STRUCTURES

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CRN: 29476 / 53471
1
School of Science, Engineering & Environment
TRIMESTER TWO EXAMINATION
PROGRAMMES:
BEng (Hons) Civil Engineering
BEng (Hons) Civil & Architectural Engineering
MEng (Hons) Civil Engineering
MEng (Hons) Civil & Architectural Engineering
BEng (Hons) Civil Engineering Degree Apprenticeship (Part Time)
STRUCTURES E1 P1
Start: Monday 10 May 2021 at 09.00 Finish – Friday 28 May 2021 at 16:00
___________________________________________________________________
Instructions to Candidates
There are four questions on this examination paper. Full marks may be obtained for
correct answers to THREE questions. Marks for parts of questions are shown in
brackets.
Each candidate has their own individual parameters for each question. Please use
the parameters particular to you. Using parameters attributed to another candidate
will result in zero marks for the attempted question. Check the provided named list for
your individual parameters.
Start your answer to each question on a new sheet of the answer script and write your
individual parameters at the top.
Approved electronic calculators may be used but NOT in text storage mode.
Structural mechanics and Eurocode design formula sheets are attached.
CRN: 29476 / 53471
2
1. A building roof truss is formed by a statically determinate pin jointed, triangulated
framework, and loaded by two variable actions as shown in Figure 1. The structural
elements are manufactured from mild steel circular hollow sections (CHS). The section
designation and steel grade of the CHS sections is an individual parameter. Please note
that the section designation will depend upon whether a member is subjected to tension
or compression forces under the described load case of Figure 1.
(a) State the three Laws of Static Equilibrium and calculate the values of the three
reactions.
(4 Marks)
(b) Stating your sign convention, obtain the sense and magnitude of the internal forces
in all the framework elements. Show your answer diagrammatically. The applied
forces, FB and FC are individual parameters. The geometry of the truss in Figure
1 are individual parameters as well.
(12 Marks)
(c) Use Castigliano’s Theorem to determine the vertical deflection at joint F.
(10 Marks)
(d) State whether you consider the stiffness of this framework to be adequate for the
building roof truss, giving reasons for your decision.
(2 Marks)
(e) For the loading direction shown, determine whether element DE is adequate for
strength according to EN1993. Assume that joints E and D are positions of lateral
restraint about the y-y axis and z-z axis. To check strength a load partial safety
factor of 1.5 should be applied to the internal forces calculated in (b) above.
(5 Marks)
Total 33 Marks
Figure 1. Arrangement and loading of a simply supported, pin jointed framework. Please
note that this Figure is indicative and may not be exact with consideration of individual
parameters
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2. The beam shown in Figure 2 is being checked for capacity using stress analysis. The
UB beam section designation and steel grade of the beam are individual parameters.
The loads and geometry in Figure 2 are also individual parameters.
(a) Draw the axial force, shear force and bending moment diagrams showing the
principal values. (12 Marks)
Considering a small element at point C, just to the right of the uniform load, where the
web meets the top flange:
(b) Calculate the axial stress assuming the axial force is applied at the section centroid.
State whether the stress is tension or compression.
(3 Marks)
(c) Calculate the bending stress. State whether the stress is tension or compression.
(5 Marks)
(d) Assuming the uniform loading spreads at 450 through the section, calculate the
bearing stress, stating whether it is tension or compression.
(5 Marks)
(e) Calculate the shear stress in the web, assume that clockwise is positive.
(5 Marks)
(f) Summarise the stresses on a stress element diagram.
(3 Marks)
Total 33 Marks
Figure 2. Beam loading arrangement. Please note that this Figure is indicative and may
not be exact with consideration of individual parameters
FVB (kN) FVE (kN)
UDLBC (kN/m)
FHE (kN)
A
E
LDE (m)
L (m)
D
LAB (m) LCD (m)
B
LBC (m)
C
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3. A fully lateral restrained, uniform, isotropic steel beam is subject to a point load and
uniformly distributed load, as shown in Figure 3. The loadings in Figure 3 are
individual parameters as well as the geometrical lengths. The beam is a mild steel UB
section. The section designation and steel grade of the beam are also individual
parameters.
(a) Calculate the reactions.
(3 Marks)
(b) Draw the axial force, shear force and bending moment diagrams showing the
principal values.
(8 Marks)
(c) Use Macaulay’s method to produce a general formula for deflection of the beam
(10 Marks)
(d) Calculate the deflection at point C and hence sketch the deflected shape of the
beam.
(4 Marks)
(e) State whether you consider this beam section to be adequately stiff, giving reasons
for your decision. Use the output from (d) with the assumption that the loads are
factored at the appropriate limit state.
(2 Marks)
(f) Determine whether the beam is adequate for strength according to EN1993.
Assume that the compression flange is laterally restrained throughout its whole
length. Use the output from (b) with the assumption that the loads are factored at
the appropriate limit state
(6 Marks)
Total 33 Marks
Figure 3. Arrangement and loading of a simply supported beam. Please note that this
Figure is indicative and may not be exact with consideration of individual parameters
B
D
UDLCD (kN/m)
FHD (kN)
A
C
FVB (kN)
x
L (m)
LAB (m) LBC (m) LCD (m)
 
CRN: 29476 / 53471
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4.
(a)
Draw diagrams of the four standard cases of effective length, giving theoretical and
design values of effective length factors.
(6 Marks)
A mild steel section forms a diagonal compression brace in a steelwork building
structure. The compression brace is restrained by a tubular brace at mid-length. All the
connections are pinned, about both axes. The section designation and steel grade of the
compression brace is an individual parameter. The geometry, X and H as well as the
lateral wind load, W, in Figure 4 are also individual parameters.
(b)
Sketch the possible deflected shapes of the brace for buckling about each axis. State
the effective length factors and calculate the effective lengths.
(6 Marks)
(c)
Define slenderness ratio and calculate the slenderness ratio about both axes. State
which axis the column is likely to buckle about.
(5 Marks)
Calculate the Euler, Rankine and Perry-Robertson buckling loads.
(d)
(8 Marks)
(e) Determine whether the Compression brace is adequate for strength according to
EN1993. Please note you will need to check compression capacity of the section
for the most critical slenderness value only.
(6 Marks)
(f)
You are further required to state the efficiency of the section under the described
loading in the value engineering process.
(2 Marks)
Total 33 Marks
Figure 4. A building structure bracing bay.
X
CRN: 29476 / 53471
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Structural Mechanics Formulae
Simple bending:
E R
f z
M I
= =
Shear flow :
Ib
VA’z’
 =
Simple torsion:
L I p
G
T r
 
= =
Castigliano’s Theorem : = L EI M WU =  AE FL WU

 . .
Differential equation of flexure : M
dx
d z
EI = –
2
2
Theorem of the Parallel Axis : I NA yy = I yy +  Ah2 where

=
A
Az
z
Euler Buckling: 2
2 L
EI
PE

=
Rankine Capacity: 2
1 a
f A
P y
R
+
= where a  0.0001
Perry-Robertson Stress: fc = fy + fcr( + )- fy + fcr(1+ )2 – fy fcr
1 4
1
1 2
 
where
2
2 
 E
fcr = and  = 0.003
Unsymmetrical bending:
y
I I I
M I M I
x
I I I
M I M I
I I I Sin I
I I I Sin I
I I
I
Tan
yy zz yz
yy zz zz yz
yy zz yz
zz yy yy yz
z
vv zz yz yy
uu yy yz zz
yy zz
yz




+




=
= + +
= – +

= –
2 2
2 2
2 2
cos 2 sin
cos 2 sin
2
2

  
  

Principal Stress :
x z
Tan xz
 



=
2
2
,
2
1 2
max
 


=
1 ( )2 4 2
1 2
2 x z xz
x z
  
 
 + – +
+
= and 2 ( )2 4 2
1 2
2 x z xz
x z
  
 
 – – +
+
=
CRN: 29476 / 53471
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Standard Analysis Cases
Centroids of Area
Shape
A
Iyy
Izz
Rectangle
Triangle
Circle
b.d
12
.
b d 3 3
12
d.b
.d 2
b
36
.
b d 3
4
.
 D2
64
.
 D4
8
wL2
L 2
w
w (per metre)
L
EI
wL
384
5 4
max =
L
Pab
L
Pa
L
Pb
P
a b




 = –
3 2
max 4 3
48 L
a
a L
EI
PL
4
PL
P 2
P 2
P
L 2
L 2
EI
PL
48
3
max =
Deflection BMD SFD
b 2
d 2
b 3
d 3
R
D
CRN: 29476 / 53471
8
Hot Rolled Steel Section Properties
UNIVERSAL BEAMS
Depth
of
Section
Width
of
Section
Thick
ness of
Web
Thick
ness of
Flange
Second
Moment of
Area
Radius of
Gyration
Elastic
Modulus
Plastic
Modulus
Area of
Section
Serial size
type
h
b
tw
tf
r
I yy
I zz
i y
i z
Wel, y
Wel, z
Wpl, y
Wpl, z
A
mm
mm
mm
mm
mm
cm4
cm4
cm
cm
cm3
cm3
cm3
cm3
cm2
203x133x25
UB
203.2
133.2
5.7
7.8
7.6
2360
310
8.56
3.1
230
46.2
258
70.9
32
203x133x30
UB
206.8
133.9
6.4
9.6
7.6
2890
384
8.71
3.17
280
57.5
314
88.2
38.2
254x146x31
UB
251.4
146.1
6
8.6
7.6
4440
449
10.5
3.36
351
61.3
393
94.1
39.7
254x146x37
UB
256
146.4
6.3
10.9
7.6
5560
571
10.8
3.48
433
78
483
119
47.2
254x146x43
UB
259.6
147.3
7.2
12.7
7.6
6560
677
10.9
3.52
504
92
566
141
54.8
305x165x40
UB
303.4
165
6
10.2
8.9
8520
763
12.9
3.86
560
92.6
623
142
51.3
305x165x46
UB
306.6
165.7
6.7
11.8
8.9
9950
897
13
3.9
646
108
720
166
58.7
305x165x54
UB
310.4
166.9
7.9
13.7
8.9
11700
1060
13
3.93
754
127
846
196
68.8
457x191x74
UB
457
190.4
9
14.5
10.2
33400
1670
18.8
4.2
1458
176
1653
272
94.6
457x191x82
UB
460
191.3
9.9
16
10.2
37100
1870
18.8
4.23
1611
196
1831
304
104
457x191x89
UB
463.4
191.9
10.5
17.7
10.2
41000
2090
19
4.29
1770
218
2014
338
114
UNIVERSAL
COLUMNS
Depth
of
Section
Width
of
Section
Thick
ness
of
Web
Thick
ness of
Flange
Second
Moment of
Area
Radius of
Gyration
Elastic
Modulus
Plastic
Modulus
Area of
Section
Serial size
type
h
b
tw
tf
r
I yy
I zz
i y
i z
Wel, y
Wel, z
Wpl, y
Wpl, z
A
mm
mm
mm
mm
mm
cm4
cm4
cm4
cm
cm3
cm3
cm3
cm3
cm2
152x152x23
UC
152.4
152.2
5.8
6.8
7.6
1260
403
6.54
3.7
164
52.6
182
80.2
29.2
152x152x30
UC
157.6
152.9
6.5
9.4
7.6
1740
558
6.76
3.83
222
73.3
248
112
38.3
152x152x37
UC
161.8
154.4
8
11.5
7.6
2220
709
6.85
3.87
273
91.5
309
140
47.1
203x203x46
UC
203.2
203.6
7.2
11
10.2
4560
1540
8.82
5.13
450
152
497
231
58.7
203x203x52
UC
206.2
204.3
7.9
12.5
10.2
5260
1770
8.91
5.18
510
174
567
264
66.3
203x203x60
UC
209.6
205.8
9.4
14.2
10.2
6090
2040
8.96
5.2
584
201
656
305
76.4
203x203x71
UC
215.8
206.4
10
17.3
10.2
7650
2540
9.18
5.3
706
246
799
374
90.4
203x203x86
UC
222.2
209.1
12.7
20.5
10.2
9460
3120
9.28
5.34
850
299
977
456
110
CIRCULAR HOLLOW
SECTIONS
Diameter
of
Section
Thickness
Second
Moment
of Area
Radius of
Gyration
Elastic
Modulus
Plastic
Modulus
Area of
Section
Serial size
type
h
t
I yy
iyy
Wel, y
Wpl, y
A
mm
mm
cm4
cm
cm3
cm3
cm2
88.9×4
CHS
88.9
4
96.3
3
21.7
28.9
10.7
88.9×5
CHS
88.9
5
116
2.97
26.2
35.2
13.2
88.9×6.3
CHS
88.9
6.3
140
2.93
31.5
43.1
16.3
114.3×4
CHS
114.3
4
211
3.9
36.9
48.7
13.9
114.3×5
CHS
114.3
5
257
3.87
45
59.8
17.2
114.3×6.3
CHS
114.3
6.3
313
3.82
54.7
73.6
21.4
139.7×4
CHS
139.7
4
393
4.8
56.2
73.7
17.1
139.7×5
CHS
139.7
5
481
4.77
68.8
90.8
21.2
139.7×6.3
CHS
139.7
6.3
589
4.72
84.3
112
26.4
ANGLES
Depth
of
Section
Width
of
Section
Thick
ness
Area
of
Section
Second Moment of Area
Radius of Gyration
Elastic
Modulus
Serial size
type
h
b
t
A
I yy
I zz
I uu
I vv
i y
i z
i u
i v
Wel, y
Wel, z
mm
mm
mm
cm2
cm4
cm4
cm4
cm4
cm
cm
cm
cm
cm3
cm3
80x80x10
EA
80
80
10
15.1
87.5
87.5
139
36.4
2.41
3.03
3.03
1.55
15.4
15.4
80x80x6
EA
80
80
6
9.35
55.8
55.8
88.5
23.1
2.44
3.08
3.08
1.57
9.57
9.57
80x80x8
EA
80
80
8
12.3
72.2
72.2
115
29.9
2.43
3.06
3.06
1.56
12.6
12.6
90x90x10
EA
90
90
10
17.1
127
127
201
52.6
2.72
3.43
3.43
1.75
19.8
19.8
90x90x12
EA
90
90
12
20.3
148
148
234
61.7
2.7
3.4
3.4
1.74
23.3
23.3
90x90x8
EA
90
90
8
13.9
104
104
166
43.1
2.74
3.45
3.45
1.76
16.1
16.1
100x100x10
EA
100
100
10
19.2
178
178
283
73.7
3.05
3.05
3.05
1.96
24.8
24.8
100x100x12
EA
100
100
12
22.7
207
207
328
85.7
3.02
3.8
3.8
1.94
29.1
29.1
100x100x15
EA
100
100
15
27.9
249
249
393
104
2.98
3.75
3.75
1.93
35.6
35.6
CRN: 29476 / 53471
9
EN 1991 Design Rules
Table 1. Variable action sensitivity factors. Table 2. Action Partial Safety Factors.
Use
combination
0
frequent
1
quasi
permanent
2
STR and GEO limits
Unfavourable
Favourable
Office
0.7
0.5
0.3
Permanent
1.35
1.0
Shopping
0.7
0.7
0.6
Variable
1.50

Storage
1.0
0.9
0.8
Accidental
1.0

Roofs
0.7


Combination equations.
Snow
0.5
0.2

ULS combination
Wind
0.5
0.2

SLS combination
 GGk +  QQk1 +  Q 0,2Qk 2 + …
Table 3. Minimum roof imposed action.
sk = 0.15 + (0.1z + 0.05)+   A525 -100   and snow on the roof
is, s = isk
Table 4. Snow action coefficients.
i
Roof slope
0o ≤  ≤
15o
15o <  ≤ 30o
30o <  < 60o
 ≥ 60o
monopitch
0.8

duopitch
0.8


 




30
60
0.8  
 




+
15
15
0.8 0.4  
 




30
60
1.2 
Basic velocity pressure,
Peak velocity pressure,
where,
qb = 0.613(Vb,0cdircseasoncaltcprob )2 calt =1+ 0.001A
qp = cece,Tqb
Table 5. External wall pressure coefficients.
d
h
coefficients Cp,e
Side
Front
back
5
-0.8
+0.8
-0.7
1
-0.8
+0.8
-0.5
-0.8
+0.7
-0.3
 0.25 Table 6. Monopitch roof pressure coefficients
Roof
angle
coefficients, Cp,e
=00
=900
A
B
C
A
B
C
< 50
-1.2
-0.7
±0.2
-1.2
-0.7
±0.2
150
-1.3
-0.9
-1.9
-0.8
-0.7
300
-1.8
-0.8
-1.5
-1.0
-0.8
450
-1.5
-0.7
-1.4
-1.0
-0.9
Gk +1,1Qk1 + 2,2Qk 2 + 2,3…
Roof slope
 < 30o
30o ≤  <
60o
 ≥ 60o
0.6


 




30
60
0.6  b
h d
d
 b
= 900
 = 00
CRN: 29476 / 53471
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CRN: 29476 / 53471
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Exposure correction factor, Ce,T (z-hdis versus distance inside town terrain in km)
Exposure factor, Ce (z-hdis versus distance upwind to shoreline in km)
Distance upwind to shoreline (km) Distance inside town terrain (km)
CRN: 29476 / 53471
12
EN 1993 Design Rules
Table 7. Local buckling classification.
Maximum Outstand element aspect ratio
Class 1
Class 2
Class 3
Class 4
UB or UC sections in
pure bending
Flange element
≤ 9
≤ 10
≤ 14
> 14
where
Web element
≤ 72
≤ 83
≤ 124
> 124
UB or UC sections in
pure compression
Flange element
≤ 9
≤ 10
≤ 14
> 14
Web element
≤ 33
≤ 38
≤ 42
> 42
fy
235
 =
Table 8. Material Partial Safety Factors.
Local failure (yielding)
m0
1.0
Member instability (buckling)
m1
1.0
Tension fracture
m2
1.1
Joint resistance
m2
1.25
Bolt friction SLS
m3,ser
1.1
Bolt friction ULS
m3
1.25
3

,
M
v y
c Rd
A f
V

=

,
M
pl y
pl Rd
W f
M

= for Class 1 and 2 sections,

,
M
el y
el Rd
W f
M

= for Class 3 sections.
1
,
M
LT y y
b Rd
W f
M


= where
96
z
LT

 = for S275 material and
85
z
LT

= for S355 material
Table 9. Lateral torsional buckling curve selection, rolled sections.
Aspect ratio
Buckling curve
General
Rolled sections
c
b
d
c
 2
h b
 2
h b
1
,
M
y
b Rd
Af
N


= where
 L
 = and
y
L
E f
 = 
Table 10. Flexural buckling curve selection, rolled sections.
Aspect ratio
Buckling axis
Buckling curve
y-y
z-z
a b
y-y
z-z
b c
 1.2
h b
 1.2
h b
1.5 1.0
, ,
,
, ,
,
, ,
, + + 
z cb Rd
z Ed
b y Rd
y Ed
b z Rd
E d
M M
M M
N N
CRN: 29476 / 53471
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Lateral Torsional Buckling Curve for f y=275N/mm2

0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
LT
LT
Flexural Buckling Curve for f y=275N/mm2

0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3


a
c
a
c

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