STRUCTURES: PURPOSE AND FUNCTION
1.10 DESIGN FOR STRUCTURAL RESPONSE
modeling is greatly aided by the use of computers. However, routine problems (that is, 98% of all problems) are still often treated by use of simple hand computations or by reference to data in handbook tables or graphs.
The purpose of this book is essentially educational, so the emphasis here is on visualization and understanding, not necessarily on efficiency of computational means. Major use is made of graphical visualization, and readers are strongly encouraged to develop the habit of using such vi- sualization. The use of sketches as learning and problem-solving aids cannot be overemphasized. Four types of graphical devices are espe- cially useful: the free-body diagram, the cut section, the exaggerated pro- file of the load-deformed structure, and the graphical plot of critical equations.
A free-body diagram consists of a picture of any isolated physical el- ement that shows the full set of external forces that operate on that ele- ment. The isolated element may be a whole structure or any fractional part of it. Consider the structure shown in Figure 1.54. Figure 1.54a shows the entire structure, consisting of attached horizontal and vertical elements (beams and columns) that produce a planar rigid frame bent.
This may be one of a set of such frames comprising a building structure.
The free-body diagram in Figure 1.54arepresents the entire structure, with forces external to it represented by arrows. The arrows indicate the location, sense, and direction of each external force. At some stage of in- vestigation, numbers may be added indicating the magnitude of these forces. The forces shown include the weight of the structure, the hori- zontal force of wind, and the net forces acting at the points of support for the frame.
Shown in Figure 1.54bis a free-body diagram of a single beam from the framed bent. Operating externally on the beam are its own weight plus the effects of interaction between the beam and the columns to which it is attached. These interactions are not visible in the free-body di- agram of the full frame, so one purpose for the diagram of the single beam is simply the visualization of the nature of these interactions. It may now be observed that the columns transmit to the ends of the beams a combination of vertical and horizontal forces plus rotational bending actions. The observation of the form of these interactions is a necessary first step in a full investigation of this beam.
Figure 1.54cshows an isolated portion of the beam length, produced by slicing vertical planes a short distance apart and removing the portion
DESIGN FOR STRUCTURAL RESPONSE 65
between them. Operating on this free body are its own weight and the ac- tions of the of the beam segments on the opposite sides of the slicing planes; that is, the effects that hold this segment in place in the uncut beam. This slicing device, called a cut section, is used to visualize the in- ternal force actions in the beam and is a first step in the investigation of the stresses that relate to the internal forces.
Figure 1.54 Free-body diagrams.
Finally, in Figure 1.54dis shown a tiny particle of the material of the beam, on which the external effects are those of the adjacent particles.
This is the basic device for visualization of stress. In the example, the particle is seen to be operated on by a combination of vertical shear (and its horizontal complement) and horizontally directed compression.
Figure 1.55ashows the exaggerated deformed profile of the same bent under wind loading. The overall form of lateral deflection of the bent and the character of bending in each member can be visualized from this figure. As shown in Figure 1.55b, the character of deformation of segments and particles can also be visualized. These diagrams are very helpful in establishing the qualitative nature of the relationships between force actions and overall shape changes or between stresses and strains.
Quantitative computations often become considerably abstract in their operation, but these diagrams are real exercises in direct visualization of behavior.
DESIGN FOR STRUCTURAL RESPONSE 67
Figure 1.55 Visualization of structural deformations.
For both visualization and quantification, considerable use is made of graphical plots of mathematical expressions in this book. Figure 1.56 shows the form of damped vibration of an elastic spring. The graph con- sists of a plot of the variation of displacement (+ or –s) of the spring from its neutral position as a function of elapsed time t. This is a plot of the equation
which describes the function mathematically but not visually. The graph helps us to literally seethe rate of decline of the vibration (damping ef- fect) and the specific location of the spring at any given point in time.
Only mathematicians can see these things from an equation; for the rest of us, the graph is a big help.
s= et P Qt R
[
+]
1 sin( )
Figure 1.56 Displacement versus elapsed time plot of a cyclic (harmonic) motion.
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2
FORCES AND FORCE ACTIONS
The preceding chapter provided an overview of the world of structural analysis as an activity for the support of design of building structures.
This chapter begins a more deliberate study of the basic applications of physics and mathematics to the real work of structural analysis. This study begins with a consideration of forces and their actions.