chapter 3 normal stresses 2

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Stress Analysis Dr. Eng. Mohamed El-Tantawy -1-

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Normal Stresses

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  • Stress Analysis

    Dr. Eng. Mohamed El-Tantawy

    -1-

  • Example:A cross section of a concrete dam with3 m breadth is shown in figure, find thelength (a) so that no tensile stresseswould occurs at its base. (= 2.5 t/m3)

    a

    1m

    h=6m

    0.5m

    3m

    -2-

  • 12

    0.5 0.252 20.5 0.5 0.25

    2 3 6

    a aX

    a a aX

    Solution:

    Straining actions:

    1

    2

    2

    0.5 7 3 2.5 26.250.5 7 3 2.5 26.250.5 6 6 3 5426.25 26.250.0

    54 2 26.25 26.25 0.252 6

    4.375 6.5625 108

    x

    y

    W tW a a tH tN aM

    a aM a

    a a

    h=6a

    1m

    0.5m

    Y

    W1 W2

    W1 W2

    a

    1.5m

    1.5m

    X

    0.5m

    4m

    2m

    H

    H

    X2X1

    -3-

  • Properties of section:

    3

    3 0.5

    3 0.512y

    A a

    aI

    Equation of stress:

    2

    3

    2

    . .

    The normal stress at point (1) must be zero26.25 26.25 4.375 6.5625 108 0.50.0 .

    3 0.5 23 0.512

    52.5 78.75 634.875 0.02.81

    yx

    x y

    MMN y xA I I

    a a a a

    a a

    a a

    a m

    Y

    a

    1.5m

    1.5m

    X

    0.5m

    -

    1

    -4-

  • Stresses due to an eccentric normal force1. Normal Force acting at a point on the Y-axis :

    X

    YN

    -

    +

    N.A

    ye

    ny

    22

    .

    . 0.0

    *. 0.0

    1. 0.0

    *

    *

    *

    :

    x

    x

    xn

    x

    yn

    x

    yn

    x

    xy n

    xy n x n

    y

    x y

    MN yA I

    MN y

    N NM N e

    The equatio

    A IN eN y

    A Ie

    yA I

    Ie

    n of the neutral axi

    yA

    ie y i y

    e

    s is

    -5-

  • 2. Normal Force acting at a point on the X-axis :

    22

    .x

    .x 0.0

    *.x 0.0

    1.x 0.0

    .

    *

    x.

    :

    x

    y

    y

    yn

    y

    xn

    y

    xn

    y

    y

    y

    x n

    yx n y n

    x

    x

    N NM N e

    The equation of the neutral axis

    MNA I

    MNA I

    N eNA I

    e

    A II

    eA

    i

    e

    s

    ie i x

    Y

    X-

    +

    N

    N.A

    xenx

    -6-

  • YX

    N

    N.A-

    +

    nx

    ny

    xe

    ye

    2 2

    2 2

    2 2

    . . 0.0

    . . 0.0

    (1 . . ) 0.0

    1 . . 0 0

    :

    .

    yxx y

    yx

    x y

    y x

    x y

    y x

    x y

    y x

    x

    x y

    y

    y x

    IIi iA A

    MMN y xA I I

    N e N eN y xA A i A i

    e eN y xA i

    N NM N e M N e

    The equa

    i

    tion of the neutral ax

    e ey

    is is

    xi i

    3. Normal Force acting at an arbitrary location :

    -7-

  • 2 2

    22

    22

    1 . . 0.0

    at 0.0 .

    at 0.0 .

    y x

    x y

    xy n x n

    y

    yx n y n

    x

    e ey xi i

    ix e y i y

    e

    iy e x i x

    e

    Notes1. The position of the N.A. does not depend upon the value of N.F.2. The N.A. lies in the quarter opposite to that the N.F. acts.3. N=0.0 N.A. passes through the C.G.4. ( , ),( , ) have oppositex n y ne x e y

    signs.

    -8-

  • Example:The hollow circular section shown in Fig.is subjected to a tensile force N = 25 t and abending moment My = 2.5 t.m, find thedistribution of normal stresses in thesection.

    Solution:

    Properties of section:

    2 2 2

    4 4 4 2 2

    (7 5.5 ) 58.91167.057 5.5 1167.05 19.8

    4 58.9y

    y y

    A cmI

    I cm i cmA

    42

    4x yrA r I I

    The eccentricity of the normal force:

    2.5 0.1 1025

    yx

    Me m cm

    N

    Straining actions:25 2.5 .yN t M m t

    Y

    XN=25t

    My=2.5m.t14cm

    11cm

    -9-

  • 2 19.8 1.9810

    yn

    x

    ix cm

    e

    Neutral axis location:

    Stress distribution:

    2 2

    2

    2

    2

    2

    (1 . . )

    25 10(1 . )58.9 19.80.4244 0.2142.0.4244 0.2142 7 1.924 /0.4244 0.2142 7 1.075 /0.4244 0.2142 5.5 1.602 /0.4244 0.2142 5.5 0.7537 /

    y x

    x y

    A

    D

    B

    C

    e eN y xA i i

    x

    x

    t cm

    t cm

    t cm

    t cm

    Y

    X

    N.A

    A B C D-

    +

    A

    1.98nx

    D

    -10-

  • Core of the section

    The core of a section is an area within which, if the point of application of thenormal force lies, the normal stresses induced in the whole section will be ofthe same kind (tensile or compressive)

    The boundary of the core is the locus of the points of application of the normalforces, whose corresponding neutral axis is tangential to the external peripheryof the section.

    N1 outside core N.A1 inside the section.

    N2 on the boundary of the core N.A2tangent to the section.

    N3 inside core N.A3 outside the section.

    X

    Y

    N1

    N2N3

    N.A1N.A2N.A3

    -11-

  • Core for the rectangular section:2 2

    2 2

    2

    2 2

    2 2

    2

    1

    2

    For12 12

    2

    0.0

    *212* 6

    2*2

    12*b 6

    0

    .

    .

    For

    0

    .

    yxx y

    n n

    yx

    n

    xy

    n

    n n

    yx

    n

    xy

    n

    II h bi iA A

    hx y

    ie

    x

    i h

    N A

    he

    y h

    bx y

    i

    N A

    b be

    x

    ie

    y

    1

    2

    Y

    X

    N.A1

    N.A2

    b

    h6h

    6b

    3

    3

    *

    12

    12

    x

    y

    A b hbhI

    hbI

    -12-

  • Core for the triangular section:3 2 3 2

    2 2

    1

    2

    2 2

    2

    2 2

    2 2

    2 236 18 48 24

    For .

    3

    0.0

    *318* 6

    For .2

    3 3*3

    24*b 8*3

    18*2 12

    yxx y

    n n

    yx

    n

    xy

    n

    n n

    yx

    n

    xy

    n

    II bh h hb bi iA bh A bh

    N Ah

    x y

    ie

    x

    i h he

    y hN Ab h

    x y

    i b be

    x

    i h he

    y h

    Y

    X

    N.A1

    N.A2

    1

    23h

    23h

    2b

    2b

    6h

    12h

    8b

    3

    3

    2

    36

    48

    x

    y

    bhA

    bhI

    hbI

    -13-

  • Core for the circular section: Y

    X

    N.A

    r

    r/4

    r/4

    4 22 2

    2

    2 2

    2

    4 4

    4 4

    0.0

    For .

    x y

    n n

    yx

    n

    xy

    n

    r ri ir

    x r y

    i r re

    x r

    iy

    N A

    e

    2

    4

    4x y

    A rrI I

    Remarks1. A symmetrical section has a symmetrical core; the axis of symmetry

    being the same for both figures.

    2. For every corner on the perimeter of the section, there is acorresponding straight line on the core.

    3. For every straight line on the perimeter of the section, there is acorresponding corner in the core.

    -14-

  • ReportQ1: The section is subject tocompressive normal force N as shownin Fig. Determine the maximum valueof N so that the max. compressive andtensile stresses do not exceed than1.2 t/cm2 and 0.5 t/cm2 respectively.

    XN5cm

    10cm

    5cm

    1015512

    Q2: Draw the normal stressdistribution at sections I, II inthe shown frame. 6m

    2t/m

    4m6m

    II

    A B0.8m

    1m

    0.4m

    Sec I Sec II

    I4t

    0.4m

    -15-

  • Q3: A cross section of a concrete damwith 4 m breadth is shown in figure;find the length (a) so that no tensilestresses would occurs in its base.(= 2.4 t/m3)

    a

    1m

    h=5m

    4m

    -16-