**Flexibility Method: Beams and Frames with More Than One Redundant, and a Truss with One Redundant****Solve the following 3 problems:****Problem 1:**Given: The following beam has fixed supports at A and B, and has a constant EI.![Diagram: A beam with fixed supports at A and B, subjected to a uniform load w along its length L.]**Required:**- Evaluate the two vertical reactions (Ay and By) and the two moment reactions (MA and MB). Neglect Ax and Bx.- Draw the Shear Diagram- Draw the Moment Diagram- Sketch the deflected shape**Hint:** In this problem, it looks like it is indeterminate to the third degree, but let’s neglect axial effects. By neglecting axial effects, we are not considering the horizontal force equilibrium equation (i.e., we neglect the sum of forces in the x-direction). Thus, we neglect the two horizontal reactions (Ax and Bx). Therefore, the problem simplifies as follows:We are left with 2 equilibrium equations (i.e., sum of forces in the y-direction and sum of moments). So, you have 2 equations and 4 unknowns: Ay, MA, By, and MB. Thus, we need to come up with 2 more equations......these are your compatibility equations! In the end, you’ll need to consider two redundants.**For this problem, please consider the following redundants:**1. The moment at B [MB] and2. The vertical reaction at B [By].Do you know what the primary structure will look like? It will be a cantilever! You will need 3 diagrams (Primary 0: primary structure with original loads; Primary 1: the primary structure with a unit VERTICAL force at B – **assume downward**; and Primary 2: the primary structure with a unit MOMENT at B – **assumed clockwise**. Each of the 3 diagrams will have a vertical deflection (movement 1) and

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Problem 1: Given: The following beam has fixed supports at A and B, and has a constant EI. A. Required: Evaluate the two vertical reactions (Ay and By) and the two moment reactions (MA and MB). Neglect Ax and Bx. Draw the Shear Diagram Draw the Moment Diagram Sketch the deflected shape

Problem 1: Given: The following beam has fixed supports at A and B, and has a constant EI. A. Required: Evaluate the two vertical reactions (Ay and By) and the two moment reactions (MA and MB). Neglect Ax and Bx. Draw the Shear Diagram Draw the Moment Diagram Sketch the deflected shape

Structural Analysis

6th Edition

ISBN:9781337630931

Author:KASSIMALI, Aslam.

Publisher:KASSIMALI, Aslam.

Chapter2: Loads On Structures

Section: Chapter Questions

Problem 1P

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Question

**Flexibility Method: Beams and Frames with More Than One Redundant, and a Truss with One Redundant**

**Solve the following 3 problems:**

**Problem 1:**

Given: The following beam has fixed supports at A and B, and has a constant EI.

![Diagram: A beam with fixed supports at A and B, subjected to a uniform load w along its length L.]

**Required:**

- Evaluate the two vertical reactions (A<sub>y</sub> and B<sub>y</sub>) and the two moment reactions (M<sub>A</sub> and M<sub>B</sub>). Neglect A<sub>x</sub> and B<sub>x</sub>.
- Draw the Shear Diagram
- Draw the Moment Diagram
- Sketch the deflected shape

**Hint:** In this problem, it looks like it is indeterminate to the third degree, but let’s neglect axial effects. By neglecting axial effects, we are not considering the horizontal force equilibrium equation (i.e., we neglect the sum of forces in the x-direction). Thus, we neglect the two horizontal reactions (A<sub>x</sub> and B<sub>x</sub>). Therefore, the problem simplifies as follows:

We are left with 2 equilibrium equations (i.e., sum of forces in the y-direction and sum of moments). So, you have 2 equations and 4 unknowns: A<sub>y</sub>, M<sub>A</sub>, B<sub>y</sub>, and M<sub>B</sub>. Thus, we need to come up with 2 more equations......these are your compatibility equations! In the end, you’ll need to consider two redundants.

**For this problem, please consider the following redundants:**

1. The moment at B [M<sub>B</sub>] and
2. The vertical reaction at B [B<sub>y</sub>].

Do you know what the primary structure will look like? It will be a cantilever! You will need 3 diagrams (Primary 0: primary structure with original loads; Primary 1: the primary structure with a unit VERTICAL force at B – **assume downward**; and Primary 2: the primary structure with a unit MOMENT at B – **assumed clockwise**. Each of the 3 diagrams will have a vertical deflection (movement 1) and

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Calculate the reactions at the supports of the following statically indeterminate beam, using the Method of Cross and applying the simplifications. Consider that the supports A and D are hinges and the supports B and C are rollers. Consider that EI is constant: