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日期：2023-09-06 05:11

CEGE0101: Structural Mechanics

LSA in place of Mid-Term Exam:

Beam Behaviour (Shear, Bending & Deflection)

UNIVERSITY COLLEGE LONDON

Department of Civil,

Environmental &

Geomatic Engineering

CEGE0101: Structural Mechanics –Beam Behaviour CW

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The aim of this work is to compare the theoretical calculations, which you learned

during the course and the numerical approach through an advanced software

commonly used in the industry (Oasys GSA).

You are going to look at shear force, bending moment, stress, strain and deflection of

a simply supported beam, loaded at two points, under different material and cross

sectional arrangement.

You will submit a report, which will be worth 20% of your total module mark.

Part 1: Theoretical Shear Force & Bending Moment in the Beam

You are going to consider a simply supported beam which is loaded at two points.

Each student will consider a different loading configuration. The spacing of the

supports and the location of the loads for each student are specified in Table 3

(Page 8). The total length (𝑙) of the beam is 1000 mm.

By drawing a free body diagram (FBD) of the system, you should find the value of

the reactions (𝑅). By making imaginary sections along the beam’s length you should

establish what the shear force (𝑉) and bending moment (𝑀) are along the beam’s

length.

Part 2: Theoretical Longitudinal Stress & Strain in the

Beam

The next step is to calculate stresses (σ) and strains (ε) in the beam which result

from the bending moment at centre point of the beam’s length (i.e. midspan 𝑙/2). In

any loading case there is an axis about which the beam is said to bend and along

which there will be zero longitudinal stress. This axis is known as the Neutral Axis

(𝑁.𝐴.). For a section in sagging, the stress will be compressive above the neutral

axis and tensile below this axis. No matter what the shape of the cross section, the

stress varies linearly with distance from the neutral axis. If we define 𝑦 as the

distance from the neutral axis, then the longitudinal stress (𝝈) is evaluated as

follows:

𝜎 =

𝑀 ∙ 𝑦

𝐼

where 𝑀 is the moment carried by the section and 𝐼 is the second moment of area.

The equation gives you the value of stress (𝜎) at all points along the dotted red line,

shown in Figure 1. The dashed black line is the neutral axis (𝑁.𝐴.). This passes

through the centroid of the section.

Figure 1. Cross Section through beam showing Neutral Axis (dashed black line)

The most significant stress points are nearly always the top and bottom of the cross

section where 𝑦 is maximum. Hence, to find the stress on each surface (top and

bottom) we can put in a value of 𝑦 = 𝑑/2 , where 𝑑 is the cross-sectional depth.

𝑦 𝑁. 𝐴.

CEGE0101: Structural Mechanics –Beam Behaviour CW

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𝐼 is the second moment of inertia (a.k.a. moment of inertia). It is a property of the

cross section and an axis. The 𝐼 value which applies here is the one about the

neutral axis (𝑁.𝐴.). For a rectangular and circular cross section this is given by:

Rectangular Cross Sections with depth 𝑑 and width 𝑏 𝐼 =

𝑏 ∙ 𝑑

3

12

Circular Cross Sections with radius 𝑟 𝐼 =

𝜋 ∙ 𝑟

4

4

𝐼 governs the ability of a section to carry a bending moment (𝑀). The bigger it is the

larger the capacity of the section to carry load. This helps with the definition of 𝑏 and

𝑑. In case of rectangular cross sections, the “depth” (𝑑) is clearly the quantity which

most significantly increases bending strength (i.e. maximum stress capacity), as it

is in the power of 3 when compared with the width (𝑏). To calculate 𝐼 and thus 𝜎

you will need to have 𝑏 and 𝑑 or 𝑟 values of the beam’s cross section, which are

given in Table 1.

Table 1 - Cross Sectional Arrangements

Cross Sectional Arrangement Width

𝒃 (mm)

Depth

𝒅 (mm)

Arrangement 1 (Rectangular) 50 10

Arrangement 2 (Rectangular) 8 50

Arrangement 3 (Circular) 15 15

Arrangement 1 Arrangement 2 Arrangement 3

Having all necessary measures, now you can calculate the stress (𝜎) expected on

the top and bottom surfaces of the beam under two 60N loads, as long as the

bending moment is measures at any point along the length of the beam.

Strain (𝜺) is literally the ratio between how much longer something has got and its

original length (𝜀 = 𝛥𝑙 / 𝑙0). Values are typically very small and are often multiplied

by × 106 and hence referred to as micro-strain (µ𝜀).

The estimated stress (𝜎) can be converted into strain (𝜀), thanks to Hooke’s law:

𝜀 =

𝜎

𝐸

where 𝐸 is the Young’s Modulus (N.mm-2

) of the beam’s material. The materials you

are using are Steel (Grade: S235) and Aluminium (Grade: 6063). You should

establish a representative value for the elastic modulus (𝐸) of these material from

the literature and online sources.

50 mm

50 mm

8 mm

10 mm

15 mm

15 mm

CEGE0101: Structural Mechanics –Beam Behaviour CW

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Part 3: Theoretical Deflection of the Beam

We can figure out the deflection (𝛥) of a beam with a known bending moment

distribution. The bending moment at each point in the beam actually causes

curvature (that’s the second derivative of deflection). The deflection (𝛥) of the beam

is the net result of all of this curvature.

For a beam with a single point load, the shear diagram, moment diagram as well as

relevant equations related to end reactions (𝑅), moments (𝑀) and deflections (𝛥)

throughout the length of the beam are shown in Figure 2.

To find the deflection (𝛥) under both loads at the same time you will need to think

about linear elastic behaviour (Hooke’s Law) and the principle of superposition,

which was discussed in the course. Calculate the value of deflection under each of

the applied 60N point loads and combine them.

Figure 2. Simply supported beam with a single concentrated load

applied at any location along the beam’s length

We now have a theoretical calculation of the stress and strain on each surface

(top and bottom) at the centre point of the beam (i.e. midspan 𝑙/2) under two 60N

loads, as well as the Deflection under each of the 60N point loads.

Part 4: Numerical Shear Force, Bending Moment & Deflection

Now you should analyse your simply supported beam model in Oasys GSA under

following loading cases for ONE of the considered materials (either Steel S235 or

Aluminium 6063) and ONE of the cross sectional arrangements, while keeping your

allocated loading distances (i.e. distance from supports according to Table 3 on

Page 8). You can choose which material and cross section arrangmemt to use.

Table 2 – Numerical Loading Conditions

Loading Cases Point Load 1 (N) Point Load 2 (N)

Case 1 30 30

Case 2 60 60

CEGE0101: Structural Mechanics –Beam Behaviour CW

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Plot, compare and discuss the following Diagrams obtained from the software under

the given loading cases, material and cross sectional arrangement.

1) Loading + Fx, Fy and Fz Reaction Diagram

2) Myy Bending Moment Diagram

3) Fz Shear Force Diagram

4) Displacement Diagram + Deformed Image

For the numerical modelling using Oasys GSA, make sure you refer to the

“Structural Modelling and Analysis” course, available on UCLeXtend. Further

information on how to access this course and how to download and instal the

software is provided on CEGE0101 Moodle. Make sure you watch the training clip

and follow the provided notes of the following topic on UCLeXtend:

 Simply Supported Beam - Linear Static Analysis

Submission: Short Professional Technical Report

You are required to write a brief technical report describing your method, results

and comparisons. As part of the report, you will need to answer the questions posed

here below. The report should be no longer than maximum 4 pages and contain

about 1000-1500 words. If necessary, you can add a single page of appendix.

Answer the following questions in your report using the results obtained from your

theoretical and numerical analysis. You can support your discussions and

reasoning by providing screen shots of your analysis results (diagrams or contours).

 Compare and discuss the theoretical calculations you did for 60N loading

case and the numerical ones obtained from the software.

 Do you see any difference when you change the material from Steel

(Grade: S235) to Aluminium (Grade: 6063)? What values and results

change ? Discuss why.

 Do you see any difference when you change the cross-section

arrangements ? What values and results change ? Discuss why.

 Do you see any difference when you change the loading values ? What

values and results change ? Discuss why.

 Why do you think 60N is the maximum load we will try at each point ?

Report Writing & Formatting

The reports need to be typed and any diagrams must be presented neatly, properly

labelled and captioned. Lack of use of a ruler (if hand drawings) and bad writing, will

lead to a reduction in marks.

Your report will be assessed against the follow criteria:

 Overall clarity of the report (i.e. flow of the report and how easy it is to read)

CEGE0101: Structural Mechanics –Beam Behaviour CW

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(guide the reader through the data presented ensuring there is a narrative /

story)

 Appropriateness of Structure

(relevance, analysis, logic and coherence from introduction to conclusion)

 Clarity of English grammar / spelling

(spelling, grammar, punctuation, paragraphing and general fluency; use of

appropriate vocabulary; detail, accuracy and completeness of citations)

 Quality of Figures and Tables presentation

(presentation quality; effectiveness in highlighting points and making an

argument; synthesis of data in original format)

 Correct referencing of Figures and Tables and other sources

(use of an appropriate range of relevant sources)

 Quality of technical “story”

(ability to relate argument to core concepts in general topic method and

theory; reasoning from problem to methods to data, to build a sustained

argument; clear statement/understanding of a research problem or question).

 Correct and well-explained calculations

 Rational answering of questions and discussion of observations

 Compliance with the required length of report

(maximum of 4 pages, about 1000-1500 words, a single page of Appendix if

necessary)

Submission

You must submit the report in .PDF format, via e-mail to t.rossetto@ucl.ac.uk

by the 7th September 2023

CEGE0101: Structural Mechanics –Beam Behaviour CW

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Table 3 - Theoretical Loading Condition and Support Arrangement

Student

Number

Point Load 1 Point Load 2

Value

(N)

Distance from

Pin Support

(mm)

Value

(N)

Distance from

Roller Support

(mm)

22228845 60 100 60 150

20129287 60 200 60 100

21095854 60 300 60 100

22027669 60 400 60 100

22052231 60 200 60 250

22156335 60 300 60 200

60 400 60 200

60 300 60 350

60 400 60 300

60 100 60 125

60 200 60 125

60 300 60 125

60 400 60 125

60 200 60 225

60 300 60 225

60 400 60 225

60 300 60 325

60 400 60 325