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Simply Supported Beam Design with Torsional Loading

Using AISC Steel Shapes v14.1 Data

Introduction

This application performs a design analysis on a simply supported beam with torsional loading for a W10X54 steel beam (as defined by the AISC Steel Shapes Database).

References:

 • Simplified Design for Torsional Loading of Rolled Steel Members, Lin, P.H., Engineering Journal, AISC, 1977
 • 2010 Specification for Structural Steel Buildings (ANSI/AISC 360-10), Fourth Printing (https://www.aisc.org/content.aspx?id=2884)

 You will need to install the AISC Shapes Database package from the MapleCloud before you can use this application.

Load the AISC Package

 > $\mathrm{with}\left(\mathrm{AISCShapes}\right):$
 > $\mathrm{with}\left(\mathrm{Units}\left[\mathrm{Standard}\right]\right):$

Data from the AISC Shapes Database for Steel Shape W10X54

 > $\mathrm{Cw}≔\mathrm{Property}\left("W10X54","Cw"\right);\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{Property}\left("W10X54","Cw","metadata"\right)\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}$
 ${\mathrm{Cw}}{≔}{2320.0}{}⟦{{\mathrm{in}}}^{{6}}⟧$
 ${"Warping constant"}$ (3.1)
 > $\mathrm{JT}≔\mathrm{Property}\left("W10X54","J"\right);\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{Property}\left("W10X54","J","metadata"\right)\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}$
 ${\mathrm{JT}}{≔}{1.82}{}⟦{{\mathrm{in}}}^{{4}}⟧$
 ${"Torsional moment of inertia"}$ (3.2)
 > $d≔\mathrm{Property}\left("W10X54","d"\right);\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{Property}\left("W10X54","d","metadata"\right)$
 ${d}{≔}{10.1}{}⟦{\mathrm{in}}⟧$
 ${"Overall depth of member, or width of shorter leg for angles, or width of the outstanding legs of long legs back-to-back double angles, or the width of the back-to-back legs of short legs back-to-back double angles"}$ (3.3)
 > $\mathrm{Sx}≔\mathrm{Property}\left("W10X54","Sx"\right);\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{Property}\left("W10X54","Sx","metadata"\right)$
 ${\mathrm{Sx}}{≔}{60.0}{}⟦{{\mathrm{in}}}^{{3}}⟧$
 ${"Elastic section modulus about the x-axis"}$ (3.4)
 > $\mathrm{Sy}≔\mathrm{Property}\left("W10X54","Sy"\right);\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{Property}\left("W10X54","Sy","metadata"\right)$
 ${\mathrm{Sy}}{≔}{20.6}{}⟦{{\mathrm{in}}}^{{3}}⟧$
 ${"Elastic section modulus about the y-axis"}$ (3.5)
 > $\mathrm{rx}≔\mathrm{Property}\left("W10X54","rx"\right);\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{Property}\left("W10X54","rx","metadata"\right)$
 ${\mathrm{rx}}{≔}{4.37}{}⟦{\mathrm{in}}⟧$
 ${"Radius of gyration about the x-axis = sqrt\left(Ix/A\right)"}$ (3.6)
 > $A≔\mathrm{Property}\left("W10X54","A"\right);\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{Property}\left("W10X54","A","metadata"\right)$
 ${A}{≔}{15.8}{}⟦{{\mathrm{in}}}^{{2}}⟧$
 ${"Cross-sectional area of member"}$ (3.7)
 > $\mathrm{Zx}≔\mathrm{Property}\left("W10X54","Zx"\right);\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{Property}\left("W10X54","Zx","metadata"\right)$
 ${\mathrm{Zx}}{≔}{66.6}{}⟦{{\mathrm{in}}}^{{3}}⟧$
 ${"Plastic section modulus about the x-axis"}$ (3.8)
 > $\mathrm{Ix}≔\mathrm{Property}\left("W10X54","Ix"\right);\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{Property}\left("W10X54","Ix","metadata"\right)$
 ${\mathrm{Ix}}{≔}{303.0}{}⟦{{\mathrm{in}}}^{{4}}⟧$
 ${"Moment of inertia about the x-axis"}$ (3.9)
 > $\mathrm{Iy}≔\mathrm{Property}\left("W10X54","Iy"\right);\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{Property}\left("W10X54","Iy","metadata"\right)$
 ${\mathrm{Iy}}{≔}{103.0}{}⟦{{\mathrm{in}}}^{{4}}⟧$
 ${"Moment of inertia about the y-axis"}$ (3.10)

Parameters

 > $w≔1.15⟦\frac{\mathrm{kipf}}{\mathrm{ft}}⟧:$

Lateral point load at the middle:

 > $F≔5⟦\mathrm{kipf}⟧:$

Torsion at mid-span:

 >

 > $P≔96⟦\mathrm{kipf}⟧:$

Beam length:

 > $L≔15⟦\mathrm{ft}⟧:$

Beam yield stress:

 > $\mathrm{Fy}≔50⟦\mathrm{ksi}⟧:$

Vertical bending unbraced length:

 > $\mathrm{Lb}≔15⟦\mathrm{ft}⟧:$

Axial vertical unbraced length:

 > $\mathrm{Lx}≔15⟦\mathrm{ft}⟧:$

Axial horizontal unbraced length:

 > $\mathrm{Ly}≔7.5⟦\mathrm{ft}⟧:$

Young's modulus and shear modulus:

 > $E≔29000⟦\mathrm{ksi}⟧:$
 > $G≔11200⟦\mathrm{ksi}⟧:$

Torsional property (Phillip, 1977):

 > $\mathrm{λ}≔\sqrt{\frac{G\cdot \mathrm{JT}}{E\cdot \mathrm{Cw}}}$
 ${0.01740610961}{}⟦\frac{{1}}{{\mathrm{in}}}⟧$ (4.1)

Determine Governing Moments at Middle of Span

Flexural moment:

 > $\mathrm{Mx}≔\frac{w{L}^{2}}{8}$
 ${32.34}{}⟦{\mathrm{foot}}{}{\mathrm{kipf}}⟧$ (5.1)
 > $\mathrm{My}≔\frac{FL}{4.0}$
 ${18.75}{}⟦{\mathrm{foot}}{}{\mathrm{kipf}}⟧$ (5.2)
 > $\mathrm{M0}≔\frac{TL}{4d}$
 ${22.72}{}⟦{\mathrm{foot}}{}{\mathrm{kipf}}⟧$ (5.3)

Philip page 101

 > $\mathrm{β}≔\frac{4{\mathrm{sinh}\left(\frac{\mathrm{λ}L}{2}\right)}^{2}}{\mathrm{λ}L\mathrm{sinh}\left(\mathrm{λ}L\right)}$
 ${\mathrm{\beta }}{≔}{0.5850278056}$ (5.4)

Torsional moment:

 > $\mathrm{MT}≔\mathrm{β}\mathrm{M0}$
 ${13.29}{}⟦{\mathrm{foot}}{}{\mathrm{kipf}}⟧$ (5.5)

Check Torsional Capacity (AISC 360-10 H3.3 & Philip page 100)

Maximum combined normal stress at the load point:

 > $\mathrm{fbx}≔\frac{\mathrm{Mx}}{\mathrm{Sx}}+\frac{2\mathrm{MT}}{\mathrm{Sy}}$
 ${21.96}{}⟦\frac{{\mathrm{kipf}}}{{{\mathrm{inch}}}^{{2}}}⟧$ (6.1)

Safety factor for compression:

 > $\mathrm{Ω}≔1.67:$
 > $\mathrm{Fnx}≔\frac{\mathrm{Fy}}{\mathrm{Ω}}$
 ${29.94}{}⟦{\mathrm{ksi}}⟧$ (6.2)
 > $\frac{\mathrm{fbx}}{\mathrm{Fnx}}$
 ${0.7333393767}$ (6.3)

This is less then 1, so it is satisfactory.

Check Combined Compression and Bending Capacity (AISC 360-10, H1)

 > $\mathrm{Mrx}≔\left(\frac{\mathrm{Mx}}{\mathrm{Sx}}+\frac{2\mathrm{MT}}{\mathrm{Sy}}\right)\mathrm{Sx}$
 ${109.78}{}⟦{\mathrm{foot}}{}{\mathrm{kipf}}⟧$ (7.1)

Effective length factor:

 > $K≔0.85:$

Elastic bucking stress:

 > $\mathrm{Fe}≔\frac{{\mathrm{π}}^{2}E}{{\left(\frac{KL}{\mathrm{rx}}\right)}^{2}}$
 ${233.50}{}⟦{\mathrm{ksi}}⟧$ (7.2)

Critical stress:

 > $\mathrm{Fcr}≔{0.658}^{\frac{\mathrm{Fy}}{\mathrm{Fe}}}\mathrm{Fy}$
 ${45.71}{}⟦{\mathrm{ksi}}⟧$ (7.3)
 > $\mathrm{Pn}≔\mathrm{Fcr}A$
 ${722.27}{}⟦{\mathrm{kipf}}⟧$ (7.4)

Allowable axial strength:

 > $\mathrm{Pc}≔\frac{\mathrm{Pn}}{\mathrm{Ω}}$
 ${432.50}{}⟦{\mathrm{kipf}}⟧$ (7.5)

This is greater than 3/4 Pr, so it is satisfactory.

Available flexural strength (Chapter F AISC 360-10):

 > $\mathrm{Mn}≔\mathrm{min}\left(\mathrm{Fy}\mathrm{Zx},\mathrm{Fy}\mathrm{Sx}\right)$
 ${250.00}{}⟦{\mathrm{foot}}{}{\mathrm{kipf}}⟧$ (7.6)
 > $\mathrm{Mcx}≔\frac{\mathrm{Mn}}{\mathrm{Ω}}$
 ${149.70}{}⟦{\mathrm{foot}}{}{\mathrm{kipf}}⟧$ (7.7)

This is greater than Mrx, so it is satisfactory.

 > $\mathrm{Mcy}≔\frac{\mathrm{Mn}}{\mathrm{Ω}}$
 ${149.70}{}⟦{\mathrm{foot}}{}{\mathrm{kipf}}⟧$ (7.8)

These should be below 1 for a satisfactory design.

 > $\frac{P}{\mathrm{Pc}}+\frac{8}{9}\cdot \left(\frac{\mathrm{Mrx}}{\mathrm{Mcx}}+\frac{\mathrm{My}}{\mathrm{Mcy}}\right)$
 ${.99}$ (7.9)

Determine Deflections

Max twist angle (Lin, p100 eq4) in degrees:

 > $\mathrm{φ}≔\frac{T}{2G\mathrm{JT}\mathrm{λ}}\cdot \left(\frac{\mathrm{λ}\cdot L}{2}-\frac{2\cdot \mathrm{sinh}\left(\frac{\mathrm{λ}\cdot L}{2}\right)}{\mathrm{sinh}\left(\mathrm{λ}\cdot L\right)}\right)\cdot \mathrm{sinh}\left(\frac{\mathrm{λ}\cdot L}{2}\right)$
 ${\mathrm{\phi }}{≔}{0.2304416908}$ (8.1)
 > $\mathrm{I3}≔\mathrm{Ix}{\mathrm{sin}\left(\frac{\left(90-\mathrm{φ}\right)\mathrm{π}}{180}\right)}^{2}+\mathrm{Iy}{\mathrm{cos}\left(\frac{\left(90-\mathrm{φ}\right)\mathrm{π}}{180}\right)}^{2}$
 ${303.00}{}⟦{{\mathrm{in}}}^{{4}}⟧$ (8.2)
 > $\mathrm{I4}≔\mathrm{Ix}{\mathrm{cos}\left(\frac{\left(90-\mathrm{φ}\right)\mathrm{π}}{180}\right)}^{2}+\mathrm{Iy}{\mathrm{sin}\left(\frac{\left(90-\mathrm{φ}\right)\mathrm{π}}{180}\right)}^{2}$
 ${103.00}{}⟦{{\mathrm{in}}}^{{4}}⟧$ (8.3)

Vertical deflection at the middle:

 >
 ${.15}{}⟦{\mathrm{in}}⟧$ (8.4)

Horizontal deflection at the middle:

 >
 ${.20}{}⟦{\mathrm{in}}⟧$ (8.5)
 >