Structure I Course Code: ARCH 208 Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 5.1,5.2

Example 1 Determine the reactions at the supports for the beam shown in Fig 5 kN/m 3 kN/m Hinge Hinge 20 m 20 m 50 m 20 m 20 m

Free Body Diagram

Next, we apply the equation of condition, which involves the summation of moments about B of all the forces acting on the portion AB.

Similarly, by applying the equation of condition, we determine the reaction Fy as follows.

The remaining two equilibrium equations can now be applied to determine the remaining two unknowns, Cy and Dy:

It is important to realize that the moment equilibrium equations involve the moments of all the forces acting on the entire structure, whereas, the moment equations of condition involve only the moments of those forces that act on the portion of the structure on one side of the internal hinge. Finally, we compute Dy by using the equilibrium equation:

Applications of the Equations of Equilibrium (Frame) The support reactions of rigid frame structures may be determined using the equations of static equilibrium, and the internal forces in the members from a free body diagram of the individual members. The internal forces on a member are most conveniently indicated as acting from the node on the member: i.e., as support reactions at the node.

Determinacy & Stability Rigid frame with no internal hinges In a rigid frame with j nodes, including the supports, 3 j equations of equilibrium may be obtained since, at each node: Each member of the rigid frame is subjected to an unknown axial and shear force and bending moment. If the rigid frame has n members and r external restraints, the number of unknowns is (3 n+ r). Thus, a beam or frame is determinate when the number of unknowns equals the number of equilibrium equations or: 3n + r = 3j

Example 2 Classify each of the frames shown in Fig. as statically determinate or statically indeterminate f statically indeterminate, report the no. of degree of indeterminacy

Determinacy & Stability Rigid frame with internal hinges or roller 3n + r = 3j + b +2s where n is the number of members, j is the number of nodes in the rigid frame before the introduction of hinges, r is the number of external restraints, b is the number of internal hinges, and s is the number of rollers introduced.

Frame is statically indeterminate ( First degree)

Example 3 Determine the horizontal and vertical components of reaction at the pins A, B, and C of the two-member frame shown in Fig..

Free Body Diagram

Example 4 Determine the support reactions and member forces in the rigid frame shown in Figure due to the applied loads indicated.

Free Body Diagram

The vertical reaction at support 4 of the frame is obtained by considering moment equilibrium about support 1. Hence:

The internal forces in member 1-2 are determined from a free body diagram of the member, as shown in Figure . Resolving forces vertically gives the internal force acting at node 2 as:

The internal forces in member 3-4 are determined from a free body diagram of the member, as shown in Figure . Resolving forces vertically gives the internal force acting at node 3 as:

The internal forces in member 23 are determined from a free body diagram of the member, as shown in Figure . Resolving forces vertically gives the internal force acting at node 2 as:

Example 5 Determine the components of reaction at the fixed support D and the pins A, B, and C of the three-member frame. Neglect the thickness of the members. kN/m m kN/m m m

Example 6 Determine the reactions at the supports A and B. The joints C and D are fixed connected.