Unit 2:
Engineering Science
Unit code:
L/601/1404
QCF Level:
4
Credit value:
15
OUTCOME 1 - TUTORIAL 1
STRESSES IN BEAMS DUE TO BENDING
1.
Be able to determine the behavioural characteristics of elements of static engineering systems
Simply supported beams: determination of shear force; bending moment and stress due to bending;
radius of curvature in simply supported beams subjected to concentrated and uniformly distributed
loads; eccentric loading of columns; stress distribution; middle third rule
Beams and columns: elastic section modulus for beams; standard section tables for rolled steel beams;
selection of standard sections e.g. slenderness ratio for compression member, standard section and
allowable stress tables for rolled steel columns, selection of standard sections
Torsion in circular shafts: theory of torsion and its assumptions e.g. determination of shear stress, shear
strain, shear modulus; distribution of shear stress and angle of twist in solid and hollow circular section
shafts
You should judge your progress by completing the self assessment exercises.
It is assumed that students doing this tutorial are already familiar with the following work
at national level:



The reaction forces in simply supported beams.
Moments of force.
First and second moments of area.
Basic stress and strain (direct and shear)
If you need to study this, you will find the material elsewhere on the web site.
CONTENTS
1.
INTRODUCTION
2. THE BENDING FORMULA
2.1
Neutral Axis
2.2
Radius of Curvature
2.3
Relationship between Strain and Radius of Curvature
2.4
Relationship between Stress and Bending Moment
2.5
Standard Sections
© D.J.DUNN
1
1
INTRODUCTION
A beam is a structure that is loaded laterally (sideways)
to its length. These loads produce bending and bending
is the most severe way of stressing a component.
Suppose you were given a simple rod or a ruler and
asked to break it. You would struggle to break it by
stretching it or twisting it but it would be easy to break
by bending it.
A beam may have point loads or a uniformly distributed load
(udl) such as might occur due to its weight or the weight of a
wall built along it.
We normally show the loads with simplified diagrams like this.
There are other ways of supporting beams as shown below but you don’t need to study these.
The loads produce shear force and bending moments that vary with position along the length. You should
already know how to construct shear force and bending moment diagrams for simply supported beams.
When a bending moment M is applied to a beam, one surface is compressed (negative stress) and the other
is stretched (tensile positive stress). The stress varies across the section from a maximum negative to a
maximum positive as shown. Somewhere in between there is a longitudinal layer that is not stressed
(neutral) and this layer lays on the NEUTRAL AXIS. The neutral axis is through the centroid for pure
bending and the centroid is at the middle of symmetrical sections. In addition the transverse forces produce
shearing on a given section as indicated.
The purpose of this tutorial is to enable you to calculate the stress due to the bending moment. We will
derive the following three part equation known as the bending equation.
M

E


I
y
R
M is the bending moment at a given point along the length.
I is the second moment of area of the sectional area about the neutral axis.
σ is the stress due to bending at a distance y from the neutral axis.
E is the modulus of elasticity.
R is the radius of curvature.
© D.J.DUNN
2
WORKED EXAMPLE No. 1
The free body diagram shown is for a simply
supported beam with point loads.
Calculate the reaction forces Ra and Rb.
MOMENTS ΣM = 0 about any point so it is convenient here to use the left end at the point where R a
acts. Since the reaction at that point cannot produce a moment, it is eliminated from the balance.
(Remember clockwise is negative)
(Rb x 5) - (60 x 4) - (20 x 2) = 0
(Rb x 5) = (60 x 4) + (20 x 2) Rb = 280/5 = 56 kN
VERTICAL FORCES ΣFy = 0 (remember up is positive)
Ra + Rb - 20 - 60 = 0
Ra + Rb = 80 kN
Ra + 56 = 80 kN
Ra = 80 - 56 = 24 kN
UNIFORMLY DISTRIBUTED LOADS (u.d.l.)
A uniform load is one which is evenly distributed along a
length such as the weight of the beam or a wall built on top of
a beam. It is depicted by a series of arrows as shown. We
usually denote the loading as w Newtons per metre.
We deal with uniform loads by replacing them with an equivalent point load.
If the load extends over a length x metres then the total load is F = wx Newton. We replace the u.d.l. with a
single point load at the middle a distance of x/2 metres from the end.
WORKED EXAMPLE No. 2
A uniform load on a beam is shown below.
It has a value of 2.5 kN per metre.
Calculate the reaction forces.
SOLUTION
If the load is 2.5 kN for each metre of length then the total load is 4 x 2.5 = 10 kN
This load is equivalent to a single point load at the middle of 10 kN. It should be apparent that in this
case, the reactions must be equal to half the total load so are both 5 kN acting up. Let’s do the
calculation anyway. Taking moments about Rb we have
(10 x 2) - (Ra x 4 ) = 0
4Ra = 20
Ra = 5 kN
Balancing vertical forces
Ra + Rb - 10 = 0
5 + Rb - 10 = 0
Rb = 5 kN
© D.J.DUNN
3
WORKED EXAMPLE No. 3
A beam 4 m long rests on simple supports and carries a uniform load of 2.5 kN/m over the first 1.5 m as
shown. Calculate the reaction forces.
The total load is now 1.5 x 2.5 = 3.75 kN
This acts at the middle of the length 0.75 m from the
end. Balancing moments about Ra we have
(Rb x 4) - (3.75 x 0.75) = 0
Rb = 2.8125/4 = 0.703 kN
Balancing vertical forces
© D.J.DUNN
Rb + Ra = 3.75
4
Ra = 3.75 - 0.703 = 3.05 Kn
2.
2.1
THE BENDING FORMULA
NEUTRAL AXIS
This is the axis along the length of the beam which remains unstressed, neither compressed nor stretched
when it is bent. Normally the neutral axis passes through the centroid of the cross sectional area. For simple
rectangular and circular sections, this is the axis along the centre line.
Consider that the beam is bent into an arc of a circle
through angle  radians. AB is on the neutral axis
and is the same length before and after bending. The
radius of the neutral axis is R.
Remember the length of an arc is radius x angle in
radians
2.2
RADIUS OF CURVATURE
Normally the beam does not bend into a circular arc. However, what ever shape the beam takes under the
sideways loads; it will basically form a curve on an x – y graph. In maths, the radius of curvature at any
point on a graph is the radius of a circle that just touches the graph and has the same tangent at that point.
2.3
RELATIONSHIP BETWEEN STRAIN AND RADIUS OF CURVATURE
The length of AB
AB = R 
Consider a layer of material distance y from the neutral axis as shown. This layer is stretched because it
must become longer and the material has stress and strain in it in a lengthwise direction as a result. (If y
was to the inside of the neutral axis it would be compressed and become shorter).
The radius of the layer is R + y.
The length of this layer is the line DC.
DC = (R + y)
 = change in length/original length
This layer is strained and strain () is defined as
Substitute AB = R  and DC = (R + y)
R  y  R
DC  AB
y


AB
R
R
The modulus of Elasticity (E) relates direct stress () and direct strain () for an elastic material and the
definition is as follows.
stress 
E

strain 
y
R
E 
Substitute   and E 

R
y
R y
It follows that stress and strain vary along the length of the beam depending on the radius of curvature.
 
We will now go on to show that the radius of curvature depends upon the bending moment M acting at any
given point along the length of the beam.
© D.J.DUNN
5
2.4
RELATIONSHIP BETWEEN STRESS AND BENDING MOMENT
Consider a beam with a consistent shape along its length. An arbitrary oval shape is shown here. Think of
the beam as being made of many thin layers of material running the length of the beam and held together by
molecular forces.
Consider one such elementary layer at a given point along the length at a distance y from the neutral axis.
When the cross section is viewed end on it appears as an elementary strip width b and thickness y.
The cross sectional area is A.
The elementary strip is a small part of the total cross sectional Area and is denoted in calculus form as A.
The strip may be regarded as a thin rectangle width b and height y so A = b y
The stress on the strip is  = Ey/R
If the layer shown is stretched, then there is a small force F pulling normal to the section trying to slide the
layer out of the material in a lengthwise direction. This force must be the product of the stress and the area
and is a small part of the total force acting on the section F.
Ey
Ey
δF  δA Substitute 
and δF 
δA
R
R
Consider that the whole beam s made up of many such layers. Some are being stretched and pull normal to
the section and some are compressed and push. The total force acting on the section is the sum of all these
small forces.
Ey
F   δF   δA
R
In the limit as y tends to zero, the number of strips to be summed tends to infinity. The small quantities y
and A become the differential coefficient dy and dA. The total force is given by the integration
top
Ey
E
F  
dA 
R
R
bottom
top
 y dA
bottom
top
The expression
 y dA
is by definition the first moment of area of the shape about the centroid.
bottom
Evaluating this expression would give zero since any first moment of area is zero about the centroid.
The centroid in this case is on the neutral axis. The areas
above and below the neutral axis are equal. Half the force
is a compressive force pushing into the diagram, and half is
tensile pulling out. They are equal and opposite so it
follows that F = 0 which is sensible since cross sections
along the length of a beam obviously are held I
equilibrium.
© D.J.DUNN
6
The diagram indicates that the two forces produce a turning moment about the neutral axis because half the
section is in tension and half in compression. This moment must be produced by the external forces acting
on the beam making it bend in the first place. This moment is called the bending moment M.
Consider the moment produced by the force on the elementary strip F. It acts a distance y from the neutral
axis so the moment produced is M = y F
In the limit as y tends to zero the total moment is found by reverting to calculus again.
M 
 yδδ
top
 ydF  

bottom
M 
E
R
top
bottom
y
Ey
dA
R
top
y
2
dA
bottom
top
The expression
y
2
dA is by definition the second moment of area about the neutral axis and this is not
bottom
top
zero but has a definite value. In general it is denoted by the symbol I.
I 
y
2
dA
bottom
We may now write the moment as M 
Combining
E
M
E

I and rearrange it to
I
R
R
E

M

E
M
E
and
we now have




I
R
R
y
I
y
R
This is called the bending equation and it has 3 parts.
My
Ey
or  
This indicates
I
R
that the stress in a beam depends on the bending moment and so the maximum stress will occur where the
bending moment is a maximum along the length of the beam. It also indicates that stress is related to
distance y from the neutral axis so it varies from zero to a maximum at the top or bottom of the section. One
edge of the beam will be in maximum tension and the other in maximum compression. In beam problems,
we must be able to deduce the position of greatest bending moment along the length.
If the stress is required at a given point along the beam we use either  
© D.J.DUNN
7
2.5
STANDARD SECTIONS
At this stage, don't have to worry about how M is found, it is covered later. For simple sections the value of
I may be determined by mathematics. The good news is that for standard engineering sections, they may be
looked up in tables.
Steel and other products used in structural engineering are manufactured with standard cross sections and
sizes and in the UK they are made to British Standard BS4. In this you will find Universal Beams, Universal
Columns and much more with the sizes and sectional properties listed. You will find ‘I’ sections, ‘T’
sections, ‘U’ sections and more. You will find names like RSJ (Rolled Steel Joists) and RSC (Rolled Steel
Columns) which refer to the method of manufacture. A sample of the table is attached for ‘I’ section beams.
The areas and second moments of area are listed in the standards and since the distance y to the edge is also
I
known they list a property called the ELASTIC MODULUS and this is defined as z  . The stress at the
y
edge of the beam is then found from the equation:
My
M
.
 

I
Z
For standard shapes the second moment of area can be calculated with the formulae shown. This is covered
in the pre-requisite tutorial on moments of area. The following formulae apply to standard shapes.
For more complex shapes such as TEE and U sections, you will need to study the pre-requisite level tutorial
in order to solve the second moment of area.
There are also many computer programmes for solving beam problems that contain the standard
information or calculate the second moment of area when the dimensions are supplied.
The Archon Engineering web site has many such programmes.
You will currently (2005) find details of steel sections at www.roymech.co.uk
© D.J.DUNN
8
WORKED EXAMPLE No.4
A beam has a rectangular cross section 80 mm wide and 120 mm deep. It is subjected to a bending
moment of 15 kNm at a certain point along its length. It is made from metal with a modulus of
elasticity of 180 GPa. Calculate the maximum stress on the section.
SOLUTION
B = 80 mm, D = 100 mm. It follows that the value of y that gives the maximum stress is 50 mm.
Remember all quantities must be changed to metres in the final calculation.
BD3
80 x 1003
I 

 6.667 x 10 6 mm 4  6.667 x 10  6 m 4
12
12
M


I
y
 
My
15 x 103 x 0.05

 112.5 x 10 6 N/m 2
6
I
6.667 x 10
WORKED EXAMPLE No.5
A beam has a hollow circular cross section 40 mm outer diameter and 30 mm inner diameter. It is made
from metal with a modulus of elasticity of 205 GPa. The maximum tensile stress in the beam must not
exceed 350 MPa.
Calculate the following.
(i) the maximum allowable bending moment.
(ii) the radius of curvature.
SOLUTION
D = 40 mm, d = 30 mm
I = (404 - 304)/64 = 85.9 x 103 mm4 or 85.9 x 10-9 m4.
The maximum value of y is D/2 so y = 20 mm or 0.02 m
M σ

I y
σI 350 x 10 6 x 85.9 x 10 -9

 1503 Nm or 1.503 M Nm
y
0.02
σ E

y R
M
R
Ey 205 x 10 9 x 0.02

 11.71 m
σ
350 x 10 6
© D.J.DUNN
9
WORKED EXAMPLE No.6
A beam is made from a universal column with an ‘I’ section to BS4. The size of the beam is 356 x 127
x 39. The modulus of elasticity of 205 GPa. The maximum tensile stress in the beam must not exceed
350 MPa.
Calculate the following.
(i) The maximum allowable bending moment.
(ii) The radius of curvature.
SOLUTION
It is normal to arrange the ‘I’ section so that it bends about the x-x axis.
From the table (at the end of the tutorial) the elastic modulus z is 576 cm3
(576 x 10-9 m3)
The depth of the section is 353.4 mm so y = 176.7 mm
M
σ
z
M  σ z  350 x 10 6 x 576 x 10 -9  201 Nm
σ E

y R
R
Ey 205 x 10 9 x 0.1767

 103.4 m
σ
350 x 10 6
SELF ASSESSMENT EXERCISE No.1
1.
A beam has a bending moment (M) of 3 kNm applied to a section with a second moment of area (I) of
5 x 10-3 m4.
The modulus of elasticity for the material (E) is 200 x 109 N/m2.
Calculate the radius of curvature. (Answer 333.3 km).
2.
The beam is Q1 has a distance from the neutral axis to the edge in tension of 60 mm. Calculate the
stress on the edge. (Answer 36 kPa).
3.
A beam under test has a measured radius of curvature of 300 m. The bending moment applied to it is 8
Nm. The second moment of area is 8000 mm 4. Calculate the modulus of elasticity for the material.
(Answer 300 GPa).
4.
An ‘I’ section universal beam made to BS4 had dimensions 610 x 305 x 238. Assuming the modulus of
elasticity is 200 GPa, calculate the stress and radius of curvature when a bending moment of 500 kNm
is applied about the x axis. (Answer 75.9 MPa and 838 m)
5.
A beam must withstand a bending moment of 360 Nm. If the maximum stress must not exceed 250
MPa, determine the elastic modulus ‘z’ and select an appropriate ‘I’ section from the table for BS3
Universal Beams. (Answer z = 1440 cm3 so select 457 x 152 x 82)
© D.J.DUNN
10
UNIVERSAL BEAMS ‘I’ SECTION
Thickness of
Designation
Mass
per
m
Depth
of
Section
Width
of
Section
Web
Flange
Root
Radius
Depth
between
Fillets
Area
of
Section
Second Moment
Area
Axis
x-x
Axis
y-y
Radius
of Gyration
Axis Axis
x-x
y-y
Elastic
Modulus
Axis
x-x
Axis
y-y
Plastic
Modulus
Axis
x-x
Axis
y-y
M
h
b
s
t
r
d
A
Ix
Iy
rx
ry
Zx
Zy
Sx
Sy
kg/m
mm
mm
mm
mm
mm
mm
cm2
cm4
cm4
cm
cm
Cm3
cm3
cm3
cm3
914x419x388
388
921
420.5
21.4
36.6
24.1
799.6
494
719635
45438
38.2
9.59
15627
2161
17665
3341
914x419x343
343.3
911.8
418.5
19.4
32
24.1
799.6
437
625780
39156
37.8
9.46
13726
1871
15477
2890
914x305x289
289.1
926.6
307.7
19.5
32
19.1
824.4
368
504187
15597
37
6.51
10883
1014
12570
1601
914x305x253
253.4
918.4
305.5
17.3
27.9
19.1
824.4
323
436305
13301
36.8
6.42
9501
871
10942
1371
914x305x224
224.2
910.4
304.1
15.9
23.9
19.1
824.4
286
376414
11236
36.3
6.27
8269
739
9535
1163
914x305x201
200.9
903
303.3
15.1
20.2
19.1
824.4
256
325254
9423
35.7
6.07
7204
621
8351
982
838x292x226
226.5
850.9
293.8
16.1
26.8
17.8
761.7
289
339704
11360
34.3
6.27
7985
773
9155
1212
838x292x194
193.8
840.7
292.4
14.7
21.7
17.8
761.7
247
279175
9066
33.6
6.06
6641
620
7640
974
838x292x176
175.9
834.9
291.7
14
18.8
17.8
761.7
224
246021
7799
33.1
5.9
5893
535
6808
842
762x267x197
196.8
769.8
268
15.6
25.4
16.5
686
251
239957
8175
30.9
5.71
6234
610
7167
959
762x267x173
173
762.2
266.7
14.3
21.6
16.5
686
220
205282
6850
30.5
5.58
5387
514
6198
807
762x267x147
146.9
754
265.2
12.8
17.5
16.5
686
187
168502
5455
30
5.4
4470
411
5156
647
762x267x134
133.9
750
264.4
12
15.5
16.5
686
171
150692
4788
29.7
5.3
4018
362
4644
570
686x254x170
170.2
692.9
255.8
14.5
23.7
15.2
615.1
217
170326
6630
28
5.53
4916
518
5631
811
686x254x152
152.4
687.5
254.5
13.2
21
15.2
615.1
194
150355
5784
27.8
5.46
4374
455
5000
710
686x254x140
140.1
683.5
253.7
12.4
19
15.2
615.1
178
136267
5183
27.6
5.39
3987
409
4558
638
686x254x125
125.2
677.9
253
11.7
16.2
15.2
615.1
159
117992
4383
27.2
5.24
3481
346
3994
542
610x305x238
238.1
635.8
311.4
18.4
31.4
16.5
540
303
209471
15837
26.3
7.23
6589
1017
7486
1574
Designation
Mass
per
m
Depth
of
Section
Width
of
Section
Thickness of
Web
Root
Radius
Depth
between
Fillets
Area
of
Section
Flange
Second Moment
Area
Axis
x-x
Axis
y-y
Radius
of Gyration
Axis Axis
x-x
y-y
Elastic
Modulus
Axis
x-x
Axis
y-y
Plastic
Modulus
Axis
x-x
Axis
y-y
M
h
b
s
t
r
d
A
Ix
Iy
rx
ry
Zx
Zy
Sx
Sy
kg/m
mm
mm
mm
mm
mm
mm
cm2
cm4
cm4
cm
cm
cm3
cm3
cm3
cm3
610x305x179
179
620.2
307.1
14.1
23.6
16.5
540
228
153024
11408
25.9
7.07
4935
743
5547
1144
610x305x149
149.2
612.4
304.8
11.8
19.7
16.5
540
190
125876
9308
25.7
7
4111
611
4594
937
610x229x140
139.9
617.2
230.2
13.1
22.1
12.7
547.6
178
111777
4505
25
5.03
3622
391
4142
611
610x229x125
125.1
612.2
229
11.9
19.6
12.7
547.6
159
98610
3932
24.9
4.97
3221
343
3676
535
610x229x113
113
607.6
228.2
11.1
17.3
12.7
547.6
144
87318
3434
24.6
4.88
2874
301
3281
469
610x229x101
101.2
602.6
227.6
10.5
14.8
12.7
547.6
129
75780
2915
24.2
4.75
2515
256
2881
400
533x210x122
122
544.5
211.9
12.7
21.3
12.7
476.5
155
76043
3388
22.1
4.67
2793
320
3196
500
533x210x109
109
539.5
210.8
11.6
18.8
12.7
476.5
139
66822
2943
21.9
4.6
2477
279
2828
436
533x210x101
101
536.7
210
10.8
17.4
12.7
476.5
129
61519
2692
21.9
4.57
2292
256
2612
399
533x210x92
92.1
533.1
209.3
10.1
15.6
12.7
476.5
117
55227
2389
21.7
4.51
2072
228
2360
356
533x210x82
82.2
528.3
208.8
9.6
13.2
12.7
476.5
105
47539
2007
21.3
4.38
1800
192
2059
300
457x191x98
98.3
467.2
192.8
11.4
19.6
10.2
407.6
125
45727
2347
19.1
4.33
1957
243
2232
379
© D.J.DUNN
11
457x191x89
89.3
463.4
191.9
10.5
17.7
10.2
407.6
114
41015
2089
19
4.29
1770
218
2014
338
457x191x82
82
460
191.3
9.9
16
10.2
407.6
104
37051
1871
18.8
4.23
1611
196
1831
304
457x191x74
74.3
457
190.4
9
14.5
10.2
407.6
94.6
33319
1671
18.8
4.2
1458
176
1653
272
457x191x67
67.1
453.4
189.9
8.5
12.7
10.2
407.6
85.5
29380
1452
18.5
4.12
1296
153
1471
237
457x152x82
82.1
465.8
155.3
10.5
18.9
10.2
407.6
105
36589
1185
18.7
3.37
1571
153
1811
240
457x152x74
74.2
462
154.4
9.6
17
10.2
407.6
94.5
32674
1047
18.6
3.33
1414
136
1627
213
457x152x67
67.2
458
153.8
9
15
10.2
407.6
85.6
28927
913
18.4
3.27
1263
119
1453
187
457x152x60
59.8
454.6
152.9
8.1
13.3
10.2
407.6
76.2
25500
795
18.3
3.23
1122
104
1287
163
457x152x52
52.3
449.8
152.4
7.6
10.9
10.2
407.6
66.6
21369
645
17.9
3.11
950
84.6
1096
133
406x178x74
74.2
412.8
179.5
9.5
16
10.2
360.4
94.5
27310
1545
17
4.04
1323
172
1501
267
406x178x67
67.1
409.4
178.8
8.8
14.3
10.2
360.4
85.5
24331
1365
16.9
3.99
1189
153
1346
237
406x178x60
60.1
406.4
177.9
7.9
12.8
10.2
360.4
76.5
21596
1203
16.8
3.97
1063
135
1199
209
406x178x54
54.1
402.6
177.7
7.7
10.9
10.2
360.4
69
18722
1021
16.5
3.85
930
115
1055
178
406x140x46
46
403.2
142.2
6.8
11.2
10.2
360.4
58.6
15685
538
16.4
3.03
778
75.7
888
118
406x140x39
39
398
141.8
6.4
8.6
10.2
360.4
49.7
12508
410
15.9
2.87
629
57.8
724
90.8
356x171x67
67.1
363.4
173.2
9.1
15.7
10.2
311.6
85.5
19463
1362
15.1
3.99
1071
157
1211
243
356x171x57
57
358
172.2
8.1
13
10.2
311.6
72.6
16038
1108
14.9
3.91
896
129
1010
199
356x171x51
51
355
171.5
7.4
11.5
10.2
311.6
64.9
14136
968
14.8
3.86
796
113
896
174
356x171x45
45
351.4
171.1
7
9.7
10.2
311.6
57.3
12066
811
14.5
3.76
687
94.8
775
147
356x127x39
39.1
353.4
126
6.6
10.7
10.2
311.6
49.8
10172
358
14.3
2.68
576
56.8
659
89.1
356x127x33
33.1
349
125.4
6
8.5
10.2
311.6
42.1
8249
280
14
2.58
473
44.7
543
70.3
Thickness of
Designation
Mass
per
m
Depth
of
Section
Width
of
Section
Web
Flange
Root
Radius
Depth
between
Fillets
Area
of
Section
Second Moment
Area
Axis
x-x
Axis
y-y
Radius
of Gyration
Axis Axis
x-x
y-y
Elastic
Modulus
Axis
x-x
Axis
y-y
Plastic
Modulus
Axis
x-x
Axis
y-y
M
h
b
s
t
r
d
A
Ix
Iy
rx
ry
Zx
Zy
Sx
Sy
kg/m
mm
mm
mm
mm
mm
mm
cm2
cm4
cm4
cm
cm
cm3
cm3
cm3
cm3
305x165x54
54
310.4
166.9
7.9
13.7
8.9
265.2
68.8
11696
1063
13
3.93
754
127
846
196
305x165x46
46.1
306.6
165.7
6.7
11.8
8.9
265.2
58.7
9899
896
13
3.9
646
108
720
166
305x165x40
40.3
303.4
165
6
10.2
8.9
265.2
51.3
8503
764
12.9
3.86
560
92.6
623
142
305x127x48
48.1
311
125.3
9
14
8.9
265.2
61.2
9575
461
12.5
2.74
616
73.6
711
116
305x127x42
41.9
307.2
124.3
8
12.1
8.9
265.2
53.4
8196
389
12.4
2.7
534
62.6
614
98.4
305x127x37
37
304.4
123.4
7.1
10.7
8.9
265.2
47.2
7171
336
12.3
2.67
471
54.5
539
85.4
305x102x33
32.8
312.7
102.4
6.6
10.8
7.6
275.9
41.8
6501
194
12.5
2.15
416
37.9
481
60
305x102x28
28.2
308.7
101.8
6
8.8
7.6
275.9
35.9
5366
155
12.2
2.08
348
30.5
403
48.5
305x102x25
24.8
305.1
101.6
5.8
7
7.6
275.9
31.6
4455
123
11.9
1.97
292
24.2
342
38.8
254x146x43
43
259.6
147.3
7.2
12.7
7.6
219
54.8
6544
677
10.9
3.52
504
92
566
141
254x146x37
37
256
146.4
6.3
10.9
7.6
219
47.2
5537
571
10.8
3.48
433
78
483
119
254x146x31
31.1
251.4
146.1
6
8.6
7.6
219
39.7
4413
448
10.5
3.36
351
61.3
393
94.1
254x102x28
28.3
260.4
102.2
6.3
10
7.6
225.2
36.1
4005
179
10.5
2.22
308
34.9
353
54.8
254x102x25
25.2
257.2
101.9
6
8.4
7.6
225.2
32
3415
149
10.3
2.15
266
29.2
306
46
254x102x22
22
254
101.6
5.7
6.8
7.6
225.2
28
2841
119
10.1
2.06
224
23.5
259
37.3
203x133x30
30
206.8
133.9
6.4
9.6
7.6
172.4
38.2
2896
385
8.71
3.17
280
57.5
314
88.2
203x133x25
25.1
203.2
133.2
5.7
7.8
7.6
172.4
32
2340
308
8.56
3.1
230
46.2
258
70.9
203x102x23
23.1
203.2
101.8
5.4
9.3
7.6
169.4
29.4
2105
164
8.46
2.36
207
32.2
234
49.8
178x102x19
19
177.8
101.2
4.8
7.9
7.6
146.8
24.3
1356
137
7.48
2.37
153
27
171
41.6
152x89x16
16
152.4
88.7
4.5
7.7
7.6
121.8
20.3
834
89.8
6.41
2.1
109
20.2
123
31.2
127x76x13
13
127
76
4
7.6
7.6
96.6
16.5
473
55.7
5.35
1.84
74.6
14.7
84.2
22.6
© D.J.DUNN
12