Determine the deflection function of a beam subjected to uniformly distributed bending moment shown in the Figure. Take the shear deformation into account. One end of the beam is hinged the other is built-in. Solve the differential equation system
a) analytically, see Example 25,
b) using the force method (see Example 27),
c) *analytically, using Eq.(3-61) (See Example 110),
Give the maximum rotation and the maximum deflection with the following data: length of the beam is L= 2 m, uniformly distributed moment load is m = 10.0 kNm/m. Shear stiffness of the cross section is S = 3.5×105 kN, the bending stiffness is EI = 1.71×104 kNm2.
Solve Problem
Maximum deflection, vmax[mm]=Check displacement vector. Rotation at the hinged support, χ(L) [×10-5]
Solve
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Steps
Solution a) Step 2. Express the deformation functions from the material equations. Step 3. Derive the displacement functions from the geometrical equations. Step 4. Determine constants C1, C2, C3, C4 from boundary conditions. Step 5. Substitute the constants to obtain the displacement functions. Calculate maximum displacements. The function of the rotation of the cross section is It reaches its maximum value at the hinged support: The deflection function is The maximum locates where the first derivative is equal to 0: Solution b) The primary structure is chosen to be a cantilever, thus the redundant is the support reaction of the hinge. Step 2. Write the deflection function of the primary structure from the original load and from the redundant, respectively. Displacement function from the moment load is Displacement function from the unit load is Step 3. Determine redundant from the compatibility condition. Compatibility condition is written to the end of the cantilever, where there is no deflection of the original structure: Step 4. Give the displacement functions. Step 5. Calculate maximum displacements. The maximum rotation occurs at the hinged support: The maximum deflection and its location is given below: Solution c) Step 1. Write the differential equation system of the Timoshenko beam for the given moment load. Step 2. Find a particular solution. In a particular solution the constants can be chosen arbitrarily, thus let us set all constants equal to zero: Step 3. Assume the solution of the homogeneous equation system in exponential form. Step 3. Introduce the i-th displacement function into the homogeneous differential equation system. Solve the eigenvalue problem. The above problem results in four zero eigenvalues: Because of the multiplicity of the roots, the solution function results in Now the vector of the solution functions is substituted into the homogeneous differential equation: Which is equivalent to the following equations: The above equations are fulfilled by any value of x only when all the coefficients of the polynomials are zero. These conditions result in the following relationship between the constants: Step 4. Write the general solution. Step 5. Determine constants of the displacement functions from the boundary conditions. At the built-in end of the beam: At the hinged end of the beam: After straightforward algebraic manipulation the four conditions result in: Step 6. Give the displacement functions. Calculate the maximum deflections. Substituting the constants to the general solution the displacement vectors can be rearranged in the same form, as they are derived in Solutions b): The maximum rotation occurs at the hinged support: The maximum deflection and its location is given below:Step by step
Show Steps of Solution a)
Deflection function of the beam can be expressed from the equilibrium, geometrical and material equations.
Step 1. Determine the shear force and the bending moment functions from the equilibrium equations.
Check internal forces
Check deformations
Check displacements
Check boundary conditions
Check maximum displacements
Show Steps of Solution b)
Step 1. Choose a primary structure.
Show primary structure
Check deflections
Check redundant
Check displacement functions
Check maximum displacements
Show Steps of Solution c)
Check differential equation system
Check particular solution
Check assumed displacement functions
Check eigenvalue problem
Check general solution
Check boundary conditions
Check results
Results
Solution a) The deformation functions are expressed from the material equations: The displacement functions are derived from the geometrical equations. where constants C1, C2, C3, C4 can be determined from boundary conditions. It reaches its maximum value at the hinged support: The deflection function is The maximum locates where the first derivative is equal to 0: Solution b) The primary structure is chosen to be a cantilever, thus the redundant is the support reaction of the hinge. The deflection functions of the primary structure from the original load and from the redundant are given below: Displacement function from the moment load is Displacement function from the unit load is The redundant is determined from the compatibility condition.The compatibility condition is written to the end of the cantilever, where there is no deflection of the original structure: Thee displacement functions are Maximum displacements and their locations are given below. The maximum rotation occurs at the hinged support: The maximum deflection and its location is given below: Solution c) The differential equation system of the Timoshenko beam for the given moment load is written in the following form: First a particular solution is given: In a particular solution the constants can be chosen arbitrarily, thus let us set all constants equal to zero: The solution of the homogeneous equation system is assumed in exponential form. Introducing the i-th displacement function into the homogeneous differential equation system we get an eigenvalue problem. The above problem results in four zero eigenvalues: Because of the multiplicity of the roots, the solution function results in Now the vector of the solution functions is substituted into the homogeneous differential equation: Which is equivalent to the following equations: The above equations are fulfilled by any value of x only when all the coefficients of the polynomials are zero. These conditions result in the following relationship between the constants: The general solution is Constants of the displacement functions are determined from the boundary conditions. At the built-in end of the beam: At the hinged end of the beam: After straightforward algebraic manipulation the four conditions result in: Substituting the constants to the general solution the displacement vectors can be rearranged in the same form, as they are derived in Solutions b): The maximum rotation occurs at the hinged support: The maximum deflection and its location is given below:Show worked out solution
Show Solution a)
Deflection function of the beam can be expressed from the equilibrium, geometrical and material equations.
First the shear force and the bending moment functions are determined from the equilibrium equations.
By substituting the constants we obtain the displacement functions. The function of the rotation of the cross section is
Show Solution b)
Show Solution c)