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This problem deals with finding the load that the column in the two member frame shown in

the figure can support without buckling. From structural mechanics the critical load Pcr is given

by

Pcr =  zL 2

E I

(1)

where z is smallest positive solution to the nonlinear equation

tan z =

z

1 + z2/4

(2)

The quantity EI is the beam stiffness and L is the length of the column. The value of EI is

1.13 x 107 newtons (meter)2  and the length L is 11.5 meters.

BA C L L P

1. (10 points) modify the task1.m from the file directory to estimate the smallest positive root of equation (2). Estimate the root to two decimal places.

2. (20 points) Use Excel to program the bisection method for finding the smallest positive value satisfying equation (2). Find the root to 5 decimal places. Find the critical load. Find the critical load. (You may put the computation of the critical load on your spreadsheet.) How many steps did it take for the bisection method to obtain the 5 decimal place accuracy?

3. (20 points) Use Excel to program the secant method for finding the root of equation (2). For

the initial two values, use the initial endpoints that you used for problem 2. Find the root to 5

decimal places. Find the critical load.

How many steps did it take for the secant method obtain the 5 decimal place accuracy?

The idea of the trapezoid rule is displayed below. The area under the curve is approximated by

the area of the trapezoids formed by connection points (xj

, f(xj

) ) by straight line.

x1 x2

h h h h h h h

In simplified form, the equation for trapezoid rule is in the from

AB

f(x) dx ≈ h

  12

f(x0

) + 

j = 1

N-1

f(xj

) +

12

f(xN

)

(3)

Another method is the midpoint rule. The idea is illustrated below.

A = x0 xN = B x1 x2 h h h h h h h

The formula for the midpoint rule is

AB

f(x) dx ≈ h

  

j = 1

N

f(xj-1/2)

(4)

Let’s apply the formula (3) and (4) to approximate an integral for

-1212 e

sin x dx

(5)

4. (25 points) Using Excel to implement the trapezoid rule in equation (3) to equation (5). You

may want to write down the several terms in the summation to verify the notation. You can

evaluate the two end points, x0

xN

, separately. Try several different values of N until you see

the value converges.

5. (25 points) Using the midpoint rule in equation (4) to equation (5). Use N as a variable to

divide the interval h, and then evaluate the function at the midpoint. Calculate the individual

area and then sum the area of the pieces. Try several different value of N until you see the

value converges.

Discuss the value of Ns in the trapezoid rule and midpoint rule.