Statics - Problem 6.15

  

From Chapter 6, Structural Analysis

In (Figure 1), a = 10 ft.$\mathrm{<span\; style="color:\; \#333333;\; font-family:\; arial;"> Members\; AB\; and\; BC </span>}$can each support a maximum compressive force of 950 lb, and members AD, DC, and BD can support a maximum tensile force of 

▼ PART A

Determine the greatest load  P the truss can support.

Express your answer to three significant figures and include the appropriate units.

 STEP 1  

Start by taking calculating the reaction forces about both support points.

Take moment abt. point A (which is a pin support.)

Solve for Cy;

Now, equate all vertical forces.

We have already solved for Cy, plug in its value;

 STEP 2 

Now, Create your FBD for the truss surrounding joint A.

Force AB is the force in member AB

& Force AD is the force in member AD.

Calculate the angles subtended by BAx & DAx.

BAx calculation:

*SOH CAH TOA

Use tangent for this angle calculation since we know the height (a) and the base (a).

DAx calculation:

*SOH CAH TOA

Use tangent for this angle calculation since we know the height (a/4) and the base (a).

 STEP 3 

Now, equate all horizontal forces.

Equate the forces along the vertical direction.

= 0.647P (tension)

 STEP 4 

Calculate the force in member AB:

= 0.943P (compression)

Due to symmetrical shape of ABD and BDC, the tensions in both members are equivalent.

= 0.647P

= 0.943P

Statics - Problem 4.47

 From Chapter 4, Force System Resultants

The cable exerts a 130-N force on the telephone pole shown in (Figure 1)

Determine the moment of this force about point A. Enter the components of the moment separated by commas. Express your answer in newton-meters to three significant figures.

 STEP 1  

Gather all presented data, as you will use it to your advantage in vector-type problems. Firstly, gather co-ordinates in i,j,k format of all the presented points.

→ A (0,0,0)

→ B (

Statics - Problem 2.24

 Statics, Chapter 2 - Force Vectors

In (Figure 1), the resultant F_R of the two forces acting on the jet aircraft should be directed along the positive x axis and have a magnitude of 11 kN,

▼ PART A

Determine the angle θ of the cable attached to the truck at B so that F_B is a minimum. Express your answer in degrees to three significant figures.

▼ PART B

What is the magnitude of force in cable  B when this occurs? Express your answer to three significant figures and include the appropriate units.

▼ PART C

What is the magnitude of force in cable  C when this occurs? Express your answer to three significant figures and include the appropriate units.

F_R = resultant, which is directed along the x-axis and = 11 kN.

θ = ? to have to be minimum...

since F_y = 0, F_c sin(20) = F_b sin(θ)

F_x = F_R = F_c cos(20) + F_b cos(θ) = 11 kN

Let's solve for F_c using the equation for Fy

Now simply the 1/sin(20) and plug F_c  in the equation for F_R

Now, take the derivative of 2.92sin(θ) + cos(θ) in order to get the maximum value that compliments the equation

Set the right hand side = 0, and solve the equation.

Divide every term by cos(θ).

Using trigonometric identities, evaluate -sin(θ)/cos(θ) = -tan(θ)

Take the inverse tan of both sides; arctan(2.92) = 71.1

Part A answ = 71.1°

Now plug in 71.1° in this equation from above...

... to get our value for F_b or F_b(min)

when doing so, our Part B answer is, F_b = 11/3.09 or 3.56 kN

Now for Part C, we can plug in what we now know to find F_c re-using the equation below...

F_c = 9.84 kN

Statics - Problem 10.27

 Statics, Chapter 10 - Moments of Inertia

Consider the "Z" section in (Figure 1) with h = 7 in.

▼ PART A

Determine the moment of inertia of the "Z" section with respect to the x centroidal axis. Express your answer in inches to the fourth power to three significant figures.

▼ PART B

Determine the moment of inertia of the "Z" section with respect to the y centroidal axis. Express your answer in inches to the fourth power to three significant figures.

 STEP 1  

Firstly, Gather all the information you have on the object. Make a FBD if needed. This Z section - if taken apart - can be constructively made out of three individual rectangles. Let's get the area of all three.

A₁ = 7 x 1 = 7 in^2 ........ (longer piece) 

A₂ = 5 x 1 = 5 in^2 ........ (twin piece) 

A₃ = 5 x 1 = 5 in^2 ........ (twin piece) 

 STEP 2  

Next, we need to find the second moment of area, or the moment of inertia for all three plane areas. Every basic shape has a specific formula. This table from fxsolver.com has a great one for two dimensional objects here.

I_x & I_y calculation for piece A₁: since it is coinciding with both axises, no need to for usage of parallel axis theorem.

 STEP 3  

Now, for the twin pieces. We can find the I_x & I_y of just one piece and multiply the end result by two since both pieces are of the same length and width and also the same distance from both the x and y axises.

 STEP 4  

For Part A, combine all I_x

answ = 189 in^4

For Part B, combine all I_y

answ = 61.4 in^4

centroidal axiExpress your answer in inches to the fourth power to three significant figures.

Locating ȳ of area, using calculus

 

Statics, Chapter 9 - Center of Gravity and Centroid

Consider the area in (Figure 1). (Figure 1).

Locate the centroid ˉy of the area. 

Express your answer to three significant figures and include the appropriate units.

 STEP 1  

Consider the equation for the area of the differential element (dA), and that its equation is: 

dA = ydx, or 

dA = xdy 

(in this instance since we will use dA = ydx since we are given the equation for y.)

Consider the equation for a centroid of the y co-ordinate as well. It is:

ˉy = y/2

 

 STEP 2  

Now, we will use the formula for the center of a side length (y-bar) to calculate the ˉy using the centroid formula.

dA =  ydx or (1-1/4x^2), and ˉy = y/2 or 1/2*(1-1/4x^2)

The integral is taken on the area range -2 to 2.

 STEP 3  

All that is left to do is anti-differentiate.

 

and simplify.

Answer is (2/5) m, or 0.400 m

Statics, Chapte