29 Method of Joints Concept Overview
METHOD OF JOINTS | CONCEPT OVERVIEW The TOPIC of METHOD OF JOINTS can be referenced on page 68 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
CONCEPT INTRO: The METHOD OF JOINTS is the technique of solving for the internal forces of each truss member by representing each joint as a free body diagram, and using the equations of equilibrium to solve for any forces acting on that joint. As shown in the free body diagram of the overall truss system below, it is important that you are able to identify the support constraints, members, and joints of a truss system.
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By drawing a free body diagram of a joint, we are able to treat it is a body at rest, and analyze all the forces acting on it, including the internal forces produced by the members, as well as any external forces that make be acting on the joint. In the free body diagram below, we have drawn a free body diagram for joint A in the truss system on the previous page.
The forces acting on the joint can be treated as a concurrent force system for which only two equations of equilibrium are available. Therefore, we need to realize that we can only solve for two unknowns at each joint, and may need to analyze multiple joints if we are looking to solve more than 2 unknown variables.
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The topic of CONCURRENT FORCES can be referenced under the topic of STATICALLY DETERMINATE TRUSS on page 68 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. A CONCURRENT FORCE SYSTEM is a system of two or more forces whose lines of action ALL intersect at a common point. All of the individual vectors might not actually be in contact with the common point, but have lines of action that intersect with a common point. These are the simplest force systems to resolve with any one of many graphical or algebraic options.
The method of joints is based on the fact that any joint in a truss is in static equilibrium. The process of analyzing a structure using the Method of Joints starts with analyzing a joint on which there are no more than two unknown forces. The process is then continued by working your way through the structure, pin by pin, until all unknown forces have been determined. The formulas for EQUILIBRIUM REQUIREMENTS FOR THE METHOD OF JOINTS can be referenced under the topic of STATICALLY DETERMINATE TRUSS on page 68 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Made with
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The method of joints uses the summation of forces at joint to solve for the force in any member connected to that joint. It does not use the moment equilibrium equation to solve for the problem. When drawing the free body diagram, you should use the vector notation of a compressive force towards the joint, and tensile forces away from the joint. β πΉπ₯ = 0 and β πΉπ¦ = 0 Where: β’ πΉπ₯ is the horizontal forces and member components β’ πΉπ¦ is the vertical forces and member components At the initial stages of analysis, it is acceptable to assume tension for each member of a joint. If you get a negative answer, this tells you the member is actually in compression, and you can adjust your approach from there. Once a force is known at one end of a member, the force can be concluded for the other end of that particular member rotated by 180 degrees, to account for the change in line of action for the member
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. PROCEDURE FOR THE METHOD OF JOINTS: Using a Free Body Diagram, the unknown forces at each joint are determined until all unknown forces for all members in the entire structure are defined. The process is as follows: 1. Define the coordinate system of the of the truss system. For the purpose of the FE Exam, we should always use a Cartesian Coordinate System when working with truss systems. As there will be varying angles in each triangular group of members, we should not incline the coordinate system in any scenario.
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2. Draw a Free Body Diagram of the overall truss system and find the external support reactions, if required. This may require you to use the equations of equilibrium for the overall truss system, and sum the moments about a particular support constraint or joint.
You will typically find that a truss system is
supported by a roller and pin, so you will want to take the moment about the pin support constraint to account for the horizontal and vertical support reactions. 3. Select a joint to evaluate and isolate it from the rest of the truss system. You typically want to choose a joint with only 2 unknowns, as we can only write two equations for each joint. If a joint has more than 2 unknowns, then we will need to evaluate another joint or support nearby to write additional equations that we can use to substitute unknowns into. 4. Draw a Free Body Diagram of that joint and solve for the forces acting on the joint. Assume the unknown member forces to be tensile, unless otherwise stated in the problem. Forces drawn as acting towards the joint are compressive (-), and Made with
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forces drawing as away from the joint are tensile (+). Positive-values of a force indicate a correct assumption of the direction of forces. 5. Solve for the unknown forces in the members acting on the joint. If there are more than two unknowns, then you will need to set up equations that you will substitute in the equations of equilibrium for other joints that share common members. 6. Update the overall truss free body diagram as you solve for the forces in each member until you are able to solve for the desired forces in a particular member. NCEES will typically ask you to solve for the forces in 1 or 2 members, so you should not be solving for the forces in every member unless explicitly stated to do so. 7. Repeat steps 3-6 until you solve for all of the unknowns being asked in the problems statement. It is important to keep your work organized, and clearly identify the equations of equilibrium associated with joint being evaluated.
CONCEPT EXAMPLE: Given the simply supported truss system below, what are the internal forces in members AC and AD?
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A. π΄πΆ: 1,541.25 (π); π΄π·: 1725 (C) B. π΄πΆ: 1,541.25 (πΆ); π΄π·: 1725 (T) C. π΄πΆ: 4,507.51 (π); π΄π·: 479.45 (T) D. π΄πΆ: 4,507.51 (πΆ); π΄π·: 479.45 (C)
SOLUTION: The topic of METHOD OF JOINTS can be referenced under the topic of STATICALLY DETERMINATE TRUSS on page 68 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
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We are looking to solve for the forces in members AC and AD of the truss system. The first step in this problem is to define the coordinate system. As we will be working with trusses, we will use a Cartesian Coordinate System to define the xβ and π¦ β ππ₯ππ .
By visual inspection, we see that members AC and AD share joint A with a pinned support constraint. In order to solve for the forces in members AC and AD, we will need to calculate the support reactions generated by the pinned constraint at joint A.
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In order to solve for the support reaction at joint A, we will draw a free body diagram for overall truss system. As there is a pinned support at joint A, there are support reactions to restrain against translation vertically and horizontally. At joint B, there is a roller that will only provide support against translation vertically. Made with
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Now that we have a free body diagram of the overall truss system, we can solve for the restraining forces generated at the support constraints at joints A and B. It is important to remember that as we have 3 unknowns, we need to write at least 3 equations in order to solve for all of the unknowns. The formulas for EQUILIBRIUM REQUIREMENTS FOR THE METHOD OF JOINTS can be referenced under the topic of STATICALLY DETERMINATE TRUSS on page 68 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We will first sum the forces of the overall truss system along the π¦ β ππ₯ππ : Made with
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β πΉπ¦ = 0: π΄π¦ + π΅π¦ β 2000 ππ = 0
π΄π¦ + π΅π¦ = 2000 ππ We then sum the forces of the overall truss system along the π₯ β ππ₯ππ :
β πΉπ₯ = 0: π΄π₯ + 800 ππ = 0
π΄π₯ = β800 ππ We will then sum the moments around the support constraint at point A to establish the third equation of equilibrium. Point A is a good place to sum the moments about, as it will allow us to write an equation of equilibrium that has no terms associated with the support reactions generated by the pinned support constraint at point A. ππ΄ = 0: (2,000 ππ)(15 ππ‘) + (800 ππ)(20 ππ‘) β (π΅π¦ )(60 ππ‘) Solving for the vertical support reaction at point B, π΅π¦ , we find: π΅π¦ = 767 ππ We can then plug in the calculated value for the vertical support reaction at point B into the equation we derived by summing the forces about the π¦ β ππ₯ππ , and solve for the vertical support reaction at point A.
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π΄π¦ + π΅π¦ = 2,000 ππ π΄π¦ + 767 ππ = 2,000 ππ π΄π¦ = 1,233 ππ
Next, we will update our overall free body diagram to include the support reactions that we calculated. It is important that we maintain our overall free body diagram so we keep our equations organized, and do not mislabel or incorrectly write our equations of equilibrium. As we many need to evaluate multiple joints in the truss system in a given Made with
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problem, it is imperative that your work stay organized as you continue on with the problem.
As we are solving for the forces in members AC and AD, a good joint to evaluate for the missing forces is joint A. Looking at the free body diagram for joint A below, we see that there are two support reactions and two members acting at joint A as each joint can provide two equations of equilibrium, and we have only two unknowns, we can solve for the desired forces by simply looking at joint A. Made with
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Looking at the free body diagram for joint A as shown above, we notice that we have all of the information we need to solve for the force in member AC, except the angle that the member makes with member AD. As a truss is comprised of triangle, we draw a vertical bisector at joint C, to split triangle ACD in half, and create a triangle with a 90 Β° angle that we can use our trigonometric functions on.
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We now have the horizontal and vertical distance of our triangle using the vertical bisector at joint C to divide the horizontal distance of the original triangle in half. We can now use the tangent trigonometric function to calculate the angle that both members make at joint A.
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tan(π) =
πππππ ππ‘π π΄πππππππ‘
Plugging in the values of 30 π for the opposite side of the triangle and 40 π for the adjacent side of the triangle, we are able to calculate the angle of the incline as:
π = tanβ1 (
20 ) = 53.13Β° 15
We can then plug in the calculated angle value into our free body diagram for joint A. The free body diagram now has all of the relevant information we need to solve for the forces in member AC and AD.
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We can now redraw the free-body diagram of joint A with all of the forces labeled. When representing a member on the free body diagram, we will assume it is in tension (+), and its line of action is acting away from the joint. If the member is actually in compression, then our answer at the end will have an opposite sign, such that if we Made with
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assume a member is in tension to the right (+), then at the end our calculations, the member should be drawn with the line of action to the left to indicate compression (-).
Moving on the next step of the problem, we write the equations of equilibrium for joint A to solve for the forces in members AC and AD. It is important to realize that the force in member AC acts on joint A with an angle of 53.13Β°, so we will need to break that force into components when analyzing the vertical and horizontal planes.
We will first sum the forces of joint A along the π₯ β ππ₯ππ :
β πΉπ₯ = 0: 0 = β800 + πΉπ΄π· + πΉπ΄πΆ cos(53.13Β°) Made with
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We then sum the forces of joint A along the π¦ β ππ₯ππ : β πΉπ¦ = 0: 0 = 1,233 + πΉπ΄πΆ sin(53.13Β°)
Using the expression written for forces about the π¦ β ππ₯ππ , we are now able to solve for force in member AC as: πΉπ΄πΆ = β1,541.25 πππ (πππππππ π πππ) If you calculated the following answer for the force in member AC, then your calculator is in radians mode, and your answer is incorrect. You need to always make sure that it is in degree mode when working statics problems, as angles will always be given in degrees not radians. πππππ πππ π€ππ: πΉπ΄πΆ = β4,507.51 πππ (πππππππ π πππ) We can then plug in the calculated value for the force in member AC into the equation we derived by summing the forces about the π₯ β ππ₯ππ :
β πΉπ₯ = 0: 0 = β800 + πΉπ΄π· + (β1,541.25) cos(53.13Β°)
Solving for the force in member AD, we find: πΉπ΄π· = 1724.75 Made with
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Therefore, the correct answer choice is B. π¨πͺ: π, πππ. ππ (πͺ); π¨π«: ππππ (π»)
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CONCEPT INTRO: The METHOD OF JOINTS is the technique of solving for the internal forces of each truss member by representing each joint as a free body diagram, and using the equations of equilibrium to solve for any forces acting on that joint. As shown in the free body diagram of the overall truss system below, it is important that you are able to identify the support constraints, members, and joints of a truss system.
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by Prepineer | Prepineer.com
By drawing a free body diagram of a joint, we are able to treat it is a body at rest, and analyze all the forces acting on it, including the internal forces produced by the members, as well as any external forces that make be acting on the joint. In the free body diagram below, we have drawn a free body diagram for joint A in the truss system on the previous page.
The forces acting on the joint can be treated as a concurrent force system for which only two equations of equilibrium are available. Therefore, we need to realize that we can only solve for two unknowns at each joint, and may need to analyze multiple joints if we are looking to solve more than 2 unknown variables.
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The topic of CONCURRENT FORCES can be referenced under the topic of STATICALLY DETERMINATE TRUSS on page 68 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. A CONCURRENT FORCE SYSTEM is a system of two or more forces whose lines of action ALL intersect at a common point. All of the individual vectors might not actually be in contact with the common point, but have lines of action that intersect with a common point. These are the simplest force systems to resolve with any one of many graphical or algebraic options.
The method of joints is based on the fact that any joint in a truss is in static equilibrium. The process of analyzing a structure using the Method of Joints starts with analyzing a joint on which there are no more than two unknown forces. The process is then continued by working your way through the structure, pin by pin, until all unknown forces have been determined. The formulas for EQUILIBRIUM REQUIREMENTS FOR THE METHOD OF JOINTS can be referenced under the topic of STATICALLY DETERMINATE TRUSS on page 68 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Made with
by Prepineer | Prepineer.com
The method of joints uses the summation of forces at joint to solve for the force in any member connected to that joint. It does not use the moment equilibrium equation to solve for the problem. When drawing the free body diagram, you should use the vector notation of a compressive force towards the joint, and tensile forces away from the joint. β πΉπ₯ = 0 and β πΉπ¦ = 0 Where: β’ πΉπ₯ is the horizontal forces and member components β’ πΉπ¦ is the vertical forces and member components At the initial stages of analysis, it is acceptable to assume tension for each member of a joint. If you get a negative answer, this tells you the member is actually in compression, and you can adjust your approach from there. Once a force is known at one end of a member, the force can be concluded for the other end of that particular member rotated by 180 degrees, to account for the change in line of action for the member
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by Prepineer | Prepineer.com
. PROCEDURE FOR THE METHOD OF JOINTS: Using a Free Body Diagram, the unknown forces at each joint are determined until all unknown forces for all members in the entire structure are defined. The process is as follows: 1. Define the coordinate system of the of the truss system. For the purpose of the FE Exam, we should always use a Cartesian Coordinate System when working with truss systems. As there will be varying angles in each triangular group of members, we should not incline the coordinate system in any scenario.
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2. Draw a Free Body Diagram of the overall truss system and find the external support reactions, if required. This may require you to use the equations of equilibrium for the overall truss system, and sum the moments about a particular support constraint or joint.
You will typically find that a truss system is
supported by a roller and pin, so you will want to take the moment about the pin support constraint to account for the horizontal and vertical support reactions. 3. Select a joint to evaluate and isolate it from the rest of the truss system. You typically want to choose a joint with only 2 unknowns, as we can only write two equations for each joint. If a joint has more than 2 unknowns, then we will need to evaluate another joint or support nearby to write additional equations that we can use to substitute unknowns into. 4. Draw a Free Body Diagram of that joint and solve for the forces acting on the joint. Assume the unknown member forces to be tensile, unless otherwise stated in the problem. Forces drawn as acting towards the joint are compressive (-), and Made with
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forces drawing as away from the joint are tensile (+). Positive-values of a force indicate a correct assumption of the direction of forces. 5. Solve for the unknown forces in the members acting on the joint. If there are more than two unknowns, then you will need to set up equations that you will substitute in the equations of equilibrium for other joints that share common members. 6. Update the overall truss free body diagram as you solve for the forces in each member until you are able to solve for the desired forces in a particular member. NCEES will typically ask you to solve for the forces in 1 or 2 members, so you should not be solving for the forces in every member unless explicitly stated to do so. 7. Repeat steps 3-6 until you solve for all of the unknowns being asked in the problems statement. It is important to keep your work organized, and clearly identify the equations of equilibrium associated with joint being evaluated.
CONCEPT EXAMPLE: Given the simply supported truss system below, what are the internal forces in members AC and AD?
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A. π΄πΆ: 1,541.25 (π); π΄π·: 1725 (C) B. π΄πΆ: 1,541.25 (πΆ); π΄π·: 1725 (T) C. π΄πΆ: 4,507.51 (π); π΄π·: 479.45 (T) D. π΄πΆ: 4,507.51 (πΆ); π΄π·: 479.45 (C)
SOLUTION: The topic of METHOD OF JOINTS can be referenced under the topic of STATICALLY DETERMINATE TRUSS on page 68 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
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We are looking to solve for the forces in members AC and AD of the truss system. The first step in this problem is to define the coordinate system. As we will be working with trusses, we will use a Cartesian Coordinate System to define the xβ and π¦ β ππ₯ππ .
By visual inspection, we see that members AC and AD share joint A with a pinned support constraint. In order to solve for the forces in members AC and AD, we will need to calculate the support reactions generated by the pinned constraint at joint A.
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In order to solve for the support reaction at joint A, we will draw a free body diagram for overall truss system. As there is a pinned support at joint A, there are support reactions to restrain against translation vertically and horizontally. At joint B, there is a roller that will only provide support against translation vertically. Made with
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Now that we have a free body diagram of the overall truss system, we can solve for the restraining forces generated at the support constraints at joints A and B. It is important to remember that as we have 3 unknowns, we need to write at least 3 equations in order to solve for all of the unknowns. The formulas for EQUILIBRIUM REQUIREMENTS FOR THE METHOD OF JOINTS can be referenced under the topic of STATICALLY DETERMINATE TRUSS on page 68 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We will first sum the forces of the overall truss system along the π¦ β ππ₯ππ : Made with
by Prepineer | Prepineer.com
β πΉπ¦ = 0: π΄π¦ + π΅π¦ β 2000 ππ = 0
π΄π¦ + π΅π¦ = 2000 ππ We then sum the forces of the overall truss system along the π₯ β ππ₯ππ :
β πΉπ₯ = 0: π΄π₯ + 800 ππ = 0
π΄π₯ = β800 ππ We will then sum the moments around the support constraint at point A to establish the third equation of equilibrium. Point A is a good place to sum the moments about, as it will allow us to write an equation of equilibrium that has no terms associated with the support reactions generated by the pinned support constraint at point A. ππ΄ = 0: (2,000 ππ)(15 ππ‘) + (800 ππ)(20 ππ‘) β (π΅π¦ )(60 ππ‘) Solving for the vertical support reaction at point B, π΅π¦ , we find: π΅π¦ = 767 ππ We can then plug in the calculated value for the vertical support reaction at point B into the equation we derived by summing the forces about the π¦ β ππ₯ππ , and solve for the vertical support reaction at point A.
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π΄π¦ + π΅π¦ = 2,000 ππ π΄π¦ + 767 ππ = 2,000 ππ π΄π¦ = 1,233 ππ
Next, we will update our overall free body diagram to include the support reactions that we calculated. It is important that we maintain our overall free body diagram so we keep our equations organized, and do not mislabel or incorrectly write our equations of equilibrium. As we many need to evaluate multiple joints in the truss system in a given Made with
by Prepineer | Prepineer.com
problem, it is imperative that your work stay organized as you continue on with the problem.
As we are solving for the forces in members AC and AD, a good joint to evaluate for the missing forces is joint A. Looking at the free body diagram for joint A below, we see that there are two support reactions and two members acting at joint A as each joint can provide two equations of equilibrium, and we have only two unknowns, we can solve for the desired forces by simply looking at joint A. Made with
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Looking at the free body diagram for joint A as shown above, we notice that we have all of the information we need to solve for the force in member AC, except the angle that the member makes with member AD. As a truss is comprised of triangle, we draw a vertical bisector at joint C, to split triangle ACD in half, and create a triangle with a 90 Β° angle that we can use our trigonometric functions on.
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We now have the horizontal and vertical distance of our triangle using the vertical bisector at joint C to divide the horizontal distance of the original triangle in half. We can now use the tangent trigonometric function to calculate the angle that both members make at joint A.
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tan(π) =
πππππ ππ‘π π΄πππππππ‘
Plugging in the values of 30 π for the opposite side of the triangle and 40 π for the adjacent side of the triangle, we are able to calculate the angle of the incline as:
π = tanβ1 (
20 ) = 53.13Β° 15
We can then plug in the calculated angle value into our free body diagram for joint A. The free body diagram now has all of the relevant information we need to solve for the forces in member AC and AD.
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We can now redraw the free-body diagram of joint A with all of the forces labeled. When representing a member on the free body diagram, we will assume it is in tension (+), and its line of action is acting away from the joint. If the member is actually in compression, then our answer at the end will have an opposite sign, such that if we Made with
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assume a member is in tension to the right (+), then at the end our calculations, the member should be drawn with the line of action to the left to indicate compression (-).
Moving on the next step of the problem, we write the equations of equilibrium for joint A to solve for the forces in members AC and AD. It is important to realize that the force in member AC acts on joint A with an angle of 53.13Β°, so we will need to break that force into components when analyzing the vertical and horizontal planes.
We will first sum the forces of joint A along the π₯ β ππ₯ππ :
β πΉπ₯ = 0: 0 = β800 + πΉπ΄π· + πΉπ΄πΆ cos(53.13Β°) Made with
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We then sum the forces of joint A along the π¦ β ππ₯ππ : β πΉπ¦ = 0: 0 = 1,233 + πΉπ΄πΆ sin(53.13Β°)
Using the expression written for forces about the π¦ β ππ₯ππ , we are now able to solve for force in member AC as: πΉπ΄πΆ = β1,541.25 πππ (πππππππ π πππ) If you calculated the following answer for the force in member AC, then your calculator is in radians mode, and your answer is incorrect. You need to always make sure that it is in degree mode when working statics problems, as angles will always be given in degrees not radians. πππππ πππ π€ππ: πΉπ΄πΆ = β4,507.51 πππ (πππππππ π πππ) We can then plug in the calculated value for the force in member AC into the equation we derived by summing the forces about the π₯ β ππ₯ππ :
β πΉπ₯ = 0: 0 = β800 + πΉπ΄π· + (β1,541.25) cos(53.13Β°)
Solving for the force in member AD, we find: πΉπ΄π· = 1724.75 Made with
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Therefore, the correct answer choice is B. π¨πͺ: π, πππ. ππ (πͺ); π¨π«: ππππ (π»)
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