statically indeterminate structurescivil.seu.edu.cn/_upload/article/files/7e/86/28414db44e... ·...

of 49/49
Statically Indeterminate Structures [email protected]

Post on 24-Apr-2020

18 views

Category:

Documents

0 download

Embed Size (px)

TRANSCRIPT

  • Statically Indeterminate Structures

    [email protected]

  • • Statically Determinate Problems(静定问题)

    • Statically Indeterminate Problems(超静定问题)

    • Degrees of Indeterminacy(超静定次数)

    • Advantages & Disadvantages of Indeterminate Structures

    (超静定结构的优缺点)

    • Redundancy & Basic Determinate System(冗余约束与基本静定系)

    • General Ideas for Analyzing Indeterminate Structures

    (超静定问题一般解法)

    Contents

    2

  • • Force Method Solution Procedure(力法)

    • Statically Indeterminate Bars(超静定拉压杆)

    • Assembly Stress(装配应力)

    • Thermo Stress(热应力)

    • Statically Indeterminate Shafts(超静定扭转轴)

    • Statically Indeterminate Beams(超静定弯曲梁)

    • Moment-Area Theorems with Indeterminate Beams(图乘法求解超静定梁)

    • Combined Indeterminate Structures (组合超静定结构)

    Contents

    3

  • • Problems can be completely solved via static equilibrium

    • Number of unknowns (forces/moments) = number of

    independent equations from static equilibration

    0 ?0 ?

    ?0

    x Ax

    y Ay

    ByA

    F FF F

    FM

    Statically Determinate Problems

    4

    A B

  • A BC

    • Problems cannot be solved from static equilibrium alone

    • Number of unknowns (forces/moments) > number of

    independent equations from static equilibration

    ?0 ?

    0 ?

    ?0?

    Ax

    x Ay

    y By

    CxA

    Cy

    FF F

    F F

    FMF

    Statically Indeterminate Problems

    5

  • • For a co-planer structure, there are at most three

    equilibrium equations for each portion of the structure. If

    there is a total of n portions and m unknown reaction

    forces from supports:

    3 Statically determinate

    3 Statically

    - 3 Degrees of indetermi

    indet

    n

    erminate

    acy

    m n

    m n

    m n

    Statically determinate Degrees of Indeterminacy = 2

    Degrees of Indeterminacy of Structures

    6

  • • Advantages:

    Advantages & Disadvantages of Indeterminate Structures

    - Redistribution of reaction forces / internal forces

    - Smaller deformation

    - Greater stiffness as a whole structure

    • Disadvantages:

    - Thermal and residual stresses due to temperature change

    and fabrication errors

    7

  • • Redundancy: unnecessary restraints without which the

    static equilibrium of a structure still holds.

    • Basic determinate system: the same structure as of a

    statically indeterminate system after replacing redundant

    restraints with extra constraining loads

    Redundancy & Basic Determinate System

    8

  • • Basic determinate structure: obtained via replacing

    redundant restraints with extra constraining loads.

    How to Analyze Statically Indeterminate Structures?

    • Equilibrium: is satisfied when the reaction forces at

    supports hold the structure at rest, as the structure is

    subjected to external loads

    • Deformation compatibility: satisfied when the various

    segments of the structure fit together without intentional

    breaks or overlaps

    • Deformation-load relationship: depends on the manner

    the material of the structure responds to the applied loads,

    which can be linear/nonlinear/viscous and elastic/inelastic;

    for our study the behavior is assumed to be linearly elastic

    9

  • • The method of consistent deformations or force method

    was originally developed by James Clerk Maxwell in 1874

    and later refined by Otto Mohr and Heinrich Műller-

    Breslau.

    Christian Otto Mohr

    (1835 – 1918)

    James Clerk Maxwell

    (1831 – 1879)

    Heinrich Műller-Breslau

    (1851 - 1925)

    Force Method (Method of Consistent Deformations)

    10

  • • Making the indeterminate structure determinate

    Force Method Solution Procedure

    • Obtaining its equations of equilibrium

    • Analyzing the deformation compatibility that must be

    satisfied at selected redundant restraints

    • Substituting the deformation-load relationship into

    deformation compatibility, resulting in complementary

    equations

    • Joining of equilibrium equations and complementary

    equations

    • Types of Problems can be addressed include (a) bar

    tension/compression; (b) shaft torsion; and (c) beam

    bending11

  • Statically Indeterminate Bars

    • For an axially loaded bar with both ends fixed, determine

    the reaction forces.

    12

    1

    2

    0BB

    B

    A

    P R LR LL

    EA EA

    LR P

    L

    LR P

    L

    • Solution:

    12

  • • Determine the reaction forces at the bar ends.

    6

    3

    3 64

    31

    6

    3

    6

    400 10

    600 10

    150 10 400 100

    600 10

    250 10

    900 10

    250 10

    577kN

    900 577 323kN

    B

    B

    i i

    i i B

    B

    B

    A

    R

    R

    F LL

    EA E R

    R

    R

    R

    • Solution:

    Sample Problem

  • • Solution:

    Sample Problem

    • Determine the reaction forces, if any, at the bar ends. E = 200 Gpa.

    43

    1

    6

    3

    3 6

    9 3

    6

    3

    6

    4.5 10

    400 10

    600 10

    150 10 400 10

    200 10 600 10

    250 10

    900 10

    250 10

    115.4kN

    900 115.4 784.6kN

    i i

    i i

    B

    B

    B

    B

    B

    A

    F LL

    EA

    R

    R

    R

    R

    R

    R

    14

  • Sample Problem

    • What is the deformation of the

    rod and tube when a force P is

    exerted on a rigid end plate as

    shown?

    • Solution:

    1 2

    1 21 2

    1 1 2 2

    1 1 2 21 2

    1 1 2 2 1 1 2 2

    1 1 2 2

    ,

    P P P

    PL P LL L

    E A E A

    E A P E A PP P

    E A E A E A E A

    PLL

    E A E A

    15

  • • Given: tension/compression rigidities E1A1 = E2A2 = EA,

    E3A3, external load F. Find: internal forces in bars.

    1. Make the truss determinate by

    replacing extension/contraction

    restraint along Bar 3 with FN3F

    A

    1 23

    l

    1 2

    1 2 3

    0

    0 ( )cos

    x N N

    y N N N

    F F F

    F F F F F

    AF

    FN3FN1 FN2

    Sample Problem

    • Solution:

    2. Static equilibrium at joint A

    16

  • 4. Displacement-force relationship

    3. Displacement compatibility at joint A

    Due to symmetry

    21 ll

    cos31 ll

    11

    cos

    NF llEA

    33

    33

    AE

    lFl N

    A

    1 23

    l

    A*

    17

  • 5. Complementary equation

    21 31 3 1 3

    1 1 3 3 3 3

    cos cos coscos

    N NN N

    F l F l EAl l F F

    E A E A E A

    6. Joint of force equilibrium and complementary equations

    1 23 3

    1 2 2

    1 2 3

    323

    1 3

    3 33 3

    2coscos

    ( )cos

    cos 1 2 cos

    N N

    N N

    N N N

    N

    N N

    FF F

    E AF F

    EAF F F F

    FFEA EAF F

    E A E A

    18

  • • Given: tension/compression rigidities E1A1, E2A2, E3A3,

    external load F. Find: internal forces in bars.

    F

    A

    1 23

    l

    Sample Problem

    • Solution:

    1. Make the truss determinate by

    replacing extension/contraction

    restraint along Bar 3 with FN3

    1 2

    1 2 3

    sin sin 1

    cos cos 0 2

    N N

    N N N

    F F F

    F F F

    2. Static equilibrium at joint A

    FN3FN1 FN2

    FA

    19

  • 3. Displacement compatibility at joint A

    1l

    2

    A*2l

    3l

    32 1

    2 1 3

    2sin sin tan

    2 cos

    ll l

    l l l

    4. Complementary equation

    1 2 31 2 3

    1 1 2 2 3 3

    2 1 3

    21 2 3

    1

    2 2 1 1 3

    1 2 2 3 3

    3

    2 cos 0

    , , cos cos

    2 coscos cos

    3

    N N N

    N

    N

    N N

    N N

    F l F l F ll l l

    E A E A E A

    F l F l F l

    E A E A E A

    F F F

    E A E A E A

    5. Joint of force equilibrium and complementary equations

    A

    1 23

    l

  • • Residual stresses exist in statically indeterminate

    structures, originated from assembling / fabrication errors

    of structural elements

    A

    1l 2l

    3l

    A

    FN1 FN3 FN2

    A

    1 23

    l

    • Force equilibrium:

    • Displacement compatibility:

    N1 N2

    N3 N1 N2( )cos

    F F

    F F F

    cos

    1

    3

    ll

    • Bar 3 is δ shorter than required

    Assembly Stress

    21

  • • Given: AB rigid; E1A1 = E2A2 = EA; Bar 1 is δ shorter than required. Find: internal forces in Bar 1 and 2 after assembly.

    • Solution:

    Bar 1 lengthened & Bar 2

    shortened:

    2. Displacement compatibility:

    1 1 22

    22

    l al l

    l a

    BA

    21

    1l

    a a

    2l

    BA

    F1 F2FA

    1 20 2AM aF aF

    1. Basic determinate structure and force equilibrium of AB:

    Sample Problem

    22

  • 3. Displacement-force relationship:

    1 21 2;

    Fl F ll l

    EA EA

    4. Complementary equation:

    1 21 22( ) 2

    Fl F ll l

    EA EA

    1 2 1

    1 2

    2

    42

    5

    2 2

    5

    EAaF aF F

    lFl F l

    EAFEA EA

    l

    5. Joint of force equilibrium and complementary equations

    23

  • • A temperature change results in a change in length or

    thermal strain. There is no stress associated with the

    thermal strain unless the elongation is restrained by

    the supports.

    thermal expansion coefficient

    T P

    PLT L

    AE

    • Treat the additional support as redundant and apply

    the principle of superposition.

    0

    0

    AE

    PLLT

    PT

    P AE T

    PE T

    A

    • The thermal deformation and the deformation from

    the redundant support must be compatible.

    Thermo Stress

    • Example (low-carbon steel)

    6 0200 GPa 12.5 10 2.5 MPaC T T 24

  • Coefficient of Thermal Expansion

    25

  • • Extension / contraction knots:

    • Extension / contraction slot:

    Addressing Thermo Stresses

    26

  • Statically Indeterminate Shafts

    • Determine (a) the reactive torques at the ends, (b) the maximum shearing stresses in each segment of the bar, and (c) the angle of rotation at the cross section where the load T0 is applied.

    • Solution:

    0

    /

    0 0

    max max

    0

    ,

    2 2,

    B AB BB A

    pB pA

    A pB B pA

    B A

    B pA A pB B pA A pB

    B B A A B B A ABC AC C

    pB pA pB pA

    T T LT L

    GI GI

    L I L IT T T T

    L I L I L I L I

    T d T d T L T L

    I I GI GI

    27

  • Statically Indeterminate Shafts

    • For the special case of dA = dB:

    0 0 0 0

    0 0

    max max

    0

    ,

    2 2,

    2 2

    A pB B pAA BB A

    B pA A pB B pA A pB

    B AA BBC AC

    p p p p

    A BB B A AC

    p p p

    L I L IL LT T T T T T

    L I L I L L I L I L

    T d T dL dT L dT

    I LI I LI

    L L TT L T L

    GI GI LGI

    28

  • Sample Problem

    • What is the deformation of the

    rod and tube when a torque T is

    exerted on a rigid end plate as

    shown?

    1 2

    1 21 2

    1 1 2 2

    1 1 2 2

    1 2

    1 1 2 2 1 1 2 2

    1 2

    1 1 2 2

    ,

    p p

    p p

    p p p p

    p p

    T T T

    T L T L

    G I G I

    G I T G I TT T

    G I G I G I G I

    TL

    G I G I

    • Solution:

    29

  • Sample Problem

    • Determine the maximum shearing stress in each of

    the two shafts with a diameter of 1.5 in.

    • Solution:

    4max

    max max

    4 12 600

    2 12 0

    2 2 4 4

    120 lb ft, 240 lb ft, 1440 lb

    2 240 12 1.5 24.35 ksi

    1.5 32

    2 12.17 ksi

    2

    A

    B

    B AF E

    p p

    A B

    B

    BD

    p

    A

    AC BD

    p

    T F

    T F

    T L T L

    GI GI

    T T F

    T d

    I

    T d

    I

    30

  • • Make the beam determinate by replacing the restraint at B with FB

    l

    q

    A B

    qA B

    FB

    qBA

    MA• Make the beam determinate by replacing the rotational restraint at A with MA

    • A propped cantilever beam

    Statically Indeterminate Beams

    31

  • q

    AB

    l

    • Known: q and l. Find: reaction forces at the supports.

    Sample Problem

    32

  • • Method 1:

    A B

    q

    FB

    3 4

    0

    03 8

    3

    8

    B

    B

    B

    w

    F l ql

    EI EIql

    F

    • Deformation compatibility requires:

    • Take the deflection restraint at B as the redundant restraint

    33

    A B

    q

    l

  • • Take the rotational restraint at A as the redundant restraint

    q

    BA

    MA

    3

    2

    0

    03 24

    8

    A

    A

    A

    M l ql

    EI EI

    qlM

    • Deformation compatibility requires:

    A B

    q

    l

    • Method 2:

    34

  • PA

    B

    C D

    aa

    • Cantilever beam AB is enhanced by another cantilever

    beam CD with the same EI. Find (1) contact force at D;

    (2) ratio of deflections at B with and without enhancement;

    (3) ratio of the maximum bending moments in AB with and

    without enhancement.

    Sample Problem

    35

  • • Contact force at DP

    A

    B

    CFD

    FD

    • Deflection compatibility

    at D requires D Dw AB w CD

    3

    3

    DD

    F aw CD

    EI

    CD

    FD

    • Solution:

    36

  • 2 32 35

    3 3 26 6 3 6

    D DD

    F a F aPa Paw AB a a a a

    EI EI EI EI

    P

    AB

    D1

    FD

    AB

    D

    FD

    P

    AB

    D

    3 335 5

    3 6 3 4

    D DD D D

    F a F aPaw AB w CD F P

    EI EI EI

    37

  • • Ratio of deflections at B with and without the enhancement

    3 3

    0

    2 33 3 3

    1

    3 3

    1

    0

    2 8

    3 3

    58 8 393 2

    6 3 6 3 24

    3939 8

    24 3 64

    B

    D DB

    B

    B

    P a Paw

    EI EI

    F a F aPa Pa Paw a a

    EI EI EI EI EI

    w Pa Pa

    EI EIw

    38

  • • Ratio of the maximum bending moments in AB with and

    without the enhancement.

    0 0max 0 2AM x Px M M Pa

    • Without:

    • With:

    1

    1max 1

    2D

    D

    Px x aM x

    Px F x a a x a

    M M Pa

    PA B

    x

    P

    ABD

    FD

    x

    39

  • Exercise

    • Find the reaction forces for the following continuous beam.

    MMqqa

    a2a

    AB D

    C

    a

    q qa

    a2a

    AB D

    C

    a

    • Solution

    40

  • Moment-Area Theorems with Indeterminate Beams

    • Reactions at supports of statically indeterminate

    beams are found by designating a redundant

    constraint and treating it as an unknown load which

    satisfies a displacement compatibility requirement.

    • The (M/EI) diagram is drawn by parts. The

    resulting tangential deviations are superposed and

    related by the compatibility requirement.

    • With reactions determined, the slope and deflection

    are found from the moment-area method.

    41

  • • For the coplanar structure shown below, the flexural

    rigidity of AB is EI, tension rigidity of CD is EA; P, L and

    a are given. Find FCD.

    P

    A BC

    D

    a

    2L

    2L

    Combined Indeterminate Structures

    42

  • • Solution

    3

    3

    3

    48

    48

    C CD

    C CC CD

    C

    w l

    P F L F aw l

    EI EA

    ALF P

    Ia AL

    D

    a

    C

    P

    A BC

    2L

    2L

    CF

    • Deformation compatibility requires:

    43

    Combined Indeterminate Structures

  • More Examples

    44

  • More Examples

  • More Examples

    46

  • More Examples

    47

    Consider

    only the

    effects of

    bending.

  • • Statically Determinate Problems(静定问题)

    • Statically Indeterminate Problems(超静定问题)

    • Degrees of Indeterminacy(超静定次数)

    • Advantages & Disadvantages of Indeterminate Structures

    (超静定结构的优缺点)

    • Redundancy & Basic Determinate System(冗余约束与基本静定系)

    • General Ideas for Analyzing Indeterminate Structures

    (超静定问题一般解法)

    Contents

    48

  • • Force Method Solution Procedure(力法)

    • Statically Indeterminate Bars(超静定拉压杆)

    • Assembly Stress(装配应力)

    • Thermo Stress(热应力)

    • Statically Indeterminate Shafts(超静定扭转轴)

    • Statically Indeterminate Beams(超静定弯曲梁)

    • Moment-Area Theorems with Indeterminate Beams(图乘法求解超静定梁)

    • Combined Indeterminate Structures (组合超静定结构)

    Contents

    49