# Is static determinacy regardless of the number of reactions?

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I assume those supports in the first figure are pins so there are 4 reactions in total. And as shown in the second figure, there are 3 reactions in total.

I thought for a structure in plane to be statically determinate, it must have only 3 reactions, but these are quite contradictory to me.

• Jan 30, 2021 at 9:36

We usually describe structures as either statically determinate or indeterminate.

However, this determinacy is actually two-fold: there's internal and external determinacy.

External static determinacy describes whether the supports can be determined via static equilibrium equations. This is what you were thinking of when you are looking at the number of supports. However, both examples are actually externally statically determinate: the simple cantilever truss has four supports, but it also has a hinge, which allows us to perform bending moment calculations to one side of the hinge, granting us a fourth equilibrium equation; and the truss frame only has three supports and is therefore trivially easy to solve for its reactions.

Internal static determinacy describes whether the internal forces of the members can also be determined via static equilibrium equations. This is what the textbook is actually talking about in this case. As described by @kamran's answer, this is a function of the number of members, reactions, and joints (a statically determinate truss has $$m + R \leq 2j$$).

The first truss satisfies this equation ($$2 + 4 \leq 2\cdot3$$), while the second doesn't ($$6 + 3 \leq 2\cdot4$$). Therefore the first truss is internally statically determinate while the second is internally statically indeterminate.

An externally statically indeterminate structure will always be internally statically indeterminate. However, an externally statically determinate structure can be either internally statically determinate or indeterminate.

A truss is determinate when all its member forces and reactions can be determined using only static equations. without any consideration to section properties.

For 2D trusses this means the number of members plus the number of support reactions must not exceed the number of joints times 2, m+R = 2j.

Your 2nd structure has 6 members 4 joints and 3 ractions.

6+3 >2*4

Therefore it is indeterminate.

Intuitively by just looking at the two diagonal members we can tell their section properties play a role in the behavior of the truss.

Imagine the one ending at upper right corner has a stiffness 4 times bigger than the other diagonal. It attracts most of the layeral forces and of course more vertical force as well.