3 Engineering Example 2

3.1 The web-flange junction

Introduction

In problems of torsion, the torsion constant , $J$ , which is a function of the shape and structure of the element under consideration, is an important quantity.

A common beam section is the thick $I$ -section shown here, for which the torsion constant is given by $J=2J_1+J_2+2\alpha D^4$ where the $J_1$ and $J_2$ terms refer to the flanges and web respectively, and the $D^4$ term refers to the web-flange junction. In fact $\alpha =min\left[\frac{t_f}{t_w},\frac{t_w}{t_f}\right]\left(0.15+0.1\frac{r}{t_f}\right)$

where $t_f$ and $t_w$ are the thicknesses of the flange and web respectively, and $r$ is the radius of the concave circle element between the flange and the web. $D$ is the diameter of the circle of the web-flange junction.

As $D$ occurs in the form $D^4$ , the torsion constant is very sensitive to it. Calculation of $D$ is therefore a crucial part of the calculation of $J$ .

Problem in words

Find $D$ , the diameter of the circle within the web–flange junction as a function of the other dimensions of the structural element.

Mathematical statement of problem

1. Find $D$ , the diameter of the circle, in terms of $t_f$ and $t_w$ (the thicknesses of the flange and the web respectively) in the case where $r=0$ . When $t_f$ = 3cm and $t_w$ = 2cm, find $D$ .
2. For $r\ne 0$ , find $D$ in terms of $t_f$ , $t_w$ and $r$ . In the special case where $t_f$ = 3 cm, $t_w$ = 2 cm and $r$ = 0.4 cm, find $D$ .

Mathematical analysis

1. Consider a co-ordinate system based on the midpoint of the outer surface of the flange.

   

   The centre of the circle will lie at $\left(0,-R\right)$ where $R$ is the radius of the circle, i.e. $R=D∕2$ . The equation of the circle is

   $x^2+{\left(y+R\right)}^2=R^2$ (1)

   In addition, the circle passes through the ‘corner’ at point $A$ $\left(t_w∕2,-t_f\right)$ , so

   ${\left(\frac{t_w}{2}\right)}^2+{\left(-t_f+R\right)}^2=R^2$ (2)

   On expanding

   $\frac{t_w^2}{4}+t_f^2-2R{t}_f+R^2=R^2$

   giving

   $2R{t}_f=\frac{t_w^2}{4}+t_f^2\Rightarrow R=\frac{\left(t_w^2∕4\right)+t_f^2}{2t_f}=\frac{t_w^2}{8t_f}+\frac{t_f}{2}$

   so that

   $D=2R=\frac{t_w^2}{4t_f}+t_f$ (3)

   Setting $t_f$ = 3 cm, $t_w$ = 2 cm gives

   $D=\frac{2^2}{4\times 3}+3=3.33\text{ cm}$

2. Again using a co-ordinate system based on the mid-point of the outer surface of the flange, consider now the case $r\ne 0$ .

   

   Point $B$ $\left(t_w∕2+r,-t_f-r\right)$ lies, not on the circle described by (1), but on the slightly larger circle with the same centre, and radius $R+r$ . The equation of this circle is

   $x^2+{\left(y+R\right)}^2={\left(R+r\right)}^2$ (4)

   Putting the co-ordinates of point $B$ into equation (4) gives

   ${\left(\frac{t_w}{2}+r\right)}^2+{\left(-t_f-r+R\right)}^2={\left(R+r\right)}^2$ (5)

   which, on expanding gives

   $\frac{t_w^2}{4}+t_{w}r+r^2+t_f^2+r^2+R^2+2t_{f}r-2t_{f}R-2rR=R^2+2rR+r^2$

   Cancelling and gathering terms gives

   $$\begin{array}{rcll}\frac{t_w^2}{4}+t_{w}r+r^2+t_f^2+2t_{f}r& =& 4rR+2t_{f}R& \\ & =& 2R\left(2r+t_f\right)& \end{array}$$

   so that

   $2R=D=\frac{\left(t_w^2∕4\right)+t_{w}r+r^2+t_f^2+2t_{f}r}{\left(2r+t_f\right)}$

   $\text{so}D=\frac{t_w^2+4t_{w}r+4r^2+4t_f^2+8t_{f}r}{\left(8r+4t_f\right)}$ (6)

   Now putting $t_f=3$ cm, $t_w=2$ cm and $r=0.4$ cm makes

   $D=\frac{2^2+\left(4\times 2\times 0.4\right)+\left(4\times 0.4^2\right)+\left(4\times 3^2\right)+\left(8\times 3\times 0.4\right)}{\left(8\times 0.4\right)+\left(4\times 3\right)}=\frac{53.44}{15.2}=3.52\text{ cm}$

Interpretation

Note that setting $r=0$ in Equation (6) recovers the special case of $r=0$ given by equation (3). The value of $D$ is now available to be used in calculations of the torsion constant, $J$ .

3.2 The parabola

The standard form of the parabola is shown in Figure 5. Here the $x$ -axis is the line of symmetry of the parabola.

Figure 5

It can be shown that light rays parallel to the $x$ -axis will, on reflection from the parabolic curve, come together at the focus. This is an important property and is used in the construction of some kinds of telescopes, satellite dishes and car headlights.

Task!

Sketch the curve $y^2=8x$ . Find the position of the focus and confirm its light-focusing property.

Answer

By changing the equation of the parabola slightly we can change the position of the parabola along the $x$ -axis. See Figure 6.

Figure 6 :

We can also have parabolas where the $y$ -axis is the line of symmetry (see Figure 7). In this case the standard equation is

$x^2=4ay\text{or}y=\frac{x^2}{4a}$

Figure 7

Task!

Sketch the curves $y^2=x$ and $x^2=2\left(y-3\right).$

Answer

The focus of the parabola $y^2=4a\left(x-b\right)$ is located at coordinate position $\left(a+b,0\right)$ . Changing the value of $a$ changes the convexity of the parabola (see Figure 8).

Figure 8

The standard form of the hyperbola is shown in Figure 9(a).

This has standard equation

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

The eccentricity, $e$ , is defined by

$e^2=1+\frac{b^2}{a^2}\left(e>1\right)$

Figure 9

Note the change in sign compared to the equivalent expressions for the ellipse. The lines $y=\pm \frac{b}{a}x$ are asymptotes to the hyperbola (these are the lines to which each branch of the hyperbola approach as $x\to \pm \infty$ ).

If light is emitted from one focus then on hitting the hyperbolic curve it is reflected in such a way as to appear to be coming from the other focus. See Figure 9(b). The hyperbola has fewer uses in applications than the other conic sections and so we will not dwell here on its properties.

3.3 General conics

The conics we have considered above - the ellipse, the parabola and the hyperbola - have all been presented in standard form:- their axes are parallel to either the $x$ - or $y$ -axis. However, conics may be rotated to any angle with respect to the axes: they clearly remain conics, but what equations do they have?

It can be shown that the equation of any conic , can be described by the quadratic expression

$A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$

where $A,B,C,D,E,F$ are constants.

If not all of $A,B,C$ are zero (and $F$ is a suitable number) the graph of this equation is

1. an ellipse if $B^2<4AC$    (circle if $A=C$ and $B=0$ )
2. a parabola if $B^2=4AC$
3. a hyperbola if $B^2>4AC$

Example 3

Classify each of the following equations as ellipse, parabola or hyperbola:

1. $x^2+2xy+3y^2+x-1=0$
2. $x^2+2xy+y^2-3y+7=0$
3. $2x^2+xy+2y^2-2x+3y=6$
4. $3x^2+2x-5y+3y^2-10=0$

Solution

1. Here $A=1,B=2,C=3$ $\therefore$ $B^2<4AC$ . This is an ellipse.
2. Here $A=1,B=2,C=1$ $\therefore$ $B^2=4AC$ . This is a parabola.
3. Here $A=2,B=1,C=2$ $\therefore$ $B^2<4AC$ also $A=C$ but $B\ne 0$ . This is an ellipse.
4. Here $A=3,B=0,C=3$ $\therefore$ $B^2<4AC$ . Also $A=C$ and $B=0$ . This is a circle.

Task!

Classify each of the following conics:

1. $x^2-2xy-3y^2+x-1=0$
2. $2x^2+xy-y^2-2x+3y=0$
3. $4x^2-y+3=0$
4. $-x^2-xy-y^2+3x=0$
5. $2x^2+2y^2-x+3y=7$

Answer

Exercises

1. The equation $9x^2+4y^2-36x+24y-1=0$ represents an ellipse. Find its centre, the semi-major and semi-minor axes and the coordinate positions of the foci.
2. Find the equation of a circle of radius 3 which has its centre at $\left(-1,2.2\right)$
3. Find the centre and radius of the circle $x^2+y^2-2x-2y-5=0$
4. Find the position of the focus of the parabola $y^2-x+3=0$
5. Classify each of the following conics
   1. $x^2+2x-y-3=0$
   2. $8x^2+12xy+17y^2-20=0$
   3. $x^2+xy-1=0$
   4. $4x^2-y^2-4y=0$
   5. $6x^2+9y^2-24x-54y+51=0$
6. An asteroid has an elliptical orbit around the Sun. The major axis is of length $5\times 10^8$ km. If the distance between the foci is $4\times 10^8$ km find the equation of the orbit.

Answer