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## Q. 4.2

Classify each of the plane trusses shown in $Fig. 4.15$ as unstable, statically determinate, or statically indeterminate. If the truss is statically indeterminate, then determine the degree of static indeterminacy.  ## Verified Solution

(a) The truss shown in Fig. 4.15(a) contains $17$ members and $10$ joints and is supported by $3$ reactions. Thus, $m + r = 2j$. Since the three reactions are neither parallel nor concurrent and the members of the truss are properly arranged, it is statically determinate.                             Ans.

(b) For this truss, $m = 17 , j = 10$, and $r = 2$. Because $m + r < 2j$, the truss is unstable.                          Ans.

(c) For this truss, $m = 21 , j = 10$, and $r = 3$. Because $m + r > 2j$, the truss is statically indeterminate, with the degree of static indeterminacy $i = (m + r) – 2j = 4$. It should be obvious from Fig. 4.15(c) that the truss contains four more members than required for stability.                                Ans.

(d) This truss has $m = 16 , j = 10$, and $r = 3$. The truss is unstable, since $m + r < 2j$.                       Ans.

(e) This truss is composed of two rigid portions, $AB$ and $BC$, connected by an internal hinge at $B$. The truss has $m = 26 , j = 15$, and $r = 4$. Thus, $m + r = 2j$. The four reactions are neither parallel nor concurrent and the entire truss is properly constrained, so the truss is statically determinate.

(f ) For this truss, $m = 10 , j = 7$, and $r = 3$. Because $m + r < 2j$, the truss is unstable.                       Ans.

(g) In Fig. 4.15(g), a member $BC$ has been added to the truss of Fig. 4.15(f ), which prevents the relative rotation of the two portions $ABE$ and $CDE$. Since m has now been increased to $11$, with $j$ and $r$ kept constant at $7$ and $3$, respectively, the equation $m + r = 2j$ is satisfied. Thus, the truss of Fig. 4.15(g) is statically determinate.                                 Ans.

(h) The truss of Fig. 4.15(f ) is stabilized by replacing the roller support at $D$ by a hinged support, as shown in Fig. 4.15(h). Thus, the number of reactions has been increased to $4$, but $m$ and $j$ remain constant at $10$ and $7$, respectively. With $m + r = 2j$, the truss is now statically determinate.                              Ans.

(i) For the tower truss shown in Fig. 4.15(i), $m = 16 , j = 10$, and $r = 4$. Because $m + r = 2j$, the truss is statically determinate.                                             Ans.

( j) This truss has $m = 13 , j = 8$, and $r = 3$. Although $m + r = 2j$, the truss is unstable, because it contains two rigid portions $ABCD$ and $EFGH$ connected by three parallel members, $BF , CE$, and $DH$, which cannot prevent the relative displacement, in the vertical direction, of one rigid part of the truss with respect to the other.                                          Ans.

(k) For the truss shown in Fig. 4.15(k), $m = 19 , j = 12$, and $r = 5$. Because $m + r = 2j$, the truss is statically determinate.                                      Ans.