김경종
246 lines
40 KiB
Markdown
246 lines
40 KiB
Markdown
<!-- source-page: 771 -->
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<details>
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<summary>text_image</summary>
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m₁
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f₁ᵧ
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L
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m₂
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f₂ᵧ
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</details>
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Positive nodal force conventions
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Table D-1 Single element equivalent joint forces $f_0$ for different types of loads
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<table><tr><td></td><td> $f_{1y}$ </td><td> $m_1$ </td><td>Loading case</td><td> $f_{2y}$ </td><td> $m_2$ </td></tr><tr><td>1.</td><td> $\frac{-P}{2}$ </td><td> $\frac{-PL}{8}$ </td><td><img src="images/c4eb5dff151ef3deb8c3420f3ece01736029aa82c6c7bbc6af979361ee9a83d4.jpg"/></td><td> $\frac{-P}{2}$ </td><td> $\frac{PL}{8}$ </td></tr><tr><td>2.</td><td> $\frac{-Pb^2(L+2a)}{L^3}$ </td><td> $\frac{-Pab^2}{L^2}$ </td><td><img src="images/5053486e0fca699d48bcff3679e866f6be99fce0677bb3302f3f8d376ec87ab8.jpg"/></td><td> $\frac{-Pa^2(L+2b)}{L^3}$ </td><td> $\frac{Pa^2b}{L^2}$ </td></tr><tr><td>3.</td><td> $-P$ </td><td> $-\alpha(1-\alpha)PL$ </td><td><img src="images/9130703c8cb5feed39e9b58927c54a0cb48acf7d34ed9972c9754d2ece10a859.jpg"/></td><td> $-P$ </td><td> $\alpha(1-\alpha)PL$ </td></tr><tr><td>4.</td><td> $\frac{-wL}{2}$ </td><td> $\frac{-wL^2}{12}$ </td><td><img src="images/cab456f64f38393cc0c0f22b7e8805d5d8febc54ccad17e95714368388c66df7.jpg"/></td><td> $\frac{-wL}{2}$ </td><td> $\frac{wL^2}{12}$ </td></tr><tr><td>5.</td><td> $\frac{-7wL}{20}$ </td><td> $\frac{-wL^2}{20}$ </td><td><img src="images/951bc67dd69575a18f8d7a2474f1aa1c7a8c8ecf6411134195e1b864af00d7c1.jpg"/></td><td> $\frac{-3wL}{20}$ </td><td> $\frac{wL^2}{30}$ </td></tr><tr><td>6.</td><td> $\frac{-wL}{4}$ </td><td> $\frac{-5wL^2}{96}$ </td><td><img src="images/dccf562900909d0dbea6b6b1d77aaa13fc09bae30a68ac04f9a3562fb4d473fa.jpg"/></td><td> $\frac{-wL}{4}$ </td><td> $\frac{5wL^2}{96}$ </td></tr><tr><td colspan="6">(Continued)</td></tr></table>
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<!-- source-page: 772 -->
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Table D-1 (Continued)
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<table><tr><td></td><td> $f_{1y}$ </td><td> $m_1$ </td><td>Loading case</td><td> $f_{2y}$ </td><td> $m_2$ </td></tr><tr><td>7.</td><td> $\frac{-13wL}{32}$ </td><td> $\frac{-11wL^2}{192}$ </td><td><img src="images/007c91d8c550ab1346a402841af95f49d2baea05557bc586f15d801a8c901ec3.jpg"/></td><td> $\frac{-3wL}{32}$ </td><td> $\frac{5wL^2}{192}$ </td></tr><tr><td>8.</td><td> $\frac{-wL}{3}$ </td><td> $\frac{-wL^2}{15}$ </td><td><img src="images/5f9f69663c9123d2abdf7794be5872bd81f878ba301b112b1578b46c399660b3.jpg"/></td><td> $\frac{-wL}{3}$ </td><td> $\frac{wL^2}{15}$ </td></tr><tr><td>9.</td><td> $\frac{-M(a^2+b^2-4ab-L^2)}{L^3}$ </td><td> $\frac{Mb(2a-b)}{L^2}$ </td><td><img src="images/ec68638e5fcde82b96cb54f5df6775f772652d270c2fd6214324e36b6924ea1b.jpg"/></td><td> $\frac{M(a^2+b^2-4ab-L^2)}{L^3}$ </td><td> $\frac{Ma(2b-a)}{L^2}$ </td></tr></table>
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<!-- source-page: 773 -->
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In this appendix, we will use the principle of virtual work to derive the general finite element equations for a dynamic system.
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Strictly speaking, the principle of virtual work applies to a static system, but through the introduction of D’Alembert’s principle, we will be able to use the principle of virtual work to derive the finite element equations applicable for a dynamic system.
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The principle of virtual work is stated as follows:
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If a deformable body in equilibrium is subjected to arbitrary virtual (imaginary) displacements associated with a compatible deformation of the body, the virtual work of external forces on the body is equal to the virtual strain energy of the internal stresses.
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In the principle, compatible displacements are those that satisfy the boundary conditions and ensure that no discontinuities, such as voids or overlaps, occur within the body. Figure E–1 shows the hypothetical actual displacement, a compatible (admissible) displacement, and an incompatible (inadmissible) displacement for a simply supported beam. Here dv represents the variation in the transverse displacement function v. In the finite element formulation, dv would be replaced by nodal degrees of freedom $\delta d _ { i }$ . The inadmissible displacements shown in Figure E–1(b) are the result when the support condition at the right end of the beam and the continuity of displacement and slope within the beam are not satisfied. For more details of this principle, consult structural mechanics references such as Reference [1]. Also, for additional descriptions of strain energy and work done by external forces (as applied to a bar), see Section 3.10.
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Applying the principle to a finite element, we have
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$$
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\delta U ^ {(e)} = \delta W ^ {(e)} \tag {E.1}
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$$
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where $\delta U ^ { ( e ) }$ is the virtual strain energy due to internal stresses and $\delta W ^ { ( e ) }$ is the virtual work of external forces on the element. We can express the internal virtual strain
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<!-- source-page: 774 -->
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<details>
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<summary>text_image</summary>
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v
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Actual displacement v
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x
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Virtual displacement δv
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Admissible displacement v + δv
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</details>
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(a)
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<details>
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<summary>text_image</summary>
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v
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</details>
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(b)
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Figure E–1 (a) Admissible and (b) inadmissible virtual displacement functions
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<details>
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<summary>text_image</summary>
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ρdd V
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ρdd V
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ρdd V
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dz
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dy
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dx
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</details>
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<details>
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<summary>text_image</summary>
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y, v, v̇
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x, u, u̇
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z, w, ẅ
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</details>
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Figure E–2 Effective forces acting on an element
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energy using matrix notation as
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$$
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\delta U ^ {(e)} = \iint_ {V} \delta \underline {{\varepsilon}} ^ {T} \underline {{\sigma}} d V \tag {E.2}
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$$
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From Eq. (E.2), we can observe that internal strain energy is due to internal stresses moving through virtual strains $\delta \varepsilon .$ . The external virtual work is due to nodal, surface, and body forces. In addition, application of D’Alembert’s principle yields effective or inertial forces ru€ $d V .$ ; rv€ $d V ,$ and $\rho \ddot { w } d V$ , where the double dots indicate second derivatives of the translations $u , v ,$ and w in the $x , y ,$ and z directions, respectively, with respect to time. These forces are shown in Figure E–2. According to D’Alembert’s principle, these effective forces act in directions that are opposite to the assumed positive sense of the accelerations. We can now express the external virtual work as
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$$
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\delta W ^ {(e)} = \delta \underline {{d}} ^ {T} \underline {{P}} + \iint_ {S} \delta \underline {{\psi}} _ {s} ^ {T} \underline {{T}} d S + \iiint_ {V} \delta \underline {{\psi}} ^ {T} (\underline {{X}} - \rho \ddot {\underline {{\psi}}}) d V \tag {E.3}
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$$
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where $\delta \underline { { d } }$ is the vector of virtual nodal displacements, $\delta \underline { { \psi } }$ is the vector of virtual displacement functions $\delta u , \delta v ,$ and $\delta w , \delta \underline { { \psi } } _ { s }$ is the vector of virtual displacement functions acting over the surface where surface tractions occur, $\underline { { P } }$ is the nodal load matrix, $\underline { T }$ is the surface force per unit area matrix, and $\underline { { \boldsymbol { X } } }$ is the body force per unit volume matrix.
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<!-- source-page: 775 -->
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Substituting Eqs. (E.2) and (E.3) into Eq. (E.1), we obtain
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$$
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\iint_ {V} \delta \underline {{\varepsilon}} ^ {T} \underline {{\sigma}} d V = \delta \underline {{d}} ^ {T} \underline {{P}} + \iint_ {S} \delta \underline {{\psi}} _ {s} ^ {T} \underline {{T}} d S + \iint_ {V} \delta \underline {{\psi}} ^ {T} (\underline {{X}} - \rho \ddot {\underline {{\psi}}}) d V \tag {E.4}
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$$
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As shown throughout this text, shape functions are used to relate displacement functions to nodal displacements as
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$$
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\underline {{\psi}} = \underline {{N}} \underline {{d}} \quad \underline {{\psi}} _ {s} = \underline {{N}} _ {s} \underline {{d}} \tag {E.5}
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$$
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$\underline { { N } } _ { s }$ is the shape function matrix evaluated on the surface where traction T occurs.
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Strains are related to nodal displacements as
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$$
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\underline {{\varepsilon}} = \underline {{B}} \underline {{d}} \tag {E.6}
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$$
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and stresses are related to strains by
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$$
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\underline {{\sigma}} = \underline {{D}} \underline {{\varepsilon}} \tag {E.7}
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$$
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Hence, substituting Eqs. (E.5), (E.6), and (E.7) for $\underline { { \psi } } , \underline { { \varepsilon } } ,$ and $\underline { { \sigma } }$ into Eq. (E.4), we obtain
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$$
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\iint_ {V} \delta \underline {{d}} ^ {T} \underline {{B}} ^ {T} \underline {{D}} \underline {{B}} \underline {{d}} d V = \delta \underline {{d}} ^ {T} \underline {{P}} + \iint_ {S} \delta \underline {{d}} ^ {T} \underline {{N}} _ {s} ^ {T} \underline {{T}} d S + \iint_ {V} \delta \underline {{d}} ^ {T} \underline {{N}} ^ {T} (\underline {{X}} - \rho \underline {{N}} \ddot {\underline {{d}}}) d V \tag {E.8}
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$$
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Note that the shape functions are independent of time. Because d $( \mathrm { o r } \underline { { d } } ^ { T } )$ is the matrix of nodal displacements, which is independent of spatial integration, we can simplify Eq. (E.8) by taking the $\underline { { \dot { d } } } ^ { T }$ terms from the integrals to obtain
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$$
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\delta \underline {{d}} ^ {T} \iint_ {V} \underline {{B}} ^ {T} \underline {{D}} \underline {{B}} d V \underline {{d}} = \delta \underline {{d}} ^ {T} \underline {{P}} + \delta \underline {{d}} ^ {T} \iint_ {S} \underline {{N}} _ {s} ^ {T} \underline {{T}} d S + \delta \underline {{d}} ^ {T} \iint_ {V} \underline {{N}} ^ {T} (\underline {{X}} - \rho \underline {{N}} \ddot {\underline {{d}}}) d V \tag {E.9}
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$$
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Because $\delta \underline { d } ^ { T }$ is an arbitrary virtual nodal displacement vector common to each term in Eq. (E.9), the following relationship must be true.
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$$
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\iint_ {V} \underline {{B}} ^ {T} \underline {{D}} \underline {{B}} d V \underline {{d}} = \underline {{P}} + \iint_ {S} \underline {{N}} _ {s} ^ {T} \underline {{T}} d S + \iint_ {V} \underline {{N}} ^ {T} \underline {{X}} d V - \iint_ {V} \rho \underline {{N}} ^ {T} \underline {{N}} d V \ddot {\underline {{d}}} \tag {E.10}
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$$
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We now define
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$$
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\underline {{m}} = \iint_ {V} \rho \underline {{N}} ^ {T} \underline {{N}} d V \tag {E.11}
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$$
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$$
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\underline {{k}} = \iint_ {V} \underline {{B}} ^ {T} \underline {{D}} \underline {{B}} d V \tag {E.12}
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$$
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$$
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\underline {{f}} _ {s} = \iint_ {S} \underline {{N}} _ {s} ^ {T} \underline {{T}} d S \tag {E.13}
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$$
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$$
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\underline {{f}} _ {b} = \iint_ {V} \underline {{N}} ^ {T} \underline {{X}} d V \tag {E.14}
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$$
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<!-- source-page: 776 -->
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Using Eqs. (E.11)–(E.14) in Eq. (E.10) and moving the last term of Eq. (E.10) to the left side, we obtain
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$$
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\underline {{m}} \ddot {\underline {{d}}} + \underline {{k}} \underline {{d}} = \underline {{P}} + \underline {{f}} _ {s} + \underline {{f}} _ {b} \tag {E.15}
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$$
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The matrix m in Eq. (E.11) is the element consistent-mass matrix [2], k in Eq. (E.12) is the element stiffness matrix, $\underline { { f } } _ { s }$ in Eq. (E.13) is the matrix of element equivalent nodal loads due to surface forces, and $\underline f b$ in Eq. (E.14) is the matrix of element equivalent nodal loads due to body forces.
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Specific applications of Eq. (E.15) are given in Chapter 16 for bars and beams subjected to dynamic (time-dependent) forces. For static problems, we set $\ddot { \underline { d } }$ equal to zero in Eq. (E.15) to obtain
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$$
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\underline {{k}} \underline {{d}} = \underline {{P}} + \underline {{f}} _ {s} + \underline {{f}} _ {b} \tag {E.16}
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$$
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Chapters 3–9, 11 and 12 illustrate the use of Eq. (E.16) applied to bars, trusses, beams, frames, and to plane stress, axisymmetric stress, three-dimensional stress, and plate-bending problems.
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# References
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[1] Oden, J. T., and Ripperger, E. A., Mechanics of Elastic Structures, 2nd ed., McGraw-Hill, New York, 1981.
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[2] Archer, J. S., ‘‘Consistent Matrix Formulations for Structural Analysis Using Finite Element Techniques,’’ Journal of the American Institute of Aeronautics and Astronautics, Vol. 3, No. 10, pp. 1910–1918, 1965.
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<!-- source-page: 777 -->
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# Properties of Structural Steel and Aluminum Shapes
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<details>
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<summary>text_image</summary>
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Y
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← t_w
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d X —— X
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Y
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← b_f
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t_f
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</details>
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Wide Flange Shapes (W Shapes)\*: Theoretical Dimensions and Properties for Designing
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<table><tr><td rowspan="2">Section Number</td><td rowspan="2">Weight per Foot (lb)</td><td rowspan="2">Area of Section A (in.2)</td><td rowspan="2">Depth of Section d (in.)</td><td colspan="2">Flange</td><td rowspan="2">Web Thickness $t_w$ (in.)</td><td colspan="3">Axis X-X</td><td colspan="3">Axis Y-Y</td></tr><tr><td>Width $b_f$ (in.)</td><td>Thick-ness $t_f$ (in.)</td><td> $l_x$ (in.4)</td><td> $S_x$ (in.3)</td><td> $r_x$ (in.)</td><td> $l_y$ (in.4)</td><td> $S_y$ (in.3)</td><td> $r_y$ (in.)</td></tr><tr><td rowspan="5">W36 ×</td><td>300</td><td>88.3</td><td>36.74</td><td>16.655</td><td>1.680</td><td>0.945</td><td>20,300</td><td>1,110</td><td>15.2</td><td>1,300</td><td>156</td><td>3.83</td></tr><tr><td>280</td><td>82.4</td><td>36.52</td><td>16.595</td><td>1.570</td><td>0.885</td><td>18,900</td><td>1,030</td><td>15.1</td><td>1,200</td><td>144</td><td>3.81</td></tr><tr><td>260</td><td>76.5</td><td>36.26</td><td>16.550</td><td>1.440</td><td>0.840</td><td>17,300</td><td>953</td><td>15.0</td><td>1,090</td><td>132</td><td>3.78</td></tr><tr><td>245</td><td>72.1</td><td>36.08</td><td>16.510</td><td>1.350</td><td>0.800</td><td>16,100</td><td>895</td><td>15.0</td><td>1,010</td><td>123</td><td>3.75</td></tr><tr><td>230</td><td>67.6</td><td>35.90</td><td>16.470</td><td>1.260</td><td>0.760</td><td>15,000</td><td>837</td><td>14.9</td><td>940</td><td>114</td><td>3.73</td></tr><tr><td rowspan="7">W36 ×</td><td>210</td><td>61.8</td><td>36.69</td><td>12.180</td><td>1.360</td><td>0.830</td><td>13,200</td><td>719</td><td>14.6</td><td>411</td><td>67.5</td><td>2.58</td></tr><tr><td>194</td><td>57.0</td><td>36.49</td><td>12.115</td><td>1.260</td><td>0.765</td><td>12,100</td><td>664</td><td>14.6</td><td>375</td><td>61.9</td><td>2.56</td></tr><tr><td>182</td><td>53.6</td><td>36.33</td><td>12.075</td><td>1.180</td><td>0.725</td><td>11,300</td><td>623</td><td>14.5</td><td>347</td><td>57.6</td><td>2.55</td></tr><tr><td>170</td><td>50.0</td><td>36.17</td><td>12.030</td><td>1.100</td><td>0.680</td><td>10,500</td><td>580</td><td>14.5</td><td>320</td><td>53.2</td><td>2.53</td></tr><tr><td>160</td><td>47.0</td><td>36.01</td><td>12.000</td><td>1.020</td><td>0.650</td><td>9,750</td><td>542</td><td>14.4</td><td>295</td><td>49.1</td><td>2.50</td></tr><tr><td>150</td><td>44.2</td><td>35.85</td><td>11.975</td><td>0.940</td><td>0.625</td><td>9,040</td><td>504</td><td>14.3</td><td>270</td><td>45.1</td><td>2.47</td></tr><tr><td>135</td><td>39.7</td><td>35.55</td><td>11.950</td><td>0.790</td><td>0.600</td><td>7,800</td><td>439</td><td>14.0</td><td>225</td><td>37.7</td><td>2.38</td></tr><tr><td colspan="13">(Continued)</td></tr></table>
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\*A W section is designated by the letter W followed by the nominal depth in inches and the weight in pounds per foot.
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<!-- source-page: 778 -->
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Wide Flange Shapes (W Shapes)\*: Theoretical Dimensions and Properties for Designing (Continued )
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<table><tr><td rowspan="2">Section Number</td><td rowspan="2">Weight per Foot (lb)</td><td rowspan="2">Area of Section A (in.2)</td><td rowspan="2">Depth of Section d (in.)</td><td colspan="2">Flange</td><td rowspan="2">Web Thickness ${t}_{w}$ (in.)</td><td colspan="3">Axis X-X</td><td colspan="3">Axis Y-Y</td></tr><tr><td>Width ${b}_{f}$ (in.)</td><td>Thick-ness ${t}_{f}$ (in.)</td><td> ${l}_{x}$ (in.4)</td><td> ${S}_{x}$ (in.3)</td><td> ${r}_{x}$ (in.)</td><td> ${l}_{y}$ (in.4)</td><td> ${S}_{y}$ (in.3)</td><td> ${r}_{y}$ (in.)</td></tr><tr><td rowspan="3">W33 ×</td><td>241</td><td>70.9</td><td>34.18</td><td>15.860</td><td>1.400</td><td>0.830</td><td>14,200</td><td>829</td><td>14.1</td><td>932</td><td>118</td><td>3.63</td></tr><tr><td>221</td><td>65.0</td><td>33.93</td><td>15.805</td><td>1.275</td><td>0.775</td><td>12,800</td><td>757</td><td>14.1</td><td>840</td><td>106</td><td>3.59</td></tr><tr><td>201</td><td>59.1</td><td>33.68</td><td>15.745</td><td>1.150</td><td>0.715</td><td>11,500</td><td>684</td><td>14.0</td><td>749</td><td>95.2</td><td>3.56</td></tr><tr><td rowspan="4">W33 ×</td><td>152</td><td>44.7</td><td>33.49</td><td>11.565</td><td>1.055</td><td>0.635</td><td>8,160</td><td>487</td><td>13.5</td><td>273</td><td>47.2</td><td>2.47</td></tr><tr><td>141</td><td>41.6</td><td>33.30</td><td>11.535</td><td>0.960</td><td>0.605</td><td>7,450</td><td>448</td><td>13.4</td><td>246</td><td>42.7</td><td>2.43</td></tr><tr><td>130</td><td>38.3</td><td>33.09</td><td>11.510</td><td>0.855</td><td>0.580</td><td>6,710</td><td>406</td><td>13.2</td><td>218</td><td>37.9</td><td>2.39</td></tr><tr><td>118</td><td>34.7</td><td>32.86</td><td>11.480</td><td>0.740</td><td>0.550</td><td>5,900</td><td>359</td><td>13.0</td><td>187</td><td>32.6</td><td>2.32</td></tr><tr><td rowspan="3">W30 ×</td><td>211</td><td>62.0</td><td>30.94</td><td>15.105</td><td>1.315</td><td>0.775</td><td>10,300</td><td>663</td><td>12.9</td><td>757</td><td>100</td><td>3.49</td></tr><tr><td>191</td><td>56.1</td><td>30.68</td><td>15.040</td><td>1.185</td><td>0.710</td><td>9,170</td><td>598</td><td>12.8</td><td>673</td><td>89.5</td><td>3.46</td></tr><tr><td>173</td><td>50.8</td><td>30.44</td><td>14.985</td><td>1.065</td><td>0.655</td><td>8,200</td><td>539</td><td>12.7</td><td>598</td><td>79.8</td><td>3.43</td></tr><tr><td rowspan="5">W30 ×</td><td>132</td><td>38.9</td><td>30.31</td><td>10.545</td><td>1.000</td><td>0.615</td><td>5,770</td><td>380</td><td>12.2</td><td>196</td><td>37.2</td><td>2.25</td></tr><tr><td>124</td><td>36.5</td><td>30.17</td><td>10.515</td><td>0.930</td><td>0.585</td><td>5,360</td><td>355</td><td>12.1</td><td>181</td><td>34.4</td><td>2.23</td></tr><tr><td>116</td><td>34.2</td><td>30.01</td><td>10.495</td><td>0.850</td><td>0.565</td><td>4,930</td><td>329</td><td>12.0</td><td>164</td><td>31.3</td><td>2.19</td></tr><tr><td>108</td><td>31.7</td><td>29.83</td><td>10.475</td><td>0.760</td><td>0.545</td><td>4,470</td><td>299</td><td>11.9</td><td>146</td><td>27.9</td><td>2.15</td></tr><tr><td>99</td><td>29.1</td><td>29.65</td><td>10.450</td><td>0.670</td><td>0.520</td><td>3,990</td><td>269</td><td>11.7</td><td>128</td><td>24.5</td><td>2.10</td></tr><tr><td rowspan="3">W27 ×</td><td>178</td><td>52.3</td><td>27.81</td><td>14.085</td><td>1.190</td><td>0.725</td><td>6,990</td><td>502</td><td>11.6</td><td>555</td><td>78.8</td><td>3.26</td></tr><tr><td>161</td><td>47.4</td><td>27.59</td><td>14.020</td><td>1.080</td><td>0.660</td><td>6,280</td><td>455</td><td>11.5</td><td>497</td><td>70.9</td><td>3.24</td></tr><tr><td>146</td><td>42.9</td><td>27.38</td><td>13.965</td><td>0.975</td><td>0.605</td><td>5,630</td><td>411</td><td>11.4</td><td>443</td><td>63.5</td><td>3.21</td></tr><tr><td rowspan="4">W27 ×</td><td>114</td><td>33.5</td><td>27.29</td><td>10.070</td><td>0.930</td><td>0.570</td><td>4,090</td><td>299</td><td>11.0</td><td>159</td><td>31.5</td><td>2.18</td></tr><tr><td>102</td><td>30.0</td><td>27.09</td><td>10.015</td><td>0.830</td><td>0.515</td><td>3,620</td><td>267</td><td>11.0</td><td>139</td><td>27.8</td><td>2.15</td></tr><tr><td>94</td><td>27.7</td><td>26.92</td><td>9.990</td><td>0.745</td><td>0.490</td><td>3,270</td><td>243</td><td>10.9</td><td>124</td><td>24.8</td><td>2.12</td></tr><tr><td>84</td><td>24.8</td><td>26.71</td><td>9.960</td><td>0.640</td><td>0.460</td><td>2,850</td><td>213</td><td>10.7</td><td>106</td><td>21.2</td><td>2.07</td></tr><tr><td rowspan="5">W24 ×</td><td>162</td><td>47.7</td><td>25.00</td><td>12.955</td><td>1.220</td><td>0.705</td><td>5,170</td><td>414</td><td>10.4</td><td>443</td><td>68.4</td><td>3.05</td></tr><tr><td>146</td><td>43.0</td><td>24.74</td><td>12.900</td><td>1.090</td><td>0.650</td><td>4,580</td><td>371</td><td>10.3</td><td>391</td><td>60.5</td><td>3.01</td></tr><tr><td>131</td><td>38.5</td><td>24.48</td><td>12.855</td><td>0.960</td><td>0.605</td><td>4,020</td><td>329</td><td>10.2</td><td>340</td><td>53.0</td><td>2.97</td></tr><tr><td>117</td><td>34.4</td><td>24.26</td><td>12.800</td><td>0.850</td><td>0.550</td><td>3,540</td><td>291</td><td>10.1</td><td>297</td><td>46.5</td><td>2.94</td></tr><tr><td>104</td><td>30.6</td><td>24.06</td><td>12.750</td><td>0.750</td><td>0.500</td><td>3,100</td><td>258</td><td>10.1</td><td>259</td><td>40.7</td><td>2.91</td></tr><tr><td rowspan="4">W24 ×</td><td>94</td><td>27.7</td><td>24.31</td><td>9.065</td><td>0.875</td><td>0.515</td><td>2,700</td><td>222</td><td>9.87</td><td>109</td><td>24.0</td><td>1.98</td></tr><tr><td>84</td><td>24.7</td><td>24.10</td><td>9.020</td><td>0.770</td><td>0.470</td><td>2,370</td><td>196</td><td>9.79</td><td>94.4</td><td>20.9</td><td>1.95</td></tr><tr><td>76</td><td>22.4</td><td>23.92</td><td>8.990</td><td>0.680</td><td>0.440</td><td>2,100</td><td>176</td><td>9.69</td><td>82.5</td><td>18.4</td><td>1.92</td></tr><tr><td>68</td><td>20.1</td><td>23.73</td><td>8.965</td><td>0.585</td><td>0.415</td><td>1,830</td><td>154</td><td>9.55</td><td>70.4</td><td>15.7</td><td>1.87</td></tr><tr><td rowspan="2">W24 ×</td><td>62</td><td>18.2</td><td>23.74</td><td>7.040</td><td>0.590</td><td>0.430</td><td>1,550</td><td>131</td><td>9.23</td><td>34.5</td><td>9.80</td><td>1.38</td></tr><tr><td>55</td><td>16.2</td><td>23.57</td><td>7.005</td><td>0.505</td><td>0.395</td><td>1,350</td><td>114</td><td>9.11</td><td>29.1</td><td>8.30</td><td>1.34</td></tr><tr><td rowspan="5">W21 ×</td><td>147</td><td>43.2</td><td>22.06</td><td>12.510</td><td>1.150</td><td>0.720</td><td>3,630</td><td>329</td><td>9.17</td><td>376</td><td>60.1</td><td>2.95</td></tr><tr><td>132</td><td>38.8</td><td>21.83</td><td>12.440</td><td>1.035</td><td>0.650</td><td>3,220</td><td>295</td><td>9.12</td><td>333</td><td>53.5</td><td>2.93</td></tr><tr><td>122</td><td>35.9</td><td>21.68</td><td>12.390</td><td>0.960</td><td>0.600</td><td>2,960</td><td>273</td><td>9.09</td><td>305</td><td>49.2</td><td>2.92</td></tr><tr><td>111</td><td>32.7</td><td>21.51</td><td>12.340</td><td>0.875</td><td>0.550</td><td>2,670</td><td>249</td><td>9.05</td><td>274</td><td>44.5</td><td>2.90</td></tr><tr><td>101</td><td>29.8</td><td>21.36</td><td>12.290</td><td>0.800</td><td>0.500</td><td>2,420</td><td>277</td><td>9.02</td><td>248</td><td>40.3</td><td>2.89</td></tr><tr><td rowspan="5">W21 ×</td><td>93</td><td>27.3</td><td>21.62</td><td>8.420</td><td>0.930</td><td>0.580</td><td>2,070</td><td>192</td><td>8.70</td><td>92.9</td><td>22.1</td><td>1.84</td></tr><tr><td>83</td><td>24.3</td><td>21.43</td><td>8.355</td><td>0.835</td><td>0.515</td><td>1,830</td><td>171</td><td>8.67</td><td>81.4</td><td>19.5</td><td>1.83</td></tr><tr><td>73</td><td>21.5</td><td>21.24</td><td>8.295</td><td>0.740</td><td>0.455</td><td>1,600</td><td>151</td><td>8.64</td><td>70.6</td><td>17.0</td><td>1.81</td></tr><tr><td>68</td><td>20.0</td><td>21.13</td><td>8.270</td><td>0.685</td><td>0.430</td><td>1,480</td><td>140</td><td>8.60</td><td>64.7</td><td>15.7</td><td>1.80</td></tr><tr><td>62</td><td>18.3</td><td>20.99</td><td>8.240</td><td>0.615</td><td>0.400</td><td>1,330</td><td>127</td><td>8.54</td><td>57.5</td><td>13.9</td><td>1.77</td></tr><tr><td rowspan="3">W21 ×</td><td>57</td><td>16.7</td><td>21.06</td><td>6.555</td><td>0.650</td><td>0.405</td><td>1,170</td><td>111</td><td>8.36</td><td>30.6</td><td>9.35</td><td>1.35</td></tr><tr><td>50</td><td>14.7</td><td>20.83</td><td>6.530</td><td>0.535</td><td>0.380</td><td>984</td><td>94.5</td><td>8.18</td><td>24.9</td><td>7.64</td><td>1.30</td></tr><tr><td>44</td><td>13.0</td><td>20.66</td><td>6.500</td><td>0.450</td><td>0.350</td><td>843</td><td>81.6</td><td>8.06</td><td>20.7</td><td>6.36</td><td>1.26</td></tr></table>
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All printed with permission of American Institute of Steel Construction
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Wide Flange Shapes (W Shapes)\*: Theoretical Dimensions and Properties for Designing (Continued )
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<table><tr><td rowspan="2">Section Number</td><td rowspan="2">Weight per Foot (lb)</td><td rowspan="2">Area of Section A (in.2)</td><td rowspan="2">Depth of Section d (in.)</td><td colspan="2">Flange</td><td rowspan="2">Web Thickness $t_w$ (in.)</td><td colspan="3">Axis X-X</td><td colspan="3">Axis Y-Y</td></tr><tr><td>Width $b_f$ (in.)</td><td>Thick-ness $t_f$ (in.)</td><td> $l_x$ (in.4)</td><td> $S_x$ (in.3)</td><td> $r_x$ (in.)</td><td> $l_y$ (in.4)</td><td> $S_y$ (in.3)</td><td> $r_y$ (in.)</td></tr><tr><td rowspan="5">W18 ×</td><td>119</td><td>35.1</td><td>18.97</td><td>11.265</td><td>1.060</td><td>0.655</td><td>2,190</td><td>231</td><td>7.90</td><td>253</td><td>44.9</td><td>2.69</td></tr><tr><td>106</td><td>31.1</td><td>18.73</td><td>11.200</td><td>0.940</td><td>0.590</td><td>1,910</td><td>204</td><td>7.84</td><td>220</td><td>39.4</td><td>2.66</td></tr><tr><td>97</td><td>28.5</td><td>18.59</td><td>11.145</td><td>0.870</td><td>0.535</td><td>1,750</td><td>188</td><td>7.82</td><td>201</td><td>36.1</td><td>2.65</td></tr><tr><td>86</td><td>25.3</td><td>18.39</td><td>11.090</td><td>0.770</td><td>0.480</td><td>1,530</td><td>166</td><td>7.77</td><td>175</td><td>31.6</td><td>2.63</td></tr><tr><td>76</td><td>22.3</td><td>18.21</td><td>11.035</td><td>0.680</td><td>0.425</td><td>1,330</td><td>146</td><td>7.73</td><td>152</td><td>27.6</td><td>2.61</td></tr><tr><td rowspan="5">W18 ×</td><td>71</td><td>20.8</td><td>18.47</td><td>7.635</td><td>0.810</td><td>0.495</td><td>1,170</td><td>127</td><td>7.50</td><td>60.3</td><td>15.8</td><td>1.70</td></tr><tr><td>65</td><td>19.1</td><td>18.35</td><td>7.590</td><td>0.750</td><td>0.450</td><td>1,070</td><td>117</td><td>7.49</td><td>54.8</td><td>14.4</td><td>1.69</td></tr><tr><td>60</td><td>17.6</td><td>18.24</td><td>7.555</td><td>0.695</td><td>0.415</td><td>984</td><td>108</td><td>7.47</td><td>50.1</td><td>13.3</td><td>1.69</td></tr><tr><td>55</td><td>16.2</td><td>18.11</td><td>7.530</td><td>0.630</td><td>0.390</td><td>890</td><td>98.3</td><td>7.41</td><td>44.9</td><td>11.9</td><td>1.67</td></tr><tr><td>50</td><td>14.7</td><td>17.99</td><td>7.495</td><td>0.570</td><td>0.355</td><td>800</td><td>88.9</td><td>7.38</td><td>40.1</td><td>10.7</td><td>1.65</td></tr><tr><td rowspan="3">W18 ×</td><td>46</td><td>13.5</td><td>18.06</td><td>6.060</td><td>0.605</td><td>0.360</td><td>712</td><td>78.8</td><td>7.25</td><td>22.5</td><td>7.43</td><td>1.29</td></tr><tr><td>40</td><td>11.8</td><td>17.90</td><td>6.015</td><td>0.525</td><td>0.315</td><td>612</td><td>68.4</td><td>7.21</td><td>19.1</td><td>6.35</td><td>1.27</td></tr><tr><td>35</td><td>10.3</td><td>17.70</td><td>6.000</td><td>0.425</td><td>0.300</td><td>510</td><td>57.6</td><td>7.04</td><td>15.3</td><td>5.12</td><td>1.22</td></tr><tr><td rowspan="4">W16 ×</td><td>100</td><td>29.4</td><td>16.97</td><td>10.425</td><td>0.985</td><td>0.585</td><td>1,490</td><td>175</td><td>7.10</td><td>186</td><td>35.7</td><td>2.52</td></tr><tr><td>89</td><td>26.2</td><td>16.75</td><td>10.365</td><td>0.875</td><td>0.525</td><td>1,300</td><td>155</td><td>7.05</td><td>163</td><td>31.4</td><td>2.49</td></tr><tr><td>77</td><td>22.6</td><td>16.52</td><td>10.295</td><td>0.760</td><td>0.455</td><td>1,110</td><td>134</td><td>7.00</td><td>138</td><td>26.9</td><td>2.47</td></tr><tr><td>67</td><td>19.7</td><td>16.33</td><td>10.235</td><td>0.665</td><td>0.395</td><td>954</td><td>117</td><td>6.96</td><td>119</td><td>23.2</td><td>2.46</td></tr><tr><td rowspan="5">W16 ×</td><td>57</td><td>16.8</td><td>16.43</td><td>7.120</td><td>0.715</td><td>0.430</td><td>758</td><td>92.2</td><td>6.72</td><td>43.1</td><td>12.1</td><td>1.60</td></tr><tr><td>50</td><td>14.7</td><td>16.26</td><td>7.070</td><td>0.630</td><td>0.380</td><td>659</td><td>81.0</td><td>6.68</td><td>37.2</td><td>10.5</td><td>1.59</td></tr><tr><td>45</td><td>13.3</td><td>16.13</td><td>7.035</td><td>0.565</td><td>0.345</td><td>586</td><td>72.7</td><td>6.65</td><td>32.8</td><td>9.34</td><td>1.57</td></tr><tr><td>40</td><td>11.8</td><td>16.01</td><td>6.995</td><td>0.505</td><td>0.305</td><td>518</td><td>64.7</td><td>6.63</td><td>28.9</td><td>8.25</td><td>1.57</td></tr><tr><td>36</td><td>10.6</td><td>15.86</td><td>6.985</td><td>0.430</td><td>0.295</td><td>448</td><td>56.5</td><td>6.51</td><td>24.5</td><td>7.00</td><td>1.52</td></tr><tr><td rowspan="2">W16 ×</td><td>31</td><td>9.12</td><td>15.88</td><td>5.525</td><td>0.440</td><td>0.275</td><td>375</td><td>47.2</td><td>6.41</td><td>12.4</td><td>4.49</td><td>1.17</td></tr><tr><td>26</td><td>7.68</td><td>15.69</td><td>5.500</td><td>0.345</td><td>0.250</td><td>301</td><td>38.4</td><td>6.26</td><td>9.59</td><td>3.49</td><td>1.12</td></tr><tr><td rowspan="6">W14 ×</td><td>730</td><td>215</td><td>22.42</td><td>17.890</td><td>4.910</td><td>3.070</td><td>14,300</td><td>1,280</td><td>8.17</td><td>4,720</td><td>527</td><td>4.69</td></tr><tr><td>665</td><td>196</td><td>21.64</td><td>17.650</td><td>4.520</td><td>2.830</td><td>12,400</td><td>1,150</td><td>7.98</td><td>4,170</td><td>472</td><td>4.62</td></tr><tr><td>605</td><td>178</td><td>20.92</td><td>17.415</td><td>4.160</td><td>2.595</td><td>10,800</td><td>1,040</td><td>7.80</td><td>3,680</td><td>423</td><td>4.55</td></tr><tr><td>550</td><td>162</td><td>20.24</td><td>17.200</td><td>3.820</td><td>2.380</td><td>9,430</td><td>931</td><td>7.63</td><td>3,250</td><td>378</td><td>4.49</td></tr><tr><td>500</td><td>147</td><td>19.60</td><td>17.010</td><td>3.500</td><td>2.190</td><td>8,210</td><td>838</td><td>7.48</td><td>2,880</td><td>339</td><td>4.43</td></tr><tr><td>455</td><td>134</td><td>19.02</td><td>16.835</td><td>3.210</td><td>2.015</td><td>7,190</td><td>756</td><td>7.33</td><td>2,560</td><td>304</td><td>4.38</td></tr><tr><td rowspan="13">W14 ×</td><td>426</td><td>125</td><td>18.67</td><td>16.695</td><td>3.035</td><td>1.875</td><td>6,600</td><td>707</td><td>7.26</td><td>2,360</td><td>283</td><td>4.34</td></tr><tr><td>398</td><td>117</td><td>18.29</td><td>16.590</td><td>2.845</td><td>1.770</td><td>6,000</td><td>656</td><td>7.16</td><td>2,170</td><td>262</td><td>4.31</td></tr><tr><td>370</td><td>109</td><td>17.92</td><td>16.475</td><td>2.660</td><td>1.655</td><td>5,440</td><td>607</td><td>7.07</td><td>1,990</td><td>241</td><td>4.27</td></tr><tr><td>342</td><td>101</td><td>17.54</td><td>16.360</td><td>2.470</td><td>1.540</td><td>4,900</td><td>559</td><td>6.98</td><td>1,810</td><td>221</td><td>4.24</td></tr><tr><td>311</td><td>91.4</td><td>17.12</td><td>16.230</td><td>2.260</td><td>1.410</td><td>4,330</td><td>506</td><td>6.88</td><td>1,610</td><td>199</td><td>4.20</td></tr><tr><td>283</td><td>83.3</td><td>16.74</td><td>16.110</td><td>2.070</td><td>1.290</td><td>3,840</td><td>459</td><td>6.79</td><td>1,440</td><td>179</td><td>4.17</td></tr><tr><td>257</td><td>75.6</td><td>16.38</td><td>15.995</td><td>1.890</td><td>1.175</td><td>3,400</td><td>415</td><td>6.71</td><td>1,290</td><td>161</td><td>4.13</td></tr><tr><td>233</td><td>68.5</td><td>16.04</td><td>15.890</td><td>1.720</td><td>1.070</td><td>3,010</td><td>375</td><td>6.63</td><td>1,150</td><td>145</td><td>4.10</td></tr><tr><td>211</td><td>62.0</td><td>15.72</td><td>15.800</td><td>1.560</td><td>0.980</td><td>2,660</td><td>338</td><td>6.55</td><td>1,030</td><td>130</td><td>4.07</td></tr><tr><td>193</td><td>56.8</td><td>15.48</td><td>15.710</td><td>1.440</td><td>0.890</td><td>2,400</td><td>310</td><td>6.50</td><td>931</td><td>119</td><td>4.05</td></tr><tr><td>176</td><td>51.8</td><td>15.22</td><td>15.650</td><td>1.310</td><td>0.830</td><td>2,140</td><td>281</td><td>6.43</td><td>838</td><td>107</td><td>4.02</td></tr><tr><td>159</td><td>46.7</td><td>14.98</td><td>15.565</td><td>1.190</td><td>0.745</td><td>1,900</td><td>254</td><td>6.38</td><td>748</td><td>96.2</td><td>4.00</td></tr><tr><td>145</td><td>42.7</td><td>14.78</td><td>15.500</td><td>1.090</td><td>0.680</td><td>1,710</td><td>232</td><td>6.33</td><td>677</td><td>87.3</td><td>3.98</td></tr><tr><td rowspan="2">W14 ×</td><td>132</td><td>38.8</td><td>14.66</td><td>14.725</td><td>1.030</td><td>0.645</td><td>1,530</td><td>209</td><td>6.28</td><td>548</td><td>74.5</td><td>3.76</td></tr><tr><td>120</td><td>35.3</td><td>14.48</td><td>14.670</td><td>0.940</td><td>0.590</td><td>1,380</td><td>190</td><td>6.24</td><td>495</td><td>67.5</td><td>3.74</td></tr></table>
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(Continued )
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Wide Flange Shapes (W Shapes)\*: Theoretical Dimensions and Properties for Designing (Continued )
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<table><tr><td rowspan="2">Section Number</td><td rowspan="2">Weight per Foot (lb)</td><td rowspan="2">Area of Section A (in.2)</td><td rowspan="2">Depth of Section d (in.)</td><td colspan="2">Flange</td><td rowspan="2">Web Thickness $t_w$ (in.)</td><td colspan="3">Axis X-X</td><td colspan="3">Axis Y-Y</td></tr><tr><td>Width $b_f$ (in.)</td><td>Thick-ness $t_f$ (in.)</td><td> $l_x$ (in.4)</td><td> $S_x$ (in.3)</td><td> $r_x$ (in.)</td><td> $l_y$ (in.4)</td><td> $S_y$ (in.3)</td><td> $r_y$ (in.)</td></tr><tr><td></td><td>109</td><td>32.0</td><td>14.32</td><td>14.605</td><td>0.860</td><td>0.525</td><td>1,240</td><td>173</td><td>6.22</td><td>447</td><td>61.2</td><td>3.73</td></tr><tr><td></td><td>99</td><td>29.1</td><td>14.16</td><td>14.565</td><td>0.780</td><td>0.485</td><td>1,110</td><td>157</td><td>6.17</td><td>402</td><td>55.2</td><td>3.71</td></tr><tr><td></td><td>90</td><td>26.5</td><td>14.02</td><td>14.520</td><td>0.710</td><td>0.440</td><td>999</td><td>143</td><td>6.14</td><td>362</td><td>49.9</td><td>3.70</td></tr><tr><td>W14 ×</td><td>82</td><td>24.1</td><td>14.31</td><td>10.130</td><td>0.855</td><td>0.510</td><td>882</td><td>123</td><td>6.05</td><td>148</td><td>29.3</td><td>2.48</td></tr><tr><td></td><td>74</td><td>21.8</td><td>14.17</td><td>10.070</td><td>0.785</td><td>0.450</td><td>796</td><td>112</td><td>6.04</td><td>134</td><td>26.6</td><td>2.48</td></tr><tr><td></td><td>68</td><td>20.0</td><td>14.04</td><td>10.035</td><td>0.720</td><td>0.415</td><td>723</td><td>103</td><td>6.01</td><td>121</td><td>24.2</td><td>2.46</td></tr><tr><td></td><td>61</td><td>17.9</td><td>13.89</td><td>9.995</td><td>0.645</td><td>0.375</td><td>640</td><td>92.2</td><td>5.98</td><td>107</td><td>21.5</td><td>2.45</td></tr><tr><td>W14 ×</td><td>53</td><td>15.6</td><td>13.92</td><td>8.060</td><td>0.660</td><td>0.370</td><td>541</td><td>77.8</td><td>5.89</td><td>57.7</td><td>14.3</td><td>1.92</td></tr><tr><td></td><td>48</td><td>14.1</td><td>13.79</td><td>8.030</td><td>0.595</td><td>0.340</td><td>485</td><td>70.3</td><td>5.85</td><td>51.4</td><td>12.8</td><td>1.91</td></tr><tr><td></td><td>43</td><td>12.6</td><td>13.66</td><td>7.995</td><td>0.530</td><td>0.305</td><td>428</td><td>62.7</td><td>5.82</td><td>45.2</td><td>11.3</td><td>1.89</td></tr><tr><td>W14 ×</td><td>38</td><td>11.2</td><td>14.10</td><td>6.770</td><td>0.515</td><td>0.310</td><td>385</td><td>54.6</td><td>5.88</td><td>26.7</td><td>7.88</td><td>1.55</td></tr><tr><td></td><td>34</td><td>10.0</td><td>13.98</td><td>6.745</td><td>0.455</td><td>0.285</td><td>340</td><td>48.6</td><td>5.83</td><td>23.3</td><td>6.91</td><td>1.53</td></tr><tr><td></td><td>30</td><td>8.85</td><td>13.84</td><td>6.730</td><td>0.385</td><td>0.270</td><td>291</td><td>42.0</td><td>5.73</td><td>19.6</td><td>5.82</td><td>1.49</td></tr><tr><td>W14 ×</td><td>26</td><td>7.69</td><td>13.91</td><td>5.025</td><td>0.420</td><td>0.255</td><td>245</td><td>35.3</td><td>5.65</td><td>8.91</td><td>3.54</td><td>1.08</td></tr><tr><td></td><td>22</td><td>6.49</td><td>13.74</td><td>5.000</td><td>0.335</td><td>0.230</td><td>199</td><td>29.0</td><td>5.54</td><td>7.00</td><td>2.80</td><td>1.04</td></tr><tr><td>W12 ×</td><td>190</td><td>55.8</td><td>14.38</td><td>12.670</td><td>1.735</td><td>1.060</td><td>1,890</td><td>263</td><td>5.82</td><td>589</td><td>93.0</td><td>3.25</td></tr><tr><td></td><td>170</td><td>50.0</td><td>14.03</td><td>12.570</td><td>1.560</td><td>0.960</td><td>1,650</td><td>235</td><td>5.74</td><td>517</td><td>82.3</td><td>3.22</td></tr><tr><td></td><td>152</td><td>44.7</td><td>13.71</td><td>12.480</td><td>1.400</td><td>0.870</td><td>1,430</td><td>209</td><td>5.66</td><td>454</td><td>72.8</td><td>3.19</td></tr><tr><td></td><td>136</td><td>39.9</td><td>13.41</td><td>12.400</td><td>1.250</td><td>0.790</td><td>1,240</td><td>186</td><td>5.58</td><td>398</td><td>64.2</td><td>3.16</td></tr><tr><td></td><td>120</td><td>35.3</td><td>13.12</td><td>12.320</td><td>1.105</td><td>0.710</td><td>1,070</td><td>163</td><td>5.51</td><td>345</td><td>56.0</td><td>3.13</td></tr><tr><td></td><td>106</td><td>31.2</td><td>12.89</td><td>12.220</td><td>0.990</td><td>0.610</td><td>933</td><td>145</td><td>5.47</td><td>301</td><td>49.3</td><td>3.11</td></tr><tr><td></td><td>96</td><td>28.2</td><td>12.71</td><td>12.160</td><td>0.900</td><td>0.550</td><td>833</td><td>131</td><td>5.44</td><td>270</td><td>44.4</td><td>3.09</td></tr><tr><td></td><td>87</td><td>25.6</td><td>12.53</td><td>12.125</td><td>0.810</td><td>0.515</td><td>740</td><td>118</td><td>5.38</td><td>241</td><td>39.7</td><td>3.07</td></tr><tr><td></td><td>79</td><td>23.2</td><td>12.38</td><td>12.080</td><td>0.735</td><td>0.470</td><td>662</td><td>107</td><td>5.34</td><td>216</td><td>35.8</td><td>3.05</td></tr><tr><td></td><td>72</td><td>21.1</td><td>12.25</td><td>12.040</td><td>0.670</td><td>0.430</td><td>597</td><td>97.4</td><td>5.31</td><td>195</td><td>32.4</td><td>3.04</td></tr><tr><td></td><td>65</td><td>19.1</td><td>12.12</td><td>12.000</td><td>0.605</td><td>0.390</td><td>533</td><td>87.9</td><td>5.28</td><td>174</td><td>29.1</td><td>3.02</td></tr><tr><td>W12 ×</td><td>58</td><td>17.0</td><td>12.19</td><td>10.010</td><td>0.640</td><td>0.360</td><td>475</td><td>78.0</td><td>5.28</td><td>107</td><td>21.4</td><td>2.51</td></tr><tr><td></td><td>53</td><td>15.6</td><td>12.06</td><td>9.995</td><td>0.575</td><td>0.345</td><td>425</td><td>70.6</td><td>5.26</td><td>95.8</td><td>19.2</td><td>2.48</td></tr><tr><td>W12 ×</td><td>50</td><td>14.7</td><td>12.19</td><td>8.080</td><td>0.640</td><td>0.370</td><td>394</td><td>64.7</td><td>5.18</td><td>56.3</td><td>13.9</td><td>1.96</td></tr><tr><td></td><td>45</td><td>13.2</td><td>12.06</td><td>8.045</td><td>0.575</td><td>0.335</td><td>350</td><td>58.1</td><td>5.15</td><td>50.0</td><td>12.4</td><td>1.94</td></tr><tr><td></td><td>40</td><td>11.8</td><td>11.94</td><td>8.005</td><td>0.515</td><td>0.295</td><td>310</td><td>51.9</td><td>5.13</td><td>44.1</td><td>11.0</td><td>1.93</td></tr><tr><td>W12 ×</td><td>35</td><td>10.3</td><td>12.50</td><td>6.560</td><td>0.520</td><td>0.300</td><td>285</td><td>45.6</td><td>5.25</td><td>24.5</td><td>7.47</td><td>1.54</td></tr><tr><td></td><td>30</td><td>8.79</td><td>12.34</td><td>6.520</td><td>0.440</td><td>0.260</td><td>238</td><td>38.6</td><td>5.21</td><td>20.3</td><td>6.24</td><td>1.52</td></tr><tr><td></td><td>26</td><td>7.65</td><td>12.22</td><td>6.490</td><td>0.380</td><td>0.230</td><td>204</td><td>33.4</td><td>5.14</td><td>17.3</td><td>5.34</td><td>1.51</td></tr><tr><td>W12 ×</td><td>22</td><td>6.48</td><td>12.31</td><td>4.030</td><td>0.425</td><td>0.260</td><td>156</td><td>25.4</td><td>4.91</td><td>4.66</td><td>2.31</td><td>0.848</td></tr><tr><td></td><td>19</td><td>5.57</td><td>12.16</td><td>4.005</td><td>0.350</td><td>0.235</td><td>130</td><td>21.3</td><td>4.82</td><td>3.76</td><td>1.88</td><td>0.822</td></tr><tr><td></td><td>16</td><td>4.71</td><td>11.99</td><td>3.990</td><td>0.265</td><td>0.220</td><td>103</td><td>17.1</td><td>4.67</td><td>2.82</td><td>1.41</td><td>0.773</td></tr><tr><td></td><td>14</td><td>4.16</td><td>11.91</td><td>3.970</td><td>0.225</td><td>0.200</td><td>88.6</td><td>14.9</td><td>4.62</td><td>2.36</td><td>1.19</td><td>0.753</td></tr><tr><td>W10 ×</td><td>112</td><td>32.9</td><td>11.36</td><td>10.415</td><td>1.250</td><td>0.755</td><td>716</td><td>126</td><td>4.66</td><td>236</td><td>45.3</td><td>2.68</td></tr><tr><td></td><td>100</td><td>29.4</td><td>11.10</td><td>10.340</td><td>1.120</td><td>0.680</td><td>623</td><td>112</td><td>4.60</td><td>207</td><td>40.0</td><td>2.65</td></tr><tr><td></td><td>88</td><td>25.9</td><td>10.84</td><td>10.265</td><td>0.990</td><td>0.605</td><td>534</td><td>98.5</td><td>4.54</td><td>179</td><td>34.8</td><td>2.63</td></tr><tr><td></td><td>77</td><td>22.6</td><td>10.60</td><td>10.190</td><td>0.870</td><td>0.530</td><td>455</td><td>85.9</td><td>4.49</td><td>154</td><td>30.1</td><td>2.60</td></tr><tr><td></td><td>68</td><td>20.0</td><td>10.40</td><td>10.130</td><td>0.770</td><td>0.470</td><td>394</td><td>75.7</td><td>4.44</td><td>134</td><td>26.4</td><td>2.59</td></tr><tr><td></td><td>60</td><td>17.6</td><td>10.22</td><td>10.080</td><td>0.680</td><td>0.420</td><td>341</td><td>66.7</td><td>4.39</td><td>116</td><td>23.0</td><td>2.57</td></tr><tr><td></td><td>54</td><td>15.8</td><td>10.09</td><td>10.030</td><td>0.615</td><td>0.370</td><td>303</td><td>60.0</td><td>4.37</td><td>103</td><td>20.6</td><td>2.56</td></tr><tr><td></td><td>49</td><td>14.4</td><td>9.98</td><td>10.000</td><td>0.560</td><td>0.340</td><td>272</td><td>54.6</td><td>4.35</td><td>93.4</td><td>18.7</td><td>2.54</td></tr></table>
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All printed with permission of American Institute of Steel Construction
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