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Documentation Structural Mechanics

ap er

Cross-Sectional Properties of Areas

2.1 Introduction

Engineering analysis and design often uses properties of plane sections in calculations. For example, in stress analysis of a beam under bending

and torsional loads, you use the cross-sectional properties to determine the stress and displacement distributions in the beam cross section. In

calculating the natural frequencies and mode shapes of a machine element, you also need to know the area, centroid, and various moments ofinertia of a cross section. You can use the package SymCrossSectionProperties, designed toact as of an electronic handbook, to calculate

properties of cross sections parametrically. Then you can use the functions in the package NumCrossSectionPropertiesto compute these

attributes numerically.

This chapter deals with the cross-sectional properties of plane sections, both symbolically and numerically. The advantage of a symbolic output is

that analytical solutions can be expressed in closed form, which enables further mathematical manipulations of the result. In the cases for which a

symbolic result is not needed and/or is dif ficult to obtain, a numerical approach based on a triangulation scheme is used.

With the functionality of the packages SymCrossSectionPropertiesand NumCrossSectionProperties, you can compute various

cross-sectional properties, such as area, centroid, and moment of inertia. For symbolic results, the package includes standard cross sections, such

as T-sections, I-sections, and channel sections, as well as basic domain objects, such as rectangular and triangular shapes, circular and elliptical

sectors, and sectors of circular and elliptical annuli. A simple procedure to combine these subdomain objects to form a complex cross section is

also introduced. Using this procedure, you can easily in clude many practical shapes, such as L-sections, Z-sections, tampered I-sections, T-

sections, and channel sections, in the calculations. The design of Structural Mechanics makes it possible to add a set of definitions for new

subdomain objects to the functionality of the package. When computing the cross-sectional properties of a complex domain, numerical techniques

are preferred. The numerical computations are based on a triangulation technique and an integration-in-a-triangle routine.

2.2 Symbolic Computations

2.2.1 General Remarks

The main advantage of having symbolic expressions for the cross-sectional properties of a section is that they enable the employment of powerful

mathematical techniques in later stages of engineering computations. As a result of a symbolic expression, instead of finding one specific solution,

you can obtain a general solution to a class of p roblems. With this general solution, you can evaluate a wide range o f design possibilities and,

consequently, reach a better design.

In Structural Mechanics, the plane sections are represented by using the function Domain.

Each cross section is treated as an object or a set of objects created using the Domainfunction. For example, Domain[CircleSector, r, {0, /2}]

represents a data object, one-fourth of a circle with radius r,on which you can perform various operations, including SectionArea, which

calculates the area of this sector. Note that angles are in radians, and the sign of an angle is described by the traditional counterclockwise direction

convention. An angle described counterclockwise from the positive part of the x axis is considered positive. If it is described clockwise, the angle

is considered negative. You can obtain the usage message for a domain object by typing ?objectname. For instance, ?CircleSector

generates the usage message for the domain object CircleSector.

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The area of one-fourth of a circle with radius r is calculated by the function SectionArea, which will be described in detail shortly.

In[1]:=

In[2]:=SectionArea[Domain[CircleSector,r,{0, /2}]]

Out[2]=

Using the Structural Mechanics' graphics primitive DomainGraphics,you can plot the shape of this sector.

In[3]:=Show[DomainGraphics[Domain[CircleSector,1,{0, /2}]]];

You can generate the same plot by using the function CrossSectionPlot.

In[4]:=

You can view the basic domains introduced in Structural Mechanicsusing the variable name BasicDomainList.

In[5]:=BasicDomainList

Out[5]=

The listSectionListcontains built-in data objects for common cross sections generated by using domains in the list BasicDomainList.

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In[6]:=SectionList

Out[6]=

The cross sections in both lists form the collection of predefined cross sections of the package SymCrossSectionProperties. As shown in

the following sections, you can easily add a new cro ss section to this list as either a basic domain object or a section obj ect. When you add a new

object to the package, the lists BasicDomainListand SectionListmust be edited accordingly.

You can depict the area of a sector with the radius 1 unit, the origin at (1, 1), and the sweeping angle from /4 to .

In[7]:=

In[8]:=p1=Show[DomainGraphics[dmn],

Axes->True,

PlotRange->{{-4,4},{-4,4}}];

To compute the area of the following elliptical domain object, first, create a domain object for the ellipse sector. It is important to n ote that the

arguments { , } in Domain[ EllipseSector, { , }, { , } ]are the actual polar angles as opposed to the

parametrization variable sinx= andy=

In[9]:=els=Domain[EllipseSector,{1,1},{3,2},{0,3 /2}];

A number of option sets graphically represent this object.

In[10]:=p2=Show[DomainGraphics[els],

Axes->True,

PlotRange->{{-4,4},{-4,4}}];

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As already noted, the other three basic built-in domain objects included in Structural Mechanicsare RectangularSection,

RightTriangle, and Parallelogram. Here are the usage messages for the domain objects created using these section names.

As as with the ellipse sector earlier, you can generate plots of these basic domain objects.

In[11]:=tri=DomainGraphics[Domain[RectangularSection,{0,0},1,1]];

rtri=DomainGraphics[Domain[RightTriangle,{0,0},0.5,1]];

par=DomainGraphics[Domain[Parallelogram,{1,1},{4,1},{5,3},{0,3}]];

Show[GraphicsArray[{tri,rtri,par}]];

Many useful cross sections, such as hollow-rectangular, annulus, T-sections, and I-sections, can be created by arranging these basic domains as

building blocks. For example, you can create the section IsoscelesTriangledomain by using two right triangles.

In[15]:=Domain[IsoscelesTriangle, b, h]

Out[15]=

In[16]:=iso=DomainGraphics[Domain[IsoscelesTriangle,{0,0},0.5,0.5]];

This shows the domain objectiso.

In[17]:=Show[iso,Axes->True];

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You can create the domain objects HollowCircleand HollowRectangleby removing the inner domain from the outer domain of a

relevant domain object.

The domain HollowCircleis represented as the difference between two circular domains.

In[18]:=Domain[HollowCircle,{xo,yo},r1,r2]

Out[18]=

The graphical representation of composite domains with the function DomainGraphicsis supported as well.

In[19]:=p3=Show[DomainGraphics[Domain[HollowCircle,{1,1},1,2]],

Axes->True,

PlotRange->{{-4,4},{-4,4}} ];

Similarly, you can use the domain HollowEllipsefor elliptical hollow areas.

This graphically represents the domain HollowEllipse.

In[20]:=Show[DomainGraphics[Domain[HollowEllipse,{1,1},{2,1},{3,2}]],

Axes->True,

PlotRange->{{-4,4},{-4,4}}];

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In[27]:=tp1=ShowDimensions[TSection];

Note that by using Structural Mechanics, in addition to symmetric T-sections, you can also consider asym metric T-sections by changing the value

of the arguments. Unlike most handbooks and reference books, this feature provides you with a more flexible tool to deal with more general

sections.

With this depiction of the dimensions of a T-section, you can accurately place the arguments in the domain object without any confusion.

In[28]:=tsec=Domain[TSection,2.8,0.6,1.0,6.8,3.0];

In[29]:=Show[DomainGraphics[tsec]];

I-Section

Another type of section commonly used in engineering and design is the I-section, for wh ich Structural Mechanicscontains the built-in domain

object ISection.

An I-section consists of three rectangular domain objects.

In[30]:=Domain[ISection,e1,e2,a,b,c,t1,t2,t3]

Out[30]=

Again, the function ShowDimensionsprovides a graphical representation of the arguments for characteris