XXXX

00

2

20222121

10212111

=+⋅+⋅=+⋅+⋅

=

δδδδδδ

:equations onal)(deformati Canonicalsnacy indetermin statical of Degree Loading states Bending moments

E

E

XX

EE

E

EE

54000

2625

5250

0

9450014850035,1

543622

626

72272

004,0

132

6

72452

004,0

1

3))3

11(5,05,11(

3

3122(

004,0

1

3

35,15,1

002,0

2)

3

622

3

322(

004,0

1

10

2112

20222121

−

=+⋅+⋅

=+−=⋅⋅+⋅⋅

+⋅⋅⋅+⋅+⋅⋅+⋅⋅−=

=⋅+⋅+⋅+⋅⋅⋅==

=⋅⋅⋅+⋅⋅+⋅⋅==

δδδ

δδ

δδδ

2211

:tscoefficien naldeformatio ofn Calculatio


Solution of example 8.3, diagrams of bending moments for partial loading states

E

E

E

EE

5,5006235,1

6

5,55875,372

002,0

2148500

35,16002,0

20 −

−

=⋅⋅+⋅⋅+−=

==⋅⋅⋅

δ


equationslinear of System

XX

XX

05,5006252502625

00,5400026255250 21

=−⋅+⋅=−⋅+⋅

Loading states Bending moments

:equations of system theofSolution

kNXkNX

XX

857,5,357,7

05,5006252502625

21

21

==

=−⋅+⋅


Solution of example 8.3, diagrams of bending moments for partial loading states

Example 8.3

Beam Left beam Right beam


Resulting reactions and internal forces diagrams for the grid of Example 8.3

Symmetry of a grid

(c) (b) (a)

:acyindetermin statical of Degree

n n n

,n,n,nSs

Ss

Ss

sss

123

966

===

===

19 / 32Utilization of the symmetry of a grid

Simple, double and quadruple symmetry of the regular ceiling grid

Symmetry of a grid


Utilization of the symmetry of the ceiling grid

Example 8.4

bar each oflenght a

has (c) pict. toaccording grid

symmetricQuadruply

:Problem

.12ml =

Loading states Bending moments

picturein given is Loading

bar each oflenght a .12ml =


Solution of example 8.4, bending moment diagrams for partial loading states

(c) Picture


XSsn

01

10111

=

=+⋅:tscoefficien naldeformatio ofon Caluculati

then symmetry, utilizingWhen

:equation Cannonical

:Solution

δδ

kNX 469,61863

288

288

1863

66

)3()108812(24

)636

)81108481(

6

33)8125,472(2(2

3

63324)363

3

3332(2

1

10

11

=−=

=⋅−⋅+⋅⋅⋅

+⋅⋅+⋅++⋅⋅+⋅⋅⋅=

=⋅⋅⋅⋅+⋅⋅+⋅⋅⋅⋅=

−

:tscoefficien naldeformatio ofon Caluculati

δδ


Solution of example 8.4, bending moment diagrams for partial loading states

Example 8.4


Resulting reactions and internal forces diagrams for the grid of Example 8.4

Samples of grids

24 / 32

Ceiling grid, hypermarket Tesco, Ostrava - Třebovice

Dvojkloubový oblouk s táhlem

Samples of grids

25 / 32

Detail of ceiling grid, hypermarket Tesco, Ostrava - Třebovice

Grid

Samples of grids

26 / 32

Grid roof structure, railway station Ostrava - Svinov

Grid

Samples of grids

27 / 32

Detail of the grid roof structure, railway station Ostrava - Svinov

Grid

Samples of grids

28 / 32


Grid

Samples of grids

29 / 32


Grid

Samples of grids

30 / 32

Detail of the grid roof structure, railway station Ostrava - Svinov

Grid

Samples of grids

31 / 32

Detail of support of the grid roof structure, railway station Ostrava - Svinov

Grid

Samples of grids

32 / 32

Grid foundation structure, Centre of advanced technologies, VŠB-TU Ostrava

Grid

statics of building structures i., erasmus

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