In this video, we look into a three-hinged arch with tie. This problem is extracted from Structural Analysis book by RC Hibbeler. In this question, we are required to solve for the reaction forces and tension in tie. This can be achieved by using solely equilibrium equations.
Next, we extend our scope to determine the internal forces and produce diagrams for shear force, bending moment and axial force. The internal force functions, namely shear force and axial force are dependent of the inclination angle of tangent at one point. Knowing this, we need to develop curve function for the arch. In this problem, we take this as a parabolic arch and develop the function. Using this function, we can calculate the inclination angle. The segmentization of arch follows the same rule for beam and frame. Then, we apply equilibrium equations for each segment to obtain the internal force functions. After that, we are good to determine the internal force at any point on the arch.
The following shows the procedure to analyse an arch. Watch the video above for full details.
Arch problem overview
A 6.5m span arch with height to crown of 2m is subjected to two concentrated loads
Reaction Cy
By taking moment about point A, reaction Cy is determined using equilibrium equation.
Reaction Ay
By taking equilibrium in y direction, reaction Ay is determined.
Tie force T
By taking moment about point B, tie force T is calculated by focusing on right part of three-hinged arch.
Hinge force Bx
By taking equilibrium in x direction for right part of arch, force Bx is determined.
Hinge force By
By taking equilibrium in y direction for right part of arch, force By is determined.
Reaction Ax
By taking equilibrium in x direction for left part of arch, force Ax is determined.
Result overview
The reactions and tie tension are determined and shown as follows.
Curve function
The function describing the geometry of curve is determined based on the base length and height of arch.
Segmentization of arch
The arch is segmentized into five parts, break at the points where concentrated loads act and the location of internal hinge.
Segment 1
Segment 2
Segment 5
Segment 4
Segment 3
Internal forces
Internal forces for every 0.5m ordinate (horizontal) are calculated.
Axial force diagram
Shear force diagram
Bending moment diagram
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