static Equilibrium and Elasticity

12.4 Elastic Properties of Solids

In the previous discussion we have mengansumsikan that the object will remain rigid whenthere are external forces acting on it. But in fact, all objects can change shape.

Stress is the external force acting on the body per unit cross-sectional area / cross.

Strain is the result of Stress

The Stress required to produce a particular strain depends on the state of the material is pressed.

Comparison between stress and strain, strain or stress unity, called the elastic modulus.

The  elastic modulus  in  general relates what  is  done  to  a  solid  object  (a  force  is  applied) to how that object responds (it deforms to some extent). The greater the elastic modulus, the greater the stress needed for a particular strain. We consider three types of deformation and define an elastic modulus for each:

Young’s modulus, which measures the resistance of a solid to a change in its length

Shear modulus, which measures the resistance to motion of the planes within a solid parallel to each other

Bulk modulus, which the resistance of solids or liquids to changes in thei volume.

Young’s Modulus: Elasticity in Length

Consider  a  long bar of  cross-sectional  area A and  initial  length Li that  is  clamped  at one end, as  in Figure 12.14. When an external  force  is applied perpendicular  to  the cross  section,  internal  forces  in  the  bar  resist  distortion  (“stretching”),  but  the  bar reaches an equilibrium  situation  in which  its final  length Lf is greater  than Li and  in which the external force is exactly balanced by internal forces. In such a situation, the bar is said to be stressed. We define the tensile stress as the ratio of the magnitude of the external force F to the cross-sectional area A. The tensile strain in this case is defined  as  the ratio  of  the  change  in  length

to  the  original  length  Li. We  define Young’s modulus by a combination of these two ratios:  

 

                                                                       

Shear Modulus: Elasticity of Shape

Another type of deformation occurs when an object is subjected to a force parallel to one of its faces while the opposite face is held fixed by another force (Fig. 12.16a). The stress in this case  is called a shear stress. If the object  is originally a rectangular block, a shear stress results in a shape whose cross section is a parallelogram. A book pushed sideways, as shown  in Figure 12.16b,  is an example of an object subjected to a shear stress. To a first approximation (for small distortions), no change in volume occurs with this deformation. We define the shear stress as F/A, the ratio of the tangential force to the area A of the face being sheared. 

The shear strain is defined as the ratio /h, where   is the horizontal distance  that  the  sheared  face moves and h is  the height of  the object.  In terms of these quantities, the shear modulus is      

 Values of  the  shear modulus  for  some representative materials are given  in Table 12.1. Like Young’s modulus, the unit of shear modulus is the ratio of that for force to that for area.  

Bulk Modulus: Volume Elasticity

Bulk modulus  characterizes  the  response  of  an  object  to  changes  in  a  force  of uniform  magnitude  applied  perpendicularly  over  the  entire  surface  of  the  object,  as shown  in  Figure  12.17.  (We  assume  here  that  the  object  is made  of  a  single  substance.) As we  shall  see  in Chapter 14,  such  a uniform distribution of  forces occurs when an object  is  immersed  in a fluid. An object  subject  to  this  type of deformation undergoes a change in volume but no change in shape. The volume stress is defined as the ratio of the magnitude of the total force F exerted on a surface to the area A of the surface. The quantity P =F/A is called pressure, which we will study in more detail  in Chapter  14. If  the  pressure  on  an  object  changes  by  an  amount  =/A,then  the object will experience a volume change . The volume strain is equal  to the change in volume   divided by the initial volume Vi . Thus, from Equation 12.5, we  can  characterize  a  volume  (“bulk”)  compression  in  terms of  the bulk modulus,  which is defined as  

Prestressed Concrete

If the stress on a solid object exceeds a certain value, the object fractures. The maximum stress  that can be applied before  fracture occurs depends on  the nature of  the material  and  on  the  type  of  applied  stress.  For  example,  concrete  has  a  tensile strength  of  about  2 x 10⁶ N/m²,  a  compressive  strength  of  20 x 10⁶ N/m²,  and  a shear  strength of 2 x 10⁶ N/m².  If  the  applied  stress  exceeds  these  values,  the  concrete  fractures. It  is common practice  to use  large safety  factors  to prevent  failure  

in concrete structures.

Concrete  is normally  very brittle when  it  is cast  in  thin  sections. Thus, concrete slabs tend to sag and crack at unsupported areas, as shown in Figure 12.18a. The slab can be strengthened by the use of steel rods to reinforce the concrete, as illustrated in Figure 12.18b. Because concrete is much stronger under compression (squeezing) than under  tension  (stretching) or  shear,  vertical  columns of  concrete  can  support very heavy  loads, whereas horizontal beams of concrete tend to sag and crack. However, a significant increase in shear strength is achieved if the reinforced concrete is prestressed,  as  shown  in  Figure  12.18c.  As  the  concrete  is  being  poured,  the  steel rods are held under tension by external forces. The external forces are released after the concrete cures; this results in a permanent tension in the steel and hence a compressive  stress  on  the  concrete.  This  enables  the  concrete  slab  to  support  a much heavier load.