AHSANULLAH UNIVERSITY OF SCIENCE AND TECHNOLOGY

DEPARTMENT OF CIVIL ENGINEERING

PRESENTATION ON SOLVING STATICALLY INDETERMINATE STRUCTUE:MOMENT CO EFFICENT METHOD

COURSE TEACHER: Munshi Galib Muktadir

Sabreena Nasrin

SUBMITTED BY: KAZI RIYADH AL SAIF AHMED

I.D.: 10.01.03.128 SECTION:C 4th Year 2nd Semester..

STATICALLY INDETERMINATE STRUCTURE:MOMENT COEFFICIENTMETHOD

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 law of motoin, the equilibrium equations available for a two-dimensional body are the vectorial sum of the forces acting on the body equals zero

This translates Σ H = 0: the sum of the horizontal components of the forces equals zero; 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.

Reference: http://en.wikipedia.org/wiki/Statically_indeterminate

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 (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. Considerations in the material properties and compatibility in deformations are taken to solve statically indeterminate systems or structures.

Reference: http://en.wikipedia.org/wiki/Statically_indeterminate

In the beam construction on the right, the four unknown reactions are VA, VB, VC and HA. The equilibrium equations are:

Free Body Diagram of a statically indeterminate Beam

Moment co efficient method:

The ACI / SBC approximate method (also called coefficient method) is used for the analysis of continuous beams, ribs and two-way slabs.

. It allows for various load patterns where live load is applied on selected spans and maximum shear force and bending moment values are obtained by the envelope curves. This simplified and approximate method allows also for the real rotation restraint at external supports, where the real moment is not equal to zero.

Elastic analysis gives systematic zero moment values at all external pin supports. The coefficient method is thus more realistic but is only valid for standard cases. It is advised to use this method whenever its conditions of application are satisfied. Elastic analysis should be used only if the conditions of the code method are not satisfied.

The Moment Coefficient Method included for the first time in 1963 ACI Code is applicable to two-way slabs supported on four sides of each slab panel by walls, steel beams relatively deep, stiff, edge beams (h = 3hf).

 

Although, not included in 1977 and later versions of ACI code, its continued use is permissible under the ACI 318-08 code provision (13.5.1). Visit ACI 13.5.1.

 

two way slab moment co efficient method: casesdepending on the support conditions several case are possible…

Another typical case from moment co efficient method:

The method makes use of tables of moment coefficient for a variety of conditions. Thesecoefficients are based on elastic analysis but also account for inelastic redistribution.

 

 .C.S = column strip; M.S = middle strip

  Figure: Elements of two- way slab with beam by moment efficient method

Moments: M aneg.= Ca neg. Wula2

M b neg.=Cb negWl2

M a pos.= Ca pos. * . Wula2 + Ca pos *Wulb2

M b pos.= Cb pos. * . Wula2 + Cb pos *Wulb2

Where Ca and Cb are tabulated moment co efficient

Wu= ultimate uniform load la and lb = lenth of clear spans in short and long

directions respectively

Maximum Spacing and Minimum Reinforcement Requirement

l Maximum spacing (ACI 13.3.2):  smax = 2 hf in each direction.

  Minimum Reinforcement (ACI 7.12.2.1):

 Asmin = 0.0018 b hf for grade 60.

 Asmin = 0.002 b hf for grade 40 and 50

Advantages and Disadvantages of moment co efficient method:

In some respects, when estimating parameters of a known family of probability distributions, this method was superseded by Fisher's method of maximum likelihood, because maximum likelihood estimators have higher probability of being close to the quantities to be estimated.

The coefficients give more exact analysis. Significant economy can be achieved by making a more precise analysis. There should be no reversal of moments at the critical design sections near midspan or at the support faces.

References

  CRSI Design Handbook  

ACI 318  

Design of Concrete Structures 13th Ed. by Nilson, Darwin and Dolan.