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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

CRN: 29476 / 53471

3

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

CRN: 29476 / 53471

4

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

5

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

6

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

7

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

10

CRN: 29476 / 53471

11

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

13

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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