STRUCTURES: PURPOSE AND FUNCTION
1.5 INTERNAL FORCES
the supports, the column bottoms can be held completely in their original positions, as shown in Figure 1.13c.
The combination of loads and support reactions constitutes the total external effort on a structure. This system is in some ways independent of the structure; that is, the external forces must be in equilibrium, re- gardless of the materials, strength, and so on, of the structure. For exam- ple, the task for a beam can be totally defined in terms of effort without reference to what the beam actually consists of.
With its tasks defined, however, it becomes necessary to consider the response developed by the structure. This means moving on to consider what happens inside the structure in terms of internal force effects.
ample, the principal cause of the structure’s deformation is bending re- sistance, called internal resistive bending moment.
The stresses associated with the internal force action of bending mo- ment are horizontally directed compression in the upper portion of the plank and horizontally directed tension in the lower portion. Anyone could have predicted that the plank would sag when the person stepped on it. But we can also predict the deformation as an accumulation of strains, resulting in the shortening of the upper portion and the lengthen- ing of the lower portion of the plank. Thus, the stress condition can be in- ferred from observed deformation, but likewise the deformation can be predicted from known stress conditions.
For the relatively thin wooden plank, the bending action and strain ef- fects are quite apparent. If the plank is replaced by a thick wooden beam, the sag will not be visually apparent with a light load and a short span.
However, the internal bending still occurs and the sag—however slight—
does exist. For the investigation of structural behaviors, visualization of internal forces is aided by considering an exaggerated deformation of the structure, assuming it to be much more flexible than it really is.
INTERNAL FORCES 29
Figure 1.14 Internal bending.
ently stable, must have adequate strength for an acceptable margin of safety, and must have a reasonable stiffness for resistance to deformation.
These three basic characteristics—stability, strength, and stiffness—are the principal functional requirements of structures.
Stability
Stability has both simple and complex connotations. In the case of the wooden plank, it is essential that there be two supports and that the per- son stand between the supports. As shown in Figure 1.15, if the plank ex- tends over one support, and a person stands on the extended end, disaster will certainly occur unless a counterweight is placed on the plank or the plank is anchored to the opposite support. In this case, either the coun- terweight or the anchorage is necessary for the stability of the structure—
unrelated to the strength or stiffness of the plank.
A slightly different problem of stability is illustrated by another ex- ample. Suppose you have a sore foot and want to use a walking stick to assist your travel. You are offered a 3⁄⁄4-in. round wooden stick and a 1⁄⁄4- in. round steel rod, each 3 ft long. After handling both, you would prob- ably choose the wooden stick, since the steel rod would buckle under your weight. This buckling action can be visualized, demonstrated, and measured. The essential property of a structure that determines its buck- ling potential is its slenderness.
In engineering analysis, the geometric property of slenderness used to establish the likelihood of buckling is the slenderness ratio, also called the relative slenderness, expressed as
L/r in which
L= length of the compression member over which there is no lateral bracing to prevent buckling
r= a geometric property of the member cross section called the radius of gyration
The geometric property rcan be expressed as
In this formula,
A= the member cross-sectional area
I= a property called the second moment of the area or the moment of inertia
r I
= A
1 2/
FUNCTIONAL REQUIREMENTS OF STRUCTURES 31
Figure 1.15 Developing stability.
has an L/rof 192, while the ⁄⁄4-in. steel rod has an L/rof 576. If we take the steel and flatten it out and roll it up to produce a cylinder with a 3⁄⁄4in.
diameter, the area remains the same, but the Ivalue is significantly in- creased. Furthermore, the rvalue is thus also increased, so that the L/r now becomes 136. As long as the cylinder wall is not made too thin, the pipe-shaped stick represents a major improvement in buckling resistance.
Figure 1.16 shows the three cross sections and the corresponding L/r values.
Bending and buckling stiffness are also affected by the stiffness of the material. Thus, a 1⁄4in. rod of wood would be even less stiff than the one of steel, since wood is considerably less stiff than steel. For a single, very slender, compression member, the compression force required to produce buckling is expressed by the Euler formula, shown in the plot of com- pression failure versus length in Figure 1.17. As the member is short- ened, buckling becomes less critical, and the limiting effect becomes simple compressive crushing of the material. At very short lengths, there- fore, the compression limit is determined by the stress resistance of the
Figure 1.16 RelativeL /rvalues.
material. At the other end of the graph, the curve becomes that of the Euler formula, in which the index of the member resistance is stiffness—
of both the member cross section (I) and the material (E, which is the stiffness modulus of the material). Between the limits, the curve slowly changes from one form to the other, and the buckling phenomenon con- tains some aspect of both types of failure.
Stability can be a problem for a single structural member, such as a single column, or it can be a problem for a whole structural assemblage.
The eight-element framework shown in Figure 1.18 may be stable in re- sisting vertical gravity loads, but it must be braced in some manner against any horizontal forces, such as those caused by wind or earth- quakes. The illustrations in Figure 1.18 show the three principal means
FUNCTIONAL REQUIREMENTS OF STRUCTURES 33
Figure 1.17 Compression load limit versus member slenderness. Eis a factor that indicates the stiffness of the material.
for achieving this stability: by using rigid joints between members, by using truss bracing in the wall planes, or by using rigid panels in the wall planes, called infilling.
Strength
Strength is probably the most obvious requirement for a structure. Even though it is stable, the plank in Figure 1.14 is not strong enough to hold the weight of ten people. This has to do partly with the material—if the plank were made of steel, it might do the job. It also has to do with the form and orientation of the plank cross section—if the wood plank were turned on its edge, like a floor joist, it would probably also support ten people.
Material strength often depends on the type of stress that the material must sustain. Steel is adaptable and capable of major resistance to tension, compression, shearing, twisting, and bending with equal dexterity. Wood, however, has different strengths depending on the direction of the stress with reference to the wood grain. As shown in Figure 1.19, the develop-
Figure 1.18 Means of stabilizing a frame structure.
ment of major stresses perpendicular to the wood grain direction can cause the wood to fail easily. Reforming the wood, either by glue lamina- tion or by pulverising the wood and using the wood fiber to produce com- pressed fiber panels, is a way of overcoming the grain limitation.
Stone, concrete, and fired clay are examples of materials that have varying strengths for different stresses. All are relatively strong in resist- ing compression, but are much less strong in resisting tension or shear.
This requires caution in their use in structures to avoid these stresses or to compensate for them—such as by using steel reinforcement in con- crete structures.
Attention must be given both to the form and nature of elements and to their uses. A cable assembled from thin steel wires has little resistance to compression or bending or to anything but the single task for which it is formed—resisting tension. This is so despite the fact that the steel, as a material, has other stress potentials.
A stack of bricks with no bonding in the joints has the capability of sup- porting a compressive load applied directly downward on the top of the stack. Picking the unbonded stack up by lifting the top brick or turning the stack sideways to create a spanning structure, as shown in Figure 1.20, is obviously not possible. Thus, joint formation of elements in an assembled structure is also a concern for strength.
FUNCTIONAL REQUIREMENTS OF STRUCTURES 35
Figure 1.19 Effect of orientation to load.
Figure 1.20 Effect of orientation to load.
Stiffness
All structures change shape and move when subjected to forces (see Figure 1.21). The relative magnitude of these changes determines a qual- ity of the structure called rigidity or stiffness. The degree of stiffness de- pends on the material of the of the structure, on the configuration of its parts, and—for assemblages—on the arrangement of the assembled members. It may also depend on the connections between parts and on the type of restraint offered by supports. The presence or absence of bracing may also be a factor.
Although stiffness is usually not as critical to the safety of a structure as strength and stability, it is frequently important for use of the structure.
If a slammed door rocks the whole building, or if floors bounce when walked on, the users of the building will probably not be satisfied with the structure.
Equilibrium of Structures
Most structures act as transfer elements, receiving certain forces and transferring them to other points. This transfer capability is dependent on the internal strength and stability of the structure. As shown in Figure 1.22, a thin sheet of aluminum may be easily buckled, a block of wood may be easily split along its grain, and a rectangular framework with loose, single-pin joints may be easily collapsed sideways. All of these structures fail because of an inability to maintain internal equilibrium through lack of strength, or because of the lack of some inherent stabil- ity, or for both reasons.
The complete static equilibrium of a structure requires two separate balances: that of the external forces and that of the internal forces. Ex- ternally sufficient reaction components must be developed by the sup- ports. Internally, there must be an inherent capability for stability and
Figure 1.21 Deformation of structures under load.
sufficient strength to do the work of transferring the applied loads to the supports.
As shown in Figure 1.23, there are three possible conditions for exter- nal stability. If support conditions are insufficient in type or number, the structure is externally unstable. If support conditions are just adequate, the structure is stable. If the supports provide an excess of the necessary con- ditions, the structure is probably stable, but may be indeterminate—not necessarily a bad quality, just a problem for achieving a simple investiga- tion of structural behavior.
For internal stability, the structure must be formed, arranged, and fas- tened together to develop the necessary resistance. In the examples shown in Figure 1.22, the aluminum sheet was too thin for its size, the wood block had weak shear planes, and the frame lacked the necessary arrangement of members or type of joints. All three could be altered to make them more functional. As shown in Figure 1.24, the aluminum sheet can be braced with stiffening ribs, the solid-sawn wood block can be replaced with a laminated piece with alternate plies having their grain
FUNCTIONAL REQUIREMENTS OF STRUCTURES 37
Figure 1.22 Lack of internal resistance.
Figure 1.23 Stability analysis.
Figure 1.24 Alteration of internal conditions to improve structural resistance.
directions perpendicular to each other, and the frame can be stabilized by adding a diagonal member.