Homework #7 Solution
ME 311-Winter 2016
Due Date 03/11/2016
1
5.10. Find the fully plastic moment MP and the location of the corresponding
neutral axis for the beam section of Figure P5.10, if the uniaxial yield stress is
200 MPa.
25
25
25
5
M
5
20
5
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(a)
(b)
Code:
function [] = HW03TK01()
clear all; close all; clc;
a = 1;
% (a)
f1 = @(x) tan(a*x);
figure(1), hold on, box on, grid on
h11 = ez
Point mass m at a distance .r from the axis of rotation.
A point mass does not have a moment of inertia around its own axisr lJut
_ _ . . . . I = rm2
usmg the parallel aXIS theorem a moment of Inertia around a distant aXlS
of rotation is achie
1
4.3. A cantilever beam of rectangular cross section is loaded by a force F directed along the diagonal AC of the section as shown in Figure P4.3. Show that
the neutral axis in this case coincides with the other diagonal BD for all values
of the dimensio
1
4.18. A beam of circular cross-section, radius 3a has two eccentric holes of radius a as shown in Figure P4.18.
y
a
3a
2
3a
a
x
O
3a
2
Figure P4.18
Find the centroid of the section and the second moments of area, Ix , Iy , Ixy
about centroidal axes.
Den
1
6.9. The beam of Figure P6.1 is fabricated from a channel and two bars by
welding, details of the weld being shown in Figure P6.9. If the minimum weld
thickness t = 1 mm, find the shear stresses in the weld.
A
3
1
3
B
all dimensions in mm
Figure P6.9
If
1
8.3. A cylindrical tank of height h, radius a and wall thickness t is supported
from the top, as shown in Figure P8.3. It is just full of oil of density . Find the
membrane stresses (i) at the top of the tank and (ii) just above the flat bottom
of the t
1
6.45. The T-beam of Figure P6.45 transmits a shear force of 20 kN with the line
of action shown. Find the maximum shear stress and the twist per unit length,
if the material is steel with G = 80 GPa.
20 kN
10
10
100
70
100
all dimensions in mm
Figure P6
1
5.10. Find the fully plastic moment MP and the location of the corresponding
neutral axis for the beam section of Figure P5.10, if the uniaxial yield stress is
200 MPa.
25
25
25
5
M
5
20
5
all dimensions in mm
Figure P5.10
The neutral axis must divide t
1
4.34. Make a copy of the beam section of Figure P4.17 and use the least squares
fit method to estimate the location of the principal axes. Label the stiff and
flexible axes 1 and 2 respectively.
Estimates for the location of the principal axes are shown
1
3.17. The pin-jointed structure of Figure P3.17 consists of three rigid links each
of length L constrained by two springs on the diagonals, whose stiffnesses and
original lengths are k1 , L1 and k2 , L2 respectively.
B
C
k1
k2
L
A
D
L
Figure P3.17
If no
Homework #12 Solution
ME 311-Winter 2016
Due Date 04/18/2016
1
12.7. The uniform beam in Figure P12.7 has a circular cross section of diameter
d and length L. It is subjected to an axial compressive force P which is lower
than the first critical force for
Homework #6 Solution
ME 311-Winter 2016
Due Date 02/26/2016
1
4.37. A beam with the cross section of Figure P4.37 is loaded by a bending moment about the vertical axis as shown. Show on the figure your best estimate of
(i) the location of the principal ax
Homework #1 Solution
ME 311-Winter 2016
Due Date 01/15/2016
1
1.6. A wooden plank is 200 mm wide, 25 mm thick and 1.5 m long. It is simply
supported at the two ends. How far will it deflect if you stand on it at the centre?
Depending on how heavy you are,
Homework #2 Solution
ME 311-Winter 2016
Due Date 01/22/2016
1
3.15. Figure P3.15 shows a mechanism consisting of a set of pin-jointed rigid
bars, constrained by a spring of stiffness k connecting the points BC. A force F is
applied at E. Find a kinematic
Homework #8 Solution
ME 311-Winter 2016
Due Date 03/18/2016
1
6.28. Find the torsional stiffness K for the closed thin-walled beam section of
Figure P6.28. The wall thickness is everywhere equal to t.
a
a
a
a
a
t
Figure P6.28
The perimeter of the section
Homework #3 Solution
ME 311-Winter 2016
Due Date 01/29/2016
1
3.27. A complete circular ring of radius R and flexural rigidity EI is subjected
to two equal and opposite forces F on a diameter, as shown in Figure P3.27. Use
the Rayleigh-Ritz method with a
Homework #5 Solution
ME 311-Winter 2016
Due Date 02/19/2016
1
4.5. The C200 20 channel section of Figure P4.5 has second moments of area
Ix = 15 106 mm4 , Iy = 0.637 106 mm4 . It is loaded by the bending moments
Mx = 2200 Nm and My = 350 Nm. Find the magn
Homework #8 Solution
ME 311-Winter 2016
Due Date 03/18/2016
1
6.3. The wide flange I-beam of Figure P6.3 has second moment of area Ix =
7a3t/12 and is loaded by a vertical shear force Vy . Find (i) the shear stress at A,
(ii) the shear stress at O and (ii
Homework #4 Solution
ME 311-Winter 2016
Due Date 02/05/2016
1
3.45. Figure P3.45 shows a beam of length L and flexural rigidity EI, which is
simply supported at B and supported by a spring of stiffness k at A. Find the
slope of the beam at A when the beam
Homework #10 Solution
ME 311-Winter 2016
Due Date 04/01/2016
1
8.1. A spherical fuel oil tank of radius R sits on an annular ring support, as
shown in Figure P8.1. The oil density is and the oil level is a distance bR from
the crown of the tank.
bR
o
R
45
Examples of modeling, analyisis, and control
Analytical solutions of differential equations and
stability
Analytical solutions of differential equations and
stability
Linearization, particle dynamics
Rigid body dynamics
Elastic and dissipative elements in
Homework #11 Solution
ME 311-Winter 2016
Due Date 04/08/2016
1
8.14. Figure P8.14 shows a spherical roof shell whose thickness t varies with ,
being given by
t = t0 (4 3 cos ) ,
where t0 is a constant.
t
a
o
45
Figure P8.14
Calculate the membrane stresses
ME 360
Modeling, Analysis and Control of
Dynamic Systems
FALL 2015
Department of Mechanical Engineering
University of Michigan, Ann Arbor
Instructors: Prof Gbor Orosz & Ms Jin Ge
Model-Based Analysis of Systems
Metal Cutting
video
Suspension Design
Shimmy
1
1.4. A 1.5 m wooden beam of 25 mm square cross-section is built-in at one end.
If you put all your weight on the other end of the beam will it break? What is
the longest beam of this cross-section that would be strong enough to support
your weight?
Prob