Beam Analysis Using the Stiffness Method

When subdividing a beam into finite elements for stiffness method analysis, where should nodes generally be located?

For the stiffness method applied to beams, why are transformation matrices, which are required for trusses, generally not needed?

In the stiffness method for a beam, how many degrees of freedom are typically considered at each node if both bending and shear effects are included?

What is the value of the stiffness influence coefficient k_NyNy in the beam-member stiffness matrix, which relates the shear force at the near end (Ny) to a vertical displacement at the near end (Ny)?

What is the physical meaning of the elements in the second column of the beam-member stiffness matrix k presented in Equation 15-1?

How is the structure stiffness matrix K assembled from the individual member stiffness matrices k?

What does the partitioned matrix equation Qk = K11*Du + K12*Dk represent in the stiffness method?

When analyzing a beam with a distributed load using the stiffness method, what is the purpose of the fixed-end loading matrix, q0?

For a beam element of length L = 4 units with constant EI, what is the value of the stiffness coefficient k_NzFz, which relates the moment at the near end (Nz) to a rotation at the far end (Fz)?

For a prismatic beam element of length L and properties E and I, what is the value of the stiffness influence coefficient k_FyFy, which relates the shear force at the far end (Fy) to a vertical displacement at the far end (Fy)?

According to the procedure for analysis, what should be done with the fixed-end loadings from an intermediate load on an element when establishing the known external loads vector Qk?

A beam element has a length of 6 meters, and constant properties E and I. What is the stiffness coefficient relating the moment at the far end (qFz) to the vertical displacement at the near end (dNy)?

What does a negative value in the final calculated member force vector, q, indicate?

In a beam with an internal hinge at node 3 connecting elements 2 and 3, how does the kinematic indeterminacy change compared to a continuous beam at that node?

For a beam element with L=2 and constant EI, what is the value of the stiffness coefficient k_FzFz, which relates the moment at the far end (Fz) to the rotation at the far end (Fz)?

What is the order (size) of the structure stiffness matrix K for a beam that has been assigned a total of eight degrees of freedom (code numbers 1 through 8)?

For a beam element subject to a uniform distributed load w over its length L, what are the fixed-end shear reactions, (q0)_Ny and (q0)_Fy, used in the stiffness method?

In the general structure stiffness equation Q = KD, what do the column matrices Q and D represent?

For a beam element of length L=5 with constant EI, what is the value of the stiffness coefficient relating the shear force at the far end (qFy) to the rotation at the far end (dFz)?

What is the final equation used to determine the internal shear and moment, q, at the ends of a beam element that has an intermediate loading?

Consider a compound beam with an internal pin connecting two members. A single downward vertical load is applied at the pin. For the stiffness method analysis, what is the primary reason for placing a node at the pin?

In a beam analysis using the stiffness method, a node is located at a roller support. Which degrees of freedom at this node are typically constrained and which are unconstrained?

For a beam element with length L=1, and constant properties E and I, what is the value of the stiffness influence coefficient k_NyFz, which relates the shear at the near end (Ny) to the rotation at the far end (Fz)?

What is the primary assumption about each finite element of a beam used in the stiffness method derivation?

In the partitioned stiffness equation Qu = K21*Du + K22*Dk, what does Qu represent?

A beam member has a length of 10 ft. Its material has E = 29000 ksi and its cross-section has I = 510 in4. To use these values in the stiffness matrix from Equation 15-1, what must first be done?

What property must the member and structure stiffness matrices, k and K, always have as a partial check of calculations?

A continuous beam over two 2-meter spans is fixed at the left end, has a roller in the middle, and a pin at the right end. Using the stiffness method, how many unconstrained degrees of freedom does this beam have?

For a beam element of length L with constant EI, which stiffness coefficient relates the shear force at the near end (qNy) to the rotation at the near end (dNz)?

What is the kinematic indeterminacy of a propped cantilever beam (one end fixed, one end roller supported)?

A simply supported beam of length L is subjected to a single concentrated load P at its midspan. To analyze this using the stiffness method, how many elements and nodes are required at a minimum?

In the fixed-end loading column matrix q0 for a uniformly loaded beam element, the signs for the two moments are the same (both positive, meaning counter-clockwise). Why is this?

What does the stiffness influence coefficient K23 represent in the structure stiffness matrix K?

For a beam member with L=3 and constant EI, calculate the value of the stiffness coefficient k_NyFy relating the shear force at the near end (Ny) to the vertical displacement at the far end (Fy).

A continuous beam has two elements and three nodes. The first element has stiffness matrix k1 and the second has k2. If node 2 is common to both elements, how is the structural stiffness K at the degrees of freedom for node 2 determined?

The solution to a beam problem using the stiffness method yields a displacement vector Du. What is the next step to find the support reactions?

A beam element with constant EI has a length of 10 units. A positive (upward) vertical displacement dFy' = 1 is imposed at the far end, while all other displacements are held at zero. What is the resulting moment at the near end, qNz'?

In Example 15.2, a compound beam with an internal pin is analyzed. What is the size of the assembled structure stiffness matrix K?

When a support on a beam settles (e.g., a roller support moves down by 2 mm), how is this handled in the stiffness method formulation Q = KD?

In Example 15.1, two beam elements of length 2 m and constant EI are used. For the first element, k1, what is the value of the stiffness coefficient relating shear at the near end to rotation at the near end?

The sign convention for the beam-member stiffness matrix in Chapter 15 defines positive moments (qNz, qFz) as being what direction?

In Example 15.4, a beam with a distributed load is analyzed. Why are the values 144 and 1008 placed in the known load matrix Qk?

After solving for the unknown displacements Du and support reactions Qu, what is the final step in the 'Procedure for Analysis' outlined in Chapter 15?

What is the kinematic indeterminacy of a beam that is fixed at both ends?

In Example 15.5, the final calculated vertical displacement at the point of the load is D1 = -1.667P/EI. What does the negative sign indicate?

The member stiffness matrix k in Equation 15-1 relates which two quantities for a single beam element?

In Example 15.3, a support settlement of 1.5 mm is given for a beam where E=200 GPa and I=22(10^-6) m4. Why is it important to use consistent units for these values?

What is the primary difference in modeling between a beam with a uniformly distributed load and a beam with a triangular distributed load using the stiffness method?

A beam element with constant EI and length L=4 is fixed at the near end and a positive (counter-clockwise) rotation dFz = 1 is imposed at the far end. What is the value of the reaction moment at the fixed near end, qNz?

Why is the beam member stiffness matrix k in Equation 15-1 the same in both local and global coordinates?