11.1 Introduction

A mechanical member subjected to a force system perpendicular to its axis is called beam. The moment of force system at any point on the beam is perpendicular to the axis of the beam. Stress developed in the beam due to bending moment is called bending stress. The bending stress developed on any section of the beam is normal to the section. The stress developed is such that the resultant reaction on the section is zero.

In general, the shear force and bending moment both are present at a section of a beam. In particular, if the shear force is absent and only bending moment is present on a beam, the beam is said to be in pure bending. Due to pure bending, the beam bends in the form of a circular arc as shown in Figure 11.1. The bending moment, which develops curvature upwards, is also called as sagging moment (Figure 11.1b). Similarly, the moment, which produces curvature downwards is called hogging moment (Figure 11.1a). It is clear from Figure 11.1 that the fibres toward the center of curvature are compressed and the outer fibres are elongated. The stresses developed in both the regions are shown in the Figure 11.1.

We conclude with a discussion of two critical issues: global terrorism and global warming. Both of them threaten to harm huge numbers of people, and both have roots far back in the human past. The greatest recent terrorist threat, that of al-Qaeda, derives directly from the imperial domination of the Middle East in the nineteenth and twentieth centuries, but al-Qaeda's supporters find their inspiration in the founding of Islam in the seventh century. Yet only the globalization of the late twentieth century made al-Qaeda's actions possible.

Global warming is a more subtle, but equally dangerous trend which, if nothing is done to avert it, will bring catastrophe to hundreds of millions of vulnerable people. It is a direct result of global industrialization since the nineteenth century. Scientists have carefully documented the warming of the planet over the past century, but the nations of the world have so far only taken very small steps to address this vital threat to human existence.

GLOBAL TERRORISM

On September 11, 2001, members of the terrorist group al-Qaeda hijacked four transcontinental airliners taking off from Boston’s Logan airport. They crashed two of them into the two World Trade Center towers in New York City (see Figure 1). The explosion of the gasoline in the airplanes incinerated the twin towers and their occupants. The third plane crashed into the Pentagon. Passengers on the fourth plane, probably intended for the White House, brought it down in a field in Pennsylvania. Nearly 3,000 people died in these attacks. The boldness of the al-Qaeda attacks stunned and horrified the world. Never before had so many civilians been killed by a deliberate attack on American soil. President George W. Bush vowed to make a War on Terror the theme of his administration.

Between 1850 and 1914 both commerce and coercion expanded on a more global scale than ever before in human history. The technologies of the Second Industrial Revolution and the advance of the consolidated state powered this expansion. The Second Industrial Revolution was a North Atlantic revolution. The new centrality of Canada and the USA in the industrial order and the beginnings of industrial development in Japan were already laying the ground for a further shift toward the Pacific. Its technologies made possible the imperialist conquests characteristic of the era. At the same time, the model of the consolidated state that had emerged in the Age of Revolution spread throughout Europe and North America and then around the world. Existing states increasingly sought to unify and homogenize their national territories, while national minorities resisting such assimilation advocated the establishment of their own unified, homogeneous independent states. One important response to the expansion of industry and centralized state power was global popular protest. Worldwide protest evolved new forms and these innovative tactics showed similarities from Hankou to Chicago; in one guise or another, social spread throughout most of the world.

Between 1850 and 1914, in Western and Central Europe, North America, and Japan, a Second Industrial Revolution occurred whose impact on the daily life of men and women around the world was greater than that of the First Industrial Revolution. This revolution produced the large factory, the great corporation, and the factory worker, transforming steel production, machine construction, chemicals, and rubber. The Second Industrial Revolution depended heavily on fossil fuels, on coal and oil, and the geography of production and distribution were heavily affected by access to these resources.

World War II left Eurasia in ruins, and two huge continental superpowers, the USA and the USSR, dominating the world. Of the two, the USA had by far the bigger economy and superior military force. The wartime alliance of the two powers did not last long: Within two years, they had begun a bitter global conflict, the Cold War. Because both sides had nuclear weapons, this war threatened to exterminate humanity itself. Fortunately, the conflict quickly stabilized in Europe, dividing it into two spheres of influence, and both powers stepped back from nuclear confrontation over Cuba. But in the rest of the world, local upheavals threatened to draw in the superpowers, making each civil war and rebellion in the Third World a test of the superpowers’ “credibility.”

The two most destructive wars took place in Korea and Vietnam. In neither one did either superpower gain total victory. Korea remains a divided nation, and in Vietnam revolutionary nationalists successfully expelled French colonialists, their Vietnamese collaborators, and their American supporters. The Soviet Union, emboldened by American defeat, moved into Afghanistan, but faced its own Vietnam there. By 1980, the superpowers no longer seemed to dominate the world.

Despite the superpower confrontation, the postwar world economy grew at an unprecedented rate for thirty years.

3.1 Introduction

A mechanical member is, in general, subjected to different types of loads. The mechanical members, depending on the type of loading, have been given different names in technical literature. Some technical names and loadings are given as follows:

(a) Column: Member subjected to compressive force along its axis (Figure 3.1a). Visible effect: Shortening in axial length of the member.

(b) Shaft: Member subjected to moments along the axis. Such a moment is called torque or twisting moment (Figure 3.1b). Visible effect: The relative rotation of the sections about the axis.

(c) Beam: Member subjected to forces and/or moments perpendicular to the axis (Figure 3.1c). Visible effect: The axis becomes curved. This is called bending and the moment responsible for it is called bending moment.

3.2 Terminology

(a) On the basis of supports, the beams are classified as:

(i) Simply supported: Both the supports are at the end and are either pin or roller (Figure 3.2a).

(ii) Cantilever: Fixed at one end and free at the other (Figure 3.2b).

(iii) Overhanging: Beam is larger than the distance between the supports, and is projected beyond the supports (Figure 3.2c).

(iv) Fixed beam: Both the ends are fixed (Figure 3.2d).

(v) Continuous beam: Beams having more than two supports (Figure 3.2e).

In 1682, the boyars of Russia chose the vigorous 10-year-old boy Peter as their new tsar, with his mother Natalia as regent. Unfortunately, Peter's accession was immediately challenged by relatives of his older half-sister Sophia. A mob of dissatisfied musketeers broke into the Kremlin square, slaughtered the top boyars, murdered Peter's brother, but swore loyalty to the new tsar. Peter and his mother, still fearing for their lives, fled the capital. For the rest of his life, Peter hated Moscow, the Kremlin, and all it represented: intrigue, violence, superstition, and anarchy. He resolved to build an orderly state that insisted on honest service from all its subjects. As a youngster growing up in the suburbs of Moscow, he played with foreigners who taught him how to train his own soldiers. He learned to build his own ships. When he grew up, Peter forced the Russian people into the world of Western European states. But Russia also expanded to the borders of China, where it met the equally dynamic Kangxi emperor. The Kangxi emperor also came to the throne as a young man, challenged his elder relatives, and made China both strong and actively engaged with Western European powers. By 1700, these two vigorous rulers had made their states the dominant powers of Eurasia.

Lu Xun (1881–1936), China’s most famous modern writer, was born in the small market town of Shaoxing, near Shanghai. In 1901 he went to Japan, intending to become a doctor. In his medical school class, he saw a slide of apathetic Chinese bystanders watching the execution of a Chinese man by Japanese soldiers during the Russo-Japanese war of 1905. Shocked by their passivity, he concluded that China’s most deadly disease afflicted the spirit, not the body. He returned to China, resolving to become a writer to rouse his people from their deadly slumber. In “Call to Arms,” his first short story collection, he described with great sympathy and insight the foibles of ordinary Chinese folk following time-honored customs, nearly oblivious of the worldwide crisis that surrounded them. In “Diary of a Madman,” whose title is borrowed from the Russian writer Nikolai Gogol, the writer suddenly realizes that the basic principle of China’s classic civilization is “eat people.” Ah Q, Lu Xun’s most famous character, blithely walks to his own execution without ever knowing why he joined the cause of revolutionary nationalism. Lu Xun and his colleagues particularly stressed the need to free women from the straitjacket of traditional morality so that they could participate actively in making the new nation. Lu Xun organized the League of Left Wing Writers to mobilize Chinese writers in the service of revolutionary nationalism. Lu Xun’s mood constantly oscillated between high hopes and black despair. He died in 1936, hoping for China’s national unification based on radical social revolution, defying his own repressive government and the imminent threat of Japanese invasion.

In 1712, the Manchu official Tulisen left Beijing for the shores of the lower Volga River, in Russia, to visit a Mongolian khan. It was a distance of over 3,000 miles, and it took him nearly three years to get there and back. The emperor of China had sent him to explore Russian territory and look for an alliance against other Mongolian rivals. He was not invited to see the tsar (the ruler of all Russia), but he wrote a detailed account of the topography, ethnography, and history of all the regions he had crossed.

Seven years later, John Bell, a Scotsman in the service of the Russian tsar, set out from St. Petersburg for Beijing, covering much the same route as Tulisen in the opposite direction. He, too, reported accurately on the region’s geography, politics, and history, gathering scientific knowledge and military intelligence at the same time. Others followed them, like Ivan Unkovski, a Russian officer, who visited the Mongolian khan in Zungharia in 1722, and the French Jesuit Gerbillon, who accompanied the Chinese emperor on his military campaigns in the middle of the century. At the end of the century, the Englishman George Lord Macartney arrived by sea in Beijing in 1793 to negotiate the opening of formal trade relations between Britain and China.

The life of Ana Maria Ribeiro di Garibaldi illustrates for us the changing nature of national identities in the mid nineteenth century. Born into a family of fishermen in 1817 in Morinhos, a small town on the coast of Brazil, she married another village fisherman named Manuel in 1835. It was apparently not a happy marriage. Only four years later, a rebel ship showed up off the shore of Morinhos. The captain caught sight of her and rowed to shore, saying “You will be mine.” She got in the rowboat and went off with him for the rest of her life.

The captain was Giuseppe Garibaldi, a failed revolutionary who had been involved in a revolt in Piedmont (what is now northwestern Italy). Sentenced to death, he fled, arriving in South America in 1836. He became a mercenary for a small state, Rio Grande del Sul, which was trying to secede from Brazil. So it was that he was captaining the Rio Pardo schooner when it stopped off Morinhos in 1839. During the following year, the Brazilian navy closed in on him, and he left the service of the rebel state, which seemed increasingly doomed.

He and Anita then drove a herd of cattle to Montevideo and tried to enter civilian life without much success. They probably married in 1842. She maintained that Manuel had died, but many have cast doubt on this claim. She was in any case accepted as Garibaldi’s wife, and she bore him several children. Three lived.

9.1 Introduction

Vibration is a very common phenomenon in machines. In certain situations, the control of vibration is very essential. To control the vibration, proper understanding of the phenomenon is desired. The essence of vibration is discussed below.

A mechanical system is said to vibrate if it executes a simple harmonic motion or oscillation.