Book Review: Richard Hofstader's The Paranoid Style in American Politics and Other Essays

Political analysis and punditry can date awfully quickly.  Nevertheless–and this is especially true in a place like the United States, whose structural continuity is exceptional–some pieces of political prose, especially when written in a historical context, can have relevance for several generations.

One of those analyses–actually a series of essays not originally intended to be a corpus–is Richard Hofstader’s The Paranoid Style in American Politics: And Other Essays.  It’s one of those books, for better or worse, whose influence has not waned in the years since its first publication in the mid-1960’s (some of the essays go back further than that).  It’s one of those works, for the left at least, that leaves you wondering why anyone bothers writing on the subjects again, except that a) the names change and b) they need the money.

The book is made up of six of Hofstader’s essays, divided into two sets of three.  The first are “Studies in the American Right”, which leaves the reader no doubt where Hofstader’s sympathies lie.  The first essay is, of course, “The Paranoid Style in American Politics”, and the other two concern the campaign and defeat of Barry Goldwater in 1964, along with an analysis of the “pseudo-conservative” (to use Hofstader’s distinction of the Goldwater right from the moderates in the Republican party who opposed him) movement.  And this is where the relevance comes in: depending upon how you look at it, Hofstader either nails the right so well that everyone who comes after him cribs him to be correct, or that his view of “flyover country” is so congenial to the left that they no longer have to think of another model to use (which is why, I think, liberals often refer to this piece in their own analyses).

All of the usual actors are in place: the militant anti-communists, the religious right, the small business people, the less educated, the emotionalist–all of them find a place in Hofstader’s knave’s gallery of which he presents a sweeping view.  Hofstader, to be fair, does not believe that the paranoid style is the exclusive franchise of the right.  But he hardly spends enough time on their left-wing counterparts to leave any doubt of where he thinks the home of paranoia in American politics lies.  One thing he consoles himself with is a) Goldwater managed to alienate such a large part of the electorate and b) their commitment to their cause without regard for political consequences indicates that their 1964 performance with Goldwater would be repeated in the future.

Or would it? In the second series of essays, “Some problems in the Modern Era”, he deals with three topics which don’t get a good deal of treatment these days.  The first concerns the Spanish-American War and our entry into imperialism, one which was uncharacteristic (for foreign imperialism at least) up to that time.  The last is devoted to William “Coin” Harvey, his “Financial School” and the subject of bimetallism, where he is able to capture a conspiracy style of thinking that has resurfaced in differing forms.  But his treatment of the intricacies of bimetallism, something that is difficult for those of us who are products of fiat money times may find difficult to grasp, is one of the strong points of the book.

But it was in the middle essay, on the subject of anti-trust, that the possible future (at that point) of the post-Goldwater Republican right surfaced:

In politics, of course, it is the right-wingers who really count–it is they who have the numbers, the money, the political leverage.  They can also invoke the old American pieties and can appeal to the kind of old-fashioned American who believes that federal fiscal policy is just like the family budget.  Much of our conservative writing echoes with concern over the decline of the older kind of economic morale, which it identifies with smaller entrepreneurship.  But conservatives understandably fear to make the large corporation the object of their criticism; this smacks too much of subversion.  They have a safer and more congenial outlet against the organisation of modern life in the form of denunciations of big government…

In this regard Hofstader was prescient, for the conservatives, roundly defeated in 1964, and the Republican Party, trashed by Nixon in Watergate, looked like a lost cause by the mid-1970’s.  But conservatives’ organisation and rootedness in the country’s core ethic–plus finding a more congenial leader in Ronald Reagan who knew “when to hold ’em and when to fold ’em”–put them back in the driver’s seat in the 1980’s and pretty much for the next quarter century.  So how else has Hofstader’s whole model held up?

Hofstader was right in pointing out the paranoid style in American politics, but he was blind to its relevance on the left.  It wouldn’t be long before that came roaring out in the 1960’s campus revolts which were in part a revolt against liberalism’s own quiescent acceptance of large corporations (something Hofstader also identified).  If there’s one thing the unwashed students didn’t think much of, it was working for “the man”.  Such raw emotionalism resulted in the nervous breakdown that the U.S. experienced in the early 1970’s.  (A more balanced view of American emotionalism can be found here).

But that, in turn, leads to a more contemporary question: if Barack Obama manages to crush the Republican Party the way he wants to (and, for those who can still count, that would lead to a one-party system) and at the same time break the right once and for all, would we have a better country for it?  Or, to put it another way that would be a fair question Hofstader doesn’t address, how is it that a nation of insane crackpots has been as successful as we have?  The 60’s and post-60’s left in this country hasn’t shown that it understands how to have a growing economy (let alone really wanting one) or a great nation, let alone one with a more even distribution of income (one that existed, by the way, in Hofstader’s day).  They are good at going through their process.  But will that carry the day when the competition is Asia?

Although Hofstader makes many astute observations about this country, its past and for him its present, one gets the impression that he’s like Biblical scholars who really don’t believe the truth content of the book they’ve devoted their lives to studying but don’t have the grit to find another line of work.  One another level one can only wish that those who pine to make us like Europe would just move there, but that’s another post.  In the meanwhile The Paranoid Style in American Politics: And Other Essays is a “must read” for those who really want to understand where liberals have come from for a long time.

"Proud Mary" No Longer Proud to be an American

Tina Turner calls its quits on the U.S.:

US pop legend Tina Turner, who has been living in Switzerland since 1995, will soon receive Swiss citizenship and will give up her US passport, Swiss media reported Friday.

“I’m very happy in Switzerland and I feel at home here. … I cannot imagine a better place to live,” Turner told German language daily Blick.

Those of us who lived through this era would be hard pressed to forget her and Ike Turner’s performance of “Proud Mary”.  But evidently all the “progress” we’ve made here since then wasn’t enough.  Not even the election of Barack Obama was enough to keep her from renouncing her U.S. citizenship and living in what is still a fabulous place.

The truth is that, as those who follow Rubin on Tax know, this country has done just about everything it knows how to make life difficult and expensive for expatriates, which is why the rate of citizenship renunciation has increased of late.  As our government closes in on the “one percent” same will find other shores for themselves and their money, so they won’t have to carry the burden of being an “American person”.  And those of us left will be the poorer.

The Animals Finally Lose it in South Florida

I guess any creature can only take so much:

Two lemurs were captured early Monday morning after they escaped from their cages and one scratched a 2-year-old girl.

About 2 a.m., North Miami Beach Police received the call about the lemurs which were loose near 2049 NE 173rd Street.

“This is the first time I’ve been dispatched to a call like this,” said North Miami Beach Police Sgt. Richard Rand.

Regulars of this blog know that I refer to South Florida as the place “where the animals are tame and the people run wild”.  This is why the call to the police was the first of its kind.  South Florida cops are used to people acting this way, but lemurs…that’s something of a novelty.

But face it: if you were caged up and had to live with a bunch of South Floridians, wouldn’t you crack up sooner or later?

The Two Questions Many Churches Never Stop Answering I Never Asked

Today is Fake Inauguration Day in the United States.  Our President, coming from a triumph many said would never happen, was actually sworn in yesterday.  If I were him, and I wanted to stick it in the eye again to serious Christians, I would have done the thing on Sunday, the way other places do elections.  But wonders never end…

Despite that opening, this piece is only tangentially about politics.  What it is about is the purposes of churches, how they present themselves to the society around them, and why that presentation doesn’t always resonate with portions of the population.  There’s a message in our changing political system for churches, and I’ll try to tie the two together.

The final purpose of our system of faith is twofold: to explain the meaning of the world around us, and ultimately to give us a good way out when physical life ceases.  However, having sat in Evangelical and Pentecostal pews for many years, neither of these gets a lot of “air time” in the pulpit, to say nothing of Christian television.  Today churches spend an inordinate about of time on two questions that never occurred to me as significant, but seem to be an obsession to others.

The first of these is this: how can I be prosperous? Coming from a family with long multi-generational success and growing up in Palm Beach, this is definitely a question I didn’t need any church to answer.  I knew how to be prosperous and successful: what eluded me was how to be happy in this life and the next.  My reading of the New Testament told me that, to do either, I would have to bypass much of the prosperity methodology that I had been taught.

What churches–especially full gospel churches–present these days is polar opposite of this, but it isn’t what you might expect.  You’ll see from time to time liberals marking the passing of the “Protestant work ethic” but the truth is that non-Catholic Christianity in this country has dumped this concept for a long time.  In its place is a system of positive confession and plentiful giving without enough time given to encourage thrift or “consistent daily doing” to achieve success.  This, I think, is the result of the centre of American Christianity shifting towards the Scots-Irish.

Although there are those who have benefited from this, the general result of this has not been as advertised.  But anyone who might put forth an alternative idea to either personal or collective success isn’t going to get much of a hearing in this environment.

The second question is related to the first: how can I be a somebody? Again coming from Palm Beach, I didn’t need much help on that either.  I saw an elegant but brutal social system in action at the top of American society.  Again my reading of the New Testament taught me that a) those who followed God would be significant in a real sense and b) seeking significance in the world ran against what Christianity was all about.

These lessons also have fallen on deaf ears beyond the island.  No one wants to admit it, but many churches are trying to build themselves on the concept that you can be a somebody here.  It’s tempting for “Main Line” churches to sniff at this, but they do so at their own peril.  The Episcopal Church, for example, more than doubled its membership in a third of a century.  How did they do this?  One way was the not so subtle message that, if you join up with us, you’ll have the ne plus ultra of respectable religions and be better than anyone else.

When Main Line Christianity began its unedifying collapse, Evangelicals (especially those on the full gospel end of things) rushed in to fill the gap.  But they, coupling this with prosperity teaching, discovered something else: respectability, especially in the nouveau riche environment we’ve been in until very recently, is expensive.  Much of the physical plant–ecclesiastical and otherwise–that churches and church people struggle to pay for these days was in fact a projection of their own success and not ancillary to the ministry.

I think that both, to some extent, have seen some abatement since our economy went into what is its (I think) enduring funk.  I think that it’s not fast enough, not for financial reasons, not for fundamental faith reasons, and not for the reason we get out of this fake inauguration.

I read a long time ago that the upper classes live in the past, the lower classes live in the present, and the middle classes live in the future.  Part of the United States as a middle class country is the deep-rooted aspirationalism of its people.  Christian churches, especially Evangelical ones, have tapped into that, moulding themselves by and around that desire to move up in this world and ultimately tying upward movement in this life and the next.  Personally I think that this is ill-advised; it doesn’t cut it theologically, the results are in bad taste, and the payments are horrendous.

Barack Obama was elected once and again by an “upstairs-downstairs” coalition, which means his support came primarily from the top and the bottom of society.  That should give a clue where the middle class stands these days.  The hue and cry these days is to resolve income inequality, but the political system we have virtually guarantees that never happens.  As long as the upstairs keeps the downstairs happy with patronage, life is good for these people.

American churches need to wake up to the fact that our people are not, in general, as aspirational as they used to be.  Much of what used to pass for successful church doesn’t work any more.  Much is made of the social stances of churches driving people away, but I think a deeper problem is the simple fact that our upwardly-mobile message doesn’t resonate with people who are focused on just staying even, and often not even that.  As I explained to Jonathan Stone about our own church:

It’s not the first time that American Christianity has equated being a Christian with economic prosperity.  And it does attract people for whom moving up is a big deal.  But it’s a Faustian bargain, and with the increasingly rigid class stratification of American society Mephistopheles is coming to fulfil the contract.  It is there that the “Great Collapse” that Jonathan Stone speaks of will come, not only to the Church of God but to Evangelical Christianity in general.

In the meanwhile our secular society in general moves towards it own collapse, burdened by a society increasingly content to go on the dole, an elite with no idea as to how to make it productive again, and a ballooning debt to service with a weakened economy.  How the Church of God responds to that confluence of events will ultimately determine whether the Church of God has a future or not.

Pulling the Plug on Canterbury

DSCN4183Ever since the hapless Rowan Williams began his exit from the stage of Lambeth Palace, and the former oil executive Justin Welby began measuring the curtains, there has been a great deal of optimism about the future course of the Church of England.  Would there be a way of putting the Communion back together again?  Would the revisionists be sent packing from Albion’s state church?  Would the Evangelicals be triumphant in the end, as their numbers show they should?  Once again we see hopeful signs.

However, in the Anglican Communion, which has in the eyes of much of Christianity the reputation of boring, repetitious liturgy and even more boring homiletics, there’s never a dull moment, and even before the mitre was officially passed the word got out that Canon Jeffrey Johns, the poster child for full inclusion of LGBT people in the life of the Church of England, was being considered to succeed Welby++ in the see of Durham.  This has generated the predictable reaction from the Central African provinces and a cloud for those in the U.S. for whom recognition by Canterbury is the silver lining of ecclesiastical life.

Let’s start with the obvious: does anyone really believe that the relationship between Canon Johns and his partner is celibate?  The LGBT community is about a lot of things these days, but celibacy isn’t one of them: not for themselves, not for anyone else either.  If the Church of England seats him on the same throne that once held up N.T. Wright, it would only do so on that representation of celibacy.  What’s going to happen if, after this is done, he comes up with the lame “I lied” admission that’s so fashionable these days?

Even if that event doesn’t happen, the transformation of the Church of England into an irreproachable repository of the orthodox faith has more obstacles than somewhat.

The Church of England is, after all, a state church.  The brouhaha over women bishops brought threats that the state would take action to override the vote of the laity in the matter.  The debate over same-sex marriage in churches has brought similar threats, although these may be abated for the moment.  But face it: the Church of England was formed so that it would do the sovereign’s unBiblical bidding about a divorce, so why not do some more unBiblical biddings?

As far as Evangelicals and their numbers are concerned, back home in Palm Beach it wasn’t the number of people you knew, it was whether you knew the right people.  The LGBT community has, implicitly or explicitly, worked the system with that ethic, and it didn’t hurt that they were economically privileged in the first place.  Our political and other systems, awash in cash and influence-peddling, only have the form of popular choice; the deal is rigged far more than most will realise or admit.

This whole blather about the Evangelicals’ numbers doesn’t matter: the right people have decided, the rest of us just have to deal with the consequences.  Has everyone forgotten that the revisionists started out in TEC as a minority, well placed in seminaries? (That, BTW, is the message behind the Louis Giglio fiasco, which has led to responses like this. Personally I think it’s for the best; Christians don’t need to give either the UK or the US any more cred than they have to).

DSCN4185Anglicans are and have been for some time in a place to respond meaningfully to this, because they have a new orthodox centre in Africa and elsewhere.  The old colonial powers are doing their best on more than one level to curb their “progeny”, but the world is changing.  It’s time for Anglicans to pull the plug on Canterbury, to leave ethnocentric dreams behind and live in a world where the only blood line they’re concerned about is the one started by Jesus Christ himself.  And wasn’t that the idea to start with?

The Geniuses Commit Suicide

One the serious disadvantages of getting a secondary education in the late 1960’s and early 1970’s is that one was exposed to the first wave of “hippie radicals” to escape from our institutions of higher learning with a diploma.  In the intervening years the subsequent waves have been merrily doing their damage to the system and the students.  Only now are these people either retiring and/or dying in their sins, but the damage is done, as our culture’s current state will attest.

One of those was my freshman and sophomore English teacher.  He was, sad to say, enormously influential in my life.  I had him for two years as a student and one semester as a coach, and it took about five years for the damage to be undone.  (One silver lining: he did introduce me to Tolkien). I knew that what he was saying was depressing, but in the climate of the time it was hard to refute, especially in the milieu I was in.  Only when I left that milieu could I make my escape good.

My parents had a far lower impression of this man, and much of that came from the first parent-teacher conference they went to.  The basic problem (although he wouldn’t put it this way) was that I was insufficiently deconstructionistic to suit his fancy.  Somehow he conveyed this to my parents, who came back with their idea that I was very intelligent and did well elsewhere.  His response: yes, but geniuses commit suicide.

I am sure that many of my friends, especially on the left, find me frustrating in that I’m not an activist.  That was in part the result of my years in prep school; stuff like this will definitely sour you on the left.  The other part is that, in the following years, my contemporaries haven’t done anything but go from one unintrospective volte-face to another, and the subject of human intelligence is one of those.

One of the persistent whines we’ve had to listen to is that our schools have dumbed themselves down and are not producing what they should.  I read somewhere that the decline in SAT scores which started in the mid-1960’s wasn’t due to some instantaneous recess in our educational system, but due to the really smart people/achievers basically gave up due to social pressure.

As I’ve documented before, the 1960’s were a revolt against many things.  They were anti-scientific, anti-intellectual, certainly anti-“establishment”, and they were also anti-achievement.  The basic problem with human intelligence–and to some extent with motivation–is that is isn’t evenly distributed throughout society, and thus is inegalitarian as a result.  In a country where all men are created equal, that’s a problem, which explains much of the anti-intellectualism that has existed in our culture. That fact has also bedevilled the left’s quest for equality here and elsewhere, but social pressure is a powerful tool, and you can beat the nerds down if you can get a critical mass of people to do the beating.  Those conditions certainly existed in those times.

Sometime in the late 1970’s another critical mass of Boomers came to the realisation that their slouchy ways would guarantee future poverty, something they weren’t ready for.  Jimmy Carter had the bad taste to be in office then; they threw him out, elected Ronald Reagan, began the long march of deregulation and tax cuts, and the gifted gravitated towards the financial sector where they proved they were too smart for their own good and everyone else’s.

Today the pseudo-egalitarians have the upper hand, but instead of clarity we have more duplicity.  One the one hand we have the endless quest for “equality”: same-sex civil marriage, flattening of income distribution, and our endless fixation with socialisation, which is a way of levelling and controlling people without having to do any work.  On the other hand we’re obsessed with raw intelligence, its cultivation in people, and packing those with it off to the “right” schools (assuming they can swing the student loans) so they can use this intelligence to lead a people that no longer have to be led but herded.

We as Americans have never figured out whether we value intelligence or not, so we send mixed signals.  Today, the question is whether it’s worth the effort to really apply intelligence and achieve in an honest way in the face of the Byzantine legal, regulatory and “gotcha” culture.  That’s one reason why we have the mediocre political leadership we have.  And that’s why I, as much now as in the day, find it supremely problematic to make progress in our political and social systems given their equivocal nature and potential for unintended (and potentially disastrous) consequences given the duplicitous nature of the system itself.

Well, there’s one good thing now.  Geniuses don’t have to consider suicide any more.  They can just go on the dole.

Local Stiffness Matrix for Combined Beam and Spar Element With Axial and Lateral Linear Resistance

Our objective is to develop an element with the following characteristics:

  • Two-dimensional, two node element
  • Euler-Bernoulli beam theory
  • Axial stiffness (“spar” type element)
  • “Beam on elastic foundation” characteristic
  • Axial elastic resistance

Although it is doubtless possible to start with a single weak-form equation and develop the stiffness matrix, it is more convenient to develop the axial and bending local stiffness matrices separately and then to put them together with superposition.

Both spar and beam elements generally use two nodes, one at each end. For this derivation all of the constants (beam elastic modulus, moment of inertia, cross-sectional area and spring constants) will be assumed to be uniform the full length of the element. If one desires to model non-uniform beams, one can either develop an element with the desired non-uniformity or use more elements, and we see both in finite element analysis.

Let us start with the bending portion. The weak form of the equation for the fourth-order Euler-Bernoulli beam element is
\int_{x_{{1}}}^{x_{{2}}}\!{\it E1}\,{\it XI1}\,\left({\frac{d^{2}}{d{x}^{2}}}v(x)\right){\frac{d^{2}}{d{x}^{2}}}w(x)+gv(x)w(x)-v(x)t{dx}=0

where E1 is the Young’s modulus of the material and XI1 is the moment of inertia of the beam. The variable g represents the continuous spring constant along the length of the beam relative to the displacement of that beam, the “beam on elastic foundation.” The variable t is a uniform load along the beam. The equations were derived using Maple with the idea of the results used on FORTRAN 77, thus the naming convention of some of the variables. An explanation of the weak form, its derivation and the significance of w\left(x\right) and v\left(x\right) can be found in a finite element text such as this.

At this point we need to select appropriate weighting functions for the equation. For beam elements we choose weighting functions to satisfy the Hermite interplation of the two primary variables at local nodes 1 and 2, to wit

\Delta_{{2}}=-C_{{2}}-2\, C_{{3}}x_{{1}}-3\, C_{{4}}{x_{{1}}}^{2}
\Delta_{{4}}=-C_{{2}}-2\, C_{{3}}x_{{2}}-3\, C_{{4}}{x_{{2}}}^{2}

where \Delta_{1},\,\Delta_{3} are the “displacements” for nodes
1 and 2 and \Delta_{2},\,\Delta_{4} are the first derivative slopes
at these nodes.

This can be expressed in matrix form as follows:

\left[\begin{array}{cccc} 1 & x_{{1}} & {x_{{1}}}^{2} & {x_{{1}}}^{3}\\ \noalign{\medskip}0 & -1 & -2\, x_{{1}} & -3\,{x_{{1}}}^{2}\\ \noalign{\medskip}1 & x_{{2}} & {x_{{2}}}^{2} & {x_{{2}}}^{3}\\ \noalign{\medskip}0 & -1 & -2\, x_{{2}} & -3\,{x_{{2}}}^{2} \end{array}\right]\left[\begin{array}{c} C_{{1}}\\ \noalign{\medskip}C_{{2}}\\ \noalign{\medskip}C_{{3}}\\ \noalign{\medskip}C_{{4}} \end{array}\right]=\left[\begin{array}{c} \Delta_{{1}}\\ \noalign{\medskip}\Delta_{{2}}\\ \noalign{\medskip}\Delta_{{3}}\\ \noalign{\medskip}\Delta_{{4}} \end{array}\right]

Inverting the matrix, we have

\left[\begin{array}{c} C_{{1}}\\ \noalign{\medskip}C_{{2}}\\ \noalign{\medskip}C_{{3}}\\ \noalign{\medskip}C_{{4}} \end{array}\right]=\left[\begin{array}{cccc} {\frac{\left(3\, x_{{1}}-x_{{2}}\right){x_{{2}}}^{2}}{{x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}}} & {\frac{x_{{1}}{x_{{2}}}^{2}}{{x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}}} & {\frac{\left(x_{{1}}-3\, x_{{2}}\right){x_{{1}}}^{2}}{{x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}}} & {\frac{{x_{{1}}}^{2}x_{{2}}}{{x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}}}\\ \noalign{\medskip}-6\,{\frac{x_{{1}}x_{{2}}}{{x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}}} & -{\frac{\left(2\, x_{{1}}+x_{{2}}\right)x_{{2}}}{{x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}}} & 6\,{\frac{x_{{1}}x_{{2}}}{{x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}}} & -{\frac{\left(x_{{1}}+2\, x_{{2}}\right)x_{{1}}}{{x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}}}\\ \noalign{\medskip}3\,{\frac{x_{{1}}+x_{{2}}}{{x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}}} & {\frac{x_{{1}}+2\, x_{{2}}}{{x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}}} & -3\,{\frac{x_{{1}}+x_{{2}}}{{x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}}} & {\frac{2\, x_{{1}}+x_{{2}}}{{x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}}}\\ \noalign{\medskip}-2\,\left({x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}\right)^{-1} & -\left({x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}\right)^{-1} & 2\,\left({x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}\right)^{-1} & -\left({x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}\right)^{-1} \end{array}\right]\left[\begin{array}{c} \Delta_{{1}}\\ \noalign{\medskip}\Delta_{{2}}\\ \noalign{\medskip}\Delta_{{3}}\\ \noalign{\medskip}\Delta_{{4}} \end{array}\right]
Multiplying the result, we have for the coefficients
\left[\begin{array}{c} C_{{1}}\\ \noalign{\medskip}C_{{2}}\\ \noalign{\medskip}C_{{3}}\\ \noalign{\medskip}C_{{4}} \end{array}\right]=\left[\begin{array}{c} {\frac{\left(3\, x_{{1}}-x_{{2}}\right){x_{{2}}}^{2}\Delta_{{1}}}{{x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}}}+{\frac{x_{{1}}{x_{{2}}}^{2}\Delta_{{2}}}{{x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}}}+{\frac{\left(x_{{1}}-3\, x_{{2}}\right){x_{{1}}}^{2}\Delta_{{3}}}{{x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}}}+{\frac{{x_{{1}}}^{2}x_{{2}}\Delta_{{4}}}{{x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}}}\\ \noalign{\medskip}-6\,{\frac{x_{{1}}x_{{2}}\Delta_{{1}}}{{x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}}}-{\frac{\left(2\, x_{{1}}+x_{{2}}\right)x_{{2}}\Delta_{{2}}}{{x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}}}+6\,{\frac{x_{{1}}x_{{2}}\Delta_{{3}}}{{x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}}}-{\frac{\left(x_{{1}}+2\, x_{{2}}\right)x_{{1}}\Delta_{{4}}}{{x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}}}\\ \noalign{\medskip}3\,{\frac{\left(x_{{1}}+x_{{2}}\right)\Delta_{{1}}}{{x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}}}+{\frac{\left(x_{{1}}+2\, x_{{2}}\right)\Delta_{{2}}}{{x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}}}-3\,{\frac{\left(x_{{1}}+x_{{2}}\right)\Delta_{{3}}}{{x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}}}+{\frac{\left(2\, x_{{1}}+x_{{2}}\right)\Delta_{{4}}}{{x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}}}\\ \noalign{\medskip}-2\,{\frac{\Delta_{{1}}}{{x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}}}-{\frac{\Delta_{{2}}}{{x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}}}+2\,{\frac{\Delta_{{3}}}{{x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}}}-{\frac{\Delta_{{4}}}{{x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}}} \end{array}\right]

The weighting function in its complete form is thus

w={\frac{\left(3\, x_{{1}}-x_{{2}}\right){x_{{2}}}^{2}\Delta_{{1}}}{{x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}}}+{\frac{x_{{1}}{x_{{2}}}^{2}\Delta_{{2}}}{{x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}}}+{\frac{\left(x_{{1}}-3\, x_{{2}}\right){x_{{1}}}^{2}\Delta_{{3}}}{{x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}}}+{\frac{{x_{{1}}}^{2}x_{{2}}\Delta_{{4}}}{{x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}}}+

-6\,{\frac{x_{{1}}x_{{2}}\Delta_{{1}}}{{x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}}}x-{\frac{\left(2\, x_{{1}}+x_{{2}}\right)x_{{2}}\Delta_{{2}}}{{x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}}}x+6\,{\frac{x_{{1}}x_{{2}}\Delta_{{3}}}{{x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}}}x-{\frac{\left(x_{{1}}+2\, x_{{2}}\right)x_{{1}}\Delta_{{4}}}{{x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}}}x

3\,{\frac{\left(x_{{1}}+x_{{2}}\right)\Delta_{{1}}}{{x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}}}{x}^{2}+{\frac{\left(x_{{1}}+2\, x_{{2}}\right)\Delta_{{2}}}{{x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}}}{x}^{2}-3\,{\frac{\left(x_{{1}}+x_{{2}}\right)\Delta_{{3}}}{{x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}}}{x}^{2}+{\frac{\left(2\, x_{{1}}+x_{{2}}\right)\Delta_{{4}}}{{x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}}}{x}^{2}

-2\,{\frac{\Delta_{{1}}}{{x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}}}{x}^{3}-{\frac{\Delta_{{2}}}{{x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}}}{x}^{3}+2\,{\frac{\Delta_{{3}}}{{x_{{1}}}^{3}-3\,{x_{{1}}}^{2}x_{{2}}+3\, x_{{1}}{x_{{2}}}^{2}-{x_{{2}}}^{3}}}{x}^{3}-{\frac{\Delta_{{4}}}{{x_{{2}}}^{2}-2\, x_{{1}}x_{{2}}+{x_{{1}}}^{2}}}{x}^{3}

This breaks down in to weighting functions for each independent variable as follows:
\Phi_{1}=1-3\,{\frac{{\it \bar{x}}^{2}}{{\it he}^{2}}}+2\,{\frac{{\it \bar{x}}^{3}}{{\it he}^{3}}}

\Phi_{2}=2\,{\frac{{\it \bar{x}}^{2}}{{\it he}}}-{\it \bar{x}}-{\frac{{\it \bar{x}}^{3}}{{\it he}^{2}}}

\Phi_{3}=3\,{\frac{{\it \bar{x}}^{2}}{{\it he}^{2}}}-2\,{\frac{{\it \bar{x}}^{3}}{{\it he}^{3}}}

\Phi_{4}=-{\frac{{\it \bar{x}}^{3}}{{\it he}^{2}}}+{\frac{{\it \bar{x}}^{2}}{{\it he}}}

additionally assuming that


If we substitute these weighting functions into the weak form of the governing equations, perform the appropriate substitution, differentiation, integration and algebra, the first term results in the following stiffness matrix:

M=\left[\begin{array}{cccc} 12\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{3}}} & -6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}} & -12\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{3}}} & -6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}}\\ \noalign{\medskip}-6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}} & 4\,{\frac{{\it E1}\,{\it XI1}}{{\it he}}} & 6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}} & 2\,{\frac{{\it E1}\,{\it XI1}}{{\it he}}}\\ \noalign{\medskip}-12\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{3}}} & 6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}} & 12\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{3}}} & 6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}}\\ \noalign{\medskip}-6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}} & 2\,{\frac{{\it E1}\,{\it XI1}}{{\it he}}} & 6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}} & 4\,{\frac{{\it E1}\,{\it XI1}}{{\it he}}} \end{array}\right]

The second term (for the “elastic foundation”) yields the following stiffness matrix:
N=\left[\begin{array}{cccc} {\frac{13}{35}}\,{\it he}\, g & -{\frac{11}{210}}\,{\it he}^{2}g & {\frac{9}{70}}\,{\it he}\, g & {\frac{13}{420}}\,{\it he}^{2}g\\ \noalign{\medskip}-{\frac{11}{210}}\,{\it he}^{2}g & {\frac{1}{105}}\,{\it he}^{3}g & -{\frac{13}{420}}\,{\it he}^{2}g & -{\frac{1}{140}}\,{\it he}^{3}g\\ \noalign{\medskip}{\frac{9}{70}}\,{\it he}\, g & -{\frac{13}{420}}\,{\it he}^{2}g & {\frac{13}{35}}\,{\it he}\, g & {\frac{11}{210}}\,{\it he}^{2}g\\ \noalign{\medskip}{\frac{13}{420}}\,{\it he}^{2}g & -{\frac{1}{140}}\,{\it he}^{3}g & {\frac{11}{210}}\,{\it he}^{2}g & {\frac{1}{105}}\,{\it he}^{3}g \end{array}\right]

The combined local stiffness matrix for bending only is

K_{b}=\left[\begin{array}{cccc} 12\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{3}}}+{\frac{13}{35}}\,{\it he}\, g & -6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}}-{\frac{11}{210}}\,{\it he}^{2}g & -12\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{3}}}+{\frac{9}{70}}\,{\it he}\, g & -6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}}+{\frac{13}{420}}\,{\it he}^{2}g\\ \noalign{\medskip}-6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}}-{\frac{11}{210}}\,{\it he}^{2}g & 4\,{\frac{{\it E1}\,{\it XI1}}{{\it he}}}+{\frac{1}{105}}\,{\it he}^{3}g & 6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}}-{\frac{13}{420}}\,{\it he}^{2}g & 2\,{\frac{{\it E1}\,{\it XI1}}{{\it he}}}-{\frac{1}{140}}\,{\it he}^{3}g\\ \noalign{\medskip}-12\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{3}}}+{\frac{9}{70}}\,{\it he}\, g & 6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}}-{\frac{13}{420}}\,{\it he}^{2}g & 12\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{3}}}+{\frac{13}{35}}\,{\it he}\, g & 6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}}+{\frac{11}{210}}\,{\it he}^{2}g\\ \noalign{\medskip}-6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}}+{\frac{13}{420}}\,{\it he}^{2}g & 2\,{\frac{{\it E1}\,{\it XI1}}{{\it he}}}-{\frac{1}{140}}\,{\it he}^{3}g & 6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}}+{\frac{11}{210}}\,{\it he}^{2}g & 4\,{\frac{{\it E1}\,{\it XI1}}{{\it he}}}+{\frac{1}{105}}\,{\it he}^{3}g \end{array}\right]

The FORTRAN 77 code for this is as follows:

K(1,1) = 12/he**3*E1*XI1+13.E0/35.E0*he*g
K(1,2) = -6/he**2*E1*XI1-11.E0/210.E0*he**2*g
K(1,3) = -12/he**3*E1*XI1+9.E0/70.E0*he*g
K(1,4) = -6/he**2*E1*XI1+13.E0/420.E0*he**2*g
K(2,1) = -6/he**2*E1*XI1-11.E0/210.E0*he**2*g
K(2,2) = 4/he*E1*XI1+he**3*g/105
K(2,3) = 6/he**2*E1*XI1-13.E0/420.E0*he**2*g
K(2,4) = 2/he*E1*XI1-he**3*g/140
K(3,1) = -12/he**3*E1*XI1+9.E0/70.E0*he*g
K(3,2) = 6/he**2*E1*XI1-13.E0/420.E0*he**2*g
K(3,3) = 12/he**3*E1*XI1+13.E0/35.E0*he*g
K(3,4) = 6/he**2*E1*XI1+11.E0/210.E0*he**2*g
K(4,1) = -6/he**2*E1*XI1+13.E0/420.E0*he**2*g
K(4,2) = 2/he*E1*XI1-he**3*g/140
K(4,3) = 6/he**2*E1*XI1+11.E0/210.E0*he**2*g
K(4,4) = 4/he*E1*XI1+he**3*g/105

The vector for the last term is

T=\left[\begin{array}{c} 1/2\,{\it he}\, t\\ \noalign{\medskip}-1/12\,{\it he}^{2}t\\ \noalign{\medskip}1/2\,{\it he}\, t\\ \noalign{\medskip}1/12\,{\it he}^{2}t \end{array}\right]

and the FORTRAN for this is

te(1,1) = he*t/2
te(2,1) = -he**2*t/12
te(3,1) = he*t/2
te(4,1) = he**2*t/12

Now let us turn to the spar element part of the stiffness matrix. The weak form equation for this is
{\it E1}\, A\int_{0}^{{\it he}}\!\left({\frac{d}{dx}}w(x)\right){\frac{d}{dx}}y(x){dx}+w(x){\frac{d}{dx}}y(x)+\int_{0}^{{\it he}}\! w(x)cy(x){dx}-\int\! w(x)q{dx}=0

Here A is the cross-sectional area of the beam, c is a distributed axial spring constant along the spar, and q is a distributed axial force along the element. To integrate from 0 to he is no different than doing so from x_{1} to x{}_{2}, only the coordinates change.

In this case we select linear weighting functions, to wit

W_{1}=1-{\frac{x}{{\it he}}}
W_{2}={\frac{x}{{\it he}}}

If as before we do the substitutions and integrations, we end up with a local stiffness matrix for the spar element only as follows:
K_{s}=\left[\begin{array}{cc} {\frac{{\it E1}\, A}{{\it he}}}+1/3\, c{\it he} & -{\frac{{\it E1}\, A}{{\it he}}}+1/6\, c{\it he}\\ \noalign{\medskip}-{\frac{{\it E1}\, A}{{\it he}}}+1/6\, c{\it he} & {\frac{{\it E1}\, A}{{\it he}}}+1/3\, c{\it he} \end{array}\right]

FORTRAN code for this is

K1(1,1) = E1*A/he+c*he/3
K1(1,2) = -E1*A/he+c*he/6
K1(2,1) = -E1*A/he+c*he/6
K1(2,2) = E1*A/he+c*he/3

The right hand side vector is as follows:
T=\left[\begin{array}{c} 1/2\,{\it he}\, q\\ \noalign{\medskip}1/2\,{\it he}\, q \end{array}\right]

and the code for this is

fe1(1,1) = he*q/2
fe1(2,1) = he*q/2

Now we need to combine these. We note that there are three variables:

  • x displacement (spar element only)
  • y displacement (beam element only)
  • rotation (beam element only)

We thus construct a 6\times6 element with the rows and columns in the above order, repeated twice each way for the two nodes. Doing this results in the following local stiffness matrix:
K=\left[\begin{array}{cccccc} {\frac{{\it E1}\, A}{{\it he}}}+1/3\, c{\it he} & 0 & 0 & -{\frac{{\it E1}\, A}{{\it he}}}+1/6\, c{\it he} & 0 & 0\\ \noalign{\medskip}0 & 12\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{3}}}+{\frac{13}{35}}\,{\it he}\, g & -6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}}-{\frac{11}{210}}\,{\it he}^{2}g & 0 & -12\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{3}}}+{\frac{9}{70}}\,{\it he}\, g & -6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}}+{\frac{13}{420}}\,{\it he}^{2}g\\ \noalign{\medskip}0 & -6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}}-{\frac{11}{210}}\,{\it he}^{2}g & 4\,{\frac{{\it E1}\,{\it XI1}}{{\it he}}}+{\frac{1}{105}}\,{\it he}^{3}g & 0 & 6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}}-{\frac{13}{420}}\,{\it he}^{2}g & 2\,{\frac{{\it E1}\,{\it XI1}}{{\it he}}}-{\frac{1}{140}}\,{\it he}^{3}g\\ \noalign{\medskip}-{\frac{{\it E1}\, A}{{\it he}}}+1/6\, c{\it he} & 0 & 0 & {\frac{{\it E1}\, A}{{\it he}}}+1/3\, c{\it he} & 0 & 0\\ \noalign{\medskip}0 & -12\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{3}}}+{\frac{9}{70}}\,{\it he}\, g & 6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}}-{\frac{13}{420}}\,{\it he}^{2}g & 0 & 12\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{3}}}+{\frac{13}{35}}\,{\it he}\, g & 6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}}+{\frac{11}{210}}\,{\it he}^{2}g\\ \noalign{\medskip}0 & -6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}}+{\frac{13}{420}}\,{\it he}^{2}g & 2\,{\frac{{\it E1}\,{\it XI1}}{{\it he}}}-{\frac{1}{140}}\,{\it he}^{3}g & 0 & 6\,{\frac{{\it E1}\,{\it XI1}}{{\it he}^{2}}}+{\frac{11}{210}}\,{\it he}^{2}g & 4\,{\frac{{\it E1}\,{\it XI1}}{{\it he}}}+{\frac{1}{105}}\,{\it he}^{3}g \end{array}\right]

or in code

K2(1,1) = E1*A/he+c*he/3
K2(1,2) = 0
K2(1,3) = 0
K2(1,4) = -E1*A/he+c*he/6
K2(1,5) = 0
K2(1,6) = 0
K2(2,1) = 0
K2(2,2) = 12/he**3*E1*XI1+13.E0/35.E0*he*g
K2(2,3) = -6/he**2*E1*XI1-11.E0/210.E0*he**2*g
K2(2,4) = 0
K2(2,5) = -12/he**3*E1*XI1+9.E0/70.E0*he*g
K2(2,6) = -6/he**2*E1*XI1+13.E0/420.E0*he**2*g
K2(3,1) = 0
K2(3,2) = -6/he**2*E1*XI1-11.E0/210.E0*he**2*g
K2(3,3) = 4/he*E1*XI1+he**3*g/105
K2(3,4) = 0
K2(3,5) = 6/he**2*E1*XI1-13.E0/420.E0*he**2*g
K2(3,6) = 2/he*E1*XI1-he**3*g/140
K2(4,1) = -E1*A/he+c*he/6
K2(4,2) = 0
K2(4,3) = 0
K2(4,4) = E1*A/he+c*he/3
K2(4,5) = 0
K2(4,6) = 0
K2(5,1) = 0
K2(5,2) = -12/he**3*E1*XI1+9.E0/70.E0*he*g
K2(5,3) = 6/he**2*E1*XI1-13.E0/420.E0*he**2*g
K2(5,4) = 0
K2(5,5) = 12/he**3*E1*XI1+13.E0/35.E0*he*g
K2(5,6) = 6/he**2*E1*XI1+11.E0/210.E0*he**2*g
K2(6,1) = 0
K2(6,2) = -6/he**2*E1*XI1+13.E0/420.E0*he**2*g
K2(6,3) = 2/he*E1*XI1-he**3*g/140
K2(6,4) = 0
K2(6,5) = 6/he**2*E1*XI1+11.E0/210.E0*he**2*g
K2(6,6) = 4/he*E1*XI1+he**3*g/105

As long as all of the elements line up along the x-axis, we are done. But we know that this cannot always be the case. So we need to effect a rotation of the local stiffness matrix. Since each element can be either oriented differently, of different length or both, we need
to rotate the local stiffness matrix before inserting it into the global one. The rotation matrix is
G=\left[\begin{array}{cccccc} {\it cosine} & {\it sine} & 0 & 0 & 0 & 0\\ \noalign{\medskip}-{\it sine} & {\it cosine} & 0 & 0 & 0 & 0\\ \noalign{\medskip}0 & 0 & 1 & 0 & 0 & 0\\ \noalign{\medskip}0 & 0 & 0 & {\it cosine} & {\it sine} & 0\\ \noalign{\medskip}0 & 0 & 0 & -{\it sine} & {\it cosine} & 0\\ \noalign{\medskip}0 & 0 & 0 & 0 & 0 & 1 \end{array}\right]

where sine and cosine are the angles of the elements from the x-axis. To effect a rotation, we need to first premultiply the matrix $K$ by the inverse of G and then postmultiply the result by G. That process is somewhat simplified by the fact that G is orthogonal; thus, its inverse and transpose are identical. Going through that process, the rotated local stiffness matrix is (in code only; we have overwhelmed WordPress’ LaTex conversion capability):

Kglobal(1,1) = cosine**2*(E1*A/he+c*he/3)+sine**2*(12/he**3*E1*XI1
Kglobal(1,2) = cosine*(E1*A/he+c*he/3)*sine-sine*(12/he**3*E1*XI1+
Kglobal(1,3) = -sine*(-6/he**2*E1*XI1-11.E0/210.E0*he**2*g)
Kglobal(1,4) = cosine**2*(-E1*A/he+c*he/6)+sine**2*(-12/he**3*E1*X
Kglobal(1,5) = cosine*(-E1*A/he+c*he/6)*sine-sine*(-12/he**3*E1*XI
Kglobal(1,6) = -sine*(-6/he**2*E1*XI1+13.E0/420.E0*he**2*g)
Kglobal(2,1) = cosine*(E1*A/he+c*he/3)*sine-sine*(12/he**3*E1*XI1+
Kglobal(2,2) = sine**2*(E1*A/he+c*he/3)+cosine**2*(12/he**3*E1*XI1
Kglobal(2,3) = cosine*(-6/he**2*E1*XI1-11.E0/210.E0*he**2*g)
Kglobal(2,4) = cosine*(-E1*A/he+c*he/6)*sine-sine*(-12/he**3*E1*XI
Kglobal(2,5) = sine**2*(-E1*A/he+c*he/6)+cosine**2*(-12/he**3*E1*X
Kglobal(2,6) = cosine*(-6/he**2*E1*XI1+13.E0/420.E0*he**2*g)
Kglobal(3,1) = -sine*(-6/he**2*E1*XI1-11.E0/210.E0*he**2*g)
Kglobal(3,2) = cosine*(-6/he**2*E1*XI1-11.E0/210.E0*he**2*g)
Kglobal(3,3) = 4/he*E1*XI1+he**3*g/105
Kglobal(3,4) = -sine*(6/he**2*E1*XI1-13.E0/420.E0*he**2*g)
Kglobal(3,5) = cosine*(6/he**2*E1*XI1-13.E0/420.E0*he**2*g)
Kglobal(3,6) = 2/he*E1*XI1-he**3*g/140
Kglobal(4,1) = cosine**2*(-E1*A/he+c*he/6)+sine**2*(-12/he**3*E1*X
Kglobal(4,2) = cosine*(-E1*A/he+c*he/6)*sine-sine*(-12/he**3*E1*XI
Kglobal(4,3) = -sine*(6/he**2*E1*XI1-13.E0/420.E0*he**2*g)
Kglobal(4,4) = cosine**2*(E1*A/he+c*he/3)+sine**2*(12/he**3*E1*XI1
Kglobal(4,5) = cosine*(E1*A/he+c*he/3)*sine-sine*(12/he**3*E1*XI1+
Kglobal(4,6) = -sine*(6/he**2*E1*XI1+11.E0/210.E0*he**2*g)
Kglobal(5,1) = cosine*(-E1*A/he+c*he/6)*sine-sine*(-12/he**3*E1*XI
Kglobal(5,2) = sine**2*(-E1*A/he+c*he/6)+cosine**2*(-12/he**3*E1*X
Kglobal(5,3) = cosine*(6/he**2*E1*XI1-13.E0/420.E0*he**2*g)
Kglobal(5,4) = cosine*(E1*A/he+c*he/3)*sine-sine*(12/he**3*E1*XI1+
Kglobal(5,5) = sine**2*(E1*A/he+c*he/3)+cosine**2*(12/he**3*E1*XI1
Kglobal(5,6) = cosine*(6/he**2*E1*XI1+11.E0/210.E0*he**2*g)
Kglobal(6,1) = -sine*(-6/he**2*E1*XI1+13.E0/420.E0*he**2*g)
Kglobal(6,2) = cosine*(-6/he**2*E1*XI1+13.E0/420.E0*he**2*g)
Kglobal(6,3) = 2/he*E1*XI1-he**3*g/140
Kglobal(6,4) = -sine*(6/he**2*E1*XI1+11.E0/210.E0*he**2*g)
Kglobal(6,5) = cosine*(6/he**2*E1*XI1+11.E0/210.E0*he**2*g)
Kglobal(6,6) = 4/he*E1*XI1+he**3*g/105

The use of “sine” and “cosine” for the trigonometric functions makes it possible to compute these once for each matrix, thus speeding up computations.

One possible application of such a element is with driven piles or deep foundations in soil; the element can be used for both axial and flexural loads. The biggest problem is that the soil response is never linear, so they cannot be used in a “straightforward” fashion, but iteratively.

Some Thoughts on the "Three Streams" Business

I’ve thought about writing this for some time, but Stand Firm in Faith is doing what they do best: standing firm, in this case against the “Three Streams” concept of Anglicanism.  Since I have, indirectly, been accused of holding this idea–and more recently gotten myself bogged down in an unedifying debate on the subject of the origins and nature of Anglicanism–and at the risk of starting the blogger’s equivalent of Groundhog Day again, I’d like to say a few things about this.

First, I don’t think that the composite nature of Anglicanism is the result of a conscious effort on the part of its founders.  The whole beginning of the Church of England is a messy, complicated affair that does nothing for the self-proclaimed role of the English-speaking peoples as the human race’s guardians of liberty.  It was in fact a brutal, zig-zag process which cost many of its participants on both sides their lives.  The result was a church basically Reformed in doctrine but with a number of residual “outs” that would either enrich it or come back to haunt it, depending upon how you look at the situation.

Second, the lack of human intentionality doesn’t preclude the fact that God is working in a process even when it looks to us to be flawed or not in a “straight line”.  Too much of the discussion in the Christian world centres around how this or that tradition, institution or doctrinal system is “seamlessly” descended both from above and from the origins of the faith.  The Orthodox, for example, would like for you to think that the Apostles were crossing themselves with three fingers and chanting the Thrice-Holy Hymn (with or without the additions of the Peter the Fuller) before Acts 2 ends, but we know things are more complicated than that.  Part of the nature of the creation is that created beings are imperfect, but that imperfection is the source of their free will.  In this time where everything is a “perfect package deal” that gets lost, but it doesn’t change reality.

Third, the adoption of an episcopal form of government was a strategic mistake from the standpoint of having a truly Reformed church.  Calvin himself commended the Presbyterian form of government, not only because it squared with his concept of what the church was, but also because it represented a clean break with Roman Catholicism, a break that he, a Frenchman in a country where Catholicism was his major opponent, felt a greater necessity for than those in a country where the Roman Catholic church had been effectively nationalised.  I think he knew that, if the bishops were allowed to hang around, sooner or later someone would get the idea that we should start drifting towards Rome.  It took three centuries for that to happen, but happen it did.  Calvin may have been wrong about many things, but he wasn’t stupid.

There were those in England who agreed with Calvin, and much of the unrest in the seventeenth century leading up to Cromwell stemmed from that agreement.  There was also the example of the Scots, who did adopt Calvin’s model.  But if there’s one thing the English hate just about more than anything else, it’s being upstaged by the Scots.

Turning from that unpleasant thought, we need to consider another aspect of the “three streams” analogy that gets overlooked: what happens when the streams overflow their banks.  Much of the impact of Anglicanism on Christianity in general takes place outside of the Anglican confine, and in turn those influences have come back to the Anglican world in one fashion or another.  Other than the residual influence of non-Anglican churches which come out of an Anglican culture (I think of the West Indies at this point) the biggest overflow is the Wesleyan movement.  John Wesley never intended to be anything than an Anglican, and his idea is deeply rooted in Anglicanism, but ultimately it had to go outside of the confines of the Anglican world to flower.  When it did, Protestant Christianity found the strongest alternative to a purely Reformed construct that it has ever had, and of course modern Pentecost came out of the Wesleyan tradition.

Finally, most discussions of streams and what centre on doctrine when the key question is ecclesiology.  I’ve written about this before, but the question we must answer is this: is the church a formal mediator between man and God?  And can the church ultimately make divinely authoritative pronouncements on the meaning and definition of our faith? These, more than anything, are the questions that separate churches which profess and call themselves Protestant and Roman Catholicism.  Anglo-Catholicism is little more than a formalistic spirituality if it does not affirm that it too can bind and loose in the same sense as the Roman Catholic Church, and Anglo-Catholics are not univocal on this.

So there we are.  How to sort this out in one ecclesiastical “structure” where revisionists really make a mess out of things and have the upper hand in many high places is the ongoing challenge in the Anglican-Episcopal world.  If we keep our eyes on the essentials, and realise that the messy origins neither prevent progress nor offer perfect uniformity, we’d go a long way towards realising God’s plan for the Anglican world, if not ours.

Civil Marriage Gets It In The End

The top end of the income scale, that is:

A 39.6% rate applies to income above a certain threshold (specifically, income in excess of the “applicable threshold” over the dollar amount at which the 35% bracket begins). The applicable threshold is $450,000 for joint filers and surviving spouses, $425,000 for heads of household,  $400,000 for single filers,  and $225,000 (one-half of the otherwise applicable amount for joint filers) for married taxpayers filing separately. These dollar amounts are inflation-adjusted for tax years after 2013.

COMMENT: There is  a MAJOR marriage penalty here. Two single people living together would get two $400,000 exemptions (one each). A married couple gets hit when combined income exceeds $450,000. Perhaps some of those same-sex couples that are married under state law will not be happy now if the Supreme Court rules that they should be subject to the same tax rules as other married persons.

Looks like some high rolling gay and lesbian couples may be in for a rude awakening if they get “victory” in court, as will their heterosexual counterparts.

This, however, illustrates a point I’ve made for a long time: the “rights” of civil marriage are dicey at best, and no place are they dicier than the tax code.  It doesn’t take much for the “benefits” of civil marriage to turn into liabilities.  It’s entirely possible that those in the higher income brackets will start looking at civil marriage like those at the bottom generally do, i.e., something they can’t afford.

Stuff like this is why I’ve advocated abolishing civil marriage for a long time.  If people would look at the law less romantically and more realistically, our political life would be vastly different–and, IMHO, better.  Until then, we’ll never have change we really can believe in.

Going Over a Lower Cliff

Rubin nails it on this one:

For those waking up this morning (January 2) wondering if the House passed the Senate bill, the answer is yes, they did. Thus, the country goes over the tax portion of the fiscal cliff, albeit a lower cliff than it was a few days ago.

Other fiscal cliffs remain – the debt ceiling cliff is coming in a month or two, the sequester cliff in March (the current bill puts off the automatic sequester cuts for two months), the farm bill cliff in September, and the expiration of jobless benefits in December. So the spectacle continues.

Chairman Mao stirred things up to foment a perpetual revolution.  The system we have is designed to foment a perpetual crisis.