Remembering the Anti-Moon Luddites

Today, of course, is the fiftieth anniversary of Neil Armstrong’s first steps on the moon–“one giant leap for mankind,” to be sure.  It was a great accomplishment and deserves to be remembered.

It’s easy to forget, however, that at the time there were many–especially on the left–who believed that the whole enterprise was a mistake, that the money we spent to put Armstrong, Buzz Aldrin (and of course, Michael Collins, commemorated the following year by Jethro Tull in their album Benefit) would have been better spent on feeding the poor and rectifying social injustices.

And in a sense the years that followed this achievement were the time when real science died in this country.  As I noted earlier this year:

But by the time Armstrong and Aldrin set foot on the moon, the mood had changed. The 1960’s were a decidedly Luddite time; technology was blamed for despoiling the environment and creating the “few minutes to midnight” atmosphere of the Cold War. Those who plied their trade in technology were “nerds.” The space program collapsed and the aerospace industry went with it. A new generation turned away from technology to more “relevant” (and easier way up) professions such as law and finance. Instead of landing on Mars in 1986, we were in angst (something we’ve gotten good at) over the explosion of the Challenger.


Our home in Palm Beach. It was located on the old “Dodge Estate,” one of the last of the large estates to be broken up (Donald Trump’s Mar-a-Lago is an example of one that is still intact.) Built in the late 1950’s, it survived the hurricanes that were reasonably frequent during the years we lived in Palm Beach (we experienced two the first summer we lived there.) All of the windows were fitted with shutters (as shown here) or had a metal shield that could be fitted for a blow. This obviated the need to strip forests for plywood every time a hurricane arrived. Note also the ficus hedge running along the street. Using a hedge to both close in the yard and to obscure the view of the property (they’re generally higher now than they were then) is fairly common in Palm Beach. After living with this, being forced into the “open yard” mould so common in the U.S. (especially in the South) just doesn’t quite cut it.

The space program had many technological spinoffs that enhanced life here on earth.  But when we have the same old “zero-sum” mentality about this, we’ll end up getting nowhere, and in the long run shortchanging those we profess to help.

And where was I when the first step was taken?  In Palm Beach, of course.  Behind the balcony of our house (right) was my brother’s room, where we witnessed history on his black and white television.


NASA Glenn Research Center in Cleveland, Ohio USA — Construction and architecture

NASA John H. Glenn Research Center at Lewis Field is a NASA center, located within the cities of Brook Park and Cleveland between Cleveland Hopkins International Airport and the Cleveland Metroparks’s Rocky River Reservation, with a subsidiary facility in Sandusky, Ohio. Glenn Research Center is one of ten major NASA field centers, whose primary mission is to develop science and technology for use in aeronautics and […]

via NASA Glenn Research Center in Cleveland, Ohio USA — Construction and architecture

A State of Being


Helene Tuchbreiter Portrait by Antonio Sereix Photo Reproduction by John Haynesworth

Helene Tuchbreiter was one of Palm Beach’s most prominent socialites in her time.  She made an impact both on the social scene and on the people immediately around her.

My own recollection of her, however, was more prosaic: she was one of my mother’s best friends during the years we lived in Palm Beach.  This was made more real by an experience they both shared: the founding of the Church Mouse thrift shop.

Both of them were members at Bethesda-by-the-Sea Episcopal Church and also members of the St. Mary’s Ladies’ Guild. (or circle, as it’s called in some churches.)  Each year the guild had a “Rummage Mart” at the church to raise money for its charitable activities.  Helene was the Guild’s president. In the Spring of 1968 the Vestry of the church informed the Guild that it wasn’t right to have the sale on church property.  In support of their position they cited such Bible verses as “Jesus went into the Temple Courts and began to drive out those who were selling, Saying as he did so: “Scripture says–‘My House shall be a House of Prayer’; but you have made it ‘a den of robbers.’” (Luke 19:45-46)  This didn’t sit well with the ladies of the Guild, who thought all along they were doing something good.  The Vestry, flush with its new found knowledge of the Scriptures (they probably really wanted to keep the riff-raff off of the church grounds) stuck to its guns.  So the Guild had to find an alternative.

Helene, unwilling to be sidetracked in the mission to do good (and probably unwilling to be outdone by sellers of shirts), took the initiative and led the Guild to start a thrift shop off of the church grounds.  The Rector, Dr. Hunsdon Cary, was sceptical about the concept; he told Helene that they would probably end up poor as church mice. Beyond the absurdity of anything in Palm Beach experiencing such poverty, this doubtless spurred Helene and the others to make it work.


Left to right: Roy Tuchbreiter, my mother, father, and Helene, at a Christmas fête in Palm Beach in the late 1960’s. (Photo courtesy Bert and Richard Morgan, Palm Beach) Note: Some of the information for this article was taken from Helene Tuchbreither’s own copy of The Bulletin of Bethesda-by-the-Sea, 9 April 1991, complete with her own tart comments.

Dr. Cary’s quip also gave the enterprise its name; in March 1970 the Church Mouse thrift shop opened. It was soon forced to move because the building was being torn down for the Publix market in Palm Beach (that was another great controversy as well.) It moved to its second location at 101 North County Road. My mother kept the books for the Church Mouse during its early days.

In 1987 it moved to two locations, one at 374 South County Road and the other in West Palm Beach. The store was and is to this day a success, as good example of any of taking lemons and making lemonade.

Years later, while we were preparing for my mother’s estate sale, the portrait reproduction above was found in my mother’s things.  The individual who was organising the sale asked me the question, “What did she do?”  My response to him was, “In Palm Beach, it isn’t a matter of what you do, it’s what you are,” and went through the story you have here.

This state of affairs, however, is not unique to socialites.  When God called Moses to lead his people out of bondage, Moses asked the obvious question:

“Behold, I am going to the sons of Israel, and I shall say to them, ‘The God of your fathers has sent me to you.’ Now they may say to me, ‘What is His name?’ What shall I say to them?” And God said to Moses, “I AM WHO I AM”; and He said, “Thus you shall say to the sons of Israel, ‘I AM has sent me to you.'” And God, furthermore, said to Moses, “Thus you shall say to the sons of Israel, ‘The LORD, the God of your fathers, the God of Abraham, the God of Isaac, and the God of Jacob, has sent me to you.’ This is My name forever, and this is My memorial-name to all generations.” (Exod 3:13-15 NAS)

This abstract sounding answer had then and has now an important point.  We as human beings have a habit of defining everyone and everything by what they do.  The gods of ancient peoples were a reflection of that; every one of them had a speciality task.  But the God who commissioned Moses and later sent his own Son is beyond that: he is not defined by what he does, but by what he is and moreover that he exists: “For in him was created all that is in Heaven and on earth, the visible and the invisible–Angels and Archangels and all the Powers of Heaven. ”  (Col 1:16)  Jesus himself underscored his own nature and that of his Father when “Jesus said to them, ““In truth I tell you,” replied Jesus, “before Abraham existed I was.”” (John 8:58 NAS)

Our habit of defining ourselves and others by what we do is worse than ever in this performance based world we live in.  Helene Tuchbreiter — the preacher’s kid from Montgomery, Alabama, who went on to Palm Beach — has gone on for the last time to meet the great “I Am.”  Of the results of this encounter, we do not know, but we do know that before we do anything else we must follow God through his Son Jesus Christ, and then what we are will far surpass anything we can do — in this life and the life to come.

For more information click here.

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.

Why the Spanish Civil War is Important


A Spartan 7W, 7W-27, NC-17658. Manufactured by the Mid-Continent Aircraft Co., Tulsa, OK.  It looks suspiciously like the fighers that would dominate the skies during World War II.

At many of the aviation events that Chet organised, there were representatives of the powers that were soon to be in World War II, looking at the technology in front of them.  For example, at the 1934 Langley Day competition, in attendance were “Lieutenant Colonel Paolo Sbernadori from the Italian Embassy and Captain G.R.M. Reid from the British Embassy.”  Sooner than that, however, the two countries were involved in a “proxy” war: the Spanish Civil War, which in my opinion encapsulated most of the conflict that dominated the last century better than any other conflict.  Almost every ideology that dominated the century was represented there, either by Spanish adherents, foreign ones, or both.  And the combination of the conflict’s intensity and the tendency of the participants to romanticise their own cause and demonise their opponents’ certainly has lessons for our own polarised society today.

Probably the best single volume work on the subject in English is Hugh Thomas’ The Spanish Civil War.  He later acted as an adviser to Margaret Thatcher.  Most of what follows is derived from this work.

The existence of Spanish Latin America, from the Rio Bravo del Norte to the Tierra del Fuego–and beyond–is a testament to Spain as a world power for three centuries.  Napoleon’s invasion, with the loss of most of the American colonies, put it into more than a century of instability, ranging from absolute monarchy to constitutional monarchy to succession disputes (the Carlists) to the dictatorship of Primo de Rivera and finally to the Spanish Republic, which was established in 1931.

Through all of this, like France and Italy, Spain was a country with a wide variety of political parties, a system which tended towards fragmentation.  On the left were the Socialists, Anarchists, Communists (whose role increased as the war progressed) and other parties supporting the Republic.  On the right were Catholic parties (CEDA,) Monarchists and Carlists, Falangists and Agrarians.  There were some parties in the centre.  Complicating the scene (then and now) were the regional parties, primarily the Catalan and Basque parties, which themselves had an ideological range.  The one thing that Spanish parties had in common was a intensity of commitment to their cause that was extremely bore-sighted, first figuratively and soon literally in the war.

Most Americans will be surprised that Anarchism was a serious political movement, associating it as a fringe terrorist group involved with the assassination of President William McKinley.  In Spain it certainly was serious; the idea that we didn’t have to have a government had traction.  As Thomas explains:

To these a great new truth seem to have been proclaimed.  The State, being based upon ideas of obedience and authority, was morally evil.  In its place, there should be self-governing bodies–municipalities, professions, or other societies–which would make voluntary pacts with each other.  Criminals would be punished by the censure of public opinion.

The last point indicates that they were waiting for the advent of social media…the Anarchists on the one hand and the Socialists and Communists on the other had a great deal of bad blood between them going back to Marx and Bakunin, and this conflict bedeviled the Republic’s war effort when crunch time came.

With a Republic came a constitution, and at this point the Republican-Socialist majority made a strategic error: they decided to make the document a political one, embodying their own idea rather than creating a document acceptable to a broad range of Spaniards.  No where was that more evident than in its anticlerical clauses regarding the Catholic Church: religious education was ended, the Jesuits were banished, no more payment of salaries to priests (which were compensation for the seizure of the Church’s lands in the last century,) etc.   Overplaying one’s hand is a hallmark of religious conflicts; that was certainly the case in France, but in Spain the shoe was on the other foot.  One tireless advocate of these measures–even in face of opposition in his own coalition–was Prime Minister Manuel Azaña, who would play such a large role in the coming civil war.

Some of Azaña’s confidence that he would succeed in his quest–a quest whose genesis came from his own bad experiences in the Catholic educational system–came from the desultory way in which Spaniards related to the Church.  In 1931 only about a third of Spaniards were practicing Catholics, this in the home country of the Inquisition.  But under that low level, Spaniards wrapped their identity as such with the Church, and same Church was an instrument of social justice in many instances.  In their hard-line anti-clerical policies Azaña and his allies made unnecessary enemies which would come back to haunt them on the battlefield.

The next four years were times of conflict and instability that rivalled France’s Fourth Republic (to say nothing of postwar Italy.)  The elections of February 1936 brought a strong majority to the Republican Popular Front.  The right felt it had been cornered.  In July, part of the military rose at two ends of the Republic: in Spanish Morocco and the Canary Islands, under Francisco Franco, and in the North, under Emilio Mola.  The Spanish Civil War had begun.

From a military standpoint, as was the case with its American counterpart, the war was the steady advance of one side (in this case the Nationalists, eventually under Franco) and the steady retreat of the other (the Republicans, with Azaña as its president at the start.  As also with that war, the details in between were complicated, and only a cursory summary can be done here.

The basic reason why the Nationalists won the Spanish Civil War was that their military organisation was superior and coherent.  The Nationalists had a real army; in the early stages, the Republicans had a collection of political militias.  Only as the war progressed did Soviet and Communist influence help to weld the Republican military together, and by then it was too late.  This was also reflected politically; the Anarchists, Socialists, Communists, Catalan and Basque nationalists and other made for a fragmented scene that consistently undermined the Republic’s attempts at a united front.  They spent a great deal of energy fighting each other, and this contributed to the Republic’s defeat.  That result is always the great “Antifa” fear, one that dominates their thinking to this day.

The Spanish Civil War became a proxy war for the various powers in Europe, themselves preparing for the much greater war that was coming.  It wasn’t a straightforward or uniform process.  Starting with the Nationalists, the one power that was “all in” for Franco was Italy, who contributed more support than just about anyone else.  Much of this support left something to be desired of; Franco, for example, wished that he could sent the Italian ground troops back, finding them as useless as Hitler shortly did.  Hitler and the Germans used the Condor Legion as a military experiment for their equipment and strategy, which they put to use in Poland and France.  Their support of the Nationalists was not entirely enthusiastic; at one point Hitler wished that the Republicans would win to crush the Catholic Church, for him a desired result.

The Republic’s foreign aid was, if anything, more desultory than the Nationalists.  The power that corresponded to Italy for the Republic was the Soviet Union, although their aid was sidetracked from time to time by events at home, namely Stalin’s purges and then the pact with Germany.  They also used that aid to forward the Communist’s status in the Republic, usually at the expense of the Anarchists.  As far as Britain and France were concerned, the 1930’s were the “decade of indecision.”  As one right-wing French paper observed, how was France (then under Leon Blum) supposed to help the Spanish Republic if they couldn’t keep the Germans from reoccupying the Rhineland?  Ultimately these two lead the Non-Intervention movement, which included Germany and Italy, and this amounted to having two foxes guard two chicken coops.  In any case their lack of support for the Republic was one cause of its defeat.

But the Spanish Civil War was the golden age of “volunteers,” from all over Europe and the US.  Not even World War II excited intellectuals and writers from these places like this conflict did, and many of them fought–and died–for the Republic.  The International Brigades were the stuff of legend, a phenomenon recently replicated in Syria (which is a good recent analogy for the brutality of the Spanish conflict, at the opposite end of the Mediterranean.)

Mentioning brutality brings up the subject of the atrocities, and there were plenty.  Most people think of Guernica, whose bombing was a complete waste in every sense of the word.  (Guernica is the sacred city of the Basques, with its tree, the way the Basques look at it echoes something out of J.R.R. Tolkien.)  The majority of the brutality, however, was more direct and personal.  The rule on both sides was to shoot first, no questions later.  The difference between the two sides was the context of the brutality.  The Republicans kicked off things with a massacre of Catholic religious and the destruction of churches.  Later the Communists would import techniques of torture and execution from the Soviet Union.   In executing most of the pre-war right-wing leadership, the Republicans did Franco a favour by clearing the field of most of his potential political rivals when the war was done.  The Nationalists did their dirty work, as with the fighting, in a more methodical manner.  The brutality of each side sickened their respective intellectuals, which is more than one could say for their foreign counterparts.

Although the Nationalists became the champions of Catholic religion in Spain, that process was not instantaneous.  Franco was indifferent to the faith (his wife, however, was not.)  The Falange was largely secular; the existence of a secular right was certainly a reality in those days and is becoming one again with the alt-right movement.  The use of Catholicism to bind the Nationalists together was a process encouraged by the conflict, another by-product of the Republic’s overreach in that regard.

Franco’s ultimate victory–just before the outbreak of World War II–was followed by his neutrality.  For all of his faults, Franco had no territorial ambitions beyond Spain and its existing colonies (Morocco had furnished him some of his toughest fighters) and was a profoundly cautious man.  Hitler tried to get him to join the Axis, but his was one of the few people who stiffed Hitler and got away with it.

After Franco’s death, Spain finally got a constitutional monarchy with a Republican political bent.  Franco got what the Romans called damnatio memoriae, he cannot be mentioned.   For the most part the social issues that helped push Spain leftward have been resolved in the modern welfare state, with the good and bad that goes with that.  But issues such as Basque and Catalan separatism–and of course the perennial issue of the Catholic Church–still remind us that the issues for which 600,000 people died are still very much with it.

And not just for Spain either.  It is hard to convey the relevance of the Spanish Civil War in a piece this short.  The polarisation, the heated rhetoric, the refusal for anyone to see the broader picture–all of these things are very much with us, and if we do not take some lessons from Spain’s experience–the most riveting single story of the Twentieth Century–than we risk having our own nation go down the same road.