deformations of statically determinate bar deformations of statically determinate bar structures.

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  • Department of Structural Mechanics

    Faculty of Civil Engineering, VŠB-Technical University of Ostrava

    Statics of Building Structures I., ERASMUS

    Deformations of statically

    determinate bar structures

  • 2 / 62

    Outline of Lecture

    Outline of Lecture

    • Term „deformation“

    • Virtual works principle

    • Deformation of bar – axial loading

    • Deformation of bar – transversal loading

    • Deformation of bar – torsional loading

    • Deformation of indirect bar

    • Deformation of curved bar

    • Deformation of plane truss structure

  • 3 / 62

    Deformation

    Term „deformation“

    Deformation:

    a) Global deformation of structure

    b) Local component of deformation in some point (displacement, rotation)

  • 4 / 62

    Deformation

    Why to calculate deformations?

    1. Usability of structure

    2. Solution of statically indeterminate structures

    3. Verifying the correctness of the calculation by measurement

    Calculation assumptions:

     Physical linearity (Hooke's law applies)

     Geometric linearity (small deformations theory)

    Consequence:

     Equilibrium conditions are formulated on the deformed structure –

    The First order Theory

     Apply the principle of superposition and the principle of

    proportionality

    Term „deformation“

  • 5 / 62

    Deformation

    Nonlinear mechanics:

     2nd order theory – equilibrium conditions formulated on deformed

    structure (small deformations)

     Physical nonlinearity (nonlinearly elastic or permanent deformations)

     Theory of big deformations

     Structures with unilateral links

     Cable structures

    Term „deformation“

  • 6 / 62

    Work of external forces and moments

    Virtual works principle

    Work of point force and point moment

     cos ce PPLWork (external) of a force at point:

    Work - scalar, units are Joules (J = N.m), kJ, MJ

    .MLe Work of a moment at point:

    Notice:

    It is assumption that  () has

    other cause than P (M).

    The work is positive when there are same

    directions of:

     vectors of force and displacement ,

     moment and rotation .

  • 7 / 62

    Work of continuous force and moment loading

    Work of continuous loading

    ( ) ( ) b

    a

    e xxwxqL d ( ) ( ) b

    a

    xe xxxmL d

    Assumption – magnitude of loading is constant during movement.

    Work of external forces and moments:

    Virtual works principle

  • 8 / 62

    Virtual work

    a) Deformational virtual work

    b) Force virtual work

    1) Real loading state

    2) Virtual loading state

    2a) Deformational virtual state

    2b) Force virtual state

    ce

    ce

    wPL

    wPL

    

    

    Deformational virtual work described by Lagrange

    to study equilibrium of structures

    Virtual works principle

    real loading state real deflection curve

    virtual deflection curve force virtual

    loading state

  • 9 / 62

    Work of internal forces

    Coordinate system of the bar

    Loaded bar in a space: N, My, Mz, Vz, Vy, T

    Virtual works principle

  • 10 / 62

    Work of internal forces

    Work of internal forces of bar

      

      

      l

    x

    l

    y

    l

    z

    l

    zz

    l

    yy

    l

    i TvVwVMMuNL  dˆdˆdddd

    Positive directions of internal forces

    Work of internal forces:

    Internal forces restrain deformations, they have opposite direction compared

    to picture below, that is reason for negative sign in calculation of Li.

    Virtual works principle

  • 11 / 62

    Virtual works principle

    0 ie LL

    Axiom:

    Total virtual work on solved structure (i.e. sum of works of

    external and internal forces) is equal to zero.

    A) Deformational principle of virtual works (principle of virtual displacements)

    B) Force principle of virtual works (principle of virtual forces)

    Virtual internal forces

    Real internal forces, causes deformations

    x EA

    N u dd 

    x GA

    V w

    z

    z dˆd *

    x EI

    M

    y

    y

    y dd 

    x GA

    V v

    y

    y dˆd

    * 

    x EI

    M

    z

    z z dd 

    x GI

    T

    t

    x dd 

    TVVMMN yzzy ,,,,,

    Virtual works principle

  • 12 / 62

    Deformational loading caused by temperature

    Uniform thermal loading and decomposition of linearly changing thermal loading

    across cross-section

    Force principal of virtual works

       

       

     

     

      

    

      

    

    l

    tztyt

    l

    ty

    yy

    z

    zz

    z

    zz

    y

    yy

    e x b

    t M

    h

    t MtNx

    GI

    TT

    GA

    VV

    GA

    VV

    EI

    MM

    EI

    MM

    EA

    NN L

    0

    21 0

    0

    ** dd 

    dxtdu

    h

    e tttt

    t

    z hdh

    

    

    0

    0 )(

     h

    dx td

    ttt

    ty

    hd

    

    

    1

    1

    

    Virtual works principle

  • 13 / 62

    Betti's theorem (1872)

    Enrico Betti

    (1823 - 1892)

    The work done by the 1st loading

    state through the displacements

    produced by the 2nd loading

    state is equal to the work done

    by the 2nd loading state the

    displacements produced by the

    2nd loading state.

     l

    y

    yy x

    EI

    MM PP

    0

    II,I,

    2211 d

     l

    y

    yy x

    EI

    MM MP

    0

    I,II,

    4433 d

    44332211  MPPP 

    Virtual works principle

    1st loading state

    2nd loading state

  • 14 / 62

    Maxwell’s theorem

    Zvláštní případ Bettiho věty, kdy v každém z obou zatěžovacích

    stavů působí na konstrukci jediná síla P nebo jediný moment M.

    James Clerk

    Maxwell

    (1831 - 1879)

    Displacement done by the first force in the place and direction of

    second force is equal to displacement done by second force in the

    place and direction of the first force.

    IIIIII  PP  PPP  III   III

    A special case of Betti's theorem. In each loading state acts only

    one force P or moment M.

    Virtual works principle

    1st state

    2nd state

  • 15 / 62

    Unit force method

    Unit force method

      .1eL  

      

    

      

    

    l

    ty

    yy

    z

    zz

    z

    zz

    y

    yy x

    GI

    TT

    GA

    VV

    GA

    VV

    EI

    MM

    EI

    MM

    EA

    NN

    0

    ** d

       

       

     

    

    l

    tztyt x b

    t M

    h

    t MtN

    0

    21 0 d

    Force loading

    Thermal loading

    Virtual works principle

  • 16 / 62

    Deformation of bar – axial loading

    Deformation of bar – axial loading

    Deformation of bar exposed to axial loading

     l

    e x A

    NN

    E u

    0

    d 1

    Nt

    l

    te AtxNtu 0 0

    0 d   

    Force loading

    Thermal loading

    EA

    A xNN

    EA u N

    l

    e

      

    0

    d 1

    Constant cross-section

    Variable cross-section

    Simpson’s rule ( ) ( ) 

    3 24d 42310

    0

    d fffffxxf

    l

    

  • 17 / 62

    Example 2.1

    xN

    R

    R

    ax

    ax

    .4,813

    kN13

    085,2.4,8

    

    

    Deformation of bar – axial loading

    Problem definition and solution of example 2.1

    A = 64 mm2,

    E = 2,1.108 kPa, t = 1,2.10 -5K-1

    Calculate horizontal displacement uc for force and thermal loading state

    Force loading state:

    m000685,0 10.4,6.10.1,2

    2,9

    d

    58

    0

    

    

     l

    N c

    EA

    A x

    EA

    NN u

  • 18 / 62

    Example 2.1

    Deformation of bar – axial loading

    Problem definition and solution of example 2

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