Notes on Chapter 2 Sections 2.3, 2.4, 2.5

Towards the end of last week I finished reading sections 2.3 through 2.5 in Chapter 2. This post will give a summary of the important topics through those specific sections.

2.3 :
Section 2.3 dealt with changing material lengths under nonuniform conditions.

When a linearly elastic material is loaded only at the ends, the change in length of the material can be obtained through the equation:

\(\partial  = \frac{{PL}}{{EA}}\)

Where P is the load on the bar, L is the length of the bar, E is the modulus of elasticity, and A is the cross sectional are.

However, 2.3 deals with this equation in a more general sense and not just solely when it is used for linearly elastic materials. 

Bars consisting of Prismatic Segments:

The change in length of the bar when loaded can be obtained from the equation:

\(\partial  = \sum\limits_{i = 1}^n {\frac{{{N_i}{L_i}}}{{{E_i}{A_i}}}} \)

Where i is the numbering index for various segments, n is the total number of segments, and \({{N_i}}\) is the internal axial force in segment i.

Bars with continuously Varying Loads or Dimensions:

Given that: \[d\partial  = \frac{{N(x)dx}}{{EA(x)}}\] 

The elongation of the entire bar is obtained by integrating over the entire length with respect to x.

\[\partial  = \int\limits_0^L {d\partial  = \int\limits_0^L {\frac{{N(x)dx}}{{EA(x)}}} } \]

2.4 :

Section 2.4 deals with statically indeterminate structures.

A structure is classified as statically determinate if its reactions and internal forces can be determined solely from free-body diagrams and equations of equilibrium (i.e \(\sum F  = 0\)). It is important to note that the forces of statically determinate structures can be found without knowing the properties of the materials.

However, most structures are more complex that a bar and their reactions and internal forces cannot be found by statics alone. These types of structures are classified as statically indeterminate. In order to analyze these structures, supplemental equations must be used in addition to displacement equations. 

2.5 :

Sections 2.5 deals with Thermal effects, Misfits, and Prestrains.

External loads are not the only sources of stresses and strains in a structure. Other sources include thermal effects, which arise from temperature changes, misfits, which result from imperfections in construction, and prestrains, produced by initial deformations. 

Thermal effects:

Changes in temperature produce expansion or contraction of structural materials, resulting in thermal strains and thermal stresses.

The equation for thermal strain \({\varepsilon _T}\) is proportional to the temperature change \(\Delta T\): 

\[{\varepsilon _T} = \alpha (\Delta T)\]

Where \(\alpha\) is the coefficient of thermal expansion.

Temperature-displacement relation calculates the increase in any dimension by the equation: 

\[\partial ={ \varepsilon_T}L= \alpha(\Delta T)L\]

Misfits:

Misfit members are members with slightly different measurements than those which were intended in their design and consequently do not fit properly in their structure. 

Prestrains:

Prestrains occur in a system when misfit members are used in construction of a given structure. These misfit members introduce strains and stresses into the structure before any loads are applied to it and are therefore called "Prestrains" and "Prestresses." 

If a structure is statically determinate, small misfits in one or more members will not produce strains or stresses in the overall structure. However, in statically indeterminate structures, small misfits do introduce strains and stresses as the overall structure is not free to adjust to those misfit members. 

Next post will pertain to my notes on sections 2.6, 2.7, 2.8, and 2.9.

Thanks for reading, 

-Nick Thompson