A beam with a larger number of reactions than can be calculated using equations of static equilibrium is said to be statically indeterminate. The number of reactions is excess of the number of equilibrium equations is called the degree of statical indeterminacy redundancy.
To determine the forces is a statically indeterminate beam, account must be taken of the deflections of the beam from which, or the basis of compatibility, supplementary equations needed to calculate the reactions are obtained.
Types of Statically Indeterminate Beams
Continuous Beam: has more than one span which is continuous internal supports.
Note that the degree of statical indeterminacy can be reduced by introducing hinges on the beam. In the general case,
DSR = 3b + r - 3j
where b = number of members
r = number of reactions
j = number of joint in beam
Note that if the beam is subjected to only vertical loads, then DSR = 2b + 4 - 2j.
Analysis of Statically Indeterminate Beams
Essentially the same procedure as for the statically determinate beams is employed. The principal objective is to determine the redundant reactions which then reduces the problem to a statically determinate one.
Method of Successive Integration
The equations needed are exactly the same as was obtained for the statically determinate beams. There are always enough BCs to obtain the constants of integration as well as the redundant reactions.
Note that judicious choice of BCs will greatly simplify/minimise the computation involved. Using the 2nd Order Differential Equation of the deflection curve also simplifies the computation by reducing the number of constants.
Example (Problem 8.2-7)
Obtain the equation of the deflection curve and the reactions RA, RB and MA for a propped cantilever beam A-B supporting a triangular load of maximum intensity q.
Use of Castigliano's Theorem to obtain Redundant Reaction (See Example 1: Page 538)
Consider a section ?-? at distance x from support B.
According to Castigliano's Theorem, the deflection ?B due to the force RB is given by
Moment-Area Method for Statically Indeterminate Structures
This method involves the use of the two-moment area theorems to obtain the supplementary equations required to calculate the redundant reactions. These supplementary equations are based as known slopes and deflections of the beam and their member will always equal the number of redundant reactions.
This procedure is based on the flexibility method. The statically indeterminate beam is split into a primary beam (released beam), which is statically determinate and is acted upon by the externally applied loads, and a secondary beam (which is the same as the primary beam but with only the redundant reaction acting). The bending moment diagrams for both the primary and secondary structures are drawn. By applying the two moment-area theorems and using values of known boundary conditions, the redundant reactions are computed.
Example (Problem 8.3-1)
Determine the reactions RA, RB and MA for a propped cantilever beam AB subjected to a uniform load of intensity q.
Example (Problem 8.3-3)
Determine the reactions Ra, Rb, and Ma for the propped cantilever beam AB loaded as shown in the figure. Also, draw the shear-force and bending-moment diagrams for the beam, labeling all critical ordinates.
The two moment-area theorems are used to obtain additional equations to represent conditions as the slope and deflection of the beam and hence obtain the redundant reactions.
The approach is very similar to the method of superposition is that the statically indeterminate beam is represented by an equivalent statically determinate primary structure acted upon by the applied loads and a secondary structure (which is the primary structure with only the redundant reaction acting on it).
The effect of the external load on the primary structure and the redundant load on the secondary structure are obtained using the moment-area method and compatibility conditions are enforced to obtain the redundant reactions. Note that deformation corresponding to the selected redundant is utilized.
Example (Problem 10.3-7)
The load on a fixed-end beam AB of length L is distributed in the form if a sine curve. The intensity of the load is given by the equation q = q0sin px/L. Beginning with the forth-order differential equation of the deflection curve (the load equation), obtain the reactions of the beam and the equation of the deflection curve.
Example (Example 10-1 pg. 685)
A propped cantilever beam AB of length L supports a uniform load of intensity q. Analyze this beam by solving the second-order differential equation of the deflection curve (the bending-moment equation). Determine the reactions, shear forces, bending moments, slopes, and deflections of the beam.
Example (10-7 pg. 709)
The continuous beam shown has three spans of equal length L and constant moment of inertia I. The first span is subject to a uniform load of intensity q and the third span supports a concentrated load P of magnitude qL. The concentrated load acts at distance 3L/4 from support 3. Determine the reactions of the beam using the three-moment equation, and then construct the shear-force and bending-moment diagrams for the beam.
Example (Problem 10.4-6)
A continuous beam ABC with two unequal spans, one length L and one length 2L, supports a uniform load of intensity q. Determine the reactions RA, RB, and RC for this beam. Also, draw the shear-force and bending-moment diagrams, labeling all critical ordinates.
Solve the problem below using Castigliano's 2nd theorem. Obtain the support reactions.
Method of Superposition
This method involves the replacement of the statically indeterminate beam by a statically determinate primary structure acted upon by the external loads and secondary structures acted upon by the redundants. The deformations (deflection or rotation) corresponding to the selected redundants are determined in both the primary R secondary structure and compatibility of deformation is enforced. The method envisages the use of Tables of Load-Deflection in specifying the deformations due to the external loading and redundant reactions.
Example (Problem 9.5-10)
A beam ABC having flexural rigidity EI = 75 kNm2 is loaded by a force P = 480 N at end C and tied down at end A by wire having axial rigidity EA = 900 kN. What is the deflection at point C when the load P is applied?
Example (Problem 9.9-7)
The cantilever beam ACB shown is subjected to a uniform load of intensity q acting between points A and C. Determine the angle of rotation qA at the free end A. (Obtain the solution by using the modified form of Castigiano's theorem.)