Chapter
Structural Analysis
The structural analysis collection includes frame analysis and also some specialised finite element and beam analysis modules.
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Quick Reference
Structural Analysis using PROKON
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Frame Analysis
3-5
Plane Stress/Strain Analysis
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Single Span Beam Analysis
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Beam on Elastic Support Analysis
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Structu ral Analysis using PROKON
Structural Analysis using PROKON
The accent of the analysis modules falls on user friendliness, speed and efficiency. The frame analysis module is ideally suited for the analysis of small to medium sized structures, not to say that the analyses of large structures are not possible. Frame has a comprehensive array of static and dynamic analysis modes. Extensive use is made of interactive graphic representations during both the input and output phases. The input modules incorporate error checking to help eliminate input errors as they occur.
Frame analysis Frame can take account of own weight, temperature changes, prescribed displacements and elastic supports. Loads are entered as load cases and grouped in load combinations at ultimate and serviceability limit states. The following static analysis modes are available: •
Linear analysis: Normal elastic frame analysis.
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Second order analysis: Models sway behaviour by incorporating P-delta effects. The solution is obtained by iterative analysis, thereby allowing for options like tension elements.
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Non – linear analysis. This takes the second order analysis a bit further. The load is applied in steps and the deflected structure at the end of each step is used to apply the next step. Material non – linearity is not yet supported.
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Buckling analysis: For calculating safety factors for str uctural instability due to buckling.
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Dynamic analysis modes available include: •
Modal analysis: Calculation of a f rame’s natural modes of vibration.
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Harmonic analysis: For determining a frame’s res ponse to harmonic loading.
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Earthquake analysis: Quasi-dynamic analysis of a frame subjected to ground acceleration. 3-3
Finite element analysis Frame allows you to use finite shell elements and solid elements alongside normal beam elements. The shell elements enables you to model the combination of plate bending and membrane action in 3D. To model plate bending in concrete slabs, you may prefer using the Finite Element Slab Design – see Chapter 6 for details. You can also use the Plane Stress/Strain Analysis module to perform a finite element analysis of any general plane geometry subjected to plane stress or strain. The module features an automated element grid generation facility to help speed up the input and analysis processes.
Beam analysis Modules are available for the analysis of si mple beams and beams on elastic supports.
Post-processing of analysis results Linear and second-order analysis output can be post-processed by the steel member design modules, Strut and Com, to evaluate and optimise section profiles. The Space Frame Analysis module can also design finite shell elements as reinforced concrete members.
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Structu ral Analysis using PROKON
Frame Analysis
Frame can be used for the analysis of the following types of structures by selecting a domain on the ‘General’ input tab: •
Plane Frames: Analysis of a frames in a vertical (X-Y) plane.
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Grillages: Analysis of a structure in a horizontal (X-Z) plane.
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Space Frames: Analysis of three-dimensional structures made up of beam and/or shell elements and design of concrete shells.
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Space Trusses: Analysis of three-dimensional trusses where only axial forces are considered.
Frame analysis results can be post-processed using the steel member design modules for axial and combined stress, namely Strut and Com.
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Frame Analysis
Theory and application The following text explains the sign conventions used and gives a brief background of the analysis techniques.
Sign conventions Frame input and output uses a mixture of global axis and local axes values. Global axes The global axis system is nearly exclusively used when entering frame geometry and loading. Global axes are also used in the analysis output for deflections and reactions. The global axes are defined as foll ows: •
For the sake of this definition, the X-axis is chosen to the right.
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The Y-axis always points vertically upward.
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Using a right-hand rule, the Z-axis points out of the screen. Note: Unlike some other 3D programs that put the Z-axis vertical, Frame take the Y-axis as being vertical.
Beam element local axes Local axes are used in the output for element forces. You can also apply loads in the direction of a beam element’s local y-axis.
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The local axes for beam elements are defined as follows: •
The local z-axis and axial force is chosen in the direction from the smaller node number to the larger node number.
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The y-axis is taken in a vertical plane perpendicular to the z-axis. The y and z-axes thus describe a vertical plane with the y-axis pointing vertically or diagonally upward.
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The x-axis is taken perpendicular to the y and z-axes, using a left-hand rule.
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One special case exists: •
In the case of a vertical member, the z-axis is taken parallel to the global Y-axis. A unique definition of the y-axis is obtained by taking it parallel to the global X-axis.
Shell element local axes For shell elements, the local axes are defined as follows: •
The local x and y-axes are chosen in the plane of the shell in such a way that the x-axis is horizontal and the y-axis lies perpendicular to and upward from the x-axis.
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Using a right-hand rule the z-axis is taken perpendicular to the shell element to point diagonally upward.
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Two special cases exist: •
Horizontal elements: The local x is chosen parallel to the global X-axis and the y-axis parallel to the negative Z-axis. The z-axis is then taken parallel to the Y-axis.
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Vertical elements: The y-axis is taken vertically upward, i.e. parallel to the global Y-axis. The x-axis is taken horizontal in the plane of the shell and z-axis is taken horizontal perpendicular to the shell. The z-axis points t owards you if the shell’s nodes are defined in an anti-clockwise directi on and away if defined clockwise.
To simplify the analysis output, the orientation of the local shell axes can sometimes be manipulated by slight rotation of the shell elements. In the case of horizontal slab, for example, the local x and y-axes (and stresses) are taken parallel to the global X and Z-axes. In the case of a circular slab, radial and concentric stresses may often be more desirable. By generating the shell elements at a slight slope towards the centre, they will not be considered as horizontal 3-8
Frame Analysis
anymore. As a result, local y-axes will point (upward) towards the centre and the x-axes taken perpendicular to that, i.e. radial and concentric respectively. The small inclination will normally have no significant effect on the analysis. Note: Rotating elements (for the sake of manipulating the local axes) can induce additional support conditions in some cases. Such manipulation should thus be performed with great care. Beam element forcesThe conventions are as follows:
sign
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The axial force, Pz, is taken in the z-direction.
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The shear forces, Vx and Vy, are given in the x and y-directions respectively.
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Torsional moment, T, is taken about the z-axis using a right-hand rule.
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The moments, Mxx and Myy, are about the x and y-axes respectively.
Note: In this manual, the global and local axes are written in uppercase and lowercase respectively. Shell element stresses Shell element stresses are given using t he local axes: •
Bending stresses: The entities Mx and M y are moment per unit width about the local x and y-axes. Mxy represents a torsional moment in the local x-y plane. The principal bending moments per unit width are represented as Mmax and Mmin.
•
Plane stresses: The stresses in the plane of a shell, Sx and Sy, are given in the directions of the local x- and y-axes. Sxy represents the shear stress in the plane of the element. Values are also given for the principal plane stresses, Smax and Smin. Note: To assist you in evaluating shell element stresses, stress contour diagrams show orientation lines at the centre of each shell element. An orientation line indicates the direction (not axis) of bending or plane stress. In a concrete shell, the orientation line would indicate the direction of reinforcement resisting the particular stress.
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Shell reinforcement axes Reinforcement is calculated in the user-defined x’ and y’-directions. Refer to page 3-64 for detail. Solid element stresses Shell element stresses are given using the global axes: Direct stresses, Sx , Sy and Sz, are given in the directions of the global X-, Y- and Z-axes. S xy represents the shear stress in the XY-plane of the element. Sxz represents the shear stress in the XZ-plane of the element. S yz represents the shear stress in the YZ-plane of the element. Values are also given for the principal plane stresses, S 1, S2, and S3. Von Mises stresses are also calculated.
Units of measurement The following units of measurement are supported: Units
Metric
Imperial
Distance
mm, m
ft, inch
Force
N, kN
lb, kip
Use the Convert Units button on the Settings page to change the units for the curr ent analysis: •
Convert Units: Changes the units and converts all numeric data from the old to the new units of measurement.
Analysis modes The following types of analysis are possible: •
Linear analysis: Basic linear elastic analysis. A linear analysis is normally sufficient for the static analysis of a frame or truss with negligible sway.
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Second order analysis: Choose this mode to include p-delta effects in the analysis. This option is recommended for structures where sway may have a marked effect on the member forces, e.g. portal frames. The second order analysis is an iterative procedure. The total strain energy of the frame is calculated after each iteration. The analysis is deemed to have converged once the total strain energy of two sequential iterations differs by less than the specified tolerance. If convergence was not possible, e.g. structural instability due to buckling of critical members, a message to that effect will be displayed.
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Non Linear analysis: Choose this mode where non-linear effects and large deflections may be expected or where second order analysis might not provide sufficient accuracy.
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Frame Analysis
Loading is applied in a series of steps and an iterative analysis is carried out at each step so that the forces in the deflected structure at that point balance with the applied loading. •
Modal analysis: For calculating the natural modes of vibration. The modal analysis is an iterative procedure during which several sets of trial vectors are selected and evaluated. The process takes relatively long to complete and it is therefore recommended that the structure size be limited to a few hundred nodes. You can specify the number of mode shapes to be calculated and other d ynamic analysis parameters.
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Harmonic analysis: Choose harmonic analysis to determine the response of the frame to harmonic loading. Load amplitudes are entered exactly like static nodal and element loads. You can enter a load frequency and phase angle for each harmonic load case. The first step of a harmonic analysis is the calculation of the frame’s natural modes of vibration. Therefore, if preceded by a modal analysis, the results of that analysis are re-used and only the harmonic response calculated. The harmonic response is taken as the sum of the square (SSRS) of the maximum modal responses, a method that is considered fundamentally sound when modal frequencies are well separated. When frequencies of major contributing modes are very close together, the SSRS method can give poor results.
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Earthquake analysis: Use this option to calculate the response of the frame to the specified a seismic acceleration parameters. Nodal and element loads entered are treated as static loads. The analysis procedure starts by calculating the frame’s natural modes of vibration. Therefore, if preceded by a modal analysis, the results of the modal analysis are re-used and only the seismic response calculated.
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Buckling analysis: Use this option to determine the buckling load factors and mode shapes for each load case or combination. Being the critical case, the first buckling mode shape is normally the only one of interest.
Finite element shells Frame allows you to use finite shell elements alongside normal beam elements. For this purpose the program uses four-node quadrilateral and three-node triangular isoparametric shell elements with plate bending and membrane behaviour. Element formulation The bending formulation of the quadrilateral shell element was derived from Kirchoff-Midlin Quadrilateral. The membrane behaviour of the element was introducing the drilling degree of freedom using an interpolation technique by result is a shell finite element that shows good plate and membrane characteristics.
the Discrete improved by Alman. The performance
Accuracy of triangular elements Both the quadrilateral and triangular elements yield accurate stiffness modelling. However, stress recovery from the triangular elements is not as accurate as is the case for quadrilateral 3-11
elements. This means that deflections calculated using triangular elements are generally quite accurate, but moments may be less accurate. Stress smoothing A reduced integration technique is used to calculate the element stiffness matrices. The stresses are calculated at the Gaussian integration points and subsequently extrapolated bi-linearly to the corner point and centre point of each element. Taking the average of all contributing stress components smooths stresses at common nodes. Element layout Consider a typical continuous flat concrete slab supported on columns or walls. To ensure accurate modelling of curvature, a minimum of about four elements should be used between bending moment inflection points. This translates to a minimum of about eight elements per span in both directions. Using more elements per span often does not yield a significant improvement in analysis accuracy. In addition, the particular finite element formulation yields its most accurate results when the element thickness does not greatly exceed its plan dimensions. For a typical concrete slab with a thickness of about one-tenth or one-fifteenth of the span length, a reasonable rule of thumb is to make the plan dimensions of the shell elements no smaller than the thickness of the slab. In other words, use a maximum of about ten to fifteen elements per span.
Finite element solids Frame also provides 8-noded hexahedral and 4-noded tetrahedral solid finite elements. The elements have 3 rotational degrees of freedom per node and are generally more accurate than more commonly used solid elements that do not have rotational degrees of freedom. Accuracy of tetrahedral elements Both the hexahedral and tetrahedral elements yield accurate stiffness modelling. However, stress recovery from the tetrahedral elements is not as accurate as is the case for hexahedral elements. This means that deflections calculated using tetrahedral elements are generally quite accurate, but stresses may be less accurate. Stress smoothing A reduced integration technique is used to calculate the element stiffness matrices. The stresses are calculated at the Gaussian integration points and subsequently extrapolated to the corner nodes of each element. Taking the average of all contributing stress components smooths stresses at common nodes. Element layout 3-12
Frame Analysis
To ensure accurate modelling of curvature, a minimum of about four elements should be used between bending moment inflection points. This translates to a minimum of about eight elements per span direction in a continuous slab.
Concrete design Frame can perform reinforced concrete design for shell elements. The Wood and Armer equations are used to transform the bending and torsional stresses to effective bending moments in the user-defined x’ and y’-directions. To allow for the effect of in-plane forces, bending moments Mx, M y and Mxy are increased to include the effects of these forces. The moments are increased by conservatively taking the inplane forces to act with a lever arm of a quarter of the section depth. The Wood and Armer equations are then evaluated in same manner as described above. Codes of practice The following concrete design codes are supported: •
ACI 318 - 1995.
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BS 8110 - 1997.
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CSA A23.3 - 1993.
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Eurocode 2 - 1992.
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SABS 0100 - 1992.
Steel member design Frame analysis results can be opened in the steel member design modules for design. The available options are: •
Steel Member Design for Axial Stress, Strut: Can design steel trusses.
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Steel Member Design for Combined Stress, Com: Can design beam members.
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Settings Settings are done on the ‘Settings’ tab:
Analysis type •
Linear analysis: Normal linear elastic frame analysis. A linear analysis is normally sufficient for the static analysis of a frame or truss with negligible sway. The linear analysis procedure is performed faster than any other type of analysis. If you need to perform a second order, buckling or dynamic analysis, it will be wise to first verify the basic integrity of the frame input by performing a linear analysis.
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Second order analysis: Models sway behaviour by incorporating P-delta effects. The solution is obtained by iterative analysis, thereby allowing for options like tension elements.
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Frame Analysis
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Non-linear analysis: This analysis is used when large deflections or non-linear behaviour are expected. Only geometric non-linearity is supported a this stage. Material non-linearity will be added in the near future. The solution is obtained by a stepped iterative analysis. Loads are added in steps. The analysis is iterated to convergence for each step so that the reactions and forces are in balance with the applied loads after each step. The deflected structure at the end of each step is then used to apply the next load step and the process is repeated until the total load has been applied.
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Modal analysis: Calculation of a frame’s natural modes of vibration. The process takes relatively long to complete and it therefore recommended that the structure size be limited to a few hundred nodes. You can specify the number of mode shapes to be calculated and other dynamic analysis parameters.
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Harmonic analysis: For determining a frame’s response to harmonic loading.
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Earthquake analysis: Quasi-dynamic analysis of a frame subjected to ground acceleration.
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Buckling analysis: For calculating safety factors for structural instability due to buckling. You can specify the number of mode shapes t o be calculated.
Analysis parameters Depending on the selected analysis type, you may need to specify additional analysis parameters: •
Concrete design parameters: Concrete and reinforcement properties. Details are given on page 3-64.
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Second order and buckling parameters: Required analysis tolerance and number of buckling mode shapes. Refer to page 3-65 for detail.
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Dynamic parameters: Values influencing modal, seismic and harmonic analysis. A detailed discussion is given on page 3-67.
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Non linear parameters: Values influencing the non-linear analysis. A detailed discussion is given on page 3-67.
Own weight The own weight of the frame can be calculated using the entered cross-sectional areas and member lengths. If you specify a load case, the own weight is calculated and added to the other loads of that case. The following are points of i mportance: •
By default, the own weight of the frame is set to not be included in the analysis. Be sure to select the appropriate load case for own weight or, alternatively, to include the frame's own weight in the values of the loads entered. 3-15
The list of load cases from which you can select is based on the load cases defined on the Nodal loads, Beam loads, and Shell loads input pages. You may thus prefer to specify the own weight load case only after completing all other input for the frame. However, you can also enter the own weight load case at the start of the frame input process in which case you may ignore the warning message (that the load case does not exist).
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Tip: If you wish to use own weight in its own separate load case, you can do so by defining an empty load case. You can enter a zero load at any node number, for example, and then select that load case as the one to use for own weight. The own weight or beam elements are modelled as uniformly distributed loads along the lengths of the beams. In the case of a vertical beam element, own weight is modelled as two equal point loads at the ends of the beam, yielding a constant axial force equal to half the own weight. In the case of shell elements, own weight is modelled as point loads at the corner nodes. In the case of solids, the weight is added as point loads at the nodes. Graphics Options Click on the graphics options button to have the graphics options dialog displayed. •
Select whether you want items like node numbers and supports to be displayed.
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Choose whether you want all beam elements or only a certain type to be displayed.
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Display the structure with full 3D rendering, e.g. to verify section orientations. 3D rendering is automatically suppressed when viewing output.
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Choose quick or detailed rendering. Quick rendering is faster than the detailed method, but you may find that some surfaces are drawn incorrectly.
All surfaces are drawn as polygons. You can choose to make the surfaces transparent or have them filled and outlined. Tip: The Graphics options and 3D rendering function can also be accessed using the buttons next to the displayed picture.
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Frame Analysis
•
Views: You can save the current viewpoint and view plane. The current view’s name is displayed on the picture. To re-use a saved view, click the view name on the picture to drop down a list of saved views.
The functions described above can also be used when viewing output. Contour diagrams, for example, are drawn as polygons. You can therefore use the Graphics options setting for polygons to change their appearance. Views defined during input are also available when viewing output and vice versa.
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Input Work through the relevant Input pages to enter the frame geometry and loading: •
General input: Select the domain (Plane frame, grillage etc.) The input wizards can also be selected here. More about these wizards l ater.
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Nodes input: Frame coordinates.
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Beams input: Join nodes with beam elements.
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Beam sections input: Enter properties or read sections from the database.
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Shells input: Define shell elements.
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Solids input: Define 4- or 8- noded solid elements
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Spring elements input: For special effects, optionally enter spring elements.
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Supports input: External supports.
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Nodal loads input: Point loads and moments.
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Beam element loads input: Uniform distributed, triangular, trapezium and point loads on beams.
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Shell loads input: Apply uniform distributed loads to shells.
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Load combinations input: Group dead, live and wind loads in load combinations.
Alternative methods of generating frame analysis input are discussed on page 3-61.
Viewing the structure You may want to enlarge portions of the picture of the structure or rotate it on the screen. Several zoom and rotate functions, all of which are described in detail in Chapter 2, are available to help you using pictures of t he structure: •
Use the Zoom buttons to zoom into a part of the structure or view it from another angle.
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Use the View Point Control to set a new viewpoint or camera position.
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Use the View Planes Control to view a slice of the structure.
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Frame Analysis
General input Wizards The wizards are suitable for the rapid generation of complete input files for some typical structures. Because the resulting input data is presented in the normal way on the input pages, you are free to edit and append to the data as necessary. Input generated this way can optionally be appended to existing data – you can therefore repeatedly use the wizards to generate complicated structures.
Note: The frame analysis modules are not limited to modelling only those frames generated by the wizards. Any general two or three-dimensional frame can be collectively built up. The wizards merely serve to simplify input of typical frames. Adding input data to the Calcsheets You can append the input tables (as they appear on the screen) to the Calcsheets by clicking the Add input tables to Calcsheets button. You can add a picture from any input pages to the Calcsheets by clicking the Add to Calcsheets button next to the picture i n question. Title Enter a descriptive name for the frame. It should not be confused with the file name you use when you save the input data.
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Nodes input Use as many lines as necessary to enter the nodes defining the frame. A unique number must be assigned to each node. The node number is entered in the No column, followed by the X, Y and Z-coordinates in the X, Y and Z columns. If you leave X, Y or Z blank, a value of zero is used. You are allowed to skip node numbers to simplify the definition of the frame. You may also leave blank lines in the input to improve readability. If a node number is defined more than once, the last definition will be used. Note: Most of the examples given in this section show 3D co-ordinates as would be applicable if the domain is set to Space Frame or Space Truss. If the domain is set to Plane Frame or Grillage use the X-Y and X-Z planes res pectively.
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Frame Analysis
Error checking The program checks for nodes lying at the same position. If a potential error is detected, an Error list button will appear. Generating additional nodes When defining a node, you can have additional nodes generated at regular intervals. Example:
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The Y-coordinate of node 4 is left blank. Therefore, node 4 is put at the coordinate (0.805,0,14.614).
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The No of is set to ’2’, meaning that two additional nodes must be generated.
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Setting Increment to ’7’ means that the node numbers are incremented by seven. Therefore, node 4 is copied to node 11 and node 11 is copied t o node 18.
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The values in the X-inc, Y-inc and Z-inc columns set the distance between copied nodes. The coordinates 4 to 18 are horizontally spaced at 1.140 m and 0.472 m along the X and negative Z-axis respectively. The coordinates of the additional nodes are thus (1.945,0,14.142) and (3.085,0,13.670).
An alternative method to generate equally spaced nodes is to use the Inc to End option. This method allows you to define two n odes and then generate a number of nodes in-between: •
Use the same procedure as above to defi ne the first node’s coordinates.
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Set the values of X-inc, Y-inc and Z-inc to the total coordinate difference to the last node and enable the Inc to End option. The last node’s coordinates are then first calculated and the specified number of intermediate nodes t hen generated.
Second order generation Once you have defined one or more nodes in the table, you can copy that relevant line’s nodes by entering a '–' character in the No column of the next line. Then enter the number of additional sets of nodes to be generated in the No of column and the coordinate increments in the X-inc, Y-inc and Z-inc columns.
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Second order generation example:
The following nodes are generated: No
X
Y
Z
15
0.00
5.12
0.00
16
2.00
5.12
0.10
17
4.00
5.12
0.20
18
0.00
5.62
1.00
19
2.00
5.62
1.10
20
4.00
5.62
1.20
Block generation A group of nodes can be repeated by entering a ’B’ in the No column followed by the first and last table line numbers in which the nodes were defined. Separate the line numbers with a ’–'. ’–'. Block generation example:
The nodes defined in lines 11 to 26 are copied twice. Node numbers are incremented by thirty for each copy. The X, Y and Z-coordinate increments are 10 m, zero and zero respectivel y. To copy one line only, simply omit the end line number, e.g. 'B10' to copy line 10 only. Tip: The current line number is displayed in the status bar at the bottom left of the program's window. The block generation function may be used recursively. That means that the lines specified may themselves contain further block generation statements. Moving nodes To move a group of nodes to a new location without generating any new nodes, use the block generation function and set No-of to '1' and Inc to Inc to '0'. Arc generation A group of nodes can be repeated on an arc by entering an 'A' in the No column, followed by the start and end line numbers. Enter the centre of the arc in the X, Y and Z columns and use the X-inc, X-inc, Y-inc or Z-inc column to specify the angle increment about the X, Y or Z-axis respectively. If the program domain is set to Plane Frame or Grillage, Grillage, the angle increment should be entered in the last column. Rotation will be about the Z and Y-axis respectively. 3-22
Frame Analysis
Example:
All nodes defined in lines 5 to 9 of the table will be repeated eleven times on an imaginary horizontal arc. The centre point of the arc is located at the coordinate (10,0,1.5). The node number increment is set to 5, i.e. node number 3 becomes node 8, etc. The rotation angle between the generated groups of nodes is 30 degrees about the Y-axis, i.e. anti-clockwise using a right-hand rule. To copy one line only, simply omit the end line numbers, e.g. ’A12’ to copy line 12 only. Note: The arc generation function may be used r ecursively. Rotating nodes To rotate a group of existing nodes without generating any new nodes, use the arc generation function and set the No-of to ’1’ and Increment to Increment to ’0’. Deleting nodes Nodes can be deleted by entering ’Delete’ in the Inc to end column. This can be especially handy if you have generated a large group of nodes and then need to remove some of them again. Example:
Nodes 15 and the additional nodes 18 and 21 are deleted. Graphical input The following graphical input functions are available on the left hand side of the screen. The toolbar containing the graphics options buttons can also be dragged and docked on any side of the picture. To delete nodes click the ‘Delete nodes’ button and then select the nodes to be deleted on the screen using the mouse. Click ‘Done’ when finished. Clicking ‘Undo’ will undo the deletions in reverse order. To block delete nodes click the ‘Block delete nodes’ button. Select a 3-23
rectangle on the screen with the mouse. All nodes inside the rectangle will be deleted. Press ‘Done’ when finished. Pressing ‘Undo’ will undo the deletions in reverse order, one by one. Pressing ‘Undo All’ will undo all deletions done with this function. Click the ‘Explode nodes’ button to explode the node input. This results in the list of nodes being written, each on a separate line without block & arc generations etc. Once done, it cannot be reversed.
Click the ‘Join loos structure segments’ button to have all duplicate nodes (having the same co-ordinates) that are not connected by spring elements, deleted. This will ensure that loose pieces of the structure become connected. The function is not reversible.
Click on ‘Delete loose nodes’ to have all nodes not connected to elements deleted. The program will ignore loose nodes in t he analysis, but the input is neater and easier to interpret if unwanted nodes are removed. The function is not reversible. Deleted nodes can however be undeleted if the bottom of the input table is edited and the nodes entered as ‘deleted’ are removed.
Beam elements input A beam or frame element is defined by entering the node numbers at each end, separated with a '–'. For example, '3–9' is the element linking nodes 3 and 9. The elements themselves are not numbered. A series if elements can be input in a string, e.g. '2-6-10-14-18-22-24'. If the node number increment of a series is constant, you can replace intermediate nodes with two '–' characters. In the string above, nodes 2 to 22 have a constant increment of four. Therefore, the string can be rewritten as '2-6- - 22-24'. The node increment of four is derived from '2-6'.
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Frame Analysis
An element definition must include a section number entered in the Section Name column. The section name is used to identify the relevant section. The actual section properties for each section number defined on the Beam Sections input page.
Section orientation in a 3D analysis In 3D analysis, the local y-z plane of an element is taken as vertical by default. The principle can be illustrated by considering an I-section in its normal orientation. For this case, the web will always be considered to be in a vertical plane. If the element is aligned vertically, i.e. a column, the web will be in a vertical plane anyway. For this special case, the local y-axis is aligned with the global X-axis, i.e. the web is taken in the global X-Y plane. 3-25
An element can be rotated about its axis by entering a beta angle. The beta angle is measured about the z-axis, taking the default orientation as 0°. Instead of entering a beta angle, you may also enter a reference node – the beta angle is then taken in t he plane described by the element's nodes and the reference point. To use a reference point, first define a node with the relevant coordinate and then enter 'N' followed by the node number i n the Beta column Tip: Enable full 3D rendering in the Graphics options to view the true beam orientation. Section orientation in a 2D analysis In the case of a 2D analysis, the local y-z plane of an element is taken in the global X-Y plane. The principle can be illustrated by considering an I-section in its normal orientation. For this case, the web will always be considered to be in a vertical plane. Tip: Enable full 3D rendering in the Graphics options to view the true beam orientation. A section can be rotated through ninety degrees by selecting the alternative orientation when reading it from the section database. Note: In a space truss analysis the section orientation is of no importance. The analysis results are influenced by the secti on area and not by it's second moment of inertia. End fixity The fixity at each end of an element, i.e. continuous or pinned, must also be defined in the Fixity columns. Pins are modelled on the element itself and not on the node. External pinned supports should be defined on the supports input table. External supports are described in the next section. The following types of end fixities can be specified: •
Fixed: Specify 'F' to provide full rotational continuity. If you leave the field blank, 'F' is assumed.
•
Pinned: Use 'P' to for no rotational restraint, i.e. a ball-joint.
•
Torsional fixity: Use 'T' to provide restraint for rotation about the element axis only. This option is only available in the Grillage domain and the Space Frame domain. Note: To retain compatibility with the Dos version, you may also use '0', '1' or '2' instead of 'F', 'P' or 'T' respectively.
Entered fixities are applied at an element's lower node number (designated as the 'left' end) and higher node number (the 'right' end). The order of the node numbers entered in the first column of the table has no bearing on the application of the fixity c odes. To define a pin only at the two remote ends of a group of elements, enable the Group fix option by entering a 'Y'. In this case, the normal convention of smaller and larger node 3-26
Frame Analysis
numbers does not apply. Instead, pins are put at the remote ends in the same order that the nodes have been entered. Example:
The group of elements from node 42 to 24 is continuous except for the pins used at nodes 42 and 24. Note: Do not use an internal pin on an element to model an external support that allows free rotation. Rather allow the beam to be fixed to the node and define a simple support on the Support input page. If the Group fix is left blank or ’N’ is entered, the normal individual element fixity mode is assumed. Tip: Element fixity can be displayed graphically on the screen. For this, edit the Graphics options to disable the Elements Continuous option. When using pins, especially in the Space Frame domain, you should take care to ensure overall stability of the frame. Consider two elements on a straight line with pins at all three relevant nodes, for example. The centre node will be unrestrained for rotation about the element axis, resulting in instability during the analysis. Note: When performing a second order analysis, you can use tension elements to model bracing, for example. For this, special settings need be made on the Beam Sections input page. Refer to page 3-33 for detail. Tapered beams A beam can be made to taper between by entering two or three section names, separated with commas: •
Use two sections, e.g. ’Rafter,Haunch’ or ’1,2’, to make the program vary the section properties linearly along the length of the beam element. The first and second sections are taken at the lower and higher node numbers respectively.
•
For a more accurate non-linear variation, enter three section names, e.g. ’Rafter,Middle,Haunch’. The first, second and third sections are taken at the at the lower node number, the centre of the element and at higher node number respectively.
The procedure the enter haunches is described on page 3-32.
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Rigid links You can use rigid links to rigidly offset sub-structures, e.g. slabs with downstand beams. To define a rigid link, enter ’R’ in the Section Name column. Rigid links are modelled as very stiff beams. The stiffness of a rigid link is determined by multiplying the maximum area and bending stiffness of the other beams with a factor, typically one thousand. The rigid multiplication factors can be adjusted using the Advanced option on the Beam sections input page. Refer to page 3-33 for detail. Rigid link example:
Rigid links are defined between nodes 12 and 24, 14 and 26 and 16 and 26. Generating additional elements You can generate additional elements with the same section and fixity code values using the No of extra and Node No Inc columns. Example:
The elements between nodes 251 and 266 are copied ten times with the node numbers decrementing by five with each copy. Block generation A group of elements can be repeated by entering a ’B’ in the No column. Then enter the first and last table line numbers in which the elements were defined, separated with a ’–'.
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Frame Analysis
Block generation example:
All elements defined in lines 11 to 26 will be copied twice with a node number increment of thirty. The copied elements will use the same section number and fixity codes as the original elements. To copy one line only, simply omit the end line number, e.g. ’B11’ to copy line 11 only. Tip: The current line number is displayed in the status bar at the bottom left of the program’s window. The block generation function may be used rec ursively. The group of lines r eferenced may thus contain block generation statements. Tip: When entering a complicated structure it may help to leave a few blank lines between groups of elements. Not only will it improve readability, but it will also allow you to insert additional nodes at a later sta ge without upsetting block and arc generations. Deleting elements Beam elements can be deleted by entering a special section name ’Delete’. This can be especially handy if you have generated a large group of elements at regular increments and need to remove some of them a gain. Example:
Elements 25-27-29 and 35-37-39 are deleted. Note: The display of selected beam element groups can be activated or suppressed by editing the Graphics options. Error checking The program checks for duplicate elements and elements with zero length. It also checks that a section number is assigned to each element. If an error is detected, an Error list button will be displayed.
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Graphical input The following graphical input functions are available on the left hand side of the screen. The toolbar containing the graphics options buttons can also be dragged and docked on any side of the picture. To add beams click the ‘Add beams’ button. Enter the section name, ß angle and fixities for the beams. By clicking two successive nodes, a beam will be inserted between them. If ‘follow on’ is checked, the last node of the previous beam is taken as the first node of the next beam. If ‘Link end nodes only’ is checked, only one beam is placed between the last two nodes entered. If not, all nodes between the last two nodes are also added into the beam string. The nodes must lie within a certain tolerance from the straight line between the end nodes to be included. This tolerance can be entered in the Tolerance (%) field. If e.g. 1% is entered, any node closer than 1% of the distance between the end nodes from the line joining them is included. Pressing ‘Undo’ will delete the beams in reverse order, in which they were entered, one by one. To delete beams click the ‘Delete beams’ button and then select the beams to be deleted on the screen using the mouse. Click ‘Done’ when finished. Clicking ‘Undo’ will undo the deletions in reverse order. To block delete beams click the ‘Block delete beam elements’ button. Select a rectangle on the screen with the mouse. All beams inside the rectangle will be deleted. Press ‘Done’ when finished. Pressing ‘Undo’ will undo the deletions in reverse order one by one beam. Pressing ‘Undo All’ will undo all deletions done with this function.
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Frame Analysis
Click the ‘Change beam properties’ button to change beam properties. Enter the desired properties for the beam(s) on the dialog. One can also use the ‘Get properties’ button to do this. Click the button and then click on a beam. The beam’s properties will then be transferred to the dialog. Clicking on beams will now change their properties to those specified on the dialog. The ‘Undo’ button will undo the changes in reverse order.
Click the ‘Explode beams’ button to explode the beam input. This results in the list of beams being written, each on a separate line without block generations etc. Once done, it cannot be reversed.
On the beams input page one can also delete nodes and groups of nodes in the same way as on the nodes input page. Refer to page 3-24 f or details
Beam sections input Section properties should be assigned to all section names used on the Beam elements input page. The following properties are required for all sections: •
Cross sectional area, A.
•
Second moment of area about the local x-axis, Ix (not required for Space Truss analysis).
•
Second moment of area about the local y-axis, Iy (Space Frame domain only).
•
Torsional moment of inertia, J (Grillage and Space Frame domain only).
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Each section should also have an associated material selected. If no section or material properties are entered, the values applicable to t he previous line in the table are used. Reading sections from the database Use the Section database function to display and select sections from database. You can add your own sections, e.g. plate girders, to the database using the procedures described in Chapter 2. Entering haunches Haunches are entered by appending the haunch depth to the section designation. To add a haunch of 180 mm to a ’305x102x66’ BS taper flange I-section, enter ’305x102x66 (0.280h)’. The overall depth is then taken t o be 305 mm + 280 mm = 585 mm. Tip: You can verify your definition of haunches by enabling 3D rendering. Refer to page for 3-18 more detail.
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Frame Analysis
Tension members When performing a second order analysis, you can designate members to have tension stiffness only, e.g. slender bracing members. To make a member be ignored during the analysis when it would act in compression: 1.
Enter the member’s section properties in the usual manner.
2.
Edit the value for the cross sectional area and change its sign to negative.
Note: The program uses the absolute value of the cross sectional area. The negative sign entered merely enables the tension-only behaviour for beams of the given section group. Own weight If a material’s definition includes a density value, the own weight of a member is calculated automatically and added to the load case specified on the Settings page. Selecting materials Each section should have an associated material. To add one or more materials to a frame analysis data file, click Materials. Open the relevant material type screen and select the materials that are required for the current frame input. After adding the selected materials to the input, you can select them by clicking the Material column to drop down a list.
Adding materials to the global database The procedure to permanently add more materials to the database is described in Chapter 2.
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Advanced section options Clicking the Advanced button allows you to configure the behaviour of rigid links. A rigid link is modelled as a very stiff beam of which the area, second moment of inertia and modulus of elasticity are taken as the maximum corresponding properties all other beams multiplied with the specified factors.
Refer to page 3-27 for detail on defining rigid links.
graphical input The following graphical input functions are available on the left hand side of the screen. The toolbar containing the graphics options buttons can also be dragged and docked on any side of the picture. Click the ‘Change beam properties’ button to change beam properties. Enter the desired properties for the beam(s) on the dialog. One can also use the ‘Get properties’ button to do this. Click the button and then click on a beam. The beam’s properties will then be transferred to the dialog. Clicking on beams will now change their properties to those specified on the dialog. The ‘Undo’ button will undo the changes in reverse order.
Shell elements input You can use finite shell elements alongside beam elements, except in the Space truss domain. Shell elements can be optionally designed as reinforced concrete members. Elements are defined by referring to corner nodes, four in the case of quadrilaterals and three for triangles. You should enter the node numbers in sequence around the perimeter, either clockwise or anti-clockwise, in the Node 1 to Node 4 columns. Leave Node 4 blank to define a triangular element. 3-34
Frame Analysis
Note: Quadrilateral elements generally yield more accurate analysis results than triangular elements. Refer to page 3-11 for more detail.
An element definition must include a thickness and material type. Refer to page 3-33 for more detail on using materials. Own weight If a material’s definition includes a density value, the own weight of a member is calculated automatically and added to the load case specified on the Settings page. Tip: When entering a complicated slab, it may help to leave a few blank lines between groups of elements. Not only will it improve readability, but it will also allow you to insert additional elements at a later s tage without upsetting block and arc generations. Error checking
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The program checks for duplicate elements and nodes not connected to elements. It also checks that a group number is assigned to each element. If an error is detected, an Error list button list button will be displayed. Generating additional elements You can generate additional elements with the same group number using the Number of Extra and Node no Inc columns. Inc columns. Block generation You can use the block generation to copy shells you defined earlier in the table. A group of shells can be repeated by entering a ’B’ in the Node 1 column followed by the first and last table line numbers in which the nodes were defined. Separate the line numbers with a ’–'. ’–'. To copy a single line only, simply omit t he end line number, e.g. 'B11' to c opy line 11 only. Tip: The current line number is displayed in the status bar at the bottom left of the program's window. The block generation function may be used recursivel y. The group of lines ref erenced may thus contain block generation statements. Deleting elements Shell elements can be deleted by entering 'Delete' in the Material column. This can be useful if you have generated a large group of elements and need to remove some of them again. Example:
The element 15-16-26-25 and the generated element 18-19-29-28 are deleted.
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Frame Analysis
Graphical input The following graphical input functions are available on the left hand side of the screen. The toolbar containing the graphics options buttons can also be dragged and docked on any side of the picture. To add shells click the ‘Add shell elements’ button. Enter the material, thickness and shell type (triangular or quadrilateral). The easiest way to enter shells is to define a plane in which they lie. Click 3 nodes to indicate the plane. The three node numbers can also be typed directly into the dialog. Also enter the thickness of the plane. Only the nodes lying in the plane and half of the thickness on either side are now displayed. If one now moves the mouse across the picture, possible shells are shown in purple. Click the mouse to have each shell entered into the input table. If you do not want to use a plane in this way, click ‘Don’t use plane’. Pressing ‘Undo’ will delete the shells in the reverse order in which they were entered, one by one. To delete shells click the ‘Delete shells’ button and then select the shells to be deleted on the screen using the mouse. Click ‘Done’ when finished. Clicking ‘Undo’ will undo the deletions in reverse order. To block delete shells click the ‘Block delete shell elements’ button. Select a rectangle on the screen with the mouse. All shells inside the rectangle will be deleted. Press ‘Done’ when finished. Pressing ‘Undo’ will undo the deletions in reverse order one by one. Pressing ‘Undo All’ will undo all deletions done with this function.
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Click the ‘Change shell element properties’ button to change shell properties. Enter the desired properties for the shell(s) on the dialog. One can also use the ‘Get properties’ button to do this. Click the button and then click on a shell. The shell’s properties will then be transferred to the dialog. Clicking on shells will now change their properties to those specified on the dialog. The ‘Undo’ button will undo the changes in reverse order. Click the ‘Explode shells’ button to explode the shell input. This results in the list of shells being written, each on a separate line without block generations etc. Once done, it cannot be reversed.
On the shells input page one can also delete nodes and groups of nodes in the same way as on the nodes input page. Refer to page 3-24 f or details
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Frame Analysis
Solid elements input You can use solid elements alongside beam and shell elements, if the domain is set as Space frame. Elements are defined by referring to corner nodes, eight in the case of hexahedrons and four for tetrahedrons. You should enter the node numbers in sequence around the perimeter, either clockwise or anti-clockwise, in the Node 1 to Node 8 columns. First define the back face going either clockwise or anti-clockwise in the node 1 to node 4 columns. Next define the front face starting with node 5 above the node defined as node 1 and moving around in the same direction as nodes 1 to 4. For tetrahedrons the four nodes can be entered in any order in columns 1 to 4. Note: Hexahedral elements generally yield very accurate analysis results. The tetrahedrons should only be used as filler el ements where it is not possible to use hexahedrons.
An element definition must include a material type. Refer to page 3-33 for more detail on using materials. 3-39
Own weight If a material’s definition includes a density value, the own weight of a member is calculated automatically and added to the load case specified on the Settings page. Tip: When entering a complicated model, it may help to leave a few blank lines between groups of elements. Not only will it improve readability, but it will also allow you to insert additional elements at a later stage without upsetting block and arc generations. Error checking The program checks for duplicate elements and nodes not connected to elements. It also checks that a group number is assigned to each element. If an error is detected, an Error list button will be displayed. Generating additional elements You can generate additional elements with the same group number using the Number of Extra and Node no Inc columns. Block generation You can use the block generation to copy shells you defined earlier in the table. A group of solids can be repeated by entering a ’B’ in the Node 1 column followed by the first and last table line numbers in which the nodes were defined. Separate the line numbers with a ’–'. To copy a single line only, simply omit t he end line number, e.g. 'B11' to c opy line 11 only. Tip: The current line number is displayed in the status bar at the bottom left of the program's window. The block generation function may be used recursivel y. The group of lines ref erenced may thus contain block generation statements. Deleting elements Solid elements can be deleted by entering ' Delete' in the Material column. This can be useful if you have generated a large group of elements and need to remove some of them again. Example:
The element 201-202-152-151-226-227-177-176 and the generated element 251-252-201-201276-277-227-226 are deleted.
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Frame Analysis
Graphical input The following graphical input functions are available on the left hand side of the screen. The toolbar containing the graphics options buttons can also be dragged and docked on any side of the picture. To add solids click the ‘Add solids elements’ button. Enter the material and solid type (tetrahedral, wedges or hexahedra). At the time of writing only hexahedra were available for this function. The easiest way to enter solids is to define a plane in which they lie. Click the ‘set up plane’ button to do this. Click three nodes to indicate the plane. The three node numbers can also be typed directly into the dialog. Also enter the thickness of the plane. Only the nodes lying in the plane and half of the thickness on either side are now displayed. If one now moves the mouse across the picture, possible solids are shown in purple. Click the mouse to have each solid entered into the input table. If you do not want to use a plane in this way anymore, click ‘Clear Plane’. Pressing ‘Undo’ will delete the solids in the reverse order in which they were entered, one by one. To delete solids click the ‘Delete solid elements’ button and then select the solids to be deleted on the screen using the mouse. Click ‘Done’ when finished. Clicking ‘Undo’ will undo the deletions in reverse order. To block delete solids click the ‘Block delete solid elements’ button. Select a rectangle on the screen with the mouse. All solids inside the rectangle will be deleted. Press ‘Done’ when finished. Pressing ‘Undo’ will undo the deletions in reverse order one by one. Pressing ‘Undo All’ will undo all deletions done with this f unction. 3-41
Click the ‘Change solid element properties’ button to change solid element properties. Enter the desired material for the solid(s) on the dialog. One can also use the ‘Get properties’ button to do this. Click the button and then click on a solid. The solid’s material property will then be transferred to the dialog. Clicking on solids will now change their material to that specified on the dialog. The ‘Undo’ button will undo the changes in reverse order. Click the ‘Explode solids’ button to explode the solids input. This results in the list of solids being written, each on a separate line without block generations etc. Once done, it cannot be reversed.
On the solids input page one can also delete nodes and groups of nodes in the same way as on the nodes input page. Refer to page 3-24 f or details
Spring elements input You can use spring elements to provide elastic links between sub-structures, e.g. to model elastomeric bearings between a slab and supporting walls. In theory, two nodes connected with a spring element should have the same coordinates. The program will warn if this is not the case and still allow you to continue. Enter linear spring constants in the Kx, Ky and Kz columns and rotational spring constants in the Rx, Ry and Rz columns. The orientation of a spring element is defined by entering a bearing between any two nodes that do not necessarily need to be connected to the same or other spring elements as well. The directions of the axes are defined as f ollowed: •
A spring element's x-axis is taken i n the direction of the orientating nodes.
•
The y-axis defined in the same way as for a normal beam element, i.e. perpendicular to spring element in a vertical plane.
•
The z-axis is taken perpendicular to the x and y-axes using aright-hand rule.
Spring element example:
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Frame Analysis
Spring elements are defined between nodes 16 and 116, 17 and 117 up to 19 and 119. The spring elements are aligned parallel t o the imaginary line joining nodes 3 and 4. Tip: Spring elements can also be made "rigid" to force two nodes to have the same translation and/or rotation. In the above example, a very large value for K x would cause nodes 16 and 116 t o have identical displacements in the direction described by nodes 3 and 4. Graphical input The following graphical input functions are available on the left hand side of the screen. The toolbar containing the graphics options buttons can also be dragged and docked on any side of the picture. To add springs click the ‘Add spring elements’ button. Enter the spring stiffnesses and orientation nodes on the dialog.. Now click the mouse on nodes to have springs entered into the input table. Pressing ‘Undo’ will delete the springs in the reverse order in which they were entered, one by one.
To delete springs click the ‘Delete spring elements’ button and then select the springs to be deleted on the screen using the mouse. Click ‘Done’ when finished. Clicking ‘Undo’ will undo the deletions in reverse order. To block delete springs click the ‘Block delete spring elements’ button. Select a rectangle on the screen with the mouse. All springs inside the rectangle will be deleted. 3-43
Press the ‘Done’ button when finished. Pressing ‘Undo’ will undo the deletions in reverse order, one by one. Pressing ‘Undo All’ will undo all deletions done with this function. Click the ‘Change spring element properties’ button to change spring element properties. Enter the desired spring stiffnesses and orientation on the dialog. One can also use the ‘Get properties’ button to do this. Click the button and then click on a spring. The spring’s properties will then be transferred to the dialog. Clicking on springs will now change their properties to those specified on the dialog. The ‘Undo’ button will undo the changes in reverse order.
Click the ‘Explode springs’ button to explode the spring input. This results in the list of springs being written, each on a separate line without block generations etc. Once done, it cannot be reversed.
On the springs input page one can also delete nodes and groups of nodes in the same way as on the nodes input page. Refer to page 3-24 f or details
Supports input Frames require external supports to ensure global stability. Supports can be entered to prevent any of the six degrees of freedom at a node, i.e. translation in the X Y and Z-directions and rotation about the X, Y and Z-axes. You can also define elastic supports, e.g. an elastic soil support, and prescribed displacements, e.g. foundation settlement. Enter the node number to be supported in the Node No column. In the next column a combination of the letters 'X', 'Y' and 'Z' can be entered to indicate the direction of fixity. Use capitals and lowercase to define restraint of translation and rotation respectively, e.g. 'XYZy' means fixed against movement in the X, Y and Z-direction and rotation about the Y-axis. 3-44
Frame Analysis
Note: The use of lowercase for rotational restraints should not be confused with the convention of using lowercase for local element a xes.
Tip: To enter a simple support with no moment restraint, one would typically enter a ’XYZ’ or ’Y’. Avoid using a pin on an element to model an external hinge. If you want to repeat the supports defined on the previous line of the table, you need only enter the node number, i.e. you may leave the Fixity column blank. If the XYZxyz column is left blank, the supports applicable to the previ ous line will be used automatically. Prescribed displacements and elastic supports Use the X, Y, Z, Rx, Ry and Rz columns to enter prescribed displacements and rotations in the direction of and about the X, Y and Z-axes. Being a global support condition, the effect of the prescribed displacement is not considered to be a separate load case. Instead, the effect
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of prescribed displacements is added once only to the analysis results of each load case and load combination. Elastic supports, or springs, are defined by entering spring constants in the X, Y, Z, x, y and z columns. The spring constant is defined as the force or moment that will cause a unit displacement or rotation in the relevant direction. Enter an ’S’ in the P/S column to indicate that an entered value is a spring constant rather than a prescribed displacement. If you leave the P/S column blank, the entered values are taken as prescribed displacements. Note: The display of supports can be activated or suppressed by editing the Graphics options. Error Checking The program does a basic check on the structural stability of the frame. If a potential error is detected, an Error list button will appear. Note: You cannot define an elastic support and a prescribed displacement at the same node because it will be a contradiction of principles. Generating additional supports Additional supports and prescribed displacements can be generated using the Number of extra and Node number inc columns. The procedure is similar to that described on page 3-21 for generating additional nodes.
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Frame Analysis
Graphical input The following graphical input functions are available on the left hand side of the screen. The toolbar containing the graphics options buttons can also be dragged and docked on any side of the picture. To add supports click the ‘Add supports’ button. Enter the fixities, prescribed displacements, spring constants and fixity type on the dialog. Now click the mouse on nodes to have supports entered into the input table. Pressing ‘Undo’ will delete the supports in the reverse order in which they were entered, one by one.
To delete supports click the ‘Delete supports’ button and then select the supports to be deleted on the screen using the mouse. Click ‘Done’ when finished. Clicking ‘Undo’ will undo the deletions in reverse order. To block delete supports click the ‘Block delete supports’ button. Select a rectangle on the screen with the mouse. All supports inside the rectangle will be deleted. Press the ‘Done’ button when finished. Pressing ‘Undo’ will undo the deletions in reverse order, one by one. Pressing ‘Undo All’ will undo all deletions done with this f unction.
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Click the ‘Change support properties’ button to change support properties. Enter the desired support fixities, type, prescribed displacements and spring constants on the dialog. One can also use the ‘Get properties’ button to do this. Click the button and then click on a support. The support’s properties will then be transferred to the dialog. Clicking on supports will now change their properties to those specified on the dialog. The ‘Undo’ button will undo the changes in reverse order.
Click the ‘Explode supports’ button to explode the support input. This results in the list of supports being written, each on a separate line without block generations etc. Once done, it cannot be reversed.
On the supports input page one can also delete nodes and groups of nodes in the same way as on the nodes input page. Refer to page 3-24 f or details
Nodal loads input Loads on beam elements are categorised as nodal loads, i.e. loads at node points, and element loads, i.e. loads between nodes. Uniform distributed loads can be applied to shell elements. All loads are organised in load cases, e.g. 'DL' for own weight, 'ADL' for additional dead loads, 'LL' for live load, etc. Load cases apply equally to the various load input screens, meaning that you can build up a load case using different types of loads.
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Frame Analysis
To define a load case, type a descriptive name for each load case in the Load Case column. Use up to six characters to describe each load case. If the load case name is not entered, the load case applicable to the previous line in the t able is used. The load case at the cursor position is displayed graphically. Press Enter or Display to update the picture. A nodal load can, as its name implies, only be applied at a node. If a point load is required on an element, use the Beam loads input table instead. Sign conventions Nodal loads are applied parallel to the global axes – an explanation of the sign conventions are given on page 3-7.
Tip: For a typical steel or timber frame or roof truss, it may be easiest to define a node at each purlin position. Roof loads transferred via the purlins can then be entered as nodal loads. Error checking
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The program checks that specified nodes have indeed been defined in the Nodes input table. If an error is detected, an Error list button will appear.
Generating additional nodal loads Additional nodal loads can be generated using the Number of extra and Node number inc columns respectively. Block generation of nodal loads You can use the block function to copy blocks of nodal loads. The procedure is similar to that for generating additional nodes – see page 3-22 for more detail. Graphical input The following graphical input functions are available on the left hand side of the screen. The toolbar containing the graphics options buttons can also be dragged and docked on any side of the picture. To add nodal loads click the ‘Add nodal loads’ button. Enter the forces, moments and the load case on the dialog. Now click the mouse on nodes to have nodal loads entered into the input table. Pressing ‘Undo’ will delete the nodal loads in the reverse order in which they were entered, one by one.
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Frame Analysis
To block add nodal loads click the ‘Block add nodal loads’ button. Enter the forces, moments and the load case on the dialog. Select a rectangle on the screen with the mouse. All nodes inside the rectangle will have a nodal load added. Pressing ‘Undo’ will delete the nodal loads in the reverse order in which they were entered, one by one.
To delete nodal loads click the ‘Delete nodal loads’ button and then select the nodal loads to be deleted on the screen using the mouse. The load case for which loads are to be deleted should also be entered on the dialog. Click ‘Done’ when finished. Clicking ‘Undo’ will undo the deletions in reverse order.
To block delete nodal loads click the ‘Block delete nodal loads’ button. Select a rectangle on the screen with the mouse. All nodal loads inside the rectangle will be deleted. Press the ‘Done’ button when finished. Pressing ‘Undo’ will undo the deletions in reverse order, one by one.
Click the ‘Explode nodal loads’ button to explode the nodal loads input. This results in the list of nodal loads being written, each on a separate line without block generations etc. Once done, it cannot be reversed.
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Beam element loads input Distributed loads and point loads on beam elements are all referred to as element loads. The Nodal loads input page provides the easiest way of applying point loads and moments at nodes. Use up to six characters to enter a descriptive name for each load case in the Load Case column. Then enter the element string of nodes in the Beam element definition column. Entering the beam element definition follows the same convention used as for the Elements input table – see page 3-24 for detail. Sign conventions Depending on the selected load direction, beam loads are applied parallel to the global axes or parallel to the local y-axis – the definitions of the global and local axes are given on page 3-7 and 7 respectively. The load direction is entered in the Direction column. Enter a global direction 'X', 'Y' or 'Z'. Element loads are applied to the relevant projected le ngth of the elements. Therefore, if a 'Y' load is entered for a vertical element, for example, the resulting load will therefore be zero. You can also load a beam element parallel to its local y-axis by setting the load direction to 'L' – refer to page 3-7 f or an explanation of the local axis c onvention used.
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Frame Analysis
Types of beam loads The following loads can be entered: •
A point load’s magnitude i s entered in the P column and its position from the smaller node number in the a column.
•
For a distributed load, entered in the load intensity at the smaller and larger node numbers in the W-begin and W-end columns respectively. If the load is constant over the length of the element, W-end may be left blank.
•
To define a temperature load, enter the temperature difference in the dT column. A temperature change is used with the temperature expansion coefficient of the relevant material used. Note: Positive vertical loads act upward and negative loads act downward.
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Error checking The program checks that element definitions match previously defined elements. If an error is detected, an Error list button will appear. Generating additional element loads The No of extra and Node number Inc columns can also be used to generate additional element loads. Block generation of beam loads You can use the block function to copy blocks of beam loads. The procedure is similar to that used to generating additional beam elements – see page 3-28 for detail. Graphical input The following graphical input functions are available on the left hand side of the screen. The toolbar containing the graphics options buttons can also be dragged and docked on any side of the picture. To add beam loads click the ‘Add beam loads’ button. Enter the loads, direction and temperature change on the dialog. The load case also needs to be entered. Now click the mouse on beams to have beam loads entered into the input table. Pressing ‘Undo’ will delete the beam loads in the reverse order in which they were entered, one by one.
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Frame Analysis
To block add beam loads click the ‘Block add beam loads’ button. Enter the loads, direction and temperature change on the dialog. The load case also needs to be entered. Select a rectangle on the screen with the mouse. All beams inside the rectangle will have a beam load added. Pressing ‘Undo’ will delete the beam loads in the reverse order in which they were entered, one by one.
To delete beam loads click the ‘Delete beam loads’ button and then select the beam loads to be deleted on the screen using the mouse. The load case for which loads are to be deleted should also be entered on the dialog. Click ‘Done’ when finished. Clicking ‘Undo’ will undo the deletions in reverse order. To block delete beam loads click the ‘Block delete beam loads’ button. Select a rectangle on the screen with the mouse. All beam loads inside the rectangle will be deleted. Press the ‘Done’ button when finished. Pressing ‘Undo’ will undo the deletions in reverse order, one by one.
Click the ‘Explode beam loads’ button to explode the beam loads input. This results in the list of beam loads being written, each on a separate line without block generations etc. Once done, it cannot be reversed.
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Shell loads Distributed loads can be applied on shell elements. Enter a load case description in the Load case column followed by the relevant element numbers in the Shell numbers column. The program automatically assigns numbers to all shell elements in the sequence they are defined on the Shells input page. A series of elements can be entered by separating the first and last element numbers by a ’–' character, e.g. '1–6' to define elements 1 up t o 6. Tip: If the shell element numbers are not visible in the picture, edit the graphics options to enable detailed rendering and disable the full 3D view. Refer to page 3-18 for detail on changing the graphics options. Sign conventions Shell loads are applied parallel to the element's local z-axes – an explanation of the local axes of shell elements are given on page 3-7. Types of shell loads The following shell loads can be entered: •
For a distributed load, entered in the load intensity in the UDL column.
•
To define a temperature load, enter the temperature difference in the dT column. A temperature change is used with the temperature expansion coefficient of the relevant material used. Note: Positive vertical loads act upward and negative loads act downward.
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Frame Analysis
Error checking The program checks that the entered element numbers are valid. If an error is detected, an Error list button will appear. Generating additional element loads The No of extra and Node number Inc columns can also be used to generate additional shell loads. Block generation of shell loads You can use the block function to copy blocks of beam loads. The procedure is similar to that used to generating additional shell elements – see page 3-36 for detail.
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Graphical input The following graphical input functions are available on the left hand side of the screen. The toolbar containing the graphics options buttons can also be dragged and docked on any side of the picture. To add shell loads click the ‘Add shell loads’ button. Enter the UDL, direction and temperature change on the dialog. The load case also needs to be entered. Now click the mouse on shells to have shell loads entered into the input table. Pressing ‘Undo’ will delete the shell loads in the reverse order in which they were entered, one by one.
To block add shell loads click the ‘Block add shell loads’ button. Enter the UDL, direction and temperature change on the dialog. The load case also needs to be entered. Select a rectangle on the screen with the mouse. All shells inside the rectangle will have a shell load added. Pressing ‘Undo’ will delete the shell loads in the reverse order in which they were entered, one by one.
To delete shell loads click the ‘Delete shell loads’ button and then select the shell loads to be deleted on the screen using the mouse. The load case for which loads are to be deleted should also be entered on the dialog. Click ‘Done’ when finished. Clicking ‘Undo’ will undo the deletions in reverse order.
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Frame Analysis
To block delete shell loads click the ‘Block delete shell loads’ button. Select a rectangle on the screen with the mouse. All shell loads inside the rectangle will be deleted. Press the ‘Done’ button when finished. Pressing ‘Undo’ will undo the deletions in reverse order, one by one.
Click the ‘Explode shell loads’ button to explode the shell loads input. This results in the list of shell loads being written, each on a separate line without block generations etc. Once done, it cannot be reversed.
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Load combinations input
You can model practical scenarios by grouping load cases together in load combinations. Enter the load combination name in the Load comb column; followed by the load case name and relevant load factors. If the Load comb column is left blank, the load combination is taken to be the same as for the previous line of the table. The load cases to consider in a load combination are entered one per line in the Load case column. Enter the relevant ultimate and serviceability limit state load factors in the ULS factor and SLS factor columns. Tip: You may leave one or more blank lines between load combination definitions to improve readability. The ultimate and serviceability limit states are used as follows: • 3-60
Deflections are calculated using the entered SLS loads. A set of reactions is also calculated at SLS for the purposes of evaluating support stability and bearing pressures. Frame Analysis
Element forces and a second set of reactions are determined using the entered ULS f orces.
•
Tip: If you plan to use a working stress method to design the frame members, e.g. steel design according to SABS 0162 - 1984, you may use the same load factors at ULS and SLS. Error checking The program only checks that valid load cases are specified. It has no knowledge of the design code that will be used in the member design and therefore does not check the validity of the entered load factors.
Alternative frame input methods Alternative means of frame input are available: •
Input Wizards: Modules are available for the rapid generation of input for typical frame structures.
•
External Graphical input: Structures can be drawn in Padds or another CAD system and converted to frame analysis input.
Input Wizards A number of typical frames can be input by entering a number of parameters. The Input Wizards do most of the data input. See page 3-19 for more detail on the Input wizards. Note: The frame analysis modules are not limited to modelling only those frames generated by the input wizards modules. Instead, the input wizards merely serve to simplify input of some typical frames.
External Graphical input In some situations, it may be easier to define a frame’s geometry graphically. With Padds you can draw a frame and then generate a frame analysis input file. Using Dos Padds for frame input To use Dos Padds to define a frame’s geometry: 1.
Use Padds to draw the frame. Alternatively, import a drawing from another CAD system.
2.
The frame should be drawn to scale using millimetres as unit. Identify different beam sections by using different pen numbers.
3.
Use the Generate input command on the Macro to display the drawing conversion options. Choose the target frame analysis module and press OK to start the conversion procedure. 3-61
The resultant frame analysis input file will be compatible with both the Dos and Windows versions of the frame analysis modules. The file is saved in the working folder as a last file, e.g. ’Lastsf.a03 ’ for Frame Analysis. 4.
Close Dos Padds.
Tip: To see a graphical input example, open ’..\prokon\data\demo\inputgen.pad’ in Padds. Importing drawings You can also use your favourite CAD system to save a frame’s geometry in a 2D or 3D Dxf/Dwg format drawing and then use the Import -> Dxf/Dwg files command on the File menu to convert it to frame i nput.
The same basic rules apply as gi ven above: •
The drawing should be to scale.
•
You should use millimetre units.
•
Different pen numbers should be used for different beam sections.
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Frame Analysis
When importing 3D .Dxf & .Dwg drawings, you can optionally interchange the Z- and Y-axes. This option is given to correctly import a drawing where the Z-axis is taken as vertical, into Frame Analysis where the Y-axis is vertical. Typical problems experienced include the following: •
Polylines may not be recognised correctly. Break or explode polylines into single lines before saving the .Dxf file.
•
Blocks may not import correctly and may need to be broken or exploded into individual entities.
•
Using AutoCAD, lines colours set ’by layer’ translates to the default pen number. Rather set colours using pen numbers to ensure correct section numbering.
•
If you experience problems importing a DXF file saved using a brand new version of your CAD system, it may help saving the file as an older DXF file version, e.g. version 12.
CIMsteel The Space Frame Analysis module can import complete frame models, including geometrical and loading data, defined in the CIMsteel (Computer Integrated Manufacturing for Constructional Steelwork) integration standard. Modelling packages that can create CIMsteel files include Intergraph Frameworks and Microstation Structural Triforma. Note that at the time of writing this feature had not been fully developed.
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Settings The Settings page allows you to set t he parameters relevant to the analysis met hod.
Analysis type Select the type of analysis to be performed. Refer to page 3-14 for a description of the various analysis modes.
Concrete design parameters input It is generally impractical to design reinforcement to resist torsional moments in slabs. Reinforcement is usually fixed in two directions approximately, but not necessarily, perpendicular to each other. This justifies the use of transformed moments to calculate reinforcement. Frame uses the Wood and Armer theory to convert calculated bending and torsional moments to transformed bending moments. More detail is given on page 3-13. Please note that these parameters only apply to shell elements and not to beams or solids.
The required concrete design parameters are: • 3-64
Enter the concrete and reinforcement material characteristics, f cu and f y. Frame Analysis
•
Define the orientation for the ’main’ and ’secondary’ reinforcement, i.e. the x’ and y’-axis. Looking from the top, the x’-axis is measured anti-clockwise from the local x-axis to the reinforcement x’-axis. The y’-axis is in turn measured anti-clockwise from the local x’-axis. The direction of the local x-axis of a shell element is explained on page 3-7.
•
Define the reinforcement levels in the slab by entering the concrete cover values for the top and bottom reinforcement in both directi ons.
•
Optionally incorporate membrane behaviour by including the effect of in-plane stresses.
Reinforcement contours can be displayed on the Reinforcement tab under the View Output age. Values are calculated for reinforcement at the top and bottom of shell elements. In this context, top and bottom are defined as f ollows: •
The top of a shell element is taken on the side towards which its local z-axis points.
•
For a non-vertical element, the top side is the visible side when looking down on the element.
•
For a vertical element, the top side is the visible side if the nodes (as entered on the Shell elements input page) defining the element are orientated anti-clockwise.
Second order parameters input If you want to perform a second order analysis of the frame, you need to set the relevant analysis parameters: •
Second order analysis tolerance: The second order analysis is an iterative procedure. The analysis is deemed to have converged once the total strain energies of two sequential iterations differ by less than the specified tolerance
Buckling parameters input •
Number of buckling mode shapes: Set the number of buckling mode shapes to be calculated when determining the load factor for buckling for each load case or combination. The first mode shape is often critical since it represents the most likely buckling failure mechanism. For practical reasons, only the first mode shape would normally be of interest. However, the accuracy of the analysis improves with the number of mode shapes analysed. It is therefore suggested that you allow at least four mode shapes to be calculated. The program supports a maximum number of forty mode shapes. Note that internally the program actually 3-65
calculates quite a few more mode shapes than specified to improve the accuracy of the output for the modes that are specified. Because of this, the model should not contain too few nodes. If no convergence is obtained, split the members into smaller portions by adding extra nodes.
Non-linear analysis parameters input Number of load steps: Each load case is applied to the structure in a number of steps. The accuracy increases with the number of steps entered. Ten steps should be adequate for most problems. One must be aware that for each step, the structure is completely analysed a number of times for each load case to obtain convergence for that step. The analysis time can therefore become very long if a high number of steps are specified on a large structure with many load cases. It is suggested that one use say 3-5 steps initially, to ensure that the solution progresses properly, before increasing the number of steps for a final analysis.
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Frame Analysis
Dynamic analysis parameters input Edit the dynamic analysis parameters if you will be performing a modal, harmonic or earthquake analysis. Some parameters should be entered in all cases while others are specific to harmonic or earthquake analysis. The graphic below shows all the possible input fields, although some of them might not be visible, depending on the type of dynamic analysis selected.
Number of mode shapes Enter the number of natural modes of vibration to be calculated. You can use the following guidelines when deciding on the number of mode shapes to be calculated: •
The number of mode shapes should preferably not exceed the number of degrees of freedom in the structure divided by twelve.
•
In a harmonic analysis, use enough mode shapes so that the highest natural frequency is at least 50% higher than the applied l oading frequency.
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•
For an earthquake analysis, the first three or four mode shapes are normally sufficient to obtain the probable maximum combined effects.
•
Do not use too few nodes. If convergence is not obtained when calculating the natural frequencies, add more nodes and split the eleme nts. See also the first point above.
Add axial force effects to stiffness Compression forces in a frame’s members reduce the effective stiffness of the frame. The frequencies of vibration are therefore reduced with corresponding changes in the mode shapes. If the option is enabled, the eff ect of axial force is i ncorporated as follows: •
A static analysis is first performed with the mass of the structure, i.e. own weight and the Y-components of the specified load cases, being the only load case.
•
The resulting axial forces are then incorporated when calculating the modified geometric stiffness matrix of the structure. The geometric stiffness matrix is subsequently used to modify the global stiffness matrix to account for the axial force effects.
Include torsional modelling This applies to beam elements. If included, the torsional modes of the beams will also be included. Include load cases as mass A frame’s frequencies of vibration and mode shapes depend on its inertia or mass distribution in relation to its stiffness distribution. Any loads on the structure will therefore influence the natural frequencies and therefore its response to dynamic loadings. The user must select the loads to be included as mass in the right hand side list. Only the Y-components (vertical) of the loads are added as mass. The program uses the following components for compiling t he mass matrix: •
The own weight of the frame, calculated using the density values entered in the Section properties input table.
•
The loads cases entered by the user. Note: In a harmonic or seismic analysis, all dead and live load that move with the structure during its dynamic response will influence the dynamic response of the structure. Such loads should be considered as masses for the purpose of determining the mode shapes.
Own weight and distributed vertical loads are added using the consistent mass matrix formulation. Point loads are added as lumped masses at the relevant nodes. Note: In a seismic analysis all loads not selected as masses will be excluded from the analysis. 3-68
Frame Analysis
Average damping ratio The dynamic response of a frame will include some viscous effects. The average damping ratio depends on the type and condition of the structure. Typical values, taken from TMH7 - Part 2, are tabled below: Stress level
Working stress no more than about half of yield stress
Working stress near yield point
Type and condition of structure
Damping ratio
Welded steel. Prestressed concrete, well reinforced concrete (slight cracking)
2%
Reinforced concrete with considerable cracking
3% to 5%
Bolted and riveted steel
5% to 7%
Welded steel, prestressed concrete under full pre-stress (slight cracking)
5%
Prestressed concrete in cracked state
7%
Reinforced concrete
7% to 10%
Bolted and/or riveted steel
10% to 15%
Design ground acceleration (seismic analysis only) 3-69
For an earthquake analysis, specify the maximum ground accelerations. TMH7 - Part 2 gives the following typical values according to a modified Mercalli classification: Modified Mercalli intensity at epicentre
Design ground acceleration
ii – iii
0.003 g
iv – v
0.01 g
Vi
0.03 g
vii – viii
0.1 g
Ix
0.3 g
x – xi
1.0 g
Note: Refer to local seismic data for relevant acceleration data. TMH 7 - Part 2, when considering the influence of seismic disturbances, gives the following guidelines: •
Symmetrical structures: The influence of seismic disturbances should be considered along both principal axes of symmetry.
•
Non-symmetrical structures: It would normally be sufficient to consider the seismic effects along any two arbitrarily chosen orthogonal axes.
•
For structures, symmetrical and non-symmetrical, the seismic effects along two orthogonal axes may normally be considered independently of each other.
Design load factor (seismic analysis only) Seismic action is normally not considered in combination with other secondary loads. For this reason, the following approach is used: •
The vertical components of selected load cases are included as masses when determining the mode shapes. Refer to page 3-68 for detail.
•
All other effects of all load cases and combinations are ignored.
•
The dynamic response of the structure to the applied ground accelerations is amplified with the specified design load factor.
The design load factor for forces and moments should be chosen in accordance with the relevant loading code. At ultimate limit state, an average factor between 1.3 and 1.5 is normally applied. 3-70
Frame Analysis
Spectrum reduction factor for vertical direction (seismic analysis only) The relevant resonance frequencies of a structure in the vertical direction are generally quite different from those in the horizontal direction. If it is deemed necessary to consider the effects of vertical seismic motions, t he average vertical design spectrum can often be re duced. For typical bridge structures, TMH7 - Part 2 allows the average vertical design spectrum to be taken as two-thirds the average horizontal seismic response spectrum. Ductility (seismic analysis only) The ductility of a structure is a function of the type and arrangement of the elements resisting lateral forces. A unity value corresponds to perfect elastic behaviour and a value greater than unity elastic-plastic behaviour. Typical ductility values, taken from TMH7 - Part 2, are given below. Type or arrangement of resisting ele ments Un-braced structural steel structures with elements adequately designed to resist the total later al forces in bending
Structural ductility factor
4
Structures with braced flexural vertical members of structural steel Slender reinforced columns carrying superstructure of braced structural steel, reinforced concrete or prestressed concrete designed to resist the total lateral force Vertical elements of shear wall proportions with superstructure in monolithic reinforced or prestressed concrete designed to resist the total lateral force
3
2
If you enter a value greater than one, the elastic response spectrum will be adjusted to obtain an appropriate inelastic response spectrum. The program allows you to adjust the design response spectrum as required.
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Foundation soil type (seismic analysis only) Additional factors not covered above can be accounted for by adjusting this load factor. TMH7 - Part2, for example, gives an additional factor relating to the type of founding material. Description of founding material Rock, dense and very dense coarse-grained soils, ver y stiff and hard fine-grained soils Compact coarse-grained soils, and firm and stiff fine-grained soils from 0 to 1 5 m deep Compact coarse-grained soils, firm and stiff fine grained soils with a depth greater than 15 m Very loose and loose coarse-grained soils and ver y soft and soft fine-grained soils from 0 t o 15 m deep Very loose and loose coarse-grained soils and ver y soft and soft fine-grained soils with depths greater than 15 m
Foundation factor f
1.0
1.2
1.5
Acceleration response spectrum (seismic analysis only) Acceleration response spectra are empirically derived definitions for typical responses of structures to a range of ground accelerations. Choose an applicable response spectrum: •
SABS 0160 - 1989.
•
TMH7 - Part2.
•
UBC 1194.
Alternatively, define a custom response spectrum by entering the ground acceleration for damping ratios of 0.5%, 3%, 5% and 10% for nine different periods. Any period values can be entered, including those for the acceleration/velocity and velocity/displacement bound regions dividing points. If the ductility value greater than unity is specified, the elastic response spectrum is adjusted to obtain an inelastic response spectrum. The elastic spectrum, for any given damping ratio, is modified along the displacement bound region by multiplying it with a factor 1/ µ. Along the acceleration bound region the elastic spectrum is multiplie d by a factor 1/(2µ−1)½..
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Frame Analysis
Analysis To analyse the structure, open the Analysis page and press Start Analysis:
The analysis progress is displayed to help you judge the time remaining to complete the analysis. After a successful analysis, the deflected shape is displayed for the first load case or load combination or, in the case of modal or buckling analysis, the first mode shape. Error checking during analysis During the input phase, the frame geometry and loading data is checked for errors. Not all reported errors are necessarily serious. To define duplicate elements between two nodes, for example, could be an accidental error on your side. However, the program can deal with a situation like this and will allow the analysis procedure to continue. Other input errors could be serious enough to prevent an analysis from being completed successfully. Nodes with no elements, for example, have no restraints and will cause numeric instability during the analysis. 3-73
The first step of any analysis is the final verification of the input data. In the case of critical errors still present, a warning message will be displayed. If you then choose to not proceed with the analysis, you will be taken to the input table with the error. However, choosing to proceed and ignore the warning will have an unpredictable result.
Fixing errors that occurred during the analysis Even if all input data seems valid, numeric errors may still occur during an analysis. For example, if you entered incorrect section properties, such as a very small E-value, the mistake may go by unnoticed. However, the analysis will then yield an invalid value in the stiffness matrix or extremely large deflections. The same applies to the stability of the frame. Although the frame may appear stable, some combinations of internal hinges may result in some nodes being unstable.
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Frame Analysis
If an error was detected during the analysis, a warning will be displayed. The cause of the error should become clear when studying the output file: •
The text at the end of the output file normally gives the reason for the error.
•
If the output file seems complete, the problem will require more careful attention. Scan all output tables for excessively large or s mall values.
Second order and buckling analysis problems The following points should be kept i n mind when trouble-shooting a second order analysis: •
During a second-order analysis, an element is removed from the frame as soon as its axial force exceeds its Euler buckling load. If one or more elements have been removed from the frame in this way, structural stability cannot necessarily be guaranteed any more. The removal of a single element may cause a chain reaction of elements failing. If the remaining members do not constitute a stable structure, the structure will fail. All member forces in the output file will then be shown as ’∗∗∗∗∗∗∗∗’.
•
A second order analysis fails in instability can be verified using a buckling analysis. You can examine the buckling mode shapes to easily locate problem areas in the structure that may require stiffening.
•
Slender non-structural elements often buckle before major structural members, distorting analysis results. Such elements should preferably be e xcluded from the model.
•
If the linear and second order analysis results show negligible differences, the structure is likely not sensitive to p-delta effects for the given loading. This is often true for truss-type frames. Tip: Perform a buckling analysis to get an indication of the general stability of the frame under the entered loads. A frame with a buckling load factor less than unity normally experiences the same local or global stabilit y problems during a second order analysis.
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Design Links Once the analysis is complete, Frame can link up with other Prokon modules for further postprocessing and design. This is done on the ‘Design links’ tab:
Selecting a design link The following links are available: •
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Steel connections: Select this option on the left hand side of the screen. Now click on any node in the structure. A dialog with possible connections will appear as shown above. The program will look at the types of members meeting at the node. These members must be steel members selected from the Prokon database. There should also be at least two members meeting at the node. If for example, there are no hollow sections at the node, the ‘Hollow section connection’ option will remain greyed. For the ‘Apex connection’ the members must be I- or H-sections. Check the type of connection that you want. Sometimes there is more than one possible connection of the chosen type at the node. The desired connection can be chosen from the drop-down(s) to the right of the connection type
Frame Analysis
chosen. Pressing ‘OK’ will convert the output data and transfer it to the relevant connection design module. •
Concrete columns: Select ‘rectangular’ or ‘circular’ column on the left hand side of the screen. Now select a member on the screen with the mouse. The data will be transferred from Frame and the relevant column design module will be opened.
•
Concrete base: Select ‘Concrete base’ on the left hand side of the screen. Now select a support node on the screen with the mouse. The data will be transferred from Frame and the base design module will be opened. Note that only support nodes can be selected.
•
Member design modules: Separate buttons are provided to call up ‘Member design for axial stress’, ‘Member design for combined stress’ and ‘Timber member design’. Each of these modules will open with t he current structure and output, read y for further processing.
•
Important note regarding load combinations: If the analysis was a linear analysis, the load combinations are broken up using the load factors entered in Frame. In the case of a second-order analysis, the results of a combination are not necessarily equal to the sum of the load cases times their load factors, due to the secondary effects taken into account. The program will calculate appropriate load factors, which will be similar to the ones entered, but adjusted so that the sum of load case forces times load factors gives the final forces calculated by Frame.
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Viewing output The analysis results can be viewed graphically or in tabular format. Output data, including graphics and tabled values, can be selectively appended to the Calcsheets using the Add to Calcsheets function on each output page.
Viewing output graphics Diagrams can be displayed for deflection, member forces and stress and shell reinforcement of any load case: •
Deflections: Deflections are generally small in relation to dimensions of the structure. To improve the visibility of the deflection diagram, you can enter a screen magnification factor. You can optionally display the deflected shape without the original geometr y.
•
Mode shapes: In the case of a dynamic or buckling analysis, you can display the mode shapes one-by-one. Use the Animate function to bring a mode shape to life. If you tick the ‘record animation’ box, the animation is stored as an animated .gif file as shown below.
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Frame Analysis
Note: Mode shapes should not be confused with deflections. Mode shapes represent the natural dynamic characteristics of a structure. Values are normalised with the maximum "displacement" given as one thousand. •
Reactions: The reactions forces and moments at all supported nodes are displayed. The arrowheads points in the direction of each reacti on.
•
Beam element forces:
•
Axial forces: The force is shown as expanded red and blue lines. Compression forces are shown in red and tension forces in blue. The distance of a line from the element centre line is in proportion to the size of the axial force.
•
Moments: Bending moments about the local x and y-axes. A plot factor can also be entered to enlarge or reduce the bending moment diagram on the frame.
•
Shear: Shear force diagrams are drawn for the local y and x-directions. A beam element’s shear force diagram is constructed by viewing it with its local z-axis pointing to the right. Since the direction of the z-axis depends on the node numbers, irregular numbering of nodes can result in apparent irregular signs used in the shear
3-79
force diagrams. Refer to page 3-8 for detail on the sign conventions used for beam element forces. • •
Beam element force envelopes: •
3-80
Torsion: The torsional moment about the z-axis, i.e. element axis.
Envelopes: Enter a series of elements and select the load case and combinations to include in the envelopes. Envelopes are drawn using the values as tabulated from the output file. Positive moments, for example, are drawn below the line and negative above. Because members of different orientations can be included in the same envelope, no simple distinction is made between tension and compression faces of members.
Frame Analysis
•
In-plane stresses in shells:
•
Stresses in x, y and xy directions: Display contours of the membrane stresses in the local x and y-directions and the shear stresses. Refer to page 3-7 for detail on the local axis convention for shell elements.
•
Maximum and minimum stresses: These correspond to the largest tensile and compression stresses respectively. Positive values indicate compression and negative values tension.
•
Von Mises stresses: The Von Mises stresses give a graphical indication of a yield criterion, i.e. a general indication of the combined effect of all stresses. The Von Mises stresses takes into account in-plane stresses as well as bending stresses and is presented for both the top and bottom faces of the shell elements..
3-81
•
3-82
Bending moments and shear forces in shells:
•
The x, y and xy bending moments: The bending moments about the local x and yaxes and the torsional moment. The direction (not axis) of bending is shown as a small line on each shell element.
•
Maximum and minimum bending moments: The principal bending moments.
•
The x, y and maximum shear forces: The shear forces are in local x and y-axes. The maximum shear stress is also given and is obtained by dividing the maximum shear force by the thickness of the element.
Frame Analysis
•
Reinforcement in shells:
Contours of the required reinforcement in the top and bottom faces in the x’ and y’directions are given. The corresponding ‘Wood & Armer’ moments from which the reinforcement was calculated is also given. The reinforcement direction is shown as a small line on each shell. Refer to page 3-64 for the definition of the reinforcement directions. Tip: If the lines indicating the direction of bending or of the reinforcement is not clearly visible, enable detailed rendering under the graphics options. Refer to page 3-18 for instructions.
Tip: Shell element stress contours are drawn on the deflected shape of the structure. Careful choice of the deflection magnification fact or can enhance contour diagrams.
3-83
•
Stresses in solids:
The stresses presented are the direct stresses in the X, Y and Z directions, the shear stresses in the XY, XZ and YZ planes, the 3 principal stresses 1,2 and 3 as well as the maximum shear stress and the von Mises stress. For solids all the stresses are plotted in the global co-ordinates.
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Frame Analysis
Viewing output tables
Open the Output file page for a tabular display of the frame analysis output file. You can filter the information sent to the calcsheets by enabling or disabling the relevant sections. The Find heading function allows you to quickly locate an y main section of the output file. If you right click on the output, various editing functions are available. For example, you can search for any string by pressing Ctrl and F.
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Calcsheets Frame analysis output can be grouped on a calcsheet for printing or sending to Calcpad. To include a particular component of the output in the calcsheets, view the relevant output information and then click Add to Calcsheets.
Recalling a data file The Data File is automatically included in the calcsheet sent to Calcpad. You can later recall the frame by double-clicking the relevant object in Calcpad. A data file embedded in Calcpad is saved as part of a project and therefore does not need to be saved in the frame analysis module as well.
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Frame Analysis
3-87
Plane Stress/Strain Analysis
The Plane Stress/Strain Analysis module can be used to perform a finite element analysis of a membrane of any general geometry subjected to plane stress or strain. An automated element grid generation facility helps speeding up the input and analysis processes.
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Plane Stress/Strain Analysis
Theory and application The following text gives an overview of the finite element analysis theory and its application.
Scope The program analyses membrane structures of any general shape, including openings. The cases of plane stress and plane strain are both supported. Element grids are automatically generated with a customisable grid size.
Plane strain and plane stress The conditions of plane stress and plane strain are two similar two-dimensional states of stress and are subsequently often confused. However, their definitions set them apart as two very distinct conditions. When forces are applied to a thin two-dimensional plate in its own plane, the state of stress and deformation in the plate is called plane stress. A typical example would be a shear wall that, due to it being a thin plate, will experience mainly in-plane stresses. No restraint is provided for out-of-plane deformation. On the other hand, a prismatic solid subjected to a constant loading normal to its axis can be analysed as an infinite length of two-dimensional slices of unit thickness experiencing plane strain. A dam wall, for example, would typically be subjected to hydrostatic and soil pressures normal to its surface. A slice is taken from the wall will be restrained from deforming out-ofplane.
Finite element modelling The program uses eight-noded quadrilateral and six-noded triangular isoparametric finite elements formulated for plane stress and plane strain. A mesh of isoparametric quadrilateral or triangular is automatically generated and optimised during analysis. You can specify the grid spacing in the X and Y-directions as part of the analysis parameters. A finer grid will often improve accuracy. However, the time taken to perform an analysis is a function of the number of finite elements – the finer the grid, the longer the analysis time.
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Sign conventions When entering coordinates and forces, the following sign convention is used: •
Positive Y-coordinates and vertical forces are taken upward, i.e. parallel to the Y-axis.
•
Positive X-coordinates and horizontal forces are taken to the right, i.e. parallel to the X-axis.
In the analysis results, deflections are measured along the Y-axis. A positive deflection therefore denotes an upward movement.
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Plane Stress/Strain Analysis
Input Use the four input tables, i.e. Nodes, Supports, Material properties and Loads, to define the structure’s geometry and loading.
Nodes input A structure is defined by entering one or more shapes. A shape may comprise straight lines and arcs. When more than one shape is entered, the shapes will accumulate and form one structure. Often, a complicated section is easier defined using more than one shape. Note: Shapes must be entered in an a nti-clockwise order. An explanation of the node input table i s given below: •
The Mat. No. column is used for categorise the data that follows in the next columns: 1 to 9 : The start of a new polygon with the specified material property. An absolute reference coordinate must be entered in t he X and Y columns. If you leave X or Y blank, a value of zero is used. 0 : Start of an opening. An absolute reference coordinate must be entered in the X and Y columns. R : If you enter an ’R’ or leave the Mat. No. column blank, a line is drawn using relative coordinates, i.e. measured from the previous coordinate. A : Enter an ’A’ in the Mat. No. column blank to make the coordinate absolute.
•
The X and Y columns are used for entering coordinates: X : Absolute or relative X coordinate. Values are taken positive to the right and negative to the left. Y : Absolute or relative Y coordinate. Values are taken positive upward and negative downward.
•
Use the Bulge column to define an arc of a specified radius. Consider an imaginary line joining the previous coordinate and the coordinate entered in the X and Y columns. A bulge greater than zero then defines an arc to the right of the line. Similarly, a negative bulge draws an arc to the left of the line. If no bulge is specified, a zero value is used, i.e. a straight line.
You do need to close the polygon defining a shape – the starting coordinate is automatically used as the ending coordinate. If two polygons intersect, the geometry of the last polygon takes preference and the previous polygon is clipped. A hole in a structure can, for example, be entered on top of previously entered shapes.
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Nodes are automatically numbered as they are input. You can later use the node numbers to position supports and loads.
Supports input You can define point supports, distributed support and prescribed displacements anywhere along the edges of the structure. Supports are entered as follows: Nodes : Enter a single node number for a point support or a range of nodes for a distributed support, e.g. ’2’ for node 2 only and ’2-5’ for the zone described by the straight line joining nodes 2 and 5. XY support : Enter ’X’ and/or ’Y’ for horizontal and/or vertical support. Displ. : Specify the value of any horizontal or vertical prescribed displacement in the relevant column (m). Note: Point supports invariably result in localised stress concentrations, with the effect increasing for smaller element grids. It is therefore recommended to avoid point supports and rather distribute each support over as large a width as possible.
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Plane Stress/Strain Analysis
Material properties The following material properties must be input for every material property code used in the Code column of the Nodes input table: t : Thickness (m). 3 E : Modulus of elasticity (kN/m ). If left blank, the value for the preceding material type is used. Typical values are tabled below.
Poisson : Poisson ratio. If left blank, the value for entered in the previous row is used. Typical values are tabled below. 3 Dens : Density on (kN/m ). Typical values are tabled below.
Material
E modulus (kPa)
Poison’s ratio
Density 3 (kN/m )
Masonry
10E6
0.20
20 - 25
Concrete (normal strength)
25E6 to 35E6
0.20
± 24
Aluminium
± 70E6
0.16
± 27
Structural steel
± 205E6
0.30
± 78
Loads input Point loads and distributed loads can be defined anywhere along the edges of the structure. Use as many lines as necessary to define the loads. Loads are entered as follows: Nodes : Enter a single node number for a point load or a range of nodes for a distributed load, e.g. '2' for node 2 only and '2-5' for the zone described by the straight line joining nodes 2 and 5. X : The load direction can be either 'X' or 'Y' for horizontal or vertical respectively. Wleft : Distributed load intensity at the smaller node number (kN/m). Wright : Distributed load intensity at the larger node number (kN/m).
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a : Distance from first node to beginning of distributed load (m). A value of zero is used if field is left blank. b : Length in m, of distributed load. The load is taken up to the ending node if this field is left blank. Note: Positive forces are taken to work upward and to the right.
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Plane Stress/Strain Analysis
Analysis On completing the input, go to the Analyse page to analyse the structure. Following a successful analysis, use the View page the display the analysis results. The following text describes the analysis options that are available and gives information on finding and fixing analysis problems.
Analysis options During the analysis, the program generates a rectangular grid of nodes in which rectangular and, where necessary, triangular finite elements are placed. The grid spacing can be set independently in the horizontal and vertical directions. Choose Settings to set the grid s pacing and other analysis options: •
Finite element size: Horizontal and vertical grid spacings (m).
•
Type of analysis: Enter ’E’ for plane stress or ’A’ for plane strain.
•
Angle increment: The program models arcs as straight lines at the specified angle increment. Although a smaller angle would yield a smoother modelling of an arc, the resulting increase in modes will mean that more elements will be used. Generally, an angle increment between 5° and 15° would yield good results.
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Analysis results The analysis results can be viewed and printed in tabular or graphical format: •
Elastic deflections.
•
Maximum principal stresses, i.e. the largest tensile stresses. Positive values indicate tension and negative values compression.
•
Minimum principal i.e. the largest compression Negative values compression.
•
The Von Mises yield condition if often used to determine whether a material is behaving elastically under combined stress. According to the Von Mises theory, the total elastic energy comprises volumetric changes and shearing distortions. By considering only the shearing distortion at yield in simple tension in relation to that under combined stress, the yield criterion can be established. The Von Mises theory assumes a ductile isotropic material.
•
Principal stress vectors, with compression stresses in red and tension stresses in blue.
•
You can also inspect the results in tabular format by displaying the output file, named PS.OUT
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stresses, possible stresses. indicate
Plane Stress/Strain Analysis
Fixing errors that occurred during the analysis Some common problems during plane stress and plain str ain analyses are: •
If the program is unable to analyse the structure, there may be errors in the input. A common mistake is the definition of shapes in a clockwise direction – the program expects anti-clockwise input.
•
Stress concentrations will be present at positions of point loads and point supports. Such concentrations are further exaggerated when using finer element grids. In practice, loads and supports rarely act at points, but rather on small areas. It is likewise recommended to spread all point loads and supports over s mall lengths.
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Calcsheets The finite element analysis output can be grouped on a calcsheet for printing or sending to Calcpad. Various settings can be made with regards to the inclusion of design results and pictures.
Tip: You can embed the Data File in the calcsheet for easy recalling from Calcpad.
Recalling a data file If you enable the Data File option before sending a calcsheet to Calcpad, you can later recall its design by double-clicking the relevant object in Calcpad. A data file embedded in Calcpad is saved as part of a project and therefore does not need to be saved in the analysis module as well.
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Plane Stress/Strain Analysis
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Single Span Beam Analysis
The Single Span Beam Analysis module can be used to quickly analyse a beam. The beam can be a single span beam or a single span taken from a continuous beam with the appropriate end conditions. The analysis results of steels beams can be post-processed with the steel member design module for combined stress, Com.
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Single Span Beam Analysis
Sign conventions When entering forces and moments, the f ollowing sign convention is used: •
Positive vertical forces act downward, i.e. parallel to the negative Y-axis.
•
Positive moments work anti-clockwise.
•
In the analysis results, the following applies:
•
Deflections are measured along the Y-axis. A positive deflection indicates uplift and negative deflection a downward movement.
•
Shear forces are measured along the Y-axis. A positive shear force at the left edge of the beam, for example, represents an upward vertical reaction.
•
Moments are drawn on the tension face of the beam.
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Input The beam definition has two main i nput components: •
Geometry and material properties.
•
Loads.
Beam input The following values must be entered: Length : The overall length of the beam (m). M Left : The applied moment at the left-hand end (kNm). If you leave the field blank, a zero moment is used, e.g. the beam is simply supported. You can al so fix an end by entering an ’F’ or make it a free cantilever end with a ’C’. M Right : The applied moment at the right-hand end (kNm).
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Single Span Beam Analysis
4 I : The second moment of inertia of beam (m ). 6 E : Young’s modulus of the beam (kPa). A value of 205x10 kPa, i.e. steel, is used if this field is left blank. Typical values are:
Material
E modulus (kPa)
Timber
5E6 to 15E6
Concrete (normal strength)
25E6 to 35E6
Aluminium
± 70E6
Structural steel
± 205E6
Section input The moment of inertia of a standard steel section can be selected from the section database. You can also define your own sections or remove sections from the database. Own weight On selecting a steel section form the database, the own weight is automatically entered as a uniform distributed load.
Loads input Use as many lines as needed to enter any general loading on the beam i n the Loads input table: W Left : Distributed load intensity (kN/m) applied at the left-hand starting position of the load. If you do not enter a value, the program will use a value of zero. W Right : Distributed load intensity (kN/m) applied on the right-hand ending position of the load. If you leave this field blank, the value is made equal to Wleft, i.e. a uniformly distributed load is assumed. P : Point load (kN). M : Moment (kNm).
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a : The start position of the distributed load, position of the point load or position of the moment (m). The distance is measured from the left-hand edge of the beam. If you leave this field blank, a value of zero is used, i.e. the load is taken to start at the left-hand edge of the beam. b : The end extent of the distributed load, measured from the start position of the load (m). Leave this field blan k if you want the load t o extend up to the ri ghthand edge of the beam. Note: Positive forces and moments are taken to work downward and taken anti-clockwise respectively.
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Single Span Beam Analysis
Analysis Press Analyse to display the analysis results: •
Elastic short-term deflections.
•
Bending moment diagram.
•
Shear force diagram.
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Calcsheet The beam analysis results can be grouped on a calcsheet for printing or sending to Calcpad. Various settings can be made with regards t o the inclusion of design results and pictures.
Tip: You can embed the Data File in the calcsheet for easy recalling from Calcpad.
Recalling a data file If you enable the Data File option before sending a calcsheet to Calcpad, you can later recall the beam analysis by double-clicking the relevant object in Calcpad. A data file embedded in Calcpad is saved as part of a project and therefore does not need to be saved in the beam analysis module as well.
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Single Span Beam Analysis
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Beam on Elastic Support Analysis
The Beam on Elastic Support Analysis module can be used to quickly analyse a beam or slab on an elastic foundation. The beam cross-section may vary along its length and the elastic foundation can include gaps and rigid supports.
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Beam on Elastic Support Analysis
Theory and application The following text gives a background into the analysis technique used.
Sign conventions and units of measurement When entering forces and moments, the following sign convention is used: •
Positive vertical forces act downward, i.e. parallel to the negative Y-axis.
•
Positive moments work anti-clockwise.
In the analysis results, the following ap plies: •
Deflections are measured along the Y-axis. A positive deflection indicates uplift and negative deflection a downward movement.
•
Bearing pressure is also measured along the Y-axis. A positive bearing pressure denotes an upward reaction.
•
Shear forces are measured along the Y-axis. A positive shear force at the left edge of the beam, for example, represents an upward vertical reaction.
•
Moments are drawn on the tension face of the beam.
Analysis procedure The program performs a linear analysis in which the beam is modelled as a two-dimensional frame on a series of least fifty closely spaced springs. Rigid supports are put at the specified positions and gaps in the elastic support where the supporting width is set to zero. Nodes are taken at close intervals along the le ngth of the beam. A node is also introduced at every support and load position. If negative soil pressures are not allowed, i.e. uplift is allowed, springs with negative reactions are removed and the analysis repeated. Likewise, previously removed springs are restored if downward deflections are calculated at the points concerned. The analysis procedure is repeated until the iteration converges to a stable solution. A beam is considered unstable, i.e. to overturn under the applied load, if the analysis yields less than two springs with compressive forces.
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Input The beam definition has several input components: •
Geometry and material properties.
•
Supports input
•
Loads.
Beam input The beam is defined as one or more segments, each with its own properties. The following values must be entered: Lsec : The length of a beam segment with a specified stiffness and support width (m). You may enter more than one segment to define a beam varying section or an elastic medium of varying stiffness. Each additional beam se gment entered is added to the right-hand side of beam. Isec : The stiffness of the beam segment, express as the second moment of inertia of the relevant cross section (m). The value of Young’s modulus, applicable to the whole beam, is entered under the analysis settings. Bsec : The support width of the beam segment (m). This beam width is multiplied by the foundation modulus of the soil, Km, t o obtain the support stiffness per unit length of the beam. Enter a zero value f or no foundation stiffness, i.e. a gap in the elastic medium.
Supports input Use the Support input columns to enter rigid supports i n the elastic medium: Position : A rigid support position, measured from the left-hand side (m). Support : You can set the support type to vertical and/or rotational: Support
Description
Y
Vertical support, e.g. a solid rock intr usion in the elastic medium
R
Rotational support, e.g. a rigid column above built into the beam
YR
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Vertical and rotational support, e.g. a ri gid pile below that is built into the beam.
Beam on Elastic Support Analysis
Loads input Use as many lines as needed to enter an y general loading on the beam in the Loads input table: W Left : Distributed load intensity (kN/m) applied at the left-hand starting position of the load. If you do not enter a value, the program will use a value of zero. W Right : Distributed load intensity (kN/m) applied on the right-hand ending position of the load. If you leave this fiel d blank, the value is made equal to Wleft, i.e. a uniformly distributed load is assumed. P : Point load (kN). M : Moment (kNm). a : The start position of the distributed load, position of the point load or position of the moment (m). The distance is measured from the left-hand edge of the beam. If you leave this field bl ank, a value of zero is used, i .e. the load is taken to start at the left- hand edge of the beam. b : The end position of the distributed load, measured from the start position of the load (m). Leave this field blank if you want the load to extend up to the righthand edge of the beam. Note: Positive forces and moments are taken to work downward and taken anti-clockwise respectively.
Analysis settings Press Settings to edit the material c onstants and other parameters to be used in the analysis: •
E modulus: Young’s modulus for the beam (kPa). Values for typical building materials are tabled below: Material
E modulus (kPa)
Timber
5E6 to 15E6
Concrete (normal strength)
25E6 to 35E6
Aluminium Structural steel
± 70E6
± 205E6
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•
K modulus: Foundation modulus or modulus of subgrade reaction (kN/m3). Typical empirical values derived suggested by Bowles are given below: Soil type
3
K modulus (kN/m )
Loose sand
4 800 to 16 000
Medium dense sand
9 600 to 80 000
Dense sand
64 000 to 128 000
Clayey medium dense sand
32 000 to 80 000
Silty medium dense sand
24 000 to 48 000
Clayey soil: qa ≤ 200 kPa
12 000 to 24 000
200 kPa ≤ qa ≤ 800 kPa
24 000 to 48 000
qa > 800 kPa
> 48 000
The foundation modulus, K, is a conceptual relationship between the soil pressure and deflection of the beam. Because the beam stiffness is usually ten or more times as great as the soil stiffness as defined by K, the bending moments in the beam and calculated soil pressures are normally not very sensitive to the value used for K. Recognizing this, Bowles suggests that the value of K can be approximated from the serviceability limit state bearing capacity, qa, as being 40 × qa (kN/m3) or 12 × qa (k/ft3). •
Allow negative pressure: Enter 'Y' to enable full adhesion between beam and elastic medium. Enter 'N' to allow uplift, i.e. zero adhesion between beam and elastic medium.
Note: The foundation modulus, Km, is multiplied with the support width to obtain the support stiffness per unit length of the beam. Enter a zero value for no foundation stiffness, i.e. a gap in the elastic medium.
∗
'Foundation Analysis and Design, Fifth Edition', by Joseph E. Bowles, published by McGraw Hill 3-112
Beam on Elastic Support Analysis
Example The sketch shows an 800 mm wide by 300 mm deep beam is modelled on an elastic foundation: •
The first fourteen meters of its length is supported on very stiff clay. The foundation modulus is set to 40 000 kN/m 3.
•
The beam crosses a rock intrusion ten meters fr om the left that provides vertical support.
•
The beam then spans four meters over a ditch, i.e. no support. This is modelled by entering a zero section width.
•
On the other side support is provided on a strip of hard clay, two meter wide. The hard clay is modelled by increasing the support width to 1.2 m. The resulting effective foundation modulus is then given by 1.2 / 0.8 x 40 000 = 60 000 kN/m 3.
•
The beam is loaded with a long trapezoidal distributed load, twelve meters long, a point load and a moment at its ri ght-hand end.
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Analysis Press Analyse to display the analysis results: •
Bearing pressure.
•
Bending moment.
•
Shear force diagram.
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Beam on Elastic Support Analysis
