Back when I left university (when dinosaurs etc. …) I joined
a consultancy, with my degree certificate clutched in my sweaty little palm,
thinking that I knew everything. On Day 1, they asked me to work out the
deflections of a three span continuous bridge, with integral sloping piers …
and I found out exactly how much I knew about being a real engineer.
For a very long time, I thought that my lack of knowledge
about one of the fundamental tools in engineering was a) because I did general
engineering, b) computer analysis was cutting edge and my university was behind
the times or c) I must have overslept and missed the lecture.
After many years of explaining the basics, I have found out
that the answers are a) I get the same problem with grads who specialised in
civil engineering b) they even have bridge design games on the internet now so
it cannot be that and c) probably not.
It seems that at least some
universities think that using a computer analysis programme is beneath their
contempt academically – if you cannot do a PhD about it, it is work for a mere
technician and they do not bother to teach it. I was taught a lot of guff about
matrix manipulation, which may have been academically interesting, but of no
use unless I actually wanted to write computer software. I do apologise, most
sincerely, to those universities who do teach the basics, but to those who do
not, may I please beg that you at least run a few lectures as an elective ?
Aside: before some pedant writes in to say that computer
programme should actually be spelt with …am at the end, that is only common
practice because Microsoft played such a large part in developing computers and
that is how they spelt it. It was originally spelt …amme even for computer
programmes. I am English, believe that the term American English is an
oxymoron, and if my pitiful solo stand annoys you, you can always stop reading J. Whilst
I am ranting, the Americans did not capture the first Enigma machine from a
German U-boat, as portrayed in U571, it was the British sub HMS Bulldog before
America joined the war – U571 was actually sunk in 1944 by a flying boat from
the Royal Australian Air Force. Time to lie
down in a darkened room again …
My normal approach in the chatty guide is to start off with
the simple cases and build on them: this is, however, a total rewrite of such
an attempt because it became so convoluted to explain what was not
involved in a simple case that even I could not follow it. I will therefore
start with what is theoretically the most complex case (3-Dimensional, or 3D,
space frame) and later explain why other types (plane structures or trusses)
are almost exactly the same but with a few bits missed out. Where there are
fundamental changes possible in the simpler cases, I will try to signpost them
with a little comment.
There are a large number of computer analysis programmes on
the market, so I will try to keep this as widely applicable as possible. I do
have a favourite programme, which does almost everything that I could ever
want, and allows me to hack the input to do things in any number of ways. Some
programmes are both proscriptive and prescriptive (I got confused and had to
check that they were different, although both actually apply) and I am sure
that you will sometime hear the phrase “That is not how the industry does
things”: what they actually mean is that is not how THEY want you to do it. I
will therefore try to keep to the things that are common, whilst trying to
explain the advantages of particular approaches. I am somewhat odd, being a
hard core engineer in my 50’s, but that means that I get a lot of the fun jobs
to do and I need to have a large and varied toolbox, and do not like being
dictated to by analysis programme snobs (sadly they do exist).
Fundamental: I will mainly be chatting about computer modelling
programmes. That is not a precise definition, but what I mean are pieces of
software that take information about the geometry, section properties and
elasticity of a structure and calculate how a series of loads will affect that
structure. In my totally arbitrary terms, any programme that then goes on to
calculate the strength of the structure, for instance by calculating the
effective length of a member, then its limiting stresses etc. is acting as a
bolt on to the end of the modelling programme. I am making a generalisation
here, especially with regard to some non-linear analysis, but that is advanced
and this is a lesson on the basics.
How much detail should you put in
Enough.
Well that was useful (sounds of foot tapping on floor) ….
Oh all right then.
It depends ….
Without wanting to go into too much detail, a computer
analysis creates a large matrix that models the behaviour of particular points
on the structure using equations that represent the links between those points
(i.e. the structural bits). You need to break the structure down into enough
pieces that a) the equations are going to be accurate enough, and b) that you
get the results in the places that you want.
a)
Most beam equations are rigorous and you do not
need to put in more members than physically exist to get the correct behaviour.
Finite element equations make more assumptions and, up to a point, using a
finer mesh will give a more accurate result, especially in locations where the
results are changing fast such as around a stress concentration.
b)
But unless you are using beams and working with
graphical results (amateur !), you will want results at lots of locations along
a member. Most programmes give results at the ends of members, so if you want
lots of results, then you will need lots of members.
As a rough guide, on a bending structure, like a simply
supported beam or a continuous frame, you will want results at the ends,
midspan and a few unpredictable locations in between. Most engineers will split
a physical member into 8 to 12 model members to get a wide enough range of
results, since there will usually be a wide variation in effects along a beam.
On a triangulated, or truss, structure (more of which later)
the axial forces will not vary much between panel points (where the members
come together) especially if the loads are applied at the panel points. Any
moments that occur will generally vary linearly from one end to the other. In
this situation, you only need results at the ends of the physical members and
you will generally only need to have one model member to represent one physical
member.
With finite elements – learn to walk before you run (then
have a look at the end of this post).
Fundamental: the analysis works by creating a large matrix and
inverting it. The matrix values will depend on the stiffness of the members in
between. They not only includes the section properties, but also the length.
Having members that range from mm to m will vary by a factor of 1,000 in length
but a factor of 1,000,000 in bending stiffness from length alone. This means
that the computer will have to do some very involved, often repetitive
calculations on numbers that are orders of magnitude different and that is
never good. A matrix with too wide a range of values is called “Ill-conformed”.
Some engineers try to be clever by inserting a short 1mm long member to offset
one member from another, to make the model more accurate, not realising that
this messes up the calculations and may give a totally wrong result. This is
especially so if the calculation is non-linear and the same numbers are being
worked over again and again. Keep things simple and try not to have a factor of
more than 10 between the longest and shortest members. You could cope with much
more, but why make trouble for yourselves.
What you are trying to achieve
It may sound silly, but
the aim of computer modelling is to create something that is an acceptable
approximation of your real life structure. Obviously, nothing will ever be a
perfect model but you are trying to get close. One of the things that you will
learn over time is what parts of the analysis are important and which can be
given a lower priority, but it is important to remember that the model is
supposed to represent a real structure. If your model does something unusual
then it can be a) you made a mistake (very common) b) the computer programme
has a bug (not very common) or c) you have discovered something that will
improve your understanding of structures. With 30 years’ experience, the
unusual results are nearly always my mistake, but sometimes I learn something
new - I do have to thoroughly think the problem through for a few hours, and bounce
my thoughts off of my colleagues, before I am happy with something new though.
The dots
All modelling programmes start from a series of points in
space. These are usually called nodes or joints, but if you find another term
then please substitute that in your own mind. I will use the term joint here.
In a 3D analysis, joints are positioned in three-dimensional
space. This 3D space uses what is called the Global Coordinate System, and (odd
as it may seem now for me to say this, but it will make sense later) all joints
use the same Global Coordinate System. Although most programmes can use
cylindrical or polar coordinates, most modelling uses the Cartesian coordinate
system X, Y and Z, with two axes at right angles in the horizontal plane and
one vertical. If you are not entering data in the Cartesian system, your
programme will probably just be translating the data into Cartesian anyway.
Fundamental: Please note that
different programmes use different axes for “Up”, and up is very important as
you will find out when we discuss the axes used by members. But whatever
programme you use, the global axes will follow the right hand screw rule, where
turning a screw from the global X axis to the global Y axis will drive the
screw into the positive Z direction.
Aside: If you have never driven a
screw into a piece of wood, or tightened up a bolt, then this will mean nothing
to you, but then I would ask if you are in the right profession J
For the purposes of
this discussion, I will always assume that Z is “Up”: if your programme uses a
different system then please adapt.
In order to visualise this, try thinking of the origin, i.e.
X=0, Y=0, Z=0, or (0,0,0) as the front left hand corner of your (hopefully
rectangular) desk. The global X axis runs from left to right along the front of
the desk, global Y runs along the left hand edge from front to back, and global
Z is vertically up.
PLEASE NOTE that I am using uppercase letters for the Global
Coordinate System.
The joints can usually be entered into the model in a number
of ways, and the possibilities are programme specific. Typically, though, you
might be able to enter them individually, generate a series of joints in a line
or a grid, use Excel to generate and then import them in (easy in good
programmes, next to impossible in others), copy existing joints at regular
intervals, mirror or rotate existing joints etc. etc etc. However you get
there, your model will have a number of joints that define its position in
space.
Joining the dots (joints)
Having defined your joints, you now need to join the dots
together with some bits of structure. The types of structural bits will depend
on what you are trying to do, as well obviously as by what your programme can
cope with, but they can be most simply (childishly ??) thought of as rods,
plates or blocks.
A rod will have two ends, a plate will (typically) have
three or four corners (although some programmes are too clever by half and have
extra nodes that are automatically generated) and a block will typically have
eight corners (if you are sensible) but could actually have anything from four
to eight (six is OK if an extruded triangle but others only if you like pain).
What happens now depends on what you are trying to do. For
this initial discussion, I will just be talking about “Rods” aka beams,
members, struts, whatever… The rod in the model will generally run along the
centroid of physical element and the joints are the best fit of where the
sticks intersect.
Fundamental: The basic rule is that, unless you do something to
tell the programme otherwise, the ends of any member move in the same way as
the joints to which they are attached, and those attachments are rigid for both
force and moment. Hence, unless you rule otherwise, two or more members that
are joined to the same joint will be rigidly connected at that location for
both force and moment. Only if you over ride this with a specific member
command, or with a global analysis command, will they act as though they are
not rigidly connected.
How can the joints move ?
The following bit is actually very easy, but you need to
keep your wits about you. As I have said before, there are lots of methods used
in modelling to make your life easier, unfortunately they can also make it more
complicated until you work out what is going on (at which point it becomes easy
– honest guv).
In a 3D model, any joint in a model can translate (i.e. move
along) any of the three global axes X, Y or Z. These joints can also rotate
ABOUT those same three axes. Again, rotation about a global axis follows the
right hand screw rule with which you should be familiar from pre-university
physics. Hence in the general case, there are six possible ways that a joint
can move. In typical “Blind you with BS” mode, lots of people with pointy heads
will say that each joint has “Six Degrees of Freedom” or 6 DoF: this means that
the programme has to keep track of six pieces of information for each node and
that section properties, supports etc have to have enough information to keep up.
More of this later when I get on to simpler model forms.
Local member axes (local coordinate system)
I used the somewhat silly term “Rod” because the general term
is “Member” and this apparently causes some sniggers in some parts of the world,
where it is used as a euphemism. Unfortunately I now have to bite the bullet
and refer to members (no, titter ye not ! - with apologies to the late Frankie
Howerd) since that is what the programmers usually call them.
Unlike joints, which all use the same Global Coordinate
System, every single member (or element or block or whatever) has a different
Local Coordinate System, or local axis system.
When you define a member, it will run between two joints and
you will have to specify one of them first, followed by the second joint: I
will call these End 1 and End 2 (although some manuals use other other terms).
The LOCAL x axis (please note the lower case x for a local axis) runs from End
1 to End 2 regardless of how the member
is orientated in the global coordinate system. If you are particularly
masochistic, and had a member parallel to the global X axis, with End 1 at a
higher value of X than End 2, then the local x axis could actually be running
in the opposite direction to the global X axis.
Figure 39
- direction of local x axis
|
FOR THE VERY PARTICULAR CASE WHEN THE MEMBER LIES ALONG THE
GLOBAL X AXIS AND END 2 X > END 1 X, and only when it does, the set of three
local axes, referred to as the x, y and z axes, are parallel to the respective
global X, Y and Z axes.
And now for the more interesting local axes
The remaining local axes are not quite so straightforward,
when you look at the definition, but are actually quite sensible when you look
at what they are trying to do.
Unless the member is running in a vertical direction, the
local axes should follow the following process (but read your manual to check).
A vertical plane is created which contains the member under
consideration. The first image in Figure 40
shows a plan view of the member (red) and the plane (black). Section marks are
also shown on the first image that define the direction of the second view,
which is looking horizontally and perpendicular to the plane.
Taking the global Z axis as “Up” (and modify to suit if it
is not), the local z axis will be drawn on the vertical plane perpendicular to
the local x axis. There are two directions in which it could be drawn, and the
programme will point it in the direction that is most upwards. In other words,
all the programme is doing is setting the local z perpendicular to the local x,
and then making it point as far upwards as it possibly can.
The local y axis is then defined perpendicular to both x and
z axes following the right hand screw rule (slightly odd here since we have to
turn from x to y when y does not exist, but you probably have the idea).
In the very special case of the member being vertical (often
taken to be when it lies within 1O of the global “Up” axis” – but
check your manual) then there are an infinite number of vertical planes that
could contain the member. The local “Up” axis is then usually taken as parallel
to the NEGATIVE global X axis. This makes a lot of sense in practice since a
lot of analysis lie within the single plane containing the global X and Z axes.
If you take the horizontal member, running parallel to X which then has z
upwards, and rotate it upwards about the local y axis, then the local z axis
will rotate away from global Z towards negative global X. As the local x axis
reaches global Z, the local z will naturally fall onto the negative X axis.
This is only really a problem if your main plane of members is in the global Y
Z plane, when the axes will suddenly turn 90 degrees as you approach vertical,
But as the old joke goes “Doctor, Doctor, my shoulder hurts when I do this” –
“Well don’t do it then”, we can also choose to orientate our major members
along the global X axis and the problem goes away.
Member forces and moments
These are always specified relative to the local axes. Axial
forces, tension or compression, always act along the local x axis. Shearing
forces are defined by the direction in which one part of the member is trying
to move, relative to the adjacent section. If the member is being sheared such
that one part is trying to move in the local z direction, relative to the
adjacent part of the same member, then this is a shear in the z direction
(surprise !).
In a similar manner, if you hold the left hand end still and
rotate the right hand end about the local x axis of a ruler, it will twist.
Hence rotations about the local x axis causes torsion moments, or “x” moments.
Holding the ruler flat, and rotating both ends in opposite
senses about the local y axis will cause either hogging or sagging bending
moments, referred to as “y” moments. Turning it on its side and rotating about the
local z axis will cause bending in the other direction and I will leave it to
you to guess what they are called.
Bored now ! says the poor reader, who is wondering why I am
prattling on about something so obvious.
Section properties
The members are generally prismatic, i.e. they have the same
cross section along their length and straight (some programmes do allow tapered
or curved members but that is more specialist). The properties that are entered
for a member are those for the relevant cross section.
MAJOR POTENTIAL PITFALL COMING UP !
As discussed above, the local axes are x along, with, say, z
up and y sideways on the cross section. However, the various codes, such as
BS5400 or the Eurocodes, quite often use both x and y axes on the cross
section. This is shown in Figure 41.
The two sign conventions are in no way related and it is quite common for the
member y axis to be equivalent to the code x and member z to code y. But when
you are modelling, you need to use the modelling conventions and then convert
the axes into the code conventions. I am sorry about this, but I would like to
stress that I wrote neither the codes nor the analysis programmes.
I started off by describing how analysis works for 3D frames
but mentioned that I would later get onto simpler forms of structures.
In the most complex 3D form, as I wrote previously, each
node can translate (move) in three directions and rotate around three axes,
giving 6 Degrees of Freedom. With this form of structure, you can put loads in
any direction, but require information to cover all forms of member deflection,
and the structural matrix that is created is slightly smaller than 6n x 6n
where there are n nodes in the structure.
But if you are only
applying loads in the plane of the frame, i.e. your frame sits in the global XZ
plane and the loads all sit inside the global XZ plane, then the various joints
on the frame will only move in the X and Z directions, and rotate about the
global Y direction. So if you are working on a problem that requires only these
limited types of movement then the programme can ignore movement in the Y
direction, and rotation about X and Z axes. For this problem you could use a 2D
frame analysis rather than the 3D frame. The advantages are that the computer
only needs to keep track of three pieces of information per joint and since the
member can only stretch in one direction, shear in another and bend about the
third, only three member section properties are required to describe the member
rather than the normal 6. This type of analysis is referred to as having joints
with Three Degrees of Freedom, or 3 DoF. Of course, you could do the analysis
with all 6 DoF, but it is harder work for both you and the computer.
If you have loads that are acting out of plane, i.e. in the
global Y direction, then you will need to use the full 3D frame analysis, with
6 DoF and 6 member section properties.
There is another form of analysis that only uses 3 DoF,
which is the Floor, or Grillage, analysis. These are used when the loads are only perpendicular to the plane of the
members, hence the “Floor”. They are typically used for deck type analyses,
where you might have a slab, or a series of beams and slabs, that can
distribute load in two directions. The relevant DoF here are movement in Z and
rotation about X and Y. This form of analysis gives results of bending moment
and shear, as well as torsion, but does not consider in plane forces (leading
to axial forces) at all.
In both the plane frame and the grillage models, the default
position is that members that come into a joint are moment connected, as well
as force connected (for axial and/or shear force as relevant).
Truss analysis
Please listen very carefully, I will say this only once !
(more references to dodgy English sitcoms, and actually very unlikely for
someone who repeats himself as much as I do).
Truss structures, in the physical or real world, are triangulated
skeletal structures that predominantly
resist load in axial tension and compression. Being a truss, in the real world,
does not mean that the
structure will not have any
bending effects, just that they are not the dominant behaviour.
Truss structures, in modelling, are triangulated skeletal structures
that only resist load in
axial tension and compression. They should strictly be referred to as
pin-jointed trusses because they assume that the ends of the members are pinned
i.e. they can rotate relative to each other.
Plane trusses, where the loads only occur in the plane of
the structure, only need to keep track of two displacements and no rotations.
They are therefore 2 DoF structures, with smaller matrices to solve, and the
only section property required is axial area – simples !
3D trusses are similar to 2D trusses, in that they have no
moments, and require no rotations, just movement. They are therefore referred
to as 3 DoF problems. They do, however, require triangulation in three
dimensions or they will just fall over.
But just because a structure is physically a truss in the
real world does not mean that it should necessarily be modelled as a truss -
that is an informed choice that you need to make. Railway bridges are often
physical trusses, and the vertical and diagonal members are often treated as
pin ended truss members because the connections are not sufficiently strong to
restrain rotation, and the resulting plastic yield will release any moments.
But the top and bottom chords, and often the end verticals/diagonals, are
usually very robust, and the moments that they pick up may have very nasty
local effects on parts of the members (which often have unstiffened plates that
could buckle locally). In this situation, it is quite common to model the
“Truss” as a frame (to give continuity around the robust members) and then to
release the ends of the internal diagonals and verticals so that they act as
pin ended truss members.
Overall, if you need to model a mixture of 2D and 3D
members, or a mixture of bending and truss members, then your overall model
will need to accommodate the most complex parts, and the rest of the structure
will need to be modified to suit (e.g. by pinning the ends of those members
that you want to act as truss members).
Aside: for the purists amongst you, there is also a form of
“Truss” called a Vierendeel truss that is not triangulated and relies on local
bending of the members. Vierendeel trusses CANNOT be analysed as pin jointed
trusses. They are much beloved by my dear, dear friends the architects, who
love “… the clean lines and elegant flow …” of the structural type, but they
are very expensive to build and only an architect would think that the form
followed function. For more info, please refer to Wikipedia (another annoying
abuse of the English language but I will forgive them because it is so useful).
Section properties
The section properties that an analysis requires are those
of the cross section i.e. the section that is perpendicular to the centroid of
the beam. For the beam in Figure 41,
that is the bit at the I section shown at the end with a thicker line. Please
refer to the cross section in Figure 41,
marked “Typical analysis programme”, for the axes that I am using, and remember
that “y” and “z” will swap if global axes have Y up.
Ax = axial area = total area of (flanges + web)
Ay = shear area sideways = total area of flanges
Az = shear area vertically = total area of web (sometimes
including the projected bit of flanges as well, depending on code)
Ixx (which should theoretically be called J) = torsional
stiffness, which is complicated.
If it is a closed thin shell section (e.g. a hollow box)
then the stiffness = 4A2G/∮(ds/t) where A is the area enclosed by the midline
of the shell, s is the distance around the midline, which is split into short
sections ds, with ds/t integrated around the entire section. Typically, closed
sections are torsionally stiff.
If the section is not closed, e.g. an I section, a channel
or an angle, or even an almost square hollow section where a plate is bent to
form an almost continuous section but not welded to close the corner, then the
section is broken into a series of rectangular plates. Each of these plates is
treated separately, of width b and thickness t, and torsional stiffness is
calculated as 1/3(1‑0.63t/b).bt3. The results for the sections are
then simply summed. Relative to a closed section, open sections have almost no
torsional stiffness (although you have to put something in or the computer
might have a conniption fit).
Iyy = vertical bending stiffness calculated using the
parallel axis theorem
Izz = horizontal bending stiffness calculated using the
parallel axis theorem.
On an I section Iyy >> Izz.
But if you use the beam on its side then everything
switches.
Fundamental: if you are analysing something that is not skeletal
then different practices often apply. If you are analysing concrete bridge
decks, I would recommend Bridge Deck Behaviour by Dr. Edmund Hambly. This
covers items like the interaction between two rectangles of concrete in
torsion, when they are joined instead of separate (the torsion stiffness
changes by a factor of 2 but I forget which way J ) and analysing the transverse shear stiffness of a
concrete deck with top and bottom flanges with widely spaced webs (it acts
predictably in bending but the shear depth might be 5mm for a 1.5m deep deck).
Please remember that this is a starter document to get you going and is not
intended to make you an expert.
Material properties
Typically these will only be Young’s Modulus, the shear
modulus and density, but you might need coefficient of expansion if you are
looking at temperature effects and sometimes a few others.
Supports
This is an interesting one, which causes more grief than
anything else, but just requires a bit of common sense.
The programme is NOT thinking the same way that you are.
Consider a plane (i.e. 2D) frame analysis. If you are working with a horizontal
simply supported beam, and your only loads are vertical, then there will be no
horizontal reactions, and you might think that you do not need vertical
restraint (i.e. restraint against movement in the Z direction aka fixing in the
Z direction) – wrong. The programme works in a way that can accommodate any
loads, whether you apply them or not. If you specify a structure with just
vertical supports then the computer will think that the beam can slide off
sideways and the analysis will fail.
Aside: I do apologise to the purists for my anthropomorphism,
because the programme does not think, and it is actually coming back with an
error because the structural matrix is not soluble, but we all find ourselves
having discussions with our pet programmes as though they are our human
nemeses.
You will therefore need to put in a horizontal restraint as
well i.e. fixing in the X direction. Although, for reasons that I will not go
into, the vertical bending behaviour would not be changed if you fixed both
ends, it is normal to use the least number of restraints that will keep the
structure stable AND that will accurately reflect the behaviour.
If we stretch the same structure to 3D, then one end would
normally be fixed against movement in X, Y and Z and the other end would also
be fixed in Y and Z. HOWEVER, this structure would also fail because a 3D
analysis allows torsion about local x and the beam would be able to rotate like
a prop shaft unless you fix one end against rotation as well (this is one
reason why you normally only use the dimensions that you need to).
Similarly, a grillage model (2D with all loads perpendicular
to the plane) does not require horizontal restraint because it does not
consider horizontal movement. But converting this to a 3D space frame, which
does include horizontal movement, would require a number of horizontal
fixities.
The important thing to consider is that your model is representing
a real structure (something that some people do have a problem with). If they
try to balance a sheet of card on its edge, they understand that it will fall
over but then cannot see why a plane frame would also fall over if they
modelled it in 3D without rotational restraint.
If you are lucky enough to use a good programme then it
might detect a problem, warn you, and put in a weak notional spring to get some
sort of solution, this then allows you to see what is going on and solve the
problem. But unfortunately, not all programmes are created equal.
Loads
You will need some of these.
Typically individual loads are applied in groups called load
cases. Nearly all programmes will then allow you to combine load cases into load
combinations e.g. 1.1 x load case 1 + 1.5 x load case 2 etc. This is always
very tempting but (and I know that I am banging my head against a brick wall
and you will probably ignore this) please, please, please, please, please, please
do not do it.
Reason the First. One major problem with computers, and
where they differ greatly from my graduates, is that they do exactly what you
tell them.
Aside: Ok so some graduates do what you tell them, but usually
only when my instruction has not been precise enough and I am sure that they
are wilfully misinterpreting me in the most embarrassing cases J.
If you apply a series of loads then some of them will make a
problem worse (adverse effects) and some will make a problem better (relieving
effects). Most programmes will simply add the results together and will not
consider that you are reducing a bending moment by adding a load that may or
may not be there all of the time. Even for loads that are always there, adverse
loads might require a factor of 1.2 whilst relieving loads would require a
factor of 1.0 (I have never been clear why this should not be 1/1.2 but go
figure). So unless you are very clever or very lucky, you risk getting the
wrong answer by using load combinations. Taking the results out into Excel and
then using simple rules to combine the results gives a much better result.
Reason the Second. This reason is either irrelevant, or the
most important thing since sliced bread, depending on your approach to life. If
you only look at combinations of loads then you will never get a feel for what
is important in an analysis. Everything is an approximation, but some
approximations matter more than others. If you know that a load only
contributes 0.5% of the answer, compared to 80% from another load, then it
should be fairly obvious which one has to be applied “Accurately” (there ain’t
no such animal) and which one you only need to get in the right ballpark.
One of my favourite games with a cocky graduate is to pose
them this question (and I am reducing my fun quotient if any of them read this
before I get to them) …
If you are analysing a three span continuous concrete
bridge, with spans of 16, 20 and 16m and a solid slab of concrete that is 1m
thick (plus surfacing as well), then what proportion of the total nominal load
will come from the long term loads (concrete plus surfacing) and what
proportion from the live (vehicle) loads.
The typical answer from a grad (and even quite a few grey
bearded engineers)might be vary from 20 and 50% long term with 80 to 50% live
loads.
I have seen some grads create 80 or 90 load cases to get the
worst bending and the worst shear cases at all points across the deck, trying
to squeeze a couple of extra % of load. The calculation for concrete
self-weight might be three lines.
When I then point out that a realistic answer might actually
be 75% long term load and 25% live load, they realise that they have been
concentrating all their time to optimise a few % of 25%, and ignoring that the
edge cantilever (which they might have forgotten) is actually 10% of a much
larger number. I must add that a steel/concrete composite bridge is more likely
to be 50:50 but that the surfacing might be 1/6 of the total load.
Piling everything in together will never allow you to develop
an understanding of which loads your structure is sensitive to and therefore
which loads you ought to be taking the most care with. It will also prevent you
picking up that the wrong load is governing, which might be caused by something
as simple as a decimal point in the wrong place.
Results
When you finally manage to get some of these, they will
probably be wrong – get used to it.
With 30 years’ experience, concentrating on doing the hard
numbers and minimising the management side where possible, I am considered as
being good at my job. In practice this means that I might get the model right
first time about one time in twenty (yes I did mean to write 5% of the time).
Admittedly my models tend to be more complex than most because I get all of the
fun jobs, but you have to remember that everyone makes mistakes and that the
mark of a good engineer is to spot them.
The first thing to check is that the structure is bending in
a sensible manner. My first ever bridge analysis resulted in self-weight
causing an upwards deflection of a
quarter of a million miles. This was great because, even as a fresh grad, I
could see that there was probably something slightly wrong. It turned out that
I had applied gravity as a positive force, when that programme needed a
negative (be warned, some programmes use a factor for
self-weight which might be +1 to give you a negative Z force – others might
require a -9.81) so it deflected the wrong way. I had also forgotten to
put in a Young’s modulus, and rather than giving an error, it used a value of
1E‑16 and was therefore a tad too flexible.
Deflections actually control the behaviour of any structure,
and this is a good way of picking up problems with your supports. If you have
applied too many displacement supports of a certain type, your structure might
have an unexpected rotational fixity as one works against another, and looking
at the deflected shape will hopefully highlight this.
It is also worth looking at the magnitude of the overall
deflection under full loading. There are no hard and fast rules, but an overall
deflection between span/200 and span/1000 indicates that you are at least
within the right order of magnitude.
It is very easy to make a mistake applying loads. It is also
simple to add up the reactions and compare them to what you wanted to apply. On
my favourite programme, I once made a mistake whilst running a train of loads
over the span. I was used to applying this load train to plane frames, so the
“Axle” load was actually a single point load. When I applied this to a track on
a 3D structure, with the two rails being modelled 1.5m apart, what the
programme called axle loads (which should have been divided by two to get the
wheel loads) turned out to be wheel loads: I had therefore applied one train to
each rail, not to each track, and inadvertently doubled the applied load. I
picked this up, however, when the total reaction from the trains was twice what
it should have been.
Results (really this time)
Member results from a skeletal analysis (as opposed to
finite elements, which I will touch on in a moment) are always either forces or
moments. The forces run in the direction of the relevant axis (FX is axial
force, FZ is shearing in the local z direction) and the moments obey the right
hand screw rule and act about the relevant axes (MX is the torsion moment, MY
is the moment about the local y axis).
Nice and simple …
BUT humans work with bending moments whilst computer
programmes work with moments. These seem to be the same thing and some
programmes even refer to bending moments instead of moments.
But the analysis will use a number of members that are all
joined together. These are shown in red in the blow up, and I have pulled the
ends in to make the diagram clearer. The member end actions, i.e. the actions
that happen at the end of the member (gosh darnit but Engineers think of some
complex names for things) are acting in the opposite directions at the ends of
two adjacent elements (note the +M2 and the –M2) and the
results table will show flipping of the signs on either side of the node. A
similar thing happens for the axial forces and the shear forces. You need to
decide how you do things, but generally I pick a sign convention that I am
happy with and then chose to take the results from either end 1 OR end 2 to
suit and use those. I usually have to take one result from the other end,
adjacent to a support, and reverse the signs, but it generally works out OK
(partly because I plan my models so that this will work).
Finite Elements
I am not going to talk much about finite elements, like
plates/shells or solid elements, except to say that they are just a different
implementation of member analysis. The main difference is that the equations
that form the structural matrix are more complex.
Just as you can have members that take bending only, axial
only, or both, you can have elements that can have bending only, membrane
action only or both.
Element properties are usually simpler and often consist
just of thickness, although some programmes will allow you to specify different
bending properties etc in two directions at right angles. Programmes vary, but
often the local x axis is set up in the direction between joints 1 and 2,
although the z axis depends on which way round the element the joints 1, 2, 3
(and probably 4) are ordered.
The biggest kicker, which confuses a lot of beginners, is
that although the x stress is in the x direction, the x bending is a moment that
results in a bending stress in the x direction. If the two plate axes are x and
y, then x bending in the plate is actually bending about the local y axis. Again, do not start off on me because I did
not decide this.
When creating elements, you need to keep them as flat as
possible, i.e. all nodes in a particular element must stay very close to one
plane, although that plane can be oriented in any direction and will often vary
greatly between the various elements. The shape of the element should not be too
extreme, and I always try to keep rectangular elements within a ratio of 2 to
1, although I will stretch to 4 to 1 if there is not any high rate of stress
change.
The density of the mesh matters, and needs to be fine where
stresses are changing fast. One technique that the experts use is to model the
problem with progressively finer meshes until the results do not change. Nice
idea if you have the time but I tend to look at the stress plots and start to
get worried if I get more than 3 of the typically 15 stress contours crossing
one element (not scientific but what the heck) or if there are any violent
changes in direction at stress concentrations.
Conclusion
That’s almost all for now folks.
I will, however, leave you with one reminder and that is
that the model is representing a real structure. You need to develop a feel for
how the structure will behave in real life and then make your model match that.
The real structure could not give a tinker’s cuss what you have modelled (and,
not being sentient does not realise that you are modelling it) and you should
never make the argument “But the model
does …” unless you are confident that that is reasonable behaviour.
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