DET
Civil Engineering
Structural Mechanics and Design 1
(Higher)
7529
August 2000
DET
Civil Engineering
Structural Mechanics and Design 1
Higher
Support Materials
HIGHER STILL
DET
: Civil Engineering: Structural Mechanics and Design 1 (Higher)
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CONTENTS
Overview
Teacher’s/Lecturer’s guide
Student’s guide
Tutorials
1. Loads and support reactions for statically determinate beams
2. Construction of shear force and bending moment diagrams
3. Bending and shear stresses in simple beams
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OVERVIEW
These Support Materials are provided to assist teachers/lecturers in delivery of the
Higher Civil Engineering unit Structural Mechanics and Design 1.
The Teacher’s/Lecturer’s guide contains a suggested teaching programme and a
summary of the course content. Advice is offered on use of the additional 40 hours of
the course and on induction to the unit. Essential resources are highlighted, along
with additional resources which will enhance learning and teaching.
Guidance is provided on use of the National Assessment Bank Support Materials and
on the preparation of candidates for external assessment of the Higher course.
The Teacher’s/Lecturer’s guide should be read in conjunction with the Subject Guide
for DET Construction (published by SCCC/HSDU/SFEU 1997) and any subsequent
supplements to the Guide. Reference is also made to the Arrangements Documents
for Higher Civil Engineering.
A series of tutorials, with model solutions, are provided which will assist students in
their learning, and teachers/lecturers in formative assessment and in the preparation of
course estimates. The subjects covered by the tutorials are:
1.
loads and support reactions for statically determinate beams
2.
shear force and bending moment diagrams
3.
bending and shear stresses in simple beams.
The Student’s guide provides a brief introduction to the unit, along with details of
assessment and reassessment. Information is included on use of the additional 40
hours of the course to enhance learning and for consolidation.
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: Civil Engineering: Structural Mechanics and Design 1 (Higher)
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TEACHER’S/LECTURER’S GUIDE
Possible Teaching Programme
Notes
1.
each block in the table represents a nominal 2 hour period
2.
the Internal Assessment items and associated duration are those contained in the
National Assessment Bank Support Materials
3.
those items in the table which are associated with the additional 40 hours allocated
to the course, are indicated in
italics
4.
as it may be assumed that a centre will provide a general induction to all
candidates, only those induction aspects pertaining to the Higher Civil
Engineering have been timetabled here.
Table 1 – Possible Teaching Programme
BLOCK
NUMBER
CONTENT
1 Induction to Higher Civil Engineering
2 Basic Principles of Structural Analysis and Design outcome 1
3 do outcome 1
do (30 minutes) outcome 1
4
Estimation of Loads on Simple Structures outcome 2
Internal Assessment 1 – 30 minutes
5
Estimation of Loads on Simple Structures outcome 2
6 Evaluation of Support Reactions for Statically Determinate Beams outcome 2
7 do outcome 2
Internal Assessment 2 – 45 minutes
8
Construction of Shear Force and Bending Moment Diagrams outcome 3
9 Construction of Shear Force and Bending Moment Diagrams outcome 3
10 Review of Overall Progress in Course
11 Guest Lecture in Building Control
12 Construction of Shear Force and Bending Moment Diagrams outcome 3
13 do outcome 3
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Table 1 – Possible Teaching Programme  continued
BLOCK
NUMBER
CONTENT
14 do outcome 3
15 do outcome 3
Internal Assessment 3 – 45 minutes
16
Determination of Bending and Shear Stresses in Simple Members outcome 4
17 Determination of Bending and Shear Stresses in Simple Members outcome 4
18 do outcome 4
19 do outcome 4
20 do outcome 4
do (60 minutes) outcome 4
21
Computer Solutions to Shear Force and Bending Moment Diagrams and
determination of Bending and Shear Stresses
outcome 5
22 Computer Solutions to Shear Force and Bending Moment Diagrams and
determination of Bending and Shear Stresses
outcome 5
Computer Solutions to Shear Force and Bending Moment Diagrams and
determination of Bending and Shear Stresses (60 minutes)
outcome 5
23
Additional Computing (60 minutes)
24
Additional Computing
25 Internal Assessments 4 and 5 – 80 minutes
26
Final Assessment Preparation
27
Final Assessment Preparation
Total
40 hours plus additional 13 hours
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Teaching programme content
Induction
It is assumed that the centre will provide a general overall induction to candidates
outwith
the time allocated to this unit.
Such general induction might cover areas such as:
•
role of the construction industry in 20/21st centuries
•
reference to recent major Scottish construction projects
•
study facilities within centre.
Induction areas relating particularly to the Higher Civil Engineering course and which
may be covered within the time allocated to this unit are:
•
overview of course, key learning, (60 mins)
•
course expectations and milestones, (30 mins)
•
staff support, (30 mins)
•
learning activities, standards of achievement for course units and course
assessment methods of generating evidence of achievement (60 mins).
Reference should be made to the
Possible Teaching Programmes
for the other two
course units, where time is also allocated to cover the induction areas pertaining to the
Higher Civil Engineering course.
Outcome 1
Explain the basic principles of structural analysis and design (nominally 5 hours,
including assessment).
Performance criteria
PC (a) The purposes of structural analysis and design are explained clearly and
accurately.
Content
•
the purpose of structural analysis and the concept of breaking down complex
structures into simpler elements
•
the interaction between structural analysis and design.
PC (b) The sources of loading upon structures are identified correctly.
Content
•
the meanings of dead, imposed, wind and snow loadings
•
the distinction between concentrated and uniformly distributed loads
•
sources of loading, dead (materials), imposed, snow and wind
•
partial factors of safety.
References
BS 6399 (appropriate parts), weights of common building materials.
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PC (c) The load paths and primary frames of simple structures are illustrated
correctly.
Content
•
concurrent and nonconcurrent coplanar force systems
•
the understanding of the nature and effect of hinged, roller and fixed supports on
the analysis of structural members
•
the load carrying mechanisms for structural members such as: slabs, beams,
columns, foundations and simple plane frames, subject to compressive, tensile,
bending and shear effects as appropriate
•
the tracing of load transfer paths through skeletal structures consisting of slabs,
beams, columns, foundations and frames.
Laboratory equipment
•
demonstration of effects of different supports to beams and columns
•
demonstration of loading paths in a fullscale frame.
Assessment
Referring to National Assessment Bank Unit Assessment, Question 1, parts a, b and c
may be used and should be allocated 30 minutes.
Outcome 2
Estimate the loads acting upon simple structures and evaluate the support reactions for
statically determinate beams (nominally 7 hours, including assessment).
Performance criteria
PC (a) The conditions of determinacy for beams are stated correctly.
Content
•
the equations of equilibrium (
∑
V = 0,
∑
H =0,
∑
M =0)
•
the determinacy of beams, applied to simply supported beams, cantilever beams
and simply supported beams with overhangs.
PC (b) The loading on statically determinate beams is assessed correctly.
Content
•
calculation of loadings on statically determinate beams of the forms covered in
PC (a).
Tutorial
Tutorial 1 – Determination of loads and evaluation of support reactions for
statistically determinate beams. (Questions 1, 2, 3 and 4).
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PC (c) The support reactions for statically determinate beams are evaluated
correctly.
Content
•
evaluation of support reactions for statically determinate beams of the forms
covered in PC (a).
Tutorial
Tutorial 1 – Determination of loads and evaluation of support reactions for
statistically determinate beams. (Questions 5, 6, 7, 8 and 9).
Assessment
Referring to National Assessment Bank Unit Assessment, Question 1, part d, question
2, parts a and b and question 3, parts a and b may be used and should be allocated 45
minutes.
Outcome 3
Construct shear force and bending moment diagrams for statically determinate beams
(nominally 11 hours, including assessment).
Performance criteria
PC (a) Shear force and bending moment diagrams are drawn accurately.
PC (b) Significant values of shear force and bending moment are identified
correctly.
PC (c) Shear force and bending moment values for standard applied loading cases
are identified and used correctly.
PC (d) The location of points of contraflexure for simply supported beams with
overhangs are calculated correctly.
Content
•
sign conventions for shear force and bending moment calculations
•
relationships between shear force and bending moment e.g. maximum bending
moment occurring at point of zero shear
•
points of contraflexure on appropriate beams (using a quadratic equation for
solution)
•
the importance and uses of shear force and bending moment diagrams in the
design process
•
shear force and bending moment diagrams to be drawn for simply supported
beams, cantilever beams and simply supported beams with overhangs, considering
concentrated loads, uniformly distributed loads (partly or wholly covering a span),
or a combination of concentrated and uniformly distributed loads
•
development and use of shear force and bending moment values for standard cases
viz: uniformly distributed load on simply supported and on cantilever beams,
central concentrated load on simply supported beam, end concentrated load on
cantilever beam.
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Tutorial
Tutorial 2 – Construction of shear force and bending moment diagrams.
Laboratory equipment
Demonstration of changes of longitudinal beam profile under variation of load.
Assessment
Referring to National Assessment Bank Unit Assessment, question 1, part e, question
2, part c and question 3, part c may be used and should be allocated 40 minutes.
Outcome 4
Determine the bending and shear stresses in simple members (nominally 13 hours,
including assessment).
Performance criteria
PC (a) Second moments of area of symmetrical sections are calculated accurately.
Content
•
calculation of second moment of area of solid rectangular sections and of hollow
rectangular sections in steel and in timber
•
second moment of area of standard steelwork sections obtained from appropriate
section properties.
References
Tables of section properties of standard steel sections viz: universal beams and hollow
rectangular sections.
Tutorial
Tutorial 3 – Determination of bending and shear stresses in simple members.
PC (b) The equation of simple bending is applied correctly to determine bending
stresses.
PC (c) Shear stresses at significant positions are determined correctly.
PC (d) Comparisons between evaluated and permissible bending and shear
stresses are carried out correctly.
Content
•
development of expressions
σ
= M y/I (bending) and
τ
= V A y/I b (shear) for
beams of crosssection indicated in PC (a)
•
bending and shear stress distribution over the depth of section
•
permissible stresses in timber and steel.
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Tutorial
Tutorial 3 – Determination of bending and shear stresses in simple members.
Assessment
Referring to National Assessment Bank Unit Assessment, question 1, part f, question
2, parts d, e and f and question 3, part d may be used and should be allocated 50
minutes.
Outcome 5
Solve structural problems using computerised techniques (nominally 4 hours,
including assessment).
Performance Criteria
PC (a) Statically determinate beams are analysed correctly to determine the shear
force and bending values using computer techniques.
PC (b) Bending and shear stresses for standard sections are determined accurately
using computer techniques.
PC (c) The results from computer programmes are correctly checked by manual
means.
Content
•
use of appropriate commercial software and/or inhouse packages and/or
spreadsheets to solve problems covered in Outcomes 3 and 4
•
information on range of commercial software available
•
importance of manual checks on computer output.
Assessment
Referring to National Assessment Bank Unit Assessment, an assignment may be used
and should be allocated 30 minutes.
Notes on Outcome 5
•
The Assessment for this outcome is shown in the Possible Teaching Programme
as being combined with that for Outcome 4.
•
Time for candidates to gain more experience in the use of computer software is
shown in the Possible Teaching Programme. This time is part of the additional 40
hours available in the Higher Civil Engineering Course.
General resources
It is expected that centre staff will use references to relevant text book(s) during the
delivery of the unit. Candidates may also be recommended to refer to such texts.
As there are many textbooks covering the material of the unit, it was felt that it was
not therefore necessary to indicate a specific text here.
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Assessment
Internal assessment
As indicated in the Possible Teaching Programme, time is allocated for Internal
Assessments.
The National Assessment Bank contains examples of suitable Internal Assessments
which will assess all the Performance Criteria within each Outcome of the unit.
Maximum times for the completion of each assessment is also indicated in the
National Assessment Bank.
All Internal Assessment should be undertaken under supervision to ensure that it is the
candidate’s own work.
Reassessment
Where a candidate has not attained the standard necessary to pass a particular
Outcome or Outcomes, there should be the opportunity for that candidate to be
reassessed. As a general rule, reassessment in a particular instrument should be
offered on a maximum of two occasions. While the actual timing of reassessment is
at the discretion of the individual centre, it may be beneficial to the candidate
concerned if the first opportunity for reassessment was made available, as soon as
practicable after the original assessment had been undertaken.
The second reassessment might have to be delayed until near the end of the unit, for
practical reasons.
Evidence from the original unit assessment should assist staff in identifying why an
individual candidate has failed to achieve a particular outcome and hence to plan
suitable support for learning.
The actual reassessment instrument will depend upon the extent to which the
candidate has failed to achieve the outcome concerned. It might, for example require
the candidate to repeat all or part of the original instrument, or, in other cases,
undertake a new instrument. In all cases, the original conditions under which the
instrument was undertaken should be repeated but the time allowed would be
allocated on a prorata basis.
While any overall discussion/analysis of the original assessment instrument should be
delayed until the first opportunity for reassessment has been undertaken, those
candidates requiring the reassessment should be advised of their weaknesses and
guidance should be given to them on how they can find information which would help
them in their reassessment. It may be beneficial to hold a more formal revision lesson
if there are a substantial number of candidates requiring reassessment.
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Recording and retention of evidence
It is important that records of unit assessment and evidence of assessment be retained
for external moderation and appeals purposes, in accordance with the requirements of
the Scottish Qualifications Authority.
While centres will normally have their own standardised recording systems, examples
of possible attainment and marks records and an assessment summary record are
given in the National Assessment Bank pack for this unit. Particular attention should
be made to those Instruments of Assessment for which cutoff scores are applicable.
(See later notes on course estimates).
Uses of internal assessments as teaching aids
In addition to the most obvious use of the Internal Assessments i.e. to judge the
competence of the candidate in each outcome, they can also be used in the following
ways:
•
Once the assessment and the initial reassessment have been completed, then an
overall discussion/analysis of the instrument can take place. This will provide an
opportunity for candidate feedback, consolidation of learning and any integration
with other outcomes within this unit and with other units in the course.
•
Together with the attainment and marks records of the other units of the course, a
periodic review of the course can be undertaken. (See Possible Teaching
Programme for all three units). It is suggested that such reviews should take place
twice during the year, at such times when sufficient evidence is available to allow
meaningful decisions to be made on individual candidate’s future targets,
modification to teaching programmes and techniques, etc.
External assessment
The external assessment for the course consists of a 3 hour examination paper,
externally devised and assessed.
The Paper will be in two sections viz:
Section A  five short answer and restricted response questions, of which all must be
attempted, worth 40 marks.
Section B  five extended response and/or structured questions, of which candidates
will be required to attempt three, worth 60 marks.
The questions in both sections of the examination paper will sample across the content
of the three units of the course and the spread of marks, in both sections will be
roughly in proportion to the time allocated to the teaching of each outcome.
A specimen question paper will be available to indicate the format in which the
examination paper will be set.
On the basis of this examination paper, successful candidates will be graded A, B or
C, for the course. The relative grade descriptions are given in the Arrangements
Document.
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Preparation of candidates for external assessment
As the nature and purpose of the Instruments of Internal Assessment may differ from
the those of the questions in the External Assessment examination paper, it will be
necessary to give candidates some practice in answering the relative types of question
and guidance in examination techniques.
From the Possible Teaching Programme, it can be seen that time is scheduled for this
towards the end of the unit. (See also the programmes for the other two units of the
course).
Candidate estimates
Centres will be required to furnish estimates of candidates likely performance in the
course and of their likely grading. This information will be used in the case of
appeals by the centre and to eventually establish the degree of concordance of a
centre. The estimates must be submitted by April, at the latest.
Course estimates will require to be made by considering the performance of
candidates over all three units. In the case of this particular unit, guidance is given in
the National Assessment Bank, where, in each of the Instruments, candidates can
exceed the cutoff scores which signify competence in an outcome. Thus the actual
scores can be used to provide part of a candidate’s estimate. As an alternative to this,
a prelim examination could be held.
Such a prelim will be more meaningful if held towards the end of a unit but in order to
produce the estimate by April, it will need to be undertaken by March.
Additional 40 hours
For convenience, this time is shown as being spread over the three units,
approximately 13 hours per unit (see Possible Teaching Programme). Some of the
time shown is clearly dedicated to this unit viz:
•
additional computing, which is intended to give sufficient practice to all
candidates, whatever their background and computer experience
•
external assessment practice.
The remaining time scheduled for induction, review of overall progress and guest
lecture on building control can be considered as being general and should be read in
conjunction with the general aspects shown in the Possible Teaching Programmes of
the other two units.
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STUDENT’S GUIDE
Outcomes covered in unit
1.
Explain the basic principles of structural analysis and design:
i)
The purposes of structural analysis and design.
ii)
The sources of loading upon structures.
iii)
The load paths and primary frames of simple structures.
2.
Estimate the loads acting upon simple structures and evaluate the support
reactions for statically determinate beams:
i)
The conditions of determinacy for beams.
ii)
The loading on statically determinate beams.
iii)
The support reactions for statically determinate beams.
3.
Construct shear force and bending moment diagrams for statically determinate
beams:
i)
The drawing of shear force and bending moment diagrams.
ii)
Significant values of shear force and bending moments.
iii)
Shear force and bending moment values for standard loading cases.
iv)
Location of points of contraflexure for simply supported beams with
overhangs.
4.
Determine bending and shear stresses in simple members:
i)
Second moments of area of symmetrical sections.
ii)
The equation of simple bending.
iii)
Shear stresses at significant positions.
iv)
Comparison between evaluated and permissible bending and shear stresses.
5.
Solve structural problems using computerised techniques:
i)
Shear force and bending values in statically determinate beams by computer
techniques.
ii)
Bending and shear stresses for standard sections by computer techniques.
iii)
Computer program results checked manually.
Internal assessment
The unit shall be assessed internally by means of a series of Instruments of
Assessment.
These Instruments will cover all of the Outcomes and all of the Performance Criteria
within each Outcome.
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Structural Mechanics and Design will be assessed by two Instruments, as follows:
•
Instrument of Assessment 1
–
will consist of three structured questions, each
question being based upon a particular beam. The three beams will cover a range
of loading criteria, support conditions and constituent materials. In the context of
the three beams, all of the Performance Criteria of Outcomes 1,2,3 and 4. This
will involve explaining the basic principles of structural analysis and design,
estimating the loads acting on beams, evaluating the support reactions,
constructing relevant shear force and bending moment diagrams and checking if
the beam sections allocated are satisfactory, under the given loading.
The questions undertaken will be ‘closed book’, although any relevant section
property tables will be provided. A total time allowance of 2 hours 45 minutes
will be allowed, however it is anticipated that this will be split up to allow the
relevant part(s) of each question to be undertaken at the end of the appropriate
outcome.
•
Instrument of Assessment 2
– will consist of an assignment, using a relevant
computer package or packages to cover Outcome 5. The shear force and bending
moment diagrams will be created and hence the maximum bending and shear
stresses found for one of the beams in Instrument of Assessment 1. The results
obtained will be compared with those deuced by manual calculation.
Access to the appropriate instruction manual or explanatory notes on the
program(s) will be available and a time allowance of 30 minutes will be given.
Required achievement standard in instruments of assessment
The criteria for determining whether an Outcome/Performance Criteria has been
satisfactorily completed will be clearly indicated, prior to the undertaking of the
relevant Instrument. For example, the minimum mark required to constitute a pass,
the cutoff score, will be indicated prior to that part being undertaken.
Reassessment
In the event of a partpass or nonpass being recorded, there will be an early
opportunity for reassessment in that particular Outcome. There will be some guidance
given by the teacher/lecturer on the shortcomings of the original response before the
reassessment s attempted. Depending upon the original response, the reassessment
may consist of:
•
a repeat of the original instrument
•
a new instrument requiring responses to all sections covered in the outcome
•
a repeat of the section(s) of the original instrument in which the responses had
been unsatisfactory
•
a new instrument requiring responses only in the section(s) in which the original
responses had been unsatisfactory.
If c or d above is used, then the time allocation will be on a prorata basis to the
original.
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If after this reassessment, a pass is still not attained, then a second reassessment will
be arranged, towards the end of the unit. This reassessment will follow the format a,
b, c and d above and at this stage, a reassessment will be available for any of the unit
outcomes yet to be passed.
No further opportunities for reassessment are likely to be available during that present
running of the unit.
External assessment
The external assessment for the course consists of a 3 hour examination paper,
externally devised and assessed.
The Paper will be in two sections viz:
Section A –five short answer and restricted response questions, of which all must be
attempted, worth 40 marks.
Section B –five extended response and/or structured questions, of which candidates
will be required to attempt three, worth 60 marks.
The questions in both sections of the examination paper will sample across the content
of the three units of the course and the spread of marks, in both sections will be
roughly in proportion to the time allocated to the teaching of each outcome.
On the basis of this examination paper, successful candidates will be graded A, B or
C, for the course.
Preparation of candidates for external assessment
As the nature and purpose of the Instruments of Internal Assessment may differ from
the those of the questions in the External Assessment examination paper, it will be
necessary for candidates to have some practice in answering the relative types of
question and guidance in examination techniques. It is expected that this will be
carried out for all three units towards the end of the course. (See below).
Presentation of unit
The centre will prepare a structured teaching programme for the unit. This
programme will detail the coverage of the unit material in a logical order and will
timetable the Instruments of Assessment as they are required. Structured tutorial
sheets will be issued and tutorial time will be allocated in the teaching programme.
Laboratory demonstrations/experiments will also be carried out within the
programme.
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Use of additional 40 hours
In addition to the three units of the course, which are allocated 40 hours each, an
additional 40 hours is included in the course. This additional 40 hours is intended to
be used to provide:
•
Induction – see below.
•
Contacts with industry which will enable candidates to see the relevance of what
they are learning. This is accomplished by site visits, supplemented with previsit
and postvisit discussions, guest visiting lecturers and the use of videos.
•
Periodic review of the overall course. This will probably take place at two suitable
points during the course and will give candidates the opportunity to evaluate their
progress to date and discuss any changes which they will need to make in the
remainder of the course to achieve success.
•
Additional time to gain more expertise in areas of the course with which they may
be having difficulty e.g. in this unit, there will be additional time available for
practice with the computing programs used in analysis.
•
Preparation for the external assessment.
For timetabling convenience, it is likely that the 40 hours will be split amongst the
three units, except for those areas which are directly applicable to a particular unit.
Induction
A well planned induction is intended and will provide candidates with a confident
start and will also break down the barriers to learning. The Induction should be seen
as being in two parts:
1.
General
: This is not specific to the Higher Civil Engineering Course and should
be a general induction to the centre. It should cover Study facilities available in
the centre, the concepts of Internal and External Assessment, Scottish Group
Awards, etc.
2.
Specific
: For the Higher Civil Engineering Course the following should be
covered, in all three units:
•
overview of the course, key learning points
•
course expectations and milestones
•
teaching programmes, incorporating aspects of the additional 40 hours
•
learning activities
•
standards of achievement for units and course
•
assessment and methods of generating evidence of achievement
•
reference material, mandatory and nonmandatory.
Portfolios of internally assessed work
All work submitted by candidates for Internal Assessment will be marked and kept in
individual portfolios. Although candidates will see the marked assessments for
information on progress, advice for reassessment etc., the portfolios will be retained
by the centre, during the currency of the unit. This is done for the purpose of
inspection by External Verifiers and the possible mounting of appeals in relation to
the course result.
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Candidate estimates
Centres will be required to furnish estimates of candidate’s likely performance in the
course and of their likely grading. This information will be used in the case of
appeals by the centre, and require to be submitted by April, at the latest.
Course estimates will require to be made by considering the performance of
candidates over all three units. In the case of this particular unit, candidates can
exceed the cutoff scores which signify competence in an outcome. Thus the actual
scores can be used to provide part of a candidate’s estimate. As an alternative to this,
a prelim examination could be held. Such a prelim will be more meaningful if held
towards the end of a unit but in order to produce the estimate by April, it will need to
be undertaken by March.
Guidance on this will be provided during the induction period.
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TUTORIAL 1
Loads and support reactions for statically determinate beams
(Loading on beams should be illustrated on line diagrams, where this will be clearer)
1.
For the part floor plan of the steelframed structure, shown in figure Q1, together
with the relevant design data, evaluate:
i)
The loading on Beam A, which is simply supported over a span of 4 m.
ii)
The loading on Beam B, which is simply supported over a span of 6 m.
2.
For the part floor plan of the steelframed structure, shown in figure Q2, together
with the relevant design data, evaluate the loading on Beams B1, B2 and B3.
3.
A timber – floored structure, together with the relevant design data is shown in
plan in Fig.Q3. Evaluate the loading on Beams A, B, C and D.
4.
Fig.Q4 shows, in plan, part of a steelframed structure, together with the relevant
design data. Considering Beam C, evaluate the loading on the complete length of
the beam.
5.
Considering the beam loadings found in Q1, above, evaluate:
i)
The end reaction from Beam A on Beam B.
ii)
The end reaction from Beam B on the column X.
6.
Considering the beam loadings found in Q2, above, evaluate the end reactions
from Beam B3 on columns X and Y.
7.
Considering the beam loadings found in Q3, above, evaluate:
i)
The end reactions of the five beams on Beam E.
ii)
The end reactions of Beam E on the supporting walls.
8.
Considering the loading on Beam C, found in question 4 above, evaluate the end
reactions from the beam on the supporting columns.
9.
For the beams shown in figure Q9, evaluate the end reactions.
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TUTORIAL 1: FIGURES
4
m
4
m
BEAM A
FLOOR
SPAN
X
FLOOR
SPAN
BEAM B
3 m
3 m
6 m
Figure Q1
Concrete floor slab = 250 mm thick Imposed load on slab = 5 kN/m
2
Self weight of all beams = 1.5 kN/m Brickwork on Beam B = 9kN/m
4m
4m 4m2m2m
10m
FLOOR
SPAN
FLOOR
SPAN
FLOOR
SPAN
FLOOR
SPAN
BEAM B3
BEAM B2
BRICKWORK
BEAM B1
S
T
A
I
R
W
E
L
L
S
T
A
I
R
W
E
L
L
Z
Y
X
Figure Q2
Brickwork as indicated = 9 kN/m Self weight of all beams = 1 kN/m
Concrete slab thickness = 200 mm Imposed load on floor = 4 kN/m
2
DET
: Civil Engineering: Structural Mechanics and Design 1 (Higher)
21
TUTORIAL 1 – FIGURES
500
500 500
5
00 500
5
00
5
00
3m
BEAM E
X
Y
S
UPPORTING
W
ALLS
HEAVY EQUIPMENT
NO IMPOSED ;LOAD
BENEATH  BUT 4
POINT LOADS  EACH 2kN
400400
400
1.5m
BEAM A
BEAM A
BEAM B
BEAM C
BEAM D
Figure Q3
Dead load of flooring, ceiling and Beams A, B, C and D equivalent to 1.5 kN/m
2
.
Self weight of Beam E = 0.6 kN/m. Imposed loading on floor = 2 kN/m
2
6m
3
m
1.5 m
COLUMN A
ABOVE & BELOW
FLOOR
C
OLUMN B
BELOW FLOOR
BEAM C
C
OLUMNS ABOVE
F
LOOR EACH
2
0kN LOADING
BRICKWORK
2 m
1.5 m
5 m
Figure Q4
Slab self weight = 5 kN/m
2
Brickwork = 8 kN/m
Self weight of all beams = 2 kN/m Imposed load on slab = 3 kN/m
2
DET
: Civil Engineering: Structural Mechanics and Design 1 (Higher)
22
TUTORIAL 1 – FIGURES
4kN/m
1
6kN/m
3
6kN
L
R
1
.5m3m
6
m
(
a)
1m3m
5
m
2kN
8
kN
1kN/m
L
(b)
L
R
(
c)
6m
1m
1
m
2
m
2
m
5kN
3
0kN
8
kN/m
Figure Q9
DET
: Civil Engineering: Structural Mechanics and Design 1 (Higher)
23
SOLUTIONS TO TUTORIAL 1
1.Beam A: Width of floor carried per m run = 3/2 + 3/2 = 3 m
Weight of slab per square metre = 1 x 1 x 0.25 x 24 = 6 kN
Load per m run = 1.5 + 3 x 1 x 6 + 3 x 1 x 5 = 34.5 kN
Beam B: Uniformly distributed load = 1.5 + 9 = 10.5 kN/m
Central concentrated load = 34.5 x 4/2 = 69 kN
2 Beam 1: Uniformly distributed load = 1 + 9 = 10 kN/m
Beam 2: Uniformly distributed load = 1 + 9 = 10 kN/m
Central concentrated load = 10 x 4/2 = 20 kN
Beam 3: Concentrated load at Z = 20/2 + 10 x 4/2 = 30 kN
Uniformly distributed load (XZ) = 1 + 4/2 x (1 x 1 x 0.2 x 24) + 4/2 x 4
= 18.6 kN/m
Uniformly distributed load (ZY) = 1 + 4 x(1 x 1 x 0.2 x 24) + 4 x 4
= 36.2 kN/m
3.Beam A: Uniformly distributed load = (0.5/2 + 0.5/2) x (1.5 + 2) = 1.75 kN/m
Beam B: Uniformly distributed load (not under equipment) = 1.75 kN/m
Uniformly distributed load (beneath equipment) = 0.5/2 x (1.5 + 2) + 0.5/2 x
1.5 = 1.25 kN/m
Concentrated loads at corners of equipment = 2 kN
Beams C and D:Similar to Beam B, in magnitudes but with different
combinations (best illustrated with diagrams).
4.Beam C: (6 m Span) Uniformly distributed load = 5/2 x (5 + 3) + 2 = 22 kN/m
(cantilever) Uniformly distributed load = 2 kN/m
End concentrated load = 1.5 x 8 + 5/2 x 2 + 20 + 3/2 x (5 + 3) x 2.5 = 67 kN
5.End reaction from Beam A on Beam B = 34.5 x 4/2 =69 kN
End reaction from Beam B on Column X = 69/2 + 10.5 x 6/2 = 66 kN
6.Taking moments about Y: RX x 10 = 30 x 6 + 18.6 x 4 x 8 + 36.2 x 6 x3
RX = 142.68 kN
RY = 30  18.6 x 4  36.2 x 6 + 142.68 = 178.92 kN
DET
: Civil Engineering: Structural Mechanics and Design 1 (Higher)
24
SOLUTIONS TO TUTORIAL 1
7.Reactions on Beam E:
From Beam A (2 beams) = 3/2 x 1.75 = 2.625 kN
From Beam B = (1.75 x 2.2 x 1.9 + 1.75 x 0.4 x 0.2 + 1.25 x 0.4 x 0.6 + 2 (0.8
+ 0.4))/3 = 3.385 kN
From Beam C = (1.75 x 1.5 x 2.25 + 1.75 x 0.3 x 0.95 + 1.75 x 0.4 x 0.2 +
1.25 x 0.4 x 1.3 + 1.25 x 0.4 x 0.6 + 2(0.4 + 0.8 + 1.1 +
1.5))/3 = 5.032 kN
From Beam D = (1.75 x 1.5 x 2.25 + 1.75 x 1.1 x 0.55 + 1.25 x 0.4 x 1.3 +
2(1.1 + 1.5))/3 = 4.272 kN
Taking moments about Y:
RX x 3 = 2.625 x 2.5 + 2.625 x 2 + 3.385 x 1.5 + 5.032 x 1 + 4.272 x 0.5 +
0.6 x 3 x 1.5
RX = 8.919kN
RY = 2.625 x 2 + 3.385 + 5.032 + 4.272 + 0.6 x 3  8.919 = 10.82 kN
8.Beam C:Taking moments about left hand supporting column A:
RB x 6 = 22 x 6 x 3 + 2 x 3 x 7.5 + 67 x 9
RB = 174 kN
RA = 22 x 6 + 2 x 3 + 67  174 = 31 kN
9. a) Taking moments about R:
RL x 6 = 16 x 6 x 3 + 4 x 3 x 4.5 + 36 x 1.5 thus RL = 66 kN
RR = 16 x 6 + 4 x 3 + 36  66 = 78 kN
b) RL = 3 x 1 + 8 + 2 = 13 kN
ML = 3 x 1 x 1.5 + 8 x 4 + 2 x 5 = 46.5 kNm
c) Taking moments about L:
RR x 6 = 30 x 1 + 5 x 8 + 8 x 3 x 5.5 thus RR = 33.67 kN
RL = 30 + 5 + 8 x 3  33.67 = 25.33 kN
DET
: Civil Engineering: Structural Mechanics and Design 1 (Higher)
25
TUTORIAL 2
Construction of shear force and bending moment diagrams
(The examples in this tutorial are those for which the support reactions were evaluated
in Tutorial 1).
Construct, to suitable scales, the shear force diagram and hence the bending moment
diagram, indicating salient values in all diagrams.
1.
6
9kN
3m
6
m
66kN
6
6kN
1
0.5kN/m
2.
3
0kN
1
0m
4m
1
8.6kN/m
3
6.2kN/m
178.92kN
142.68kN
3.
0.5m 0.5m
0
.5m
0
.5m
0
.5m 0.5m
3
m
2.625kN 2.625kN
3
.385kN
5
.032kN
4
.272kN
10.82kN8.919kN
0.6kN/m
DET
: Civil Engineering: Structural Mechanics and Design 1 (Higher)
26
TUTORIAL 2
4.
3
m
6m
31kN
1
47kN
67kN
22kN/m
2
kN/m
In addition, evaluate the position of the point of contraflexure.
5.
1
.5m3m
6
m
66kN 78kN
3
6kN
1
6kN/m
4
kN/m
6.
1m3m
5
m
13kN
2kN
8
kN
1kN/m
46.5kNm
7.
2m
2
m1m
1
m
6
m
25.33kN
5kN
3
0kN
8
kN/m
In addition, evaluate the position of the point of contraflexure.
DET
: Civil Engineering: Structural Mechanics and Design 1 (Higher)
27
SOLUTIONS TO TUTORIAL 2
SFD
66k N
66k N
34.5kN
34.5kN
BMD
150.75kNm
SFD
142.68kN
6
8.28kN
3
8.28kN
1
78.92kN
1.O57m
B
MD
4
21.92kNm
442.15kNm
SFD
8.919
8.619
5.994
5.694
3.069
2
.769
0
.616
0.916
5.948
6.248
10.52
1
0.82
B
MD
4
.385
k
Nm
7
.307
k
Nm
8
.767
k
Nm
8.384
kNm
5.34
kNm
DET
: Civil Engineering: Structural Mechanics and Design 1 (Higher)
28
SOLUTIONS TO TUTORIAL 2
SFD
BMD
31kN
101kN
73kN
6
7kN
1.409m
2
.818m
2
1.86kN
210kNm
P
OINT OF CONTRAFLEXURE:
3
1x = 22x
2
2
x = 2.818 m
SFD
3.375m
66kN
6
kN
18kN
54kN
7
8kN
B
MD
1
08
k
Nm
109.125
kNm
99
kNm
SFD
13kN
10kN10kN
2kN
BMD
46.5kNm
12kNm
2kNm
SFD
25.33kN
25.33kN
4.67kN
4.67kN
20.67kN
13k N
5kN5kN
BMD
5.197m
25.33kNm
11.33k Nm
14k Nm
5kNm
DET
: Civil Engineering: Structural Mechanics and Design 1 (Higher)
29
TUTORIAL 3
Bending and shear stresses in simple beams
1.
The analysis of a loaded steel beam gives the maximum shear force and the
maximum bending moment as 66 kN and 150.75 kNm, respectively. It is proposed
that a 533 x 210 x 82 Universal Beam be used for the beam.
a)
Calculate:
i)
The actual maximum shear stress on the section, using the beam dimensions from
the appropriate Section Property Tables.
ii)
The actual maximum bending stress on the section using the appropriate section
properties from tables.
b)
If the maximum permissible bending stress on the steel is 165 N/mm
2
, evaluate
the minimum second moment of area required for a suitable beam section and
hence choose the most economic Universal Beam section, from appropriate tables.
2.
The analysis of a loaded rectangular, hollow steel beam gives the maximum shear
force and the maximum bending moment as 178.92 kN and 442.15 kNm,
respectively.
The section proposed for the beam is to be 500 mm deep, 300 mm wide with webs
and flanges all 12.5 mm thick.
a)
Evaluating all the required section properties from the given section dimensions,
calculate the actual maximum shear stress and bending stress on the section and
compare the values obtained with the permissible shear and bending stresses of
115 N/mm
2
and 165 N/mm
2
, respectively.
b)
If at a later stage, it is found that the beam provided is of a higher grade material,
with a permissible bending stress of 230 N/mm
2
, calculate the maximum uniformly
distributed load which the beam could carry over a simply supported span of 10m.
3.
The analysis of a loaded rectangular timber beam gives the maximum shear force
and the maximum bending moment as 10.82 kN and 8.77 kNm respectively. The
beam is to be 50 mm wide.
a)
If the maximum permissible shear stress and bending stress for the timber is 0.66
N/mm
2
and 4.1 N/mm
2
, respectively, calculate a suitable depth for the rectangular
section.
b)
At a point in the span of the beam, where the bending moment is reduced to 4.4
kNm, the depth found in a) is reduced by 50 mm to allow for services. Check that
the reduced can carry the appropriate bending moment.
DET
: Civil Engineering: Structural Mechanics and Design 1 (Higher)
30
TUTORIAL 3
4.
a)
A timber floor is carried by 222 x 40 mm rectangular timber joists, at 500
mm spacing, spanning 3.5 m, simply supported. If the floor loading is 2.0 kN/m
2
,
inclusive of all self weights,
check the suitability of the given joist section, in both
shear and bending, assuming the permissible shear and bending stresses to be 0.67
N/mm
2
and 5.3 N/mm
2
, respectively.
b)
Considering bending only, calculate a suitable spacing of the joists if they were
increased to 250 x 50 mm in section, under the same magnitude of loading, but
now using timber with a permissible bending stress reduced to 4.1 N/mm
2
.
5.
A small floor 5 m x 4.5 m is to be supported by one main steel beam and 200 x 50
mm timber joists, spanning between the wall and the steel beam, as shown in
Fig.Q5.
a)
Calculate the safe floor load in kN/m
2
, inclusive of all self weights and
considering only bending in the timber joists. The permissible bending stress of the
timber is 4.1 N/mm
2
.
b)
Considering bending only, calculate a suitable section modulus (I/y) for the steel
beam, assuming the permissible bending stress to be 165 N/mm
2
, and hence
choose, from Section Property Tables a suitable Universal Beam section.
(To simplify analysis, the loading on the UB may be assumed as uniformly
distributed).
Check the chosen Universal Beam section for shear, assuming the permissible stress
to be 115 N/mm
2
.
2.5m
2
.5m
TIMBER JOISTS
AT 500 CENTRES
S
UPPORTING WALLS
Figure Q5
4.5 m
DET
: Civil Engineering: Structural Mechanics and Design 1 (Higher)
31
SOLUTIONS TO TUTORIAL 3
1.
a) i)
Maximum shear stress = V A y/I b where V = 66000 N, I = 47540 x 10
4
mm
4
A y = (208.8 x 13.2 x 257.55 + 250.95 x 9.6 x 250.95/2) = 1.012 x 10
6
mm
3
b = 9.6 mm
Thus maximum shear stress = 14.64 N/mm
2
ii)
Maximum bending stress = (150.75 x 10
6
x 528.3 /2)/47540 x 10
4
= 83.76 N/mm
2
b)
If the maximum permissible stress is 165 N/mm
2
, then the minimum I/y required
to carry a bending moment of 150.75 kNm is found from:
150.75 x 10
6
/I = 165/y thus I/y = 0.9136 x 10
6
mm
3
= 913.6 cm
3
I/y is the elastic Modulus about XXaxis.
Thus from Tables select a 406 x 178 x 64 UB or a 457 x 152 x 52 UB
2.
a)
I = (300 x 500
3
 275 x 475
3
)/12 = 0.66897 x 10
9
mm
4
Shear stress = 178.92 x 1000 (300 x 12.5 x 243.75 + 25 x 237.5 x 237.5/2)

0.66897x10
9
x 25
= 17.32 N/mm
2
< 115 N/mm
2
Bending stress =
442.15 x 10
6
x 250/0.66897 x 10
9
= 165.24 N/mm
2
> 165 N/mm
2
b)
Maximum Bending Moment = 230 x 0.66897 x 10
9
/250 Nmm = 615.45 kNm
= w x 10 x 10/8 thus w = 49.24 kN/m
3.
a)
Shear Force = 10.82 x 1000 = 0.667 x 0.66 x 50 x D thus D = 491.6 mm
Bending Moment = 8.77 x 10
6
= 4.1 x 50 x D
2
/6 thus D = 506.6 mm
Provide a beam 525 x 50 mm
Beam is now 475 x 50 thus bending stress is now = 4.4 x 10
6
x 6

50 x 475
2
= 2.34 N/mm
2
< 4.1 N/mm
2
DET
: Civil Engineering: Structural Mechanics and Design 1 (Higher)
32
SOLUTIONS TO TUTORIAL 3
4.
a) Loading per joist/m = 0.5 x 1 x 2 = 1 kN/m thus M = 1 x 3.5
2
/8 = 1.53 kNm
and V = 1 x 3.5/2 = 1.75 kN
Shear stress = 1.5 x 1.75 x 1000/40 x 225 = 0.292 N/mm
2
< 0.67 N/mm
2
Bending stress = 1.53 x 10
6
x 6/40 x 225
2
= 4.53 N/mm
2
< 5.3 N/mm
2
b) Moment Capacity of larger joists = 4.1 x 50 x 250
2
/6 x 10
6
kNm
= 2.135 kNm = w x 3.5 x 3.5/8
thus w = 1.394 kN/m thus joist spacing = 1.394/2 = 0.697 m say 700mm
5.
a) Moment capacity of joists = 4.1 x 50 x 200
2
/6 x 10
6
kNm = 1.37 kNm
1.37 = 0.5 x w x 2.5
2
/8 thus w = 3.5 kN/m
2
b) I/y required for UB = (2.5 x 3.5 x 4.5
2
/8) x 10
6
/ 165 = 134233 mm
3
= 134.233 cm
3
thus use 178 x 102 x 19 UB
c) Shear stress =
(2.5 x 3.5 x 4.5/2 x 1000) x (101.2 x 7.9 x 84.95 + 4.8 x 81 x 81/2)

1356 x 10
4
x 4.8
= 25.31 N/mm
2
< 115 N/mm
2
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