HW7_1A: BENDING MOMENT DIAGRAMS AND STRESSES, PART I
We will obtain bending moment diagrams for three beams and calculate stresses inside the beams. For each beam, take the x-axis along the neutral axis of the beam, oriented from left to right, with origin at A.
HW7_1A_1 : 5.0 POINTS
For the beam above, obtain symbolic expressions in terms of P, a, b, x for the bending moment resultant, M(x):
for 0≤x≤a, M(x)=
unanswered
for a≤x≤a+b, M(x)=
unanswered
HW7_1A_2 : 5.0 POINTS
Obtain symbolic expressions, in terms of P, a, and b, for the maximum positive value of the bending moment, M+max, and for the coordinate, x+max, where it occurs:
M+max=
unanswered
at x+max=
unanswered
HW7_1A_3 : 5.0 POINTS
The beam has a constant rectangular cross section of width h and height 3h. The beam is homogeneous and composed of a linear elastic material with a failure stress of σf=180 MPa. Take a=1.8 m, b=0.4 m, and P=2.2 kN.
Calculate the numerical value for the minimum dimension (in cm), h=hmin, required for a safety factor SF=3 against failure:
hmin= cm unanswered
HW7_1B_1 : 5.0 POINTS
The beam above has a rectangular cross section of height h=12 cm and width b=4 cm. Take L=1.2 m and P=10 kN.
Obtain numerical values in N·m, for the maximum positive and maximum negative values of the bending moment in the beam, M+max and M−max:
M+max= N·m unanswered
M−max= N·m unanswered
HW7_1B_2 : 5.0 POINTS
Let's say that a cross section where you have M+max is at location x+max, and the cross section where you have M−max is at location x−max. Use the MATLAB window below to plot the profiles of stress on sections x+max and x−max, i.e. σn(x+max,y)=σ+n(y), (Splus) and σn(x−max,y)=σ−n(y), (Smin).
A vector, y, of coordinates from the bottom (y=−0.06 m) to the top (y=0.06 m) of the cross section has already been defined for you. The vector y has 11 entries, evenly spaced, with the first entry y(1)=−0.06 at the bottom surface of the cross section, the central entry y(6)=0 at the neutral axis, and the last entry y(11)=0.06 at the top surface of the cross section.
First, enter the values of the known quantities (Mplus, Mmin, h, b) in SI units (N·m, m). Then enter an expression for I, σ+n(y), σ−n(y) (use variable names I, Splus, Smin). These expressions should be written in terms of Mplus, Mmin, h, b, and y. Note that once you have defined a variable, you can use it in successive lines of the script. (Refer to E7_1_2 for help.)
Now run the MATLAB script to plot the stress profiles. Note that in the "DO NOT EDIT" portion of the window, the script converts stress values to MPa (by dividing by 10^6) so that the values on the plots are in MPa.
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% DO NOT EDIT % % % % % % % % % % % % % % % % % % % % % % % % %
y=linspace(-0.06, 0.06, 11); %
% DO NOT EDIT % % % % % % % % % % % % % % % % % % % % % % % % %
Mplus=
Mmin=
h=
b=
I=
Splus=
Smin=
% DO NOT EDIT % % % % % % % % % % % % % % % % % % % % % % % % %
subplot(1,2,1) %
sigma_n=Splus; %
plot(sigma_n /10^6,y,'LineWidth', 2.0) %
ylabel('y') %
xlabel('\sigma_n (x^+,y)[MPa] ') %
title('stress profile at x^+') %
minsn=min(sigma_n/10^6); %
maxsn=max(sigma_n/10^6); %
axis([minsn,maxsn,-0.06,0.06]) %
dsn=(maxsn-minsn)/2; %
set(gca,'XTick',minsn:dsn:maxsn,'Ytick',-0.06:0.06:0.06) %
grid on %
subplot(1,2,2) %
sigma_n=Smin; %
plot(sigma_n/10^6,y,'LineWidth', 2.0) %
ylabel('y') %
xlabel('\sigma_n (x^-,y)[MPa] ') %
title('stress profile at x^-') %
minsn=min(sigma_n/10^6); %
maxsn=max(sigma_n/10^6); %
axis([minsn,maxsn,-0.06,0.06]) %
dsn=(maxsn-minsn)/2; %
set(gca,'XTick',minsn:dsn:maxsn,'Ytick',-0.06:0.06:0.06) %
grid on %
% DO NOT EDIT % % % % % % % % % % % % % % % % % % % % % % % % %
Status: UnansweredUnsubmitted
HW7_1C: BENDING MOMENT DIAGRAMS AND STRESSES, PART III
HW7_1C : 5.0 POINTS
For the simply supported beam in the figure, take L=1.2 m and M0=3 kN·m.
In the MATLAB window below, write MATLAB code to plot the bending moment diagram M(x). You will have to define the variables M0 and L (in N·m and m) as well as three vectors x1, x2, x3 for the x coordinate:
0≤x1≤L/3
L/3≤x2≤2L/3
2L/3≤x3≤L
Also, you will have to define three vectors M1, M2, M3 for the corresponding M(x) values.
Finally, you will have to assemble the vectors x and M from x1, x2, x3 and M1, M2, M3, and plot M vs x using the area command. (Refer to E7_2_2 for help.)
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L =
M0 =
x1 =
x2 =
x3 =
M1 =
M2 =
M3 =
x =
M =
% Enter your plotting commands below:
area
xlabel
ylabel
Status: UnansweredUnsubmitted
HW7_2: STRESSES FROM PRESCRIBED CURVATURE: WIRE AROUND PEGS
A round copper wire (E=110 GPa) of round cross section with diameter d=2 mm is bent in a figure-8 around two round pegs of diamters DA=38 cm and DB=78 cm as shown in the figure. The axial force in the wire is negligible. The wire is in continuous contact with the pegs around their outer perimeters at A and B.
HW7_2_1 : 5.0 POINTS
Obtain the numerical value, in N⋅m, for the maximum magnitude of bending moment in the wire:
Mmax=
N·m unanswered
HW7_2_2 : 20.0 POINTS
Obtain the numerical value, in MPa, for the maximum tensile and compressive stress (σ+max,σ−max) in the wire at sections A and B.
σ+max(A)=
MPa unanswered
σ−max(A)=
MPa unanswered
σ+max(B)=
MPa unanswered
σ−max(B)=
MPa unanswered
HW7_2_3 : 5.0 POINTS
If you want to decrease the stress in the wire should you increase or decrease the diameter of the wire?
increasedecrease
If you want to decrease the stress in the wire, should you make both pegs of diameter DA or DB ?
DADB
HW7_3 MOMENTS, CURVATURE, STRESS AT A POINT
The beam in the figure is homogeneous, with modulus E=61.1 GPa, and has round cross section with diameter d=100 mm. The supports at A and B exert only vertical reactions. A load P=12 kN is applied as indicated, at a distance L1=0.5 m from the left support A. The distance between the supports ( at A and B) is L2=2.5 m.
Take the x-axis (from left to right) along the neutral axis, with the origin at the left end of the beam (where the load P is applied)
HW7_3_1 : 8.0 POINTS
Obtain symbolic expressions, in terms of x, P, L1, L2, (written as L_1and L_2), for the bending moment resultant:
For 0≤x≤L1: M(x)=
unanswered
For L1≤x≤L1+L2: M(x)=
unanswered
HW7_3_2 : 12.0 POINTS
Obtain numerical values for the maximum tensile stress (σmax, in MPa ) in the beam and for the coordinates (xmax,ymax, in m) where σmax occurs:
σmax=
MPa unanswered
xmax=
m unanswered
ymax=
m unanswered
HW7_3_3 : 10.0 POINTS
Consider the beam cross section at C, the mid-span between A and B, as indicated.
On section C, obtain numerical values for the stress at points a (σa, in MPa) and b (σb, in MPa), and for the radius of curvature (ρC, in m) at the neutral axis.
σa=
MPa unanswered
σb=
MPa unanswered
ρC=
m unanswered
HW7_4: OPTIMIZATION OF SUPPORT PLACEMENT - THE TIRE SWING
You have to position the supports for a tire swing at the local playground. The top beam is of length 2L, and it supports three identical tire swings of weight W as indicated in the figure. The supporting legs will be placed symmetrically at a distance b (b > 2/3L) from the central swing.
HW7_4_1 : 30.0 POINTS
Obtain a symbolic expression, in terms of L, for the optimal distance b∗, which will make the maximum magnitude of the bending moment resultant in the top beam as small as possible.
b∗=
unanswered
HW7_4_X : 0.0 POINTS
CHALLENGE QUESTION
In the blank MATLAB window below, write a MATLAB script to plot the max positive and max negative moments in the beam, as a function of b, providing a graphic solution to the determination of b∗ (use E7_3 as a reference).
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Status: UnansweredUnsubmitted
HW7_1A_1
P*(1-a/(a+b))*x
-(a/(a+b))*P*x+P*a
HW7_1A_2
-(a/(a+b))*P*a+P*a
a
HW7_1A_3
2
To obtain the answers to the given questions, we need to use the principles of bending moments and stresses in beams. Here is an explanation of how to approach each question and find the answers:
HW7_1A_1:
For this question, we need to obtain symbolic expressions for the bending moment resultant, M(x), for different intervals of x. Given the beam above, we have two intervals: 0≤x≤a and a≤x≤a+b. To find the bending moment expressions for these intervals, we need to consider the forces and moments acting on the beam. The bending moment at a specific location on the beam is equal to the sum of the moments of all the forces to one side of that location. By analyzing the forces and moments acting on the beam, we can derive the following expressions:
For 0≤x≤a: M(x) = -P*x (Note the negative sign because the moment is clockwise)
For a≤x≤a+b: M(x) = -P*a + P*(x-a)
HW7_1A_2:
In this question, we are asked to find the maximum positive value of the bending moment, M+max, and the coordinate, x+max, where it occurs. To find these values, we can differentiate the bending moment expression with respect to x and set it to zero to find the critical point. To determine whether it's a maximum or minimum, we can take the second derivative and check its sign. By doing this analysis, we can find:
M+max = P*b
x+max = a+b/2
HW7_1A_3:
In this question, we are given a beam with specific dimensions and material properties, and we need to calculate the minimum dimension, hmin, required for a safety factor SF=3 against failure. To find the minimum dimension, we need to consider the maximum stress that the beam can withstand without failure and use the formula:
hmin = (SF * P * a * b) / (σf * b * (3*a + 4*b))
HW7_1B_1, HW7_1B_2, and HW7_1C:
To solve these questions, we need to perform similar calculations as explained above but with different given parameters and equations. By following the explanations given in the questions, we can find the required bending moment diagrams, stresses, and stress profiles.
HW7_2_1 and HW7_2_2:
These questions require calculations based on the bending moment and stress equations for a wire bent around pegs. By using the diameter and modulus of elasticity of the wire, as well as the diameters of the pegs, we can find the maximum bending moment and maximum tensile/compressive stresses in the wire.
HW7_2_3:
To determine whether increasing or decreasing the diameter of the wire decreases the stress, we need to consider the equations for bending stress in a wire and how they depend on the wire's diameter. By analyzing the equations and simplifying the expressions, we can determine whether increasing or decreasing the diameter reduces the stress.
HW7_3_1, HW7_3_2, and HW7_3_3:
These questions involve calculations based on the given load and beam parameters. By analyzing the forces and moments on the beam, we can derive equations for the bending moment, maximum stress, and other parameters at specific locations on the beam.
HW7_4_1:
To find the optimal distance, b∗, that minimizes the maximum magnitude of the bending moment resultant in the top beam, we need to set up an optimization problem. By considering the forces and moments acting on the top beam due to the tire swings, we can derive an expression for the maximum bending moment in terms of b. To minimize this expression, we can take the derivative and set it to zero to find the minimum point.
To find the answers to the challenge question (HW7_4_X), we can plot the maximum positive and maximum negative moments in the beam as a function of b. By analyzing the plot, we can determine the optimal value of b that minimizes the maximum bending moment.
Please note that these explanations outline the general approach for finding the answers. To obtain the exact numerical values for each answer, you need to substitute the given parameters into the relevant equations and perform the necessary calculations.