World Library  
Flag as Inappropriate
Email this Article

Statically determinate

Article Id: WHEBN0000899891
Reproduction Date:

Title: Statically determinate  
Author: World Heritage Encyclopedia
Language: English
Subject: Index of structural engineering articles
Publisher: World Heritage Encyclopedia

Statically determinate

In statics, a structure is statically indeterminate (or hyperstatic)[1] 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 equations available for a two-dimensional body are

  • \sum \vec F = 0 : 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 zero;
Σ V = 0: the sum of the vertical components of forces equals zero;
  • \sum \vec M = 0 : the sum of the moments (about an arbitrary point) of all forces equals zero.

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

Σ V = 0:

VAFv + VB + VC = 0

Σ H = 0:

HAFh = 0

Σ MA = 0:

Fv · aVB · (a + b) - VC · (a + b + c) = 0.

Since there are four unknown forces (or variables) (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.

Statically determinate

If the support at B is removed, the reaction VB cannot occur, and the system becomes statically determinate (or isostatic).[2] Note that the system is completely constrained here. The system becomes an exact constraint kinematic coupling. The solution to the problem is

H_A=F_h ,
V_C=\frac{F_v\cdot a}{a+b+c} ,
V_A=F_v-V_C .

If, in addition, the support at A is changed to a roller support, the number of reactions are reduced to three (without HA), but the beam can now be moved horizontally; the system becomes unstable or partially constrained -- a mechanism rather than a structure. In order to distinguish between this and the situation when a system under equilibrium is perturbed and becomes unstable, it is preferable to use the phrase partially constrained here. In this case, the 2 unknowns VA and VC can be determined by resolving the vertical force equation and the moment equation simultaneously. The solution yields the same results as previously obtained. However, it is not possible to satisfy the horizontal force equation unless F_h=0.

Statical indeterminacy is the existence of a non-trivial (non-zero) solution to the homogeneous system of equilibrium equations. It indicates the possibility of self-stress (stress in the absence of an external load) that may be induced by mechanical or thermal action.

See also

External links

  • Beam calculation online (Statically indeterminate)


This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Hawaii eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.