Concept explainers
Find the member end moments and reactions for the frames.

Answer to Problem 31P
The reaction at point A
The end moment at the member
Explanation of Solution
Fixed end moment:
Formula to calculate the relative stiffness for fixed support
Formula to calculate the fixed moment for point load with equal length are
Formula to calculate the fixed moment for point load with unequal length are
Formula to calculate the fixed moment for UDL is
Formula to calculate the fixed moment for UVL are
Formula to calculate the fixed moment for deflection is
Calculation:
Consider the elastic modulus E of the frame is constant.
Show the free body diagram of the entire frame as in Figure 1.
Refer Figure 1,
Calculate the relative stiffness
Calculate the relative stiffness
Calculate the distribution factor
Substitute
Calculate the distribution factor
Substitute
Calculate the distribution factor
Substitute
Check for sum of distribution factor as below:
Substitute
Hence, OK.
Calculate the distribution factor
Substitute
Calculate the distribution factor
Substitute
Check for sum of distribution factor as below:
Substitute
Hence, OK.
Calculate the distribution factor
Substitute
Calculate the distribution factor
Substitute
Check for sum of distribution factor as below:
Substitute
Hence, OK.
Calculate the distribution factor
Substitute
Calculate the distribution factor
Substitute
Calculate the distribution factor
Substitute
Check for sum of distribution factor as below:
Substitute
Hence, OK.
Show the translation
Write the expression to calculate the Fixed-end moment of the member.
Assume the Fixed-end moment of the members AC, CA, BD and DB as
Write the expression to calculate the Fixed-end moment of the member.
Assume the Fixed-end moment of the members CE, EC, DF and FD as
Show the calculation of
Show the free body diagram of the frame with unknown reaction
Consider member EC:
Calculate the horizontal reaction at the joint C by taking moment about point E.
Calculate the horizontal reaction at joint E by resolving the horizontal equilibrium.
Consider member FD:
Calculate the horizontal reaction at the joint D by taking moment about point F.
Calculate the horizontal reaction at joint F by resolving the horizontal equilibrium.
Calculate the reaction
Show the free body diagram of the frame with unknown reaction
Consider member AC:
Calculate the horizontal reaction at the joint A by taking moment about point C.
Calculate the horizontal reaction at joint C by resolving the horizontal equilibrium.
Consider member BD:
Calculate the horizontal reaction at the joint B by taking moment about point D.
Calculate the horizontal reaction at joint D by resolving the horizontal equilibrium.
Calculate the reaction
Show the translation
Write the expression to calculate the Fixed-end moment of the member.
Assume the Fixed-end moment of the members CE, EC, DF and FD as
Show the calculation of
Show the free body diagram of the frame with unknown reaction
Consider member EC:
Calculate the horizontal reaction at the joint C by taking moment about point E.
Calculate the horizontal reaction at joint E by resolving the horizontal equilibrium.
Consider member FD:
Calculate the horizontal reaction at the joint D by taking moment about point F.
Calculate the horizontal reaction at joint F by resolving the horizontal equilibrium.
Calculate the reaction
Show the free body diagram of the frame with unknown reaction
Consider member AC:
Calculate the horizontal reaction at the joint A by taking moment about point C.
Calculate the horizontal reaction at joint C by resolving the horizontal equilibrium.
Consider member BD:
Calculate the horizontal reaction at the joint B by taking moment about point D.
Calculate the horizontal reaction at joint D by resolving the horizontal equilibrium.
Calculate the reaction
Write the equation by superimposing the horizontal forces at joints C,
Write the equation by superimposing the horizontal forces at joints E,
Calculate the value of
Calculate the actual member end moments of the member AC and BD:
Substitute
Calculate the actual member end moments of the member CA and DB:
Substitute
Calculate the actual member end moments of the member CD and DC:
Substitute
Calculate the actual member end moments of the member CE and DF:
Substitute
Calculate the actual member end moments of the member EC and FD:
Substitute
Calculate the actual member end moments of the member EC and FD:
Substitute
Show the section free body diagram of the member as in Figure 8.
Consider member EF:
Calculate the vertical reaction at the joint E by taking moment about point F.
Calculate the vertical reaction at joint F by resolving the horizontal equilibrium.
Consider member CD:
Calculate the vertical reaction at the joint C by taking moment about point D.
Calculate the vertical reaction at joint D by resolving the horizontal equilibrium.
Calculate the reaction at joint A using the relation:
Calculate the reaction at joint B using the relation:
Consider member AC:
Calculate the horizontal reaction at the joint A by taking moment about point C.
Consider member BD:
Calculate the horizontal reaction at the joint B by taking moment about point D.
Show the reactions of the frame as in Figure 9.
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