Lesson 1. Introduction to Structural Analysis.pdf

Teoría de Estructuras y Construcciones Industriales. 2014-2015

CHAPTER 1. Introduction to Structural Analysis

Teoría de Estructuras y Construcciones Industriales. 2014-2015

Lesson 1. Introduction to Structural Analysis 1.1. Objectives of structural analysis. 1.2. Structural models. 1.3. Structural forms. 1.4. Simplified structural models. 1.5. Types of internal forces. 1.6. Types Types of external loading. 1.7. Types Types of supports and reactions. 1.8. Determinacy and freedom degrees. 1.9. Design hypothesis: first and second order theories.

Teoría de Estructuras y Construcciones Industriales. 2014-2015

1.1.. Obje 1.1 Obj ecti ctive ves s of str uctu r al anal an alys ysii s    could  S t r u c t u r e  could

be a load-bearing element or a group of load-bearing elements with functional

requirements.

The

main objective of structural analysis is to determine how a structure responds to specified loads and actions. 3

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1.1.. Obje 1.1 Obj ecti ctive ves s of str uctu ucturr al anal an alys ysii s

Th e str u ctur al des desi gn proc pr oce ess:  Conceptual

and preliminary stages (types of structure, materials and loading),

 Calculation

stage (internal forces and displacements),

 Final

stage (design drawings with written specifications of materials and  pertinent codes to be employed),  Construction

stage. -4-

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1.1.. Obje 1.1 Obj ecti ctive ves s of str uctu ucturr al anal an alys ysii s Structural model (elements, materials, supports)

Loading analysis

Calculation of displacements

Calculatin of internal forces

Calculation of stresses and strains

Checking limit states (design codes)

New structural design

NO

YES Final design

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1.2. Si 1.2. Si mpli f i ed str str uctu ucturr al mod mode el s (I de deali ali zation) Vari ar i ables ables to def def i ne: ne:

• Materials, • Structural forms, • Elements types, • Joints (connections) types, • Supports, • External loading, • Type  Type of calculation (static, dynamic, dynamic, etc.) • Type  Type of structural analysis

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1.3. 1. 3. Str u ctur al f or orms ms Clasi Clasi f i cation by str uctur al f unti on:

Houses, offices and industrial buildings.

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1.3. 1. 3. Str u ctur al f or orms ms

Industrial equipment (cranes,  pressure vessels, silos)

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1.3. 1. 3. Str u ctur al f or orms ms

Gateways, bridges and slabs.

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1.4. 1. 4. Si mpl mplii f i ed str u ctur al model model s Clas Cl ass si f i cati cati on of str uctur uctu r es:   Rigid and pin joint structures.  Rigid : same angle before and after deformation. Rigid Rigid j oin t 

Before deformation

 After deformation

2D and 3D Rigid joint structures (Frames) - 10 -

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1.4. 1. 4. Si mpl mplii f i ed str u ctur al model model s Pin Join Join t  : free rotation (hinge)

2D Pin joint structures (Trusses)

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Teoría de Estructuras y Construcciones Industriales. 2014-2015

1.4. 1. 4. Si mpl mplii f i ed str u ctur al model model s E l emen men t type t ypes s: 

1D elements (bar elements)



2D elements: - Membranes, - Plates, - Shells, - Shear walls.



3D elements.

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1.5.. Type 1.5 Types s of i n te terr n al f or orce ces s 1D el emen men ts: ts: bars bar s el emen men ts

Pin joint elements (Axial Force: Tension and Compression)

b)

2D and 3D Rigid elements (Axial and Shear Forces and Bending and Torsion Moments)

Cables (Axial Force: Tension)

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1.5.. Type 1.5 Types s of i n te terr n al f or orce ces s 2D el emen men ts: ts: M embr anes

Membranes examples - 14 -

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1.5.. Type 1.5 Types s of i n te terr n al f or orce ces s 2D el ements ment s: Pl ates ates

Plates examples - 15 -

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1.5.. Type 1.5 Types s of i n te terr n al f or orce ces s 2D el emen men ts: ts: Sh el l s

 membrana Membrane

 placa Plate

Shells examples - 16 -

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1.5.. Type 1.5 Types s of i n te terr n al f or orce ces s 2D el emen men ts: ts: Sh ear wall wal l s

Shear wall example

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1.5.. Type 1.5 Types of i n te terr n al f or orce ces s 3D el emen men ts

Dam example

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1.6. 1. 6. Type Types s of ex te terr n al l oad oadii n g Clas Cl ass si f i cati cati on (ac ( acti ti ons):  ons):  

Surface and volume loads.  Static and dynamic loads.  Permanent and variable loads.

Clas Cl ass si f i cati cati on (ac ( acti ti ons):  ons): 

snow load

own weight load



Point and distributed loads.  Thermal load.  Enforced displacements.  Fitting defects. wind load - 19 -

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1.7. Type Types s of su ppor pports ts an and d r eacti action ons s Support upport clasi clasi f i cati cation: on: 

Pin and roller (rocker) (r ocker) supports



Fixed or clamped (encastre) support



Guide



Spring supports: linear ad torsional springs.

Pin support

Roller support Spring support

Fixed support - 20 -

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1.7. Type Types s of su ppor pports ts an and d r eacti action ons s

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1.7. Type Types s of su ppor pports ts an and d r eacti action ons s 3D element section (6 degrees of freedom)  Restriction  Restriction of 6 FD • Fixed:  • Pin:   Restriction  Restriction of 3 FD

3 linear displacements (u, v, w ) 3 rotations (F (Fx, Fy, Fz) 6 Reactions (Unknowns) 3 Reactions (Unknowns)

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1.7. Type Types s of su ppor pports ts an and d r eacti action ons s 2D element section (3 degrees of freedom)

Pin support

2 linear displacements (u, v) 1 rotation (F ( F z)

Roller support

Fixed support

Guide

Schematic representation - 23 -

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1.7. Type Types s of su ppor pports ts an and d r eacti action ons s pr i ng supports uppor ts:  :  • Spri  partially limitation of linear and rotation  partially displacements. The reactions depend on the rigid constant (k = s m).

RH RV

Linear spring support

Torsional spring support

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1.8. 1. 8. D ete terr mi minacy nacy gr grad ade e: D G ( = GH ) = r - 3 

If DG < 0

Unstable structures (mechanisms) (No. unknowns < No. available equilibrium equations)



If DG = 0

Statically determinate structures (No. unknowns = No. available equilibrium equations)



If DG > 0

Statically indeterminate structures (No. unknowns > No. available equilibrium equations)

DG=1

DG=3 - 25 -

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1.8. 1. 8.a) a) Stati Statica call l y inde in dete terr mi minate nate r i gid j oin t str str uctur es. 

Rigid joint open structures: DG = DGext + DGint = r - 3  –  a  ao DGext = r  –  3  3  –  a  ao DGint = 0

r = No. reactions ao = bo  –  1  1 = hinge equations (open contours) 

Rigid joint closed structures:

DG = DGext = 3

DG = DGext = 2

DG = DGext + DGint = r + 3 cc - 3  –  (a  (ao + ac) DGext = r  –  3  3 - aa DGint = 3 cc  –  a  ac r = No. reactions, cc = No. closed contours, ao = bo - 1= hinge equations (open ( open contours) ac = bc  –  1  1 = hinge equations (closed contours)

DG = 3+3=6 DGext = 3

DG = 3+2=5 DGext = 3

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1.8. 1. 8.b) b) Stati Statica call l y in de dete terr mi minate nate pin j oin t str str uctu ucturr es.

DG = DGext + DGint = r + b  –  2n  2n DGext = r - 3 DGint = DG - DGext = b  –  2n  2n + 3 b = No. bars, r = No. reactions, n = No. joints b = 17 r=3 n = 10

b = 17 r=3 n = 10

DG=0

50kN

50kN

b = 18 r=3 n = 10

d)

b=5 r=4 n=4

DG=0

D

3m B

 A

C

Sm

DG=DGint=1

4m

4m

DG=DGext=1

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1.8.c) Ki ne 1.8.c) nematicall maticall y in de dete terr mi minate nate str uctur es: D egre gree es of f r eedo dom m F D (= GL ) = N R + BR BR Structures without sidesway: If AE = ∞  (axial deformations, e = 0), kinematic unknowns (freedom degrees) are rotation of joints (NR) only, only, because bar rotation are cero (BR = 0).

jB A

jC

B

FD = NR = 2 (j (jB , jC)

C

D

CIR

q

q VB

jB y AB  A

jC B UB

AE=∞ (axial (axial Structures with sidesway: If AE=∞ deformations, e deformations, e = 0), kinematic unknowns (freedom degrees) are both joint rotations (NR) and bar rotations (BR).

yBC

C

VC

FD = NR + BR = 3 (jB , jC , yAB)

UC

yBC = f (y (yAB)

yDC

D

 (y AB) yDC = f  ( - 28 -

Teoría de Estructuras y Construcciones Industriales. 2014-2015

1.9. D esi gn hypo h ypoth the esi s: F i r st and an d se second orde or derr th the eor orii es Requi Requirr ements ments of f i r st orde or derr the th eory: or y:

• Linear-elastic material behaviour. • Small strain and displacements Th ese condi condi ti ons per per mi t: • Equilibrium applied to undeformed structure geometry.

• Linear equation systems solution. • Superposition principle.

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1.9. D esi gn hypo h ypoth the esi s: F i r st and an d se second orde or derr th the eor orii es F i r st order order theory:  theory: 

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1.9. D esi gn hypo h ypoth the esi s: F i r st and an d se second orde or derr th the eor orii es Second or der der the th eory:  or y: 

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