Digital
Learning Tool
for
Engineering
Mechanics
Developed by
Dr S.T. SMITH
Department of Civil Engineering
The
INTRODUCTION
This
digital learning tool (DLT) has been created to facilitate in the learning of
the engineering mechanics component of CIVL1013 Engineering Mechanics and
Materials. CIVL1013 is one of the first courses in the structures stream that
undergraduate students study who are enrolled in the Department of Civil
Engineering at The University of Hong Kong (HKU).
Engineering
mechanics is a core course and is of fundamental importance to the discipline
of structural engineering in addition to other disciplines such as geotechnical
engineering. The fundamental engineering mechanics knowledge learnt in
CIVL1013, addition to the material learnt in ENGG1010 Foundations of
Engineering Mechanics (i.e. HKU Faculty of Engineering Common Course), will
form a solid foundation of knowledge to be used for the following structural
engineering courses. The theory will also be used in elective courses and
graduate courses in the structural discipline.
·
CIVL1007 Theory
and Design of Structures I
·
CIVL2007 Theory
and Design of Structures II
·
CIVL3007 Theory
and Design of Structures III
As
this learning tool is a working document, more tools may be added from time to
time. Please therefore check it regularly. Also, please do email me any
comments/suggestions of relevance you may have at stsmith@hku.hk. Finally, thanks are extended
to Mr Jiaqi YANG (PhD
candidate, HKU) for preparing the animations.
To facilitate your learning of
CIVL1013, please confirm all the results given in this entire learning tool
using the theory you have been presented with in class. You may wish to verify
the results contained in each sub-topic as each sub-topic is presented in
class. Reference will also be made regularly to this document throughout the
lectures.
I
hope this digital learning tool will be helpful to your learning. Enjoy your
digital learning experience!
Scott
SMITH
January
2012
COURSE TOPICS
The
mechanics component of CIVL1013 is divided into the following 5 sub-topics and
is delivered in the first half of the second semester of the academic year. The
latest version of the course is offered in Semester 2 of the 2011-12 academic year.
|
Topic |
Title |
|
A |
Bending Moment, Shear and Axial Force
Diagrams |
|
B |
|
|
C |
Deflections of Beams |
|
D |
Analysis of Stress and Strain and
Stress-Strain Transformation |
|
E |
Buckling of Columns |
DIGITAL RESOURCES
There
are several digital resources available to undertake calculations which vary
considerably in complexity. The following resource is suitable for the level of
CIVL1013. It is also freely available and hence reference will be made to it
herein.
xcalcs: http://www.xcalcs.com
It
is important to note that digital tools are merely tools. They produce an
answer based on the information that is provided. As an engineer (in training),
you need to verify that the computer results are correct. Verification for you
as a student means you need to calculate the results from first principles
using the theory that you have been taught in class. You are strongly
encouraged to supplement your knowledge with independent learning.
The
program ABAQUS, which utilizes the finite element method, has been used to
create some of the following animations. Note that finite element analysis is
outside the scope of CIVL1013 but students will have the opportunity to learn
about it in more advanced undergraduate and graduate courses. Linear-elastic behaviour has been assumed and maintained for all cases
herein.
TOPIC A:
Bending Moment, Shear and Axial Force Diagrams
Note
the following terminology:
BMD
= Bending Moment Diagram
SFD
= Shear Force Diagram
AFD
= Axial Force Diagram
Animations
The
following animations show the development of bending moment and shear force
(and axial force for one case) for a variety of beams, support conditions and
load cases for varying levels of load. Notice in the animations that the shape
of the BMD, SFD and AFD’s do not change but their
magnitude does.
·
Simply-supported
beam subjected to central load (3-point bending)
|
|
|
|
Case 1 |
Case 2 |
·
Cantilevered
beam subjected to free-end load
|
|
|
|
Case 1 |
Case 2 |
·
Simply-supported
beam subjected to uniformly distributed load (UDL)
|
|
|
|
Case 1 |
Case 2 |
·
Simply-supported
beam subjected to central couple
|
|
|
|
Case 1 |
Case 2 |
·
Simply-supported
beam with overhang subjected to point load at overhang tip
|
|
|
|
Case 1 |
Case 2 |
·
Simply-supported
beam subjected to inclined load
|
|
|
|
Case 1 |
Case 2 |
Superposition
|
|
|
|
BMD for UDL |
|
|
BMD for Point Load |
|
|
BMD for UDL + Point Load |
|
|
SFD for UDL |
|
|
SFD for Point Load |
|
|
SFD for UDL + Point Load |
|
Digital Resource
Utilising
xcalcx, analyse the following
beam scenarios. For each scenario, calculate the bending moment (BM) and shear
force (SF) values at mid-span as well as quarter span positions. In addition,
plot the distribution of BM and SF for each beam scenario.
- Simply supported
beam of span
- Simply supported
beam of span
- Cantilever beam of
span
For
each beam scenario, assume the following:
Solid
rectangular cross-section of
Once
you have completed the digital analysis, please confirm all numbers using the
theory you have been taught in CIVL1013.
Once
you are satisfied with all digital and hand-calculation results, then start to
experiment with xcalcx, i.e. experiment with different
support conditions and load types. Practice as many variations as you can and
verify them with hand-calculations. Remember that practice makes perfect!
Note: Expected Answers from xcalcs for
three beam scenarios: (all BM in kNm, all SF in kN) (on account of possible
differences in sign convention, the absolute values of BM and SF have been
given).
Case 1: BM (midspan)=5,
BM (¼ span)=2.5, SF (midspan)=5, SF(¼
span) = 5.
Case 2: BM (midspan)=5,
BM (¼ span)=3.75, SF (midspan)=0, SF(¼
span) =5.
Case 3: BM (support)=10, BM (¼
span from support)=7.5, SF (support)= 10, SF(¼ span from support) = 10.
TOPIC B:
Stresses in Beams and Shear Centre
The
following animations show development of longitudinal stress at the mid-span
position (note that +’ve = tensile and -’ve = compressive stresses). The change in bending moment
and shear force are also provided although only the bending moment information
in needed to calculate the longitudinal stresses. Please verify all results
using the theory you have been provided with in class. You may wish to
supplement your knowledge by consultation of the open literature.
·
Simply-supported
beam subjected to central load (3-point bending)
·
Simply-supported
beam subjected to two point loads (4-point bending)
·
Cantilevered
beam subjected to free-end load
·
Simply-supported
beam subjected to uniformly distributed load (UDL)
The
following animation shows the development of compressive and tensile stress at the
mid-span of a beam subjected to uniform bending. Note the following:
·
Compressive
stress act towards face (i.e. in top half of section)
·
Tensile stresses
act away from section (i.e. in bottom half of section)
·
The neutral axis
(i.e. position of zero bending) is located at the mid-depth of the section.
(note that the neutral axis is located in this position for this example
because the section is symmetric)
Mid-span section (Note: +’ve tensile
stress, -‘ve compressive stress)
Shear-span section
Shear Centre
The
following two animations show the physical reality of shear centre using a
cantilevered C-section beam as an example in which load is applied to the
free-end of the beam. Note the following:
(i) Load is applied through the centroid of the section. In this
case, the beam is shown is shown to deflect vertically downwards WITH rotation.
(ii)
Load is applied through the shear centre of the section. In this case, the beam
is shown to deflect downwards WITHOUT rotation.
Appreciation of shear centre is therefore
important for designers of structures because we would not want a member to
twist if is was supporting say a wall or window.
|
|
|
|
(i)
Load applied through centroid |
(ii) Load applied
through shear centre |
Exercise: The centroid and shear centre of a section may be in the same location
or they may not. As an exercise, calculate the positions of the centroid and shear centre of the following commonly used
structural sections. What do you notice from your calculations?
(i) C-Section
(ii) Rectangular Hollow Section (RHS)
(iii) I-Section
(note: Use dimensions of standard sections
which can be found in steel design handbooks)
TOPIC C:
Deflections of Beams
The
following animations show the relationship between vertical load and central or
free-end vertical deflection. The deflected shape of the beam is also shown. As
an exercise, please calculate all the results below using the theory that you
have been presented with in class (note that the flexural rigidity EI = 1000
kNm2).
·
Simply-supported
beam subjected to central load (3-point bending)
·
Simply-supported
beam subjected to two point loads (4-point bending)
·
Cantilevered
beam subjected to free-end load
·
Simply-supported
beam subjected to uniformly distributed load (UDL)
TOPIC D:
Analysis of Stress and Strain and Stress-Strain Transformations
The
following diagrams are provided to explain the relationship between normal and
shear stress on an element as it is rotated. Five different normal stress
states have been provided as per the following table:
|
|
Maximum Principal
Stress (MPa) |
Minimum Principal
Stress (MPa) |
|
Case 1 |
3 |
1 |
|
Case 2 |
2 |
0 |
|
Case 3 |
1 |
-1 |
|
Case 4 |
-2 |
0 |
|
Case 5 |
-3 |
-1 |
Note:
+’ve tensile normal stress, -‘ve
compressive normal stress
·
Case 1
(Maximum
Principal Stress = 3 MPa, Minimum Principal Stress = 1 MPa)
·
Case 2
(Maximum
Principal Stress = 2 MPa, Minimum Principal Stress = 0 MPa)
·
Case 3
(Maximum
Principal Stress = 1 MPa, Minimum Principal Stress = -1 MPa)
·
Case 4
(Maximum
Principal Stress = -2 MPa, Minimum Principal Stress = 0 MPa)
·
Case 5
(Maximum
Principal Stress = -3 MPa, Minimum Principal Stress = -1 MPa)
TOPIC E:
Buckling of Columns
Buckling Modes
The
following animations illustrate the concepts of (i)
flexural buckling, (ii) torsional buckling, and (iii)
flexural-torsional buckling. A pinned-pinned
C-section strut is used as the example of which an axial load is applied. Note
that the flexural buckling of (i) and (iii) is
occurring about the minor (2-2) axis. As the 2-2 axis has a smaller second
moment of area compared to the major (1-1) axis and as there are not
intermediate restraints, the 2-2 axis is the critical buckling axis.
|
|
|
|
|
(i) Flexural Buckling |
(ii) Torsional Buckling |
(iii) Flexural-Torsional
Buckling |
Load versus Lateral Deflection Responses
The
qualitative relationship between axial load (P) and lateral deflection of the
following four column scenarios is provided herein. The first animation for each
case represents a column with no initial (lateral) imperfection while the
second animation represents the same column with initial imperfection. Initial
imperfections will always exist and they can be induced during the
manufacturing and installation stages of the column.
·
Buckling of
Pinned-Pinned Column
|
|
|
|
Without Imperfection |
With Imperfection |
·
Buckling of
Fixed-Free Column
|
|
|
|
Without Imperfection |
With Imperfection |
·
Buckling of
Fixed-Fixed Column
|
|
|
|
Without Imperfection |
With Imperfection |
·
Buckling of
Fixed-Pinned Column
|
|
|
|
Without Imperfection |
With Imperfection |
Euler Buckling Modes
The
number of buckling half-wavelengths (n) which can be developed in a column will
influence the critical Euler buckling load of the column as per Pcr=n2p2EI/Le2.
The following three animations show the first three buckling modes and the
corresponding Euler buckling loads.
|
|
|
|
|
Pcr=p2EI/Le2 (1 Buckling Half
Wavelength: n = 1) |
Pcr=4p2EI/Le2 (2 Buckling Half
Wavelengths: n = 2) |
Pcr=9p2EI/Le2 (3 Buckling Half
Wavelengths: n = 3) |
Worked Example E-2 (Critical Buckling Stress) of Lecture Notes
The following I-section columns are pin
supported at each end (i.e. pinned-pinned). In addition, the mid-height
position is supported about the minor axis (2-2 axis) direction but not about
the major axis (1-1 axis). Buckled shapes about the major and minor axes are
shown in figures (i) and (ii), respectively, while
buckling is prevented in figure (iii).
|
|
|
|
|
(i)
Buckling about Major Axis (i.e. 1-1 Axis) |
(ii) Buckling about
Minor Axis (i.e. 2-2 Axis) |
(iii) Yield of
Material (no buckling) |
