1 Annex Influence of accessories on bending beam moment For the ...

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15 Νοε 2013 (πριν από 3 χρόνια και 7 μήνες)

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1


Annex Influence of accessories on bending beam moment

For the reference beams it was sometimes necessarily to use help pieces (adapters) at the
supports in order to obtain a height of 50 mm being the standard height. If these help pieces
are very tight connected to the beam they will have a stiffening effect. At the clamps these
accessories will contribute to the bending of the beam (providing that the accessories follow
the deflection profile of the beam) and this will increase the bending beam moment EI at the
clamps. As a consequence the effective total bending beam moment EI will increase and the
deflection will decrease. In figures 1 and 2 a schematic overview is given.











E
adaptor















E
beam








































Figure 1. Modeling of the cross section at the clamp.



X=
1



X=L/2













































X=0
X=A-
2

X=A
X=A+
2


2



An Excel program has been written to investigate the possible influence of these help pieces
on the bending deflection V
b
. The distance from the heart of the outer support to the centre is
divided in 5 sections:

Section 1: 0 < x < Ls with an EI value denoted by E
1
I
1
according to equation 1.
Section 2: Ls < x < A-Ls with an EI value E
2
I
2
=E
beam
I
beam

Section 3: A-Ls < x < A with an EI value denoted by E
3
I
3
according to equation 1.
Section 4: A < x < A+Ls with an EI value denoted by E
4
I
4
=E
3
I
3
according to equation 1.
Section 5: A+Ls < X < L/2 with an EI value E
5
I
5
=E
beam
I
beam

The equations for the deflection V
b
on these 5 intervals are given by equations 2 to 6.

3
3
1 1 1 1
1 1
3
3
2 2 2 2 2
3
3
3 3 3 3 3
3 3
4 4
{ } (2)
12
{ } (3)
12
{ } (4)
12
{ }
b
b
beam beam
b
b
x x FL
V x with
L L E I
x x FL
V x with
L L E I
x x FL
V x with
L L E I
x
V x
L
  
   
   

 
   
= + = 
 
   
   
 
 
 
   
= + + = 
 
   
   
 
 
 
   
= + + = 
 
   
   
 
 
 
=
 
 
2
2
4 4 4
4 4
2
2
5 5 5 5 5 5
(5)
4
{ } & 1 (6)
4
b
beam beam
x FAL
with
L E I
x x FAL
V x with
L L E I
  
    
 
 
+ + = 
 
 
 
 
 
 
   
= + + =  = 
 
   
   
 
 


At the crossings of the sections the deflection and its first derivative have to be continuous.
Thus also the following term has to be continuous at the crossing x
c

{ } { } { } { }
c c
b b b b
x x x x
d d
x V x V x x V x V x
dx dx
 
   
 = 
   
   
(7)

Starting with the crossing between section 1 and se ction 2 the constants 
I
can be calculated.
After calculating 
5
the continuity in the deflection V
b
is used for the determination of 
4
etc.
Finally the deflection V
b
{L/2} can be calculated. The ratio of the deflectio n V
b
{L/2} without
using the help pieces and this last deflection gives an indication of the overestimation made in
the back calculation of the E value when using the accessories. Examples are given in table 1.

Table 1. Maximal overestimation of the back calculated E v alue in % when the help pieces
are used at all supports.

H = 0.035 m H = 0.030 m H = 0.025 m

= 0.005 m
7.2 7.4 7.6

= 0.010 m 15.3 15.9 16.3


3


Remark

The figures given in table 1 are based on the assum ption that all the help pieces (accessories)
are very tight connected to the beam and can be see n as a part of the beam when it is bent. As
shown by the figures in the last row (  = 0.01 m) the lengths of the help pieces (2  ) is of
large influence specially at the inner supports. If the help pieces are not used at the inner
supports the possible overestimation is mu ch smaller (see also Annex Design of reference beam).
If no help pieces/adaptors are used at the inner clamp the induced errors are much smaller as indicated
in table 2.

Table 2. Maximal overestimation of the back calculated E v alue in % when the help pieces
are used only at the outer supports.

H = 0.035 m H = 0.030 m H = 0.025 m

= 0.005 m
0.001 0.001 0.001

= 0.010 m
0.01 0.01 0.01