NATIONAL UNIVERSITY OF SINGAPORE
Department of Mechanical Engineering
ME2113 Mechanics of Materials I
Tutorial 1
Analysis of Stress and Strain
1. An element on a loaded body is found to have the following stresses acting on it:x = 28 MN/m2 ; y = - 14 MN/m
ME2113 Mechanics of Material I Part II
Tutorials Energy Methods
KO KO NAING
National University of Singapore
Ko Ko [email protected]
ME2113 Mechanics of Material I Part II
Part II Q9 Energy Methods
da db
L
=
A(x) =
da dx
x
d2
x
4
=
4
da
Strain Energy U
ME2113 Tutorial 3
1. Determine the maximum slope and deflection in the following beams;
state the magnitude and direction of the slopes and deflections with respect to the
horizontal x-axis (the flexural stiffness EI is constant in all the beams).
(a)
Mxz
Solutions To : Tutorial 2
Q1
(a)
y
5 mm
z
40 mm
For the straight strip of the above dimensions,
3
1
1
1
I z bh 3
40 10 3 5 10 3 m 4
5 10 9 m 4
12
12
12
9
E 210 GPa 210 10 Pa
Now, recall that M xz
Since R 2.5 m ,
M
EI z
M
E
, xx y xz y y
I
R
I
R
z
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mechanical Engineering
ME2113 Mechanics of Materials I
Tutorial 2
Stresses in Beams
1. A straight strip 5 mm thick and 40 mm wide is bent around a cylindrical wine cask of
5 m diameter and the ends are connec
ME2113 Tutorial 1
1. An element on a loaded body is found to have the following stresses acting on it:x = 28 MN/m2; y = -14 MN/m2; xy = 0
Using Mohrs circle or otherwise determine the normal and shear stresses acting on a
plane whose normal is inclined at
ME2113 Mechanics of Material I Part II
Tutorials Q6 to Q8
KO KO NAING
National University of Singapore
Ko Ko [email protected]
ME2113 Mechanics of Material I Part II
Part II Q6
Axial Force: F =
Shear Force: V =
P
2
P
2
cos
0 < < 90
sin
0 < cos < 1
CHAPTER3
STRESSESINLOADEDBEAMS
National University of
Singapore
STRESSES IN LOADED BEAMS
3.1 PURE BENDING
Consider a straight horizontal beam with a cross-section which is
symmetrical about the vertical axis; this beam experiences a bending
moment which i
Chapter 4
DEFLECTION OF BEAMS
4.1 STATICALLY DETERMINATE BEAMS
From beam bending theory
d 2V d 2v
EI 2EI M M
dx dx 2
(1)
The deflection v can thus be found by double integration of Eq (1).
This method is known as the Macaulays Method.
Hence
V
1
Mdxdx
EI
SESSION 2012-13 Semester 1
ME2113 Mechanics of Materials I
Tay C. Tay
A/P C.J.J. Office EA-05-13
[email protected]
Tel: 65162557
Applied Mechanics Group
Applied Mechanics Lab
NUS Staff Web Page
National University of Singapore
Chapter 1
ANALYSIS OF STRE
ME2113 Mechanics of Material I Part II
Tutorials Q1 to Q5
KO KO NAING
National University of Singapore
Ko Ko [email protected]
ME2113 Mechanics of Material I Part II
Part II Q1 Tresca Criterion
Tmax = ?
J
T = xy R
Tresca Criterion: max min = yield
1,2 =
Solutions To : Tutorial 2
Q1
(a)
y
5 mm
z
40 mm
For the straight strip of the above dimensions,
3
1
1
1
I z bh 3
40 10 3 5 10 3 m 4
5 10 9 m 4
12
12
12
9
E 210 GPa 210 10 Pa
Now, recall that M xz
Since R 2.5 m ,
M
EI z
M
E
, xx y xz y y
I
R
I
R
z
EC1301 Principles of Economics
Semester 1 AY2015/2016
Problem Set 8
Question 1
Draw a graph showing a 45-degree line and an aggregate expenditure line.
(a) Choose a point where real GDP is less than aggregate expenditure and label it .
Explain what will h
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ME2113 Tutorial 3 Deection of Beams
Formulae Summary
Ko Ko Naing [email protected]
1
Methods for Deection
1.1
Macaulays Method
(1) Get Moment equation either by singularity or Cutting section methods.
2
d
(2) M (x) = EI dxv = EIv
2
(4) slope = EIv =
Assignment no.1 (Chapter 2) August 2008
In designing a mercury-in-bulb glass thermometer, as shown in the figure below, the
device comprises a spherical bulb of volume Vb1 and a hollow stem of area A1. At a
reference temperature (T1), the mercury fills th
ME2113 Part II
Formulae Summary
Ko Ko Naing [email protected]
1
Yield
1.1
1.1.1
Failure Criteria
Yield Criteria
1. Tresca Criterion:
max min
yield
n
(1)
For plane stresses max = 1
If 2 > 0 min = 0
If 2 < 0 min = 2
2. Mises Criterion:
yield 2
1
(1 2 )2
Frequently Asked Questions
CONTENTS
SECTION (G)
SECTION I
SECTION II
SECTION III
SECTION IV
General Questions
Analysis of Stress and Strain
Bending of Beams
Stresses in beams
Deflections of beams
1
SECTION (G)
General Questions
1) What is the weightage of
Ko Kos Formulae Sheet for ME2113 Tutorial 2
Formulae sheet
Load Intensity, Shear Force and Bending Moment Relationship:
Singularity
(i)
Uniformly distributed load acting downwards
(negative downwards)
(ii)
Linearly-increasingly load intensity (negative
do
Slide 76 of notes
Example
T
T
Bending and torsion moment diagrams - T
600Nm
A
300Nm
C
D
T
300Nm
B
E
-300Nm
-600Nm
How do you get the torsion at different points in the bar?
76
Lecture 3
Explanation
T
To see why the torsion in any section between D
and E
CHAPTER 2
BENDING OF BEAMS
SINGULARITY FUNCTIONS AND SIGN CONVENTION
Examples for the use of beams:
Supporting floors of buildings, bridges
Automotive axles, leaf springs
Airplane wings, brackets.
BENDING OF BEAMS
Singularity Functions
Singularity Func
Chapter 2
BENDING OF BEAMS
(SINGULARITY FUNCTIONS AND SIGN
CONVENTION)
2.1 SINGULARITY FUNCTIONS
Singularity Functions can be used to describe the load
intensity w(x), shear force Fxy and bending moment
Mxz distributions along a beam. These functions are
Chapter 1
ANALYSIS OF STRESS AND STRAIN
1.1 STRESS
S2
n is normal to dA
dF
P
S1 , S2 are tangential
(in plane)
dA
S1
n
Apply general force dF on dA
dFn
Normal
n lim
dA0
Defn:
dA
dFS1
S1 lim
dA0 dA
dFS 2
S 2 lim
dA0 dA
Shear
As dA 0, stress state is
Example
Find horizontal displacement at the free end of the thin curved beam
below due to load P
P
R
E, I
122
Lecture 5
Example
d
P
R
dl
Rsin
R
Ft
2
2
M PR sin
(a)
Strain energy is due to M, thus
Lecture 5
(c)
P 2 R 3 sin 2 d P 2 R 3
2 EI
8EI
0
From the
Chapter 3
STRESSES IN LOADED BEAMS
3.1 PURE BENDING
Bending Deformations
Beam with a plane of symmetry in pure
bending:
member remains symmetric
bends uniformly to form a circular arc
cross-sectional plane passes through arc center
and remains planar
Founded 1905
NATIONAL UNIVERSITY OF
SINGAPORE
Department of Mechanical Engineering
ME2113
MECHANICS OF
MATERIALS I
Course Lecturer:
A/P CJ TAY
i
Founded 1905
SESSION 2012-13
Semester 1
ME2113 Mechanics of Materials I
Modular Credits: 3
Part I Lecture Note
Why Ushear is small for slender beams
1
W P
2
M Px
Recall Example on slide 121
L
U bending
Example
M2
dx
2 EI
0
L
Find the deflection of the free end
of a long thin cantilever beam of
length L subjected to load, P.
P2 x2
P 2 L3
dx
2 EI
6 EI
0
1
P 2 L3