Application of the Equations of Equilibrium CE 213
Occasionally, the members of a structure are connected together in such a
way that the joints can be assumed as pins. Building frames and trusses
are typical examples that are often constructed in this manner. Provided a
pin-connected coplanar structure is properly constrained and contains no
more supports or members than are necessary to prevent collapse, the
forces acting at the joints and supports can be determined by applying the
three equations of equilibrium to each
member. Understandably, once the forces at the joints are obtained, the
size of the members, connections, and supports can then be determined
on the basis of design code specifications.
To illustrate the method of force analysis, consider the three-member
frame shown in Fig. 2–26a, which is subjected to loads and The
free-body diagrams of each member are shown in Fig. 2–26b. In total
there are nine unknowns; however, nine equations of equilibrium can be
written, three for each member, so the problem is statically determinate.
For the actual solution it is also possible, and sometimes convenient, to
consider a portion of the frame or its entirety when applying some of
these nine equations. For example, a free-body diagram of the entire
frame is shown in Fig. 2–26c. One could determine the three reactions
and on this “rigid” pin-connected system, then analyze
any two of its members, Fig. 2–26b, to obtain the other six unknowns.
Furthermore, the answers can be checked in part by applying the three
equations of equilibrium to the remaining “third”member.To summarize,
this problem can be solved by writing at most nine equilibrium equations
using free-body diagrams of any members and/or combinations of
connected members. Any more than nine equations written would not
be unique from the original nine and would only serve to check the
results.
Ax, Ay, Cx
P1 P2.
1©Fx = 0, ©Fy = 0, ©MO
Consider now the two-member frame shown in Fig. 2–27a. Here the
free-body diagrams of the members reveal six unknowns, Fig. 2–27b;
however, six equilibrium equations, three for each member, can be
written, so again the problem is statically determinate. As in the previous
case, a free-body diagram of the entire frame can also be used for part
of the analysis, Fig. 2–27c. Although, as shown, the frame has a tendency
to collapse without its supports, by rotating about the pin at B, this will
not happen since the force system acting on it must still hold it in
equilibrium. Hence, if so desired, all six unknowns can be determined by
applying the three equilibrium equations to the entire frame, Fig. 2–27c,
and also to either one of its members.
The above two examples illustrate that if a structure is properly
supported and contains no more supports or members than are necessary
to prevent collapse, the frame becomes statically determinate, and so the
unknown forces at the supports and connections can be determined
from the equations of equilibrium applied to each member. Also, if the
structure remains rigid (noncollapsible) when the supports are removed
(Fig. 2–26c), all three support reactions can be determined by applying
the three equilibrium equations to the entire structure. However, if the
structure appears to be nonrigid (collapsible) after removing the supports
(Fig. 2–27c), it must be dismembered and equilibrium of the individual
members must be considered in order to obtain enough equations to
determine all the support reactions.
way that the joints can be assumed as pins. Building frames and trusses
are typical examples that are often constructed in this manner. Provided a
pin-connected coplanar structure is properly constrained and contains no
more supports or members than are necessary to prevent collapse, the
forces acting at the joints and supports can be determined by applying the
three equations of equilibrium to each
member. Understandably, once the forces at the joints are obtained, the
size of the members, connections, and supports can then be determined
on the basis of design code specifications.
To illustrate the method of force analysis, consider the three-member
frame shown in Fig. 2–26a, which is subjected to loads and The
free-body diagrams of each member are shown in Fig. 2–26b. In total
there are nine unknowns; however, nine equations of equilibrium can be
written, three for each member, so the problem is statically determinate.
For the actual solution it is also possible, and sometimes convenient, to
consider a portion of the frame or its entirety when applying some of
these nine equations. For example, a free-body diagram of the entire
frame is shown in Fig. 2–26c. One could determine the three reactions
and on this “rigid” pin-connected system, then analyze
any two of its members, Fig. 2–26b, to obtain the other six unknowns.
Furthermore, the answers can be checked in part by applying the three
equations of equilibrium to the remaining “third”member.To summarize,
this problem can be solved by writing at most nine equilibrium equations
using free-body diagrams of any members and/or combinations of
connected members. Any more than nine equations written would not
be unique from the original nine and would only serve to check the
results.
Ax, Ay, Cx
P1 P2.
1©Fx = 0, ©Fy = 0, ©MO
Consider now the two-member frame shown in Fig. 2–27a. Here the
free-body diagrams of the members reveal six unknowns, Fig. 2–27b;
however, six equilibrium equations, three for each member, can be
written, so again the problem is statically determinate. As in the previous
case, a free-body diagram of the entire frame can also be used for part
of the analysis, Fig. 2–27c. Although, as shown, the frame has a tendency
to collapse without its supports, by rotating about the pin at B, this will
not happen since the force system acting on it must still hold it in
equilibrium. Hence, if so desired, all six unknowns can be determined by
applying the three equilibrium equations to the entire frame, Fig. 2–27c,
and also to either one of its members.
The above two examples illustrate that if a structure is properly
supported and contains no more supports or members than are necessary
to prevent collapse, the frame becomes statically determinate, and so the
unknown forces at the supports and connections can be determined
from the equations of equilibrium applied to each member. Also, if the
structure remains rigid (noncollapsible) when the supports are removed
(Fig. 2–26c), all three support reactions can be determined by applying
the three equilibrium equations to the entire structure. However, if the
structure appears to be nonrigid (collapsible) after removing the supports
(Fig. 2–27c), it must be dismembered and equilibrium of the individual
members must be considered in order to obtain enough equations to
determine all the support reactions.
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