Support Reactions

 Knowledge of support reactions generated by a support is very helpful in analyzation of a structure. Support reactions  generation knowledge is necessary to draw free body diagram of a structure . In calculation of indeterminacy also support reaction knowledge is very much helpful.

Support reactions are basically developed due to restraint of deformation provided by a support. In general any restrain on translation of a body provides a force support reaction in the opposite direction of translation. Similarly restrain on rotation of body provides a  moment support reaction.

SUPPORT REACTIONS ON A SUPPORT SUBJECTED TO 2-D FORCE-SYSTEM

SUPPORT REACTIONS ON A SUPPORT SUBJECTED TO 3-D FORCE-SYSTEM

DETERMINANT, INDETERMINANT STRUCTURES AND INDETERMINACY OF STRUCTURE

DETERMINATE AND INDETERMINATE STRUCTURE DEFINITION

 In structural analysis our main motive is to find unknown forces or moments in different component of structures. 

If these unknown forces or moments can be determined only by the use of equation of statics, then the structure is said to be determinant structure and if not then it is said to be indeterminant structure. So, for a determinate structure ,no any additional equation is required for calculation of unknown while for the indeterminate structure some compatibility equations ( additional equations) which relates applied loads and reactions to the displacement known at some points in the structure are required.

So, the definition of determinate structure will be that- it is the type of structure in which the unknown forces and moments will be calculated by only using the statics equations. 

And the definition of indeterminate structure will be that- Indeterminate structure are the type of structure in which the unknown forces and moments are calculated by some compatibility equations in addition to those of equation of statics. 

INDETERMINACY OF STRUCTURE

For indeterminate structures, the difference between the total number of unknowns and the available number of static equilibrium equation is known as indeterminacy of structure (or static indeterminacy).

Ds = (total unknown forces in member or at support reactions)-(available equilibrium equation)

EXTERNAL AND INTERNAL INDETERMINACY

External indeterminacy is difference between total number of support reactions and the number of available static equilibrium equations.

∴ Dse = (total number of support reactions) - (available equilibrium equation)

Internal indeterminacy is indeterminacy in excess of external indeterminacy.

∴ Dsi = (total indeterminacy) - (external indeterminacy)

STATIC EQUILLIBRIUM EQUATION

• For PLANAR STRUCTURE i.e. 2D structure

1. ∑ Fx =0
2. ∑ Fy =0
3. ∑ M =0

• For SPACE STRUCTURE i.e. 3D structure 
  

1. ∑ Fx =0
2. ∑ Fy =0
3. ∑ Fz =0
4. ∑ Mx =0
5. ∑ My =0
6. ∑ Mz =0

VARIATION OF AIR DENSITY WITH ALTITUDE

 As fluid properties such as pressure and temperature varies non uniformly with the variation of the altitude, in result the density of air also varies nonuniformly. In lower atmosphere with increase in altitude the temperature and pressure of air decreases but as altitude increases the temperature and pressure varies non uniformly and so the density varies with altitude. The variation  of properties follows following Atmospheric model:-

(ENGLISH UNIT)

H- altitude(ft) ,  T- temperature(℉)  , P - pressure(lbs/sq.ft) , ρ -density(slugs/cu.ft)

             ρ= P/(1718*(T+459.7))

1. TROPOSPHERE REGION (H<36152 ft)

      T = 59-0.00356H  i.e. Temperature decreases linearly

      P = 2116*((T+459.7)/518.6)^ 5.256 i.e. Pressure decreases exponentially

2. LOWER STRATOSPHERE REGION (36152<H<82345)

      T = -70 i.e. Temperature remains constant

      P = 473.1 * e^(1.73-0.000048H) i.e. Pressure decreases exponentially

3.UPPER STRATOSPHERE REGION(H>82345)

    T= -205.05+0.00164H i.e. Temperature increases linearly

    P= 51.97*((T+459.7)/389.98)^(-11.388) i.e. Pressure decreases exponentially

(SI UNITS)

H- altitude(m) ,  T- temperature(℃)  , P - pressure(Kg/cu.m) , ρ -density(K-Pa)

ρ= P/(0.2869*(T+273.1))

1. TROPOSPHERE REGION (H<11000)

T=15.04 - 0.00649H

P= 101.29*((T+273.1)/288.08)^5.256

2. LOWER STRATOSPHERE REGION (11000<H<25000)

T= -56.46

P = 22.65*e^(1.73-0.000157H)

3.UPPER STRATOSPHERE REGION(H>25000)

T= -131.21 + 0.00299H

P = 2.488* ( (T+273.1)/216.6) ^(-11.388)