Moment of inertia: 2 channels and 2 plates

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Question statement

A built-up beam is constructed by welding two channels separated by a distance $b = \var{b}$ to two ({w} $\times$ {h}) plates.

{geogebra_applet('rgft4vzy',["b":ggb(b),'bf': ggb(bf),'w': ggb(w) ,'d': ggb(d),'t': ggb(t),'h': ggb(h)])}

Channel properties:

$\begin{align}\var{desc}\\\text{Area}&= \var{A_C} \\D &= \var{d} \\b_f &= \var{bf}\\\bar{x} &= \var{xbar}\\\bar{I}_{xx} &= \var{Ibarx_C} \\\bar{I}_{yy} &= \var{Ibary_C}\end{align}$

Give any introductory information the student needs.

			Name	Type	Generated Value

data	list	List of 32 items
C_data	dict	Dictionary with 10 entries
Desc	string	C180$\times$14.6
units	quantity	quantity(1, "mm")
Ibary_c	quantity	quantity(398000, "mm^4")
Ibarx_c	quantity	quantity(8820000, "mm^4")
A_C	quantity	quantity(1850, "mm^2")
xbar	quantity	quantity(13.699999999999999, "mm")
d	quantity	quantity(178, "mm")
bf	quantity	quantity(53.100000000000001, "mm")
t	quantity	quantity(9.300000000000001, "mm")

			Name	Type	Generated Value

Ibarx_r	quantity	quantity(247395.8333333333333333333333333333333333, "mm^4")
A_r	quantity	quantity(4750, "mm^2")
d_y	quantity	quantity(101.5, "mm")
Ix_R	quantity	quantity(49183083.33333333333333333333333333333333, "mm^4")
Ix	quantity	quantity(116006166.6666666666666666666666666666667, "mm^4")
A_t	quantity	quantity(13200, "mm^2")
kx	quantity	quantity(93.74617837328602055977760935619822025827, "mm")
Ibary_R	quantity	quantity(14289583.33333333333333333333333333333333, "mm^4")
d_x	quantity	quantity(48.699999999999999, "mm")
Iy_C	quantity	quantity(4785626.49999999981981000000000000185, "mm^4")
Iy	quantity	quantity(38150419.66666666630628666666666667036667, "mm^4")
ky	quantity	quantity(53.76042511067791076109392481667496529169, "mm")

			Name	Type	Generated Value

unit	integer	1
index	integer	18
b	quantity	quantity(70, "mm")
w	quantity	quantity(190, "mm")
h	quantity	quantity(25, "mm")

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
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    "DESC": [
      "C4$\\times$4.5",
      "C100$\\times$6.7"
    ],
    "W": [
      4.5,
      6.7
    ],
    "A": [
      1.34,
      865
    ],
    "d": [
      4,
      102
    ],
    "bf": [
      1.52,
      38.6
    ],
    "tw": [
      0.125,
      3.18
    ],
    "tf": [
      0.296,
      7.52
    ],
    "x": [
      0.473,
      12
    ],
    "Ix": [
      3.53,
      1.47
    ],
    "Iy": [
      0.265,
      0.11
    ]
  },
  {
    "DESC": [
      "C3$\\times$6",
      "C75$\\times$8.9"
    ],
    "W": [
      6,
      8.9
    ],
    "A": [
      1.76,
      1140
    ],
    "d": [
      3,
      76.2
    ],
    "bf": [
      1.6,
      40.6
    ],
    "tw": [
      0.356,
      9.04
    ],
    "tf": [
      0.273,
      6.93
    ],
    "x": [
      0.455,
      11.6
    ],
    "Ix": [
      2.07,
      0.862
    ],
    "Iy": [
      0.3,
      0.125
    ]
  },
  {
    "DESC": [
      "C3$\\times$5",
      "C75$\\times$7.4"
    ],
    "W": [
      5,
      7.4
    ],
    "A": [
      1.47,
      948
    ],
    "d": [
      3,
      76.2
    ],
    "bf": [
      1.5,
      38.1
    ],
    "tw": [
      0.258,
      6.55
    ],
    "tf": [
      0.273,
      6.93
    ],
    "x": [
      0.439,
      11.2
    ],
    "Ix": [
      1.85,
      0.77
    ],
    "Iy": [
      0.241,
      0.1
    ]
  },
  {
    "DESC": [
      "C3$\\times$4.1",
      "C75$\\times$6.1"
    ],
    "W": [
      4.1,
      6.1
    ],
    "A": [
      1.2,
      774
    ],
    "d": [
      3,
      76.2
    ],
    "bf": [
      1.41,
      35.8
    ],
    "tw": [
      0.17,
      4.32
    ],
    "tf": [
      0.273,
      6.93
    ],
    "x": [
      0.437,
      11.1
    ],
    "Ix": [
      1.65,
      0.687
    ],
    "Iy": [
      0.191,
      0.0795
    ]
  },
  {
    "DESC": [
      "C3$\\times$3.5",
      "C75$\\times$5.2"
    ],
    "W": [
      3.5,
      5.2
    ],
    "A": [
      1.09,
      703
    ],
    "d": [
      3,
      76.2
    ],
    "bf": [
      1.37,
      34.8
    ],
    "tw": [
      0.132,
      3.35
    ],
    "tf": [
      0.273,
      6.93
    ],
    "x": [
      0.443,
      11.3
    ],
    "Ix": [
      1.57,
      0.653
    ],
    "Iy": [
      0.169,
      0.0703
    ]
  }
]

Description

AISC data for channels

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `list`

[ [ "DESC": [ safe("C15$\\times$50"), safe("C380$\\times$74") ], "W": [ 50, 74 ], "A": [ 14.7, 9480 ], "d": [ 15, 381 ], "bf": [ 3.72, 94.5 ], "tw": [ 0.716, 18.2 ], "tf": [ 0.65, 16.5 ], "x": [ 0.799, 20.3 ], "Ix": [ 404, 168 ], "Iy": [ 11, 4.58 ] ], [ "DESC": [ safe("C15$\\times$40"), safe("C380$\\times$60") ], "W": [ 40, 60 ], "A": [ 11.8, 7610 ], "d": [ 15, 381 ], "bf": [ 3.52, 89.4 ], "tw": [ 0.52, 13.2 ], "tf": [ 0.65, 16.5 ], "x": [ 0.778, 19.8 ], "Ix": [ 348, 145 ], "Iy": [ 9.17, 3.82 ] ], [ "DESC": [ safe("C15$\\times$33.9"), safe("C380$\\times$50.4") ], "W": [ 33.9, 50.4 ], "A": [ 10, 6450 ], "d": [ 15, 381 ], "bf": [ 3.4, 86.4 ], "tw": [ 0.4, 10.2 ], "tf": [ 0.65, 16.5 ], "x": [ 0.788, 20 ], "Ix": [ 315, 131 ], "Iy": [ 8.07, 3.36 ] ], [ "DESC": [ safe("C12$\\times$30"), safe("C310$\\times$45") ], "W": [ 30, 45 ], "A": [ 8.81, 5680 ], "d": [ 12, 305 ], "bf": [ 3.17, 80.5 ], "tw": [ 0.51, 13 ], "tf": [ 0.501, 12.7 ], "x": [ 0.674, 17.1 ], "Ix": [ 162, 67.4 ], "Iy": [ 5.12, 2.13 ] ], [ "DESC": [ safe("C12$\\times$25"), safe("C310$\\times$37") ], "W": [ 25, 37 ], "A": [ 7.34, 4740 ], "d": [ 12, 305 ], "bf": [ 3.05, 77.5 ], "tw": [ 0.387, 9.83 ], "tf": [ 0.501, 12.7 ], "x": [ 0.674, 17.1 ], "Ix": [ 144, 59.9 ], "Iy": [ 4.45, 1.85 ] ], [ "DESC": [ safe("C12$\\times$20.7"), safe("C310$\\times$30.8") ], "W": [ 20.7, 30.8 ], "A": [ 6.08, 3920 ], "d": [ 12, 305 ], "bf": [ 2.94, 74.7 ], "tw": [ 0.282, 7.16 ], "tf": [ 0.501, 12.7 ], "x": [ 0.698, 17.7 ], "Ix": [ 129, 53.7 ], "Iy": [ 3.86, 1.61 ] ], [ "DESC": [ safe("C10$\\times$30"), safe("C250$\\times$45") ], "W": [ 30, 45 ], "A": [ 8.81, 5680 ], "d": [ 10, 254 ], "bf": [ 3.03, 77 ], "tw": [ 0.673, 17.1 ], "tf": [ 0.436, 11.1 ], "x": [ 0.649, 16.5 ], "Ix": [ 103, 42.9 ], "Iy": [ 3.93, 1.64 ] ], [ "DESC": [ safe("C10$\\times$25"), safe("C250$\\times$37") ], "W": [ 25, 37 ], "A": [ 7.35, 4740 ], "d": [ 10, 254 ], "bf": [ 2.89, 73.4 ], "tw": [ 0.526, 13.4 ], "tf": [ 0.436, 11.1 ], "x": [ 0.617, 15.7 ], "Ix": [ 91.1, 37.9 ], "Iy": [ 3.34, 1.39 ] ], [ "DESC": [ safe("C10$\\times$20"), safe("C250$\\times$30") ], "W": [ 20, 30 ], "A": [ 5.87, 3790 ], "d": [ 10, 254 ], "bf": [ 2.74, 69.6 ], "tw": [ 0.379, 9.63 ], "tf": [ 0.436, 11.1 ], "x": [ 0.606, 15.4 ], "Ix": [ 78.9, 32.8 ], "Iy": [ 2.8, 1.17 ] ], [ "DESC": [ safe("C10$\\times$15.3"), safe("C250$\\times$22.8") ], "W": [ 15.3, 22.8 ], "A": [ 4.48, 2890 ], "d": [ 10, 254 ], "bf": [ 2.6, 66 ], "tw": [ 0.24, 6.1 ], "tf": [ 0.436, 11.1 ], "x": [ 0.634, 16.1 ], "Ix": [ 67.3, 28 ], "Iy": [ 2.27, 0.945 ] ], [ "DESC": [ safe("C9$\\times$20"), safe("C230$\\times$30") ], "W": [ 20, 30 ], "A": [ 5.87, 3790 ], "d": [ 9, 229 ], "bf": [ 2.65, 67.3 ], "tw": [ 0.448, 11.4 ], "tf": [ 0.413, 10.5 ], "x": [ 0.583, 14.8 ], "Ix": [ 60.9, 25.3 ], "Iy": [ 2.41, 1 ] ], [ "DESC": [ safe("C9$\\times$15"), safe("C230$\\times$22") ], "W": [ 15, 22 ], "A": [ 4.4, 2840 ], "d": [ 9, 229 ], "bf": [ 2.49, 63.2 ], "tw": [ 0.285, 7.24 ], "tf": [ 0.413, 10.5 ], "x": [ 0.586, 14.9 ], "Ix": [ 51, 21.2 ], "Iy": [ 1.91, 0.795 ] ], [ "DESC": [ safe("C9$\\times$13.4"), safe("C230$\\times$19.9") ], "W": [ 13.4, 19.9 ], "A": [ 3.94, 2540 ], "d": [ 9, 229 ], "bf": [ 2.43, 61.7 ], "tw": [ 0.233, 5.92 ], "tf": [ 0.413, 10.5 ], "x": [ 0.601, 15.3 ], "Ix": [ 47.8, 19.9 ], "Iy": [ 1.75, 0.728 ] ], [ "DESC": [ safe("C8$\\times$18.75"), safe("C200$\\times$27.9") ], "W": [ 18.75, 27.9 ], "A": [ 5.51, 3550 ], "d": [ 8, 203 ], "bf": [ 2.53, 64.3 ], "tw": [ 0.487, 12.4 ], "tf": [ 0.39, 9.91 ], "x": [ 0.565, 14.4 ], "Ix": [ 43.9, 18.3 ], "Iy": [ 1.97, 0.82 ] ], [ "DESC": [ safe("C8$\\times$13.75"), safe("C200$\\times$20.5") ], "W": [ 13.75, 20.5 ], "A": [ 4.03, 2600 ], "d": [ 8, 203 ], "bf": [ 2.34, 59.4 ], "tw": [ 0.303, 7.7 ], "tf": [ 0.39, 9.91 ], "x": [ 0.554, 14.1 ], "Ix": [ 36.1, 15 ], "Iy": [ 1.52, 0.633 ] ], [ "DESC": [ safe("C8$\\times$11.5"), safe("C200$\\times$17.1") ], "W": [ 11.5, 17.1 ], "A": [ 3.37, 2170 ], "d": [ 8, 203 ], "bf": [ 2.26, 57.4 ], "tw": [ 0.22, 5.59 ], "tf": [ 0.39, 9.91 ], "x": [ 0.572, 14.5 ], "Ix": [ 32.5, 13.5 ], "Iy": [ 1.31, 0.545 ] ], [ "DESC": [ safe("C7$\\times$14.75"), safe("C180$\\times$22") ], "W": [ 14.75, 22 ], "A": [ 4.33, 2790 ], "d": [ 7, 178 ], "bf": [ 2.3, 58.4 ], "tw": [ 0.419, 10.6 ], "tf": [ 0.366, 9.3 ], "x": [ 0.532, 13.5 ], "Ix": [ 27.2, 11.3 ], "Iy": [ 1.37, 0.57 ] ], [ "DESC": [ safe("C7$\\times$12.25"), safe("C180$\\times$18.2") ], "W": [ 12.25, 18.2 ], "A": [ 3.59, 2320 ], "d": [ 7, 178 ], "bf": [ 2.19, 55.6 ], "tw": [ 0.314, 7.98 ], "tf": [ 0.366, 9.3 ], "x": [ 0.525, 13.3 ], "Ix": [ 24.2, 10.1 ], "Iy": [ 1.16, 0.483 ] ], [ "DESC": [ safe("C7$\\times$9.8"), safe("C180$\\times$14.6") ], "W": [ 9.8, 14.6 ], "A": [ 2.87, 1850 ], "d": [ 7, 178 ], "bf": [ 2.09, 53.1 ], "tw": [ 0.21, 5.33 ], "tf": [ 0.366, 9.3 ], "x": [ 0.541, 13.7 ], "Ix": [ 21.2, 8.82 ], "Iy": [ 0.957, 0.398 ] ], [ "DESC": [ safe("C6$\\times$13"), safe("C150$\\times$19.3") ], "W": [ 13, 19.3 ], "A": [ 3.82, 2460 ], "d": [ 6, 152 ], "bf": [ 2.16, 54.9 ], "tw": [ 0.437, 11.1 ], "tf": [ 0.343, 8.71 ], "x": [ 0.514, 13.1 ], "Ix": [ 17.3, 7.2 ], "Iy": [ 1.05, 0.437 ] ], [ "DESC": [ safe("C6$\\times$10.5"), safe("C150$\\times$15.6") ], "W": [ 10.5, 15.6 ], "A": [ 3.07, 1980 ], "d": [ 6, 152 ], "bf": [ 2.03, 51.6 ], "tw": [ 0.314, 7.98 ], "tf": [ 0.343, 8.71 ], "x": [ 0.5, 12.7 ], "Ix": [ 15.1, 6.29 ], "Iy": [ 0.86, 0.358 ] ], [ "DESC": [ safe("C6$\\times$8.2"), safe("C150$\\times$12.2") ], "W": [ 8.2, 12.2 ], "A": [ 2.39, 1540 ], "d": [ 6, 152 ], "bf": [ 1.92, 48.8 ], "tw": [ 0.2, 5.08 ], "tf": [ 0.343, 8.71 ], "x": [ 0.512, 13 ], "Ix": [ 13.1, 5.45 ], "Iy": [ 0.687, 0.286 ] ], [ "DESC": [ safe("C5$\\times$9"), safe("C130$\\times$13") ], "W": [ 9, 13 ], "A": [ 2.64, 1700 ], "d": [ 5, 127 ], "bf": [ 1.89, 48 ], "tw": [ 0.325, 8.26 ], "tf": [ 0.32, 8.13 ], "x": [ 0.478, 12.1 ], "Ix": [ 8.89, 3.7 ], "Iy": [ 0.624, 0.26 ] ], [ "DESC": [ safe("C5$\\times$6.7"), safe("C130$\\times$10.4") ], "W": [ 6.7, 10.4 ], "A": [ 1.97, 1270 ], "d": [ 5, 127 ], "bf": [ 1.75, 44.5 ], "tw": [ 0.19, 4.83 ], "tf": [ 0.32, 8.13 ], "x": [ 0.484, 12.3 ], "Ix": [ 7.48, 3.11 ], "Iy": [ 0.47, 0.196 ] ], [ "DESC": [ safe("C4$\\times$7.25"), safe("C100$\\times$10.8") ], "W": [ 7.25, 10.8 ], "A": [ 2.13, 1370 ], "d": [ 4, 102 ], "bf": [ 1.72, 43.7 ], "tw": [ 0.321, 8.15 ], "tf": [ 0.296, 7.52 ], "x": [ 0.459, 11.7 ], "Ix": [ 4.58, 1.91 ], "Iy": [ 0.425, 0.177 ] ], [ "DESC": [ safe("C4$\\times$6.25"), safe("C100$\\times$9.3") ], "W": [ 6.25, 9.3 ], "A": [ 1.84, 1190 ], "d": [ 4, 102 ], "bf": [ 1.65, 41.9 ], "tw": [ 0.247, 6.27 ], "tf": [ 0.296, 7.52 ], "x": [ 0.453, 11.5 ], "Ix": [ 4.19, 1.74 ], "Iy": [ 0.374, 0.156 ] ], [ "DESC": [ safe("C4$\\times$5.4"), safe("C100$\\times$8") ], "W": [ 5.4, 8 ], "A": [ 1.58, 1020 ], "d": [ 4, 102 ], "bf": [ 1.58, 40.1 ], "tw": [ 0.184, 4.67 ], "tf": [ 0.296, 7.52 ], "x": [ 0.457, 11.6 ], "Ix": [ 3.85, 1.6 ], "Iy": [ 0.312, 0.13 ] ], [ "DESC": [ safe("C4$\\times$4.5"), safe("C100$\\times$6.7") ], "W": [ 4.5, 6.7 ], "A": [ 1.34, 865 ], "d": [ 4, 102 ], "bf": [ 1.52, 38.6 ], "tw": [ 0.125, 3.18 ], "tf": [ 0.296, 7.52 ], "x": [ 0.473, 12 ], "Ix": [ 3.53, 1.47 ], "Iy": [ 0.265, 0.11 ] ], [ "DESC": [ safe("C3$\\times$6"), safe("C75$\\times$8.9") ], "W": [ 6, 8.9 ], "A": [ 1.76, 1140 ], "d": [ 3, 76.2 ], "bf": [ 1.6, 40.6 ], "tw": [ 0.356, 9.04 ], "tf": [ 0.273, 6.93 ], "x": [ 0.455, 11.6 ], "Ix": [ 2.07, 0.862 ], "Iy": [ 0.3, 0.125 ] ], [ "DESC": [ safe("C3$\\times$5"), safe("C75$\\times$7.4") ], "W": [ 5, 7.4 ], "A": [ 1.47, 948 ], "d": [ 3, 76.2 ], "bf": [ 1.5, 38.1 ], "tw": [ 0.258, 6.55 ], "tf": [ 0.273, 6.93 ], "x": [ 0.439, 11.2 ], "Ix": [ 1.85, 0.77 ], "Iy": [ 0.241, 0.1 ] ], [ "DESC": [ safe("C3$\\times$4.1"), safe("C75$\\times$6.1") ], "W": [ 4.1, 6.1 ], "A": [ 1.2, 774 ], "d": [ 3, 76.2 ], "bf": [ 1.41, 35.8 ], "tw": [ 0.17, 4.32 ], "tf": [ 0.273, 6.93 ], "x": [ 0.437, 11.1 ], "Ix": [ 1.65, 0.687 ], "Iy": [ 0.191, 0.0795 ] ], [ "DESC": [ safe("C3$\\times$3.5"), safe("C75$\\times$5.2") ], "W": [ 3.5, 5.2 ], "A": [ 1.09, 703 ], "d": [ 3, 76.2 ], "bf": [ 1.37, 34.8 ], "tw": [ 0.132, 3.35 ], "tf": [ 0.273, 6.93 ], "x": [ 0.443, 11.3 ], "Ix": [ 1.57, 0.653 ], "Iy": [ 0.169, 0.0703 ] ] ]

→ Used by:

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Parts

Gap-fill

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Determine the moment of inertia and radii of gyration of the composite beam with respect to the $x$- and $y$-axes.

$I_x = $Ix $\qquad k_x = $kx

$I_y = $Iy $\qquad k_y = $ky

Advice

Determine moment of inertia with respect to the x-axis

For the channel, since the x-axis passes through its centroid:

$(I_x)_C = \bar{I}_{xx} = \var{Ibarx_C} $

For the plate, the x-axis does not pass through its centroid so the parallel axis theorem must be used.

$(I_x)_R = [\bar{I} + A d^2] $ where,

$\bar{I} = \dfrac{b h^3}{12} = \dfrac{(\var{w})(\var{h})^3}{12} = \var{siground(Ibarx_R,4)}$

$A = b h = (\var{w})\,(\var{h}) = \var{A_r}$

$d = D/2 + h/2 = \dfrac{\var{d}+ \var{h}}{2} = \var{d_y} $

$(I_x)_R = \var{disp(Ix_R)}$

For the composite shape,

$A = 2 [A_C + A_R] = \var{disp(A_T)}$

$I_x = 2 [ (I_x)_C + (I_x)_R ] = \var{disp(Ix)}$, and $k_x = \sqrt{\dfrac{I_x}{A}} = \var{disp(kx)}$

Determine moment of inertia with respect to the y-axis

For the rectangle, since the y-axis passes through its centroid:

$(I_y)_R = \dfrac{h b^3}{12} = \dfrac{(\var{h}) (\var{w})^3}{12} = \var{disp(Ibary_r)}$

For the channel, the y-axis does not pass through its centroid so the parallel axis theorem must be used.

$(I_y)_C = [\bar{I} + A d^2] $ where,

$\bar{I} = \bar{I}_{yy} = \var{Ibary_C}$

$A = A_C = \var{A_C}$

$d = b/2 + \bar{x} = \dfrac{\var{b}}{2} + \var{xbar}= \var{d_x} $

$(I_y)_C = \var{disp(Iy_C)}$

For the composite shape,

$A = 2 [A_C + A_R] = \var{disp(A_T)}$

$I_y = 2 [ (I_y)_C + (I_y)_R ] = \var{disp(Iy)}$, and $k_y = \sqrt{\dfrac{I_y}{A}} = \var{disp(ky)}$

Give a worked solution to the whole question.

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This question is used in the following exams:

Chapter 10 Exercises by Michael Proudman in Engineering Statics.
42. Standard Sections by Michael Proudman in Engineering Statics.
41. Standard Sections by William Haynes in Engineering Statics.
Chapter 10 Exercises by William Haynes in Engineering Statics.

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Taxonomy: mathcentre

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Taxonomy: Context

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Checkpoint description

William Haynes 4 years, 2 months ago

William Haynes 4 years, 4 months ago

William Haynes 4 years, 7 months ago

William Haynes 4 years, 7 months ago

William Haynes 4 years, 7 months ago

William Haynes 4 years, 7 months ago

William Haynes 5 years, 9 months ago

William Haynes 5 years, 10 months ago

William Haynes 5 years, 10 months ago

William Haynes 5 years, 10 months ago

William Haynes 5 years, 10 months ago

William Haynes 5 years, 10 months ago

Error in variable testing condition

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Generated value: `list`

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Moment of inertia: 2 channels and 2 plates

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

William Haynes 4 years, 2 months ago

William Haynes 4 years, 4 months ago

William Haynes 4 years, 7 months ago

William Haynes 4 years, 7 months ago

William Haynes 4 years, 7 months ago

William Haynes 4 years, 7 months ago

William Haynes 5 years, 9 months ago

William Haynes 5 years, 10 months ago

William Haynes 5 years, 10 months ago

William Haynes 5 years, 10 months ago

William Haynes 5 years, 10 months ago

William Haynes 5 years, 10 months ago

Error in variable testing condition

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Generated value: list

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Generated value: `list`