Chapter XX: Part I: Outline of the Theory of Structures

§ 2. _Support of Structures._--Every structure, as a whole, is
maintained in equilibrium by the joint action of its own _weight_, of
the _external load_ or pressure applied to it from without and tending
to displace it, and of the _resistance_ of the material which supports
it. A structure is supported either by resting on the solid crust of
the earth, as buildings do, or by floating in a fluid, as ships do in
water and balloons in air. The principles of the support of a floating
structure form an important part of Hydromechanics (q.v.). The
principles of the support, as a whole, of a structure resting on the
land, are so far identical with those which regulate the equilibrium
and stability of the several parts of that structure that the only
principle which seems to require special mention here is one which
comprehends in one statement the power both of liquids and of loose
earth to support structures. This was first demonstrated in a paper
"On the Stability of Loose Earth," read to the Royal Society on the
19th of June 1856 (Phil. _Trans._ 1856), as follows:--

Let E represent the weight of the portion of a horizontal stratum of
earth which is displaced by the foundation of a structure, S the
utmost weight of that structure consistently with the power of the
earth to resist displacement, [phi] the angle of repose of the earth;
then

S /1 + sin[phi]\²
--- = ( ------------ ).
E \1 - sin[phi]/

To apply this to liquids [phi] must be made zero, and then S/E = 1, as
is well known. For a proof of this expression see Rankine's _Applied
Mechanics_, 17th ed., p. 219.

§ 3. _Composition of a Structure, and Connexion of its Pieces._--A
structure is composed of _pieces_,--such as the stones of a building
in masonry, the beams of a timber framework, the bars, plates and
bolts of an iron bridge. Those pieces are connected at their joints or
surfaces of mutual contact, either by simple pressure and friction (as
in masonry with moist mortar or without mortar), by pressure and
adhesion (as in masonry with cement or with hardened mortar, and
timber with glue), or by the resistance of _fastenings_ of different
kinds, whether made by means of the form of the joint (as dovetails,
notches, mortices and tenons) or by separate fastening pieces (as
trenails, pins, spikes, nails, holdfasts, screws, bolts, rivets,
hoops, straps and sockets.)

§ 4. _Stability, Stiffness and Strength._--A structure may be damaged
or destroyed in three ways:--first, by displacement of its pieces from
their proper positions relatively to each other or to the earth;
secondly by disfigurement of one or more of those pieces, owing to
their being unable to preserve their proper shapes under the pressures
to which they are subjected; thirdly, by _breaking_ of one or more of
those pieces. The power of resisting displacement constitutes
stability, the power of each piece to resist disfigurement is its
_stiffness_; and its power to resist breaking, its _strength_.

§ 5. _Conditions of Stability._--The principles of the stability of a
structure can be to a certain extent investigated independently of the
stiffness and strength, by assuming, in the first instance, that each
piece has strength sufficient to be safe against being broken, and
stiffness sufficient to prevent its being disfigured to an extent
inconsistent with the purposes of the structure, by the greatest
forces which are to be applied to it. The condition that each piece of
the structure is to be maintained in equilibrium by having its gross
load, consisting of its own weight and of the external pressure
applied to it, balanced by the _resistances_ or pressures exerted
between it and the contiguous pieces, furnishes the means of
determining the magnitude, position and direction of the resistances
required at each joint in order to produce equilibrium; and the
_conditions of stability_ are, first, that the _position_, and,
secondly, that the _direction_, of the resistance required at each
joint shall, under all the variations to which the load is subject, be
such as the joint is capable of exerting--conditions which are
fulfilled by suitably adjusting the figures and positions of the
joints, and the _ratios_ of the gross loads of the pieces. As for the
_magnitude_ of the resistance, it is limited by conditions, not of
stability, but of strength and stiffness.

§ 6. _Principle of Least Resistance._--Where more than one system of
resistances are alike capable of balancing the same system of loads
applied to a given structure, the _smallest_ of those alternative
systems, as was demonstrated by the Rev. Henry Moseley in his
_Mechanics of Engineering and Architecture_, is that which will
actually be exerted--because the resistances to displacement are the
effect of a strained state of the pieces, which strained state is the
effect of the load, and when the load is applied the strained state
and the resistances produced by it increase until the resistances
acquire just those magnitudes which are sufficient to balance the
load, after which they increase no further.

This principle of least resistance renders determinate many problems
in the statics of structures which were formerly considered
indeterminate.

§ 7. _Relations between Polygons of Loads and of Resistances._--In a
structure in which each piece is supported at two joints only, the
well-known laws of statics show that the directions of the gross load
on each piece and of the two resistances by which it is supported must
lie in one plane, must either be parallel or meet in one point, and
must bear to each other, if not parallel, the proportions of the sides
of a triangle respectively parallel to their directions, and, if
parallel, such proportions that each of the three forces shall be
proportional to the distance between the other two,--all the three
distances being measured along one direction.

Considering, in the first place, the case in which the load and the
two resistances by which each piece is balanced meet in one point,
which may be called the _centre of load_, there will be as many such
points of intersection, or centres of load, as there are pieces in the
structure; and the directions and positions of the resistances or
mutual pressures exerted between the pieces will be represented by the
sides of a polygon joining those points, as in fig. 86 where P1, P2,
P3, P4 represent the centres of load in a structure of four pieces,
and the sides of the _polygon of resistances_ P1 P2 P3 P4 represent
respectively the directions and positions of the resistances exerted
at the joints. Further, at any one of the centres of load let PL
represent the magnitude and direction of the gross load, and Pa, Pb
the two resistances by which the piece to which that load is applied
is supported; then will those three lines be respectively the diagonal
and sides of a parallelogram; or, what is the same thing, they will be
equal to the three sides of a triangle; and they must be in the same
plane, although the sides of the polygon of resistances may be in
different planes.

According to a well-known principle of statics, because the loads or
external pressures P1L1, &c., balance each other, they must be
proportional to the sides of a closed polygon drawn respectively
parallel to their directions. In fig. 87 construct such a _polygon of
loads_ by drawing the lines L1, &c., parallel and proportional to, and
joined end to end in the order of, the gross loads on the pieces of
the structure. Then from the proportionality and parallelism of the
load and the two resistances applied to each piece of the structure to
the three sides of a triangle, there results the following theorem
(originally due to Rankine):--

_If from the angles of the polygon of loads there be drawn lines (R1,
R2, &c.), each of which is parallel to the resistance (as P1P2, &c.)
exerted at the joint between the pieces to which the two loads
represented by the contiguous sides of the polygon of loads (such as
L1, L2, &c.) are applied; then will all those lines meet in one point
(O), and their lengths, measured from that point to the angles of the
polygon, will represent the magnitudes of the resistances to which
they are respectively parallel._

When the load on one of the pieces is parallel to the resistances
which balance it, the polygon of resistances ceases to be closed, two
of the sides becoming parallel to each other and to the load in
question, and extending indefinitely. In the polygon of loads the
direction of a load sustained by parallel resistances traverses the
point O.[2]

§ 8. _How the Earth's Resistance is to be treated_.... When the
pressure exerted by a structure on the earth (to which the earth's
resistance is equal and opposite) consists either of one pressure,
which is necessarily the resultant of the weight of the structure and
of all the other forces applied to it, or of two or more parallel
vertical forces, whose amount can be determined at the outset of the
investigation, the resistance of the earth can be treated as one or
more upward loads applied to the structure. But in other cases the
earth is to be treated as _one of the pieces of the structure_, loaded
with a force equal and opposite in direction and position to the
resultant of the weight of the structure and of the other pressures
applied to it.

§ 9. _Partial Polygons of Resistance._--In a structure in which there
are pieces supported at more than two joints, let a polygon be
constructed of lines connecting the centres of load of any continuous
series of pieces. This may be called a _partial polygon of
resistances_. In considering its properties, the load at each centre
of load is to be held to _include_ the resistances of those joints
which are not comprehended in the partial polygon of resistances, to
which the theorem of § 7 will then apply in every respect. By
constructing several partial polygons, and computing the relations
between the loads and resistances which are determined by the
application of that theorem to each of them, with the aid, if
necessary, of Moseley's principle of the least resistance, the whole
of the relations amongst the loads and resistances may be found.

§ 10. _Line of Pressures--Centres and Line of Resistance._--The line
of pressures is a line to which the directions of all the resistances
in one polygon are tangents. The _centre of resistance_ at any joint
is the point where the line representing the total resistance exerted
at that joint intersects the joint. The _line of resistance_ is a line
traversing all the centres of resistance of a series of joints,--its
form, in the positions intermediate between the actual joints of the
structure, being determined by supposing the pieces and their loads to
be subdivided by the introduction of intermediate joints _ad
infinitum_, and finding the continuous line, curved or straight, in
which the intermediate centres of resistance are all situated, however
great their number. The difference between the line of resistance and
the line of pressures was first pointed out by Moseley.

§ 11.* The principles of the two preceding sections may be illustrated
by the consideration of a particular case of a buttress of blocks
forming a continuous series of pieces (fig. 88), where aa, bb, cc, dd
represent plane joints. Let the centre of pressure C at the first
joint aa be known, and also the pressure P acting at C in direction
and magnitude. Find R1 the resultant of this pressure, the weight of
the block aabb acting through its centre of gravity, and any other
external force which may be acting on the block, and produce its line
of action to cut the joint bb in C1. C1 is then the centre of pressure
for the joint bb, and R1 is the total force acting there. Repeating
this process for each block in succession there will be found the
centres of pressure C2, C3, &c., and also the resultant pressures R2,
R3, &c., acting at these respective centres. The centres of pressure
at the joints are also called _centres of resistance_, and the curve
passing through these points is called a _line of resistance_. Let all
the resultants acting at the several centres of resistance be produced
until they cut one another in a series of points so as to form an
unclosed polygon. This polygon is the _partial polygon of resistance_.
A curve tangential to all the sides of the polygon is the _line of
pressures_.

§ 12. _Stability of Position, and Stability of Friction._--The
resistances at the several joints having been determined by the
principles set forth in §§ 6, 7, 8, 9 and 10, not only under the
ordinary load of the structure, but under all the variations to which
the load is subject as to amount and distribution, the joints are now
to be placed and shaped so that the pieces shall not suffer relative
displacement under any of those loads. The relative displacement of
the two pieces which abut against each other at a joint may take place
either by turning or by sliding. Safety against displacement by
turning is called _stability of position_; safety against displacement
by sliding, _stability of friction_.

§ 13. _Condition of Stability of Position._--If the materials of a
structure were infinitely stiff and strong, stability of position at
any joint would be insured simply by making the centre of resistance
fall within the joint under all possible variations of load. In order
to allow for the finite stiffness and strength of materials, the least
distance of the centre of resistance inward from the nearest edge of
the joint is made to bear a definite proportion to the depth of the
joint measured in the same direction, which proportion is fixed,
sometimes empirically, sometimes by theoretical deduction from the
laws of the strength of materials. That least distance is called by
Moseley the _modulus of stability_. The following are some of the
ratios of the modulus of stability to the depth of the joint which
occur in practice:--

Retaining walls, as designed by British engineers 1:8
Retaining walls, as designed by French engineers 1:5
Rectangular piers of bridges and other buildings, and
arch-stones 1:3
Rectangular foundations, firm ground 1:3
Rectangular foundations, very soft ground 1:2
Rectangular foundations, intermediate kinds of ground 1:3 to 1:2
Thin, hollow towers (such as furnace chimneys exposed
to high winds), square 1:6
Thin, hollow towers, circular 1:4
Frames of timber or metal, under their ordinary or
average distribution of load 1:3
Frames of timber or metal, under the greatest
irregularities of load 1:3

In the case of the towers, the _depth of the joint_ is to be
understood to mean the _diameter of the tower_.

§ 14. _Condition of Stability of Friction._--If the resistance to be
exerted at a joint is always perpendicular to the surfaces which abut
at and form that joint, there is no tendency of the pieces to be
displaced by sliding. If the resistance be oblique, let JK (fig. 89)
be the joint, C its centre of resistance, CR a line representing the
resistance, CN a perpendicular to the joint at the centre of
resistance. The angle NCR is the _obliquity_ of the resistance. From R
draw RP parallel and RQ perpendicular to the joint; then, by the
principles of statics, the component of the resistance _normal_ to the
joint is--

CP = CR · cos PCR;

and the component _tangential_ to the joint is--

CQ = CR · sin PCR = CP · tan PCR.

If the joint be provided either with projections and recesses, such as
mortises and tenons, or with fastenings, such as pins or bolts, so as
to resist displacement by sliding, the question of the utmost amount
of the tangential resistance CQ which it is capable of exerting
depends on the _strength_ of such projections, recesses, or
fastenings; and belongs to the subject of strength, and not to that of
stability. In other cases the safety of the joint against displacement
by sliding depends on its power of exerting friction, and that power
depends on the law, known by experiment, that the friction between two
surfaces bears a constant ratio, depending on the nature of the
surfaces, to the force by which they are pressed together. In order
that the surfaces which abut at the joint JK may be pressed together,
the resistance required by the conditions of equilibrium CR, must be a
_thrust_ and not a _pull_; and in that case the force by which the
surfaces are pressed together is equal and opposite to the normal
component CP of the resistance. The condition of stability of friction
is that the tangential component CQ of the resistance required shall
not exceed the friction due to the normal component; that is, that

CQ [/>] f · CP,

where f denotes the _coefficient of friction_ for the surfaces in
question. The angle whose tangent is the coefficient of friction is
called _the angle of repose_, and is expressed symbolically by--

[phi] = tan^-1 f.

Now CQ = CP · tan PCR;

consequently the condition of stability of friction is fulfilled if
the angle PCR is not greater than [phi]; that is to say, if _the
obliquity of the resistance required at the joint does not exceed the
angle of repose_; and this condition ought to be fulfilled under all
possible variations of the load.

It is chiefly in masonry and earthwork that stability of friction is
relied on.

§ 15. _Stability of Friction in Earth._--The grains of a mass of loose
earth are to be regarded as so many separate pieces abutting against
each other at joints in all possible positions, and depending for
their stability on friction. To determine whether a mass of earth is
stable at a given point, conceive that point to be traversed by planes
in all possible positions, and determine which position gives the
greatest obliquity to the total pressure exerted between the portions
of the mass which abut against each other at the plane. The condition
of stability is that this obliquity shall not exceed the angle of
repose of the earth. The consequences of this principle are developed
in a paper, "On the Stability of Loose Earth," already cited in § 2.

§ 16. _Parallel Projections of Figures._--If any figure be referred to
a system of co-ordinates, rectangular or oblique, and if a second
figure be constructed by means of a second system of co-ordinates,
rectangular or oblique, and either agreeing with or differing from the
first system in rectangularity or obliquity, but so related to the
co-ordinates of the first figure that for each point in the first
figure there shall be a corresponding point in the second figure, the
lengths of whose co-ordinates shall bear respectively to the three
corresponding co-ordinates of the corresponding point in the first
figure three ratios which are the same for every pair of corresponding
points in the two figures, these corresponding figures are called
_parallel projections_ of each other. The properties of parallel
projections of most importance to the subject of the present article
are the following:--

(1) A parallel projection of a straight line is a straight line.

(2) A parallel projection of a plane is a plane.

(3) A parallel projection of a straight line or a plane surface
divided in a given ratio is a straight line or a plane surface divided
in the same ratio.

(4) A parallel projection of a pair of equal and parallel straight
lines, or plain surfaces, is a pair of equal and parallel straight lines,
or plane surfaces; whence it follows

(5) That a parallel projection of a parallelogram is a parallelogram,
and

(6) That a parallel projection of a parallelepiped is a parallelepiped.

(7) A parallel projection of a pair of solids having a given ratio
is a pair of solids having the same ratio.

Though not essential for the purposes of the present article, the
following consequence will serve to illustrate the principle of
parallel projections:--

(8) A parallel projection of a curve, or of a surface of a given
algebraical order, is a curve or a surface of the same order.

For example, all ellipsoids referred to co-ordinates parallel to any
three conjugate diameters are parallel projections of each other and
of a sphere referred to rectangular co-ordinates.

§ 17. _Parallel Projections of Systems of Forces._--If a balanced
system of forces be represented by a system of lines, then will every
parallel projection of that system of lines represent a balanced
system of forces.

For the condition of equilibrium of forces not parallel is that they
shall be represented in direction and magnitude by the sides and
diagonals of certain parallelograms, and of parallel forces that they
shall divide certain straight lines in certain ratios; and the
parallel projection of a parallelogram is a parallelogram, and that of
a straight line divided in a given ratio is a straight line divided in
the same ratio.

The resultant of a parallel projection of any system of forces is the
projection of their resultant; and the centre of gravity of a parallel
projection of a solid is the projection of the centre of gravity of
the first solid.

§ 18. _Principle of the Transformation of Structures._--Here we have
the following theorem: If a structure of a given figure have stability
of position under a system of forces represented by a given system of
lines, then will any structure whose figure is a parallel projection
of that of the first structure have stability of position under a
system of forces represented by the corresponding projection of the
first system of lines.

For in the second structure the weights, external pressures, and
resistances will balance each other as in the first structure; the
weights of the pieces and all other parallel systems of forces will
have the same ratios as in the first structure; and the several
centres of resistance will divide the depths of the joints in the same
proportions as in the first structure.

If the first structure have stability of friction, the second
structure will have stability of friction also, so long as the effect
of the projection is not to increase the obliquity of the resistance
at any joint beyond the angle of repose.

The lines representing the forces in the second figure show their
_relative_ directions and magnitudes. To find their _absolute_
directions and magnitudes, a vertical line is to be drawn in the first
figure, of such a length as to represent the weight of a particular
portion of the structure. Then will the projection of that line in the
projected figure indicate the vertical direction, and represent the
weight of the part of the second structure corresponding to the
before-mentioned portion of the first structure.

The foregoing "principle of the transformation of structures" was
first announced, though in a somewhat less comprehensive form, to the
Royal Society on the 6th of March 1856. It is useful in practice, by
enabling the engineer easily to deduce the conditions of equilibrium
and stability of structures of complex and unsymmetrical figures from
those of structures of simple and symmetrical figures. By its aid, for
example, the whole of the properties of elliptical arches, whether
square or skew, whether level or sloping in their span, are at once
deduced by projection from those of symmetrical circular arches, and
the properties of ellipsoidal and elliptic-conoidal domes from those
of hemispherical and circular-conoidal domes; and the figures of
arches fitted to resist the thrust of earth, which is less
horizontally than vertically in a certain given ratio, can be deduced
by a projection from those of arches fitted to resist the thrust of a
liquid, which is of equal intensity, horizontally and vertically.

§ 19. _Conditions of Stiffness and Strength._--After the arrangement
of the pieces of a structure and the size and figure of their joints
or surfaces of contact have been determined so as to fulfil the
conditions of _stability_,--conditions which depend mainly on the
position and direction of the _resultant_ or _total_ load on each
piece, and the _relative_ magnitude of the loads on the different
pieces--the dimensions of each piece singly have to be adjusted so as
to fulfil the conditions of _stiffness_ and _strength_--conditions
which depend not only on the _absolute_ magnitude of the load on each
piece, and of the resistances by which it is balanced, but also on the
_mode of distribution_ of the load over the piece, and of the
resistances over the joints.

The effect of the pressures applied to a piece, consisting of the load
and the supporting resistances, is to force the piece into a state of
_strain_ or disfigurement, which increases until the elasticity, or
resistance to strain, of the material causes it to exert a _stress_,
or effort to recover its figure, equal and opposite to the system of
applied pressures. The condition of _stiffness_ is that the strain or
disfigurement shall not be greater than is consistent with the
purposes of the structure; and the condition of _strength_ is that the
stress shall be within the limits of that which the material can bear
with safety against breaking. The ratio in which the utmost stress
before breaking exceeds the safe working stress is called the _factor
of safety_, and is determined empirically. It varies from three to
twelve for various materials and structures. (See STRENGTH OF
MATERIALS.)