DESCRIPTION OF SAMPLE BRIDGE
3.3. MODELLING OF THE SAMPLE BRIDGE
L6M19, L6M20, L8M20, L8M21, L10M21, L10M22, L12M22, L12M23, L14M23, L14M24, L16M24, L16M17)
[NB: Similar types of members for another truss as well]
18
Portal Bracing
(P2P5, P2U16, P3U16, P3P6) [NB: Similar types of members for
another portal frame as well]
5.88E–03 6.99E–07 1.28E–05 1.60E–05
19
Portal Bracing
(P1P2, P2P3, P3P4, P2U16, P3U16) [NB: Similar types of members for
another portal frame as well]
3.36E–03 3.72E–07 2.32E–06 3.38E–06
joint along the rail level plane connecting members like “Vertical”, “Bottom Chord”,
“Diagonal” and “Rail Level - Cross Girder” is shown in Fig. 3.6. Further, a joint along the road level plane connecting members as “Vertical” and “Road Level - Cross Girder” is shown in Fig. 3.7. Finally, a joint along the top level is displayed in Fig. 3.8. Finally, the structural members modelled as 3D frame elements are identified as follows:
(a) Rail Level - Cross Girder (b) Rail Level - Stringer (c) Road Level - Cross Girder (d) Road Level - Stringer (e) Strut
(f) Portal Strut (g) Vertical
Fig. 3.6. View of a joint along the rail level
On the other hand, the structural members, modelled as 3D truss elements, are termed as follows:
(a) Bottom Chord (b) Top Cord
Vertical (M8L8) Diagonal (L8M21)
Cross Girder (L8L8′) Diagonal (L8M20)
Bottom Cord (L7L8)
(c) Diagonal
(d) Bottom Lateral Bracing (e) Top Lateral Bracing (f) Portal Bracing
Fig. 3.7. View of a joint along the road level
Fig. 3.8. View of a joint along the top level
The modelling of the bridge is done with a combination of truss and frame elements as appropriate to simulate the behaviour of the bridge. The local coordinate orientation for a
Vertical (M8L8) Cross Girder (M8M8′)
Strut (U9U9′) Top Cord (U8U9)
Diagonal (U9M22) Top Cord (U9U10) Diagonal (U9M21)
beam element is shown in Fig. 3.9. The default local coordinate system as considered in SAP2000 is briefly mentioned here. Local 1 axis is along the length of frame element with positive direction from first node to second node. The default orientation of local 2 and 3 axes is determined by the relationship between the local 1 axis and the global +Z or +3 axis.
The local 1-2 plane is taken to be vertical, i.e. parallel to the global Z or 3 axis. The exceptional case while the frame element is parallel to global Z axis, the local 2 axis is taken to be globally horizontal along global +X or +1 direction. Now the local 1 and 2 axes are set, suppose in form of unit vectors v1 and v2 respectively. The unit vector of local 3 axis can be computed with the cross product as v3 = v1 × v2. The direction cosines of local axes unit vectors v1, v2 and v3 can be referred as (l1, m1, n1), (l2, m2, n2) and (l3, m3, n3) respectively.
The transformation matrix [T] to compute the global stiffness matrix is considered as
0 0 0
0 0 0
[ ] 0 0 0
0 0 0
R T R
R R
(3.1)
where, matrix [R] is represented as
1 1 1
2 2 2
3 3 3
[ ]
l m n
R l m n
l m n
(3.2)
The global stiffness matrix [Kfg] of frame element based on the global DOF is computed as
T
fg f
K T K T
(3.3)
Where, Kf is the local stiffness matrix of a frame element.
Fig. 3.9. DOF assigned at two joints of a frame elements in local coordinate
3.3.2. Other structural modelling issues
An important aspect of the structural modelling of the bridge is the modelling of the road- level deck-slab. The deck-slab is placed over cross-girders and stringers at roadway level without monolithic bonding. Such a non-monolithic type of bonding may be modelled with in-plane high (finite) stiffness. Diaphragm modelling associated with infinite in-plane stiffness may not be considered appropriate in this case. The in-plane high stiffness action can be considered using inplane diagonal bracings with higher stiffness taking into account the mass of deck-slab in appropriate way. This approach is followed in the modelling of the deck-slab as shown in Fig. 3.5(c) where the diagonal bracings are shown representing deck- slab. The diagonal-bracings are modelled using link element (CSI 2010). The link-element is a two noded element with user defined stiffness properties. First, link-elements evaluate relative deformations (between two joints) corresponding to 6 DOFs and subsequently use a (6×6) force-deformation relation matrix to relate force and deformation. In the present study, link elements are used with uncoupled behaviour having diagonal force-deformation relation matrix with possible 6 non-zero stiffness coefficients. In the modelling, only axial force- deformation relation with only 1 non-zero stiffness coefficient (ka) is considered. Link
1 2 3 4
5
6
7 8 9
10 11
12 X
Y
Z
elements with zero mass having only stiffness property are considered. In this regard, the mass of the deck-slab is assigned as dead-load distributed over the DOFs along road-level plane. The total mass for the deck-slab for one main-span is 4000 kN. Finally, the work remains to find a suitable value of ka based on agreement with experimental findings and this exercise is carried in Chapter 6 in Section 6.3.1. It may be mentioned that the value of ka is observed to have significant effect on the major natural frequencies of this bridge structure.
In case of modelling of frame-joints, it is observed that centre-lines/neutral-axes of the associated frame members don’t coincide in some situations. In such conditions, rigid- links are usually incorporated in modelling. A rigid link creates a new joint known as master joint and the kinematics of the intersection points are governed by the kinematics of the new master joint. This also reduces the size of dynamic system matrices (mass, stiffness and damping). Initially, in the modelling of joints connecting members having non-coincident centre-lines/neutral-axes, rigid-links were incorporated. However, it is observed that modelling without using rigid-links doesn’t make any significant difference (e.g. in terms of modal parameters) compared that using rigid-links. In view of this, modelling is finally considered without using rigid-links for avoiding modelling complexity. This simplification in modelling doesn’t have any adverse effect since the final studies (design of passive control system) are intended to be carried out based on direct-updated FE model.
3.3.3. Modelling of joint mass
In a dynamic analysis, the mass of the structure is used to compute inertial forces. Normally, the mass is obtained from the elements using the mass density of the material and the volume of the element. SAP2000 automatically produces lumped (uncoupled) masses at the joints.
The element mass values are equal for each of the three translational degrees of freedom. No mass moments of inertia are assigned for the rotational degrees of freedom. In many cases,
such lumping approach is observed to be adequate for modal or time-history analyses.
Further, joint-masses from the non-structural elements (e.g. deck slab, rail tracks, sleepers etc) are assigned equally along three translational DOFs in a manner similar to element mass.