What is Statically Indeterminate Structure:
In statics, a structure is statically indeterminate when the static
equilibrium equations are insufficient for determining the internal
forces and reactions on that structure.
Based on Newton's laws of motion, the equilibrium
available for a two-dimensional body are
: the vectorial sum of the forces acting on the body equals
zero. This translates to
Σ H = 0: the sum of the horizontal components of the forces equals
Σ V = 0: the sum of the vertical components of forces equals zero;
: the sum of the moments (about an arbitrary point) of all
forces equals zero.
In the beam, the four unknown reactions are VA, VB, VC and HA. The
equilibrium equations are:
Σ V = 0:
VA − Fv + VB + VC = 0
Σ H = 0:
HA − Fh = 0
Σ MA = 0:
Fv · a − VB · (a + b) - VC · (a + b + c) = 0.
Since there are four unknown forces (or variable)
(VA, VB, VC and HA) but only three equilibrium equations,
this system of simultaneous equations does not have a
unique solution. The structure is therefore classified
as statically indeterminate.
CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS
Equilibrium equations could Equilibrium equations could
be directly solved, and thus
be solved only when
forces could be calculated
coupled with physical law
in an easy way
and compatibility equations
Not survivable, moderately
used in modern aviation
(due to damage tolerance
Survivable, widely used in
(due to damage tolerance
Easy to manufacture
Hard to manufacture
What is Slope Deflection Method?
In the slope-deflection method, the relationship is established
between moments at the ends of the members and the
corresponding rotations and displacements. This method was
developed by Axel Bendexon in Germany in 1934. This method
is applicable for the analysis of statically indeterminate beams
or rigid frames.
Slope Deflection Equation
Consider a beam segment AB having end relations θA & θB
and relative displacement Δ as shown below-
Slope deflection equation relates the moment acting on
the ends of a member with the end rotations and relative
MAB = MFAB + 2EI/L (2θA + θB + 3Δ/L)
MBA = MFBA + 2EI/L (2θB + θA + 3Δ/L)
If relative displacement Δ is zero, then –
MAB = MFAB + 2EI/L (2θA + θB)
MBA = MFBA + 2EI/L (2θB + θA)
1. Express the fixed end moment due to loads
2. Express the end moment in terms of the end rotations
and relative displacement.
3. Consider the condition of equilibrium of the joint
4. Solve for unknown rotations and displacements
5. Find the end moments from slope deflection equation.