## Structural Analysis Examples

Slope & Deflections Using Double Integration Method

**Example 1 - Calculating maximum deflection of simply supported beam with uniform loading using double integration method**

Calculate the maximum deflection of the simply supported beam shown below with the given loads using the double integration method. EI is constant.

**Example 2 - Calculating the location of maximum deflection for simply supported beam with triangular loading using double integration method**

Calculate the location of maximum deflection of the simply supported beam shown below with the given loads using the double integration method. EI is constant.

**Example 3 - Calculating maximum deflection of cantilever beam with uniform loading using double integration method**

Calculate the maximum deflection of the cantilevered beam shown below with the given loads using the double integration method. EI is constant.

**Example 4 - Calculating maximum deflection and slope of cantilever beam with concentrated load at end using double integration method**

Calculate the maximum deflection and slope of the cantilevered beam shown below with the given loads using the double integration method. EI is constant.

**Example 5 - Deriving equation of deflection curve for simply supported beam with concentrated load at midspan using double integration method**

Find the equation of the deflection curve for the simply supported beam shown below using the double integration method. EI is constant.

**Example 6 - Calculating maximum deflection of cantilevered beam with uniformly increasing load using double integration method**

Calculate the maximum deflection of the cantilever beam with uniformly increasing load using the double integration method. EI is constant.

**Example 7 - Calculating the slope at a support of a simply supported beam with loads increasing uniformly to the center using the double integration method**

Calculate the slope at point A of the simply supported with loads increasing uniformly to the center using the double integration method. EI is constant.

**Example 8 - Deriving deflection equation of a simply supported beam with end moment**

Find the deflection equation of the simply supported beam with moment at support A using the double integration method. EI is constant.

**Example 9 - Calculating the deflection at the free end of a simply supported beam with overhanging span subjected to concentrated load using double integration method**

Calculate the deflection at the free end of the simply supported beam with overhanging span under a concentrated load using the double integration method. EI is constant.

**Example 10 - Calculating the slope and deflection at the free end of a cantilever beam with uniform load on a portion of the beam using double integration method**

Find the slope and deflection at the free end of the cantilevered beam with uniform loading acting on a portion of the beam using the double integration method. EI is constant.

Slope & Deflections Using Moment Area Theorem / Method

**Example 1 - Calculating the slope at the midpoint and free end for a cantilever beam with concentrated end load using moment area theorem method**

Find the slope at the midpoint and at the free end of the cantilever beam with a concentrated end load using the moment area theorem method.

Young's Modulus, E = 29,000 ksi

Moment of Inertia, I = 600 in^4

**Example 2 - Calculating the slope at the quarter point for a simply supported beam with concentrated center load using moment area theorem method**

Calculate the slope at the quarter point of the simply supported beam with a concentrated center load using the moment area theorem method.

Young's Modulus, E = 200 GPa

Moment of Inertia, I = 6x10^6 mm^4

**Example 3 - Calculating the deflection at the midpoint and free end for a cantilever beam with end moment using moment area theorem method**

Calculate the deflection at the midpoint and at the free end for the cantilever beam with end moment using the moment area theorem method.

Young's Modulus, E = 200 GPa

Moment of Inertia, I (A to B) = 6x10^6 mm^4

Moment of Inertia, I (B to C) = 3x10^6 mm^4

**Example 4 - Calculating the deflection at the midpoint of a simply supported beam with end moment using moment area theorem method**

Calculate the deflection at the midpoint of the simply supported beam with end moment on the pin support using the moment area theorem method.

Young's Modulus, E = 29,000 ksi

Moment of Inertia, I = 20 in^4

**Example 5 - Calculating the slope at a support and displacement at the midpoint for a simply supported beam with concentrated center load using moment area theorem method**

Calculate the slope of the deflected shape at the pin support and the displacement at the midpoint for the simply supported beam with concentrated center load using the moment area theorem method.

EI is constant.

**Example 6 - Calculating the slope and deflection at the free end of a simply supported beam with over hanging span using moment area theorem method**

Calculate the slope and displacement at the free end of the simply supported beam with over hanging span using the moment area theorem method.

EI is constant.

Slope & Deflections Using Conjugate Beam Method

**Example 1 - Calculating the slope and deflection at the free end of a cantilever beam with a concentrated load at the midpoint using the conjugate beam method**

Calculate the slope and deflection at the free end of the cantilever beam with a concentrated load at the center using the conjugate beam method.

Young's Modulus, E = 29,000 ksi

Moment of Inertia, I = 100 in^4

**Example 2 - Calculating the slope and deflection at the free end of a cantilever beam with a concentrated load at the end using the conjugate beam method**

Calculate the slope and deflection at the free end of the cantilever beam with a concentrated load at the free end using the conjugate beam method.

EI is constant

**Example 3 - Calculating the slope at the support and the deflection at the midpoint of a simply supported beam with a concentrated load in the center using conjugate beam method**

Calculate the slope at the support and deflection at the midpoint of the simply supported beam with a concentrated load in the center using the conjugate beam method.

EI is constant

**Example 4 - Calculating the slope at the support and the deflection at the midpoint of a simply supported beam with two symmetrical concentrated loads using conjugate beam method**

Calculate the slope at the support and deflection at the midpoint of the simply supported beam with two symmetrical concentrated loads using the conjugate beam method.

EI is constant

**Example 5 - Calculating the slope at the support and the deflection at the midpoint of a simply supported beam with concentrated end moment using conjugate beam method**

Calculate the slope at the support and deflection at the midpoint of the simply supported beam with concentrated end moment using the conjugate beam method.

EI is constant

**Example 6 - Calculating the slope at the support and the deflection at free end of a simply supported beam with concentrated end loads on overhanging spans using conjugate beam method**

Calculate the slope at the support and deflection at free end of the simply supported beam with concentrated end loads on the overhanging spans using the conjugate beam method.

EI is constant

Deflections Using Virtual Work Method

**Example 1 - Calculating the displacement at a truss joint using virtual work method**

Find the vertical displacement at Joint C of the steel truss shown below using the virtual work method.

Cross-sectional area of each member = 0.5 in^2

Young's Modulus = 29,000 ksi

**Example 2 - Calculating the displacement at a truss joint using virtual work method**

Calculate the vertical displacement at Joint C if a 5 kN force is applied to Joint C of the truss using the virtual work method.

Cross-sectional area of each member (A) = 500 mm^2

Young's Modulus = 200 GPa

**Example 3 - Calculating the displacement at the free end of a cantilever beam subjected to uniform loading using virtual work method**

Find the displacement at Joint B for the cantilever steel beam using virtual work method .

Moment of Inertia, I = 500(10^6) mm^4

Young's Modulus, E = 200 GPa

**Example 4 - Calculating the slope at the midpoint of a cantilever beam subjected to a concentrated end load using virtual work method**

Find the slope at Joint B for the cantilever steel beam using the virtual work method.

Moment of Inertia, I = 50(10^6) mm^4

Young's Modulus, E = 200 GPa

**Example 5 - Calculating the slope and displacement of simply supported beam subjected to uniform loading using virtual work method**

Find the slope at support A and the displacement at the midpoint, Point C, for the simply supported beam shown below using the virtual work method.

EI is constant