Plane Frame Analysis Using the Stiffness Method

In the stiffness method for plane frame analysis, which of the following represents the six load-displacement relations for a member in its local coordinate system?

What is the purpose of the displacement transformation matrix, T, in the analysis of plane frames?

How is the member's global stiffness matrix, k, formulated using the local stiffness matrix, k', and the displacement transformation matrix, T?

What does each column of the 6x6 global stiffness matrix k for a frame member represent?

In the 'Procedure for Analysis' for frames, how are the direction cosines lambda_x and lambda_y determined?

In Example 16.1, what are the direction cosines (lambda_x, lambda_y) for Member 1, which is horizontal and connects node 1 (at origin) to node 2 (at x=20, y=0)?

In Example 16.1, what are the direction cosines (lambda_x, lambda_y) for Member 2, which is vertical and connects node 2 (at x=20, y=0) to node 3 (at x=20, y=-20)?

In Example 16.1, with E = 29(10^3) ksi, I = 500 in^4, and L = 20(12) inches, what is the calculated value of the stiffness term 12EI/L^3?

After solving for displacements in Example 16.1, the support reaction Q6 is calculated. The relevant row from the partitioned stiffness matrix is [0, -12.6, 1510.4, 0, 1510.4] and the displacement vector D is [0.696, -1.55(10^-3), -2.488(10^-3), 0.696, 1.234(10^-3)]. What is the value of Q6?

What is the physical meaning of the internal loading q2 = 1.87 k calculated for member 1 at node 2 in Example 16.1?

In Example 16.2, why is the analysis for the frame in Fig. 16-5b later modified by the loads shown in Fig. 16-5c?

In Example 16.2, for Member 1, what are the calculated direction cosines lambda_x and lambda_y given its geometry?

For Member 2 in Example 16.2, which is horizontal, what are the direction cosines lambda_x and lambda_y?

In Example 16.2, the structure stiffness matrix is assembled. What is the value of the term K(2,2) in the final matrix?

After solving the equations in Example 16.2, what is the calculated value for the displacement D2, which corresponds to the vertical displacement at node 2?

What is the final bending moment at the right end of member 2 (node 3) in Example 16.2 after accounting for superposition?

What are the three main types of data input required for a structural analysis program as described in the chapter?

In the general procedure for frame analysis using the stiffness method, which numbers are used to identify the unconstrained degrees of freedom?

What does the 3x3 submatrix in the top-left corner of the local stiffness matrix k' (Eq. 16-1) represent?

What is the correct relationship between the global forces Q, the local forces q, and the transformation matrix T?

According to the final computed results in Example 16.2, what is the value of the support reaction Q5?

In the local stiffness matrix for a frame member (Eq. 16-1), what is the value of the coefficient relating the near-end moment (qNz') to the far-end rotation (dFz')?

In the 'Procedure for Analysis', what is the final step to compute the internal loadings q at the ends of the members?

For the inclined member in Example 16.2 with AE/L = 1160 k/in, 12EI/L^3 = 7.73 k/in, and direction cosines lx=0.8, ly=0.6, what is the value of the global stiffness matrix term k1(1,1)?

What does a negative sign on a calculated unknown quantity, such as a displacement or reaction, indicate in the stiffness method analysis?

In Example 16.1, after calculating support reactions, Q8 is found to be 1.87 k. What does this value represent physically?

In the general expression for the global stiffness matrix of a frame member (Eq. 16-10), what does the element k(3,1) represent?

The final assembled structure stiffness matrix K, as seen in Examples 16.1 and 16.2, should always have what property?

In the local stiffness matrix for a frame member (Eq. 16-1), what is the value of the coefficient relating the near-end axial force (qNx') to the far-end axial displacement (dFx')?

What is the final vertical shear force at the left end of member 2 (node 2) in Example 16.2 after accounting for superposition?

In the displacement transformation matrix T for a plane frame member (Eq. 16-3), what does the element T(2,1) represent?

In the force transformation matrix TT (Eq. 16-5), what does the element at row 1, column 2, represent?

What is the primary reason for extending the analysis of beams to frames using transformation matrices, as mentioned in the introduction to Chapter 16?

In Example 16.1, what is the value of the internal axial force in member 2?

In the local stiffness matrix for a frame member (Eq. 16-1), what does the coefficient k(2,5) represent?

Based on Example 16.2, what is the value of the term 4EI/L for Member 1, where L=25(12) inches, E=29(10^3) ksi, and I=600 in^4?

In Example 16.2, a 3 k/ft distributed load over 20 ft on the horizontal member is replaced by equivalent end moments and shears. What is the value of the equivalent end moment?

After computing the displacements in Example 16.2, what is the final calculated support reaction Q7?

For the horizontal member (Member 2) in Example 16.2, with L=20(12) in, E=29(10^3) ksi, and I=600 in^4, what is the value of the stiffness term 12EI/L^3?

How is the internal loading `q` for member 1 computed in the final step of Example 16.1?

What is the primary difference between a plane frame member and a plane truss member in the context of stiffness analysis?

In Example 16.1, what is the calculated value of the support reaction Q9?

In Example 16.2, what does the value -1200 in the Qk vector represent?

What does the term k(6,6) in the local stiffness matrix for a frame member (Eq. 16-1) represent?

For the inclined member (Member 1) in Example 16.2, with lx=0.8, ly=0.6, and 6EI/L^2 = 1160 k, what is the value of the global stiffness matrix term k1(3,2)?

What is the physical interpretation of the displacement D3 = -0.00217 rad calculated in Example 16.2?

In the general expression for the global stiffness matrix (Eq. 16-10), what is the formula for the term k(4,1), which relates the far-end x-force to the near-end x-displacement?

Based on the FBD for member 2 in Figure 16-4d, what is the internal shear force at the bottom end (node 3)?

In the displacement transformation matrix T (Eq. 16-3), what is the value of the element T(3,3), which relates local rotation dNz' to global rotation DNz?

What is the final horizontal reaction at the support at node 1 in Example 16.2?