ST7-1.50.30.4 Using the Moving Load Module

Strand7 Webnotes – Modelling / Loading

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ST7-1.50.30.4 Using the Moving Load Module

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Using the Moving Load Module . n o i t u b i r t s i d e r r o f t o N . C T U M P 2 1 : 2 1 t a 4 1 0 2 b e F 1 2 n o ) 1 0 B S U E F A ( d t L ) y t P ( l a n o i t a n r e t n I g n i r e e n i g n E n o c i r f A o t

Introduction This Webnote outlines the use of the Strand7 Moving Load module for the modelling of transient loads moving along a path, the generation of load cases relevant to the possible positions of loads moving along a path, and to create static combination of loads moving along a path that minimises or maximises a specified response variable. Whilst each of these capabilities addresses diff erent analysis types, central to them all is the definition of moving loads in a Load Path Template. These templates have have functionality which addresses some or all of the above capabilities. Dialogs and windows in the Load Path Templates are introduced below, as they become relevant to the functionality being presented.

Load Types for each Vehicle

Tabs for Vehicle definitions

d e c n e c i L . d e t i m i L y t P 7 d n a r t S t h g i r y p o C

Template in Strand7. Figure 1. A basic Load Path Template in

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Moving Load in Strand7 . n o i t u b i r t s i d e r r o f t o N . C T U M P 2 1 : 2 1 t a 4 1 0 2 b e F 1 2 n o ) 1 0 B S U E F A ( d t L ) y t P ( l a n o i t a n r e t n I g n i r e e n i g n E n o c i r f A o t d e c n e c i L . d e t i m i L y t P 7 d n a r t S t h g i r y p o C

In Strand7, moving loads are represented using entities called Load Paths, whose characteristics are described in a Load Path Template in the same manner as an entity property. Load Path Templates may be created manually in the Properties/Load Paths window seen in Figure 1, or specified from a database. There are three ways in which the load path may be interpreted (as described in the Scope section of this Webnote and reiterated below): 1.

modelling of transient loads moving along a path;

2.

generation of load cases relevant to the possible positions of loads moving along a path;

3.

static combination of loads moving along a path that minimise or maximise a specified response variable.

Many aspects of the Load Path Template in Figure 1 are interpreted in the same way in each context – these features are described in the remainder of this introduction. However there are some features which are specific to just one or two of these cases or are interpreted differently in each context – such features are introduced together with their usage in the next sections.

Load Paths and Templates As seen in Figure 2, a load path is split spatially into Lanes. Furthermore in the left hand spreadsheet of the Load Path Template shown in Figure 1, Vehicles may be defined to load each lane. Vehicles are sets of loads that travel as a coordinated group along the load path Lanes. Specifically, they are repositioned to any one of a discrete number of positions along the length of the Lane. The number of positions along the lane is a load path attribute defined in Attributes/Load Path/Divisions. Each Vehicle footprint is defined relative to its own local x , y , z (longitudinal, transverse, out-of-plane) coordinate system, whose origin may occupy the red and bl ue points shown in Figure 2. This local system moves along the load path. The vehicle may consist of mechanical and thermal loads that typically travel with the local coordinate system, but can remain stationary, or roam independently throughout the lane depending on the context. y

y

x

Lane 1

Y

X

Lane 2

x Lane 3

1

2

3

…

…

n – 2

n – 1

n

Figure 2. The positioning of Vehicles in the Lanes of the load path, given three lanes and n divisions; Single lane vehicles may be centred on the blue points, and Double lane vehicles may be centred on the red p oints.


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The Vehicle Tab . n o i t u b i r t s i d e r r o f t o N . C T U M P 2 1 : 2 1 t a 4 1 0 2 b e F 1 2 n o ) 1 0 B S U E F A ( d t L ) y t P ( l a n o i t a n r e t n I g n i r e e n i g n E n o c i r f A o t d e c n e c i L . d e t i m i L y t P 7 d n a r t S t h g i r y p o C

Under the Vehicle tab of the load path template (Figure 3) in the Instance panel, the number of lanes occupied by the active (selected) vehicle in the spreadsheet may be set as: 

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Single Lane

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the vehicle is one lane wide

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it effectively moves along the lane centrelines

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the loads are centered about the blue points in Figure 2

or Double Lane

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the vehicle is two lanes wide

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it effectively moves along the lane boundaries

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the loads are centred about the red points in Figure 2.

The direction, velocity and start time can also be defined, if a transient analysis is required. The start time defines the delay from the beginning of the analysis before the vehicle enters the path, and can be used to stagger vehicles (staggering can be used on vehicles on a single path, or on vehicles over multiple paths).

Figure 3. The Vehicle tab.

The Path Tab Under the Path tab in Properties/Load Paths are settings which apply to the entire load path, such as its display Path Colour (this tab is visible in Figure 1). The Number of Lanes on the load path is set here, along with the Minimum Lane Width. Normally, the lane width is taken to be the maximum transverse extent of the vehicle loading among all the Vehicles (the transverse extent of double vehicles is halved for this purpose) – but if this is less than the minimum value specified here, then the Minimum Lane Width is used instead. The lane width is then multiplied by the number of lanes defined on the load path to determine the total load path width.

Figure 4. The Path tab


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Specifying Loads . n o i t u b i r t s i d e r r o f t o N . C T U M P 2 1 : 2 1 t a 4 1 0 2 b e F 1 2 n o ) 1 0 B S U E F A ( d t L ) y t P ( l a n o i t a n r e t n I g n i r e e n i g n E n o c i r f A o t d e c n e c i L . d e t i m i L y t P 7 d n a r t S t h g i r y p o C

Loads are specified relative to each vehicle’s local coordinate system, in the spreadsheet on the left of the Property/Load Path… window. There are three load types available for each vehicle in the tabs at the top of the spreadsheet; Point Force, Distributed Force and Heat Source. A single vehicle can contain all three types of load.

Point Force 

specifies forces at points (x , y ), defined in the local vehicle coordinate frame

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the force (F x , F y , F z) is applied in the coordinate system directions specified under Axis System

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may be one of two types, for the purposes of generating vehicle load cases

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Grouped forces move together longitudinally along the lane with Grouped/Leading/Trailing Distributed Forces, and Heat Sources. Floating forces may roam anywhere independently of any other loads in the lane. The coordinates (x ,i y ) are ignored, and this type is only valid in the context of static combination to optimise a response variable – in the other contexts it is treated as Grouped.

Distributed Force 

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specifies a force applied over a designated area defined in the local vehicle coordinate frame, with intensity expressed per unit length of lane. the force per unit length (P x, P y , P z) is applied in the coordinate system directions specified under Axis System . may be one of five basic types, for the purposes of generating vehicle load cases.

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Grouped distributed forces over rectangular patches (x : [x 1, x 2], y :[ y 1, y 2]), move together longitudinally along the lane with Point Forces, Leading/Trailing Distributed Forces , and Heat Sources.

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Full-length distributed forces are applied over the entire length and specified breadth y :[ y 1, y 2] of the lane. Note that the x -values are ignored for this type.

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Leading distributed forces are applied over the length x > 0 and breadth y :[ y 1, y 2], and move together longitudinally along the lane ahead of Grouped/Trailing Distributed Forces , Grouped Point Forces and Heat Sources. Trailing distributed forces are applied over the length x < 0 and breadth y :[ y 1, y 2], and move together longitudinally along the lane behind Grouped/Leading Distributed Forces, Grouped Point Forces and Heat Sources. Floating distributed forces are applied over rectangular patches (x : [x r + x 1, x r + x 2], y :[ y 1, y 2]), which may roam to any point (x r, 0) independently of the Grouped/Leading/Trailing Distributed Forces, Grouped Point Forces and Heat Sources in the lane. This type is only valid in the context of static combination to optimise a response variable – in the other contexts it is treated as Grouped. Additionally, there is a special case of the Grouped distributed force termed a Void distributed force. This is a rectangular patch in which other loads appearing in the lane are blanked – it is defined in the same way as a Grouped distributed force with a zero magnitude force specified.


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Heat Source . n o i t u b i r t s i d e r r o f t o N . C T U M P 2 1 : 2 1 t a 4 1 0 2 b e F 1 2 n o ) 1 0 B S U E F A ( d t L ) y t P ( l a n o i t a n r e t n I g n i r e e n i g n E n o c i r f A o t d e c n e c i L . d e t i m i L y t P 7 d n a r t S t h g i r y p o C

specifies a heat source in units of power (e.g. J/s), applied either at a point (x , y ), or over a rectangular area dx ×d y (a heat flux) centred at (x , y ) when dx and d y are non-zero.

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these Heat Sources may only move with the vehicle along the axis of travel – they are effectively Grouped loads, always.

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Adjacency and Centrifugal Loading The Point/Distributed Forces are further modified by the selection of the Adjacency and Centrifugal properties in each spreadsheet row. The Adjacency property is only valid in the context of static combination to optimise a response variable – it shall be discussed in a later section. When the Centrifugal property is selected for a force, it modifies the applied force on the l oad path to include an extra component, directed centrifugally, which is calculated from the vertical force F z or P z – which may be thought of as defining the mass that gives rise to the centrifugal force. The centrifugal force is calculated as specified in the Centrifugal tab, and applied with the same distribution over the same areas or points of application as the causative load, F z or P z. The centrifugal force F C is calculated from the vertical force F z by the equation F C  K 0  K 1 F z

(1)

which is set in the Centrifugal tab, along with options to limit K 1 and set the units in which this force is locally calculated (Strand7 will then convert these units to whatever is being used in the rest of the load template, and model). Note that K 0 and K 1 in the above equation may be expressed as functions of: 

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path radius R and path length L calculated from the load path geometry gravitational acceleration g relevant to the load case in which the load path is defined and vehicle velocity V (entered on the Vehicle tab).

This is discussed in detail in the next section. Note also that centrifugal loads are assumed to act on the surface of the path, i.e. at z=0 . Centrifugal loads do not take into account the centre of gravity of the vehicle that gives rise to them.


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Transient Moving Loads . n o i t u b i r t s i d e r r o f t o N . C T U M P 2 1 : 2 1 t a 4 1 0 2 b e F 1 2 n o ) 1 0 B S U E F A ( d t L ) y t P ( l a n o i t a n r e t n I g n i r e e n i g n E n o c i r f A o t d e c n e c i L . d e t i m i L y t P 7 d n a r t S t h g i r y p o C

An explicitly transient moving load can be generated from the load path definition using the settings in the Dynamic panel of the Vehicle tab. Note that these settings are ignored in load influence analysis, except to define a velocity for the Centrifugal properties. Similarly, in the context of a transient moving load the options to permute and factor lanes presented in the Path tab, and restrict the possible combination of vehicles in the Sets tab, are ignored. The completed model is included as ST7-1.50.30.4 Curved Concrete Bridge - Moving Load ( Final).st7.

Transient Moving Load Example 

File/Open the model ST7-1.50.30.4 Curved Concrete Bridge - Moving Load (Initial).st7, and File/Save As another filename.

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Create a new load path template in Property/Load Path.

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Under the Path tab in the new template:

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set six lanes for the load path

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and set the Min. lane width to 4 m.

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Select the tab at the bottom of the spreadsheet corresponding to the first Vehicle , as seen in Figure 1.

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Select the tab at the top of the spreadsheet corresponding to Distributed forces for this Vehicle and

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Add a new Force

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enter the following data in the new row to specify a 4×6 m patch of 10 kPa vertical load, and 1 kPa shear load.

Mobility

x1 (m)

x2 (m)

y1 (m)

y2 (m)

Px (N/m)

Py (N/m)

Pz (N/m)

Axis system

Adjacency

Centrifugal

Grouped

-3

3

-2

2

4000

0

−40000

Local

No

Yes

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Note that 10000×4 = 40000 N/m.

Under the Vehicle tab for Vehicle 1, seen in Figure 3 above:

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make it a Double lane vehicle

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set the checkboxes to activate it in lanes 1 and 5 only.

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set it to move Forward in these lanes

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enter a Velocity of 15 m/s

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enter a Start time of 0 s.

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Add a second Vehicle .

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Select the tab at the bottom of the spreadsheet corresponding to the second Vehicle .


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. n o i t u b i r t s i d e r r o f t o N . C T U M P 2 1 : 2 1 t a 4 1 0 2 b e F 1 2 n o ) 1 0 B S U E F A ( d t L ) y t P ( l a n o i t a n r e t n I g n i r e e n i g n E n o c i r f A o t d e c n e c i L . d e t i m i L y t P 7 d n a r t S t h g i r y p o C

Select the tab at the top of the spreadsheet corresponding to Point forces for this Vehicle and

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Add four new Forces

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enter the following data in the rows so created to specify four point forces at the corners of a 2.5×6 m rectangle.

Mobility

x (m)

y (m)

Grouped

3

Grouped

Fz (N)

Axis system

Adjacency

Centrifugal

1.25

−10000

Local

No

Yes

−3

−1.25

−10000

Local

No

Yes

Grouped

−3

1.25

−10000

Local

No

Yes

Grouped

3

−1.25

−10000

Local

No

Yes

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Fx (N)

Fy (N)

Under the Vehicle tab for Vehicle 2:

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make it a Single lane vehicle

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set the checkboxes to activate it in lanes 2, 3, 4 and 5 only

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set it to move Backward in these lanes

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enter a Velocity of 30 m/s

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enter a Start time of 3 s.

To finish entering the loading, go to the Centrifugal tab and

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enter (V*V)/(R*g) under K1

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elect not to limit K1 (by leaving this option unchecked)

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select units of metres and Newtons from the dropdown menus.

Figure 5. The Centrifugal tab.

Note that by specifying this factor for K1, the relationship between vertical and centrifugal loads 2

F C 

V

Rg

F z

(2)

is observed, meaning that given some total mass of the object m = F z/ g , the centrifugal force corresponds to that needed to accelerate the mass m around the path. Equation (1) affords extra flexibility to accommodate a departure from this strictly physical expectation, as such flexibility is demanded by many design codes.


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In Create/Load Path, select a Flat circular Path shape and select three points along the centre of the curved bridge model to define a circular arc for the load path to follow.

“Flat” load path

R

“Curved” load path

R

Figure 6. Flat vs Curved load paths.

Note that the Flat and the Curved options in Create/Load Path control the direction in which the load path is curved, when the circular or parabolic path shape is selected. Flat load paths are curved about the local z axis of the load path, whereas Curved load paths are curved about the local y axis – as seen above. Note also that the Divisions attribute is also initialised here (by default, to 20). 

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Check that gravity is applied in the negative global Y direction for the Load case in which the Load Path is created. Check also that the loads are being applied in the right direction.

It should be possible to see vectors representing the load condition imposed by the vehicles bearing down on the bridge. 

If the load is applied in the wrong direction, then you can either change the sign in the load path template or – as is usually more convenient – use Tools/Align/Flip Entities to flip the load path definition itself.

Note also that load paths in Strand7 are treated like elements in some ways. For example, load paths may be copied, moved and scaled after creating them. The Online Editor also offers some editing functionality for load paths. Before attempting a full transient solution, we will need an initial stress state – without this the bridge will experience a sudden loading due to gravity and the solution will be dominated by the transient behaviour as the bridge bounces due to the gravity load. 

Solver/Linear Static and solve for the deflections due to self-weight.

Note that in the context of a non-transient (static) solution, the forces on the load path are ignored so the above solution will include the gravitational effects of the load case only . In Solver/Linear Transient Dynamic : 

select Full System solution

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select Rayleigh Damping with the default values (i.e. a value of ζ = 0.01 at 1 Hz and 10 Hz)

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in Initial Conditions select From File from the dropdown menu, open the previously generated *.LSA file and select the gravitational result case therein apply the loads under Moving Load, and select the one moving load specified.


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The load path is approximately 100 m long, thus at a speed of 15 m/s the slower load will take about 6 s to reach the other side. . n o i t u b i r t s i d e r r o f t o N . C T U M P 2 1 : 2 1 t a 4 1 0 2 b e F 1 2 n o ) 1 0 B S U E F A ( d t L ) y t P ( l a n o i t a n r e t n I g n i r e e n i g n E n o c i r f A o t d e c n e c i L . d e t i m i L y t P 7 d n a r t S t h g i r y p o C

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Under Time Steps set 100 time steps of 0.06 s, and save at every step.

Refer to ST7-1.20.20.4 Transient Dynamic Analysis Time Stepping for guidance in selecting an appropriate time step. 

Click Solve.

Figure 7. Setting up the linear transient dynamic solver window. 

Results/Open Results File for the linear transient results file.

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Create a contour plot of vertical DY displacement.

By scrolling through the result cases, which are at 0.06 s intervals, with face and element attributes visible, it should be possible to see the loads move over the bridge. Note that to see the displacement contours change appreciably, it is useful to display them relative to the initial displacement at 0 s. This is because the gravity load typically causes more displacement than the moving load. 

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Results/Reference Displacement , select Specific case and choose the first result case from the dropdown menu. Create an animation by using Results/Animation and selecting Multi Case.

In the results, there is a point at around 4.38 s when the double lane and single lane vehicles overlap each other in lanes 2 and 5. When using the transient solvers, there are no restrictions on the possible combination of loads with the result that more than one vehicle can exist in the same lane. In this result case shown in Figure 8, it is also possible to see the centrifugal loading on the bridge, which is visibly larger for the faster moving vehicle. Note that the load path will be interpreted in exactly the same manner if the other transient solvers are used – it can be used with Quasi-static Analysis, Linear Transient Analysis and Nonlinear Transient Analysis .


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. n o i t u b i r t s i d e r r o f t o N . C T U

The vehicles move from division to division along the load path in transient analysis, so it is important to enter enough divisions to accurately capture the smooth transition of the vehicle as it moves over the mesh. The default value of 20 divisions may not be enough if your load path is long, or if your mesh is fine. You can change the load path divisions with Attributes/Load Path/Divisions.

M P 2 1 : 2 1 t a 4 1 0 2 b e F 1 2 n o ) 1 0 B S U E F A ( d t L ) y t P ( l a n o i t a n r e t n I g n i r e e n i g n E n o c i r f A o t

Alternatively, you can specify that the load path be automatically divided as necessary, based on the vehicle speeds and the analysis time step. This option is available in the Moving Load… dialog of the transient and quasi-static solvers.


Figure 8. Overlapping moving loads progressed along the curved bridge at 4.38 s.


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Generating Load Cases from a Load Path . n o i t u b i r t s i d e r r o f t o N . C T U M P 2 1 : 2 1 t a 4 1 0 2 b e F 1 2 n o ) 1 0 B S U E F A ( d t L ) y t P ( l a n o i t a n r e t n I g n i r e e n i g n E n o c i r f A o t d e c n e c i L . d e t i m i L y t P 7 d n a r t S t h g i r y p o C

In the previous example, the load path was effectively used to generate different load cases to load the structure in the course of a transient analysis. Naturally, it would also be possible to do this manually usingFactor vs Time tables and a set of appropriate load cases – although this would be very laborious. Nonetheless, there are occasionally situations where such a detailed approach is necessary, or the load cases themselves are required for a further more detailed analysis. To generate load cases corresponding to the individual movement of each Vehicle along each lane in which it is active, we can use Tools/Convert/Load Paths to Load Cases. This creates a separate load case for each possible position of each Vehicle on the load path, where the number of such positions is determined by Attributes/Load Path/Divisions. Note that the number of load cases generated for a given load path/lane/vehicle combination, will usually exceed the number of divisions set on the path. This is because additional load cases are automatically generated to progressively ease the vehicle onto and the off of the path. The number of extra cases generated is a function of the vehicle length, the load path length, an d the number of divisions (i.e. the path segment length). In the previous example, there were two Vehicles , six Lanes and the default 20 divisions – additionally, Vehicle 1 was only active in two lanes and Vehicle 2 was only active in four lanes. If this tool is used with the previous example, where there are 20 divisions (referring to Figure 2), the number of load cases created corresponds roughly to: 20 × 2 + 20 × 4 = 120 load cases.

However, more load cases are created than the estimate. This is because some extra load cases arise from the curvature of the load path and the interaction of the ends of it with the vehicle footprint. The algorithm adds extra vehicles where it discovers that it can fit another partial vehicle on the load path, or conversely may disregard vehicles whose loads do not intersect with any elements.


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Combining Loads on a Load Path for the Best or Worst Case . n o i t u b i r t s i d e r r o f t o N . C T U M P 2 1 : 2 1 t a 4 1 0 2 b e F 1 2 n o ) 1 0 B S U E F A ( d t L ) y t P ( l a n o i t a n r e t n I g n i r e e n i g n E n o c i r f A o t d e c n e c i L . d e t i m i L y t P 7 d n a r t S t h g i r y p o C

One of the principal uses of the load path is to define possible load configurations to determine the worst (or best) loading condition for a given Response Variable or combination of response variables. This is typically useful for determining the required live load case for structural design, particularly of bridges. Building codes based on limit-state design principles (e.g. AS5100.2 –2004 or EN1991-2: 2003) describe live loads due to traffic that are to be applied to the structure in combination, to arrive at a design load against which the structure must perform satisfactorily for a given limit state. Various loads relating to different traffic types are stipulated and rules for their application, combination and possible interaction are defined. The design load is then the combination of such loads that gives rise to the worst effects. In Strand7, the worst case combination is determined in the following way: 

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Traffic types are defined as Vehicles on a load path.

Response Variables relevant to the limit-state under consideration are defined. These quantify the effect of loading on the structure – in terms of optimisation theory, these variables form the objective function that is to be minimised or maximised. A Load Influence analysis is made on the structure to determine the influence line or surface over the structure. This quantifies the effect that a unit load (force or moment) on any part of the structure will have on the Response Variables. Then, the Load Influence Combinations post-processor may be run to determine the position and combination of Vehicles on the load path that causes the extreme (maximum or minimum) effect on the Response Variable – the worst case loading.

This load combination may then be recorded for use as a design load in subsequent analyses. Note that the analysis up to this point is necessarily linear.

Defining Rules for Vehicle Combination Many of the options in the Load Path Template relate to the manner in which the Vehicles may be combined in the Load Influence Combinations solver. One can imagine that there is a discrete number N of possible loads on the load path, arising from each Vehicle moved to a given position in the lane in which it is active. Possible combinations therefore, arise by picking a subset of such loads that may legitimately exist together. The way in which the effective vehicle load cases can be combined is limited in the following manner: 

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Each Lane may only be loaded by a single Vehicle. Thus Double Lane vehicles prevent both lanes they occupy from being loaded by any other vehicles. Not all of the Vehicles or Lanes need to be active – subsets of legitimate combinations are also legitimate. The Vehicle tab introduced in Figure 3 defines the extent (Single or Double Lane) and possible location of each vehicle. Under the Sets tab, groups of mutually exclusive vehicles are defined by setting an identical name for each mutually exclusive set or group.

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Thus only one Vehicle from each set may appear on the load path.

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Any Vehicle which does not belong to a set (its name i s not entered) may combine with all other vehicles, and itself.

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A single Vehicle assigned a unique set not shared by other vehicles, can appear only once on the load path.


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Consequently, there can be no interaction of loads from different Vehicles , except when Point Forces are specified at the extreme edges of lanes or when Floating Point Forces drift to the edges of lanes. In such cases, the loads are overlayed. Furthermore, it is also possible to apply load factors to the vehicle loads depending on the number of vehicles active on the load path, in the Path tab (seen on the right in Figure 1). Factors B i are specified for the i th lane, and the lane numbering itself may also be permuted. Note that these factors are only applied to those loads in the Vehicles spreadsheet for which the Adjacency property is checked, and their application varies with the options: 

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Figure 9. The Sets tab

All lanes same factor

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Where in a vehicle combination, n lanes are loaded, the load factor B n is applied to all the lanes.

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Note that not all the lanes need to be loaded (as mentioned previously).

Each lane different factor

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Each lane i is factored by its corresponding B i.

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The lane numbering in this respect is flexible – the numbering and subsequent factoring is adjusted to achieve the worst case.

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Typically, the worst case arises when the highest factor is applied to the lane with the greatest share of the influence line, and the greatest load; the second highest factor is applied to the second greatest load; and so on.

Load Influence Combinations Example We shall use the model of a curved concrete bridge developed in the previous section as a simple example of the recombination of vehicles on a load path to find the worst case loading. The model with the load path fully defined is supplied as ST7-1.50.30.4 Curved Concrete Bridge - Moving Load (Load Path Defined).st7. Alternatively the following steps may be taken to complete the Load Path Template. 

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Using an appropriate name, File/Save As the curved concrete bridge model with the load path containing two vehicles. In this new file, go to Global/Load and Freedom Cases and highlight-and-delete any load cases generated from the use of Tools/Convert/Load Paths to Load Cases. This should leave one load case containing the load path, in which gravity is also def ined.

Property/Load Path and set the Adjacency property for all the Point and Distributed Forces in the two existing vehicles. This will mean that the loads will be factored by the lane factors in the Path tab. In the Path tab:

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Set the option Each lane different factor .

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Leave the default values for the lane factors, B 1 = 1.0, B 2 = 1.0, B 3 = 0.9 and B >3 = 0.75.

This means that the i th Vehicle added to the load path is factored by B i – thus changing the order in which the vehicles are added will also change the factoring: the numbering is not static. If a vehicle were placed in lane 1, and then placed again in lane 2, the second lane would attract a factor of 0.9. However, if the same vehicle were placed in lane 2 first, and then placed in lane 1, the lane 1 vehicle would attract the 0.9 factoring. This rather esoteric factoring option is designed to cater for the stipulation of reduction factors specified in many structural


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codes: these factors reduce the load as more lanes are occupied because the likelihood of such a condition is reduced as the load path is filled. . n o i t u b i r t s i d e r r o f t o N . C T U M P 2 1 : 2 1 t a 4 1 0 2 b e F 1 2 n o ) 1 0 B S U E F A ( d t L ) y t P ( l a n o i t a n r e t n I g n i r e e n i g n E n o c i r f A o t

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Still in Property/Load Paths, Add a third Vehicle to the load path.



Select the Distributed Forces tab for this vehicle and Add a new Force with the properties specified below: Mobility

x1 (m)

x2 (m)

Full Length

y1 (m)

y2 (m)

-2

2

Px (N/m)

Py (N/m)

Pz (N/m)

Axis system

Adjacency

Centrifugal

−400

Local

Yes

Yes

This completes the setup of the Load Path Template. We are going to set a nodal Response Variable to monitor vertical displacement of the bridge, mid-span at the outer radius – in particular we are interested in the vertical displacement downwards here. Given the loads defined above, we expect that loads concentrated at the centre of the span and on the outer radius should serve to push this area downwards. Furthermore the distributed load posed by the third vehicle should be redundant in comparison to the effects of the second vehicle – which poses the same total load over the 100 m length of the bridge but is concentrated over a much smaller area.

The Load Influence Solver In order to determine the worst case combination, some basic linear static stress-analysis is required. Simply running the Linear Static solver for each of the possible combinations of load would be prohibitively computationally demanding for larger problems, however as long as the problem is considered to be linear – it can be considerably sped up. The first economy can be achieved by using linear superposition to combine each of the basic vehicle load cases or possible footprints. The second economy can be made by using a so-called influence line (or influence surface in two dimensions) to calculate the effects of each footprint at the Response Variable of interest. The next step in determining the worst case is therefore to perform a Load Influence analysis to examine the structural response. The Load Influence solver looks at the effect of nodal unit loads applied to the structure on a specified Response Variable , and reports the response of this (remote) variable at the location of the point unit load. The process is illustrated in Figure 10, where the application of a 1 N load at different points along a two span continuous bridge gives rise to a different stress response at the marker. Graphing this response over the span of the bridge gives rise to the “influence line” at the bottom of this figure – two and three dimensional structures are analogous.



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1N . n o i t u b i r t s i d e r r o f t o N . C T U M P 2 1 : 2 1 t a 4 1 0 2 b e F 1 2 n o ) 1 0 B S U E F A ( d t L ) y t P ( l a n o i t a n r e t n I g n i r e e n i g n E n o c i r f A o t d e c n e c i L . d e t i m i L y t P 7 d n a r t S t h g i r y p o C

pinned supports

20 MPa -5 MPa

response variable; bending stress at the point marked

1N

y

MPa /(N) in the -y direction 20

x

-5 Figure 10. A simple example of an influence line for a two span bridge.

In Strand7, Response Variables are specified in load cases, by specifying an entity attribute at the node or element Attributes/{Node, Beam, Plate, Brick}/Response Variable of interest. If the Response Variable is one of stress then the response has units of MPa and the load influence solver calculates results of MPa/(Newton) of applied force, or MPa/(Newton·metre) of applied moment. Contouring this result, in the manner of the load influence line in Figure 10, is useful as it allows the analyst to see exactly where on the structure exerting a force or moment will have a significant effect on the component or limit state of interest. Therefore only one load influence analysis is needed to provide all of the information necessary to determine the effect of every possible vehicular footprint or combination on the response variable of interest. 



Attributes/Node/Response Variable and select



the node midway along the span, on the outer radius of the deck



the node midway along the span, on the inner radius of the deck



then apply a Response Variable in the global DY direction for both.

Solver/Load Influence and



under Load Cases select the one load and freedom case defined



click Solve.

Note that this analysis takes place in the absence of other loads such as nodal forces, tractions on elements, or body forces. Because the structure is assumed to be linear elastic, the effect of such loads on the Response Variable is merely superposed – they do not affect the relationship between an input point force applied somewhere in the structure, and the output response at the Response Variable. 



Results/Open Results File and open the load influence results file. In Results/Results Settings contour the influence that a unit force in the global DY direction has on the Response Variable.


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Figure 11. Influence of 1 N in the Y direction on displacement in the Y direction on the mid-span outer radius.

There is a result case for each Response Variable defined in each of the load cases selected for solution; for this analysis, two result cases. Note how, as expected, unit forces located close to the Response Variable under examination cause greatest deflection. Additionally it is apparent that forces on the opposite edge of the bridge tend to push the DY Response Variable in the opposite direction and twist the deck, as seen in Figure 11. Therefore, there are some areas of the bridge where applying a force in a vertical direction will force the Response Variable in the opposite vertical direction. These areas are termed “relieving” areas of the response surface, as if for example the worst case downward deflection at the response variable is required, downward forces acting on these relieving areas will force the response variable upwards. Similarly, the relieving areas of the influence line in Figure 10 would correspond to the areas of negative response, were the stress to be maximised.

The Load Influence Combinations Solver The load influence combinations solver (Results/Load Influence Combinations) uses the results of the load influence analysis and the definitions in the load path template to arrive at load cases that produce extreme responses in the response variables defined in the load influence analysis. It does not perform a structural analysis as such, but combines the loads defined in the load template and integrates over the influence surface to arrive at the final response of a variable of interest. It is effectively an optimisation algorithm that selects load path configurations that produce maxima and minima for specified response variables, or summations of them. The dialog to define these combinations is shown below.


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The loads defined in the load path templates are combined such that: 







all allowable combinations of the load are considered in the optimisation problem, in accordance with section Defining Rules for Vehicle Combination. any Leading , Trailing or Full Length loads that relieve the response variable (that occur in the relieving areas of the influence line or surface) are removed from the load case,

Vehicles may not otherwise be partially removed – though they may be completely removed from the Load Path. Those properties defined under the Vehicle tab, marked Dynamic are ignored, except for



the Velocity which is used as an input to the Centrifugal tab.

In the model used so far, or the completed model ST7-1.50.30.4 Curved Concrete Bridge - Moving Load (Load Path Defined).st7, the worst case combination of Vehicles on the bridge relating to different limit states can be found. We shall look to optimise three responses: 1.

the downward deflection of the inner edge at the mid-span

2.

the downward deflection of the outer edge at the mid-span

3.

the sum of these two deflections.



Close the results file (if it is open), and open the Results/Load Influence Combinations solver.

In addition to load case combinations that excite the simple minimum or maximum response at the response variable, when there are multiple response variables it is also possible to define Multi-case variables. This allows the sum response of response variables of the same type (e.g. total plate force along a line of plates) to be maximised or minimised instead. 









Under Single Variable (Min) in the Load Influence Combinations solver spreadsheet, select both response variables defined previously to find the greatest downwards deflection of each. Under Single Variable (Max) in the next tab, uncheck both response variables – this would give the maximum deflection upwards in each case, which is not of interest at the moment. Click Add to add a new Multi Variable response to optimise. Under the new Multi Variable tab so created select both response variables and check Minimum to find the loading that maximises the sum of the downward deflections of the two edges at the mid-span. Click Generate to find the three load cases that optimise each of the three responses.

The three load cases can be solved with the Linear Static solver to examine the structure under the worst case loading conditions specified above – two of these load cases are shown in Figure 12.


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. n o i t u b i r t s i d e r r o f t o N . C T U M P 2 1 : 2 1 t a 4 1 0 2 b e F 1 2 n o ) 1 0 B S U E F A ( d t L ) y t P ( l a n o i t a n r e t n I g n i r e e n i g n E n o c i r f A o t

(a)

(b)

Figure 12. Worst case loading for (a) deflection of inner edge (b) deflection of outer edge, at midspan.



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Inspecting the loads that are generated, we see that for the deflection of the inner edge the heaviest vehicle (Vehicle 1) is moved to the inner side of the bridge at the midspan, and for the deflection of the outer edge this vehicle is moved to the outer side. As this vehicle cannot occur on the interior lanes of the bridge, the second heaviest vehicle (Vehicle 2) is applied at the mid span of the interior l anes instead. As may be expected, the loads maximising total deflection of both edges are simply the two heaviest vehicles, situated at the mid-span. Vehicle 2 is actually a similar total weight to Vehicle 3 – but as Vehicle 3’s load is spread over the ~100 m length of the bridge, it does not produce as large deflections at the midspan. Therefore, Vehicle 3 does not appear in the worst case loading, except for the load case maximising the outer edge deflection. Here it appears partially on the second lane from the inside. As can be seen in Figure 11 there exist small regions in this lane which when loaded, cause a moderate downward deflection at the outer edge. Vehicle 3 is a Full Length load – which may be blanked on relieving areas of the influence surface. This property permits its partial application in this lane, over so-called adverse areas. This behaviour of Full Length loads (often termed Uniformly Distributed Loads or UDL) is required by most codes. Note that the same does not occur for the load case maximising deflection on the inside edge, for which the influence surface has a different extent. Note also the addition of centrifugal components and the effect of the adjacency factors on the load magnitudes – this is particularly noticeable for the point forces, which may be inspected by hovering over the plates with the Shift key depressed and Plate selection turned on.

Summary The moving load module can be used in two primary ways: 1.

To generate load influence combinations, producing the worst case combination of a set of vehicles for a given response variable

2.

To produce a transient response of the structure to a load which moves along it through time with some velocity

The first usage is primarily of interest to bridge designers, as the templates provided define sets of vehicles which can be used without much effort to find the worst combination for a particular bridge. This solution uses the load influence solver. See ST7-1.40.60.5 Bridge Load Influence Combinations for an applied example. The specific use of each load path template to generate load influence combinations is discussed in the following Strand7 Webnotes: 

ST7-1.40.60.9 Applying AS 5100.2 Moving Load



ST7-1.40.60.10 Applying EN 1991-2 Moving Load



ST7-1.40.60.11 Applying BS 5400-2 Moving Load



ST7-1.40.60.12 Applying AASHTO-LRFD Moving Load

The second usage makes the load path very flexible and allows it to be used to determine how the bridge deflects and stresses as the vehicles move across it. It can be used to model a crane on a track, a weld head distributing heat to a work piece, or a steam roller moving over fresh asphalt, to name a few examples. This solution uses the transient solvers (linear and nonlinear transient dynamic and quasi static). SeeST7-1.40.60.6 Bridge Moving Load Analysis with Creep and Shrinkage for an applied example. Another example is found in ST7-1.30.10.1 Nonlinear Weld Thermal Analysis.


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ST7-1.50.30.4 Using the Moving Load Module

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