## Finite Element Analysis Reaction Forces

File: PVE-3179, Last Updated: Jan. 22, 2009, By: BV

Reaction forces are the resulting loads seen at the restraints of a model being analyzed. They can be used to ensure an analysis is restrained from rigid body motion, and is static or in balance. The reaction forces are equal and opposite to the sum of the applied loads.

This report shows typical methods used for restraining models and compares the resulting displacement and stresses of identical models both in balance and out of balance for two different FEA models.

#### Example #1: F&D Head – 15 Degree Swept Model (Checking Static Condition)

**Theoretical Reaction Force Components:**

X Reaction = -75,340.8 lb

Y Reaction = -78,079.2 lb

Z Reaction = -9,919.2 lb

Note: Component directions are generated by inspection of the pressure

Theoretical Resultant = SQRT ((-75,340.8 lb)^2 + (-78,079.2 lb)^2 + (-9,919.2 lb)^2)

Theoretical Resultant = 108,954 lb

Actual Reaction Force Components:X Reaction = -75,344 lb Y Reaction = -78,075 lb Z Reaction = -9,922 lb Actual Resultant = SQRT ((-75,344 lb)^2 + (-78,075 lb)^2 + (-9,922 lb)^2) Actual Resultant = 108,950 lbError Calculation:Error = ((Resultant Theoretical - Resultant Actual) / Resultant Actual) * 100% Error = ((108,954 lb - 108,950 lb) / 108,950 lb) * 100% Error = 0.00%

From the error calculation we can see that the actual results fall within 2% of the theoretical results. This criteria determines if a model is acceptable for analysis of stresses and displacements.

#### Example #2: Hydraulic Manifold Block

The hydraulic manifold block used in this example demonstrates how an out of balance model affects model displacement and stress results.

Three different methods of checking the model balance (unexpected displacement, unexpected stress and out of balance reactions) have all indicated the same thing: this model can not be used as is.

Reaction per Port = (0.864 in^2) * (300 lb/in^2) = 259.276 lb

Often the magnitude of the reaction force can be used to determine what is causing the imbalance.

Theoretical Reaction Summary:

Reaction X = 1008 lb

Reaction Y = Force Applied – Force Due to Exit Pressure

Reaction Y = 3 Ports * (259 lb -259 lb)

Reaction Y = 0

Reaction Z = 0

Theoretical Resultant = SQRT ((1008 lb)^2 + (0 lb)^2 + (0 lb)^2)

Theoretical Resultant = 1008 lb

**Actual Reaction Force Components:
**X Reaction = 1009.6 lb

Y Reaction = -4.03 lb

Z Reaction = -3.47 lb

Actual Resultant = SQRT ((1009.6 lb)^2 + (-4.03 lb)^2 + (-3.47 lb)^2)

Actual Resultant = 1009.6 lb

**Error Calculation:**

Error = ((Resultant Theoretical – Resultant Actual) / Resultant Actual) * 100%

Error = ((1008 lb – 1009.6 lb) / 1009.6 lb) * 100%

Error = -0.16 %

From the error calculation we can see that the actual results fall within 2% of the theoretical results. This model is in balance and can be used to calculate displacements and stresses.

### Summary

Checking the model balance is an important step for verifying that all loads are acting upon restraints correctly. An out of balanced model provides invalid result that cannot be used.